# 490d C363 — Shem lifespan and529 source-comparison synthesis

24 September 2026. Step 10 of 10 in the C354–C363 authorization.

**C354–C363 are complete. The main advance is a source-qualified connection between the primary529 side and Shem’s600-year lifespan, together with a second1250-year source realization.** These are additional finite comparisons. The original Mirror rectangle and its source-selection limits remain retained.

## The529 side reaches an actual Shem biography

The primary regular side has the partition460+598=1058. Applying the two already established local ratios to its two legs gives:

\[
460\frac{25}{23}+598\frac{300}{299}=500+600=1100.
\]

| Placement | Original source chain | Resulting source chain |
|---|---|---|
| i3 | SP Noah3706 → restored-MT Noah3246 → adjacent restored-MT Flood2648 | SP Noah3706 → SP Shem birth3206 → SP Shem death2606 |
| i5 | SP Noah3708 → restored-MT Noah3248 → adjacent restored-MT Flood2650 | SP Noah3708 → SP Shem birth3208 → SP Shem death2608 |

The two formulas meet at the same source Shem birth. With input nodes N,M,R and output nodes N,S,T, they are `N+(25/23)(x−N)` on the first segment and `S+(300/299)(x−M)` on the second. This is a continuous two-segment comparison with different slopes. Its lower input is a598-year cross-state Noah/Flood gap; its lower output is the actual600-year Shem lifespan.

The resulting total ratio550/529 and gain42 describe this construction. No global550/529 operator is adopted. A single affine map fixed at N and sending M to S would instead send R to the base-MT Shem birth, producing1150 rather than1100.

The new comparison also changes an outer endpoint: R moves to T by−42. Holding the cumulative corners fixed, one former17K cross-span becomes17K−42. It therefore does not preserve the full original rectangle. C353’s different458+600 repair continues to preserve the original2K/17K outer endpoints. Both comparisons keep their respective purpose and source states.

## The600/650 fork gives a1250 source span

The SP and MT Shem biographies fit a simple source sequence:

| Source event | First placement | Second placement |
|---|---:|---:|
| SP Shem birth |3206 |3208 |
| SP Shem death |2606 |2608 |
| MT Shem birth |2556 |2558 |
| MT Shem death |1956 |1958 |

The successive gaps are **600+50+600=1250**. Corresponding SP/MT endpoints differ by650; each biography is600. Thus the50 gap is650−600. It is also the difference between the two images of598:650 under E and600 under300/299.

This connects the joined500+600=1100 path with the earlier uniform-E500+650=1150 path. Both begin at the same SP Noah and pass through the same SP Shem birth, but end at different sourced events.

The1250 span also appears as a direct squared-E endpoint comparison:

| Fixed SP Shem birth | Input: restored next-Gear MT Shem death | Output: base MT Shem death |
|---:|---:|---:|
|3206 |2148 |1956 |
|3208 |2150 |1958 |

The input width is1058=2K and the output width1250; the slope is625/529. This is the existing Noah-held squared-E comparison translated by−500 along actual source role relationships: SP Noah→Shem birth and MT Flood→Shem death.

A single E step would land at2056/2058, absent from the declared frozen Noah/Shem/Flood register. The direct squared-E endpoint equality therefore does not establish a composable pair of source-qualified E edges. The intermediate coordinates remain unadopted. A cumulative counterpart to this translated regular comparison is unestablished.

## What the source census adds

File_18 §§2.2.1 and3.2.1 preserve Shem’s matched600-year biographies and its qualified endpoint states. File_02 §6.3 supplies598×300/299=600 as a gap conversion, explicitly outside the Residue Protocol.

Across the frozen tradition/rail/context menu, C354 finds96 matched600 biographies and64 adjacent-Gear598 comparisons. Each598 input has two existing600 repairs: move birth+2 while holding death, or move death−2 while holding birth. Sixteen598 inputs belong to the SP companion rail. This corrects any overgeneralization from the earlier companion Noah/Flood failure: the companion rule is local to its actual endpoint roles.

A separate finite pivot census finds three source placements of the460→500 E relation:

| Fixed SP endpoint | MT input | Output |
|---|---|---|
| Noah birth3706 | Restored Noah birth3246 | SP Shem birth3206 |
| Flood start3106 | Restored Flood start2646 | SP Shem death2606 |
| Shem death2606 | Restored Shem death2146 | Base MT Noah death2106 |

The second primary placement adds2 throughout. The source-determined translations are−600 and−1100. The normal form reduces to `650−130−60=460`, and the last500 gap to `650−(500+600−950)=500`. Six of384 context/rail/Gear controls support all three rows: primary SP, restored-Cainan/Terah+60 MT inputs relative to SP OFF/base, and a common sojourn frame. The six comprise three Gears in each of two sojourn frames. Extra local placements do not create new complete Mirror rectangles.

These source instances share their constants and derivation. Their contribution is a smaller description of the endpoint relations, not a new independent-witness count or a source-prescribed combined operation.

## Selection and phase limits

The local three-row structure occurs in nine admitted primary C319 controls across00,02,20,22. Its same-Gear MT Flood input differs from the rectangle’s R by b. Identifying them selects b=0, while the original2K side needs b=2; both roles must remain explicit.

The joined comparison and the Shem-held squared-E comparison each retain **02:i3/i5 and22:i3/i5**. They select the primary regular b=2 condition, while leaving cumulative a unresolved. Adding the separate17K requirement still leaves22:i3/i5. No reviewed source instruction prescribes the paired selection.

The companion joined map misses the actual SP birth by25/23 and, at b=2, the death by25/299. The small terminal error reflects cancellation and does not repair the source mapping. For the squared-E transport, Flood-close−500 would require MT companion deaths that are not admitted.

Matching inherited seasonal offsets pass four of16 squared-E phase controls and four of64 joined-map controls. A single physical-time map distorts the source pairs: squared E produces endpoint errors±24/529; the joined map’s birth-pair separation exceeds a half-year by27/1196. These are map diagnostics, not adopted calendar residues or phases.

## Completion and retained scope

Each of C354–C362 was published with its report, checkpoint and evidence ZIP before the next research step. The pre-C363 audit verified nine sealed packets,222 manifested payloads,108 certificate bindings, nine predecessor links and105 frozen source inputs. All18 isolated main replay outputs matched and27 publication receipts passed.

C363 binds those results and their sequential decisions. A separately written root-agent calculation passed475 checks using independent source joins and integer cross-products. This is a second implementation by the same agent; it is not an independent-agent review. The final completion record covers all ten sealed packets,30 individual deliverables and the recovery packages.

Actual File_18 and File_09 bytes match their retained hashes. File_02 matches its C354 frozen source. The C265 checkpoint is readable and its certificate hash matches the controlling record. Original source files and standing canonical controls remain unchanged. Older archives remain selective references; their stale next-step paragraphs do not reopen completed work.

The retained controls include C343’s original source ladder, C353’s endpoint-preserving repairs, the qualified500/100 and502/98 alternatives, SP-local person companions, and the distinction between generated coordinates and sourced events. The corrected LXX8464 observation remains supplemental awareness; AD3746/2300 remains superseded. The earlier incomplete September17 ZIP remains distinct from the subsequently verified intact original. C-series, original unprefixed reports and Engine reports remain distinct.

The strongest unresolved item is still the source-qualified connection between the regular and cumulative endpoint choices. The next bounded question should test the cumulative counterpart of the new joined comparison using the already frozen source register. **C364 is unexecuted.**


## Verification and limits

76 main checks and 61 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

Finite source identities and comparisons do not establish a source-prescribed coupled selector, historical intention, independent witnesses, or a global physical-time action. C364 is unexecuted.

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

**C364, proposed and unexecuted:** Test whether the joined460/598→500/600 comparison has a source-qualified cumulative counterpart in the frozen LXX Cainan ON/OFF Shem/Flood annual register. Require actual internal and terminal members and inherited phase ancestry; compare with C351’s existing458+600 repair. Preserve the460 Cainan lifespan and the existing annual-member domain.

The new regular source constructions leave cumulativea unresolved. Testing their cumulative counterpart targets that remaining connection directly and can reuse the existing register and prior exclusions. This is proposed only; authorization ends at C363.