# 490d C373 — Noah lifespan, paired P/J and529 synthesis

24 September 2026. Step 10 of 10 in the C364–C373 authorization.

**C364–C373 are complete. The main addition is an actual source-endpoint realization of the paired P/J relation behind 529.** At existing Flood pivots 460 years apart, P=70/69 expands 138→140 and J=300/299 expands 598→600. Both reach the same existing endpoint, two years from the input. This reproduces the retained 458+600=1058 repair while holding the original 2K and 17K outer endpoints fixed.

The relation is source-supported on the four complete cumulative rows and the two regular primary rows. It remains conditional on comparing these two anchored conversions. The reviewed passages supply the ratios and chronology controls; they do not prescribe their joint application here.

## Paired conversion and the 529 side

| Quantity | Primitive positive integer family | Source scale m=2 |
|---|---:|---:|
| P input → output |69m → 70m|138 → 140|
| J input → output |299m → 300m|598 → 600|
| Separation of the two pivots |230m|460|
| Whole side when its first source segment equals that separation |529m|1058 = 2K|
| Actual repaired side |—|458 + 600 = 1058|

The derivation is exact. If both maps have the same positive integer gain m, their inputs must be 69m and 299m. Their difference is 230m. In the retained source configurations, the first segment of the original side also equals that pivot separation, so the complete side is `(230+299)m=529m`. The actual 460-year source difference fixes m=2. Equivalently, `2×299−69=529`.

Integer inputs alone do not imply this integral family: the rational agreement family has inputs 13t and 3t with gain t/23. Integer outputs require t to be divisible by 23. The primitive scalar 230 is not an adopted chronology endpoint.

The source first-segment condition is essential. On regular companion rows, that segment is 459 while the matched Flood pivots remain 460 apart. The outer side is still 1058, but the conversion inputs are 599 and 139, and their gains differ. Thus a 529 side alone does not guarantee the paired conversion.

## Actual endpoint examples

The representative primary row i=3 makes the source identities explicit. Negative coordinates below are the retained signed cumulative comparison coordinates; they do not introduce civil year zero.

| Field | Input source | J pivot | P pivot | Common existing output |
|---|---|---|---|---|
| Cumulative LXX |OFF Shem −5885|OFF Flood −5287|Native Flood −5747|OFF Shem −5887|
| Reflected regular |Restored-MT Noah birth AD3246|Restored-MT Flood AD2648|SP Flood AD3108|Next-Gear restored-MT Noah birth AD3248|

Cumulative conversion moves the signed source coordinate by −2; regular conversion moves the reflected coordinate by +2. These are the same repaired records retained by C351/C353. The cumulative test passes all six available 598 configurations, i=0…5; four are within the complete C287 domain, i=2…5. Regular primary i=3 and i=5 pass. Regular companion i=2 and i=4 do not.

At matched companion Flood-close pivots, J misses the candidate source by 1/299 and P by 1/69. Using the SP Flood start instead lets P alone hit the source, but J still misses by 1/299. That alternative changes the pivot separation to 461 and cannot be merged with the matched-boundary result.

## Why these controls are selected

With the J pivot at oriented coordinate 0 and the P pivot at D, their images of input x differ by

`D+(70/69)(x−D)−(300/299)x = (10/897)x−D/69`.

There is exactly one agreement input: `x=13D/10`, with common output `30D/23`. For D=460 this gives 598 and 600, and the inner gap gives 138 and 140. The two maps are not equal operators.

In the fixed twenty-control C319 menu, the cumulative gap is 600−a and the regular gap is 600+e−b, where e identifies the SP companion rail. Joint source agreement retains **22 at i=3 and i=5**. Formal agreement at coordinates outside the admitted source range is not a source match.

This is the same retained intersection obtained earlier in the batch from fixed-19K Noah transport and the joined regular map. These criteria share constants and geometry; their repeated agreement is not an independent-witness count or a textual prescription selecting the controls.

All twelve tested same-offset seasonal components preserve the paired source relations when inputs, pivots and outputs carry their matched ±1/4 offsets. If the pivots instead stay fixed at their centers, the two images disagree by ±5/1794 at those input offsets. The component comparisons do not establish one physical-time affine action over the phase pair.

## Noah lifespan structure that led to the result

The earlier proposal—extending the regular joined 460/598→500/600 comparison to cumulative Shem/Flood endpoints—fails its frozen source test. None of its six internal or terminal images is an admitted member. The original 32-member register has no 500 or 1100 interval. The retained 460-year Cainan lifespan is not redefined.

The existing cumulative Noah source family supplies a different connection. Eight cumulative 950-year biography rectangles and three regular primary rectangles share the partition **460,490,460**. The cumulative Noah births are explicitly source-derived: native values agree with the File18 Lamech predecessor, and OFF values were already retained in C300; the reviewed Engine III passage identifies the Shem-birth/Noah-death seam.

Binding these rectangles to the four complete C287 rows gives a unique OFF annual index `j=k−e`. Primary rows retain 460,490,460; companion rows give 459,491,459. The latter includes a one-year annual-member difference and is not a changed Cainan lifespan or a cumulative companion operator. A center reflection would reverse the inherited phase and needs a different diagnostic constant for each row; no new reflection axis is adopted.

The already present **19K=10051** translation does carry the whole native cumulative Noah 950 interval to the reflected SP Noah interval, preserving the same-offset components. It exchanges birth and death roles: cumulative birth reaches reflected death, and the cumulative death/Shem seam reaches reflected birth. It does not carry the OFF internal vertex to restored-MT Noah birth; all 32 declared OFF candidates fail that particular image test.

The complete lifespan correspondence needs three bridge values, not one:

| Cumulative state → regular context | Primary bridge | Companion bridge |
|---|---:|---:|
| Native → SP |19K =10051|19K =10051|
| Native → restored MT |18K+69 =9591|18K+70 =9592|
| OFF → SP |18K+69 =9591|18K+70 =9592|
| OFF → restored MT |17K+138 =9131|17K+140 =9133|

The companion entries retain the annual-index adjustment. These are arithmetic remainders after division by 529, not calendar residues or new epoch anchors. Their 69/70 and 138/140 relation motivated the successful P/J source-endpoint test.

## Ten-step locator and decisions

| Step | Completed bounded result | Reason for the following step |
|---|---|---|
| C364 |Cumulative joined comparison has no sourced internal or terminal image.|Check the existing Noah lifespan field.|
| C365 |Cumulative/regular Noah 950 rectangles share 460,490,460.|Bind the shape to actual retained annual and phase ancestry.|
| C366 |Unique annual binding j=k−e; a center reversal fails same-phase transport.|Test the already present 19K translation.|
| C367 |19K carries the native 950 interval, but not its OFF interior.|Assess exactly which fixed controls the source criterion selects.|
| C368 |19K and the regular joined-map criteria intersect at 22:i3/i5.|Enumerate the full source-lifespan bridge matrix.|
| C369 |Bridge matrices expose primary69/138 and companion70/140 remainders.|Test whether P has an actual endpoint realization.|
| C370 |P and J reach the same sourced two-year repair on the stated rows.|Test uniqueness, all controls, boundaries and phases.|
| C371 |Unique agreement and full controls retain 22:i3/i5, with explicit exclusions.|Derive the complete integer grammar.|
| C372 |Equal integer gains yield the 230m/529m structure; sources fix m=2.|Consolidate and verify the completed batch.|
| C373 |Synthesis, alternate calculation and recovery checkpoint.|Propose a full-rectangle compatibility audit; leave it unexecuted.|

## Evidence and interpretation

The nine preceding sealed packets passed CRC, payload hashes, source and certificate bindings, predecessor links, portable replay and publication checks before this synthesis. A separately written root-agent implementation used source joins and integer cross-products, without importing the main calculations, and passed 341 checks. It shares the frozen sources and the same analyst; no new independent-agent review is claimed. The final completion record adds the sealed C373 replay and all ten packets’ publication coverage.

The inherited E=25/23 results remain separate: the regular joined 500+600 comparison changes an original endpoint, while the Shem-held squared-E endpoint comparison lacks an admitted single-E intermediate in its frozen menu. The new paired P/J repair preserves the existing outer endpoints but has not been shown to extend to the complete internal and cross-field structure of the Mirror rectangle. That is the next bounded question.


## Verification and limits

77 main checks and 63 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 support and exact arithmetic do not establish a source-prescribed coupling, historical intention, independent witnesses or a global physical-time action. C374 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

**C374, proposed and unexecuted:** Audit the full primary source-labelled Mirror rectangle after both two-year P/J repairs: enumerate the internal and cross-field spans, determine which inherited relationships survive, and test whether a whole-rectangle comparison follows. Hold the original2K/17K outer endpoints and inherited phase ancestry fixed; adopt no fitted bridge, new endpoint or anchor.

The local paired conversions now have source support and exact integer structure. Their compatibility with the complete Mirror configuration is the immediate unresolved question. This proposal is unexecuted; the current authorization ends at C373.