File: File_52c (tentative identifier)
Title: Rounded Whole-Span Inversions — Narrative Correspondences, the 529 Family, and the 6 BC Concentration
Status: Draft; original report pressure test retained; author-requested backbone extension verified; ten positive hits to 6 BC in 66 constructions across five targets; not marked Final
Primary domain: Rounded; Regular; Cumulative; Mirror; Statistical; Methodological
Traditions: MT; biblical narrative comparisons
Canonical source: Markdown
Primary anchors: 6 BC; 1 BC; AD 6; 966 BC (reference only); 1406 BC; 1401 BC; 1446 BC; 1846 BC; 1876 BC; 4106 BC; 12026 BC; 14006 BC; 14726 BC
Related files: File_00; File_05; File_16; File_18; File_22; File_46; File_51a; File_51b; File_52a; File_52b; File_54; 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; provisional inverse terminal-date reconstruction; Rounded Mirror coordinate-completion; exploratory target-frequency scan
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 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 | The same span recurs between 14726 and 4836 BC in the Rounded inverse study; the uninverted Rounded Creation gap remains 9900. |
| 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 only in the expressly selected cross-axis 529 constructions. |
| Key-of-23 whole-year expansion | $25/23$; $70/69$; $300/299$ | Exact conversions of declared durations; no physical Residue Protocol is opened. |
| 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 the Creation trunks and the shared 12026 / 14726 BC backbone | Third-partition source trace; whole-span and segmented inverses; Shem / Flood / Creation comparisons; 16000 conversion |
| §§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.

### 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 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 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.

## 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.

## 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.

### 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.

## 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.

### 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.

### 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.

## 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. |
| 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.

### 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, and counted backbone extension. 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 | Foundational argument-control precedent and treatment of the cumulative qualitative case. |
| 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_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 | 1876 BC Nexus and the 1846 BC affliction field. |
| File_18 | Raw chronology dependency; distinction between source dates and later interpretive constructions. |
| 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. |
| 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 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
```

## 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.