# 490d C386 — Source-qualified E/J affine square and component limits

24 September 2026. Step 3 of 10 in the C384–C393 authorization.

**The surviving E/J construction is an exact affine identity, with all four vertices and all four pivots already sourced.** For the four retained regular repairs,

`E_(N+h) ∘ J_A = J_D ∘ E_N`,

where h=2 and D=E_N(A)+26h=A−40 is the next-Gear SP/LXX Shem death. The common slope is7500/6877. Equality was verified by both affine coefficients, not just at a shared output.

For the original full430 G1→G2 example, the square is:

|Route|First existing edge|Second existing edge|
|---|---|---|
|J then moving E|MT Noah3246→3248, J held at MT Flood2648|3248→SP/LXX Shem3208, E held at Noah3708|
|E then transported J|3246→SP/LXX Shem3206, E held at Noah3706|3206→3208, J held at Shem death2608|

The target context has500 years from Noah3708 to Shem birth3208 and600 to Shem death2608. The source J input gap is598; its output gap is600. Thus the source geometry is500+600=1100. The original Noah-to-Flood side was1058=2K, and1100−1058=42. This is consistent with the previously retained C363500/600 geometry; the new result is the mixed-square identity, not a new discovery of the scalar sum. Comparing the required death pivot with the *next* original Flood pivot also gives42:2650−2608=42.

A finite component check assigned the inherited offsets±1/4 to all eight points/pivots. Across1024 assignments (256 per coordinate square), exactly eight retain all four source edges: the two common-offset assignments in each row. Each of those also retains the whole affine identity with its phase-shifted pivots. These are matched-component comparisons; tradition aliases do not multiply the arithmetic trials.

A single fixed physical-time composite has slope7500/6877 and changes a half-year separation by623/13754. It therefore cannot preserve both retained seasonal components as one physical map. No extra phase value was introduced and no generated coordinate was admitted.

Sources: C385 provides the eight labelled context lifts and unique pivots. C341 supplies the inherited component comparison discipline; C363 retains the500/600 construction. File02 and File46 retain the distinctions between the J gap conversion, E, and calendar residue displays.

## Verification and limits

26 main checks and 18 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.

The identity proves a finite source-labelled comparison, not historical intent or an author-prescribed protocol pairing. Matched-component maps differ from one physical-time action. No biography-preserving or original529-side-preserving action is inferred.

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

**C387, proposed and unexecuted:** Test the E/J square against the existing SP companion and valid502/98 Shem bindings, preserving actual Gear membership and recording the exact failures rather than treating the500/100 realization as a source selector.

The positive square uses the500-year Noah–Shem binding. Its interpretation depends on explicitly checking the alternative bindings already retained by the project.