{"id": "c408", "title": "Return-path obstruction to the missing squared-E factorization", "filename": "490d_C408_Return_Path_Obstruction_to_Missing_Squared_E_Factorization_20260924.md", "role": "Research action", "step": 408, "sha256": "1491118ee49185c85d7eec44f0cb092c086bcdbe350f92c51a675803eb8473d4", "bytes": 3736, "origins": ["C01–C479/packets/C408/490d_C408_Return_Path_Obstruction_to_Missing_Squared_E_Factorization_20260924.md"], "download": "/sources/1491118ee49185c85d7eec44f0cb092c086bcdbe350f92c51a675803eb8473d4.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Calendar Keys", "Mirror / 529"], "excerpt": "490d C408 — Return-path obstruction to the missing squared-E factorization\n\n24 September 2026. Step 5 of 10 in the C404–C413 authorization.\n\nNone 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 ", "content": "# 490d C408 — Return-path obstruction to the missing squared-E factorization\n\n24 September 2026. Step 5 of 10 in the C404–C413 authorization.\n\n**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.\n\nThe 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.\n\nTo 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.\n\n|Candidate sourced Shem path|E pivots in execution order|Effective fixed point|\n|---|---|---:|\n|2535→2533→2531|2558, 2556|61369/24|\n|3184→3182→3180|3207, 3205|76945/24|\n|3185→3183→3181|3208, 3206|76969/24|\n\nAll 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.\n\nAll48 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.\n\nThe 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.\n\n## Verification and limits\n\n59 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.\n\nNo 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.\n\nMirror/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.\n\n## Decision after this result\n\n**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.\n\nThe 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.\n"}