# 490d C369 — Noah lifespan bridge matrix and529 complements

24 September 2026. Step 6 of 10 in the C364–C373 authorization.

**The complete Noah-lifespan bridge matrix exposes69/70 complements of529.** Let d be the matched state gap:460 on primary rows and459 on the companion rows with C366’s explicit annual-index adjustment. For cumulative state α and regular context β, each0/1, the required950-preserving bridge is:

`Tαβ=19K−(α+β)d=(19−α−β)K+(α+β)(K−d)`.

| 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 |

Every entry is an actual950-year source-interval correspondence at the two retained Gear placements; both inherited same-offset components pass. The two middle entries share a duration but retain different source ancestry and target context. Four matrices contain16 source routes, all derived from the same constants.

One19K translation cannot carry the entire two-state family: each matrix needs three distinct bridge values. The companion matrix is not a pure460 Cainan switch; it includes the previously recorded annual-member difference, giving459. The primary and companion complements are therefore `529−460=69` and `529−459=70`.

These69/138 and70/140 values are arithmetic remainders after division by529. They are not calendar residues, new epoch anchors or newly discovered ratios. P=70/69 is already part of the retained grammar. Its appearance here motivates a precise source-lift test: can the138→140 relation also occur between actual nearby source endpoints, rather than only in bridge remainders?

## Verification and limits

70 main checks and 15 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.

K remainders are duration arithmetic, not calendar Residue Protocol values or adopted chronology anchors. Repeated matrix entries are dependent and retain distinct source roles.

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

**C370, proposed and unexecuted:** Test whether P=70/69 on138→140 has an actual source-endpoint realization at the existing cumulative and regular Flood pivots separated by460. Compare its result with the retained598→600 repair under300/299; add no fitted pivot or new source point.

The bridge matrix makes the retained P ratio visible arithmetically. An endpoint test is now needed to determine whether that relation has source support beyond the remainder decomposition.