# 490d C408 — Return-path obstruction to the missing squared-E factorization

24 September 2026. Step 5 of 10 in the C404–C413 authorization.

**None of the new Shem return paths supplies the missing factorization of the older squared-E comparison.** This is a new compatibility result; the−500 translated squared-E endpoints and missing2056/2058 intermediates were already established in C361/C363.

The retained target maps hold SP Shem3206 or3208 and send restored MT Shem death2148→1956 or2150→1958 with slope625/529. A complete intermediate E point remains absent.

To check every positive word in the new source graphs, first compare slopes. Prime7 forces the P count to zero; prime13 forces the J count to zero; prime5 then requires exactly two E arrows. Thus only the three admitted EE paths can possibly reproduce the target whole map. Adding longer positive circuits cannot change that conclusion.

|Candidate sourced Shem path|E pivots in execution order|Effective fixed point|
|---|---|---:|
|2535→2533→2531|2558, 2556|61369/24|
|3184→3182→3180|3207, 3205|76945/24|
|3185→3183→3181|3208, 3206|76969/24|

All six candidate/target whole-map comparisons fail. The closest is the SP-companion path3184→3182→3180, whose two pivots3207 then3205 give effective fixed point3206+1/24. Its output differs from the target held at3206 by−4/529 for every common input. This is a constant affine discrepancy, not a calendar residue or an adopted phase.

All48 combinations of the two existing±¼ candidate-pivot offsets and the target's matched±¼ component were also tested. None repairs the equality. This test even allows independently phased candidate pivots; imposing source-coherent equal phases is stricter and still cannot repair it.

The candidate paths also have different typed endpoint records from the older death-to-death map. The stronger whole-map failure shows that overlooking those labels would not solve the problem. The two constructions share E² as a scalar but retain different pivot and endpoint structures.

## Verification and limits

59 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.

No new intermediate, phase or axis is adopted. The all-word exclusion applies to positive paths in the newly enumerated source graphs, not arbitrary additional operators or unreviewed source grammars.

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

**C409, proposed and unexecuted:** Test the newly sourced Shem return loops under both existing17K source-corner exchanges, using the frozen cumulative register and reversed inherited phases. Track how the fixed−500 regular translation changes the original2K/17K rectangle, without shifting or fitting either exchange constant.

The new regular return correspondence is positive but cannot repair the older E² ladder. Its compatibility with the existing Mirror source corners is now the most direct unresolved529 question.