Completed and merged — 2026-09-12
The bounded primary-reading job and its smallest proposed calculation are delivered in #705, merged at eb89e94. This closes the assigned delivery, NOT every all-width spectral or critical-exponent problem.
The initial retrieval identified useful sources but incorrectly treated equality of two leading coefficients as sufficient for an exact finite-length two-eigenvalue identity. The reviewed delivery now gives an actual local square-site counterexample and complete solution: on every honest 2-by-m axis torus, m>=2,
M_{2,m}=tr(T^m)-(1-p^2)^m,
T=[[x,0,y],[0,x,y],[x,x,y]], x=p(1-p), y=p^2.
The full expansion is lambda_+^m+lambda_-^m+x^m-lambda_c^m. Both leading coefficients are ONE; the subleading remainder does not vanish. The cylinder crossing is the interior root of 2p^3+2p^2-1=0, matching Jacobsen Table 2 n=2. Every finite-m torus root is strictly below it; the leading displacement is explicitly exponentially small divided by m. No infinite-square p_c estimate or all-width pTL intertwiner is claimed.
The proof is row-event classification, not extrapolation from counts. Independent lifted-homology enumeration checked all 5456 configurations at 2x2,...,2x6 and every coefficient. The 2x2 polynomial and an integer matrix-power control also pass. Current code head fc19cc7 passed all four CI jobs before merge. Original retrieval history is preserved.
Read notes/p681-cylinder-sector-bridge-20260912.md, scripts/width2_cylinder_exact.py, and results/research-control-20260912/width2-cylinder-exact.json on main. Jacobsen's observed n^-4 law remains observational; the q=1 literal spin-twist specialization is not a usable bridge. Do not rerun a source survey or commission the width-two calculation again.
The genuinely different remaining question is the all-width observable closure and the uniformity needed to compare fixed-width cylinders with fixed-aspect tori. That is re-scoped in #636 as a bounded proof/CPU task, not the old width-10/12 engine purchase. #275's physical operator identification is separate.
Completed and merged — 2026-09-12
The bounded primary-reading job and its smallest proposed calculation are delivered in #705, merged at eb89e94. This closes the assigned delivery, NOT every all-width spectral or critical-exponent problem.
The initial retrieval identified useful sources but incorrectly treated equality of two leading coefficients as sufficient for an exact finite-length two-eigenvalue identity. The reviewed delivery now gives an actual local square-site counterexample and complete solution: on every honest 2-by-m axis torus, m>=2,
M_{2,m}=tr(T^m)-(1-p^2)^m,
T=[[x,0,y],[0,x,y],[x,x,y]], x=p(1-p), y=p^2.
The full expansion is lambda_+^m+lambda_-^m+x^m-lambda_c^m. Both leading coefficients are ONE; the subleading remainder does not vanish. The cylinder crossing is the interior root of 2p^3+2p^2-1=0, matching Jacobsen Table 2 n=2. Every finite-m torus root is strictly below it; the leading displacement is explicitly exponentially small divided by m. No infinite-square p_c estimate or all-width pTL intertwiner is claimed.
The proof is row-event classification, not extrapolation from counts. Independent lifted-homology enumeration checked all 5456 configurations at 2x2,...,2x6 and every coefficient. The 2x2 polynomial and an integer matrix-power control also pass. Current code head fc19cc7 passed all four CI jobs before merge. Original retrieval history is preserved.
Read
notes/p681-cylinder-sector-bridge-20260912.md,scripts/width2_cylinder_exact.py, andresults/research-control-20260912/width2-cylinder-exact.jsonon main. Jacobsen's observed n^-4 law remains observational; the q=1 literal spin-twist specialization is not a usable bridge. Do not rerun a source survey or commission the width-two calculation again.The genuinely different remaining question is the all-width observable closure and the uniformity needed to compare fixed-width cylinders with fixed-aspect tori. That is re-scoped in #636 as a bounded proof/CPU task, not the old width-10/12 engine purchase. #275's physical operator identification is separate.