File: File_52c (tentative identifier)
Title: Rounded Whole-Span Inversions — Narrative Correspondences, the 529 Family, the 6 BC Concentration, and the 12026 BC Actual Chronology Key
Status: Draft; original report pressure test retained; author-requested backbone extension verified; ten positive hits to 6 BC in 66 constructions across five targets; Actual chronology continuation at 12026 BC partially explored; author regards its major anchor role as strongly supported; not marked Final
Primary domain: Rounded; Actual; Regular; Cumulative; Mirror; Statistical; Methodological
Traditions: MT; bounded restored second-Cainan companion; bounded LXX / SP offset and SKL comparisons; Septuagint textual comparison; biblical and Enochic narrative comparisons
Canonical source: Markdown
Primary anchors: 6 BC; 1 BC; AD 6; AD 25 (civil comparison); 966 BC (reference only); 1406 BC; 1401 BC; 1400 BC; 1446 BC; 1846 BC; 1876 BC; 2458 BC; 3056 BC; 3654 BC; 4106 BC; 4114 BC; 8066 BC; 12026 BC; 14004 BC; 14006 BC; 14726 BC
Related files: File_00; File_01; File_05; File_11; File_16; File_18; File_21; File_22; File_34; File_46; File_51a; File_51b; File_52a; File_52b; File_54; File_61; existing File_52c second-order-closure reference
Major operators: placeholder-preserving decimal reversal; declared segmentwise reversal and summation; source-ending anchor selection; signed anchor offset for the uncounted Temple reference; same-side subtraction; Rounded coordinate width; explicitly selected civil cross-axis count; 25/23; 70/69; 300/299
Major modal states: Rounded regular whole-span birth/event inversion; separately referenced Temple construction at 966 BC; bounded cumulative trunk comparison; admitted segmented backbone extension at 12026 and 14726 BC; self-inverse source; bounded restored second-Cainan companion; provisional inverse terminal-date reconstruction; Rounded Mirror coordinate-completion; exploratory target-frequency scan; original-Flood-held 8066 BC construction; Actual Creation 1:4 bridge at 12026 BC; selected Jared / Noah Gear 1 / Flood Gear 2 comparison; partially explored 598-multiple anchor structure
Revision basis: 490d Repository Style Guide v2.5; Project Procedures v3.5; supplied Restart Capsule v11.54; supplied State_Vocabulary_Register v1.55, especially §D.12–§D.13; File_00 and File_54 style precedents; September 25 2026 author clarifications in Files 51a, 51b, 52a, and 52b; author decisions in the present discussion

# File_52c — Rounded Whole-Span Inverse Detailed Study

## 0. File-function

This report records the detailed study of Rounded birth and event dates under whole-span decimal inversion. Its subject includes the narrative correspondences involving Joseph, Joshua, the 1876 BC Nexus, and Creation; the selected `529 = 23²` relations; the 1846 BC hub associated with the 1876 BC Nexus; and the bounded survey of inverse dates against five comparison targets. The numerical–narrative study is wider than the count of multiples of 115.

The original whole-span result is nine positive-span hits to 6 BC among 64 anchored inversions of 32 source dates. The author now admits the major segmented Creation backbone: its completed Nativity-stage endpoint is 14726 BC, with $14726-6=14720=640\times23$. This supplies the tenth incident. Including both backbone anchor stages, 12026 BC and 14726 BC, gives an expanded set of 66 constructions. Its five positive counts are ten to 6 BC, two to 1406 BC, one to 1446 BC, two to 1 BC, and three to AD 6. Appendix A retains all original outcomes and adds both supplementary constructions, including their misses.

The importance of 14726 BC extends beyond that tenth incident. Its 14720 span to 6 BC converts to 16000, and its relations to inverse Shem, the unchanged cumulative Flood, and expanded cumulative Creation yield 11270 = 23 × 490, 9890 = 23 × 430, 9200 = 23 × 400, and 49680 = 23 × 6 × 360. Section 3.6 records the exact endpoints and explains why the author calls 12026 / 14726 BC the shared backbone of the regular and cumulative Rounded inverse constructions.

The author further proposes 14726 BC as a Creation pillar that helps interpret the original nine intersections with 6 BC. Holding this earlier endpoint during 25/23 expansion sends Terah's inverse 3226 BC to the +60-state firstborn threshold at 2226 BC, and the affliction hub 1846 BC to the schematic Samaria node at 726 BC, 720 years before 6 BC. Section 3.7 records these examples and all nine complementary spans under the same held-anchor rule.

Section 3.8 develops three further companion relations: the Entry inverse / Mirror span 22540 → 24500, compared with the earlier 9660 → 9800 conversion; the Nativity-held Creation–Shem partition 16000 = 10000 + 6000; and the Temple inverse's convergence at Rounded AD 526 with expansion from the reflected 12026 BC backbone. The operators, held endpoints, and textual comparisons are identified separately.

Section 3.9 extends that convergence: the second 25/23 expansion from AD 12026 reaches 476 BC, matching the Creation-held expansion of inverse Moses at 1616 BC. Expanding the 920 interval forward from AD 526 reaches Moses' reflected birth at AD 1526. The result is a Rounded 1000 + 1000 arrangement centered on AD 526, with the 80-year expansion gain matching Moses' birth-to-Exodus interval.

Section 3.10 develops Abraham's generated 976 BC endpoint: Entry → Exodus → Conquest → 976 BC forms 430 + 40 + 430, while the 1100-year movement from inverse Abraham at 2076 BC is bisected by Moses' birth at 1526 BC, giving 550 + 550. The 390-year continuation to 586 BC completes the existing 1290 Entry-to-Jerusalem span.

Section 3.11 develops the Flood-countdown inverse at 8516 BC. Its 6210 span from Creation expands to 6300 or 6750 under 70/69 or 25/23, producing 8426 or 7976 BC and intervals of 6900 = 15 × 460 or 6450 = 15 × 430 to Moses' birth. Its ten-member inventory records the author's organizing thesis and now links each member to a stated contextual interpretation.

Section 3.12 supplies a cursory exploration of Mahalalel, Shelah, and Eber as one joint Cainan pattern, giving the final three members an initial contextual interpretation. The omitted-Cainan connection is its main point; the author's selected observations do not constitute a thorough examination of the trio's much more complex relations. The inverse 6900 + 690 arrangement becomes 7500 + 750, or 23 × 330 → 25 × 330, while restored Cainan's 2551–2091 BC life provides the 130 + 330 comparison. The generated 601 BC midpoint, at 1950 + 1950, links Cainan's birth to reflected expanded Shelah. Civil AD 30 comparisons and the Rounded Mirror spans retain separate counting rules; the restored companion adds no statistical source.

Sections 3.13–3.15 add the author's proposed counterpart key at 12026 BC for Actual MT regular and cumulative chronology. The three starting levels introduce 8066 BC and its 6660 / 6666 relations; inverse cumulative Creation supplies 23 × 1260 and the 43000-year Exodus expansion. The Actual 14004–4114 BC Creation bridge divides at 12026 BC into a 1:4 pair, while Jared, Noah, and the Flood stand at 14, 15, and 16 multiples of 598. Their expansions, complete resulting-interval table, and selected Creation / SKL / AD 25 connections are preserved as a partially explored treatment. The author regards the evidence for the major anchor role as strong; the unfinished work concerns the fuller explanation of its details. These additions do not enlarge the statistical ledger.

The unadjusted independent-trial benchmark for ten or more hits among 66 trials at $p=1/23$ is approximately 0.0545%, or one in 1834. Section 9.6 also records ten among 65 if only the final backbone stage is added. These are conditional model calculations; the original whole-span scan and the author-requested segmented extension remain identifiable.

The 1846 BC observation belongs to the structural study in §5.6: Abram and Moses invert to dates centered on it at 230 + 230, while Terah's inverse stands 1380 earlier. The author identifies this hub as second in importance to 1876 BC in the relevant Egypt / affliction field and proposes it as a main artery for further expansion. It remains outside the statistical target list.

The report preserves the author's reason for continuing the inquiry: several different kinds of numerical relation appear appropriate to their particular persons, events, and narrative settings. The 6 BC divisibility scan supplies one tractable diagnostic within that wider inquiry. It does not assign a joint probability to the whole system.

Audit note — tentative file identity: the supplied Register §D.12 and existing File_52a / File_52b references already assign `File_52c` to inverse-of-the-inverse, second-order closure, and square / gate-field interpretation. The present report retains the author's requested tentative identifier as a separate draft. It does not replace that existing work or claim to complete its proof burden. Final numbering or integration remains an editorial decision.

### 0.1 Working state register

All numerical results below retain their stated source layer and operator. A shared date label alone does not establish identity between layers.

| State | Working values / operation | Function and boundary |
|---|---|---|
| Rounded regular source | File_51a §3.1; Creation 4106 BC; Jacob 2006 BC | Base birth chronology for the bounded survey; not a replacement of Actual MT chronology. |
| Whole-span birth/event inverse | Subtract selected anchor, invert the complete span, rebuild from that anchor | Primary operation of this report. |
| Source-ending selector | 6-ending dates use 6 or 1406 BC; 1-ending dates use 1 or 1401 BC | Two anchor families, with one applicable member per family for each source. |
| Self-inverse source | The inverted span equals its original value | Retained when the source is admitted to the manifest; a target hit still requires a positive span. |
| Temple reference outside statistics | 966 BC; explicit 480-year Exodus link; signed offsets −440 to 1406 BC and +960 to 6 BC | Reference-only inverse endpoints 966 BC and 696 BC; the author withheld statistical admission. |
| Segmented inverse backbone | Invert selected intervals separately, then sum them | File_52a / File_52b dependency; 12026 and 14726 BC admitted once each as the two-stage extension, distinct from the 64 whole-span routes. |
| Rounded cumulative comparison | Creation 14006 BC; selected 14011 BC companion; Shem 5436 BC; Flood / Shem-death boundary 4836 BC | Declared trunk and 529 examples; primary Flood backbone corroborates the admitted endpoints without adding duplicate counts or new source rows. |
| Actual Creation bridge comparison | File_22: 14004 / 4114 BC endpoint pair; 14006 / 4116 BC Year-6 pair; both separated by 9890 | Section 3.14 divides the endpoint bridge at 12026 BC: 1978 + 7912 → 2150 + 8600, preserving 1:4. The uninverted Rounded gap stays 9900. |
| Original-Flood-held Creation inverse | Regular 2456 + I(1650) = 8066 BC; cumulative 4836 + I(9170) = 12026 BC | Hold the original Flood in each branch; cumulative repetition follows from the self-inverse 3430 Conquest span. |
| Actual 12026 BC anchor key | Actual cumulative / regular Creation endpoints and selected chart dates associated with 1446 BC | Author regards the major anchor role as strongly supported; broader detailed interpretation remains partially explored. No new scan rows are admitted. |
| Jared–Noah–Flood 598 sequence | 3654 / 3056 / 2458 BC; anchor spans 14 / 15 / 16 × 598; 600 / 650 conversions | Noah is selected Gear 1; Flood is Gear 2. Preserve the chart's states without extending gear transport into the earlier genealogy. Creation output 3426 BC* is not a 299-multiple route. |
| Rounded coordinate state | $q(B\text{ BC})=1-B$; $q(\text{AD }A)=A-1$ | Signed multiples of five; symmetric display on the two sides. |
| Rounded Mirror | $q\mapsto-q$ | Coordinate-completion or explicitly qualified exploratory comparison; not digit reversal. |
| Literal civil count | $C=B+A-1$ | Used in expressly selected cross-axis 529 constructions, AD 30 comparisons in §3.12, and AD 25 continuations in §3.15.5; distinct from Rounded width. |
| Key-of-23 whole-year expansion | $25/23$; $70/69$; $300/299$ | Exact conversions of declared durations; no physical Residue Protocol is opened. |
| Creation-held pillar comparison | Hold 14726 BC; expand its span to each of the original nine inverse endpoints by 25/23 | Rebuilt Creation endpoint used as a common expansion anchor; 2226 BC firstborn and 726 BC schematic Samaria comparisons; pillar is a structural analogy, distinct from File_21's named Projective Boundary Pillars. |
| Entry Mirror duration comparison | 22540 × 25/23 = 24500; 9660 × 70/69 = 9800; difference 14700 | Two expressly different expansion ratios; comparison of converted durations, not a uniform dilation of both original segments. |
| Nativity-held Creation–Shem partition | 14726 → 5526 → 6 BC becomes 16006 → 6006 → 6 BC under 25/23 | One common affine expansion with 6 BC fixed; 10000 + 6000; Enochic confinement and Shem-lifespan comparisons remain interpretive. |
| Temple / reflected-backbone convergence | 14726 BC and Rounded AD 12026 held in their respective routes; both generate q = +525, displayed as AD 526 | Temple inverse 696 BC remains reference-only; 11500 = 5 × 2300 supplies a sanctuary-restoration comparison, not a new historical date claim. |
| Moses–Temple repeated-expansion closure | 10580 → 11500 → 12500 with AD 12026 held; inverse Moses / Temple 1616 / 696 BC separated by 920; outputs 476 BC / AD 526; forward counterpart AD 1526 | Rounded 1000 + 1000 centered on AD 526; declared switches of held anchor; related arithmetic consequences, not independent statistical witnesses. |
| Abraham expansion and Moses midpoint | Inverse Abraham 2076 BC → Creation-held endpoint 976 BC; gain 1100; Moses' birth at 1526 BC gives 550 + 550 | Same-side spans; Entry–Exodus–Conquest 430 + 40 + 430; 390 continuation to 586 BC; generated endpoint, no new historical event assignment. |
| Flood-countdown paired ratios | 2576 BC → inverse 8516 BC; 6210 from Creation; 70/69 → 8426 BC; 25/23 → 7976 BC | 6900 / 6450 to Moses' birth; matched 90-year shifts; 15 × 460 / 15 × 430; seven named Abraham–Moses generations counted inclusively. |
| Mahalalel–Shelah–Eber joint pattern | Inverses 1731 / 2421 / 9321 BC → Creation-held outputs 601 / 1351 / 8851 BC; 690 + 6900 → 750 + 7500 | Shelah is a 1:10 internal divider, not an arithmetic midpoint; 23 × 330 → 25 × 330; all three share one contextual treatment. |
| Restored second-Cainan companion | Selected regular state 2551 → 2421 → 2091 BC; 130 + 330 = 460; inverse 5521 → 6001 BC with 1 BC held | Local comparison only; regular insertion remains +130; no change to the frozen MT manifest or cumulative +460 operator. |
| Trio cross-axis comparison | Civil AD 30 widths 8880 / 1380 / 630; Rounded widths 11270 / 3900 / 1950 / 3440 / 2795 | Civil-display reflection pairs AD 30 with 31 BC; symmetric Rounded reflection pairs AD 30 with 30 BC. The local distinction is explicit in §3.12.4. |
| Inverse terminal-date reconstruction | Inverse birth plus an expressly specified lifespan operation | Provisional structural notation; excluded from the main birth/event count. |
| Structural 1846 BC hub | Abram / Moses inverse midpoint; Terah at 6 × 230; held-center expansion | Secondary to the 1876 BC Nexus in the author's reading; outside the target-frequency study. |
| Entry inverse / Mirror bridge | 1876 BC → 7816 BC → AD 7816; Rounded width 9660 to 1846 BC; 70/69 → 9800 | Declared inverse, Mirror, and Prophetic expansion sequence; structural link between the paired Egypt hubs. |
| Frequency diagnostic | Positive span divisible by 23, equivalently 115 on this grid | Original 64-route manifest plus two declared backbone constructions; five targets; exploratory statistical interpretation. |

### 0.2 File map

| Section | Function | Active state / operator |
|---|---|---|
| §§1–2 | Define the operation, freeze the source manifest, and identify the separate Temple reference | Whole-span inverse; Rounded coordinates; inclusion rules; uncounted 966 BC companion |
| §3 | Establish and interpret the shared 12026 / 14726 BC backbone | Rounded pillar and ten-member inventory; Moses–Temple, Abraham, countdown, and Cainan relations; partially explored Actual 12026 BC key, Creation bridge, and Jared–Noah–Flood 598 sequence |
| §§4–5 | Preserve Joseph / Joshua, the 1876 BC Nexus, and the 1846 BC hub | 110; 49 / 50; 230 + 230; held-center expansion; Entry inverse / Mirror 9660 → 9800 |
| §6 | Record the selected 529 family | Same-side spans; civil cross-axis counts; Rounded Mirror |
| §7 | Preserve the cumulative 62100 expansion chain | Held-anchor whole-year expansion |
| §§8–9 | Present the original and expanded counts and probability benchmarks | 64-route scan; paired backbone extension to 66; unadjusted binomial diagnostics |
| §§10–12 | Interpret, qualify, and route the findings | Claim-status; open questions; source controls |
| Appendices A–C | Supply all target outcomes, executable scan, and pressure-test findings | Original 64-route tables plus two backbone rows; residue census; sensitivity and dependence audit |

### 0.3 Claim-status control

The date transformations, factorizations, and counts are arithmetic facts under the stated conventions. The selection of particular correspondences as meaningful is a structural inference or typological reading. The binomial calculation is a statistical result of a simplified model; its application to the study remains exploratory.

The use of a modern era display and decimal reversal does not by itself establish ancient authorial intention. A theological interpretation extending beyond demonstrable human intention remains a proposed providential synchronization. Neither that interpretation nor the proposed third-axis analogy is established by a factorization alone.

The author's September 27 assessment specifically distinguishes confidence in the 12026 BC anchor from completeness of its interpretation. He regards the evidence for its major key role as strong and that role as not in doubt within his model. Sections 3.13–3.15 preserve this judgment while marking the exploration of the wider pattern as partial.

### 0.4 Machine Guards

Machine Guard [FILE_52C WHOLE-SPAN ANCHORS]:
Select the anchor from the original source date's ending digit before inversion. Use 6 / 1406 BC for dates ending in 6 and 1 / 1401 BC for dates ending in 1. Preserve the source date. The survey has two routes per source, not four. Its 1446 BC column is a comparison target, not a third inversion route.

Machine Guard [FILE_52C OPERATOR SEPARATION]:
Whole-span inversion, segmented inversion, Mirror reflection, civil elapsed counting, and subsequent Key-of-23 expansion are separate operations. Record the active operation and held anchor at each transition. Do not count a later expanded landing as a direct inverse hit.

Machine Guard [FILE_52C ROUNDED / CIVIL WIDTH]:
Use signed q arithmetic for a Rounded coordinate width. Opposite-side symmetric labels give $W_R=B+A-2$. The ordinary civil elapsed count between those printed labels is $C=B+A-1=W_R+1$. Retain the extra year only when the construction expressly chooses the civil count.

Machine Guard [FILE_52C FROZEN COUNT]:
Preserve the original 32 dates in §2.1 and their 64 whole-span routes. The expanded tally adds only the two declared segmented backbone endpoints, 12026 BC and 14726 BC, once each, giving 66 constructions. Alternative regular / cumulative derivations of these endpoints corroborate them without extra counts. Exclude zero-span overlaps from positive hits. Keep other cumulative examples, reconstructed deaths, additional variants, Mirror doubles, later expansions, and the separately referenced 966 BC Temple routes outside both tallies.

Machine Guard [FILE_52C STRUCTURAL HUB]:
Treat 1846 BC as a structural hub and proposed expansion anchor, not an additional statistical target. The fixed targets remain 6 BC, 1406 BC, 1446 BC, 1 BC, and AD 6. Retain the already admitted 1846 BC affliction source and its two routes in the 32-date manifest. The shared 115-year grid and the Abram–Moses / Terah relations belong to the narrative and structural discussion.

Machine Guard [FILE_52C CREATION-HELD EXPANSION]:
In §3.7, hold 14726 BC fixed and expand the positive span toward the later BC endpoint: $X^*=14726-(25/23)(14726-X)$. Preserve each source's original inversion anchor before this separate expansion. The 2226 BC comparison uses the +60 Terah firstborn state; the 726 BC comparison uses the schematic Samaria state. The shared residue class does not supply additional independent hits. The Creation-pillar analogy does not redefine the named SKL / Temple-Time Projective Boundary Pillars.

Machine Guard [FILE_52C ACTUAL 12026 KEY]:
Keep the Actual 14004 / 4114 BC completion endpoints separate from the Year-6 and Rounded Creation pairs. In §3.15, hold 12026 BC and compute the forward BC result as $12026-r(12026-X)$. Preserve Noah 3056 BC as selected Gear 1 and Flood 2458 BC as Gear 2; do not treat their 598 years as a replacement lifespan or extend gears into the earlier genealogy. Retain the Creation asterisk, the 2276 BC Noah identity, and the civil count for AD 25. The strong authorial anchor assessment and the partially explored interpretive scope coexist; no new statistical admission follows.

Machine Guard [FILE_52C STATISTICAL STATUS]:
The approximate 1-in-571 baseline and 1-in-1834 expanded tails assume 64 and 66 independent trials respectively, each with probability 1/23. The segmented extension changes the admitted operation set and is correlated with the existing chronology. These figures are not corrected significance levels, probabilities that the system is accidental, or probabilities against authorial or providential design.

## 1. Arithmetic conventions

### 1.1 Rounded dates and source identity

The base survey uses explicitly Rounded dates. It does not treat every clean or mod-5 value in the repository as part of the Rounded Scaffold. In particular, Joseph's 1916–1806 BC pair belongs to the present Rounded comparison. It must not overwrite the Actual Joseph chronology controlled elsewhere.

The source chronology follows File_51a §3.1 without adding Shem's +2. The survey does not open +60 Terah, the −215 Sojourn variant, restored Cainan, apparent-age overlays, or the wider Three-Gear field. Their existence elsewhere is not permission to multiply this manifest. The restored-Cainan companion expressly opened for §3.12 is outside that survey.

The cutoff is applied to the statistical source manifest: no date later than the Exodus at 1446 BC enters that list. A generated inverse endpoint may lie later than the cutoff. Thus Adam's inverse at 1406 BC remains a valid generated endpoint. The Temple at 966 BC is separately referenced in §2.5 because of its explicit 480-year Exodus synchronism; its two inverse routes remain outside the statistical manifest.

### 1.2 Placeholder-preserving decimal reversal

For a positive integer $n=10^k m$, where $m$ is not divisible by ten, define:

$$I(n)=10^k\operatorname{rev}(m).$$

The trailing-zero placeholders remain at the same magnitude. Define $I(0)=0$ for completeness.

The following examples all use this primary register.

| Original span | Core reversal | Inverse span |
|---:|---|---:|
| 120 | 12 → 21 | 210 |
| 390 | 39 → 93 | 930 |
| 400 | 4 → 4 | 400 |
| 2000 | 2 → 2 | 2000 |
| 2700 | 27 → 72 | 7200 |
| 3970 | 397 → 793 | 7930 |
| 12600 | 126 → 621 | 62100 |

This operation is digit reversal, not the reciprocal $1/n$. In this register, $I(I(n))=n$. A self-inverse span returns its source date when rebuilt from the same anchor.

Digit reversal preserves the digit sum, so the difference between a span and its inverse is divisible by 9. That elementary decimal property does not imply preservation of divisibility by 23.

The older segment-level Rounded inverse axiom remains a separate dependency. For example, a segment of 365 reverses to 563 and is normalized to 565 when that chain procedure is explicitly opened. The matched-anchor whole spans in this survey are multiples of ten and need no such normalization.

### 1.3 The two anchor families

For an admitted source date $D$ BC, select one anchor from each family according to its final digit.

| Final digit of source $D$ | Nativity-family anchor | Conquest-family anchor |
|---:|---|---|
| 6 | 6 BC | 1406 BC |
| 1 | 1 BC | 1401 BC |

For either selected anchor $A$ BC, calculate:

$$s=D-A,\qquad s'=I(s),\qquad D'=A+s'.$$

The result $D'$ BC is the rebuilt inverse date. The printed BC year number is not itself reversed.

Shelah illustrates the importance of retaining the source date and selecting the matching anchor. From 2421 BC, the Nativity-family span is $2421-1=2420$, which is self-inverse and returns 2421 BC. The Conquest-family span is $2421-1401=1020$, which reverses to 2010 and rebuilds 3411 BC.

Using 6 BC for that 1-ending source would introduce an unmatched-anchor span and a later normalization step. That is not the agreed primary operation. The selection rule introduces no 1441 BC anchor and does not re-anchor the rows of existing begetting or lifespan chains.

### 1.4 Rounded coordinates, Mirror labels, and civil counts

The September 25 author clarification controls the pure Rounded coordinate convention:

$$q(B\text{ BC})=-(B-1),\qquad q(\text{AD }A)=A-1.$$

A nonzero coordinate is displayed with numeral $|q|+1$ and the era supplied by its sign. Reflection changes $q$ to $-q$. Thus 1406 BC and AD 1406 display the coordinate pair −1405 and +1405.

For this study, the origin comparison is labeled 1 BC at $q=0$. Zero is the algebraic Mirror origin; it is not a civil year zero. The symmetric convention is not a single ordinary astronomical-year coordinate map.

The following coordinates identify the five statistical targets, the subsidiary Conquest construction anchor, and the separate structural hub at 1846 BC. The Role column distinguishes their functions.

| Label | Rounded coordinate | Role |
|---|---:|---|
| 6 BC | −5 | Main target; Nativity construction anchor for 6-ending sources |
| 1 BC | 0 | Origin target; Nativity construction anchor for 1-ending sources |
| AD 6 | +5 | Exploratory opposite-side target |
| 1406 BC | −1405 | Conquest construction anchor and comparison target |
| 1401 BC | −1400 | Conquest construction anchor for 1-ending sources |
| 1446 BC | −1445 | Exodus comparison target |
| 1846 BC | −1845 | Structural affliction hub and proposed expansion anchor; outside the target scan |

For opposite-side labels $B$ BC and AD $A$, keep the two counts distinct:

$$W_R=B+A-2,\qquad C=B+A-1=W_R+1.$$

In particular, the Rounded width from an inverse date $D'$ BC to AD 6 is $D'+4$. The corresponding ordinary civil count is $D'+5$. The AD 6 scan uses the former, as expressly requested by the author through the coordinate +5.

Earlier sources sometimes display a positive Mirror coordinate $q$ as civil AD $q$. The current symmetric Rounded display is AD $(q+1)$. An explicitly civil equation retains its stated labels and count. No global one-year shift is applied to Actual or phase-resolved chronology.

## 2. The frozen source manifest

### 2.1 The 32 admitted dates

The first 23 entries are the Rounded birth dates of File_51a §3.1. Aaron, Moses, and Joshua complete the 26 birth entries. Six event entries complete the manifest. All dates in this table are BC, and each row generates one Nativity-family route and one Conquest-family route.

| # | Source | Date BC | Node-class | Source basis |
|---|---|---|---|---|
| 1 | Adam / Creation | 4106 | Creation / Adam | File_51a §3.1 |
| 2 | Seth | 3976 | Birth | File_51a §3.1 |
| 3 | Enosh | 3871 | Birth | File_51a §3.1 |
| 4 | Kenan | 3781 | Birth | File_51a §3.1 |
| 5 | Mahalalel | 3711 | Birth | File_51a §3.1 |
| 6 | Jared | 3646 | Birth | File_51a §3.1 |
| 7 | Enoch's birth | 3486 | Birth | File_51a §3.1 |
| 8 | Methuselah | 3421 | Birth | File_51a §3.1 |
| 9 | Lamech | 3236 | Birth | File_51a §3.1 |
| 10 | Noah | 3056 | Birth | File_51a §3.1 |
| 11 | Shem | 2556 | Birth | File_51a §3.1 |
| 12 | Arphaxad / Flood | 2456 | Birth / shared Flood node | File_51a §3.1 |
| 13 | Shelah | 2421 | Birth | File_51a §3.1 |
| 14 | Eber | 2391 | Birth | File_51a §3.1 |
| 15 | Peleg | 2356 | Birth | File_51a §3.1 |
| 16 | Reu | 2326 | Birth | File_51a §3.1 |
| 17 | Serug | 2296 | Birth | File_51a §3.1 |
| 18 | Nahor | 2266 | Birth | File_51a §3.1 |
| 19 | Terah | 2236 | Birth | File_51a §3.1 |
| 20 | Abraham | 2166 | Birth | File_51a §3.1 |
| 21 | Isaac | 2066 | Birth | File_51a §3.1 |
| 22 | Jacob | 2006 | Birth | File_51a §3.1 |
| 23 | Joseph | 1916 | Birth | File_51a §3.1 |
| 24 | Aaron | 1531 | Birth | File_51a Rounded Aaron companion |
| 25 | Moses | 1526 | Birth | File_51a Moses node |
| 26 | Joshua | 1476 | Birth | File_51a §7.6 |
| 27 | 120-year Flood countdown | 2576 | Event | 2456 + 120 |
| 28 | Abraham's call / departure | 2091 | Event | 2166 − 75 |
| 29 | Jacob's entry into Egypt | 1876 | Event | 2006 − 130 |
| 30 | 400-year affliction node | 1846 | Event | 1446 + 400 |
| 31 | Exodus | 1446 | Event | Fixed Exodus anchor |
| 32 | Enoch's ascension (provisional) | 3121 | Event | 3486 − 365; direct-date convention |

Arphaxad and the Flood occupy one admitted numerical date, 2456 BC. Their shared label is counted once, not as two source trials. In this table the Flood label belongs to the selected Rounded model; it does not override the repository's other Flood states.

Aaron's 1531 BC value is the elected Rounded companion. It is not a revision of the Actual Aaron/Moses chronology. Joshua's 1476 BC birth is a model date associated with the selected 1406 BC Conquest framework; his lifespan of 110 is a textual datum, whereas an absolute BC birth date is not stated in the biblical text.

### 2.2 Inclusion and exclusion decisions

The final manifest records the author's decisions, including decisions made during the exploratory discussion.

| Item | Decision | Reason / count effect |
|---|---|---|
| Temple construction begins, 966 BC | Referenced separately; excluded from statistics | 1 Kings 6:1 supplies an explicit 480-year link to the Exodus. The author had been unsure about admission and retains it only as the reference in §2.5. |
| Joseph's elevation at 1886 BC | Excluded | Not admitted as a major event for this bounded survey. |
| Sarah 2156 BC; Ishmael 2081 BC; Reuben 1921 BC | Excluded supplementary nodes | Not admitted as major players in this test. |
| Enoch's ascension at 3121 BC | Retained provisionally | It had been examined; direct date inversion is explicitly distinguished from a reconstructed inverse ascension. |
| Abraham's call / departure at 2091 BC | Retained | A major event, although initially overlooked in the informal list. |
| 1846 BC affliction node | Retained | Its Conquest-anchor span of 440 is self-inverse; its separate span to the Exodus is 400. The agreed final list retains both standard routes. |
| 6 BC and 1 BC as source dates | Excluded | They serve as anchors / targets rather than additional admitted source rows. |
| Other deaths and terminal reconstructions | Excluded from the scan | Their inverse procedure remains provisional. |
| Cumulative Creation and other cumulative nodes | Excluded as additional source rows | Structural examples and corroborating backbone derivations; no duplicate statistical entries. |
| Segmented Creation backbone | Admitted as a separate two-construction extension | 12026 BC at the Conquest stage and 14726 BC at the Nativity stage; the latter supplies the tenth 6 BC hit. |

The author's restriction on additional round-number constructions does not remove all spans ending in zero: preserving those placeholders is essential to the inverse operation. No new rounded variants are manufactured to improve the result. Once a source is retained in the final manifest, both of its prescribed routes remain in the tables, including misses.

The four supplementary nodes considered earlier were reduced by excluding Sarah, Ishmael, and Reuben while retaining Enoch's ascension. The settled whole-span denominator is therefore 32 × 2 = 64. Earlier working counts are superseded by this manifest. The subsequent backbone extension adds two constructions of the already discussed Creation structure, not a 33rd source date (§8.7).

### 2.3 Enoch's provisional terminal node

The direct date 3121 BC follows from $3486-365=3121$. Enoch's 365 years and being taken by God are textual data (Genesis 5:23–24). The uncertainty in this study concerns the operation to be applied to that terminal event.

Direct inversion of 3121 BC produces 2131 BC from the 1 BC anchor and 4111 BC from the 1401 BC anchor. A reconstruction from an inverted birth and an inverted or retained lifespan is a different operation. The present scan uses only the direct-date version, by the author's explicit provisional inclusion. Both routes miss all five targets.

### 2.4 Routes, source dates, and unique endpoints

There are 32 distinct source dates, comprising 23 dates ending in 6 and nine ending in 1. Two routes per source give 64 comparisons with any one target.

The 64 routes yield 63 distinct numerical endpoints. Jacob at 2006 BC is self-inverse in both families and therefore occurs twice as a route endpoint. The count remains route-based, consistently with the question asked. Those two Jacob routes miss all five tested targets.

These facts also show why 64 routes should not automatically be treated as 64 independent statistical trials.

The paired backbone extension yields 66 constructions and 65 distinct endpoints. Its two new endpoints are absent from the original ledger. The Jacob duplicate remains a miss at every named target, and no repeated regular / cumulative derivation is counted again.

### 2.5 The Temple at 966 BC: explicit Exodus anchor, reference only

The author adds the beginning of Solomon's Temple construction at 966 BC to the report's references. The reason is the explicit chronological statement in 1 Kings 6:1: construction begins in Solomon's fourth regnal year, identified in the MT as the 480th year after the Exodus. Under the repository's adopted 480-year relation and primary Exodus anchor, the placement is:

$$1446-480=966\text{ BC}.$$

The textual datum is the Exodus-to-Temple synchronism; the absolute BC placement follows the adopted chronology (File_05 §1.1). This supplies a defined basis for referencing this post-Conquest event in the inverse study. The criterion is specifically its explicit span from the Exodus, rather than a general extension of the survey into later regnal events.

The author had been uncertain whether to include this event in the sample. Accordingly, it is discussed here but remains excluded from the statistics. Neither Temple route enters the original 64-route scan or the expanded 66-construction tally; the backbone amendment does not change this decision.

Both reference routes use the normal anchors for a 6-ending date. The Conquest route requires retaining the Temple's position later than its anchor. Use the magnitude for digit reversal and preserve the sign of the original BC-label offset:

$$s=D-A,\qquad D'=A+\operatorname{sgn}(s)I(|s|).$$

This signed placement is stated locally for the uncounted Temple reference. All source offsets in the frozen statistical scan remain positive, so its existing calculation and reproduction code require no change.

| Held anchor BC | Original signed offset | Inverted magnitude | Rebuilt inverse date BC |
|---|---|---|---|
| 1406 | 966 − 1406 = −440 | I(440) = 440 | 1406 − 440 = 966 |
| 6 | 966 − 6 = 960 | I(960) = 690 | 6 + 690 = 696 |

Thus the Temple date remains 966 BC under its Conquest-family inverse and becomes 696 BC under its Nativity-family inverse. The self-inverse 440 also places the Temple and the already discussed affliction node on opposite sides of the Conquest anchor:

$$1846-1406=1406-966=440.$$

This is a structural companion to the 1846 BC discussion. The explicit 480-year biblical anchor explains why the Temple is referenced; the author's uncertainty about statistical admission explains why neither Temple route enters the counted sample.

Section 3.8.3 develops the subsequent Temple comparison: the 696 BC inverse expands from the 14726 BC pillar to Rounded AD 526, also reached by expanding the reflected Exodus interval with AD 12026 held. That structural use preserves the Temple's exclusion from the statistical sample.

## 3. Creation trunks and the 6 BC intersection

### 3.1 Whole-span trunk inversions

The following table distinguishes the regular and cumulative source layers. Only the regular Creation row belongs to the frozen source manifest; the cumulative rows are contextual comparisons.

| Source state | Source date | Held anchor | Original span | Inverse span | Rebuilt inverse date |
|---|---|---|---:|---:|---|
| Rounded regular Creation | 4106 BC | 1406 BC | 2700 | 7200 | 8606 BC |
| Rounded regular Creation | 4106 BC | 6 BC | 4100 | 1400 | 1406 BC |
| Rounded cumulative Creation | 14006 BC | 1406 BC | 12600 | 62100 | 63506 BC |
| Rounded cumulative Creation | 14006 BC | 6 BC | 14000 | 41000 | 41006 BC |

The underlying cumulative-to-regular difference is $14006-4106=9900$. File_52a controls the architectural role of that transfer constant. The present report uses it as inherited context and does not reproduce the full second-order closure argument.

### 3.2 The 8600 span to 6 BC

The Conquest-anchored inverse of regular Creation supplies the example emphasized by the author:

$$8606-6=8600=4300+4300.$$

The equal division has arithmetic midpoint 4306 BC. No new event at that midpoint is introduced into the manifest.

The 4300 units are decimal-scale comparisons with the 430 Sojourn register; the text itself supplies the 430-year datum (Exodus 12:40–41). The scale comparison is a structural reading. It is not a claim that the biblical text directly gives an 8600-year span.

This illustrates why 6 BC was already an intersection of interest before the present probability calculation. The 8600 span is itself a miss in the 115 test. Its role in the broader inverse study is retained despite that miss.

### 3.3 The segmented 12026 BC convergence

Whole-span inversion is not additive in general. The following independently stated segmentations therefore remain distinct from the whole-span regular trunk $2700\rightarrow7200$.

Each row holds the Conquest anchor at 1406 BC and inverts the listed components separately.

| Source construction | Original components | Inverted components | Inverted sum | Rebuilt Creation node |
|---|---|---|---:|---|
| Regular Creation–Flood–Conquest | 1650 + 1050 | 5610 + 5010 | 10620 | 12026 BC |
| Regular Creation–Noah–Conquest | 1050 + 1650 | 5010 + 5610 | 10620 | 12026 BC |
| Regular corroborating backbone | 1050 + 600 + 1050 | 5010 + 600 + 5010 | 10620 | 12026 BC |
| Cumulative Creation–Flood–Conquest | 9170 + 3430 | 7190 + 3430 | 10620 | 12026 BC |

The cumulative components use $14006-4836=9170$ and $4836-1406=3430$. The regular components use $4106-2456=1650$ and $2456-1406=1050$. File_52a §§5–6 and §10, and File_52b §2, control these source constructions.

Their common 12026 BC endpoint provides a same-side 529 relation to the Exodus in §6.2. It is not an extra whole-span route in the 64-route survey. The author's subsequent admission of the major backbone includes this Conquest-stage endpoint once as the companion to the completed Nativity-stage endpoint below.

### 3.4 The completed backbone: 14726 BC and the tenth incident

The additional third partition is the 1406 BC → 6 BC interval. Read chronologically from Creation, the regular three-part outline is Creation → Flood → Conquest → 6 BC, with spans 1650 + 1050 + 1400. The cumulative primary outline has spans 9170 + 3430 + 1400. Their reversed sums are respectively 5610 + 5010 + 4100 and 7190 + 3430 + 4100, both equal to 14720. When rebuilding from the held 6 BC anchor, this last chronological partition is processed first.

Source trace: File_52a §7, “Christ anchor and the optional 2700 shift,” explicitly records 1406 BC → 6 BC = 1400, its inverse 4100, the resulting 2700 displacement, and 14726 BC. The author also gave the complete three-part equation in the preceding backbone amendment to this report. Thus the 1406-to-6 BC partition was already stated; the present clarification identifies its position in the chronological outline.

The regular backbone extends naturally from 6 BC through the Conquest and Flood to Creation:

$$6\text{ BC}\rightarrow1406\text{ BC}\rightarrow2456\text{ BC}\rightarrow4106\text{ BC}.$$

Its successive spans, read toward earlier dates, are 1400 + 1050 + 1650. Invert all three separately and rebuild from 6 BC:

$$I(1400)+I(1050)+I(1650)=4100+5010+5610=14720.$$

$$6+14720=14726\text{ BC},\qquad14726-6=14720=640\times23=128\times115.$$

The three rebuilt steps are explicit in the following table. The first endpoint shares the label of regular Creation, but its role here is the rebuilt inverse of the Conquest leg.

| Original leg, read from 6 BC | Original span | Inverse span | Rebuilt endpoint BC |
|---|---:|---:|---:|
| 6 BC to Conquest 1406 BC | 1400 | 4100 | 4106 |
| Conquest 1406 BC to Flood 2456 BC | 1050 | 5010 | 9116 |
| Flood 2456 BC to Creation 4106 BC | 1650 | 5610 | 14726 |

Thus 14726 BC is the natural third step of this admitted regular backbone. File_52a §7 already records the endpoint, its 14720 span to 6 BC, and the exact conversion $14720\times25/23=16000$. The construction predates its present admission to the statistical tally.

The same endpoint survives the following stated alternatives. All displayed dates in the path column are BC.

| Source path | Original successive spans | Inverted successive spans | Sum; endpoint from 6 BC |
|---|---|---|---|
| Regular: 6 → 1406 → 2456 → 4106 | 1400 + 1050 + 1650 | 4100 + 5010 + 5610 | 14720; 14726 BC |
| Regular via Noah: 6 → 1406 → 3056 → 4106 | 1400 + 1650 + 1050 | 4100 + 5610 + 5010 | 14720; 14726 BC |
| Regular via Flood and Noah: 6 → 1406 → 2456 → 3056 → 4106 | 1400 + 1050 + 600 + 1050 | 4100 + 5010 + 600 + 5010 | 14720; 14726 BC |
| Cumulative primary Flood path: 6 → 1406 → 4836 → 14006 | 1400 + 3430 + 9170 | 4100 + 3430 + 7190 | 14720; 14726 BC |

The regular chronology remains the main focus. Noah's two-leg alternative reverses the order of 1050 and 1650, leaving their inverted sum unchanged. The finer 1050 + 600 + 1050 backbone also agrees because $I(1650)=5610=600+5010$. These are specific structural agreements, not a general additivity rule for digit reversal or extra independent target hits.

The completed endpoint stands 2700 earlier than the Conquest-stage endpoint because $I(1400)-1400=4100-1400=2700$. Consequently $12026+2700=14726$. At the earlier stage of this calculation, $12026-1446=10580=5290+5290=20\times529$. The author reads this Exodus relation, the regular Noah alternatives, and the cumulative primary repetition as reinforcement of one major backbone. Only one 14726 BC incident is added to the 6 BC hit count.

Scope of the repetition: File_52a §11 also gives secondary cumulative paths through Shem or Noah that reach 13016 BC from 1406 BC. Completing their 6 BC leg would give 15716 BC. Those are distinct branches; the claim above concerns the primary cumulative Flood path and the specified regular Noah alternatives. They are referenced to delimit the repeat, not admitted as additional constructions in this regular-backbone extension.

### 3.5 Earlier Shem, Flood, and Creation evidence for the 6 BC anchor

The importance assigned to 6 BC did not begin with the count of nine whole-span hits. The earlier files connect it with Shem, the Flood, and the Creation backbone, alongside the 8600 = 4300 + 4300 relation in §3.2.

In the Rounded cumulative state, Shem's birth is 5436 BC and the end of his 600-year life is the 4836 BC Flood boundary (File_51a §24). This is a cumulative construction; it does not relocate the death of Shem in regular chronology. The Flood remains fixed under its Conquest-held inverse because its 3430 span is self-inverse. The discussion's term non-inversable means unchanged by inversion in this instance.

| Earlier backbone reference | Declared arithmetic | Result / role |
|---|---|---|
| Cumulative Shem death / Flood boundary | 5436 − 600 = 4836 | Shared cumulative boundary; birth and death remain distinguished. |
| Flood held at the Conquest | 4836 − 1406 = 3430; I(3430) = 3430 | 1406 + 3430 = 4836 BC; its span to 6 BC is 4830 = 210 × 23. |
| Cumulative Shem birth held at 6 BC | 5436 − 6 = 5430; I(5430) = 3450 | 6 + 3450 = 3456 BC; the span is 3450 = 150 × 23. |
| Cumulative Creation inverse and Prophetic expansion | 14006 − 1406 = 12600; I(12600) = 62100; 62100 × 70/69 = 63000 | 1406 + 63000 = 64406 BC. |
| Expanded Creation return to 6 BC | 64406 − 6 = 64400 = 2800 × 23; 64400 × 25/23 = 70000 | Exact Nativity-anchor return; full sequence in §7. |

The 5436 → 3456 BC result uses a single whole-span inverse from 6 BC. The 14006 → 64406 BC chain instead combines inverse and expansion; it is not literal double decimal reversal. Under the defined operation, $I(I(12600))=12600$, restoring 14006 BC when rebuilt from the same 1406 BC anchor.

These references explain why the author treats Shem, the Flood, and Creation as the backbone of the wider 6 BC pattern. They are not separate additions to the present statistical denominator. The new counted construction is the regular segmented backbone of §3.4, with its Conquest-stage companion; the cumulative primary path corroborates the same endpoints.

### 3.6 Why 12026 and 14726 BC form the shared backbone

The author assigns great importance to this pair because the primary regular and cumulative segmentations converge on the same rebuilt Creation endpoint at each anchor: 12026 BC with 1406 BC held, and 14726 BC after including the 1406-to-6 BC partition and holding 6 BC. The regular chronology remains the principal study; the cumulative primary route supplies the matching construction. The specified Noah alternatives reinforce the regular result (§§3.3–3.4).

The Nativity span of the completed backbone gives the exact Priestly conversion already recorded in File_52a §7:

$$14726-6=14720=23\times640,$$

$$14720\times\frac{25}{23}=16000=40\times400.$$

Holding 6 BC during this conversion places the expanded upper endpoint at 16006 BC. The converted quantity is the 16000-year interval. The original inverse endpoint remains 14726 BC.

The following comparisons show how that same backbone endpoint connects the inverse Shem and Flood states with expanded cumulative Creation. All endpoint labels in the table are BC; each companion retains its stated construction anchor.

| Comparison with the common backbone | Companion derivation / state | Checked span | Factorization |
|---|---|---|---|
| 14726 to 6 | Completed backbone from either the regular or cumulative primary path | 14726 − 6 = 14720 | 23 × 640; Priestly conversion to 16000 |
| 14726 to inverse cumulative Shem, 3456 | Original cumulative birth 5436; 5436 − 6 = 5430; I(5430) = 3450; 6 + 3450 = 3456 | 14726 − 3456 = 11270 | 23 × 490 |
| 14726 to inverse cumulative Flood, 4836 | 4836 − 1406 = 3430; I(3430) = 3430; held 1406 BC preserves 4836 BC | 14726 − 4836 = 9890 | 23 × 430 |
| 14726 to inverse regular Shem, 5526 | Original regular birth 2556; 2556 − 6 = 2550; I(2550) = 5520; 6 + 5520 = 5526 | 14726 − 5526 = 9200 | 23 × 400 = 4 × 2300 |
| Expanded cumulative Creation, 64406, to 14726 | 14006 − 1406 = 12600; I(12600) = 62100; 62100 × 70/69 = 63000; 1406 + 63000 = 64406 | 64406 − 14726 = 49680 | 23 × 2160 = 23 × 6 × 360 |

Label clarification: 11270 is the span to inverse cumulative Shem at 3456 BC. The Flood at 4836 BC supplies the separate 9890 span. Both calculations are exact. The 4836 BC Flood is the Conquest-held self-inverse companion; its anchor is retained when comparing it with the 6 BC-held backbone endpoint. Likewise, 5526 BC is the Nativity-family whole-span inverse of regular Shem, not a reconstruction from an inverse begetting chain.

The 9890 relation recovers the same total as the ordinary Actual cumulative-to-regular MT Creation bridge. File_22 §5.1 records its endpoint form, $14004-4114=9890$, and §7.1 its Year-6 / Adam-state form, $14006-4116=9890$. The agreement is a comparison between declared states. The original Rounded Creation gap remains $14006-4106=9900$; the inverse comparison $14726-4836=9890$ reproduces the Actual bridge's value without changing either chronology.

The 64406 BC endpoint is the expanded cumulative-Creation result discussed in §7 and referred to as the residual endpoint in the conversation. Its 49680 span reconnects that result to the common regular / cumulative inverse backbone through 23 × 6 × 360. The author reads the combined 490, 430, 400, and sixfold 360 relations, together with the 16000 Nativity conversion, as evidence of the structural importance of 14726 BC. At the Conquest stage, 12026 BC supplies the corresponding 10580 = 5290 + 5290 relation to the Exodus. These named relations explain the backbone designation more fully than the divisibility count alone.

Source-history check of the available earlier inverse files:

| Item | Earlier documentation | Status in this report |
|---|---|---|
| Third partition, 1406 BC → 6 BC; 1400 → 4100; 14726 BC | File_52a §7; also explicit in the author's preceding amendment | Existing construction, now identified expressly as the third chronological partition. |
| Regular / cumulative convergence at 12026 BC | File_52a §§5–6 and §10; File_52b §6.1 summarizes the 12026 BC relations | Existing shared backbone; completion to 6 BC is shown in §3.4. |
| 14726 − 6 = 14720; conversion to 16000 | File_52a §7 | Explicit earlier example retained and emphasized. |
| 14726 − 4836 = 9890 = 23 × 430 | File_52a §7 | Explicit earlier Flood comparison retained. |
| 14726 − 3456 = 11270; 14726 − 5526 = 9200; 64406 − 14726 = 49680 | These exact endpoint pairings were not located in the available File_52a / File_52b copies; §16.3 of File_52a already supplies the 64406 BC endpoint | Author-specified comparisons made explicit and verified here; not represented as quotations of an earlier table. |

The earlier files do contain 11270 in another relation, 12746 BC to 1476 BC (File_52a §8.1; File_52b §6.2). That occurrence is distinct from the present Shem comparison. The source-history conclusion is limited to the available copies checked here.

These are structural comparisons involving the already admitted backbone. They do not add new entries to the five-target scan: the expanded tally remains ten hits to 6 BC among 66 constructions. The endpoints' common residue class accounts for their mutual divisibility by 23; their particular quotients and source associations are the numerical–narrative relations being recorded.

### 3.7 The Creation pillar: a common anchor for the original nine hits

The author proposes that 14726 BC is a key date for interpreting the other nine incidents to 6 BC. It represents Creation within this construction because the original string 6 BC → 1406 BC → 2456 BC → 4106 BC terminates at Creation. After segmentwise inversion, the corresponding terminal endpoint is 14726 BC (§3.4). Its use as a pillar is therefore attached to an already defined node-role and shared regular / cumulative backbone.

The comparison uses a different held endpoint from the Nativity-held 16000 conversion in §3.6. Hold Creation at 14726 BC and expand the interval to a later inverse endpoint X by 25/23. In positive BC year labels:

$$X^*=14726-\frac{25}{23}(14726-X).$$

The increased interval moves the later endpoint forward in time, reducing its BC label. All dates in the following examples remain on the BC side, so ordinary same-side subtraction applies.

#### 3.7.1 Terah's inverse and the firstborn threshold

The original Nativity-held inverse is:

$$2236-6=2230\xrightarrow{I}3220,\qquad6+3220=3226\text{ BC}.$$

Now hold the Creation pillar:

$$14726-3226=11500=500\times23,$$

$$11500\times\frac{25}{23}=12500,\qquad12500-11500=1000,$$

$$14726-12500=2226\text{ BC}.$$

The generated label agrees with Terah's firstborn threshold in the +60 state: $2236+60=2296$ and $2296-70=2226$. File_18 §4A.5 explicitly records this threshold and routes its firstborn interpretation to File_69 §5B.5 and File_70 §2.3. The route compares two declared states: Terah's source birth is 2236 BC, while the resulting 2226 BC is a firstborn node in the +60 Terah state. It does not change the source birth to 2296 BC before inversion or propagate +60 through the rest of this survey.

#### 3.7.2 The affliction hub and the Samaria node

The admitted 1846 BC source is unchanged by its Conquest-family inverse because $1846-1406=440$ and $I(440)=440$. Holding the Creation pillar now gives:

$$14726-1846=12880=560\times23,$$

$$12880\times\frac{25}{23}=14000,\qquad14000-12880=1120,$$

$$14726-14000=726\text{ BC},\qquad726-6=720=2\times360.$$

Thus the same expansion rule connects the Egypt / affliction hub to the repository's schematic Samaria siege / collapse node, followed by 720 years to 6 BC. The author's SKL comparison concerns the importance of this 720 interval; File_21's 360:1 scalar and 720 Shem-variable framework provide the relevant span context. No SKL rail is moved by this calculation.

Dependency note — Samaria state: File_16 §7.1, under File_00's Samaria Node control, identifies 726 BC as the idealized siege / collapse label within the 726–721 BC envelope. The present example retains that schematic status and does not promote 726 BC to an independently established civil siege date.

#### 3.7.3 The complete nine-member complementary relation

The common grid explains why all nine original hits can be compared with the same pillar:

$$14726-6=14720=128\times115,$$

$$X-6=115k\quad\Longrightarrow\quad14726-X=115(128-k).$$

Each original inverse endpoint lies between 6 BC and 14726 BC, so both complementary spans are positive. Their 25/23 expansions are exact multiples of 125. The following table applies the single Creation-held rule to all nine entries of §8.3; it does not select only the two interpreted examples.

| Original source / node | Inverse endpoint X, BC | Span to 6 BC | Span from 14726 BC | Expanded span × 25/23 | Rebuilt endpoint X*, BC |
|---|---:|---:|---:|---:|---:|
| Mahalalel | 1731 | 1725 | 12995 | 14125 | 601 |
| Shem | 5526 | 5520 | 9200 | 10000 | 4726 |
| Shelah | 2421 | 2415 | 12305 | 13375 | 1351 |
| Eber | 9321 | 9315 | 5405 | 5875 | 8851 |
| Terah | 3226 | 3220 | 11500 | 12500 | 2226 |
| Abraham | 2076 | 2070 | 12650 | 13750 | 976 |
| Moses | 1616 | 1610 | 13110 | 14250 | 476 |
| 120-year Flood countdown | 8516 | 8510 | 6210 | 6750 | 7976 |
| 400-year affliction node | 1846 | 1840 | 12880 | 14000 | 726 |

Every row's two unexpanded spans sum to 14720. These are complementary descriptions of the existing nine incidents, so their shared divisibility is algebraically inherited. The event associations at 2226 BC and 726 BC supply the specific interpretive observations developed here; the other generated endpoints remain arithmetic outputs without newly assigned event identities.

The two possible held anchors also have a fixed relation. If $N(X)=6+(25/23)(X-6)$ denotes expansion with 6 BC held, and $P(X)=14726-(25/23)(14726-X)$ denotes expansion with the Creation pillar held, then:

$$N(X)-P(X)=\frac{2}{23}(14726-6)=1280.$$

Thus changing the held endpoint shifts every expanded BC label by the same 1280 years. The original Abram–Moses symmetry is preserved as well: their generated endpoints 976 BC and 476 BC lie 250 years on either side of the generated affliction node at 726 BC. This is the image of the 230 + 230 arrangement around 1846 BC in §5.6 under one common affine expansion.

Structural inference: 14726 BC functions as a candidate Creation pillar paired with the 6 BC Nativity anchor. Its construction, all nine complementary intervals, and the two contextual landings support examining that role. Subsequent sections develop the other members: Shem's partition in §3.8, Moses through the Temple convergence in §3.9, Abraham through the Entry–Exodus–Conquest symmetry in §3.10, the countdown through paired ratios in §3.11, and the Mahalalel–Shelah–Eber trio through Cainan in §3.12. All nine now have stated contextual interpretations, whose explanatory force remains a question for the wider study. The existing statistical tally remains ten hits to 6 BC among 66 constructions; this common-anchor interpretation adds no count entries.

### 3.8 Further pillar relations: Entry Mirror, Shem, and the Temple

The author adds three examples connecting the 14726 BC backbone with nodes already defined in the study. Each example retains its own operation and held endpoint. They supplement the structural interpretation without changing the statistical manifest.

#### 3.8.1 The Entry Mirror: 24500 = 14700 + 9800

The inverse of the 1876 BC Entry is 7816 BC. Its Rounded coordinate −7815 reflects to +7815, displayed as AD 7816 under §1.4. The author here uses AD 7815 in the earlier civil-style display. Both statements of the intended durations are exact when their counting conventions are identified:

| Comparison | Author's civil-style labels; BC + AD − 1 | Symmetric Rounded labels; BC + AD − 2 | Duration |
|---|---|---|---:|
| Affliction hub to Entry inverse / Mirror | 1846 BC to AD 7815: 1846 + 7815 − 1 | 1846 BC to AD 7816: 1846 + 7816 − 2 | 9660 |
| Creation pillar to Entry inverse / Mirror | 14726 BC to AD 7815: 14726 + 7815 − 1 | 14726 BC to AD 7816: 14726 + 7816 − 2 | 22540 |

AD 7815 and AD 7816 are not interchangeable labels under one counting rule. The present report continues with the symmetric Rounded display, preserving the author's intended durations.

The earlier hub relation is $4830+4830=9660$, followed by the Prophetic conversion $9660\times70/69=9800$ (§5.7). The larger span supplies:

$$22540=23\times980,\qquad22540\times\frac{25}{23}=24500=490\times50.$$

Comparing the two converted durations gives:

$$24500-9800=14700,$$

$$24500=14700+9800=(30\times490)+(20\times490).$$

Both the 9800 inner duration and the full 24500 therefore have exact 490-based descriptions. This connects the same Entry inverse / Mirror endpoint with the affliction hub and the Creation pillar.

Operator note: the larger duration uses 25/23, whereas the earlier 9800 result uses 70/69. The 14700 remainder is consequently a partition of separately converted durations. If both original segments were expanded uniformly by 25/23, the original partition $22540=12880+9660$ would instead become $24500=14000+10500$. The 700 difference between 10500 and 9800 accounts exactly for $14700=14000+700$. This preserves the author's relation without attributing it to a single uniform expansion.

#### 3.8.2 Creation and inverse Shem: 10000 + 6000 with 6 BC held

The original same-side partition is:

$$14726\text{ BC}\xrightarrow{9200}5526\text{ BC}\xrightarrow{5520}6\text{ BC},$$

$$9200+5520=14720.$$

Here 5526 BC is the Nativity-family whole-span inverse of Shem's Rounded regular birth at 2556 BC (§§3.6 and 8.3). Apply 25/23 to the whole arrangement with 6 BC held:

$$9200\times\frac{25}{23}=10000,\qquad5520\times\frac{25}{23}=6000,$$

$$14720\times\frac{25}{23}=16000=10000+6000.$$

The rebuilt positions are therefore:

$$16006\text{ BC}\xrightarrow{10000}6006\text{ BC}\xrightarrow{6000}6\text{ BC}.$$

Equivalently, $6+(25/23)(14726-6)=16006$ and $6+(25/23)(5526-6)=6006$. This is one common affine expansion with 6 BC fixed. It differs from the Creation-held operation in §3.7, which keeps 14726 BC fixed and sends 5526 BC to 4726 BC.

The author reads 6000 = 10 × 600 as a decimal-scale echo of Shem's 600-year life, and 10000 as an Enochic confinement comparison. The textual basis for the latter is specific: [1 Enoch 18:13–16 and 21:3–6](https://ms.augsburgfortress.org/downloads/9780800699109Chapter1.pdf) describe the imprisonment of seven transgressing heavenly stars and give a 10000-year duration. These passages belong to the Book of the Watchers and support the broad Watcher-confinement association. Their immediate subjects and duration remain distinct from the seventy-generation sentence for Shemihazah / Semjaza and his associates in [1 Enoch 10:11–12](https://en.wikisource.org/wiki/The_Book_of_Enoch_(Charles)/Chapter_10).

File_11 §9.1 already uses 9200 → 10000 in its SP cumulative-to-regular Watcher Bridge. That earlier relation has different source dates and chronological states. The present example reproduces the span mechanism within the Rounded inverse Creation–Shem arrangement; it does not equate the SP endpoints with the present MT-derived inverse endpoints. The partition is arithmetic fact; its Shem / Enochic reading is a stated numerical–narrative comparison.

#### 3.8.3 The Temple inverse and two routes to Rounded AD 526

The separately referenced Temple construction date is 966 BC, anchored to the Exodus by the adopted 480-year relation. Its Nativity-family inverse is 696 BC (§2.5): $966-6=960$, $I(960)=690$, and $6+690=696$.

Hold the Creation pillar at 14726 BC and expand toward the Temple inverse:

$$14726-696=14030=23\times610,$$

$$14030\times\frac{25}{23}=15250.$$

The rebuilt endpoint crosses the era boundary, so the report's Rounded coordinate rule must be applied explicitly:

$$q(14726\text{ BC})=-14725,$$

$$-14725+15250=525=q(\text{AD }526).$$

Thus the result is AD 526 in the symmetric Rounded display. The ordinary civil count from 14726 BC to the same printed AD 526 label would be 15251; a literal civil 15250-year interval would end at AD 525. The present result uses Rounded width throughout this route.

Now reflect both endpoints of the existing Conquest-backbone / Exodus comparison:

$$12026\text{ BC}\leftrightarrow\text{AD }12026,\qquad1446\text{ BC}\leftrightarrow\text{AD }1446.$$

The two positive Rounded coordinates are 12025 and 1445. Their same-side difference remains:

$$12026-1446=10580=5290+5290=20\times529.$$

Hold the reflected backbone at AD 12026 and expand backward from it toward the reflected Exodus:

$$10580\times\frac{25}{23}=11500,$$

$$12026-11500=526.$$

The expanded endpoint is again AD 526. Its displacement from the reflected Exodus is the expansion gain:

$$1446-526=920=11500-10580.$$

| Route | Held anchor | Original companion | Original interval | Expanded interval | Rebuilt Rounded endpoint |
|---|---|---|---:|---:|---|
| Temple inverse from Creation | 14726 BC | 696 BC | 14030 | 15250 | AD 526 |
| Reflected Exodus from the Conquest-stage backbone | AD 12026 | AD 1446 | 10580 | 11500 | AD 526 |

The two stated routes therefore meet at one coordinate, q = +525. A full-width check confirms their placement:

$$14726+12026-2=26750=15250+11500.$$

The two expansion gains also fill the original interval between the Temple inverse and reflected Exodus:

$$15250-14030=1220,\qquad11500-10580=920,$$

$$1220+920=2140=696+1446-2.$$

This makes the relation more specific than two unrelated target differences: expanding the two outer intervals closes the 2140-wide gap at the same Rounded coordinate.

The author associates the resulting $11500=5\times2300$ with sanctuary cleansing. [Daniel 8:14](https://www.biblegateway.com/passage/?search=Daniel+8%3A12-14&version=SG21%3BBDS%3BLSG%3BNIV%3BKJV) supplies the 2300 evenings-and-mornings / sanctuary-restoration theme, rendered as cleansing in the KJV. The fivefold numerical correspondence and the Temple source identity motivate this typological reading. The 11500-year derived span is not the passage's literal stated duration, and Rounded AD 526 remains a generated coordinate rather than an asserted historical Temple-cleansing event.

#### 3.8.4 Scope of the companion examples

These three examples retain distinct constructions: comparison of durations expanded by different ratios; a uniform Nativity-held expansion; and two 25/23 expansions from different fixed backbone anchors that converge at one Rounded coordinate. The Creation-pillar role connects them, while the labels and operators determine each calculation.

The Temple source and both of its inverse routes remain outside the statistics, as requested in §2.5. None of the converted spans, the Enochic comparison, or the common AD 526 endpoint adds a statistical entry. The tally remains ten positive hits to 6 BC among 66 constructions. The new material develops the interpretation of already referenced nodes and records the exact mathematical conditions of the proposed relations.


### 3.9 Moses–Temple closure: the second expansion and 1000 + 1000

The author extends the Temple convergence by applying 25/23 a second time to the interval held at the reflected backbone. This returns the 476 BC endpoint already generated from inverse Moses and adds a corresponding forward relation to Moses' reflected birth.

Era-label clarification: the opening reference in the author's amendment is read as AD 12026 to AD 1446, first expanding to AD 526. This is the reflected interval established in §3.8.3. The second expansion then crosses to 476 BC. Literal AD 12026 to 1446 BC would have Rounded width 13470, not 10580, so the BC labels on 1446 and the first 526 in that opening sentence are treated as slips. All relations below use the existing symmetric Rounded convention.

#### 3.9.1 Repeat the expansion with AD 12026 held

The original reflected interval contains the second factor of 23:

$$12026-1446=10580=5290+5290=20\times23^2.$$

Its successive exact expansions are:

$$10580\xrightarrow{\times25/23}11500\xrightarrow{\times25/23}12500,$$

$$20\times23^2\rightarrow20\times23\times25\rightarrow20\times25^2.$$

With AD 12026 fixed, q = +12025:

| Stage | Interval from the fixed reflected backbone | Endpoint coordinate | Rounded endpoint |
|---|---:|---:|---|
| Original reflected Exodus | 10580 | 12025 − 10580 = 1445 | AD 1446 |
| First 25/23 expansion | 11500 | 12025 − 11500 = 525 | AD 526 |
| Second 25/23 expansion | 12500 | 12025 − 12500 = −475 | 476 BC |

The first gain is 920 and the second is 1000. The second gain moves the endpoint from AD 526 to 476 BC by a Rounded width of 1000:

$$476+526-2=1000.$$

This is the second exact expansion already permitted by the 20 × 529 structure in §6.2, now given its explicit endpoint.

#### 3.9.2 The same 476 BC from inverse Moses

Moses' Rounded birth at 1526 BC inverts from the 1406 BC Conquest anchor to 1616 BC:

$$1526-1406=120,\qquad I(120)=210,\qquad1406+210=1616.$$

The Creation-held result from §3.7 is:

$$14726-1616=13110=570\times23,$$

$$13110\times\frac{25}{23}=14250,\qquad14726-14250=476\text{ BC}.$$

Thus the second expansion held at AD 12026 and the first expansion of inverse Moses held at 14726 BC meet at the same Rounded coordinate, q = −475.

The source connection identified by the author is exact. Inverse Moses and inverse Temple stand 920 years apart:

$$1616-696=920=40\times23.$$

A common 25/23 expansion with Creation held sends that pair to 476 BC and AD 526. Their separation becomes:

$$920\times\frac{25}{23}=1000,$$

$$q(\text{AD }526)-q(476\text{ BC})=525-(-475)=1000.$$

Moses' inverse was constructed from the Conquest anchor, while the Temple inverse was constructed from 6 BC. Those original routes remain distinct. The common Creation-held expansion operates on their already established endpoints.

#### 3.9.3 The forward counterpart reaches Moses' reflected birth

The interval from the shared AD 526 endpoint to the reflected Exodus at AD 1446 is also 920:

$$1446-526=920.$$

Now hold AD 526 and expand this interval forward by 25/23:

$$920\times\frac{25}{23}=1000,\qquad526+1000=1526.$$

The result is AD 1526, the Rounded Mirror of Moses' adopted birth at 1526 BC. The conversion adds 80:

$$1000-920=80=1526-1446.$$

That gain is exactly the birth-to-Exodus interval in the adopted Moses chronology. In the reflected display, the birth label lies later than the Exodus label; the operation preserves this stated Mirror orientation.

The resulting arrangement is:

$$476\text{ BC}\xrightarrow{1000}\text{AD }526\xrightarrow{1000}\text{AD }1526.$$

In Rounded coordinates it is the equally spaced triple −475, +525, +1525. AD 526 is its exact midpoint, and the total Rounded width is:

$$476+1526-2=2000.$$

The ordinary civil cross-axis counts between these printed labels would be one year larger. The 1000 + 1000 statement belongs to the report's declared Rounded coordinate system.

#### 3.9.4 Structural meaning and dependence

The combined construction connects four specified source roles: Moses' birth, the Temple, the reflected Exodus, and the shared regular / cumulative Creation backbone. It uses the same 25/23 ratio throughout the present section. The held endpoint is AD 12026 for the repeated backbone expansion, 14726 BC for the inverse Moses / Temple pair, and AD 526 for the forward counterpart to Moses' reflected birth.

The relation is algebraically coherent. Given the AD 526 convergence of §3.8.3 and the shared original 920 gap, the common Creation-held expansion must place inverse Moses 1000 before the expanded Temple. The second expansion of 11500 adds that same 1000, and the forward expansion of the other 920 interval places Moses' reflected birth 1000 after the center. These consequences explain how the parts fit together; they are not independent probability trials.

The author reads this person- and Temple-specific closure as strengthening the proposed role of 14726 BC as an interpretive pillar. The arithmetic is exact under the declared model. Its evidential weight remains a structural assessment of the connected construction; no new coincidence probability or statistical admission is asserted. The Temple remains reference-only, and the ten-of-66 tally is unchanged.


### 3.10 Abraham: the 430 / 470 symmetry and Moses at the midpoint

The author develops the Abraham member already present in the complete nine-member table (§3.7.3). Abraham's Rounded birth is 2166 BC. Its Conquest-family whole-span inverse is:

$$2166-1406=760\xrightarrow{I}670,\qquad1406+670=2076\text{ BC}.$$

This is one of the original nine hits to 6 BC:

$$2076-6=2070=18\times115.$$

#### 3.10.1 Creation-held expansion to 976 BC

Hold 14726 BC and expand the interval to inverse Abraham:

$$14726-2076=12650=550\times23,$$

$$12650\times\frac{25}{23}=13750=550\times25.$$

The gain is 1100 years. The later endpoint moves forward in time from 2076 BC to 976 BC:

$$13750-12650=1100,$$

$$14726-13750=2076-1100=976\text{ BC}.$$

The author's forward-time notation “2076 BC + 1100” describes this movement; subtraction is used for the positive BC year labels. The result 976 BC was already computed in the full pillar table. Its contextual relations are now made explicit.

#### 3.10.2 The Entry–Exodus–Conquest symmetry

The generated 976 BC endpoint completes two routes from the 1876 BC Entry:

| Route, all dates BC | First interval | Second interval | Total |
|---|---:|---:|---:|
| 1876 → 1446 → 976 | 430 | 470 | 900 |
| 1876 → 1406 → 976 | 470 | 430 | 900 |

The complementary order is equivalently expressed as one four-node sequence:

$$1876\text{ BC}\xrightarrow{430}1446\text{ BC}\xrightarrow{40}1406\text{ BC}\xrightarrow{430}976\text{ BC}.$$

Thus the Exodus and Conquest occupy symmetric positions between Entry and the generated Abraham endpoint. Both the outer pair and the inner pair have midpoint 1426 BC:

$$\frac{1876+976}{2}=\frac{1446+1406}{2}=1426.$$

This is an arithmetic center, with no new event assigned to 1426 BC. The 430 / 470 reversal follows the existing 40-year Exodus-to-Conquest separation.

The author's further 390-year relation is exact:

$$976-586=390.$$

It partitions the existing Entry-to-Jerusalem interval controlled by File_16:

$$1876-586=1290=900+390=430+40+430+390.$$

The 586 BC node retains its repository Jerusalem-destruction state. The generated 976 BC endpoint has a structural role within this partition; it is not identified with the separately retained 966 BC Temple construction date.

#### 3.10.3 Moses' birth bisects the 1100-year movement

The expansion's 1100-year gain has the stated midpoint:

$$2076-1526=550,\qquad1526-976=550,$$

$$2076\text{ BC}\xrightarrow{550}1526\text{ BC}\xrightarrow{550}976\text{ BC}.$$

Thus Moses' adopted birth at 1526 BC lies exactly between inverse Abraham at 2076 BC and Abraham's Creation-held expanded endpoint at 976 BC.

The arithmetic can also be read directly from the expansion factor: the input span is 550 × 23, the output is 550 × 25, and their difference is 2 × 550. The midpoint identity selects Moses' existing birth node within that movement. No further transformation of Moses' birth is used here; this midpoint differs from the inverse-Moses and reflected-birth constructions of §3.9.

#### 3.10.4 Narrative interpretation and scope

The author reads the symmetry as coherent with Abraham's role in the foretelling of his descendants' foreign sojourn, affliction, departure, and return to the land (Genesis 15:13–16). The Entry, Exodus, and Conquest are therefore contextually relevant comparison nodes for an Abraham-derived endpoint. The 430-year Entry-to-Exodus register retains its own source and adopted chronology (Exodus 12:40–41; Files 05 and 16); it is not substituted for the 400 years stated in Genesis 15:13.

The two observations are specific: the generated endpoint closes the 430 + 40 + 430 sequence, while the gain from inverse Abraham to that endpoint is centered on Moses' birth at 550 + 550. Their arithmetic is exact. Their interpretation as a connected Abraham–Moses / Egypt–land pattern is a structural and narrative reading within the model.

The 430 / 470 routes, the four-node sequence, and the 1426 BC center are related descriptions of one configuration. The 390 extension reconnects that configuration to the existing 1290 Nexus span. They do not supply separate independent statistical counts. The original Abraham hit, source manifest, and ten-of-66 tally remain unchanged.


### 3.11 The Flood countdown: 6900, 6450, and the ten-member interpretation

The author connects the inverse Abraham–Moses pair with the 120-year countdown to the Flood and develops two expansions from the same Creation pillar. This gives the countdown member of the original nine-hit table an explicit role in the proposed interpretation of the ten-member family.

#### 3.11.1 Seven named generations and the inverse 460 interval

Abraham and Moses bound the seven named generations Abraham, Isaac, Jacob, Levi, Kohath, Amram, and Moses. File_61 explicitly lists this seven-member sequence. The count is inclusive of both endpoints and contains six parent–child links.

Their Conquest-family inverses are already established:

$$2166\text{ BC}\rightarrow2076\text{ BC},\qquad1526\text{ BC}\rightarrow1616\text{ BC}.$$

The inverse endpoints stand 460 years apart:

$$2076-1616=460=20\times23.$$

The Rounded 120-year countdown source is 2576 BC, 120 before the Rounded Flood at 2456 BC. Its Conquest-family inverse is:

$$2576-1406=1170,\qquad I(1170)=7110,\qquad1406+7110=8516\text{ BC}.$$

The countdown inverse is 6900 before inverse Moses:

$$8516-1616=6900=15\times460.$$

The corresponding interval to inverse Abraham is $8516-2076=6440=14\times460$. Thus the three endpoints have the explicit partition:

$$8516\text{ BC}\xrightarrow{14\times460}2076\text{ BC}\xrightarrow{460}1616\text{ BC}.$$

These are endpoint intervals. The 460 is not divided into six equal biological generation lengths.

#### 3.11.2 Two expansions with 14726 BC held

The shared input span is:

$$14726-8516=6210=90\times69=270\times23.$$

Both named Key-of-23 ratios therefore operate exactly:

| Operation, 14726 BC held | Original span | Expanded span | Gain | Rebuilt endpoint BC | Span to Moses' birth, 1526 BC |
|---|---:|---:|---:|---:|---:|
| Prophetic 70/69 | 6210 | 6300 | 90 | 8426 | 6900 |
| Priestly 25/23 | 6210 | 6750 | 540 | 7976 | 6450 |

The Prophetic route gives:

$$6210\times\frac{70}{69}=6300,\qquad14726-6300=8426,$$

$$8426-1526=6900.$$

Its 90-year displacement matches the difference between inverse Moses and Moses' adopted birth:

$$8516-8426=1616-1526=90.$$

Consequently the pair 8516 / 1616 BC becomes 8426 / 1526 BC by the same forward-time displacement of 90 at each end, preserving the 6900 interval. Only the countdown endpoint is generated by the displayed Creation-held 70/69 expansion; the Moses endpoint is compared with its already adopted birth date.

The Priestly route gives:

$$6210\times\frac{25}{23}=6750,\qquad14726-6750=7976,$$

$$7976-1526=6450.$$

Audit correction: the author's 7926 BC is read as 7976 BC. The multiplication gives 6750 and therefore fixes the endpoint at 7976 BC; this agrees with the pre-existing full pillar table in §3.7.3 and with the author's stated 6450 span. Literal 7926 BC would instead stand 6400 before 1526 BC.

#### 3.11.3 The 460 / 430 comparison and the call-to-Exodus span

The two resulting countdown-to-Moses intervals have matching fifteenfold descriptions:

$$6900=15\times460,\qquad6450=15\times430.$$

Their difference is:

$$8426-7976=450=15\times(460-430).$$

The author's call-to-Exodus comparison also checks:

$$2091-1446=645,\qquad6450=10\times645=15\times430.$$

The 645 comparison uses the adopted 2091 BC call and 1446 BC Exodus anchors. It does not identify the whole call-to-Exodus interval with the 430-year Entry-to-Exodus register. Those two source spans remain distinguishable even though their scaled forms coincide here.

The relations connect the countdown, the inverse endpoints of the Abraham–Moses genealogy, Moses' birth, and the earlier Abraham call / Exodus interval. The shared 6210 input and held Creation pillar control both expansions; changing only the declared ratio gives the 6900 and 6450 outcomes.

#### 3.11.4 The ten-member interpretive program

The author proposes that the entire ten-member family of 6 BC intersections can be understood through the 14726 BC key. This is recorded as the organizing thesis of the continuing study. With the joint Mahalalel–Shelah–Eber treatment in §3.12, the following inventory now gives each member a stated contextual interpretation.

| Member | Original counted endpoint BC | Interpretation or current status |
|---|---:|---|
| Mahalalel | 1731 | Creation-held endpoint 601 BC; joint Cainan pattern, 1950 + 1950 Rounded midpoint, 130-year multiples, and Conquest-held 70-year continuation (§3.12). |
| Shem | 5526 | 9200 → 10000 relation to the pillar; Nativity-held 10000 + 6000 partition and Enochic / lifespan comparison (§§3.6, 3.8.2). |
| Shelah | 2421 | Self-inverse hinge of 690 + 6900; Creation-held endpoint 1351 BC; Cainan 130 + 330 context, reflected 3900 / 3440 / 2795 spans (§3.12). |
| Eber | 9321 | Creation-held endpoint 8851 BC; outer 23 × 330 → 25 × 330 pattern and Rounded 11270 = 23 × 490 to reflected original Shelah; declared civil Christological comparisons (§3.12). |
| Terah | 3226 | Creation-held endpoint 2226 BC agrees with the firstborn threshold in the +60 Terah state (§3.7.1). |
| Abraham | 2076 | Creation-held endpoint 976 BC frames the Entry–Exodus–Conquest sequence and gives Moses' 550 + 550 birth midpoint (§3.10). |
| Moses | 1616 | Creation-held endpoint 476 BC participates in the Temple / reflected-backbone closure and 1000 + 1000 arrangement (§3.9). |
| 120-year Flood countdown | 8516 | Prophetic / Priestly endpoints 8426 / 7976 BC give 6900 / 6450 to Moses' birth; 15 × 460 / 15 × 430 (§3.11). |
| Affliction hub | 1846 | Creation-held endpoint 726 BC, schematic Samaria node; transformed Abram–Moses midpoint (§§3.7.2–3.7.3). |
| Completed Creation backbone | 14726 | Tenth counted incident and common pillar; its 14720 span to 6 BC expands to 16000 with 6 BC held (§§3.4, 3.6). |

All nine original source members now have proposed contextual interpretations in these pillar sections; the Creation backbone supplies the tenth member's documented anchor role and Nativity-span interpretation. Mahalalel, Shelah, and Eber receive a cursory joint exploration centered on their connection to the omitted Cainan. This completes the inventory's coverage, while the trio's wider complexity remains unexamined. Historical intention, theological meaning, and joint statistical significance remain separate questions.

The tenth member is the common pillar itself. Its meaningful comparison here is its nonzero span to 6 BC; expanding its zero displacement from itself would add no information. Accordingly, the proposed interpretation concerns the Creation–Nativity framework and explicitly stated held-anchor operations, rather than ten identical nonzero operations with Creation held.

All new spans are arithmetic facts under the specified constructions. Their narrative coherence is the author's structural reading. The 14 + 1 partition, matched 90-year shifts, and 15 × 460 / 15 × 430 comparison are algebraically related parts of this example. Neither they nor the inventory alter the original source manifest or the ten-of-66 statistical tally.


### 3.12 Mahalalel, Shelah, and Eber: a cursory exploration of the joint Cainan pattern

The remaining three source members receive one connected, preliminary interpretation. Their Nativity-family inverses place Shelah between intervals of 690 and 6900. Expansion from the common 14726 BC pillar produces 601, 1351, and 8851 BC, while preserving that 1:10 relation. The author reads the resulting intervals through the restored second Cainan's 130 + 330 lifespan and his declared 2551–2091 BC companion state.

Scope — cursory exploration: the author closely examined the relationships that caught his attention. The trio is substantially more complex than this account, and a thorough examination would be too demanding for the present treatment. Its principal observation is the connection to Cainan, omitted in the MT genealogy and understood as expunged under the author's restoration hypothesis. Section 3.12.7 records further examples to illustrate that wider complexity.

This locally opens restored Cainan for structural comparison. It does not insert him into the frozen MT source manifest, re-date its upstream generations, or add an inverse route to the statistical tally.

#### 3.12.1 Source inversions and the common expansion

The source dates remain Mahalalel 3711 BC, Shelah 2421 BC, and Eber 2391 BC. Mahalalel and Shelah are initially $3711-2421=1290$ apart. Each uses the 1 BC anchor for the whole-span inversion:

| Source | Birth BC | Span to 1 BC | Inverted span | Rebuilt inverse BC |
|---|---:|---:|---:|---:|
| Mahalalel | 3711 | 3710 | 1730 | 1731 |
| Shelah | 2421 | 2420 | 2420 | 2421 |
| Eber | 2391 | 2390 | 9320 | 9321 |

Shelah's source is self-inverse. In chronological order, the inverse endpoints give:

$$9321\text{ BC}\xrightarrow{6900}2421\text{ BC}\xrightarrow{690}1731\text{ BC}.$$

Shelah is the internal junction of the two spans, dividing them in a 10:1 ratio when read in that order. The author's description of Shelah as “centered” is understood in this structural sense; Shelah is not their arithmetic midpoint.

Holding 14726 BC gives the three exact conversions:

| Member | Inverse endpoint BC | Span from 14726 BC | Expanded span × 25/23 | Gain | Rebuilt endpoint BC |
|---|---:|---:|---:|---:|---:|
| Eber | 9321 | 5405 = 23 × 235 | 5875 | 470 | 8851 |
| Shelah | 2421 | 12305 = 23 × 535 | 13375 | 1070 | 1351 |
| Mahalalel | 1731 | 12995 = 23 × 565 | 14125 | 1130 | 601 |

Among the three admitted ratios 25/23, 70/69, and 300/299, only 25/23 yields integral expanded year-spans for these inputs. The other two yield, respectively, $16450/3$, $37450/3$, $39550/3$ and $70500/13$, $160500/13$, $169500/13$. Thus “only” refers to exact whole-year conversion by this stated ratio set, not to uniqueness among all conceivable multipliers.

The resulting order is:

$$8851\text{ BC}\xrightarrow{7500}1351\text{ BC}\xrightarrow{750}601\text{ BC}.$$

Both gaps undergo the same dilation: $6900\to7500$ and $690\to750$. The preserved 1:10 partition follows from applying one affine expansion to all three points.

#### 3.12.2 The 330 / 460 structure and Cainan's restored state

The original inverse trio has the total width:

$$6900+690=7590=23\times330=33\times230=\frac{33}{2}\times460.$$

After expansion:

$$7500+750=8250=25\times330.$$

The gain across the outer endpoints is $8250-7590=660=2\times330$. These are related descriptions of one conversion, $23\times330\to25\times330$.

The textual comparison concerns the second, post-Flood Cainan. The MT-derived [Genesis 11:12](https://www.biblegateway.com/passage/?search=Genesis+11%3A12&version=KJV) connects Arphaxad directly with Shelah. The consulted [Septuagint translation of Genesis 11](https://ebible.org/eng-Brenton/GEN11.htm) places Cainan between them and gives him 130 years before Shelah's birth and 330 afterward. This supports the numerical partition $130+330=460$. The same age pair also occurs for Shelah in that rendering, so the numbers alone do not uniquely identify Cainan.

The author's description of Cainan as “expunged” expresses the restoration hypothesis used in this study; the textual difference by itself does not establish the historical direction or intention of the change. Shelah occupies the immediate-son position under Arphaxad in the MT line and follows Cainan in the restored companion.

Files 18 and 16 supply the selected regular restored-Cainan state:

$$2551\text{ BC}\xrightarrow{130}2421\text{ BC}\xrightarrow{330}2091\text{ BC}.$$

Here 2421 BC is Shelah's birth, and 2091 BC is Cainan's death, sharing the adopted Abraham-call label. The full lifespan is 460. The regular insertion operator remains the 130-year begetting interval; measuring Cainan's lifespan as 460 does not replace that operator with the cumulative +460 rule. The other Cainan date variants are not opened.

This supplies the proposed contextual link: the self-inverse Shelah is the junction of both Cainan's 130 + 330 life partition and the trio's 6900 + 690 arrangement, whose total carries 330 under the 23-to-25 conversion.

#### 3.12.3 Cainan's companion inverse and the 12000 Mirror width

Cainan's excluded source date can be examined separately with the same 1 BC selector:

$$2551-1=2550,\qquad I(2550)=5520,\qquad1+5520=5521\text{ BC}.$$

With 1 BC held, the 5520 span expands to 6000:

$$5520\times\frac{25}{23}=6000,\qquad1+6000=6001\text{ BC}.$$

Rounded reflection gives AD 6001, and the full symmetric width is:

$$W_R(6001\text{ BC},\text{AD }6001)=6001+6001-2=12000.$$

The civil elapsed count between those same printed labels would be 12001. The 12000 here is expressly the Rounded Mirror width.

The corresponding Conquest-family anchor relations are:

| Cainan event | Endpoint BC | Span to 1401 BC | Factorization |
|---|---:|---:|---|
| Birth | 2551 | 1150 | 50 × 23 |
| Death / adopted Abraham-call label | 2091 | 690 | 30 × 23 |

Their difference is $1150-690=460=20\times23$, preserving the lifespan. These are supporting companion calculations; Cainan remains outside the original 32-date survey and the 66-construction tally.

#### 3.12.4 Christological comparisons and the explicit counting split

For the author's selected AD 30 death-of-Christ comparison, the civil elapsed count gives:

$$C(8851\text{ BC},\text{AD }30)=8851+30-1=8880,$$

$$C(1351\text{ BC},\text{AD }30)=1380=3\times460,$$

$$C(601\text{ BC},\text{AD }30)=630.$$

The author reads 8880 as a new-creation number. That meaning and the selected AD 30 event label belong to the theological / chronological interpretation; the present calculation establishes the stated duration under civil counting.

The 31 BC comparison requires an explicit local Mirror convention. In the ordinary astronomical map $a(B\text{ BC})=1-B$ and $a(\text{AD }A)=A$, AD 30 has coordinate +30 and reflects to −30, displayed as 31 BC. Accordingly:

$$8851-31=8820=49\times180.$$

Under this report's symmetric Rounded map, however, 31 BC reflects to AD 31, while AD 30 reflects to 30 BC. The author's “31 BC Mirror of AD 30” is retained only as the stated civil-display reflection; it does not redefine the Rounded Mirror elsewhere.

The mixed-count examples are therefore recorded side by side:

| Printed endpoint pair | Civil count C | Rounded width W_R | Count used in the author's comparison |
|---|---:|---:|---|
| 8851 BC → AD 30 | 8880 | 8879 | Civil: new-creation comparison |
| 8851 BC → AD 2421 | 11271 | 11270 | Rounded: 23 × 490 to reflected original Shelah |
| 1351 BC → AD 30 | 1380 | 1379 | Civil: 3 × 460 |
| 1351 BC → AD 2551 | 3901 | 3900 | Rounded: to reflected Cainan birth |
| 601 BC → AD 30 | 630 | 629 | Civil: selected Christological endpoint |
| 601 BC → AD 1351 | 1951 | 1950 | Rounded: to reflected expanded Shelah |
| 1351 BC → AD 2091 | 3441 | 3440 | Rounded: 8 × 430 |
| 1351 BC → AD 1446 | 2796 | 2795 | Rounded: 13 × 215 |

Thus $11270=23\times490$ is exact for 8851 BC to Rounded AD 2421. The civil 8880 / 1380 / 630 comparisons retain AD 30 as supplied. They would have those same numerical widths under the Rounded convention if the positive endpoint were instead labeled AD 31, but that is a different displayed construction and is not silently substituted.

#### 3.12.5 The 1950 + 1950 midpoint and the 430 family

Mahalalel's expanded endpoint has the same-side relations:

$$601-1=600,$$

$$2551-601=1950=15\times130,$$

$$2421-601=1820=14\times130.$$

The latter two differ by Cainan's 130-year interval to Shelah. The first 1950 interval continues across the Rounded Mirror to the reflection of expanded inverse Shelah:

$$2551\text{ BC}\xrightarrow{1950}601\text{ BC}\xrightarrow{1950}\text{AD }1351.$$

In q coordinates these points are −2550, −600, and +1350. Therefore 601 BC is their exact Rounded midpoint. The outer width is 3900; reflecting the complete arrangement gives the corresponding 1351 BC → AD 601 → AD 2551 sequence, also 1950 + 1950. This is the same 3900 structure as the Shelah-to-reflected-Cainan comparison in the table, read in its reflected orientation.

Expanded Shelah also stands 1200 after Cainan's birth on the BC side:

$$2551-1351=1200.$$

Toward the reflected death / call and Exodus labels, the Rounded widths are:

$$W_R(1351\text{ BC},\text{AD }2091)=1351+2091-2=3440=8\times430,$$

$$W_R(1351\text{ BC},\text{AD }1446)=1351+1446-2=2795=\frac{13}{2}\times430=13\times215.$$

Their difference is $3440-2795=645=2091-1446$, preserving the adopted call-to-Exodus span already used in §3.11.3. The author thus connects the 130 / 330 Cainan partition with the 430 / 215 Exodus family through the expanded trio.

Rounded reflection preserves all q widths, so each complete Rounded arrangement can be read on the opposite side with its endpoint order reversed. This symmetry does not automatically transfer the separately chosen civil AD 30 calculations into the Rounded register.

#### 3.12.6 The Conquest-held 70-year continuation

The additional expansion from 601 BC changes the held anchor from Creation to Conquest:

$$1406-601=805=23\times35,$$

$$805\times\frac{25}{23}=875,\qquad875-805=70.$$

With 1406 BC fixed, the rebuilt endpoint is:

$$1406-875=531\text{ BC}.$$

Thus the stated local continuation is 601 → 531 BC, a forward-time displacement of 70 years. It is a second expansion of a newly selected 805-year interval, not a repeat of the original 12995-year Creation interval.

The author proposes a Babylonian seventy-year echo, for which [Jeremiah 29:10](https://www.biblegateway.com/passage/?search=Jeremiah+29%3A10&version=KJV) supplies the textual duration. The generated 601–531 BC pair functions here as a symbolic numerical comparison, without asserting it as the historical dating of that captivity.

#### 3.12.7 Cursory scope and place in the ten-member interpretation

Mahalalel, Shelah, and Eber receive an initial joint interpretation through Shelah and the restored-Cainan companion. Their common expansion, the 23 × 330 → 25 × 330 total, the Cainan 130 / 330 references, and the 1950 + 1950 arrangement belong to that proposed connection. The main point is the trio's relation to Cainan, omitted from the MT genealogy and described as expunged under the author's restoration hypothesis.

The following additional branches illustrate why the account remains cursory. Each expansion holds the stated later BC anchor and extends the earlier endpoint backward in time:

| Starting endpoint BC | Held anchor BC | Original span | Ratio | Expanded span | Earlier endpoint BC | Backward shift |
|---|---:|---:|---|---:|---:|---:|
| 2091 | 1401 | 690 | 25/23 | 750 | 2151 | 60 |
| 2551 | 1401 | 1150 | 25/23 | 1250 | 2651 | 100 |
| 5521 | 1 | 5520 | 70/69 | 5600 | 5601 | 80 |
| 5521 | 1 | 5520 | 25/23 | 6000 | 6001 | 480 |

With Cainan's original 2551 BC birth retained, the first branch leaves:

$$2551-2151=400.$$

The second branch changes the Rounded interval to reflected expanded Shelah from 3900 to 4000:

$$W_R(2551\text{ BC},\text{AD }1351)=2551+1351-2=3900,$$

$$W_R(2651\text{ BC},\text{AD }1351)=2651+1351-2=4000.$$

The last two branches begin with the previously established 1 BC whole-span inverse $2551\text{ BC}\to5521\text{ BC}$. Its 5520-year interval permits both 70/69 and 25/23, yielding 5601 or 6001 BC with 1 BC held. This does not change the earlier observation that only 25/23 gives integral outputs for the three original Creation-held spans.

These examples illustrate the branching complexity beyond the relationships initially noticed. The author examined selected observations closely; a thorough examination of the trio would be too demanding for the present treatment and has not been undertaken. The Christological, Babylonian, and other subsidiary associations retain their stated interpretive status.

Together with the earlier examples, each of the ten members in §3.11.4 now has an initial contextual interpretation. Completion refers to coverage of that inventory; the trio remains a cursory exploration centered on its omitted-Cainan connection. The shared affine ratios, Mirror duplicates, and equivalent partitions are dependent relations within the pattern. Cainan's companion calculations add no new counted source or independent success, and the existing ten-of-66 tally remains unchanged.


### 3.13 The 12026 BC counterpart key: strong anchor assessment, partial exploration

#### 3.13.1 Authorial assessment and the two anchor roles

The author regards 12026 BC as a major anchor node acting as a key comparable to 14726 BC. In his assessment, its importance is not in doubt within the developing model: the evidence is strong even though the detailed interpretation remains incomplete. The present addition is therefore marked **partially explored**. That qualification concerns the extent of the examination and the interpretation of its many relations; it does not dilute the author's confidence in the central anchor role.

The proposed pairing is specific. The 14726 BC pillar organizes the Rounded inverse dates associated with 6 BC (§§3.7–3.12). Its 12026 BC counterpart organizes Actual, non-Rounded MT regular dates associated with 1446 BC, with cumulative chronology supplying both the matching inversion route and the Actual Creation bridge. The principal numerical themes proposed for this second field are 1260, 13, and 666, together with Creation, the Watchers, judgment, and restoration.

The existing Exodus relation is:

$$12026-1446=10580=460\times23=20\times529.$$

For an Actual source date $X$ between these endpoints,

$$12026-X=10580-(X-1446).$$

Thus a multiple of 23 to 1446 BC also gives a multiple of 23 to 12026 BC. This common divisibility explains why the anchor can organize such dates; the specific expansion results and narrative associations supply the further interpretation. The present discussion examines the Creation and Jared–Noah–Flood examples. It does not claim that every remaining date in the supplied chart has already received an individual examination.

The source layers remain distinct. The original direct Rounded whole-span scan has zero positive hits to 1446 BC; the separately admitted segmented 12026 BC backbone contributes the one positive hit in the expanded ledger. The Actual-date study below is a structural continuation, not a change to that frozen survey.

#### 3.13.2 Three starting levels and the newly introduced 8066 BC node

The two Rounded source chains, read backward from 6 BC, are:

- Regular: 6 → 1406 → 2456 → 4106 BC, with spans 1400 + 1050 + 1650.
- Cumulative: 6 → 1406 → 4836 → 14006 BC, with spans 1400 + 3430 + 9170.

Their placeholder-preserving inverse spans are 4100, 5010, 5610, 3430, and 7190 respectively. Sections 3.3–3.4 already establish the two- and three-segment constructions. Holding the original Flood endpoint and reversing only its remaining Creation segment supplies the third starting level.

| Original starting anchor / retained segments | Regular rebuilt Creation BC | Cumulative rebuilt Creation BC |
|---|---:|---:|
| 6 BC; all three segments | 6 + 4100 + 5010 + 5610 = 14726 | 6 + 4100 + 3430 + 7190 = 14726 |
| 1406 BC; final two segments | 1406 + 5010 + 5610 = 12026 | 1406 + 3430 + 7190 = 12026 |
| Respective original Flood; final segment only | 2456 + 5610 = 8066 | 4836 + 7190 = 12026 |

For the regular Flood, inversion of its Conquest span gives $1406+I(1050)=6416$ BC. The last row nevertheless holds the **original 2456 BC Flood**, as the stated operation requires:

$$2456+I(4106-2456)=2456+5610=8066\text{ BC}.$$

For the cumulative Flood, $4836-1406=3430$ is self-inverse. Its Conquest-held inverse therefore remains 4836 BC, and:

$$4836+I(14006-4836)=4836+I(9170)=4836+7190=12026\text{ BC}.$$

This explains the repeated cumulative 12026 BC endpoint. The underlying 3430 span is unchanged by reversal; the printed date 4836 is not being declared literally incapable of digit reversal.

The newly introduced 8066 BC node has the following relations. All dates in this table are BC.

| Relation | Exact span | Context |
|---|---:|---|
| 12026 → 8066 | 3960 = 11 × 360 = 66 × 60 | First half of the regular Creation comparison |
| 8066 → 4106 | 3960 = 11 × 360 = 66 × 60 | Equal second half |
| 8066 → 1406 | 6660 = 10 × 666 | Conquest opening anchor |
| 8066 → 1400 | 6666 | Six years after 1406 BC; conquest-rest comparison |
| 8066 → Isaac's birth, 2066 | 6000 | Isaac / sixty-year contextual comparison |

Isaac was sixty at Jacob's birth ([Genesis 25:26](https://ebible.org/eng-kjv/GEN25.htm)), and his lifespan was 180 = 3 × 60 years ([Genesis 35:28](https://ebible.org/eng-kjv/GEN35.htm)). The discussion's initial “60 × 6” lifespan expression is corrected to 60 × 3; 360 would represent two such lifespans. Likewise, the 6666 interval terminates at 1400 BC, while the interval to 1406 BC is 6660.

#### 3.13.3 The 1260 / 666 theme and the 43000-year Exodus expansion

The author's proposed Revelation setting begins with the beast's forty-two months and the number 666 in Revelation 13:5, 18. Revelation 11:2–3 places forty-two months beside 1260 days; the schematic thirty-day-month relation is $42\times30=1260$. These passages supply the textual numerical context for the author's comparison, without making the newly calculated BC dates explicit dates in Revelation. The factor 13 also enters the arithmetic directly through $598=2\times299=2\times13\times23$ and the 650 / 1300 expansions below.

Source texts: [Revelation 13](https://ebible.org/eng-kjv/REV13.htm) and [Revelation 11](https://ebible.org/eng-kjv/REV11.htm).

Holding 6 BC, the cumulative Rounded Creation inverse is:

$$14006-6=14000,\qquad I(14000)=41000,\qquad6+41000=41006\text{ BC}.$$

Its relation to the 12026 BC key is:

$$41006-12026=28980=23\times1260,$$

$$28980\times\frac{25}{23}=31500=25\times1260.$$

The gain is $31500-28980=2520=7\times360=2\times1260$. The author reads this seven-year schematic-calendar total alongside Creation's seven-day / seven-year pattern. The unexpanded Rounded cumulative Creation has the companion interval $14006-12026=1980=66\times30$.

The inverse Creation also reaches the Exodus by an exact 23-multiple:

$$41006-1446=39560=23\times1720,$$

$$39560\times\frac{25}{23}=43000=100\times430.$$

Here the Exodus at 1446 BC is held while the earlier endpoint moves backward:

$$1446+43000=44446\text{ BC},$$

$$44446-6=44440=5\times8888.$$

The author associates 43000 with the adopted 430-year Egyptian-sojourn register, and contrasts the resulting 8888 multiple with the preceding 666 field. These are identified numerical–theological readings of the exact spans. The earlier study's distinction between the adopted 430-year chronology and Genesis' 400-year formulation remains in force.

The relation to the existing Exodus backbone is also transparent: $39560-28980=10580$, while the two converted durations differ by $43000-31500=11500$. This recalls the previously established 10580 → 11500 conversion; the difference of the converted durations is not an additional independently counted construction.

### 3.14 The Actual Creation bridge divided by 12026 BC

#### 3.14.1 The chart's origin and selected source states

The supplied image, “Covenantal thread of multiples of 23-years to 1446 BC (Exodus),” expresses its chronology from the 4114 BC Creation endpoint. Its printed year 1 is the origin, so an ordinal year $N$ represents $N-1$ elapsed years:

$$\text{date BC}=4114-(N-1).$$

The following selected chart nodes control the present examination. The chart's other covenantal nodes remain available for later study.

| Chart node | Printed ordinal year | Elapsed years from origin | Recovered date BC |
|---|---:|---:|---:|
| Creation / Adamic covenant origin | 1 | 0 | 4114 |
| Jared, chart “Jarod / covenant of evil” | 461 | 460 = 20 × 23 | 3654 |
| Noah's birth | 1059 | 1058 = 46 × 23 | 3056 |
| Flood covenant | 1657 | 1656 = 72 × 23 | 2458 |
| Sinai covenant / Exodus | 2669 | 2668 = 116 × 23 | 1446 |
| Conquest rest | 2715 | 2714 = 118 × 23 | 1400 |

The author explicitly selects Noah 3056 BC as Gear 1. The 2458 BC Flood is the Gear-2 / Shem-axis Flood under File_18's controlling table. Their 598-year separation is therefore a comparison between these selected states, not a replacement of Noah's stated 600 years at the Flood. Jared's chart coordinate is retained without extending gear transport into the earlier genealogy.

The Actual Creation completion endpoints used here are cumulative 14004 BC and regular 4114 BC. File_22 §5.1 controls that pair. They are distinct from the Actual Year-6 pair 14006 / 4116 BC and from the Rounded Creation pair 14006 / 4106 BC.

#### 3.14.2 One key divides the bridge in the ratio 1:4

The 12026 BC anchor divides the Actual cumulative-to-regular Creation bridge as follows:

| Original interval, BC | Original span | ×25/23 | Expanded 430-unit form |
|---|---:|---:|---:|
| 14004 → 12026 | 1978 = 86 × 23 | 2150 | 5 × 430 |
| 12026 → 4114 | 7912 = 344 × 23 | 8600 | 20 × 430 |
| 14004 → 4114, total | 9890 = 430 × 23 | 10750 | 25 × 430 |

The original total is $1978+7912=9890$. Its expanded counterpart is $2150+8600=10750$; the expanded parts are not subdivisions of an unexpanded 9890-year total.

Both before and after conversion, the division is exactly 1:4:

$$7912=4\times1978,\qquad8600=4\times2150,$$

$$12026=\frac{4\times14004+4114}{5}.$$

The Rounded pair retains the same dividing date:

$$14006-12026=1980,\qquad12026-4106=7920=4\times1980.$$

Moving from the Rounded endpoints to these Actual completion endpoints shortens the earlier part by 2 and the later part by 8. Their common 1:4 division survives, while the Actual parts 1978 and 7912 are divisible by 23. The Rounded total remains 9900; the Actual total remains 9890.

Holding 12026 BC and expanding its interval toward regular Creation gives:

$$12026-8600=3426\text{ BC}.$$

This generated Creation endpoint is marked **3426 BC*** throughout the following tables because its original span, $7912=344\times23$, is not divisible by 299. It belongs to the 25/23 comparison but not to the 299-multiple family of Jared, Noah, and the Flood.

### 3.15 Jared, Noah, and the Flood: the 598-year sequence, partially explored

#### 3.15.1 The central 14 / 15 / 16 structure

The author's main point is the exact integration of the previously derived 12026 BC key with the selected Jared–Noah–Flood sequence:

$$3654-3056=3056-2458=598=26\times23=2\times299.$$

The combined span is $598+598=1196$, while 1058 is the separate Creation-to-Noah interval $4114-3056=460+598$. These two totals must not be interchanged.

| Source date BC | Span from 12026 BC | 598-unit form | 299-unit form |
|---|---:|---:|---:|
| Jared 3654 | 8372 | 14 × 598 | 28 × 299 |
| Noah 3056, Gear 1 | 8970 | 15 × 598 | 30 × 299 |
| Flood 2458, Gear 2 | 9568 | 16 × 598 | 32 × 299 |

The anchor places these three selected dates at consecutive 14, 15, and 16 multiples of 598. Under the same ratio applied to each anchor span, the entire progression becomes consecutive multiples of 600 or 650:

| Source | ×300/299 | ×25/23 |
|---|---:|---:|
| Jared | 14 × 600 = 8400 | 14 × 650 = 9100 |
| Noah | 15 × 600 = 9000 | 15 × 650 = 9750 |
| Flood | 16 × 600 = 9600 | 16 × 650 = 10400 |

At the two-step scale, $1196\times300/299=1200=600+600$, and $1196\times25/23=1300=650+650$. The author connects the latter with the cursed-land / judgment theme of 1300 in his wider study.

Jared's anchor span also has the factorization:

$$8372=14\times598=7\times1196=28\times299=23\times364.$$

Its 25/23 expansion therefore gives $25\times364=9100$, a gain of $2\times364=728$. Its 300/299 expansion gives 8400, a gain of 28. The 460 in the chart is Creation → Jared; the 364 arises from the anchor-span factorization. Neither is being substituted for Jared's lifespan.

The narrative setting is supplied by the Watchers' descent in Jared's days on Mount Hermon (1 Enoch 6:6), Enoch's presentation of their petition (13:4–7), and the 364-day year (72:32). The comparison of 364-year blocks with that calendar is a proposed day–year echo, with the units distinguished. Source: [1 Enoch, R. H. Charles translation](https://www.ccel.org/c/charles/otpseudepig/enoch.htm).

#### 3.15.2 Complete expansion table with 12026 BC held

For every row below, the forward BC endpoint is computed as:

$$X^*=12026-r(12026-X),$$

where $X$ is the original BC date and $r$ is the declared ratio. Moving forward in BC time reduces the printed year number.

| Source date BC | Original anchor span | Ratio | Expanded span | Forward shift | Result BC |
|---|---:|---:|---:|---:|---:|
| Creation 4114 | 7912 | 25/23 | 8600 | 688 | **3426*** |
| Jared 3654 | 8372 | 300/299 | 8400 | 28 | **3626** |
| Jared 3654 | 8372 | 25/23 | 9100 | 728 | **2926** |
| Noah 3056 | 8970 | 300/299 | 9000 | 30 | **3026** |
| Noah 3056 | 8970 | 70/69 | 9100 | 130 | **2926** |
| Noah 3056 | 8970 | 25/23 | 9750 | 780 | **2276** |
| Flood 2458 | 9568 | 300/299 | 9600 | 32 | **2426** |
| Flood 2458 | 9568 | 25/23 | 10400 | 832 | **1626** |

*Creation's original 7912 span is not a multiple of 299. The other seven rows arise from the three 299-multiple source spans. There are eight construction rows but seven distinct resulting dates because 2926 BC occurs twice.*

Among these three source spans, Noah's $8970=30\times299$ alone is divisible by 69 as well as by 23 and 299. Its three integral conversions are:

$$8970\times\frac{300}{299}=9000=25\times360,$$

$$8970\times\frac{70}{69}=9100=25\times364,$$

$$8970\times\frac{25}{23}=9750.$$

The cumulative forward shifts are 30, 130, and 780 years. The successive movements between the original and the three generated Noah dates are instead:

$$3056\rightarrow3026\rightarrow2926\rightarrow2276\text{ BC}: \quad30+100+650=780.$$

The author relates 30 to the leap-month unit, 130 to the restored regular Cainan variant, and 780 to the selected LXX–MT difference. File_18's controlled Flood dates give LXX 3238 − MT 2458 = 780, SP 3108 − MT 2458 = 650, and LXX 3238 − SP 3108 = 130. These differences retain their declared source states. The 100 also recalls Shem's hundred-year begetting register near the Flood, with Genesis 11:10's two-year qualification retained.

Reconstructing SP and LXX chronology from these building blocks is outside this addition's scope. The suggestion is retained as context for the selected 30 + 100 + 650 pattern.

#### 3.15.3 Two equal-step sequences and the shared 2926 BC endpoint

Holding the same anchor produces two complete equal-step sequences:

- Under 300/299: Jared 3626 → Noah 3026 → Flood 2426 BC, with 600 + 600.
- Under 25/23: Jared 2926 → Noah 2276 → Flood 1626 BC, with 650 + 650.

The separation between a source's two branch results increases from Jared's 700 to Noah's 750 to the Flood's 800 years.

Noah's third ratio joins the two source interpretations at one endpoint:

$$8970\times\frac{70}{69}=9100=8372\times\frac{25}{23},$$

$$12026-9100=2926\text{ BC}.$$

Thus 2926 BC is both Jared's 25/23 result and Noah's 70/69 result. The equal step sizes follow from applying a common linear span ratio to the original equal 598-year intervals; they are coordinated expressions of this structure.

#### 3.15.4 The resulting blocks: orientation before fuller interpretation

The following table contains every interval between the seven distinct Result BC dates. The duplicate 2926 BC construction has zero separation from itself and repeats the same relations to the other dates.

| Earlier BC ↓ / Later BC → | 3426* | 3026 | 2926 | 2426 | 2276 | 1626 |
|---|---:|---:|---:|---:|---:|---:|
| 3626 | 200 | 600 | 700 | 1200 | 1350 | 2000 |
| 3426* | — | 400 | 500 | 1000 | 1150 | 1800 |
| 3026 | — | — | 100 | 600 | 750 | 1400 |
| 2926 | — | — | — | 500 | 650 | 1300 |
| 2426 | — | — | — | — | 150 | 800 |
| 2276 | — | — | — | — | — | 650 |

*The asterisk preserves the Creation exception defined above.*

The chronological adjacent gaps are $200+400+100+500+150+650=2000$. This inventory was produced to establish the blocks available for interpretation. The author subsequently clarified that it is preparatory; the main result remains the integration of 12026 BC with the 598-multiple sequence.

The author's Noah–Shem life-pattern context is the familiar 500 + 100 + 350 + 150 outline. Its application remains subject to the explicit two-year gear distinctions. Within the present seven-date table, 500, 100, and 150 occur as direct intervals; 350 does not. The table therefore does not yet reproduce that entire life-pattern as one continuous sequence.

The author's selected annotations are:

| Dates, all BC | Exact interval | Proposed relation |
|---|---:|---|
| 3626 → 1626 | 2000 = 1000 + 1000 | Full range from the earlier Jared result to the later Flood result; two millennium-days |
| 3426* → 2926 → 2426 | 500 + 500 = 1000 | Tentative comparison with the two twelve-hour halves of Creation's first day |
| 3426* → 2276 | 1150 = 23 × 50 | Further 23-multiple within the generated field |
| 3026 → 2926 | 100 | Noah's first two generated results |
| 2926 → 2426 → 2276 | 500 + 150 = 650 | First half of the 1300 structure |
| 2926 → 2276 → 1626 | 650 + 650 = 1300 | Jared–Noah–Flood under 25/23; proposed curse / judgment association |
| 3026 → 2926 → 2426 → 2276 → 1626 | 100 + 500 + 150 + 650 = 1400 | Noah's earlier result adds 100 to that 1300 structure |

The same 500-year interval, 2926–2426 BC, participates in both 500 + 500 = 1000 and 500 + 150 = 650.

Two different measurements underlie the author's two-year / two-day comparison. Jared's 728 = 364 + 364 is his forward displacement from 3654 to 2926 BC. The 2000 years instead run from Jared's 300/299 result at 3626 BC to the Flood's 25/23 result at 1626 BC. The latter is not a direct conversion of 728 by one of the three ratios. The interpretation associates two calendrical years of years, two thousand-year days, and the two-year steps in the Japheth–Shem–Ham gear scheme. The thousand-year-day comparison has its textual analogue in [2 Peter 3:8](https://www.biblegateway.com/passage/?search=2+Peter+3%3A8&version=KJV); the 500 + 500 first-day reading remains expressly tentative.

#### 3.15.5 Creation, Abraham, the SKL comparison, and AD 25

The expanded Creation and Flood endpoints inherit the familiar 1656-year source interval because both are produced by the same 25/23 expansion:

$$3426-1626=(4114-2458)\times\frac{25}{23}=1656\times\frac{25}{23}=1800.$$

Within that 1800-year span, the selected SKL 2886 BC anchor and Abraham's 2166 BC birth give a symmetric partition:

$$3426\rightarrow2886\rightarrow2166\rightarrow1626\text{ BC},$$

$$540+720+540=1800.$$

The overlapping intervals are $3426-2166=1260$ and $2886-1626=1260$. File_21 / File_34 identify 2886 BC as the SKL Long +30 rail. The author's local application of its ±720 macro bracket gives 3606 / 2886 / 2166 BC, placing Abraham's birth at the later member. This comparison retains the distinction between the SKL's 720-year scale and ordinary two-year gears.

The two AD 25 continuations use the civil no-year-zero count $C=B+A-1$, expressly distinct from the Rounded width $B+A-2$:

| Path | Exact component spans | Relation |
|---|---|---|
| 3426 BC → 2276 BC → AD 25 | 1150 + 2300 | 23 × 50 followed by the 2300 interval |
| 2426 BC → 1446 BC → AD 25 | 980 + 1470 = 2450 | 2 × 490 + 3 × 490 = 5 × 490 |

The 2276 BC node is **Noah's** 25/23 result, correcting the discussion's incidental Flood label. The Flood results remain 2426 and 1626 BC. The cross-axis equations are $2276+25-1=2300$, $1446+25-1=1470$, and $2426+25-1=2450$.

Within the adopted 6 BC Nativity chronology, $6+25-1=30$, so AD 25 supplies the proposed age-thirty Christological comparison. This is a relation within that adopted chronology, not a newly established historical ministry date.

#### 3.15.6 What is retained, and what remains to be examined

The author considers this a cursory exploration of a more complex pattern. The exact 14 / 15 / 16 multiples of 598, the coherent 600 / 650 expansions, the Noah–Jared convergence, the Actual Creation bridge, and the earlier cumulative relations support his strong assessment of 12026 BC as a major key. A comprehensive explanation of all the generated blocks and all the chart's other 23-multiple dates has not yet been worked out.

The Moses comparison was intentionally left to the end of the proposed investigation. The retained contextual suggestions are his small ark on the Nile (Exodus 2:3), his 120-year lifespan (Deuteronomy 34:7), its placeholder-preserving inverse $I(120)=210$, and the author's broader reading of Revelation through Mosaic, Babylonian, 1260, and 666 themes. No additional Moses-date construction is supplied by this amendment.

The date calculations above are verified; their fuller narrative interpretation remains partially explored. The author's confidence in the anchor itself is preserved as an explicit structural judgment. No new source rows, target cells, independence claims, or probability estimates are added to the existing statistical study.



## 4. Joseph and Joshua: the 110-year correspondences

### 4.1 Textual lifespans and Rounded placements

Joseph and Joshua both live 110 years in the biblical account (Genesis 50:26; Joshua 24:29). File_51a §7.6 supplies their Rounded placements.

| Figure | Rounded birth | Rounded death / terminal placement | Lifespan |
|---|---|---|---:|
| Joseph | 1916 BC | 1806 BC | 110 |
| Joshua | 1476 BC | 1366 BC | 110 |

The birth-to-birth and death-to-death separations both equal $440=4\times110$. The absolute death placements belong to the Rounded comparison layer. The 110-year lifespans retain their textual status.

### 4.2 Joseph's inverse lifespan placement at the Exodus

Joseph's whole-span Conquest-anchor inverse is:

$$1916-1406=510\xrightarrow{I}150,\qquad1406+150=1556\text{ BC}.$$

The inverse birth then stands exactly one Joseph lifespan before the Exodus:

$$1556-1446=110.$$

In the provisional terminal-reconstruction reading, $I(110)=110$, so applying the 110-year lifespan forward from inverse birth 1556 BC reaches 1446 BC. The original Rounded life pair 1916–1806 BC and the reconstructed pair 1556–1446 BC both span 110.

This is a contextual inverse correspondence outside the 115 test. The 110 span is a miss for divisibility by 115. Its narrative interest is not reduced to that test's pass/fail classification.

### 4.3 Joshua's inverse birth and the 50/51–54/55 sequence

Joshua's whole-span Nativity-anchor inverse is:

$$1476-6=1470\xrightarrow{I}7410,\qquad6+7410=7416\text{ BC}.$$

From this inverse birth, the four comparisons identified in the discussion are all exact multiples of the common lifespan 110. All endpoints in this table are BC.

| From inverse Joshua 7416 BC to | Span | Lifespan units |
|---|---:|---:|
| Joseph's Rounded birth, 1916 BC | 5500 | 50 × 110 |
| Joseph's Rounded death, 1806 BC | 5610 | 51 × 110 |
| Joshua's Rounded birth, 1476 BC | 5940 | 54 × 110 |
| Joshua's Rounded death, 1366 BC | 6050 | 55 × 110 |

The sequence reflects the two original 110-year lives and their 440-year separation. Once one starting alignment is fixed, the differences between these four values are constrained by that source structure. The four comparisons display a coherent relation; they are not four independent statistical successes.

### 4.4 The provisional 5750 bridge

Applying the self-inverse lifespan 110 to Joshua's inverse birth gives a provisional terminal placement:

$$7416-110=7306\text{ BC}.$$

Its separation from Joseph's inverse birth is:

$$7306-1556=5750=23\times250=50\times115.$$

This is a bridge between the lifespan comparison and the Key-of-23 register. It remains outside the frozen birth/event scan because 7306 BC is a reconstructed terminal endpoint, and 1556 BC is not one of the scan's five comparison targets.

### 4.5 Narrative setting

Typological note: the Joseph–Joshua comparison is anchored in more than two equal lifespan totals. Joseph's final instructions anticipate Israel's departure; Moses carries his bones from Egypt; and their burial is recorded near Joshua's death at the close of Joshua (Genesis 50:24–26; Exodus 13:19; Joshua 24:29–32). These passages connect Joseph's death, the Exodus, and settlement in the land.

The inverse placement of Joseph's 110-year terminal span at the Exodus and the Joshua lifespan sequence are therefore read against an existing narrative connection. The arithmetic is exact under the model. The inference that the inversion captures an intended or providential structure remains separate.

## 5. The 1876 BC Nexus

### 5.1 The two inverse routes

The source event is Jacob's entry into Egypt at 1876 BC. His age of 130 at the encounter with Pharaoh is a textual datum (Genesis 47:9); the Rounded birth 2006 BC gives $2006-1876=130$.

The two whole-span routes preserve the same source event and use the normal 6-ending anchors.

| Held anchor | Original span | Inverse span | Rebuilt inverse date |
|---|---:|---:|---|
| 1406 BC | 470 | 740 | 2146 BC |
| 6 BC | 1870 | 7810 | 7816 BC |

### 5.2 The 700 and the 300 + 400 division

The 2146 BC inverse stands 700 years before the Exodus:

$$2146-1446=700=14\times50.$$

The 1846 BC affliction node divides this interval into $2146-1846=300$ and $1846-1446=400$. The latter is the selected 400-year affliction construction, whose textual duration is supplied by Genesis 15:13. The absolute 1846 BC placement is a chronological construction controlled by the Egypt/affliction field, not an independently stated biblical BC date.

The same 1846 BC node stands $1846-6=1840=80\times23$ from the Nativity anchor. Its Priestly expansion is:

$$1840\times25/23=2000,\qquad6+2000=2006\text{ BC}.$$

Holding 6 BC and extending backward therefore reaches Jacob's Rounded birth. This later expansion is retained as a structural relation. It is distinct from the direct inverse event at 2146 BC.

### 5.3 The 6370 and Jacob's 130 years

The Nativity-anchored inverse gives the parallel 49-register relation:

$$7816-1446=6370=130\times49.$$

The quotient 130 corresponds to Jacob's stated age at entry. The comparison therefore joins the selected Entry event, its inverse, the Exodus anchor, and the 49-year register.

Structural inference: the two inverse routes associate the same Entry event with $14\times50$ and $130\times49$ to the Exodus. The author's interest concerns that event-specific fit as well as the integer relations themselves. No probability for this combined narrative pattern was calculated.

### 5.4 Abraham's call at 2091 BC

The 2091 BC call / departure date is retained because it is a major event, despite having been overlooked in the first informal enumeration. Its original span to the appropriate 1-ending Conquest anchor is:

$$2091-1401=690=30\times23=6\times115.$$

This is an original-date relation. The two inverse dates are 9021 BC from 1 BC and 2361 BC from 1401 BC. Neither gives a hit to 6 BC. The event stays in the manifest because its admission is based on the agreed scope, not on producing a favorable result.

### 5.5 The Nexus and the AD 6 comparison

The same inverse Entry date 7816 BC appears among the AD 6 hits:

$$W_R=7816+6-2=7820=340\times23=68\times115.$$

The 6370 span to the Exodus and the 7820 Rounded width to AD 6 are different comparisons from the same generated endpoint. Their arithmetic is preserved without treating the endpoint as a new source date.

### 5.6 The 1846 BC hub: Abram–Moses symmetry and Terah's 1380

Authorial structural reading: 1846 BC is an important hub, second to 1876 BC in the relevant Egypt / affliction field. Its existing 400-year relation to the Exodus and its relation to the Entry Nexus already give it a defined place in the chronology (§5.2; File_16). The author's purpose in mentioning its common 115-year grid with 6 BC is to identify a possible main artery from which further relations can expand. It is not a request to add another target to the statistical study.

Three already established inverse routes show the local structure. All dates and held anchors in this table are BC.

| Source | Held inversion anchor BC | Span inversion | Rebuilt inverse date BC |
|---|---|---|---|
| Abram / Abraham, 2166 BC | 1406 | 760 → 670 | 2076 |
| Moses, 1526 BC | 1406 | 120 → 210 | 1616 |
| Terah, 2236 BC | 6 | 2230 → 3220 | 3226 |

Abram and Moses place the affliction hub exactly at their inverse midpoint:

$$2076-1846=230,\qquad1846-1616=230,$$

$$\frac{2076+1616}{2}=1846,\qquad2076-1616=460=2\times230.$$

Terah's inverse lies on the same local 230-year structure:

$$3226-1846=1380=6\times230=60\times23.$$

Thus Terah lies six 230-year units before the hub; Abram lies one before it; and Moses lies one after it. The named sources matter to the structural reading: Terah and Abram belong to the ancestral call setting, while Moses belongs to the Exodus setting. The exact inverse placements and centered spans are arithmetic facts within the elected Rounded model; the interpretation as a main artery is the author's proposed reading of their connection.

The proposed expansion role has a direct bounded illustration. Hold 1846 BC fixed and apply 25/23 to each signed displacement from that center. The common units become:

$$230\times25/23=250,\qquad1380\times25/23=1500=6\times250.$$

| Inverse endpoint | Position relative to 1846 BC | Expanded position, same held center | Rebuilt endpoint |
|---|---|---|---|
| Abram: 2076 BC | 230 earlier | 250 earlier | 2096 BC |
| Moses: 1616 BC | 230 later | 250 later | 1596 BC |
| Terah: 3226 BC | 1380 earlier | 1500 earlier | 3346 BC |

The Abram–Moses pair therefore retains the same midpoint while 230 + 230 expands to 250 + 250. Terah retains the six-unit relation. These rebuilt labels illustrate the declared expansion and are not assigned new event identities. They do not enter the source manifest or target count. The general claim that a wider artery will develop from this hub remains a direction for further study.

The earlier shared-grid observation is retained in this structural setting:

$$1846-6=1840=16\times115,$$

$$D'-1846=(D'-6)-16\times115.$$

Consequently the nine inverse routes on the 115-year grid through 6 BC also belong to the grid through 1846 BC; one of those endpoints coincides with the hub itself. This congruence supplies a connection between the two centers. The midpoint and Terah relations specify why the already important 1846 BC date is worth examining as a hub in its own right.

The admitted backbone endpoint extends that shared membership: $14726-1846=12880=112\times115$. There are therefore ten members of this grid in the expanded tally, still including the one exact 1846 BC overlap. This continues the structural hub observation without adding 1846 BC to the statistical targets.

Section 3.7 develops the author's subsequent Creation-pillar reading of this 12880 span. With 14726 BC held, the 1846 BC hub moves to 726 BC and the Abram–Moses pair moves to 976 / 476 BC, retaining the expanded 250 + 250 symmetry. That operation holds Creation fixed; the earlier illustration in this section holds 1846 BC fixed.

### 5.7 The paired Entry hub: 9660 to 9800 through the Rounded Mirror

The author adds a further connection between the paired Egypt nodes, 1876 BC and 1846 BC. Their original separation is 30 years. The Nativity-family inverse of the Entry at 1876 BC is already established in §5.1:

$$1876-6=1870\xrightarrow{I}7810,\qquad6+7810=7816\text{ BC}.$$

Apply the Rounded Mirror to that inverse endpoint. Its coordinate −7815 reflects to +7815, displayed as AD 7816. The affliction hub at 1846 BC has coordinate −1845. The width between them is therefore:

$$W_R=7815-(-1845)=9660,$$

or, using the symmetric Rounded labels directly:

$$1846+7816-2=9660=140\times69.$$

The stated conversion is the Prophetic ratio 70/69:

$$9660\times70/69=9800=4900+4900.$$

The equal halves display the conversion explicitly:

$$9660=4830+4830,\qquad4830=70\times69,$$

$$4830\times70/69=4900=70^2.$$

Thus the span from the 1846 BC hub to the Mirror of the Entry inverse converts into two equal 4900-year units. The sequence retains its distinct operations and endpoints.

| Stage | Operation / retained reference | Result |
|---|---|---|
| Entry inverse | Whole-span inversion from the 6 BC anchor | 1876 BC → 7816 BC |
| Rounded Mirror | Reflect q = −7815 to q = +7815 | 7816 BC → AD 7816 |
| Hub comparison | Rounded width from 1846 BC to AD 7816 | 9660 = 140 × 69 |
| Prophetic expansion | Multiply the duration by 70/69 | 9800 = 4900 + 4900 |

The 9660 value uses the Rounded coordinate convention throughout the cross-axis comparison. The ordinary civil count between the same printed labels would be 9661. The 70/69 conversion here is also distinct from the 25/23 expansion illustrated around the 1846 BC center in §5.6.

Structural inference: the Entry source supplies the inverse / Mirror endpoint, and its paired affliction hub supplies the other end of the duration that converts to 4900 + 4900. This gives a further explicit relation linking 1876 BC and 1846 BC within the proposed hub structure. It belongs to the structural study; the converted duration is not assigned a new historical event or entered in the target-frequency scan.

Section 3.8.1 compares this 9800 result with the 24500 expansion of the longer span from the 14726 BC Creation pillar to the same Entry inverse / Mirror endpoint. It records the resulting 14700 + 9800 partition and distinguishes its two expansion ratios.

## 6. The 529 family and repeated expansion

### 6.1 The common arithmetic

The significance of `529 = 23 × 23` in this discussion is that it permits two exact successive applications of the Priestly ratio:

$$k\times23^2\xrightarrow{25/23}k\times23\times25\xrightarrow{25/23}k\times25^2.$$

The four selected durations below are discussed with their own endpoints and operators in the following subsections. Their shared expansion form is an arithmetic consequence of the factor 529.

| Multiplier $k$ | Initial duration $k\times529$ | After one 25/23 expansion | After two 25/23 expansions |
|---:|---:|---:|---:|
| 15 | 7935 | 8625 | 9375 |
| 19 | 10051 | 10925 | 11875 |
| 20 | 10580 | 11500 | 12500 |
| 104 | 55016 | 59800 | 65000 |

The two exact expansion steps are not independent confirmations of the factorization. They explain what the factorization makes possible within the stated operator system.

### 6.2 The 20 × 529 same-side relation

The segmented regular / cumulative convergence at 12026 BC (§3.3) yields:

$$12026-1446=10580=20\times529=23\times460.$$

The exact expansion chain is $10580\rightarrow11500\rightarrow12500$ under two applications of 25/23. The same-side span uses ordinary subtraction and needs no cross-axis correction.

This relation is inherited from the source architecture, where the 10580 value is already recorded as 23 × 460. Expressing it as 20 × 529 makes its second exact Priestly expansion explicit. It is not an additional hit in the 64-route whole-span scan.

The first expansion also participates in the Temple convergence of §3.8.3. After reflecting both backbone and Exodus endpoints, hold AD 12026: expanding 10580 to 11500 moves AD 1446 back by 920 to Rounded AD 526, the same coordinate obtained from the Creation-held Temple inverse.

The second expansion to 12500 reaches 476 BC in the same Rounded display (§3.9). That is also the Creation-held expansion endpoint of inverse Moses, 1616 BC. The successive gains are 920 and 1000, connecting the two exact expansions of 20 × 529 with the Moses–Temple pair.

### 6.3 The 19 × 529 civil comparison

The regular whole-span inverse Creation at 8606 BC is compared with the AD 1446 counterpart in the selected cross-axis construction.

The two possible widths must be shown together.

| Endpoints | Rounded coordinates | Rounded width $W_R$ | Literal civil count $C$ |
|---|---|---:|---:|
| 8606 BC to AD 1446 | −8605 to +1445 | 10050 | 10051 = 19 × 529 |

The factor 529 belongs to the expressly selected civil count:

$$8606+1446-1=10051\xrightarrow{25/23}10925\xrightarrow{25/23}11875.$$

It does not belong to the neighboring Rounded width 10050. The arithmetic verifies the selected civil construction; it does not erase the difference between the two conventions.

### 6.4 The 104 × 529 civil comparison and the 299 connection

The cumulative Nativity-anchor inverse is 41006 BC. The selected opposite-side companion is AD 14011, the symmetric Rounded display of coordinate +14010.

| Endpoints | Rounded coordinates | Rounded width $W_R$ | Literal civil count $C$ |
|---|---|---:|---:|
| 41006 BC to AD 14011 | −41005 to +14010 | 55015 | 55016 = 104 × 529 |

The source of the 55016 duration is therefore:

$$41006+14011-1=55016.$$

The first Priestly expansion has an additional exact factorization, which permits the stated Enochian conversion.

| Operation | Exact result |
|---|---:|
| $55016\times25/23$ | 59800 |
| $59800\times25/23$ | 65000 |
| $59800=200\times299$ | 59800 |
| $59800\times300/299$ | 60000 |

The starting date of the 55016 comparison is 41006 BC. The earlier mistaken substitution of 14006 BC is not retained.

### 6.5 Reapplying a converted duration on the Rounded grid

The author separately specified a launch from 14006 BC using the converted duration 59800. That launch has its own held starting point; it is not the original span that generated 55016.

On the Rounded grid:

$$-14005+59800=+45795\quad\longrightarrow\quad\text{AD }45796.$$

The resulting Rounded width is 59800. The literal civil elapsed count between the printed labels 14006 BC and AD 45796 is 59801.

This example records the full sequence: an expressly selected civil duration is converted, a new launch anchor is declared, and the converted duration is then applied using Rounded coordinates. The mixed construction remains explicit and provisional as a structural interpretation.

### 6.6 Seth, 7935, and its Rounded Mirror

Seth's Rounded birth at 3976 BC gives the Nativity-anchor inverse:

$$3976-6=3970\xrightarrow{I}7930,\qquad6+7930=7936\text{ BC}.$$

Its span to 1 BC is:

$$7936-1=7935=15\times529=345\times23=69\times115.$$

The corresponding Rounded coordinate is −7935. Reflection gives +7935, displayed as AD 7936. The full Rounded Mirror width is therefore:

$$7935+7935=15870=30\times529.$$

The literal civil count between the printed labels 7936 BC and AD 7936 would instead be 15871. The 15870 value is expressly a Rounded coordinate width.

The one-sided 7935 duration expands to 8625 and then 9375. Seth's 1 BC hit, its Mirror double, and its two exact expansion steps describe one underlying 529 relation. They are not additional independent hits in the scan.

### 6.7 The proposed third-axis reading

Exploratory note: the discussion considered whether inverse structure might act analogously to a further axis alongside the ordinary and Mirror displays. The examples show that distinct transformations can generate compatible numerical relations. A geometric interpretation requires a defined mapping and an account of what is preserved; the factorization of selected durations does not alone supply that account.

The present report records the analogy as a proposal. It does not promote the inverse operation to the same established coordinate status as the Mirror or replace the second-order treatment already assigned to the existing File_52c.

## 7. The cumulative 62100 chain and the return to 6 BC

### 7.1 The direct inverse holds 1406 BC

The cumulative Creation-to-Conquest span is:

$$14006-1406=12600\xrightarrow{I}62100=2700\times23.$$

The anchor remains 1406 BC, so the direct rebuilt date is $1406+62100=63506$ BC. Its span to 6 BC is $63506-6=63500$, which is not divisible by 115.

The direct 62100 inverse therefore belongs to the Conquest-held construction. It is not itself a direct inverse hit to 6 BC.

### 7.2 The stated expansion sequence

The subsequent sequence uses the Prophetic ratio 70/69 and then the Priestly ratio 25/23. All expansions in this table are whole-year conversions.

| Step | Operation | Result / held anchor |
|---|---|---|
| 1 | $62100\times70/69$ | 63000; increase 900 |
| 2 | $1406+63000$ | 64406 BC; held Conquest anchor |
| 3 | $64406-6$ | 64400 = 2800 × 23 |
| 4 | $64400\times25/23$ | 70000 |
| 5 | $6+70000$ | 70006 BC; held Nativity anchor |

Correction retained: the ratio taking 62100 to 63000 is 70/69. The reverse ratio 69/70 does not yield that result.

Terminology retained precisely: 64406 BC is an inverse-plus-expansion endpoint. Calling it a double inverse would conflate operators, since two applications of the defined reversal restore the original 12600 span and its 14006 BC endpoint.

### 7.3 Why the later span is divisible by 23

The increase of 900 combines with the 1400-year separation between the two anchors:

$$900+1400=2300=100\times23.$$

Thus the new Nativity span is $62100+2300=64400$. Once that expansion and anchor change are chosen, the divisibility follows from this identity.

The term residual in the discussion referred here to a remaining or recomputed interval. No fractional physical residue, residual day-year lift, or other File_00 Residue Protocol operation is being invoked.

This chain helps explain the author's interest in 6 BC as an intersection across more than one construction. It remains outside both the direct 64-route tally and the expanded 66-construction tally. The counted backbone addition uses segmented inversion, without this later expansion.

## 8. The original target scan and the backbone extension

### 8.1 Test definition

For each inverse endpoint $D'$ BC and each target with Rounded coordinate $q_T$, calculate:

$$W=\left|q_T-(1-D')\right|.$$

A positive hit requires $W>0$ and $23\mid W$. Every width in this scan is a multiple of five. Consequently:

$$23\mid W\quad\Longleftrightarrow\quad115\mid W.$$

Zero is mathematically divisible by 23, but a zero-span overlap is not a positive hit. No 6 BC-to-itself match enters either the original nine hits or the expanded ten. The target dates 6 BC and 1 BC are not source rows in the manifest.

A self-inverse source is a different case. It may produce a positive span to a target and is admitted under the stated rule. Shelah and the 1846 BC affliction node illustrate that distinction in the 6 BC results.

### 8.2 Original whole-span counts for all five targets

The same 64 routes are used in every row. The BC comparisons use absolute separations; the AD 6 comparison uses the Rounded coordinate +5.

| Target | Nativity hits | Conquest hits | All positive hits | Zero overlaps | Misses | Routes |
|---|---|---|---|---|---|---|
| 6 BC | 5 | 4 | 9 | 0 | 55 | 64 |
| 1406 BC | 2 | 0 | 2 | 1 | 61 | 64 |
| 1446 BC | 0 | 0 | 0 | 1 | 63 | 64 |
| 1 BC | 1 | 1 | 2 | 0 | 62 | 64 |
| AD 6 | 1 | 2 | 3 | 0 | 61 | 64 |

The first comparison of 6 BC, 1406 BC, and 1446 BC comprises 192 route–target comparisons: 11 positive hits, two zero-span overlaps, and 179 misses. Adding 1 BC and AD 6 gives 320 comparisons: 16 positive hits, two overlaps, and 302 misses. These are bookkeeping totals, not counts of independent statistical trials.

### 8.3 The original nine whole-span hits to 6 BC

All source dates, held anchors, and inverse dates in this table are BC. Each listed span is the inverse date minus 6.

| Source (BC) | Held anchor BC | Inverse date BC | Span to 6 BC | Factorization |
|---|---|---|---|---|
| Mahalalel (3711) | 1 | 1731 | 1725 | 75 × 23 = 15 × 115 |
| Shem (2556) | 6 | 5526 | 5520 | 240 × 23 = 48 × 115 |
| Shelah (2421) | 1 | 2421 | 2415 | 105 × 23 = 21 × 115 |
| Eber (2391) | 1 | 9321 | 9315 | 405 × 23 = 81 × 115 |
| Terah (2236) | 6 | 3226 | 3220 | 140 × 23 = 28 × 115 |
| Abraham (2166) | 1406 | 2076 | 2070 | 90 × 23 = 18 × 115 |
| Moses (1526) | 1406 | 1616 | 1610 | 70 × 23 = 14 × 115 |
| 120-year Flood countdown (2576) | 1406 | 8516 | 8510 | 370 × 23 = 74 × 115 |
| 400-year affliction node (1846) | 1406 | 1846 | 1840 | 80 × 23 = 16 × 115 |

Five hits arise from the Nativity family and four from the Conquest family. The nine positive spans do not include a zero-distance anchor identity, a cumulative source, a reconstructed death, or an expanded duration.

The two self-inverse entries are Shelah's Nativity-family route and the affliction node's Conquest-family route. Their inclusion preserves the agreed rule; neither is a target-to-itself match.

### 8.4 The 1406 BC and 1446 BC comparisons

The 1406 BC target has two positive hits, both from Nativity-family inversions.

| Source | Inverse date | Span to 1406 BC | Factorization |
|---|---|---:|---|
| Reu, 2326 BC | 2326 BC | 920 | 40 × 23 = 8 × 115 |
| Serug, 2296 BC | 9226 BC | 7820 | 340 × 23 = 68 × 115 |

Adam's Nativity-family inverse lands exactly at 1406 BC. That zero-span overlap is recorded separately.

The 1446 BC target has no positive 23-multiple hits in the original whole-span scan. The Exodus source remains 1446 BC under its Conquest-family inverse, producing the one zero-span overlap in that column. The separately admitted 12026 BC backbone endpoint adds one positive hit in the expanded tally (§8.7).

The low 23-multiple count does not summarize all numerical activity at these anchors. Joseph's 110 to the Exodus, the Entry inverse's 700, and its other route's 6370 remain explicit examples outside that divisibility class.

### 8.5 The 1 BC comparison

The 1 BC target gives two positive hits.

| Source | Held inversion anchor | Inverse date | Span to 1 BC |
|---|---|---|---|
| Seth, 3976 BC | 6 BC | 7936 BC | 7935 = 345 × 23 = 69 × 115 |
| Noah, 3056 BC | 1406 BC | 7016 BC | 7015 = 305 × 23 = 61 × 115 |

The count is close to the approximate expectation of 2.78. Seth's internal `15 × 529` structure supplies the separate observation developed in §6.6; it does not increase the count beyond two.

### 8.6 The AD 6 comparison and Isaac's lifespan

For the AD 6 target, $q_T=+5$ and the Rounded width is $D'+4$. Three routes qualify.

| Source | Held inversion anchor | Inverse date | Rounded width to AD 6 |
|---|---|---|---|
| Enoch's birth, 3486 BC | 1406 BC | 9426 BC | 9430 = 410 × 23 = 82 × 115 |
| Isaac's birth, 2066 BC | 1406 BC | 2066 BC | 2070 = 90 × 23 = 18 × 115 |
| Jacob's entry, 1876 BC | 6 BC | 7816 BC | 7820 = 340 × 23 = 68 × 115 |

The three-hit count is near the baseline expectation. The author nevertheless identified all three sources as central to the larger study: Enoch, Isaac, and the 1876 BC Nexus.

Isaac supplies a specific lifespan correspondence:

$$2070\times25/23=2250,\qquad2250-2070=180.$$

The increment is exactly Isaac's stated lifespan (Genesis 35:28). His Conquest-family inverse is self-inverse because $2066-1406=660$ and $I(660)=660$. The 2070 relation is therefore already present in the original Rounded placement and is preserved through this inverse route.

This is an interval-expansion correspondence. It does not resolve the separate procedure for reconstructing inverse deaths.

### 8.7 Expanded tally: the tenth incident and its companion stage

The author expressly admits the completed major backbone as the tenth incident at 6 BC. Its construction is the regular segmented path in §3.4, already present in File_52a §7. The original 32-source manifest remains intact. The extension records both anchor stages of that one backbone, so the denominator becomes 64 + 2 = 66 constructions.

This paired admission includes the 12026 BC miss to 6 BC as well as the successful 14726 BC endpoint. It also preserves the Exodus hit at the Conquest stage. Neither endpoint occurs in the original 64-route ledger.

| Additional construction | Held anchor | Endpoint | Span to 6 BC | 6 BC result | Span to 1446 BC | 1446 BC result |
|---|---|---|---:|---|---:|---|
| B1: segmented Creation backbone, Conquest stage | 1406 BC | 12026 BC | 12020 | Miss; remainder 60 mod 115 | 10580 | Hit; 92 × 115 = 5290 + 5290 |
| B2: segmented Creation backbone, completed Nativity stage | 6 BC | 14726 BC | 14720 | Hit; 128 × 115 = 640 × 23 | 13280 | Miss; remainder 55 mod 115 |

The expanded counts below use all 66 constructions against every target. Appendix A.6 supplies both additions' complete five-target outcomes, including every miss.

| Target | Original positive hits | Backbone positive hits | Expanded positive hits | Zero overlaps | Misses | Constructions |
|---|---|---|---|---|---|---|
| 6 BC | 9 | 1 | 10 | 0 | 56 | 66 |
| 1406 BC | 2 | 0 | 2 | 1 | 63 | 66 |
| 1446 BC | 0 | 1 | 1 | 1 | 64 | 66 |
| 1 BC | 2 | 0 | 2 | 0 | 64 | 66 |
| AD 6 | 3 | 0 | 3 | 0 | 63 | 66 |

The 6 BC count is now six Nativity-stage hits and four Conquest-stage hits. The first three target columns comprise 198 comparisons: 13 positive hits, two overlaps, and 183 misses. All five columns comprise 330 comparisons: 18 positive hits, two overlaps, and 310 misses. These totals describe comparisons, not independent trials.

The regular Noah alternatives and cumulative primary Flood path support the same two backbone endpoints and add no further entries. The 5290 + 5290 relation strengthens the author's structural reading but is one Exodus hit, not two. The Temple at 966 BC, expanded 64406 BC endpoint, and other referenced branches remain outside the tally.

## 9. Probability benchmark and its scope

### 9.1 Why the working density is 1/23

Among successive multiples of five, one in every 23 is divisible by 115. Equivalently, division by five turns the question into divisibility by 23. This supplies the simple working density:

$$p=\frac{1}{23}.$$

The last-digit split does not supply an additional factor of two. For a 6-ending inverse date measured to 6 BC, the span is divisible by ten and a successful quotient in units of 115 is even. For a 1-ending inverse date, the span ends in 5 and the successful quotient is odd. Each class has one qualifying position per 23 successive ten-year positions.

This is a density statement on complete residue cycles. It is not a demonstration that the historically related source dates, or the inverse outputs of this finite selected sample, are uniformly distributed among those residues.

### 9.2 Original benchmark: the nine-hit excess

Under the simple model, the expectation for one target is:

$$E[X]=\frac{64}{23}\approx2.7826.$$

The observed nine hits are $9/64=14.0625\%$ of the routes, compared with the model probability $1/23\approx4.3478\%$. The observed count is $207/64=3.234375$ times the model expectation.

If all 64 routes are additionally treated as independent Bernoulli trials with probability 1/23, the tail probability is:

$$P(X\geq9)=\sum_{k=9}^{64}\binom{64}{k}\left(\frac1{23}\right)^k\left(\frac{22}{23}\right)^{64-k}\approx0.001749932.$$

That is approximately 0.175%, or one in 571. The discussion's shorthand of about one in 500 is a reasonable rounding of this unadjusted model result. The stated probability is for nine or more hits, not exactly nine.

### 9.3 Why the estimate remains provisional

The two inversions of each source are functions of the same date. The source dates themselves are related through a genealogy and selected event intervals. One source produces duplicate numerical endpoints across its two routes. Independence is therefore an assumption of the benchmark, not an established property of the data.

The investigation was exploratory. The author had previously examined these patterns, refined the source list during discussion, retained a provisional Enoch endpoint, and later added comparison targets. The broader study also considers more numerical properties than divisibility by 115. No correction for that entire search history has been calculated.

The prior role of 6 BC as a major intersection explains why it was a natural target to examine. It does not turn the present retrospective scan into a wholly blind test.

The benchmark is therefore informative about the size of the observed concentration under one simple model. The actual statistical strength of the study requires a null model that preserves the relevant dependencies and selection procedure. The author-requested pressure test in Appendix C checks every target cell, all 23 residue classes, selected source-list sensitivities, and the dependence of the narrative constructions. It does not supply that calibrated null model. No randomized probability for the whole study is claimed.

### 9.4 Multiple targets and the descriptive comparison

The five statistical target coordinates occupy distinct residues modulo 23. A given route cannot be a positive divisibility hit at more than one of those five targets. Their counts are therefore related rather than independent repetitions of the same test. The separate 1846 BC hub discussion in §5.6 does not add a statistical target, change this comparison set, or alter the probability benchmark.

For the original three target columns, the simple expected aggregate is $192/23\approx8.35$, compared with 11 observed positive comparisons. For the original five columns, it is $320/23\approx13.91$, compared with 16 observed positive comparisons. These aggregate expectations are descriptive and do not replace a test of concentration at one specified target.

For the original 64 routes, 6 BC has nine hits, while the other four have two, zero, two, and three in the order 1406 BC, 1446 BC, 1 BC, and AD 6. In the expanded set these counts become ten, two, one, two, and three. Its simple aggregate expectation is $330/23\approx14.35$, compared with 18 positive comparisons. The residue census in Appendix C shows the 6 BC class as the unique largest in both sets: nine originally and ten after extension; the next largest remains six. No formal probability for the maximum or for the full distribution has been assigned.

### 9.5 Frequency and contextual structure

The test gives equal numerical weight to every admitted route. It does not separately score whether a quotient reproduces a person's lifespan, age at a narrative transition, a 49/50 construction, or a repeated 529 expansion.

That limitation explains the distinction made by the author. Joseph's and Joshua's 110s can be central to the numerical–narrative interpretation while failing the 115 test. Conversely, an exact multiple of 115 does not gain additional statistical weight merely because its source is a major figure.

The qualitative case must be evaluated through its specified operations, textual relationships, and dependence structure. The individual probability figures cannot simply be multiplied to produce a probability for the whole system.

### 9.6 Expanded benchmark: ten hits after admitting the backbone

Under the same illustrative $p=1/23$ model, the paired 66-construction set has expected count:

$$E[X]=\frac{66}{23}\approx2.8696.$$

Its ten hits represent $10/66\approx15.1515\%$ of the constructions. If all 66 are treated as independent Bernoulli trials, the upper-tail calculation is:

$$P(X\geq10)=\sum_{k=10}^{66}\binom{66}{k}\left(\frac1{23}\right)^k\left(\frac{22}{23}\right)^{66-k}\approx0.0005453274.$$

This is approximately 0.0545%, or one in 1834. The calculation is for ten or more hits. The following table makes the denominator choices explicit rather than inserting a tenth success into the unchanged 64-route total.

| Declared set | Constructions | Hits to 6 BC | Expected at 1/23 | Upper-tail probability | Approximate reciprocal |
|---|---:|---:|---:|---:|---:|
| Original whole-span scan | 64 | 9 | 2.7826 | 0.001749932 = 0.174993% | 1 in 571 |
| Original scan plus only completed 14726 BC stage | 65 | 10 | 2.8261 | 0.0004813051 = 0.048131% | 1 in 2078 |
| Original scan plus both 12026 and 14726 BC stages | 66 | 10 | 2.8696 | 0.0005453274 = 0.054533% | 1 in 1834 |

The third row is the expanded reporting convention adopted here: it carries both anchor stages of the backbone and includes the 12026 BC miss to 6 BC. The second row records the narrower interpretation of adding only the author's requested tenth incident. Neither row changes the original nine-of-64 result.

The endpoint's prior appearance in File_52a and its role in the Creation–Flood–Conquest backbone give a documented structural reason for considering it. Statistical admission nevertheless occurs after the result is known. Moreover, segmented inversion enlarges the operation set, its outputs depend on the same chronology and anchors, and the corroborating paths share components. The density 1/23 does not establish the success probability of this selected construction. Accordingly, the new tail is a conditional benchmark, not a calibrated improvement in the study's significance. It is not multiplied by the original tail or by the repetitions of the backbone.

## 10. Interpretation of the combined study

### 10.1 The place of 6 BC

Among the five comparison targets, 6 BC has the largest concentration of positive 23-multiple spans: nine in the original whole-span scan and ten in the expanded tally. Divisibility identifies a residue class; the particular importance of 6 BC also depends on its anchor role and the separately stated contextual relations. The 1846 BC connection is developed as a structural hub in §5.6. The role of 6 BC is also constructive: it is the primary Nativity anchor for dates ending in 6. The companion 1 BC anchor serves dates ending in 1. The Conquest family similarly uses 1406 and 1401 BC.

The author did not select 6 BC solely because the original table returned nine hits. The 8600 = 4300 + 4300 relation, the Shem / Flood references, the expanded Creation return at 64406 BC, and the previously recorded 14726 BC backbone had already made it a focal point of the study (§§3 and 7). The 115 diagnostic was chosen because its success criterion and simple baseline were easier to state than a probability for the full narrative pattern.

The coexistence of construction-anchor and comparison-target roles does not generate a counted self-hit here. All ten successful spans in the expanded tally are positive, and 6 BC is absent from the source manifest. The tenth is specifically the 14720 span of the completed backbone.

The Creation-pillar proposal in §3.7 connects the nine earlier whole-span hits with that tenth, segmented endpoint. Since 14726 BC and 6 BC share the same 115-year grid, their complementary divisibility is automatic. The proposed interpretive gain lies in the specific person / event relations developed in §§3.7–3.12: the +60 Terah firstborn threshold, schematic Samaria node, Shem partition, Moses–Temple closure, Abraham symmetry, countdown alternatives, and joint Mahalalel–Shelah–Eber / Cainan pattern. The ten-member inventory now has an initial interpretation for every member; the trio's treatment is expressly cursory, with the omitted-Cainan connection as its main point. Its dependent relations and local counting choices remain explicit.

### 10.2 Continued activity at the Exodus and Conquest anchors

The observed counts support a narrow statement: this set of inverse endpoints is more concentrated in the 6 BC 23-multiple residue than in the tested 1406 BC and 1446 BC residues.

The discussion described this as a movement or clustering toward 6 BC. The pressure test now includes a descriptive before-and-after comparison on the same 64 route positions (Appendix C.4). Before inversion, the source date is repeated once for each of its two routes. Positive counts change from four to nine for 6 BC, from four to two for 1406 BC, and from four to zero for 1446 BC. This confirms the direction of the count changes under a fixed denominator; it does not supply a formal significance test of migration.

The Conquest remains one of the two construction families. The Exodus remains active through the 110, 700, 6370, and 10580 relations developed above. Those examples belong to different declared constructions and cannot be reduced to the single 115 count.

Admitting the segmented backbone also admits the 10580 = 5290 + 5290 Exodus comparison to the expanded count. The resulting ten / two / one positive distribution across 6 BC / 1406 BC / 1446 BC preserves the concentration toward 6 BC while documenting that positive Exodus result. The earlier before-and-after analysis remains confined to its original 64 matched whole-span positions.

### 10.3 Numerical–narrative coherence

The broader study asks whether a restricted family of transformations repeatedly produces relationships appropriate to their source narratives. Its examples include a Joseph lifespan arriving at the Exodus, Joshua's comparison through the common 110-year unit, Jacob's Entry age appearing as the 130 in 130 × 49, and Isaac's lifespan appearing as the increment under 25/23.

The author regards this accumulated contextual fit as sufficient reason to continue investigating, while not treating the present probability estimate as proof. That position is preserved here. The count to 6 BC is one supporting observation within the wider inquiry.

Structural inference becomes more informative when the same stated rules account for several relationships without introducing fresh local adjustments. Correlated expressions of one underlying identity should be recognized as a shared structure, not multiplied into independent evidence. Further prediction would require the rules and evaluation criteria to be specified before the next test.

### 10.4 The 12026 BC key and the scope of partial exploration

The September 27 continuation gives the author's proposed counterpart relation a concrete form: 14726 BC organizes the Rounded inverse field associated with 6 BC, while 12026 BC organizes the examined Actual MT field associated with 1446 BC. Its primary examples are the cumulative / regular Creation bridge and the consecutive 14 / 15 / 16 multiples of 598 to Jared, Noah, and the Flood. The 8066 BC midpoint, 1260 / 666 relations, and selected SKL / AD 25 continuations supply further context.

The author considers the evidence for 12026 BC's major anchor role strong and does not regard the incomplete interpretation as casting doubt on that central role. The report preserves that assessment as his structural judgment. “Partially explored” describes the limited examination of the detailed network, including the chart's remaining dates and the deferred Moses comparison. The full pairwise interval inventory is preparatory, not a claim that every block has already been explained.

## 11. Audit notes and unresolved questions

### 11.1 Inverse terminal dates

The discussion left the general death-date procedure unsettled. Direct inversion of a death date and reconstruction from an inverted birth plus a lifespan operation can give different results. The 32-date scan is primarily a birth/event scan and does not settle that choice.

Joseph and Joshua are useful bounded examples because 110 is self-inverse, so retaining that lifespan and reversing it coincide. That agreement does not establish a general rule for all other lifespans. Enoch is separately provisional because his being taken is not described as an ordinary death and because the operational choice remains open.

### 11.2 Corrections preserved in this report

The following corrections were settled during the discussion or in the September 25 author clarification and are carried into this draft explicitly.

| Issue | Controlling reading |
|---|---|
| Source of the 55016 civil span | 41006 BC to AD 14011; not a span beginning at 14006 BC. |
| Applying 59800 from 14006 BC | Use q arithmetic; −14005 + 59800 = +45795, displayed as AD 45796. |
| 62100 to 63000 conversion | Multiply by 70/69. |
| Description of 14006 → 64406 BC | Inverse followed by 70/69 expansion, not two applications of decimal reversal. |
| Anchor of the direct 12600 → 62100 inverse | Hold 1406 BC; direct rebuilt date 63506 BC. |
| Later 64400 to 6 BC | Derived after the stated expansion; not a direct inverse hit in the scan. |
| Flood described as non-inversable | 3430 is self-inverse; rebuilding from 1406 BC retains 4836 BC. |
| Factorization at the tenth incident | 14726 BC is the endpoint; its span to 6 BC is 14720 = 640 × 23. |
| Third partition and reconstruction order | 1406 BC → 6 BC is third when read from Creation chronologically; its inverse 4100 is processed first when rebuilding from 6 BC. File_52a §7 already records the leg. |
| Shem / Flood spans to 14726 BC | 11270 belongs to inverse cumulative Shem at 3456 BC; 9890 belongs to the Conquest-held cumulative Flood at 4836 BC. |
| The 16000 conversion | Expand the 14720-year span by 25/23; with 6 BC held, the expanded endpoint is 16006 BC. |
| Recovered 9890 bridge | Same numerical total as File_22's Actual cumulative-to-regular Creation bridge; the original Rounded gap stays 9900. |
| Backbone repetition and admission | Four stated primary / regular paths reach the same endpoint; count once per anchor stage. The paired extension gives 66 constructions, not 64 with an extra success. |
| Whole-span inversion of 1-ending sources | Hold 1 or 1401 BC; preserve the original source date. |
| AD 6 scan | Target coordinate +5; width $D'+4$, not the neighboring civil count. |
| Target-to-itself identities | Zero-span overlaps are separate from positive hits. |
| Purpose of the 1846 BC observation | Structural hub secondary to 1876 BC; Abram–Moses centered 230 + 230; Terah 1380 to the hub; proposed expansion artery. It is not an added statistical target. |
| Shelah's position in the inverse trio | Internal 1:10 divider of 690 and 6900, becoming 750 and 7500; not an arithmetic midpoint. |
| Trio AD 30 and 31 BC comparisons | 8880 / 1380 / 630 use civil elapsed counting; 31 BC reflects AD 30 only under the declared ordinary astronomical map. The other listed cross-axis spans use Rounded width. |
| The 601 → 531 BC continuation | Hold 1406 BC and expand 805 to 875; the 70-year gain is a proposed Babylonian-duration echo, without a historical captivity-date claim. |
| 8066 BC to the Conquest field | 6660 to 1406 BC; 6666 to 1400 BC, six years later. |
| Isaac's sixty-year comparison | Age sixty at Jacob's birth; lifespan 180 = 3 × 60, not 6 × 60. |
| Jared–Noah–Flood factorization | Each 598 = 26 × 23 = 2 × 299; combined total 1196, not 1058. The latter is 4114 − 3056. |
| Jared's 8372 anchor span | 14 × 598 = 7 × 1196 = 28 × 299 = 23 × 364; the 300/299 gain is 28 years. |
| Actual Creation bridge totals | Original 1978 + 7912 = 9890; expanded 2150 + 8600 = 10750. |
| Chart year labels | Printed ordinal year N corresponds to N − 1 elapsed years from 4114 BC. |
| Creation's generated 3426 BC* | Original span 7912 is a multiple of 23 but not of 299; retain the asterisk. |
| Jared's 728 and the outer 2000 | 728 is the 3654 → 2926 BC displacement; 2000 is the 3626 → 1626 BC range across two expansion branches. |
| Generated 2276 BC | Noah's 25/23 result; Flood results are 2426 and 1626 BC. |
| AD 25 continuations | Civil BC + AD − 1 gives 2300 / 1470 / 2450; these are not Rounded BC + AD − 2 widths. |
| Scope of the qualitative case | Includes patterns outside the Key of 23, especially the Joseph / Joshua 110s. |

### 11.3 Open interpretation and statistical questions

The open questions concern the general terminal-date operator, the justification and scope of mixed civil-count / Rounded-coordinate constructions, the proposed third-axis analogy, and the null model needed to quantify the target concentration with dependencies preserved.

The report pressure test verifies the declared arithmetic and identifies these remaining proof burdens. It does not resolve them or establish a formal combined probability for the narrative and 529 examples. The 115 count by itself identifies a residue class. The contextual role of a particular date within that class is examined through the separately stated structural relations.

The Actual 12026 BC continuation adds an explicitly partial interpretive task: explain the remaining chart nodes and the wider relations among the generated dates, including the deferred Moses comparison. The author has expressed strong confidence in the anchor itself. The 500 + 100 + 350 + 150 life-pattern comparison remains incomplete in the displayed interval table, which has no direct 350-year pair.

### 11.4 Editorial identity and cross-file routing

The report began as an author-requested draft and now includes the expressly requested 1846 BC amendment, report pressure test, Temple reference, counted backbone extension, and partially explored Actual 12026 BC continuation. The substantive style, state, arithmetic, and claim-status controls are applied throughout. The author's clarification places 1846 BC in the structural discussion and preserves the five-target scope. The original source manifest and target tables remain intact; the two admitted backbone constructions and their recalculation are separately identified. This file has not been marked Final.

Recommended cross-file updates are limited to eventual identifier resolution, registration of the adopted report and its guards, and reciprocal references if the author chooses to integrate it. No control file, existing File_52c, File_52a, or File_52b is altered by this draft.

## 12. Source and verification register

### 12.1 Controlling sources

The following sources were read for the present drafting task. References identify the source responsibility; they do not transfer the entire proof burden of those files into this report.

| Source | Responsibility in this report |
|---|---|
| 490d Repository Style Guide v2.5 | Canonical Markdown; header and section structure; number formatting; state and claim-status controls; tables; Mirror classification. |
| Project Procedures v3.5 | Authorial-direction rule; dependency boundaries; verification / pressure-test distinction; control-file non-mutation. |
| Supplied Restart Capsule v11.54 and State_Vocabulary_Register v1.55 | Inherited repository controls; §D.12–§D.13 inverse / Rounded / Mirror routing; existing File_52c identity. |
| File_00, especially §1 and §8.5; Samaria Node controls | Foundational argument-control precedent; treatment of the cumulative qualitative case; schematic status of the 726 BC siege / collapse label, also stated in File_16 §7.1. |
| File_05 §1.1; 1 Kings 6:1 | Explicit Exodus-to-Temple synchronism and the adopted 966 + 480 = 1446 relation; grounds the uncounted Temple reference. |
| File_01 Abraham biography; File_16 Nexus / Jerusalem controls; Genesis 15:13–16; Exodus 12:40–41 | Abraham source identity and narrative context; Entry, Exodus, and land-return associations; 1290 = 1876 − 586. Preserve the distinction between Genesis' 400 and the adopted 430 Entry-to-Exodus register. |
| File_51a §3.1; §6.1; §7.6; §24; September 25 clarification | Rounded birth data; Enoch's direct 3121 BC terminal date; Joseph / Joshua placements; cumulative Shem birth / death at 5436 / 4836 BC; current anchor and coordinate conventions. |
| File_51b, September 25 clarification | Rounded Mirror / civil-count distinction and controlled cross-axis examples. |
| File_52a §2.1; §§4–7; §§10–11; §16.3; September 25 clarification | Whole-span selector; 12026 / 14726 BC backbone already recorded; primary cumulative repetition and distinct secondary 13016 BC branches; 62100 expansion to the 64406 BC pole. |
| File_52b §1.1; §2; §5.2; September 25 clarification | Technical inverse definitions and source checks; distinct inverse terminal-date notation. |
| File_16, Egypt / affliction source states; §7.1 | 1876 BC Nexus; 1846 BC affliction field; schematic Samaria Node 726–721 BC under File_00 control. |
| File_18, including §4A.5 and restored-Cainan state controls; File_16 regular-Cainan 460 comparison | Raw chronology dependency; source / derived-state distinction; explicit +60 Terah firstborn threshold 2226 = 2296 − 70; selected restored Cainan 2551–2091 BC and Shelah 2421 BC; regular +130 insertion distinguished from cumulative +460. |
| File_61, seven-member Abraham → Moses inventory | Explicit sequence Abraham, Isaac, Jacob, Levi, Kohath, Amram, Moses; inclusive seven-node count used for the inverse Abraham–Moses endpoints. No cumulative dates are imported into the regular inverse example. |
| File_21, scalar and Shem-variable controls | SKL 360:1 scale and 720 span context; named Projective Boundary Pillars remain distinct from the Creation-pillar analogy. |
| File_11 §§1 and 9.1 | Japheth / Shem / Ham gear definitions and two-year spacing; earlier SP 9200 → 10000 Watcher Bridge. Retain local gear scope and distinct source states. |
| 1 Enoch 18:13–16; 21:3–6; 10:11–12 | Primary-text basis for the 10000-year confinement of transgressing heavenly stars; distinction from the seventy-generation sentence in the Watcher judgment narrative. Checked in the publisher's translated excerpt and the R. H. Charles text. |
| Daniel 8:14 | 2300 evenings-and-mornings / sanctuary-restoration theme used for the author's typological reading of 11500 = 5 × 2300; no historical AD 526 event inferred. |
| Genesis 11:12–14, MT-derived KJV and Brenton Septuagint translation | Arphaxad–Shelah adjacency in the MT line; second Cainan between them in the consulted LXX rendering; Cainan's 130 + 330 partition, also present for Shelah in that rendering. Textual difference distinguished from the author's expunction hypothesis; linked in §3.12.2. |
| Jeremiah 29:10 | Textual seventy-year Babylonian duration for the author's proposed 601 → 531 BC numerical echo; linked in §3.12.6. |
| File_22 §§5.1 and 7.1 | Actual cumulative-to-regular Creation bridge: 14004 − 4114 = 14006 − 4116 = 9890; source for comparison with 14726 − 4836 without replacing the Rounded 9900 gap. |
| File_54 §§0–1 | Later-file precedent for claim-status restraint, state preservation, and local Mirror role. |
| September 25 author discussion | Final 32-date inclusion decisions; target sequence; Joseph / Joshua and Nexus comparisons; exploratory interpretation. |
| Author amendments and clarifications | 1846 BC structural hub; Abram–Moses symmetry and Terah's 1380; Entry inverse / Mirror bridge; 966 BC Temple withheld from statistics; Shem / Flood / Creation backbone; admission of 14726 BC as the tenth incident and recalculation with both backbone stages; prior pressure test retained. |
| File_18 §3.1.4 and §4A.3–4; File_22 §5.1 | Selected Noah 3056 / Flood 2458 BC comparison, 1656-year Creation–Flood interval, and Actual completion endpoints; MT / LXX / SP Flood offset controls. |
| File_21 / File_34, SKL Long and macro-gear controls | Selected 2886 BC Long +30 rail; ±720 macro-scale comparison, distinct from ordinary two-year gears. |
| Author-supplied chart, “Covenantal thread of multiples of 23-years to 1446 BC (Exodus)” | Ordinal year 1 at 4114 BC; selected Jared / Noah / Flood / Exodus / conquest-rest chart nodes; author's explicit Noah Gear-1 selection. Image supplied as Covenantal-theme-of-23-year-cycles-to-Exodus.bmp. |
| Genesis 25:26 and 35:28; Revelation 11:2–3 and 13:5, 18 | Isaac's sixty-year begetting and 180-year lifespan; forty-two months, 1260 days, and 666 textual context. Links in §3.13. |
| 1 Enoch 6:6; 13:4–7; 72:32; 2 Peter 3:8 | Jared / Watcher descent, Enoch's petition, 364-day calendar, and thousand-year-day analogy. Textual units remain distinct from chronological year-span interpretations. |
| Exodus 2:3; Deuteronomy 34:7 | Author's deferred Moses comparison: the Nile ark and 120-year lifespan. No new Moses-date construction is asserted. |
| September 27 author discussion and update authorization | 12026 BC counterpart-key assessment; three starting levels; Actual Creation bridge; complete selected expansion and interval tables; cursory status; strong confidence in the anchor; permission to incorporate the previously held discussion. |
| Rounded Inverse Three-Target Full Table, September 25 2026 | Prior 64-route ledger against 6, 1406, and 1446 BC; reproduced without changing its count rules. |

The controlling source-file checksum for the supplied File_51a used in the scan is:

`d8aa38afb2a2df8f3bb57937f5d3dea897c286c034a1bee7398938d30e7bd77c`

The September 25 author clarifications govern the pure Rounded display and whole-span anchor selector where older source passages retain earlier rendering conventions. General Actual / civil chronology remains under its existing controls.

### 12.2 Verification scope

Verification covers the 32 source dates, both prescribed routes for each source, reverse-again recovery, all 320 original target widths and remainders, every positive-hit and zero-overlap count, the reported binomial tail, and the displayed narrative and 529 arithmetic. An arithmetic digit-reversal implementation independently checks the report code. The original five-target tables are unchanged. The later additions received bounded arithmetic checks: the signed Temple reference in §2.5, the held-center illustration in §5.6, and the Entry inverse / Mirror bridge in §5.7. Those Temple and hub examples are separate from the target scan.

The backbone amendment adds a bounded verification of the four stated paths to 12026 / 14726 BC, the cumulative Shem / Flood references, the operator distinction at 64406 BC, all ten new target cells, the expanded five-target totals and residue counts, and exact-rational binomial tails for 10/65 and 10/66. The original manifest and all 320 prior outcome cells were preserved. The expanded ledger therefore covers 330 comparisons, including its misses. This amendment does not recast the original pressure test as a new independent or blind study.

The subsequent backbone-relations addition checks the 16000 conversion and the 11270 / 9890 / 9200 / 49680 spans in §3.6, confirms the earlier third-partition reference in File_52a §7, and checks the Actual 9890 comparison against File_22. Its source-history table distinguishes previously explicit examples from newly stated endpoint pairings. The statistical manifest, counts, and probability calculations are preserved; they are not rerun merely to add these structural examples.

The Creation-pillar addition in §3.7 checks all nine complementary spans and Creation-held 25/23 expansions, the 1000 and 1120 gains in the author's two examples, the uniform 1280 relation between held-anchor choices, and the transported 250 + 250 symmetry. Its event-state checks use File_18 §4A.5 for 2226 BC and File_16 §7.1 / File_00 for the schematic 726 BC label. A bounded diff verifies preservation of the source manifest, statistical sections, and outcome tables; the earlier probability study is not repeated.

The companion addition in §3.8 checks the civil-style / Rounded display equivalence of the intended 9660 and 22540 durations, both expansion ratios and their distinct partitions, the Nativity-held 10000 + 6000 arrangement, and both Temple routes to q = +525. It verifies the 26750 outer width and the 1220 + 920 closure of the 2140 inner gap. Primary-text checks qualify the Enochic and sanctuary-restoration comparisons. Statistical sections and all 330 outcome cells remain unchanged.

The Moses–Temple continuation in §3.9 checks the repeated 10580 → 11500 → 12500 expansion at fixed AD 12026, its endpoint q = −475 / 476 BC, agreement with the Creation-held expansion of inverse Moses, and the shared 920 → 1000 interval. It checks the forward landing at reflected birth AD 1526, the 80-year gain, and the Rounded midpoint at AD 526. The opening era-label slips are resolved explicitly against the preceding reflected construction. No statistical recomputation is performed.

The Abraham continuation in §3.10 checks 12650 → 13750, the 1100 gain and 976 BC endpoint, both 430 / 470 routes, their 430 + 40 + 430 form and 1426 BC center, the 390 continuation to 586 BC, and the 550 + 550 midpoint at Moses' 1526 BC birth. The existing Entry-to-Jerusalem 1290 relation is confirmed in File_16. Existing statistical sections and all 330 outcome cells are preserved.

The countdown continuation in §3.11 checks the 1170 → 7110 inverse, 460 / 6440 / 6900 intervals, both expansions of 6210, the corrected 7976 BC endpoint, the matched 90-year shifts, and 6450 = 15 × 430 = 10 × 645. File_61 supplies the inclusive seven-member genealogical sequence. The ten-member interpretation inventory is checked against the unchanged §8.3 hit list and the admitted backbone; §3.12 supplies the final three members' joint interpretation.

The trio continuation in §3.12 checks all three source inversions, the 1290 source gap, 690 + 6900 partition, exact-ratio eligibility, Creation-held endpoints 8851 / 1351 / 601 BC, and the total 23 × 330 → 25 × 330 conversion. It checks the separately opened Cainan inverse 5521 BC, 6001 BC expansion and Rounded 12000 Mirror width, both 1401 BC anchor intervals, all eight paired civil / Rounded widths, the local AD 30 ↔ 31 BC civil-display reflection, the 1950 + 1950 midpoint, and the Conquest-held 805 → 875 continuation to 531 BC. File_18 and File_16 control the restored companion state; primary-text checks ground the Cainan lifespan and Babylonian-duration comparisons. Exact assertions are appended to Appendix B. The source manifest, §§8–9, all 330 target outcomes, and the prior pressure-test appendix are preserved; no probability calculation is rerun.

The author's subsequent scope clarification identifies §3.12 as a cursory exploration focused on the omitted-Cainan connection. Bounded checks of the four supplied follow-up branches confirm the 2151 / 2651 / 5601 / 6001 BC endpoints, the 400 same-side interval, and the Rounded 3900 → 4000 comparison to AD 1351. These checks do not constitute a thorough examination of the trio's wider relations.

The September 27 addition receives a bounded verification of §§3.13–3.15: the three starting levels and 8066 BC construction; 3960 / 6660 / 6666 relations; 23 × 1260 and the 43000-year Exodus expansion; ordinal chart recovery; the Actual 1:4 Creation bridge; all eight expansion rows; all twenty-one distinct-date pairwise intervals; the shared 2926 BC output; the 540 + 720 + 540 partition; and the civil AD 25 continuations. Separate exact-rational assertions follow the existing code in Appendix B. A focused diff checks that the original source manifest, statistical sections, all 330 target outcomes, original reproduction code, and prior pressure-test appendix remain intact. This is arithmetic and editorial verification of the authorized addition, not a new full pressure test or exhaustive search of the Actual chart.

The report's Markdown checks cover its single H1, numbered section structure, required metadata, framed tables, standardized Machine Guards, and absence of thousands separators in numbers. The runnable code in Appendix B reproduces the full target ledger and the stated simple probability calculation.

The September 26 author-requested pressure test extends those checks to the complete 23-class census, source-list sensitivity, original-versus-inverse counts, coordinate boundaries, and claim dependencies. Appendix C records the evidence and remaining limitations. This is a pressure test of the present report and its opened dependencies, not a claim that every file in the wider repository has been pressure-tested or that the model has received an independent statistical calibration.

## Appendix A. All inverse routes, including misses

### A.1 Reading the tables

The row numbers in A.2–A.5 refer to the complete source manifest in §2.1. Every inverse date in this appendix is BC. Tables A.2 and A.3 give the original three targets. Tables A.4 and A.5 give the later origin and AD 6 comparisons for those same endpoints. Table A.6 adds the two expressly admitted segmented backbone constructions, labeled B1 and B2 rather than new source rows.

Cell notation: a bold equation is a positive hit. A cell of the form `W; rR` gives the width $W$ and its nonzero remainder $R$ after division by 115, and is a miss. `0; overlap` is recorded separately. Distances are absolute; `later` marks an inverse endpoint later than the BC target.

No source is omitted because it misses. The two route families retain the ending-digit selector of §1.3. The AD 6 column uses Rounded coordinate width, not civil elapsed years.

### A.2 Nativity-family inverses: 6 BC, 1406 BC, and 1446 BC

The held anchor is 6 BC for a source ending in 6 and 1 BC for a source ending in 1. All three target columns are same-side BC separations.

| # | Held anchor BC | Inverse date BC | To 6 BC | To 1406 BC | To 1446 BC |
|---|---|---|---|---|---|
| 1 | 6 | 1406 | 1400; r20 | 0; overlap | 40; r40; later |
| 2 | 6 | 7936 | 7930; r110 | 6530; r90 | 6490; r50 |
| 3 | 1 | 7831 | 7825; r5 | 6425; r100 | 6385; r60 |
| 4 | 1 | 8731 | 8725; r100 | 7325; r80 | 7285; r40 |
| 5 | 1 | 1731 | **1725 = 15 × 115** | 325; r95 | 285; r55 |
| 6 | 6 | 4636 | 4630; r30 | 3230; r10 | 3190; r85 |
| 7 | 6 | 8436 | 8430; r35 | 7030; r15 | 6990; r90 |
| 8 | 1 | 2431 | 2425; r10 | 1025; r105 | 985; r65 |
| 9 | 6 | 3236 | 3230; r10 | 1830; r105 | 1790; r65 |
| 10 | 6 | 5036 | 5030; r85 | 3630; r65 | 3590; r25 |
| 11 | 6 | 5526 | **5520 = 48 × 115** | 4120; r95 | 4080; r55 |
| 12 | 6 | 5426 | 5420; r15 | 4020; r110 | 3980; r70 |
| 13 | 1 | 2421 | **2415 = 21 × 115** | 1015; r95 | 975; r55 |
| 14 | 1 | 9321 | **9315 = 81 × 115** | 7915; r95 | 7875; r55 |
| 15 | 6 | 5326 | 5320; r30 | 3920; r10 | 3880; r85 |
| 16 | 6 | 2326 | 2320; r20 | **920 = 8 × 115** | 880; r75 |
| 17 | 6 | 9226 | 9220; r20 | **7820 = 68 × 115** | 7780; r75 |
| 18 | 6 | 6226 | 6220; r10 | 4820; r105 | 4780; r65 |
| 19 | 6 | 3226 | **3220 = 28 × 115** | 1820; r95 | 1780; r55 |
| 20 | 6 | 6126 | 6120; r25 | 4720; r5 | 4680; r80 |
| 21 | 6 | 6026 | 6020; r40 | 4620; r20 | 4580; r95 |
| 22 | 6 | 2006 | 2000; r45 | 600; r25 | 560; r100 |
| 23 | 6 | 1916 | 1910; r70 | 510; r50 | 470; r10 |
| 24 | 1 | 3511 | 3505; r55 | 2105; r35 | 2065; r110 |
| 25 | 6 | 2516 | 2510; r95 | 1110; r75 | 1070; r35 |
| 26 | 6 | 7416 | 7410; r50 | 6010; r30 | 5970; r105 |
| 27 | 6 | 7526 | 7520; r45 | 6120; r25 | 6080; r100 |
| 28 | 1 | 9021 | 9015; r45 | 7615; r25 | 7575; r100 |
| 29 | 6 | 7816 | 7810; r105 | 6410; r85 | 6370; r45 |
| 30 | 6 | 4816 | 4810; r95 | 3410; r75 | 3370; r35 |
| 31 | 6 | 4416 | 4410; r40 | 3010; r20 | 2970; r95 |
| 32 | 1 | 2131 | 2125; r55 | 725; r35 | 685; r110 |

### A.3 Conquest-family inverses: 6 BC, 1406 BC, and 1446 BC

The held anchor is 1406 BC for a source ending in 6 and 1401 BC for a source ending in 1. All three target columns are same-side BC separations.

| # | Held anchor BC | Inverse date BC | To 6 BC | To 1406 BC | To 1446 BC |
|---|---|---|---|---|---|
| 1 | 1406 | 8606 | 8600; r90 | 7200; r70 | 7160; r30 |
| 2 | 1406 | 8926 | 8920; r65 | 7520; r45 | 7480; r5 |
| 3 | 1401 | 8821 | 8815; r75 | 7415; r55 | 7375; r15 |
| 4 | 1401 | 9721 | 9715; r55 | 8315; r35 | 8275; r110 |
| 5 | 1401 | 2721 | 2715; r70 | 1315; r50 | 1275; r10 |
| 6 | 1406 | 5626 | 5620; r100 | 4220; r80 | 4180; r40 |
| 7 | 1406 | 9426 | 9420; r105 | 8020; r85 | 7980; r45 |
| 8 | 1401 | 3421 | 3415; r80 | 2015; r60 | 1975; r20 |
| 9 | 1406 | 5216 | 5210; r35 | 3810; r15 | 3770; r90 |
| 10 | 1406 | 7016 | 7010; r110 | 5610; r90 | 5570; r50 |
| 11 | 1406 | 6516 | 6510; r70 | 5110; r50 | 5070; r10 |
| 12 | 1406 | 6416 | 6410; r85 | 5010; r65 | 4970; r25 |
| 13 | 1401 | 3411 | 3405; r70 | 2005; r50 | 1965; r10 |
| 14 | 1401 | 2391 | 2385; r85 | 985; r65 | 945; r25 |
| 15 | 1406 | 1996 | 1990; r35 | 590; r15 | 550; r90 |
| 16 | 1406 | 1696 | 1690; r80 | 290; r60 | 250; r20 |
| 17 | 1406 | 2386 | 2380; r80 | 980; r60 | 940; r20 |
| 18 | 1406 | 2086 | 2080; r10 | 680; r105 | 640; r65 |
| 19 | 1406 | 1786 | 1780; r55 | 380; r35 | 340; r110 |
| 20 | 1406 | 2076 | **2070 = 18 × 115** | 670; r95 | 630; r55 |
| 21 | 1406 | 2066 | 2060; r105 | 660; r85 | 620; r45 |
| 22 | 1406 | 2006 | 2000; r45 | 600; r25 | 560; r100 |
| 23 | 1406 | 1556 | 1550; r55 | 150; r35 | 110; r110 |
| 24 | 1401 | 1711 | 1705; r95 | 305; r75 | 265; r35 |
| 25 | 1406 | 1616 | **1610 = 14 × 115** | 210; r95 | 170; r55 |
| 26 | 1406 | 1476 | 1470; r90 | 70; r70 | 30; r30 |
| 27 | 1406 | 8516 | **8510 = 74 × 115** | 7110; r95 | 7070; r55 |
| 28 | 1401 | 2361 | 2355; r55 | 955; r35 | 915; r110 |
| 29 | 1406 | 2146 | 2140; r70 | 740; r50 | 700; r10 |
| 30 | 1406 | 1846 | **1840 = 16 × 115** | 440; r95 | 400; r55 |
| 31 | 1406 | 1446 | 1440; r60 | 40; r40 | 0; overlap |
| 32 | 1401 | 4111 | 4105; r80 | 2705; r60 | 2665; r20 |

### A.4 Nativity-family inverses: 1 BC and AD 6

The route endpoints are the same as Table A.2. The first width is $D'-1$; the second is the Rounded width $D'+4$.

| # | Inverse date BC | To 1 BC | Rounded width to AD 6 |
|---|---|---|---|
| 1 | 1406 | 1405; r25 | 1410; r30 |
| 2 | 7936 | **7935 = 69 × 115** | 7940; r5 |
| 3 | 7831 | 7830; r10 | 7835; r15 |
| 4 | 8731 | 8730; r105 | 8735; r110 |
| 5 | 1731 | 1730; r5 | 1735; r10 |
| 6 | 4636 | 4635; r35 | 4640; r40 |
| 7 | 8436 | 8435; r40 | 8440; r45 |
| 8 | 2431 | 2430; r15 | 2435; r20 |
| 9 | 3236 | 3235; r15 | 3240; r20 |
| 10 | 5036 | 5035; r90 | 5040; r95 |
| 11 | 5526 | 5525; r5 | 5530; r10 |
| 12 | 5426 | 5425; r20 | 5430; r25 |
| 13 | 2421 | 2420; r5 | 2425; r10 |
| 14 | 9321 | 9320; r5 | 9325; r10 |
| 15 | 5326 | 5325; r35 | 5330; r40 |
| 16 | 2326 | 2325; r25 | 2330; r30 |
| 17 | 9226 | 9225; r25 | 9230; r30 |
| 18 | 6226 | 6225; r15 | 6230; r20 |
| 19 | 3226 | 3225; r5 | 3230; r10 |
| 20 | 6126 | 6125; r30 | 6130; r35 |
| 21 | 6026 | 6025; r45 | 6030; r50 |
| 22 | 2006 | 2005; r50 | 2010; r55 |
| 23 | 1916 | 1915; r75 | 1920; r80 |
| 24 | 3511 | 3510; r60 | 3515; r65 |
| 25 | 2516 | 2515; r100 | 2520; r105 |
| 26 | 7416 | 7415; r55 | 7420; r60 |
| 27 | 7526 | 7525; r50 | 7530; r55 |
| 28 | 9021 | 9020; r50 | 9025; r55 |
| 29 | 7816 | 7815; r110 | **7820 = 68 × 115** |
| 30 | 4816 | 4815; r100 | 4820; r105 |
| 31 | 4416 | 4415; r45 | 4420; r50 |
| 32 | 2131 | 2130; r60 | 2135; r65 |

### A.5 Conquest-family inverses: 1 BC and AD 6

The route endpoints are the same as Table A.3. The first width is $D'-1$; the second is the Rounded width $D'+4$.

| # | Inverse date BC | To 1 BC | Rounded width to AD 6 |
|---|---|---|---|
| 1 | 8606 | 8605; r95 | 8610; r100 |
| 2 | 8926 | 8925; r70 | 8930; r75 |
| 3 | 8821 | 8820; r80 | 8825; r85 |
| 4 | 9721 | 9720; r60 | 9725; r65 |
| 5 | 2721 | 2720; r75 | 2725; r80 |
| 6 | 5626 | 5625; r105 | 5630; r110 |
| 7 | 9426 | 9425; r110 | **9430 = 82 × 115** |
| 8 | 3421 | 3420; r85 | 3425; r90 |
| 9 | 5216 | 5215; r40 | 5220; r45 |
| 10 | 7016 | **7015 = 61 × 115** | 7020; r5 |
| 11 | 6516 | 6515; r75 | 6520; r80 |
| 12 | 6416 | 6415; r90 | 6420; r95 |
| 13 | 3411 | 3410; r75 | 3415; r80 |
| 14 | 2391 | 2390; r90 | 2395; r95 |
| 15 | 1996 | 1995; r40 | 2000; r45 |
| 16 | 1696 | 1695; r85 | 1700; r90 |
| 17 | 2386 | 2385; r85 | 2390; r90 |
| 18 | 2086 | 2085; r15 | 2090; r20 |
| 19 | 1786 | 1785; r60 | 1790; r65 |
| 20 | 2076 | 2075; r5 | 2080; r10 |
| 21 | 2066 | 2065; r110 | **2070 = 18 × 115** |
| 22 | 2006 | 2005; r50 | 2010; r55 |
| 23 | 1556 | 1555; r60 | 1560; r65 |
| 24 | 1711 | 1710; r100 | 1715; r105 |
| 25 | 1616 | 1615; r5 | 1620; r10 |
| 26 | 1476 | 1475; r95 | 1480; r100 |
| 27 | 8516 | 8515; r5 | 8520; r10 |
| 28 | 2361 | 2360; r60 | 2365; r65 |
| 29 | 2146 | 2145; r75 | 2150; r80 |
| 30 | 1846 | 1845; r5 | 1850; r10 |
| 31 | 1446 | 1445; r65 | 1450; r70 |
| 32 | 4111 | 4110; r85 | 4115; r90 |

### A.6 The two added backbone constructions: all five targets

The original 64 routes above remain unchanged. These two rows complete the expanded 66-construction ledger. The notation is the same as in A.1; the AD 6 width remains the Rounded value $D'+4$.

| Construction | Endpoint BC | To 6 BC | To 1406 BC | To 1446 BC | To 1 BC | Rounded width to AD 6 |
|---|---|---|---|---|---|---|
| B1: Conquest stage | 12026 | 12020; r60 | 10620; r40 | **10580 = 92 × 115** | 12025; r65 | 12030; r70 |
| B2: completed Nativity stage | 14726 | **14720 = 128 × 115** | 13320; r95 | 13280; r55 | 14725; r5 | 14730; r10 |

The additions supply two positive comparisons and eight misses across the five targets. The repeated regular and cumulative derivations of these endpoints are displayed in §3 rather than duplicated in this ledger.

## Appendix B. Reproduction code

The following standard-library Python code uses the exact manifest order in §2.1. It prints every route and all five target outcomes, verifies the counts, and computes the unadjusted binomial benchmark. The two reference endpoint lists preserve the prior 64-route ledger for comparison. The code reproduces the original residue census, source-list sensitivities, and paired descriptive counts used in Appendix C, then verifies the two-construction backbone extension and its exact-rational probability calculations.

```python
from collections import Counter
from fractions import Fraction
from math import comb

# Source dates in the exact order of the report's 32-row manifest.
dates = [4106, 3976, 3871, 3781, 3711, 3646, 3486, 3421,
         3236, 3056, 2556, 2456, 2421, 2391, 2356, 2326,
         2296, 2266, 2236, 2166, 2066, 2006, 1916, 1531,
         1526, 1476, 2576, 2091, 1876, 1846, 1446, 3121]
reference = {
    "Nativity": [1406, 7936, 7831, 8731, 1731, 4636, 8436, 2431,
                 3236, 5036, 5526, 5426, 2421, 9321, 5326, 2326,
                 9226, 6226, 3226, 6126, 6026, 2006, 1916, 3511,
                 2516, 7416, 7526, 9021, 7816, 4816, 4416, 2131],
    "Conquest": [8606, 8926, 8821, 9721, 2721, 5626, 9426, 3421,
                 5216, 7016, 6516, 6416, 3411, 2391, 1996, 1696,
                 2386, 2086, 1786, 2076, 2066, 2006, 1556, 1711,
                 1616, 1476, 8516, 2361, 2146, 1846, 1446, 4111],
}

def inverse(n):
    if n == 0:
        return 0
    scale = 1
    while n % 10 == 0:
        n //= 10
        scale *= 10
    return int(str(n)[::-1]) * scale

targets = {"6 BC": -5, "1406 BC": -1405, "1446 BC": -1445,
           "1 BC": 0, "AD 6": 5}
counts = {name: Counter() for name in targets}
endpoints = []
for family, base in [("Nativity", 6), ("Conquest", 1406)]:
    for i, date in enumerate(dates):
        anchor = base if date % 10 == 6 else base - 5
        endpoint = anchor + inverse(date - anchor)
        assert endpoint == reference[family][i]
        assert anchor + inverse(endpoint - anchor) == date
        endpoints.append(endpoint)
        for target, q_target in targets.items():
            width = abs(q_target - (1 - endpoint))
            assert width % 5 == 0
            assert (width % 23 == 0) == (width % 115 == 0)
            state = ("overlap" if width == 0 else
                     "hit" if width % 23 == 0 else "miss")
            counts[target][state] += 1
            print(family, i + 1, date, anchor, endpoint,
                  target, width, width % 115, state)

assert len(set(dates)) == 32
assert len(endpoints) == 64 and len(set(endpoints)) == 63
expected = {"6 BC": (9, 0, 55), "1406 BC": (2, 1, 61),
            "1446 BC": (0, 1, 63), "1 BC": (2, 0, 62),
            "AD 6": (3, 0, 61)}
for target, result in counts.items():
    assert tuple(result[s] for s in ("hit", "overlap", "miss")) == expected[target]
    print(target, dict(result))

# Five statistical targets; 1846 BC is a structural hub in section 5.6.
assert len(targets) == 5 and len({q % 23 for q in targets.values()}) == 5

# Simplified independent-trial benchmark only.
p_tail = sum(comb(64, k) * (1 / 23) ** k * (22 / 23) ** (64 - k)
             for k in range(9, 65))
print("Expected per target:", 64 / 23)
print("P(X >= 9):", p_tail, "reciprocal:", 1 / p_tail)

# Descriptive pressure-test diagnostics; no calibrated null-model probability.
route_rows = [(family, i + 1, date, endpoint)
              for family, values in reference.items()
              for i, (date, endpoint) in enumerate(zip(dates, values))]
phase_counts = Counter(((1 - endpoint) // 5) % 23 for endpoint in endpoints)
phase_totals = [phase_counts[r] for r in range(23)]
assert phase_totals == [2, 3, 2, 3, 2, 3, 4, 1, 5, 1, 1, 6,
                        1, 4, 2, 3, 2, 1, 3, 1, 4, 1, 9]
print("Residue census:", phase_totals)

def positive(endpoint, q):
    width = abs(q - (1 - endpoint))
    return width > 0 and width % 115 == 0

subsets = {
    "Frozen manifest": route_rows,
    "Omit ascension": [r for r in route_rows if r[1] != 32],
    "Omit call": [r for r in route_rows if r[1] != 28],
    "Omit both": [r for r in route_rows if r[1] not in (28, 32)],
    "Distinct endpoints": list({r[3]: r for r in route_rows}.values()),
    "Changed routes only": [r for r in route_rows if r[2] != r[3]],
}
for label, subset in subsets.items():
    hits = {name: sum(positive(r[3], q) for r in subset)
            for name, q in targets.items()}
    print("Sensitivity:", label, len(subset), hits)

for name in ("6 BC", "1406 BC", "1446 BC"):
    q = targets[name]
    pre = {(f, i) for f, i, d, e in route_rows if positive(d, q)}
    post = {(f, i) for f, i, d, e in route_rows if positive(e, q)}
    print("Before/after:", name, len(pre), len(post),
          "retained", len(pre & post), "lost", len(pre - post),
          "gained", len(post - pre))

# Author-requested segmented backbone extension; one count per anchor stage.
# Paths run from the Conquest toward Creation. The Nativity stage adds 1400.
backbone_paths = {
    "Regular via Flood": (1050, 1650),
    "Regular via Noah": (1650, 1050),
    "Regular via Flood and Noah": (1050, 600, 1050),
    "Cumulative primary via Flood": (3430, 9170),
}
for label, spans in backbone_paths.items():
    assert sum(inverse(s) for s in spans) == 10620
    assert 1406 + sum(inverse(s) for s in spans) == 12026
    assert 6 + inverse(1400) + sum(inverse(s) for s in spans) == 14726
    print("Backbone corroboration:", label, 12026, 14726)

# Secondary cumulative branches are documented but not admitted by this rule.
for spans in ((3430, 600, 8570), (3430, 1550, 7620)):
    assert 1406 + sum(inverse(s) for s in spans) == 13016
    assert 6 + inverse(1400) + sum(inverse(s) for s in spans) == 15716

extra = [("B1", "Conquest", 12026), ("B2", "Nativity", 14726)]
expanded_counts = {name: result.copy() for name, result in counts.items()}
for code, stage, endpoint in extra:
    assert endpoint not in endpoints
    for name, q in targets.items():
        width = abs(q - (1 - endpoint))
        assert width % 5 == 0
        state = ("overlap" if width == 0 else
                 "hit" if width % 23 == 0 else "miss")
        expanded_counts[name][state] += 1
        print("Extension:", code, stage, endpoint, name,
              width, width % 115, state)

expanded_endpoints = endpoints + [r[2] for r in extra]
assert len(expanded_endpoints) == 66 and len(set(expanded_endpoints)) == 65
expanded_expected = {"6 BC": (10, 0, 56), "1406 BC": (2, 1, 63),
                     "1446 BC": (1, 1, 64), "1 BC": (2, 0, 64),
                     "AD 6": (3, 0, 63)}
for name, result in expanded_counts.items():
    assert tuple(result[s] for s in ("hit", "overlap", "miss")) == expanded_expected[name]
    assert sum(result.values()) == 66
    print("Expanded count:", name, dict(result))
assert tuple(sum(r[s] for r in expanded_counts.values())
             for s in ("hit", "overlap", "miss")) == (18, 2, 310)

expanded_phases = Counter(((1 - e) // 5) % 23 for e in expanded_endpoints)
expanded_totals = [expanded_phases[r] for r in range(23)]
assert expanded_totals == [2, 3, 2, 3, 2, 3, 4, 1, 5, 1, 2, 6,
                           1, 4, 2, 3, 2, 1, 3, 1, 4, 1, 10]
print("Expanded residue census:", expanded_totals)

def binomial_tail(n, threshold):
    return Fraction(sum(comb(n, k) * 22 ** (n - k)
                        for k in range(threshold, n + 1)), 23 ** n)

single_stage = endpoints + [14726]
assert len(single_stage) == 65
assert sum(positive(e, targets["6 BC"]) for e in single_stage) == 10
for n, threshold in ((64, 9), (65, 10), (66, 10)):
    tail = binomial_tail(n, threshold)
    print("Conditional benchmark:", n, threshold, "expected", n / 23,
          "tail", float(tail), "percent", 100 * float(tail),
          "reciprocal", float(1 / tail))

# Earlier references remain outside the counted extension.
assert 5436 - 600 == 4836
assert 1406 + inverse(4836 - 1406) == 4836
assert 6 + inverse(5436 - 6) == 3456
assert 4836 - 6 == 210 * 23 and 3456 - 6 == 150 * 23
assert inverse(inverse(12600)) == 12600
assert 1406 + Fraction(inverse(12600) * 70, 69) == 64406
assert Fraction((64406 - 6) * 25, 23) == 70000
assert 12026 - 1446 == 5290 + 5290 == 20 * 529
assert 14726 - 6 == 640 * 23 == 128 * 115

# Shared-backbone companion relations; structural comparisons, no new count entries.
assert Fraction((14726 - 6) * 25, 23) == 16000
assert 14726 - 3456 == 11270 == 23 * 490
assert 14726 - 4836 == 9890 == 23 * 430
assert 14004 - 4114 == 14006 - 4116 == 9890
assert 14006 - 4106 == 9900
assert 6 + inverse(2556 - 6) == 5526
assert 14726 - 5526 == 9200 == 23 * 400
assert 64406 - 14726 == 49680 == 23 * 6 * 360

# Creation-held expansions of the original nine hits; no new statistical rows.
creation_pillar = 14726
pillar_expected = {1731: 601, 5526: 4726, 2421: 1351, 9321: 8851,
                   3226: 2226, 2076: 976, 1616: 476, 8516: 7976,
                   1846: 726}
pillar_inputs = [e for e in endpoints if positive(e, targets["6 BC"])]
assert len(pillar_inputs) == 9 and set(pillar_inputs) == set(pillar_expected)
for e in pillar_inputs:
    span = creation_pillar - e
    assert span > 0 and span % 115 == 0
    assert span + (e - 6) == 14720
    expanded_span = Fraction(span * 25, 23)
    shifted = creation_pillar - expanded_span
    assert shifted.denominator == 1 and shifted == pillar_expected[e]
    nativity_held = 6 + Fraction((e - 6) * 25, 23)
    assert nativity_held - shifted == 1280
assert 12500 - 11500 == 1000 and 14000 - 12880 == 1120
assert 2236 + 60 - 70 == 2226
assert 726 - 6 == 720 == 2 * 360
assert 976 - 726 == 726 - 476 == 250

# Pillar companion relations; preserve each display, operator, and held anchor.
q_bc = lambda b: 1 - b
q_ad = lambda a: a - 1
assert 1846 + 7815 - 1 == q_ad(7816) - q_bc(1846) == 9660
assert 14726 + 7815 - 1 == q_ad(7816) - q_bc(14726) == 22540
assert 4830 + 4830 == 9660
assert Fraction(9660 * 70, 69) == 9800
assert 22540 == 23 * 980
assert Fraction(22540 * 25, 23) == 24500 == 490 * 50
assert 24500 - 9800 == 14700 == 490 * 30
assert 9800 == 490 * 20
assert Fraction(12880 * 25, 23) == 14000
assert Fraction(9660 * 25, 23) == 10500
assert 24500 == 14000 + 10500 == 14700 + 9800
assert 10500 - 9800 == 14700 - 14000 == 700
assert 14726 - 5526 == 9200 and 5526 - 6 == 5520
assert Fraction(9200 * 25, 23) == 10000
assert Fraction(5520 * 25, 23) == 6000
assert 6 + Fraction((14726 - 6) * 25, 23) == 16006
assert 6 + Fraction((5526 - 6) * 25, 23) == 6006
assert 16006 - 6006 == 10000 and 6006 - 6 == 6000
assert 14726 - 696 == 14030 == 23 * 610
assert Fraction(14030 * 25, 23) == 15250
temple_left_q = q_bc(14726) + Fraction(14030 * 25, 23)
temple_right_q = q_ad(12026) - Fraction(10580 * 25, 23)
assert temple_left_q == temple_right_q == 525 == q_ad(526)
assert 14726 + 525 - 1 == 15250  # literal civil companion label
assert 12026 - 526 == 11500 == 5 * 2300
assert 1446 - 526 == 920 == 11500 - 10580
assert 14726 + 12026 - 2 == 26750 == 15250 + 11500
assert 15250 - 14030 == 1220
assert 1220 + 920 == 2140 == 696 + 1446 - 2

# Repeated expansion and Moses–Temple closure in Rounded coordinates.
ratio = Fraction(25, 23)
reflected_backbone_q = q_ad(12026)
reflected_exodus_q = q_ad(1446)
first_q = reflected_backbone_q + ratio * (
    reflected_exodus_q - reflected_backbone_q)
second_q = reflected_backbone_q + ratio * (
    first_q - reflected_backbone_q)
assert 10580 == 20 * 23**2
assert Fraction(10580) * ratio == 11500 == 20 * 23 * 25
assert Fraction(11500) * ratio == 12500 == 20 * 25**2
assert first_q == q_ad(526) and second_q == q_bc(476)
assert reflected_backbone_q - q_bc(1446) == 13470
assert 14726 - 1616 == 13110 == 570 * 23
assert Fraction(13110) * ratio == 14250
assert q_bc(14726) + Fraction(13110) * ratio == second_q
assert q_bc(14726) + Fraction(14030) * ratio == first_q
assert 1616 - 696 == reflected_exodus_q - first_q == 920
assert Fraction(920) * ratio == first_q - second_q == 1000
mirror_birth_q = first_q + ratio * (reflected_exodus_q - first_q)
assert mirror_birth_q == q_ad(1526)
assert 1000 - 920 == 1526 - 1446 == 80
assert first_q == (second_q + mirror_birth_q) / 2
assert mirror_birth_q - second_q == 2000 == 476 + 1526 - 2

# Abraham's Creation-held expansion, Entry symmetry, and Moses midpoint.
assert 2166 - 1406 == 760 and inverse(760) == 670
assert 1406 + inverse(760) == 2076
assert 2076 - 6 == 2070 == 18 * 115
assert 14726 - 2076 == 12650 == 550 * 23
assert Fraction(12650 * 25, 23) == 13750 == 550 * 25
assert 13750 - 12650 == 1100
assert 14726 - 13750 == 2076 - 1100 == 976
assert 1876 - 1446 == 1406 - 976 == 430
assert 1876 - 1406 == 1446 - 976 == 470
assert 1446 - 1406 == 40
assert 1876 - 976 == 900 == 430 + 40 + 430
assert Fraction(1876 + 976, 2) == Fraction(1446 + 1406, 2) == 1426
assert 976 - 586 == 390
assert 1876 - 586 == 1290 == 900 + 390
assert 2076 - 1526 == 1526 - 976 == 550
assert Fraction(2076 + 976, 2) == 1526

# Flood countdown: two named ratios, with Creation held in both branches.
assert 2576 - 2456 == 120
assert 2576 - 1406 == 1170
assert inverse(1170) == 7110 and 1406 + inverse(1170) == 8516
assert 2076 - 1616 == 460
assert 8516 - 2076 == 6440 == 14 * 460
assert 8516 - 1616 == 6900 == 15 * 460
assert 14726 - 8516 == 6210 == 90 * 69 == 270 * 23
assert Fraction(6210 * 70, 69) == 6300
assert 14726 - 6300 == 8426
assert 8516 - 8426 == 1616 - 1526 == 6300 - 6210 == 90
assert 8426 - 1526 == 6900
assert Fraction(6210 * 25, 23) == 6750
assert 14726 - 6750 == 7976 and 6750 - 6210 == 540
assert 7976 - 1526 == 6450 == 15 * 430
assert 2091 - 1446 == 645 and 6450 == 10 * 645
assert 8426 - 7976 == 450 == 15 * (460 - 430)
assert 7926 - 1526 == 6400  # reason for correcting the supplied endpoint

# Mahalalel–Shelah–Eber: bounded trio and restored-Cainan companion.
trio_rows = [
    ("Eber", 2391, 9321, 5405, 5875, 8851),
    ("Shelah", 2421, 2421, 12305, 13375, 1351),
    ("Mahalalel", 3711, 1731, 12995, 14125, 601),
]
trio_ratios = (Fraction(25, 23), Fraction(70, 69), Fraction(300, 299))
for name, source, inv, span, expanded, endpoint in trio_rows:
    assert source % 10 == 1
    assert 1 + inverse(source - 1) == inv
    assert 14726 - inv == span
    assert Fraction(span) * trio_ratios[0] == expanded
    assert 14726 - expanded == endpoint
    assert q_bc(14726) + expanded == q_bc(endpoint)
    assert [r for r in trio_ratios if (span * r).denominator == 1] == [
        Fraction(25, 23)]
assert [s * trio_ratios[1] for _, _, _, s, _, _ in trio_rows] == [
    Fraction(16450, 3), Fraction(37450, 3), Fraction(39550, 3)]
assert [s * trio_ratios[2] for _, _, _, s, _, _ in trio_rows] == [
    Fraction(70500, 13), Fraction(160500, 13), Fraction(169500, 13)]
assert 3711 - 2421 == 1290
assert 9321 - 2421 == 6900 and 2421 - 1731 == 690
assert 9321 - 1731 == 7590 == 23 * 330 == 33 * 230
assert Fraction(33, 2) * 460 == 7590
assert 8851 - 1351 == 7500 and 1351 - 601 == 750
assert Fraction(6900 * 25, 23) == 7500
assert Fraction(690 * 25, 23) == 750
assert 8851 - 601 == 8250 == 25 * 330
assert 8250 - 7590 == 660 == 2 * 330
assert 2551 - 2421 == 130 and 2421 - 2091 == 330
assert 2551 - 2091 == 460 == 130 + 330
assert 2551 - 1 == 2550 and inverse(2550) == 5520
assert 1 + inverse(2550) == 5521
assert Fraction(5520 * 25, 23) == 6000
assert 1 + Fraction(5520 * 25, 23) == 6001
assert q_ad(6001) - q_bc(6001) == 12000
assert 6001 + 6001 - 1 == 12001
assert 2551 - 1401 == 1150 == 50 * 23
assert 2091 - 1401 == 690 == 30 * 23
assert 1150 - 690 == 460 == 20 * 23

def trio_civil_width(bc, ad):
    return bc + ad - 1

def trio_rounded_width(bc, ad):
    return q_ad(ad) - q_bc(bc)

trio_cross_axis = [
    (8851, 30, 8880, 8879),
    (8851, 2421, 11271, 11270),
    (1351, 30, 1380, 1379),
    (1351, 2551, 3901, 3900),
    (601, 30, 630, 629),
    (601, 1351, 1951, 1950),
    (1351, 2091, 3441, 3440),
    (1351, 1446, 2796, 2795),
]
for bc, ad, civil, rounded in trio_cross_axis:
    assert trio_civil_width(bc, ad) == civil
    assert trio_rounded_width(bc, ad) == rounded == civil - 1
assert 1 - 31 == -30  # ordinary astronomical reflection of AD 30
assert q_bc(31) == -q_ad(31)
assert q_bc(30) == -q_ad(30) != q_bc(31)
assert 8851 - 31 == 8820 == 49 * 180
assert trio_rounded_width(8851, 2421) == 23 * 490
for bc, width in [(8851, 8880), (1351, 1380), (601, 630)]:
    assert trio_civil_width(bc, 30) == trio_rounded_width(bc, 31) == width
assert trio_civil_width(1351, 30) == 3 * 460
assert 601 - 1 == 600
assert 2551 - 601 == 1950 == 15 * 130
assert 2421 - 601 == 1820 == 14 * 130
assert q_bc(601) - q_bc(2551) == q_ad(1351) - q_bc(601) == 1950
assert Fraction(q_bc(2551) + q_ad(1351), 2) == q_bc(601)
assert q_ad(601) - q_bc(1351) == q_ad(2551) - q_ad(601) == 1950
assert trio_rounded_width(2551, 1351) == 3900 == 2 * 1950
assert trio_rounded_width(1351, 2551) == 3900
assert 2551 - 1351 == 1200
assert trio_rounded_width(1351, 2091) == 3440 == 8 * 430
assert trio_rounded_width(1351, 1446) == 2795 == Fraction(13, 2) * 430
assert 2795 == 13 * 215
assert 3440 - 2795 == 645 == 2091 - 1446
assert 1406 - 601 == 805 == 23 * 35
assert Fraction(805 * 25, 23) == 875
assert 875 - 805 == 70
assert 1406 - 875 == 531 and 601 - 531 == 70
assert q_bc(1406) + Fraction(805 * 25, 23) == q_bc(531)

# Cursory trio follow-up: only the author's four additional branches.
for source, anchor, branch_ratio, span, endpoint, gain in [
    (2091, 1401, Fraction(25, 23), 750, 2151, 60),
    (2551, 1401, Fraction(25, 23), 1250, 2651, 100),
    (5521, 1, Fraction(70, 69), 5600, 5601, 80),
    (5521, 1, Fraction(25, 23), 6000, 6001, 480),
]:
    assert (source - anchor) * branch_ratio == span
    assert anchor + span == endpoint
    assert endpoint - source == gain
assert 2551 - 2151 == 400  # original birth retained
assert 2551 + 1351 - 2 == 3900  # Rounded width
assert 2651 + 1351 - 2 == 4000  # Rounded width after the 100-year extension
```

### B.1 Bounded verification of the partially explored 12026 BC key

The following standalone assertions verify the September 27 addition without rerunning the preceding statistical survey.

```python
from fractions import Fraction
from itertools import combinations

# Bounded verification of sections 3.13-3.15 only.
# This does not rerun or enlarge the statistical scan.
def inverse_span(n):
    assert isinstance(n, int) and n > 0
    digits = str(n)
    stem = digits.rstrip("0")
    return int(stem[::-1]) * 10 ** (len(digits) - len(stem))

for original, reversed_span in [
    (1400, 4100), (1050, 5010), (1650, 5610),
    (3430, 3430), (9170, 7190), (14000, 41000), (120, 210),
]:
    assert inverse_span(original) == reversed_span

regular_parts = (1400, 1050, 1650)
cumulative_parts = (1400, 3430, 9170)
assert 6 + sum(map(inverse_span, regular_parts)) == 14726
assert 6 + sum(map(inverse_span, cumulative_parts)) == 14726
assert 1406 + sum(map(inverse_span, regular_parts[1:])) == 12026
assert 1406 + sum(map(inverse_span, cumulative_parts[1:])) == 12026
assert 1406 + inverse_span(2456 - 1406) == 6416
assert 1406 + inverse_span(4836 - 1406) == 4836
assert 2456 + inverse_span(4106 - 2456) == 8066
assert 4836 + inverse_span(14006 - 4836) == 12026
assert 12026 - 8066 == 8066 - 4106 == 3960 == 11 * 360 == 66 * 60
assert 8066 - 1406 == 6660 == 10 * 666
assert 8066 - 1400 == 6666 and 1406 - 1400 == 6
assert 8066 - 2066 == 6000
assert 180 == 3 * 60 and 360 == 2 * 180

r25, r70, r300 = Fraction(25, 23), Fraction(70, 69), Fraction(300, 299)
assert 42 * 30 == 1260
assert 6 + inverse_span(14006 - 6) == 41006
assert 41006 - 12026 == 28980 == 23 * 1260
assert 28980 * r25 == 31500 == 25 * 1260
assert 31500 - 28980 == 2520 == 7 * 360 == 2 * 1260
assert 14006 - 12026 == 1980 == 66 * 30
assert 41006 - 1446 == 39560 == 23 * 1720
assert 39560 * r25 == 43000 == 100 * 430
assert 1446 + 43000 == 44446
assert 44446 - 6 == 44440 == 5 * 8888
assert 39560 - 28980 == 12026 - 1446 == 10580 == 460 * 23 == 20 * 529
assert 43000 - 31500 == 11500

# The chart uses ordinal year 1 at 4114 BC.
for ordinal, elapsed, date_bc in [
    (1, 0, 4114), (461, 460, 3654), (1059, 1058, 3056),
    (1657, 1656, 2458), (2669, 2668, 1446), (2715, 2714, 1400),
]:
    assert ordinal - 1 == elapsed
    assert 4114 - elapsed == date_bc and elapsed % 23 == 0

assert 14004 - 12026 == 1978 == 86 * 23
assert 12026 - 4114 == 7912 == 344 * 23
assert 14004 - 4114 == 9890 == 430 * 23 == 1978 + 7912
assert 1978 * r25 == 2150 == 5 * 430
assert 7912 * r25 == 8600 == 20 * 430
assert 9890 * r25 == 10750 == 2150 + 8600 == 25 * 430
assert 7912 == 4 * 1978 and 8600 == 4 * 2150
assert Fraction(4 * 14004 + 4114, 5) == 12026
assert 14006 - 12026 == 1980 and 12026 - 4106 == 7920 == 4 * 1980
assert 14006 - 4106 == 9900
assert 14006 - 14004 == 2 and 4114 - 4106 == 8
assert 7912 % 299 != 0 and 12026 - 8600 == 3426

assert 3654 - 3056 == 3056 - 2458 == 598 == 26 * 23 == 2 * 299
assert 598 == 2 * 13 * 23
assert 3654 - 2458 == 1196 == 598 + 598
assert 4114 - 3056 == 1058 == 460 + 598
assert 4114 - 3654 == 460 == 20 * 23
assert 1196 * r300 == 1200 == 600 + 600
assert 1196 * r25 == 1300 == 650 + 650
for date_bc, units, span in [(3654, 14, 8372), (3056, 15, 8970), (2458, 16, 9568)]:
    assert 12026 - date_bc == span == units * 598 == 2 * units * 299
    assert span * r300 == units * 600
    assert span * r25 == units * 650
assert 8372 == 14 * 598 == 7 * 1196 == 28 * 299 == 23 * 364

# source, original anchor span, ratio, expanded span, forward gain, result BC
expansion_rows = [
    (4114, 7912, r25, 8600, 688, 3426),
    (3654, 8372, r300, 8400, 28, 3626),
    (3654, 8372, r25, 9100, 728, 2926),
    (3056, 8970, r300, 9000, 30, 3026),
    (3056, 8970, r70, 9100, 130, 2926),
    (3056, 8970, r25, 9750, 780, 2276),
    (2458, 9568, r300, 9600, 32, 2426),
    (2458, 9568, r25, 10400, 832, 1626),
]
for source, original, ratio, expanded, gain, result in expansion_rows:
    assert 12026 - source == original
    assert original * ratio == expanded
    assert expanded - original == source - result == gain
    assert 12026 - expanded == result

assert 8970 * r70 == 8372 * r25 == 9100 == 25 * 364
assert 8970 * r300 == 9000 == 25 * 360
assert 728 == 364 + 364 and 12026 - 9100 == 2926
assert 8970 % 23 == 8970 % 69 == 8970 % 299 == 0
assert 8372 % 69 != 0 and 9568 % 69 != 0
assert (3056 - 3026, 3026 - 2926, 2926 - 2276) == (30, 100, 650)
assert 30 + 100 + 650 == 780
assert 3238 - 2458 == 780 and 3108 - 2458 == 650 and 3238 - 3108 == 130
assert 3626 - 3026 == 3026 - 2426 == 600
assert 2926 - 2276 == 2276 - 1626 == 650
assert (3626 - 2926, 3026 - 2276, 2426 - 1626) == (700, 750, 800)

dates = [3626, 3426, 3026, 2926, 2426, 2276, 1626]
assert sorted({row[-1] for row in expansion_rows}, reverse=True) == dates
expected_triangular_rows = [
    [200, 600, 700, 1200, 1350, 2000],
    [400, 500, 1000, 1150, 1800],
    [100, 600, 750, 1400],
    [500, 650, 1300],
    [150, 800],
    [650],
]
for i, expected in enumerate(expected_triangular_rows):
    assert [dates[i] - later for later in dates[i + 1:]] == expected
all_intervals = [earlier - later for earlier, later in combinations(dates, 2)]
assert len(all_intervals) == 21 and 350 not in all_intervals
assert [a - b for a, b in zip(dates, dates[1:])] == [200, 400, 100, 500, 150, 650]
assert 200 + 400 + 100 + 500 + 150 + 650 == 2000
assert 3426 - 2926 == 2926 - 2426 == 500
assert 2926 - 2426 + 2426 - 2276 == 500 + 150 == 650
assert 3026 - 1626 == 100 + 500 + 150 + 650 == 1400
assert 3426 - 2276 == 1150 == 23 * 50
assert 3626 - 1626 == 2000 == 1000 + 1000
assert 3654 - 2926 == 728  # Displacement, not the 2000-year outer range.

assert (4114 - 2458) * r25 == 1656 * r25 == 1800 == 3426 - 1626
assert (3426 - 2886, 2886 - 2166, 2166 - 1626) == (540, 720, 540)
assert 3426 - 2166 == 2886 - 1626 == 1260
assert (2886 + 720, 2886, 2886 - 720) == (3606, 2886, 2166)

def civil_span(bc, ad):
    return bc + ad - 1

assert civil_span(2276, 25) == 2300
assert 2426 - 1446 == 980 == 2 * 490
assert civil_span(1446, 25) == 1470 == 3 * 490
assert civil_span(2426, 25) == 2450 == 5 * 490 == 980 + 1470
assert civil_span(6, 25) == 30
print("Bounded 12026 BC checks passed: eight expansion rows and 21 pairwise intervals.")
```

## Appendix C. Author-requested pressure test — September 26 2026

### C.1 Verdict and scope

The report passes the stated arithmetic and table checks. No correction to the nine positive spans at 6 BC was needed. The author's subsequent clarification restores the statistical scope to five targets and places 1846 BC in the structural study (§5.6). The source manifest and original statistical results remain unchanged.

Subsequent backbone amendment: the author admits the segmented 14726 BC endpoint as the tenth 6 BC incident. Bounded verification adds its companion 12026 BC stage, giving ten hits among 66 constructions and one positive Exodus hit. The new ledger and benchmarks are in §§8.7–9.6 and Appendix A.6; the original pressure-test results below retain their original 64-route scope unless expressly extended.

The stronger interpretation remains conditional. Divisibility identifies a target class rather than selecting 6 BC uniquely; the paired routes and chronological source dates are dependent; and the study was exploratory. The approximate 1-in-571 result remains a correctly computed benchmark for a simplified independent-trial model, not a calibrated probability for this investigation.

The original pressure test uses the frozen report and source data. It checks alternate counting boundaries and dependence without adding further source dates, favorable targets, or new factor searches. The later backbone admission is an explicit authorial amendment rather than an outcome silently added by that pressure test.

| Review layer | Check and result | Disposition |
|---|---|---|
| Source / scope | All 32 manifest dates checked; the first 23 match File_51a §3.1; event derivations and extra birth placements retained. | Pass for the elected Rounded model; model dates retain their source status. |
| Inverse arithmetic | All 64 routes recomputed by arithmetic digit reversal and compared with the string-based reproduction code; inverse-again recovery verified. | Pass; 63 unique endpoints, with Jacob duplicated across families. |
| Target tables | All 320 widths, remainders, hit states, overlaps, and direction markers checked for the five-target study. | Pass; original statistical tables remain unchanged. |
| Structural 1846 BC hub | Shared 115-year grid; 2076 − 1846 = 1846 − 1616 = 230; 3226 − 1846 = 1380. Held-center 25/23 illustration checked after clarification. | Exact within the model; the proposed wider expansion artery remains a structural reading outside the target scan. |
| Narrative / expansion arithmetic | 40 exact checks cover placeholders, Creation trunks, segmentations, 110 relations, the Nexus, all four 529 chains, 299 conversion, coordinate boundaries, and the 62100 chain. | Pass under the locally declared operations. |
| Statistical arithmetic | Binomial tail recomputed using an exact integer numerator over 23⁶⁴, then compared with the report's floating-point calculation. | Pass for the stated model; applicability remains uncalibrated. |
| Subsequent backbone amendment | Four stated paths checked at both anchors; ten new target cells, 66-construction totals, and exact-rational tails for 10/65 and 10/66 checked. | Pass; repeated derivations count once per stage, with conditional model status retained. |
| Structural / editorial | Required metadata, single H1, section nesting, table widths, Machine Guard syntax, number style, file map, and change scope checked. | Pass; Draft status and tentative file identity retained. |
| Interpretive claims | Numerical identities separated from textual premises, model placements, dependent restatements, and typological or providential readings. | Qualified; the remaining proof burdens are listed in C.6. |

### C.2 Zero spans, direction, and source admission

The five-target scan uses absolute widths and excludes zero-span overlaps from positive hits. Adam's inverse at 1406 BC and the Exodus inverse at 1446 BC supply the two overlaps. The direction of a BC comparison is retained in the tables wherever an endpoint lies later than its target.

There is no target-to-itself hit at 6 BC. The sources 6 BC and 1 BC remain excluded; the inverse endpoints contain neither label. Self-inverse admitted sources are a separate issue. Twelve routes preserve their source dates: five Nativity routes and seven Conquest routes. Two of those twelve contribute positive spans to 6 BC—Shelah and the affliction node. The other seven 6 BC hits come from routes whose inverse endpoints differ from their source dates.

The 1846 BC affliction node is retained under the agreed self-inverse exception because its span to the held Conquest anchor is 440 and I(440) = 440. Its separate 400-year span is to the Exodus at 1446 BC. Confusing those two anchors would obscure the reason for inclusion.

The audit also checks the cross-axis boundary. AD 6 uses Rounded coordinate +5 and width D′ + 4. Replacing this with the ordinary civil count D′ + 5 changes the operation and the residue test. Those conventions cannot be exchanged within a row.

### C.3 Exhaustive original residue census and the expanded change

For each inverse endpoint D′ BC, define its grid coordinate g = (1 − D′)/5 and classify it by g modulo 23. This exhausts all possible congruence classes of targets on the admitted mod-5 grid. The following table counts route membership, including any exact overlap when a particular target is later named. It is therefore a class census rather than a positive-span tally for an arbitrarily chosen representative date.

| Grid residue g mod 23 | Nativity routes | Conquest routes | Total members | Named targets |
|---|---|---|---|---|
| 0 | 1 | 1 | 2 | 1 BC |
| 1 | 1 | 2 | 3 | AD 6 |
| 2 | 1 | 1 | 2 | No named comparison target |
| 3 | 2 | 1 | 3 | No named comparison target |
| 4 | 0 | 2 | 2 | No named comparison target |
| 5 | 1 | 2 | 3 | No named comparison target |
| 6 | 0 | 4 | 4 | No named comparison target |
| 7 | 0 | 1 | 1 | No named comparison target |
| 8 | 1 | 4 | 5 | No named comparison target |
| 9 | 0 | 1 | 1 | No named comparison target |
| 10 | 0 | 1 | 1 | 1446 BC |
| 11 | 2 | 4 | 6 | No named comparison target |
| 12 | 1 | 0 | 1 | No named comparison target |
| 13 | 3 | 1 | 4 | No named comparison target |
| 14 | 2 | 0 | 2 | No named comparison target |
| 15 | 1 | 2 | 3 | No named comparison target |
| 16 | 2 | 0 | 2 | No named comparison target |
| 17 | 1 | 0 | 1 | No named comparison target |
| 18 | 3 | 0 | 3 | 1406 BC |
| 19 | 1 | 0 | 1 | No named comparison target |
| 20 | 3 | 1 | 4 | No named comparison target |
| 21 | 1 | 0 | 1 | No named comparison target |
| 22 | 5 | 4 | 9 | 6 BC |

The 6 BC class, residue 22 in this coordinate, is the unique largest class: nine members. The next largest is residue 11 with six; residue 8 has five. The mean membership is 64/23 ≈ 2.7826. No other class reaches nine in this fixed table.

The 1406 BC class has three members: two positive hits plus Adam's exact overlap. The 1446 BC class has one member, the Exodus overlap. This explains why the class census differs from the positive-count table for those targets. The structural connection of 1846 BC with the 6 BC grid is recorded in §5.6 without adding a target here.

This census strengthens the descriptive statement that the observed concentration is not just the largest among a few named comparisons. It does not calculate how often a comparable maximum would arise from a new source chronology. A random-target phase experiment would answer a different question from a probability model for the selected dates and their transformations; no new significance estimate is inferred from this table.

The paired backbone extension changes only the following classes. The full 23-class expanded census is reproduced by Appendix B.

| Grid residue | Original members | Added endpoint | Expanded members |
|---|---:|---|---:|
| 10, Exodus class | 1 | 12026 BC | 2 |
| 22, 6 BC class | 9 | 14726 BC | 10 |
| Other 21 classes combined | 54 | None | 54 |

The expanded mean is 66/23 ≈ 2.8696. The 6 BC class remains the unique largest with ten members; the next largest remains six. The Exodus class now contains one positive hit and one overlap. No new overlap is introduced by the backbone extension.

### C.4 Sensitivity and the original-versus-inverse comparison

The following rows are sensitivity checks of the original 64-route whole-span manifest, not amendments to it or recalculations of the expanded set. Every target column reports positive spans only.

| Scope | Route / endpoint count | 6 BC | 1406 BC | 1446 BC | 1 BC | AD 6 |
| --- | --- | --- | --- | --- | --- | --- |
| Frozen manifest | 64 | 9 | 2 | 0 | 2 | 3 |
| Omit provisional Enoch ascension | 62 | 9 | 2 | 0 | 2 | 3 |
| Omit initially overlooked call | 62 | 9 | 2 | 0 | 2 | 3 |
| Omit both late-settled entries | 60 | 9 | 2 | 0 | 2 | 3 |
| Distinct endpoint values | 63 | 9 | 2 | 0 | 2 | 3 |
| Changed routes only | 52 | 7 | 1 | 0 | 2 | 2 |

Enoch's provisional ascension and Abraham's initially overlooked call contribute no hits to any of the five named targets. Omitting either or both leaves every positive count unchanged. Deduplicating the repeated 2006 BC endpoint likewise removes a miss rather than a hit. These checks show that the nine positive spans to 6 BC do not depend on those late-settled inclusions or the Jacob duplicate. They do not justify retrospectively replacing the agreed denominator or reporting a more favorable probability.

Removing all self-inverse routes changes the question to changed endpoints only. It leaves 52 routes and seven positive hits to 6 BC; the two removed hits are identified in C.2. The original whole-span result continues to use the agreed 64-route denominator and retains self-inverse sources. Its later backbone extension does not replace this baseline sensitivity analysis.

The next table compares the original dates with their inverse dates on matched route positions. A source date appears twice before inversion because it has two specified routes afterward. The unique-source column makes that duplication explicit. Retained, lost, and gained refer to positive-hit status at the same target and the same route position.

| Target | Qualifying source dates | Before, 64 positions | After, 64 positions | Retained | Lost | Gained |
|---|---|---|---|---|---|---|
| 6 BC | 2 | 4 | 9 | 2 | 2 | 7 |
| 1406 BC | 2 | 4 | 2 | 1 | 3 | 1 |
| 1446 BC | 2 | 4 | 0 | 0 | 4 | 0 |

At 6 BC the two original qualifying dates are Shelah at 2421 BC and the affliction node at 1846 BC. Their four pre-inverse route positions become two retained hits and two misses; seven other routes become hits. At 1406 BC the original qualifying dates are Shem and Reu. At 1446 BC they are Seth and Noah. The inverse count changes are therefore directly checkable rather than inferred only from the final target totals.

The comparison supports the narrow description of increased 6 BC-class concentration and reduced positive counts at the other two targets. It is a deterministic paired description, not 64 independent before-and-after experiments or a calibrated test of the direction of change.

### C.5 Pressure on the 529 and narrative constructions

The arithmetic in the narrative sections survives recomputation. The subsequent Temple reference (§2.5), hub additions (§§5.6–5.7), and backbone amendment (§§3.3–3.5 and 8.7–9.6) received bounded arithmetic verification after the original pressure test. The following dependencies limit what can be inferred from that arithmetic.

| Construction | What survives the check | Dependence / interpretive boundary |
|---|---|---|
| Temple reference | 1446 − 480 = 966; the signed Conquest offset −440 remains −440, while the Nativity span 960 reverses to 690 and rebuilds 696 BC. | The MT Exodus-to-Temple synchronism supplies the textual basis. Both routes remain outside the statistical sample by the author's explicit instruction. |
| Joseph / Joshua 110s | Joseph's inverse birth is 1556 BC, exactly 110 before the Exodus; Joshua's 7416 BC inverse gives 50, 51, 54, and 55 lifespan units. | The four Joshua comparisons are constrained by the two 110-year lives and their 440-year separation; they are not four independent hits. Absolute placements remain Rounded model dates. |
| Provisional inverse deaths | Retaining 110 or applying I(110) gives the same bounded terminal reconstruction; the 5750 bridge is exact. | Direct inversion of a death date is another operation. These examples do not settle the general terminal-date rule or enlarge the primary sample. |
| Entry Nexus | The two inverses give 700 = 14 × 50 and 6370 = 130 × 49 to the Exodus, with the 300 + 400 partition preserved. | Both routes arise from the same 1876 BC source. Jacob's age of 130 is textual; the BC placements and chosen comparisons belong to the model. |
| 1846 BC hub | Abram and Moses center on 1846 BC at 230 + 230; Terah's inverse is six such units earlier. Holding the hub during 25/23 expansion preserves these relations. | Two Conquest-family inverses and one Nativity-family inverse supply the endpoints. The wider artery is proposed; the construction is outside the target-frequency study. |
| Entry inverse / Mirror bridge | The Rounded span from 1846 BC to AD 7816 is 9660, which converts by 70/69 to 9800 = 4900 + 4900. | The reflected endpoint comes from the established 1876 BC Nativity-family inverse. The span uses Rounded coordinates; its civil counterpart is 9661. This is a structural addition outside the target scan. |
| Four 529 chains | The durations 7935, 10051, 10580, and 55016 each permit two exact 25/23 expansions. | Exact expansion twice follows algebraically from divisibility by 23²; it is not a second independent success after finding the factor. |
| Civil 529 examples | 10051 = 19 × 529 and 55016 = 104 × 529 are correct for the expressly selected civil counts. | Their neighboring Rounded widths, 10050 and 55015, are not multiples of 529. A uniformly Rounded-width analysis would not retain those two 529 relations; the mixed convention needs its own structural justification. |
| Seth's Mirror | The 7935 half-span and 15870 Rounded full width are exact 529 multiples. | The full width is twice the same half-span. The ordinary civil full count is 15871 and does not have that factor. |
| Segmented 12026 BC | Four stated paths reach 12026 BC; its span of 10580 to the Exodus is 5290 + 5290 = 20 × 529. | I(2700) = 7200, while the segmentwise sum is 10620. Admitted once as the Conquest-stage extension; it remains a miss to 6 BC. |
| Completed 14726 BC backbone | Adding I(1400) = 4100 and rebuilding from 6 BC gives 14726 BC for all four stated paths; the 14720 span is 640 × 23. | One added 6 BC incident. Noah permutations and subdivisions share algebra; the cumulative repeat applies to the primary Flood path, not every secondary branch. |
| Shared-backbone companion relations | 14720 × 25/23 = 16000; spans to inverse cumulative Shem, the cumulative Flood, inverse regular Shem, and expanded cumulative Creation are 11270 / 9890 / 9200 / 49680. | Exact under the specified anchors. The common 6 BC residue class entails mutual divisibility; the named quotients and source associations supply the structural reading. No further statistical entries. |
| Earlier Shem / Flood reference | 5436 − 600 = 4836; 3430 is self-inverse at the Conquest anchor; 5430 reverses to 3450 and rebuilds 3456 BC from 6 BC. | Cumulative-state reference only; it does not add death reconstructions or new whole-span sources to the tally. |
| 62100 expansion chain | 62100 × 70/69 = 63000; the resulting 64400 span to 6 BC expands to 70000. | The direct endpoint is 63506 BC and misses 6 BC's 115 class. The later divisibility follows from the selected expansion plus 900 + 1400 = 2300. |
| Isaac's AD 6 increment | 2070 × 25/23 adds exactly 180. | Isaac's relevant route is self-inverse; the 2070 relation was already present in the original Rounded placement. |

These qualifications retain the author's numerical–narrative case while specifying which relations are inherited, repeated, convention-dependent, or still provisional. Arithmetic survival supports continued examination of a construction; it does not by itself establish historical authorship, geometric equivalence, or providential intention.

### C.6 Remaining issues after pressure testing

No unresolved arithmetic discrepancy remains in the displayed source manifest, inverse ledger, target tables, or checked expansion chains. The following issues remain substantive rather than clerical.

| Issue | Status after pressure testing | Required basis for a stronger conclusion |
|---|---|---|
| Statistical calibration | Open. The original 1-in-571 and expanded 1-in-1834 benchmarks are mathematically correct for their simplified assumptions. | A specified reference population and null model preserving paired routes, chronological dependencies, source and operation selection, target selection, the retrospective backbone admission, and the wider search history. |
| Roles of particular hubs | The 115 test identifies a residue class; §5.6 separately develops the structural role of 1846 BC. | Construction-anchor and narrative relations distinguish the roles of particular dates. A wider expansion artery from 1846 BC remains proposed. |
| Civil / Rounded composition | Exact under declared conventions; structural justification remains provisional. | A stable rule for choosing the width convention and any later launch anchor before assessing a new comparison. |
| General terminal-date procedure | Open beyond the bounded 110 examples and provisional direct Enoch event. | An adopted operation distinguishing direct date inversion from reconstructed inverse lifespans. |
| Third-axis interpretation | Exploratory analogy. | A defined mapping and preserved properties, routed to the existing second-order / closure work. |
| File identity and registration | Tentative File_52c retained as requested. | Eventual authorial integration or renumbering before registration alongside the existing File_52c. |

The appropriate resulting status is a pressure-tested Draft with qualified statistical and interpretive claims. No repository control file or external source file is changed by this report amendment.

## Revision Log

### Initial synthesis — September 25 2026

- File: File_52c, tentative identifier; separate detailed-study draft.
- Revision date: September 25 2026, author-local discussion date.
- Style Guide version: v2.5.
- Restart Capsule version: supplied v11.54; Register v1.55; September 25 local author clarifications control the stated Rounded operations.
- Passes completed: new-draft intake and composition; structural, terminology, modal-state, arithmetic, and argument checks; bounded table and reproduction-code verification. No claim of a completed formal revision cycle for the existing File_52c.
- Unresolved issues: final identifier / integration; general inverse terminal-date procedure; mixed civil / Rounded construction interpretation; third-axis proposal; statistically calibrated null model.
- Author decisions made: fixed 32-date manifest; two source-ending-selected anchor families; five comparison targets; positive spans only; provisional direct Enoch ascension; exclusion of supplementary nodes, Joseph's elevation, and post-Exodus sources; preservation of the broader non-115 narrative study.
- Authorial-direction handling: proceeded with the requested new report; resolved routine presentation choices under the control files; preserved substantive uncertainty as audit notes.
- Cross-file updates required: identifier and registration review only if this draft is adopted; no existing source or control file was changed.
- Status at initial synthesis: Draft for author review; not Final; pressure test not yet requested for that revision.

### Initial 1846 BC amendment and pressure test — September 26 2026 UTC

Audit history: the target-scope treatment in this revision was subsequently corrected by the author. The following entry records what was done, not the active scope.

- Author request: add the 1846 BC shared-grid relation, then pressure-test the report.
- Changes: six target labels in five distinct classes; 1846 BC table and all 64 additional outcomes; shared-grid Machine Guard; complete residue census; source-list sensitivities; descriptive before-and-after counts; arithmetic and interpretive pressure-test ledger.
- Counting decision: preserve the author's nine-intersection observation as grid membership; distinguish eight positive spans and one exact overlap at 1846 BC; retain nine positive spans at 6 BC.
- Verification: independent arithmetic reversal; all 384 target cells and direction markers; 40 narrative / coordinate / expansion checks; exact-rational binomial calculation; executed reproduction code; Markdown and scope diff audit. The original manifest and five-target tables remain unchanged.
- Pressure-test findings: shared class uniquely largest with nine members, next largest six; late-settled Enoch ascension and call do not affect positive counts; twelve self-inverse routes include two of the nine 6 BC hits; 1846 BC adds no independent statistical evidence; civil 529 examples require their explicitly selected width convention.
- Remaining issues: calibrated null model, absolute-date interpretation, mixed-width justification, general terminal operator, third-axis proposal, and tentative file identity; see C.6.
- Cross-file updates: none applied. Identifier resolution and registration remain recommendations if this draft is adopted.
- Status at that revision: Draft; author-requested report pressure test completed with qualifications; not marked Final.

### Authorial clarification: structural hub — September 25 2026 local / September 26 UTC

- Author correction: 1846 BC was mentioned because it is an important hub secondary to 1876 BC, with a proposed expansion role comparable to 6 BC; it was not requested as an additional statistical target.
- Scope correction: removed the extra 1846 BC target row, its full miss table, its sensitivity column, and its target-code entry. The study returns to five targets and 320 comparisons: 16 positive hits, two overlaps, and 302 misses. The 32-source / 64-route manifest and nine 6 BC hits are unchanged.
- Structural addition: §5.6 records Abram's Conquest-family inverse at 2076 BC, Moses' at 1616 BC, their 230 + 230 center at 1846 BC, and Terah's Nativity-family inverse at 3226 BC, 1380 = 6 × 230 before the hub. The shared nine-route 115-year grid is retained in that structural context.
- Bounded illustration: holding 1846 BC during 25/23 expansion converts the centered 230 + 230 into 250 + 250 and Terah's 1380 into 1500. Resulting labels illustrate the operation and are not assigned new event identities.
- Verification: executed the five-target reproduction code; checked the new inverse routes, centered spans, signed expansion placements, and unchanged statistical tables; checked Markdown structure and active scope references. Prior completed pressure-test findings are retained without claiming a new full pressure test.
- Current status: Draft; authorial structural interpretation preserved; statistical scope corrected; existing source and control files unchanged.

### Entry inverse / Mirror bridge — September 25 2026 local / September 26 UTC

- Author addition: pair the 1876 BC Entry with the 1846 BC hub through the Entry inverse at 7816 BC and its Rounded Mirror at AD 7816.
- New section: §5.7 records the Rounded width 9660 = 140 × 69 and the Prophetic conversion 9660 × 70/69 = 9800 = 4900 + 4900; each original half 4830 converts to 4900 = 70².
- Verification: checked the inverse, Mirror coordinates, Rounded / civil distinction, exact ratio, and equal-half decomposition; bounded addition check only. Existing statistical sections, tables, and reproduction code are unchanged.
- Current status: Draft; structural addition integrated with the paired Egypt hubs; prior pressure-test findings retained; no new statistical target or event identity introduced.

### Temple construction reference — September 26 2026

- Author addition: reference the beginning of Temple construction at 966 BC because 1 Kings 6:1 explicitly connects it to the Exodus through the 480-year synchronism. The author had been unsure about admitting this later event and expressly excludes it from the statistics.
- Added §2.5: the adopted 1446 − 480 = 966 placement; preservation of the later-than-Conquest signed offset −440; self-inverse return to 966 BC; and the Nativity route 960 → 690, rebuilding 696 BC.
- Structural relation: 1846 − 1406 = 1406 − 966 = 440. The 1846 BC source remains admitted under its self-inverse exception; the Temple reference does not change that decision.
- Verification: checked both directional inverse routes and the 480 / 440 / 960 spans; confirmed the counted manifest, statistical discussion, complete target tables, and reproduction code are unchanged. No new full pressure test or statistical computation is claimed.
- Current status: Draft; Temple referenced outside the sample; 32 sources, 64 routes, five targets, and nine 6 BC hits retained.

### Major backbone admission and recalculation — September 26 2026

- Author addition: explain the prior importance of 6 BC through the Shem / Flood / Creation backbone and admit the completed 14726 BC construction as the tenth incident. File_52a §7 already records 14726 BC and its 14720 = 640 × 23 span.
- Construction: 6 → 1406 → 2456 → 4106 BC gives 1400 + 1050 + 1650; segmentwise inversion gives 4100 + 5010 + 5610 = 14720, rebuilt as 14726 BC. The stated regular Noah alternatives and cumulative primary Flood path reproduce it. The 12026 BC Conquest stage retains the 10580 = 5290 + 5290 Exodus relation.
- Counting convention: preserve the 32-source / 64-route whole-span baseline and add both backbone anchor stages once each. Expanded total: 66 constructions, with positive counts 10 / 2 / 1 / 2 / 3 to 6 BC / 1406 BC / 1446 BC / 1 BC / AD 6. All 330 comparisons are represented, including 310 misses and two overlaps. Repeated derivations are not extra entries.
- Recalculation: conditional 10-of-66 upper tail ≈ 0.0005453274, or one in 1834. The narrower one-stage 10-of-65 convention gives ≈ 0.0004813051, or one in 2078. Original 9-of-64 benchmark retained; no calibrated significance claim or multiplication of these dependent benchmarks.
- Earlier references: cumulative Shem 5436 → 4836 BC across 600 years; the self-inverse Flood span 3430; Shem's 5430 → 3450 span rebuilding 3456 BC; and 14006 → 64406 BC through inversion followed by 70/69 expansion. Literal double reversal restores the original span. Secondary cumulative Noah / Shem branches remain distinguished from the primary repeat.
- Verification: executed the updated reproduction code; checked the original manifest and 320 outcome cells for preservation; verified the four backbone paths, ten additional target cells, expanded residue census, and exact-rational probability calculations; checked Markdown structure and numerical notation. This is a bounded amendment following the recorded pressure test.
- Current status: Draft; ten 6 BC incidents in the declared expanded tally; 1846 BC remains a structural hub outside the statistical target list; the 966 BC Temple and other referenced constructions remain outside the sample; source and control files unchanged.

### Third-partition source trace and shared-backbone relations — September 26 2026

- Author clarification: identify the earlier 1406 BC → 6 BC third partition and develop why 14726 BC, together with 12026 BC, forms the backbone of the primary regular and cumulative Rounded inverse constructions.
- Source trace: File_52a §7 explicitly contains the 1400 → 4100 leg, 14726 BC endpoint, 14720 → 16000 conversion, and 9890 span to the cumulative Flood. The author also stated the three-part equation in the preceding amendment. The chronological third partition is processed first when rebuilding from 6 BC.
- Added §3.6: 14726 − 3456 = 11270 = 23 × 490; 14726 − 4836 = 9890 = 23 × 430; 14726 − 5526 = 9200 = 23 × 400; and 64406 − 14726 = 49680 = 23 × 6 × 360. The 11270 label belongs to inverse cumulative Shem, while the Flood supplies 9890.
- Earlier-file distinction: the exact 11270, 9200, and 49680 pairings with 14726 BC were not located in the available earlier inverse copies. They are made explicit here without implying earlier textual documentation. File_52a §16.3 already supplies the expanded 64406 BC endpoint.
- Cross-state comparison: the recovered 9890 matches File_22's Actual Creation bridge in both endpoint and Year-6 forms. The original Rounded Creation gap remains 9900. All companion dates retain their own construction anchors and operators.
- Verification: checked the added spans, factors, conversion, source references, and Markdown structure. Existing statistical sections and all 330 outcome cells were preserved; no repeated probability calculation or additional statistical hit was introduced.
- Current status: Draft; shared-backbone significance expanded; ten hits to 6 BC among 66 constructions retained; earlier source and control files unchanged.

### Creation-pillar interpretation of the original nine hits — September 26 2026

- Author observation: 14726 BC represents Creation at the end of the inverted backbone and may function as a pillar, paired with 6 BC, for interpreting the original nine incidents.
- Added §3.7: with 14726 BC held, Terah's inverse 3226 BC has span 11500 → 12500 and moves to 2226 BC; the affliction node 1846 BC has span 12880 → 14000 and moves to 726 BC, 720 before 6 BC.
- Source states: 2226 BC is the firstborn threshold of Terah's +60 state, as explicitly recorded in File_18 §4A.5; 726 BC is the schematic Samaria siege / collapse label under File_00 and File_16 §7.1. Original source states and SKL rail controls remain intact.
- Common rule: 14726 − 6 = 128 × 115 gives complementary spans for all nine original hits. The full table records every resulting expansion. The two held-anchor expansions differ uniformly by 1280, and the Abram–Moses pair retains 250 + 250 symmetry around the transformed 726 BC hub.
- Interpretation: Creation pillar is a proposed structural role; the seven other generated endpoints are not assigned new event identities. Named SKL Projective Boundary Pillars retain their existing meaning.
- Bounded verification: exact arithmetic and local source-state checks passed; existing source manifest, statistical sections, and all 330 outcome cells preserved. The tally remains ten hits to 6 BC among 66 constructions, with no new probability calculation.
- Current status: Draft; author-directed structural addition only; no source or control-file revision.

### Entry Mirror, Shem partition, and Temple convergence — September 26 2026

- Added §3.8 for the author's three further examples of the 14726 BC key: 22540 → 24500 and 14700 + 9800; the Nativity-held 16006 → 6006 → 6 BC arrangement with 10000 + 6000; and the two routes to Rounded AD 526.
- Arithmetic notation: 4830 + 4830 equals 9660 before its 70/69 expansion to 9800. The author's AD 7815 civil-style count gives the same intended durations as the draft's AD 7816 symmetric Rounded display. These labels retain their respective counting rules.
- Operator distinction: 24500 uses 25/23 and the earlier 9800 uses 70/69. A uniform 25/23 partition would give 14000 + 10500; the stated 14700 + 9800 compares the two separately converted durations.
- Shem example: hold 6 BC throughout. The 6000 span is the author's 10 × 600 lifespan echo; the 10000 comparison is grounded specifically in 1 Enoch 18:16 and 21:6, with the seventy-generation Watcher sentence kept distinct. File_11's SP 9200 → 10000 precedent has different endpoints.
- Temple example: 14726 BC to inverse Temple 696 BC expands from 14030 to 15250, ending at q = +525 / Rounded AD 526. The reflected 12026-to-1446 interval expands from 10580 to 11500 and reaches the same coordinate. The gains 1220 + 920 close the original 2140 gap. The sanctuary-restoration reading uses 11500 = 5 × 2300 as a typological comparison.
- Bounded verification: added exact-rational arithmetic and coordinate checks passed; source references checked; the source manifest, statistical sections, and all 330 outcome cells preserved. The Temple remains excluded from the statistics, and no probability recalculation was introduced.
- Current status: Draft; structural companion examples added; ten hits to 6 BC among 66 constructions retained; source and control files unchanged.

### Moses–Temple second expansion and reflected birth — September 26 2026

- Added §3.9: repeat 25/23 with AD 12026 held, giving 10580 → 11500 → 12500 and Rounded endpoints AD 1446 → AD 526 → 476 BC.
- Era clarification: the author's opening references to 1446 BC and 526 BC are understood as AD 1446 and AD 526 in the already established reflected stage. The second expansion genuinely crosses to 476 BC. This follows the supplied arithmetic and preceding construction.
- Common endpoint: the Creation-held expansion of inverse Moses, 1616 BC, also reaches 476 BC. The inverse Moses–Temple interval is 1616 − 696 = 920, which expands to the 1000 Rounded width from 476 BC to AD 526.
- Forward counterpart: hold AD 526 and expand its 920 interval to reflected Exodus AD 1446; the result is AD 1526, Moses' reflected birth. The expansion gain is 80, and AD 526 is the exact Rounded midpoint of 476 BC and AD 1526 at 1000 + 1000.
- Interpretation: the connected person / Temple / backbone relations supply structural corroboration. Their algebraic dependence is stated, and no independent coincidence probability is assigned.
- Bounded verification: exact-rational and coordinate assertions passed; existing source manifest, target tables, statistical sections, and prior pressure-test appendix preserved. Current status remains Draft; the Temple remains outside the ten-of-66 statistical tally.

### Abraham's 976 BC endpoint and paired symmetries — September 26 2026

- Added §3.10: the original Abraham route 2166 BC → 2076 BC remains one of the nine whole-span hits. Holding 14726 BC expands 12650 to 13750, moving inverse Abraham forward by 1100 years to 976 BC.
- Forward-time notation is rendered by subtraction of BC labels: 2076 − 1100 = 976. No source value or operator is changed.
- Entry symmetry: 1876 → 1446 → 976 BC gives 430 + 470; 1876 → 1406 → 976 BC gives 470 + 430. The combined sequence is 430 + 40 + 430, centered arithmetically at 1426 BC.
- Moses midpoint: 2076 → 1526 → 976 BC gives 550 + 550, with the adopted birth of Moses at the center of the 1100-year movement.
- Jerusalem extension: 976 − 586 = 390, completing the existing 1876 − 586 = 1290 relation as 900 + 390.
- Interpretation: the author connects the symmetry with Abraham's foretelling of his descendants' sojourn and return. The 400 and 430 source registers remain distinct, and 976 BC receives no new historical event identity.
- Bounded verification: all added arithmetic passed; statistical sections, manifest, and all 330 target outcomes preserved. The linked structural descriptions add no independent statistical entries. Current status remains Draft.

### Flood countdown, paired ratios, and ten-member interpretation inventory — September 26 2026

- Added §3.11: the inverse Abraham–Moses endpoints are 460 apart; countdown inverse 8516 BC is 15 × 460 before inverse Moses and 14 × 460 before inverse Abraham.
- Common input: 14726 − 8516 = 6210 = 90 × 69 = 270 × 23. With Creation held, 70/69 gives 6300 and endpoint 8426 BC; 25/23 gives 6750 and endpoint 7976 BC.
- Audit correction: supplied 7926 BC is corrected to 7976 BC, as required by the multiplication, the existing pillar table, and the author's intended 6450 interval. Literal 7926 would give 6400 to Moses' birth.
- Connections: 8426 − 1526 = 6900 = 15 × 460; 7976 − 1526 = 6450 = 15 × 430 = 10 × (2091 − 1446). The Prophetic route and inverse-to-birth Moses comparison both shift forward by 90.
- Source count: File_61 explicitly supplies Abraham, Isaac, Jacob, Levi, Kohath, Amram, Moses as seven named generations counted inclusively, with six parent–child links.
- Interpretation inventory: recorded the author's proposal to explain all ten incidents individually through the Creation–Nativity framework. Six original source members have developed contextual examples, the tenth backbone has its anchor / Nativity-span interpretation, and Mahalalel, Shelah, and Eber remain individually unresolved. The pillar's own span to 6 BC is distinguished from a zero self-span.
- Bounded verification: exact arithmetic and inventory checks passed; existing statistical sections, source manifest, target tables, and prior pressure-test appendix preserved. No statistical recalculation or new count entry. Current status remains Draft.

### Mahalalel–Shelah–Eber and restored-Cainan completion — September 26 2026

- Added §3.12: the remaining three original source members receive one joint interpretation through self-inverse Shelah and a separately opened restored-Cainan companion.
- Common expansion: Creation-held spans 5405 / 12305 / 12995 become 5875 / 13375 / 14125, yielding Eber 8851 BC, Shelah 1351 BC, and Mahalalel 601 BC. Among the three admitted ratios, only 25/23 gives integral outputs for these spans.
- Structure: Shelah divides the inverse gaps 690 / 6900 in a 1:10 ratio; the expanded gaps are 750 / 7500. The total converts from 7590 = 23 × 330 to 8250 = 25 × 330. “Centered” is recorded as an internal junction, not a midpoint.
- Cainan companion: 2551 → 2421 → 2091 BC supplies 130 + 330 = 460. Its 1 BC inverse is 5521 BC, expanding to 6001 BC and a Rounded Mirror width of 12000. The 1150 / 690 spans to 1401 BC preserve the 460 lifespan. The source manifest remains Cainan-excluded.
- Counting control: 8880 / 1380 / 630 to AD 30 are civil counts; the other tabulated Mirror widths are Rounded. The supplied 31 BC reflection of AD 30 uses an expressly local ordinary astronomical map. Both numerical columns are shown, with no global date-label shift.
- Connected intervals: 601 BC is the Rounded midpoint of 2551 BC and AD 1351 at 1950 + 1950; the 130-year multiples, 1200 same-side span, 11270 = 23 × 490, 3440 = 8 × 430, and 2795 = 13 × 215 are checked.
- Local continuation: hold 1406 BC for 805 → 875, moving 601 → 531 BC by 70. The Babylonian-duration and new-creation readings retain interpretive status. The textual Cainan difference is distinguished from the author's historical expunction hypothesis.
- Inventory update: all nine original source members and the tenth backbone now have stated contextual interpretations. The final three form one connected case; completion of the inventory does not establish independent statistical witnesses or prove intention.
- Bounded verification: exact-rational, Mirror, civil-count, and inventory checks passed; original manifest, §§8–9, all 330 target outcomes, existing reproduction code, and prior pressure-test appendix preserved. New assertions appended; no statistical recalculation. Current status remains Draft.

### Author clarification: cursory trio scope — September 26 2026

- The Mahalalel–Shelah–Eber discussion is expressly a cursory exploration of selected observations. Its wider complexity has not received a thorough examination; the main point is the connection to Cainan omitted from MT, understood as expunged under the author's restoration hypothesis.
- Added illustrative branches in §3.12.7: with 1401 BC held, 2091 → 2151 BC leaves 400 to the retained 2551 BC birth; 2551 → 2651 BC changes the Rounded span to AD 1351 from 3900 to 4000. With 1 BC held, inverse 5521 BC expands to 5601 by 70/69 or 6001 by 25/23.
- Qualified the opening, ten-member inventory, and interpretation summary: inventory coverage is complete at an initial level, while the trio remains cursory. Only the supplied additional arithmetic was checked; no broader exploration or statistical recalculation was undertaken.

### Actual 12026 BC key: author-authorized partial exploration — September 27 2026

- Incorporated the discussion held since the author's no-update instruction, following explicit authorization to update the file. This is a bounded addition to the existing draft, not a new full-file revision or Finalization.
- Preserved the author's strong assessment that 12026 BC is a major anchor key comparable to 14726 BC. Marked the detailed investigation partially explored / cursory; broader interpretation, remaining chart nodes, and the Moses continuation remain open.
- Added the three starting levels, original-Flood-held 8066 BC node, 3960 + 3960 midpoint, 6660 / 6666 and Isaac comparisons, 23 × 1260 expansion, 43000-year Exodus result, and 5 × 8888 relation.
- Added the Actual 14004–4114 BC Creation bridge, its 1:4 division at 12026 BC, and the expanded 2150 + 8600 total; retained the 3426 BC Creation asterisk.
- Added the 14 / 15 / 16 × 598 Jared–Noah–Flood structure, all eight declared expansion rows, Noah's three ratios and 30 + 100 + 650 increments, the shared 2926 BC result, and all twenty-one distinct-date intervals.
- Preserved the selected two-year / two-day, Noah–Shem, curse, Creation / Abraham / SKL, and AD 25 interpretations with their source-state and counting qualifications. Recorded the arithmetic and label corrections explicitly.
- Updated the file-function, working states, file map, guard, interpretation, audit, source register, and bounded reproduction code. The original statistical ledger, probability discussion, and September 26 pressure test remain unchanged.