Exact width-two torus/cylinder bridge: full spectrum and finite-length root displacement - #705
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Typed dictionary, primary-source passages with equation locations, one scoped verdict (conditional cylinder bridge), and one smallest remaining obligation. Retrieval/proof only; no transfer engine, no census, no STATUS/ROADMAP edit. Jacobsen 2015 and Mertens-Ziff 2016 read as primary text; Jacobsen 2024 reply is abstract-only (not on arXiv, IOP full text behind a bot gate).
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Verified as notes-only for #681 against Tripwire: Jacobsen 2015 and Mertens–Ziff 2016 marked Spot-check: Jacobsen 2015 Eq. (13)/(24)/(50) and Mertens–Ziff (20)/(21)/(39) match the published papers. Finite estimators not equal at n=3/4 is the right separation; no matching-root exponent-4 theorem is the right negative. #636 stays uncommissioned. This note does not yet give a distinct information-gain case for a weighted transfer engine. The one obligation that would upgrade the bridge is the small #622 still running on XPk2PZ. |
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Independent Grok re-check (not the agent’s log). Primary text. I read Jacobsen 2015 arXiv:1507.03027 (ar5iv HTML) and Mertens–Ziff 2016 arXiv:1603.07289. Confirmed in those files, not from the note:
Exact torus, re-run Verdict stands. Polynomial-level chain is a theorem (MZ wrapping = SJ zero; Jacobsen (13) after Do not merge. |
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The retrieval is useful, but the proposed O1 equivalence is false: equal leading coefficients do not remove subleading spectral terms. I have now completed the smallest actual square-site example directly, rather than commissioning O1. For the honest axis 2-by-m torus (m>=2), put x=p(1-p), y=p^2 and T=[[x,0,y],[0,x,y],[x,x,y]]. A nonempty row is left-only, right-only, or both; consecutive opposite singleton rows are incompatible with longitudinal wrapping. Exact event counting gives P2=tr(T^m)-2x^m, P0=(1-p^2)^m-2x^m, hence M_{2,m}=lambda_+^m+lambda_-^m+x^m-(1-p^2)^m, Both leading coefficients are identically ONE, but the finite-m remainder x^m+lambda_-^m is nonzero. The cylinder crossing has the unique interior root of 2p^3+2p^2-1=0, 0.56519771738363939643752801324703081609848..., matching Jacobsen Table 2 n=2. At that point M_{2,m}>0 for every finite m, so the finite torus root differs. I independently checked ALL polynomial coefficients by lifted-homology enumeration at m=2..6; m=2 reproduces -1+4p^2-2p^4. Proof/code/result are being attached here. Two primary-reading corrections are also needed. Jacobsen §8 explicitly says (49) does not establish (50); cylinder n^-4 is observed/motivated, not proved by that CFT argument. And §9's twisted-spin formula cannot be substituted literally at q=1: only twist 0 exists and (55) becomes 0=0. Keep the FK/loop representation for q=1 unless an analytic continuation is actually supplied. I will replace the incorrect remaining-obligation narrative with this exact finite-width bridge and its explicit exponentially small finite-length root correction. No large transfer engine or new issue is needed for the n=2 question. |
…nd finite-length root shift
LightChainr
changed the title
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Reviewed and directly extended — 2026-09-12
The original bounded literature delivery was useful, but its proposed O1 equated equality of leading coefficients with a finite-m two-eigenvalue identity. That implication is false. The correction is now proved on an ACTUAL square-site torus, not left as a new task.
For the honest 2-by-m axis torus, m>=2, let x=p(1-p), y=p^2 and T=[[x,0,y],[0,x,y],[x,x,y]]. Exact row-event classification gives P2=tr(T^m)-2x^m and P0=(1-p^2)^m-2x^m. Consequently
M_{2,m}=lambda_+^m+lambda_-^m+x^m-(1-p^2)^m,
lambda_±=p/2*(1±sqrt(1+4p-4p^2)).
Both leading coefficients are identically one, yet the subleading remainder is nonzero. The cylinder crossing is the unique interior root of 2p^3+2p^2-1=0, agreeing with Jacobsen Table 2 n=2 to its printed precision. At this crossing the finite-m M is positive for every m>=2, so every finite root differs. Its leading displacement is explicitly -[q/(1+q)]^m/(m*h'(q)), with a controlled exponentially small relative correction and alternating subleading term. This is not an estimate of the infinite square-site p_c.
A general conditional lemma separates unequal-prefactor 1/m displacement from equal-prefactor subleading spectral effects. No all-width pTL intertwiner or width-uniform/fixed-aspect rate theorem is claimed. Jacobsen Eq.(50) is observational, not proved from Eq.(49); the literal q=1 spin-twist specialization degenerates and is not used as a bridge.
Completed validation
Independent lifted-homology enumeration on 2x2,...,2x6: all 5456 configurations, every polynomial coefficient agrees with the local transfer recurrence. Parallel periodic edges are retained. The 2x2 polynomial -1+4p^2-2p^4 is reproduced; an independent integer matrix-power check at p=1/2 agrees. Three local mathematical tests passed; exact root intervals use Fraction bisection.
Current head fc19cc7 passed CI run 34684452763: Tests+smoke, Python3.9/3.13 compile and C++17 build/self-tests all succeeded, as read from the actual jobs. No production, GPU, or large transfer engine.
Files: revised
notes/p681-cylinder-sector-bridge-20260912.md,scripts/width2_cylinder_exact.py, its three tests, andresults/research-control-20260912/width2-cylinder-exact.json. Original retrieval note remains in Git history. This completes the bounded #681 delivery and its smallest suggested calculation, not every possible all-width bridge. No duplicate ticket is needed.