# 490d C323 — Source-qualified529 final synthesis

24 September 2026. Step 10 of the ten-step authorization C314–C323.

The ten-step batch C314–C323 is complete. Its main gain is a source-qualified connection between the 529 framework, complete Noah/Shem biographies, and two distinct cumulative Cainan states. The work also identifies the selections required for the all-K form and preserves the cases where composition fails.

No new global axis, historical endpoint, canonical chronology or source-selection law has been established. C324 is unexecuted.

## What is now established

With K=529 and T=17K=8993, the derived LXX_OFF field gives

- cumulative Shem p=−5890+i;
- cumulative Flood q=−5290+i;
- reflected regular Flood f=p+T=3103+i;
- reflected regular Noah n=q+T=3703+i.

Thus q−p=n−f=600, while f−p=n−q=17K. The complete fixed-register census admits six coordinate rectangles, at i2…7. They carry six SP labels and three co-located LXX Gear-labelled Noah/Flood pairs, all in Cainan OFF/Terah0/full430. The three LXX Flood-close aliases do not have a matching LXX Noah endpoint. Source labels and coordinates are counted separately.

The native LXX_ON field shifts both cumulative corners by−460. It supports three corresponding MT rectangles, at i3/5/7, in restored Cainan ON/Terah60/full430. The regular counterpart is the same-Gear composition −650+130+60=−460. Both17K paths, both600 sides, and exact phases survive.

That primary correspondence extends to all five already admitted biography endpoints:

| Gear | SP OFF/T0/full430: Noah, Shem, Flood start, Noah death, Shem death (BC) | MT restored ON/T60/full430 (BC) |
|---|---|---|
|G1|3706, 3206, 3106, 2756, 2606|3246, 2746, 2646, 2296, 2146|
|G2|3708, 3208, 3108, 2758, 2608|3248, 2748, 2648, 2298, 2148|
|G3|3710, 3210, 3110, 2760, 2610|3250, 2750, 2650, 2300, 2150|

All15 endpoints and30 exact components have zero residual under the460-year comparison. The ordered spans remain500+100+350+150; Noah's950 and Shem's600 remain intact.

**This success uses the Noah-bound/Flood-bound500/100 modal state.** File_18 §1.3.5 also retains the Arphaxad-bound/textual state. Restored MTG2 Noah3248 paired with G1 Shem2746 gives502, not500. Neither binding corrects the other. C322 rejected its initial assumption that the frozen same-Gear MT records had502, after checking the actual register and source passage. The record includes that correction; no source dates were altered.

SP companions remain local. Their translated MT Flood closes are admitted, but the required MT Noah births are absent. The three primary successes do not authorize a universal SP→MT biography map.

## How the all-K form is related

The retained C287 configuration uses

`S=p−458`, `F=q`, `R=f−458`, `N=n`.

The two altered corners have explicit, different source derivations:

| Cumulative corner | Regular corner |
|---|---|
|OFF Shem_i →native ON Shem_i: −460|SP→MT, same Gear and boundary: −650|
|Select ON Shem_(i+2): +2|Regular Cainan restoration: +130|
|Net−458|Terah restoration: +60; next Gear: +2; net−458|

This yields600+458=1058=2K and the cross-width matrix[[17K,19K],[15K,17K]]. It is the same dependency as598+460=2K together with8395−460=15K. The cumulative annual choice is not a Gear operation.

These final corners are admitted only at i3…5 within the five-case C314 domain. C287's full retained domain remains i2…5. At the upper boundary the needed cumulative ON members8/9 and regular Gear4 are absent.

The four-case sensitivity test is decisive about selection. Omitting both+2 choices leaves a source-valid1060-year rectangle and still preserves both17K paths. Applying both choices gives1058=2K. Mixed choices break one17K path by two years. Therefore **two17K paths alone do not select the all-K subdivision**. The22 choice is unique only within the declared four-choice menu after imposing the all-K criterion.

The two parallel17K equations alone have rank2. Adding one600 side gives rank3; the other600 side and remaining durations follow. The six pairwise duration equations still have rank3, with common-translation freedom. Repeated paths, annual members and coordinate aliases are not independent historical confirmations.

## Reversed500 correspondence and the surviving limits

For the five inherited normalized SP Shem cases j3…7, exact source500 is restored by

`i=10−j`, with opposite cumulative ancestry phases.

All five biographies and ten components pass. This is a finite local involution, with five component two-cycles and no fixed exact component. Its derived local center does not replace the global Mirror axis.

The same-ancestry failures remain:

- C313's same cumulative component does not produce an exact500 relation; the successful correspondence explicitly reverses ancestry and usually the annual index.
- C315's common-ancestor17K and8395 paths differ598, not600. Their source-Flood mismatch remains two years in every phase.
- Selecting an existing cumulative Flood antecedent h=i−2 permits the8395 route to reach the coherent boundary. Its starting node differs from C271's recorded cumulative **Shem_i** ancestry via17K. Equal outputs do not merge those paths or identities.

The omitted SPG1 companion at i2 has a valid biography and a valid C287 row. The reversed500 construction alone requires absentj8. This is a boundary of that construction, not a rejection of the biography.

## Exact joint support

| Condition | Admitted annual indices i |
|---|---|
|Direct OFF600 coordinate rectangle|2,3,4,5,6,7|
|NativeON600/MT primary counterpart|3,5,7|
|Reversed SP500 partner|3,4,5,6,7|
|C287 complete at i|2,3,4,5|
|C287 complete at reversed j=10−i|5,6,7|
|First four conditions together|3,5|
|All five conditions together|5|

The central i=j5 is unique only after adding every listed condition, including C287 at both indices. That final requirement is an analytical choice, not a condition imposed by the ordinary source biography. Different configurations coexist at that index without becoming one source identity.

## Claim and dependency ledger

| Claim | Status |
|---|---|
|Finite17K600 rectangles and exact phases|Verified against the held source register|
|Three complete460-year primary biography correspondences|Verified in the stated Noah-bound modal state|
|Reversed500 relation|Verified conditional index/phase correspondence|
|Paired458 link to C287|Verified typed composition with finite failures|
|All-K selection follows from17K alone|False in the tested menu;00 is a counterexample|
|A source-prescribed annual/context selector|Not established by this batch; C311's bounded absence remains|
|A universal operator, preferred absolute chronology, or statistical rarity|Not established|

The source ingredients, event roles, modal states and discrete selections are premises. The rectangle's dependent edges, the two bridge factorizations and the joint-set intersections are consequences. Verification inventories below count checks, not witnesses.

## Reading and recovery

Use the C323 checkpoint and this synthesis first. The reading index locates all ten reports, checkpoints and evidence packages. The compact restart kit contains the latest replayable evidence and frozen immediate dependencies; the continuation archive contains all ten sealed packets. Older archives remain selective reference libraries. Current scope and corrections override old next-step paragraphs.

Separate mathematical and source reviews accompany every step. The recovery audit checks the completed C314–C322 ZIP payloads, certificate chain and isolated portable replays; the final publication record adds C323 and all30 individual deliverables. No prior batch was rerun as research.


## Verification and limits

69 main assertions and 54 frozen-input/predecessor checks passed. A second main replay produced byte-identical DATA and RESULTS. Separate review scope and agreement are recorded in the evidence. All frozen originals and standing controls remain unchanged. These are verification counts, not independent-witness counts.

This is a consolidation of finite source-qualified results, not newhistoricalattestation. The500/502modal distinction, dependent paths, source identities and failed boundaries remain explicit.

Mirror/529 remains the focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction x=1 January AD2; no civil year zero. Regular Gears remain Noah/Shem/Flood only. SP minus-one companion is local to SP Noah/Shem. No cumulative/upstream Gear, new endpoint or phase, fitted anchor, global axis, z-law, canonical or graph edit. Capture now; canonize later. Source state, event identity, generated coordinate and phase ancestry remain distinct. Exact arithmetic and source membership do not establish intention or independent historical witnesses.

## Decision after this result

**C324, proposed and unexecuted:** Audit only relevant supplied-source passages for an already documented endpoint/dependency rule connecting the verified460-year SP→MT context correspondence with25/23 and the Noah-bound500-year relation. Distinguish a prescribed path from the scalar width identity; do not fit an anchor or rerun C290.

The new full-biography correspondence makes this a concrete529-related source question. An explicit rule would add something not supplied by the existing arithmetic; another rearrangement of the same equations would not. C324 requires a new user instruction.