490d C333 — Source-qualified Noah lifespan and529 synthesis

24 September2026. Step10 of10 in the C324–C333 authorization.

C324–C333 are complete. The principal addition is a source-qualified Noah-lifespan dissection of the existing 529 structure: 2K = 1058 = 950 + 108. It holds across all four complete SP comparison paths in C287. The detailed 490 partition belongs to their primary states; the companion states retain different actual MT intermediates.

The new endpoint connection

Let N be SP Noah birth, M the same-Gear restored MT Noah birth, D the matched SP Noah death, and R the already admitted next-Gear MT Flood boundary. SP is OFF/Terah0/full430; MT is restored ON/Terah60/full430. These are comparisons between source contexts, with their existing reflected-coordinate convention retained.

Every row preserves the matched SP lifespan N−D = 950 and D−R = 108. Thus the existing 2K edge accommodates the complete Noah lifespan through an already admitted death endpoint. No death date was fitted to the desired total. Primary death retains File_18's inclusive/harmonic qualification; the companion death retains its ordinary-companion qualification.

For primary paths, MT Noah→MT Flood is 598 = 490 + 108. For companion paths it is 599 = 491 + 108. The hypothetical MT births3245/3247 needed to impose 460/490 on the companion cases are absent. Flood close remains an event boundary, not an MT person companion. Co-located LXX Noah labels remain preserved without a newly asserted LXX death-path study.

C10 already established the adjacent-Gear 1058 relation and its primary 598+460 dissection; C287 retained it. The new work makes the death endpoint and its 108 remainder explicit. This is an additional source-labelled reading of an existing relation, not another independent numerical witness. The bounded prior-art lookup does not establish archive-wide novelty.

What survives the controls

The complete five-role census gives positive residuals40,140,490,640 after the selected 460 context leg. These come from source intervals500,600,950,1100; the two500 intervals and two600 intervals retain different event meanings. The 1100 spans Noah birth to Shem death and is not a lifespan.

The 40 remainders change to42 under the named502 binding; the 140 lifespan/Noah-age remainders remain. These controls qualify the positive finding rather than treating valid source modes as corrections of one another.

The repeated equal-interval fields also have a known dependency. The Noah/Shem birth field and Flood/Shem-death field are600 apart at each corresponding vertex, through Noah's age at the Flood and Shem's lifespan. Those are distinct source facts with equal numerical values. In each named mode,13 biography/context equations on ten coordinates have rank9; adding all25 residual equations adds no rank. This is coordinate dependence, not a historical witness count or a new global operator.

How this relates to the local460→500 prescriptions

C324 positively located File_09's declared local cumulative460→500 endpoint completions, the AXIS report's declared-anchor example, and File_46's scoped expansion rules. Those positive source prescriptions remain intact. No explicit dependency connecting the particular regular SP→MT context translation, E=25/23 and the Noah-bound500 relation was located in the reviewed passages. That is a bounded source-audit result, not a repository-wide absence claim or a global ban on expansion.

The admitted regular four-point field has gaps460|40|460, overlap40 and outer width960. File_09's bilateral completion field has gaps40|460|40, overlap460 and outer width540. Their three-point500 subpatterns agree, while their complete metric configurations and source/generated endpoint classes differ. A common affine E map also cannot equal a biography-preserving translation across all endpoints. This does not invalidate File_09's local construction or authorize fitting a new anchor.

What the selection test establishes

In C319's already frozen menu, a is the cumulative annual choice and b the regular Gear-coordinate choice, each0 or2. With matched SP death D=N−950, the death-to-Flood remainder is110−b. Requiring108 therefore selects b=2 and restates the right2K edge. Requiring actual MT-Noah→SP-death490 additionally selects the primary rail.

Those two conditions retain both02 and22 at i3 andi5. Their N,M,D,R source endpoints are identical; only S changes. Adding the existing two17K requirement imposes a=b, so the combined conditions select22 at those two indices. That conditional uniqueness is valid; an independent source prescription for coupling the choices remains unestablished.

C332 tests only C319's i3…7 slice. C287's separately admitted i2 companion remains valid and is retained in C331. C323's stricter conditional i=j5 result also remains tied to its additional joint-support requirements; this batch does not erase those requirements or turn a two-row subset into a universal domain.

Completion and recovery

The nine predecessor steps were sealed and published incrementally, each with a report, checkpoint and evidence ZIP. Their independent archive audit verified all nine ZIPs, payloads, certificate chains and frozen sources, reproduced18/18 main outputs and9/9 independent outputs, and confirmed27 publication receipts and write markers. C333 verifies the scientific synthesis and predecessor bindings from frozen inputs. The final completion/integrity record covers C333's sealed replay and all30 individual deliverables, followed by the compact restart kit and continuation archive.

All source and standing-control hashes remain unchanged, including File_18, File_09, the C265 checkpoint/certificate and the canonical graph controls. No new endpoint, phase, fitted anchor, global axis, canonical or graph change was adopted. Original unprefixed reports, Engine reports and C-series remain distinct. The corrected LXX8464 supplement remains awareness only:1564→1700,2484→2700 and920→1000; AD3746/2300 is superseded. The earlier incomplete September17 local ZIP must not be confused with the later verified intact original.

Verification and limits

69 main checks and 66 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. Independent review and its scope are recorded in the evidence. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

Synthesis of completed finite results only; no C334 research. Repeated source stations, phase instances, equations and successful checks are not independent witnesses. Conditional selection is distinct from a source-prescribed rule.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

Decision after this result

C334, proposed and unexecuted: Conduct a bounded source-dependency audit of whether the cumulative ON-Shem annual choice i+2 is explicitly bound to the admitted next-Gear MT Flood boundary, or whether the sources only permit those two choices separately. Use the current endpoint ledger and targeted File_09/File_18 passages and their immediate cited dependencies; add no dates, anchors or numerical search.

The new death partition fixes the regular side condition while leaving the cumulative choice open. The remaining question concerns an explicit source binding, so another numerical census would not resolve it. This is a proposal only: the ten-step authorization ends at C333.