# 490d — C206–C265: coupled 11- and 13-fold Mirror expansion

23 September 2026 · All sixty authorized bounded steps completed, continuing C205.

**The principal addition is a source endpoint connection between the 11- and 13-fold families:**

\[
11\times625=6875=13\times529-2.
\]

Holding each native SP Creation member fixed, two applications of 25/23 to its native 11×529 Conquest-mirror interval yield **six retained MT Flood endpoints** across the seven allowed input pairs. The intermediate 6325 endpoints remain generated within the held Noah/Shem/Flood register. This adds source support to the final endpoint without making the entire three-stage 11-fold ladder sourced.

The two fully sourced C01 chains remain the starting point. The regular Gear restriction remains **Noah/Shem/Flood only**. Creation and Conquest are selected native field members, not additional Gears.

## 1. The original 13-fold chains have explicit local maps

For signed Creation center a, use Pₐ(t)=a+(25/23)(t−a). Each original C01 chain has its own anchor:

| Chain | Creation BC base | Right endpoints: reflected MT Flood → MT Noah → SP Noah | Interval centers |
|---|---|---|---|
| 3 | 4419 | 2458 → 3056 → 3706 | -1961/2, -1363/2, -713/2 |
| 5 | 4417 | 2460 → 3058 → 3708 | -1957/2, -1359/2, -709/2 |

Their widths are **6877→7475→8125 = 13×(529→575→625)**. Right-endpoint gains 598 and 650 give interval-center gains **299 and 325**. The 26-year change between these gains is half C06’s inherited 52 residual; it is not independent evidence.

The named first right-endpoint midpoint is SP companion Noah death at 2757/2759. Its affine image is 3381/3383, which has no match in the held 48-row regular register. Thus midpoint geometry survives while named-event status does not. A further endpoint iterate is nonannual.

The two maps are conjugate by a two-year translation. Using one unchanged map on the other source chain creates a 4/23 mismatch. The cumulative 17-fold expansion in C04 has a different anchor p, so its centers move **−69, −75**, while the fixed-Creation centers move **+299, +325**. Equal ratios do not make these the same endpoint operation.

References: C206–C235. C01/C04/C06 endpoint results and C18’s earlier 868 provenance are retained as inherited dependencies.

## 2. Seven native 11-fold inputs, six sourced final endpoints

The native input width is 5819. With Creation fixed, its successive widths are **5819→6325→6875**.

| Native SP Creation | Native Conquest mirror | First image AD base | Second image AD base | Second-image MT source |
|---|---|---|---|---|
| 4421t/4420n BC | AD 1399t/1400n | 1905 | 2455 | G1 Flood close |
| 4420t/4419n BC | AD 1400t/1401n | 1906 | 2456 | G1 Flood start |
| 4419t/4418n BC | AD 1401t/1402n | 1907 | 2457 | G2 Flood close |
| 4418t/4417n BC | AD 1402t/1403n | 1908 | 2458 | G2 Flood start |
| 4417t/4416n BC | AD 1403t/1404n | 1909 | 2459 | G3 Flood close |
| 4416t/4415n BC | AD 1404t/1405n | 1910 | 2460 | G3 Flood start |
| 4415t/4414n BC | AD 1405t/1406n | 1911 | 2461 | No match |

An AD base A denotes **AD At/(A+1)n**. Thus the endpoint 2459 means AD 2459t/2460n, not an unqualified single-season date.

The seven intermediate centers **1905–1911** have no match in the held 48-row ordinary MT/SP Noah/Shem/Flood register. This bounded negative result does not exclude a separately sourced fixed post-Flood event. No such additional search was undertaken in this batch.

For the two complete original chains, the final 11-fold image is the Flood start one Gear before the original 6877 endpoint: G1 instead of G2, and G2 instead of G3. In the wider field, companion Noah aligns with Flood close and primary Noah with Flood start.

The closing native input producing 2461 remains unsupported. C09’s conditional 300→299 full-week compression is unnecessary for the six admitted results and is not promoted here.

References: C241–C253.

## 3. The coupled interval centers reproduce the carrier ladder

At a common expansion stage, the 13-fold and 11-fold intervals share their left anchor. Their right endpoints are separated by twice the carrier, so their interval centers differ by exactly the carrier:

| Stage | 11-fold width | 13-fold width | Right-endpoint separation | Interval-center separation |
|---|---|---|---|---|
| 0 | 5819 | 6877 | 1058 | 529 |
| 1 | 6325 | 7475 | 1150 | 575 |
| 2 | 6875 | 8125 | 1250 | 625 |

This is a direct **529→575→625** center ladder. In the two complete C01 chains both right endpoints are sourced at stages zero and two. At stage one the 11-fold endpoint remains generated. The second 11-fold interval’s center is one year before the initial 13-fold center because its width is two years shorter with the same left endpoint.

References: C242–C244.

## 4. A source-supported subdivision of the existing 17-fold interval

The six final endpoints fit an explicit route:

**cumulative LXX OFF Flood BC → native SP Creation BC → Conquest mirror AD → MT Flood mirror AD → SP Noah mirror AD.**

Its consecutive segment lengths are:

\[
\boxed{868+5819+1056+1250=8993=17\times529.}
\]

| Selector s | Cumulative LXX OFF BC base | SP Creation BC base | Conquest AD base | MT Flood AD base | SP Noah AD base |
|---|---|---|---|---|---|
| 2 | 5288 | 4420 | 1399 | 2455 | 3705 |
| 3 | 5287 | 4419 | 1400 | 2456 | 3706 |
| 4 | 5286 | 4418 | 1401 | 2457 | 3707 |
| 5 | 5285 | 4417 | 1402 | 2458 | 3708 |
| 6 | 5284 | 4416 | 1403 | 2459 | 3709 |
| 7 | 5283 | 4415 | 1404 | 2460 | 3710 |

Every displayed endpoint belongs to its declared source field. AD entries are reflected source boundaries, not claims of historical AD occurrences. These are six placements of one dependent subdivision of the existing 17-fold outer interval; they are not six independent 17-fold witnesses. The historical graph is unchanged.

The original exact 13-fold route used tails **1058+1248**. The new double-11 selection uses **1056+1250**, redistributing two years while preserving the total. Its 1250 tail selects the same Noah/Flood Gear, whereas the old 1058-tail route selected the next Flood Gear and had only four exact source-qualified placements.

Inserting the generated 8K BC preimage of Conquest retains the user's 4232 framework:

\[
868+3K+8K+(2K-2)+(2K+192)=17K.
\]

The inserted preimage is generated; the unsplit five-point route above has source-field endpoints throughout.

References: C252–C256.

## 5. Integration with the prior residual and center grammar

For each admitted row, add the retained alternate MT Flood counterpart M=F′+190 and cumulative ON Shem member U=17K−M. The equal square sides are **1060=2K+2**, and the cross-width matrix is:

\[
\begin{pmatrix}17K&19K+2\\15K-2&17K\end{pmatrix}.
\]

This is C201’s inherited **δ=2** geometry. The selected 6875 interval lies between its two 17-fold interval centers, splitting their gap as **869+191=1060**. The new contribution is the source selection supplied by the double-expanded native 11-fold family, not a newly discovered independent 191 or 1060 relation.

P6 now has a valid **6875** interval ending at MT G3 Flood close 2459. Its missing **6877** endpoint at 2461 stays missing. The sourced G3 start at 2460 is a distinct δ=1 case. Likewise the sixth row remains unsupported for the older cumulative-MT prefix requiring 5297, despite supporting the present route.

N04’s original interval −4417→2460 and P6’s new interval −4416→2459 are concentric at −1957/2. One uses a different native Creation member and a different Flood boundary. No Creation Gear or compression is inferred from their shared center.

References: C248–C249, C256–C260.

## 6. Peleg: exact contacts, with an explicit interpretation limit

Two contacts arise:

- **8993−6875=2118=4K+2**, matching the retained MT Peleg-death BC base.
- **191=2357−2166**, matching the retained Peleg-birth to Abram-birth span.

The new route contains no Peleg or Abram endpoint in either BC or reflected AD coordinates. Its 2118 is the sum of separated flanks **868+1250**, with a 6875 interval between them. Its 191 is an interval-center separation. These are exact numerical associations; they do not establish a new contiguous Peleg event path or an independent source witness. Earlier Peleg/Isaac/Moses relations retain their own evidence and qualifications.

References: C261–C262; fixed event nodes from C127, inherited from C117.

## 7. Verification and restart state

All **60 reports C206–C265** have exact evidence records and predecessor-certificate hashes. The batch passes **727 step-specific assertions**, including 177 certificate/status/scope checks, plus 1140 repeated frozen-input checks. Six ten-step checkpoints receive isolated byte-identical replay of their reports, certificates and results.

A separate final audit reconstructs the key native Creation, Flood, Noah and cumulative fields from File_18 table cells and verifies the six endpoint matches independently of the report selector formulas. File_18 and File_09 still match the attached controls byte for byte. The C127 graph remains frozen at 208 nodes. No closed primer, failed Shem join or historical campaign was reopened. Original Reports 01–10 remain previously verified; interrupted original Reports 11–25 remain unestablished.

Read the checkpoint after C265 for continuation. The evidence ZIP contains all sixty reports, their exact results, source snapshots, replay records, the final source audit, calculation code and a SHA256 artifact manifest. Files were saved after each completed ten-step block.

**Best next bounded step — C266:** compare the seven generated intermediate centers **1905–1911** with a finite, fixed post-Flood MT event register. Test source profile and endpoint role before interpreting a numerical match. Regular Gears remain Noah/Shem/Flood only. **C266 is proposed, not executed.**