# 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.