C266 — The 5+1+1 completion and the 529 local-radius scaffold
24 September 2026 · One bounded continuation after C265, redirected by Dean’s comprehensive 4232 / 8464 / 12696 report.
Result: the proposed 5+1+1+6(+1) has a complete, source-labeled realization in the inherited central cumulative Flood/Shem construction. Its expanded form is 5+1+1+5+1+1=14. The same finite source fields also supply 5-, 6-, and 7-year radii about one local junction. At the declared 1000:1 comparison scale, those radii correspond exactly to the 5000 / 6000 / 7000 scaffold.
This establishes a specific relationship between the small central construction and the new radius diagram. It does not identify every large converted endpoint with a historical event, turn the diagram into a specified three-dimensional geometry, or make the 7500 branch a disjoint fifteen-year central interval.
Numbering: C266 now names this authorized endpoint-ledger task. The former proposed C266 search of AD bases 1905–1911 was superseded before execution and remains unexecuted. C01–C265, their certificates, the 208-node graph, and canonical source files remain unchanged.
1. Scope, sources, and notation
The finite domain is the MT cumulative ON, SP cumulative OFF, and LXX cumulative OFF Flood and Shem envelopes, with the translations explicitly supplied in the earlier Mirror handoffs. Their complete seven-year fields are retained. There is no cumulative Gear operation. The standing regular-Gear restriction remains Noah/Shem/Flood only; the quarantined primers and upstream regular Gear transport remain closed.
The principal inputs are:
- [A] Dean’s new Comprehensive 4232 / 8464 / 12696 Mirror Axis Report, especially §§1–2, 9–18.
- [H] Mirror Cumulative–Regular MT/LXX/SP Comprehensive Handoff, 20 September, §§3, 8–10, 12–15.
- [B] Mirror: Bilateral Integration and the 8395 Correspondence, 20 September, §§6–7.
- [18] Current File_18, §§6A–6D, and [09] File_09’s phase/Mirror controls.
[H] and [B] were recovered selectively through the verified archive chain. Their earlier central completion is recovered evidence, not a new discovery. This step makes the sequence’s endpoint ledger explicit, derives its connection to the selected 5/6/7 radii, and separates the remaining comparison tasks. No claim of corpus-wide novelty is made.
For continuity with C265, write K=529, E=25/23, P=70/69, and ρ for a radius. This avoids the new branch report’s different use of K for the conversion ratio.
Let x be the signed pair-center coordinate relative to January AD 1:
| Center | Ordered phase pair |
|---|---|
| −b | (b+1)t/bn BC |
| 0 | 1t BC / AD 1n |
| +a | AD at/(a+1)n |
Each ordered pair has proof coordinates x−1/4 and x+1/4. There is no civil year zero. Global reflection is M(x)=−x; translation is T_w(x)=x+w. A transformed coordinate retains its source label and operation history. Coordinate co-location does not identify the original historical events.
2. The full source-to-landing ledger
The cumulative inputs are directly controlled by [18] and [H, §3]. MT ON adds 460 to MT OFF; LXX OFF subtracts 460 from native LXX ON. SP retains its official Terah-145 execution and downstream Joshua-death anchor 1296 BC. Shem’s corresponding cumulative field is 600 earlier than the Flood field.
| Source profile | Flood BC bases | Flood translation | Translated Flood F | Shem BC bases | Shem translation | Translated Shem H | Reflected Shem M(H) |
|---|---|---|---|---|---|---|---|
| MT restored ON | 5296→5289 | 5290 | [−6,1] | 5896→5889 | 5888 | [−8,−1] | [1,8] |
| SP native OFF | 4716→4709 | 4715 | [−1,6] | 5316→5309 | 5313 | [−3,4] | [−4,3] |
| LXX derived OFF | 5290→5283 | 5290 | [0,7] | 5890→5883 | 5888 | [−2,5] | [−5,2] |
Intervals in the last columns are pair-center intervals, not unsuffixed historical AD dates. Their exact paired displays follow from the table in §1. Every field has eight integer boundaries spanning seven matching-season years.
The translation values are inherited inputs. Their numerical relationships do not establish that the translations were independently forced by the source chronology. In particular, this ledger preserves the source-specific SP translation rather than replacing all three translations with one common value.
3. The previously implied +1 has explicit endpoints
Read the following source-projected boundaries chronologically:
| Center x | Exact phase pair | Source/operation giving this boundary | Gap from previous |
|---|---|---|---|
| −6 | 7t/6n BC | MT ON Flood opening 5296, translated 5290 | — |
| −1 | 2t/1n BC | SP OFF Flood opening 4716, translated 4715 | 5 |
| 0 | 1t BC / AD 1n | LXX OFF Flood opening 5290, translated 5290 | 1 |
| 1 | AD 1t/2n | MT ON Flood closing 5289, translated 5290 | 1 |
| 6 | AD 6t/7n | SP OFF Flood closing 4709, translated 4715 | 5 |
| 7 | AD 7t/8n | LXX OFF Flood closing 5283, translated 5290 | 1 |
| 8 | AD 8t/9n | MT ON Shem source opening 5896, translated 5888 and reflected | 1 |
Hence:
Suppressing the internal SP closing boundary at x=6 merges the last two gaps before x=7:
Thus the final +1 is AD 7t/8n → AD 8t/9n, not an unnamed extra year.
The continuous supporting intervals are MT Flood [−6,1] followed by reflected MT Shem [1,8]. At x=1, the latter begins with the reflection of the source Shem closing boundary 5889. It coincides with the translated Flood closing boundary in both seasonal components. The SP and LXX boundaries subdivide the continuous construction; they do not erase the distinction between the source traditions.
The join is January AD 2, represented by AD 1t/2n. Global Mirror zero remains January AD 1. [H, §§8–9,14–15; B, §6]
4. One local reflection governs all three traditions
For every profile, the source Shem field is 600 earlier, while its translation is only 598 greater. Therefore:
Reflecting H therefore applies J₁(x)=2−x to F, with endpoint order reversed:
J₁ is a reflection about x=1. Its three interval pairings are:
| Translated Flood | Reflected translated Shem | Shared local reflection |
|---|---|---|
| MT [−6,1] | [1,8] | J₁ |
| SP [−1,6] | [−4,3] | J₁ |
| LXX [0,7] | [−5,2] | J₁ |
The endpoint set of these six intervals is invariant under J₁. In particular, three source-labeled outer pairs supply:
| Radius about x=1 | Left endpoint | Right endpoint | Source lineage |
|---|---|---|---|
| 5 | x=−4: 5t/4n BC | x=6: AD 6t/7n | Reflected SP Shem / translated SP Flood, both originating at their source closing members |
| 6 | x=−5: 6t/5n BC | x=7: AD 7t/8n | Reflected LXX Shem / translated LXX Flood, both originating at their source closing members |
| 7 | x=−6: 7t/6n BC | x=8: AD 8t/9n | Translated MT Flood / reflected MT Shem, both originating at their source opening members |
This is the strongest structural result of the ledger: the signed 5/6/7 scaffold can be assigned to the existing transformed endpoints of all three traditions. No new event date, Gear, or additional source selector is needed for these particular radius pairs.
5. The explicit 529 connection: 575 separates 5 from 1
The two inherited Flood translations differ by:
The corresponding source differences then give the central subdivision:
Thus the MT/LXX six-year separation is subdivided as 580−575=5 and 575−574=1. The first converted 529 carrier participates directly in the source/translation accounting.
Normalize SP to cumulative ON by adding 460 to both its source field and translation. The same first difference becomes:
Consequently 580=120+460 and 575=115+460. This connects the new 529 vocabulary with the inherited 120/115/Cainan construction. These are equivalent reductions of the same selected configuration, not independent confirmations or an authorization to propagate the cumulative SP offset to Creation.
6. How the small radii correspond to the 7000 diagram
Define an explicitly local comparison chart:
Its purpose is to compare pair-center distances around the central join with the new report’s radius labels. The existing source endpoints give:
| Selected central centers | Relative to the join | Comparison-chart labels |
|---|---|---|
| −6, −5, −4, 1, 6, 7, 8 | −7, −6, −5, 0, 5, 6, 7 | −7000, −6000, −5000, 0, 5000, 6000, 7000 |
The local reflection is respected exactly:
This supplies a relational realization of the 5000/6000/7000 scaffold at the stated symbolic scale. The new branch separately supplies 4232E²=5000, 5520E=6000, and 6348EP=7000, with their declared source and conversion histories. The ledger establishes the matching radius arrangement; it does not collapse those histories into the original small-scale event identities.
Three distinctions are essential:
- Local chart and global axis: η sends the local junction x=1 to its own zero. It sends the global Mirror origin x=0 to −1000. This is an explicit comparison between local coordinates, not a relocation of January AD 1 or a newly proved physical y/z embedding.
- Radial traversal and chronological traversal: the radial subset reads 1+1+5+5+1+1 from left to right. The chronological subset in §3 reads 5+1+1+5+1+1. They use different negative-side boundaries from the same admitted configuration. Applying η to the latter yields −7000, −2000, −1000, 0, 5000, 6000, 7000.
- Centers and seasonal components: η compares centers and their distances. Applying the same dilation to full proof coordinates would multiply quarter-year offsets to 250. Restoring ±1/4 around a transformed center is a separate convention. Neither operation may be silently substituted for the other.
7. The cumulative-Creation connection: what is already joined
The named dependency supplies a complete additional path for six admitted selectors i=2,…,7. One example is:
| Source boundary | Paired coordinate | Next interval |
|---|---|---|
| SP cumulative Shem member | 5315t/5314n BC | 598 |
| SP cumulative Flood member | 4717t/4716n BC | 3312 |
| Conquest member | 1405t/1404n BC | 5520 |
| Reflected regular MT Creation member | AD 4116t/4117n | 9890 |
| Reflected cumulative MT Creation OFF member | AD 14006t/14007n | — |
This is a source-qualified connected route linking the Flood/Shem fields to Creation. It has its own unshifted and reflected endpoints. It is not already the central 14-year path, and its displayed scaled decomposition is not the proposed 5+1+1+6+1 subdivision. [H, §§11–13; B, §7]
For example, the SP Flood opening P₄₇₁₆ appears in both the support path and the central construction, but centrally it has been translated by 4715. This records a common source ancestor and a named transformation, rather than literal co-location of the two displayed endpoints. Transporting the complete support route through a new coordinate/scale assignment would be an additional declared operation.
The cumulative Creation ON/OFF comparison in [A, §13] also supplies the verified generated chain:
It uses 322E=350 and 138P=140, followed by endpoint-calibrated transport of the original seven-year envelopes. It is not uniform dilation of every envelope member. Its addition to 26×490 remains a composite 27×490 total unless a common serial placement is separately specified.
8. Macro joins and the two complementary completions
The bounded checks retain these distinctions from [A]:
| Comparison | Ledger disposition |
|---|---|
| MT/LXX regular Creation → reflected Conquest | Two source-qualified branches with the same Conquest field; widths 5520 and 6900. They are not consecutive edges merely because they share a target. |
| Cumulative ON Flood landing minus 6000 | Comparator Q₁₄₀₀→Q₁₄₀₇, one year after the primary reflected Conquest envelope. |
| Cainan landing minus 6440; Shelah landing minus 6900 | Common comparator Q₁₃₉₈→Q₁₄₀₅, one year before the primary Conquest envelope; two years before the Flood comparator above. |
| 6440E=6900P=7000 | Equal converted durations. Endpoints also coincide only after an explicit shared anchoring rule. Mixed ratios do not define one common injective dilation of both source endpoints. |
| 6900E=7500 | Distinct admitted conversion branch; retain its own source, anchor, and theological role. |
| Central 5/6/7 radii ↔ 5000/6000/7000 | Exact pair-center correspondence under η, with source histories retained separately. |
| Whole converted-radius diagram ↔ one continuous historical Creation–Flood path | Not established by the present ledger. The positive source paths and radius correspondence above are more specific results. |
The small central matching-season arms are seven years each. Their outer Tishri-to-Nisan envelopes are 7.5 years each and overlap by half a year at the shared paired boundary:
At 1000:1 the corresponding sum is 15000 and union 14500. The selected outer pair centers are radius 7 from the join; the most distant individual seasonal components are radius 7.25. There is no supplied pair-center radius 7.5 in this central construction.
This preserves Dean’s Year 14 completion/restoration and Year 15 renewed-age/eighth-day readings with their stated quantities. The zero-centered ±7500 diagram has a 15000 outer span of its own; identifying it with the central seasonal union would change the construction. Likewise the older fixed-landing bridges give 7000+4+7000=14004 and 7500+4+7500=15004, while the new zero-centered radius diagram has no four-year central gap. [A, §15; H, §§9–10]
9. Verification, outcome, and checkpoint
The accompanying standard-library verifier passes 207 exact checks. It reads the controlling cumulative table cells, constructs every annual member of the selected full envelopes, checks both seasonal components and reflected order, records source-tagged endpoints, checks the chronological and radial traversals, verifies the 575 reduction, and preserves the macro companion/envelope distinctions. The count includes input-integrity checks and describes verification assertions, not independent historical witnesses.
An independent review checked the local reflection, endpoint assignments, source restrictions, and overlap accounting. An isolated replay reproduces the result/ledger files byte for byte. This step does not rerun the new report’s referenced 855-check campaign or any C01–C265 campaign.
Retained result: the final +1 is sourced within the transformed construction; the three traditions participate in a common local reflection; their selected endpoints supply exact 5/6/7 radii; and the 575=E(529) identity explains the 5/1 subdivision through the existing source/translation differences.
Remaining limit: the demonstrated center correspondence and existing Creation support route do not yet assign every converted radius marker to one common source-event path. The 7000 and 7500 interpretations remain complementary, with distinct phase and span accounting.
Best next bounded step — C267: test the remaining converted radius markers 2500 and 6250, together with the 7500 outer branch, against this now-fixed local comparison chart. Use only the already admitted transformed source fields and distinguish pair centers, seasonal components, interval midpoints, and envelope lengths. This will determine how much of the full four-radius 529 ladder has the same source-labeled realization without inventing endpoints. High is sufficient.
The former AD 1905–1911 search remains parked. No canonical revision or new z-law is authorized by this checkpoint.