490d C341 — Exact phase and affine-map limits of the ladder

24 September2026. Step8 of10 in the C334–C343 authorization.

The primary ladder is exact on each inherited matching seasonal component. It does not define one affine dilation of physical time that preserves both seasonal components or Noah's950-year lifespan.

All16 assignments of the existing ±1/4 component offsets were tested for each of C340's five source chains:80 controls in total. Exactly four complete ladders remain—the two primary same-Gear Shem paths, each on its two matching components. No mismatched component repairs a companion or502/98 failure. The double-E endpoint equation needs N,R,F on one matching component; H is unused in that equation, accounting for20 control assignments. These counts describe finite checks, not witnesses.

For a selected physical component e, the duration identity may be written

f_e(x) = (N+e) + (25/23)·[x−(N+e)].

It sends R+e to H+e and H+e to F+e for the primary paths. Its fixed endpoint is the existing N+e. This is a diagnostic expression of the interval matches; no new source anchor is assigned.

A single center-fixed affine map instead sends an offset e to E·e. Relative to the retained target component, the errors are−1/46 and+1/46. Equivalently, it enlarges a half-year component separation by1/23. A phase-preserving calculation on pair centers with the original suffix components retained is therefore a different mathematical description from a single affine map on physical time.

The same componentwise map sends the950-year Noah lifespan to 950×25/23=23750/23. It does not fix the source death D; its diagnostic image is outside the retained quarter-year endpoint lattice and is not adopted. Thus the source dissections950+108,950+200,950+300 must not be read as an unchanged biography under E. They are overlapping intervals from separately verified source endpoints.

This leaves a final evidential question before synthesis: does the added regular E ladder constrain the unresolved cumulative choice, and how much of the apparent multiplicity comes from the same source inputs?

Verification and limits

176 main checks and 16 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.

Existing phases only. A center/phase representation is not equated with a physical-time affine dilation. No diagnostic image or global map is adopted.

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

C342, proposed and unexecuted: Establish the ladder’s minimal source dependency and selector status. Check whether adding its full regular endpoint realization distinguishes C319 a=0 froma=2, and reconcile the exact source realization with C89 and File_46 without claiming new scalar discovery or independent witnesses.

The arithmetic, source membership, modes and phases are now bounded. The remaining pre-synthesis task is to state exactly which source facts support the result and whether it closes the original coupling gap.