Reproducibility note and two wording corrections (28 October 2026). The note’s appendix promises check_dissipation_law.py, which version 1 did not carry. It has been written and run: 109 checks, 109 pass, with 53 derived from the C60 adjacency alone, 45 from the stated premises, 6 internal-consistency, 4 Monte-Carlo and 1 using a reconstructed input. The MEASURED set is empty, and that is itself a check: the note’s own flow-position paragraph says it compares no kernel number with measured data, and that was confirmed rather than assumed. Definitions are tagged as such and are never counted as derivations. Every entry of Table 1 and every number in §3, §4, §5.3, §6.1 and §6.2 reproduces. 1. §3.1’s “fails by at least 19%” is a property of one draw, not a bound. On the setup described, the smallest control deviation is 40.6%, so the floor holds there comfortably. But across 500 random cases (25 draws × 5 β × 4 t) the minimum falls to 10.1%, with 0.6% of cases below 19%. The substantive content is far stronger than the stated floor and should be given instead: every one of the 500 control cases exceeds its own KMS error by a factor of 3.5×10¹³ or more. Suggested wording: “fails by factors of 10¹³ or more above the machine-precision residual.” Relatedly, §3.1’s “relative error 3×10⁻¹⁵” is the typical value, not an attainable bound: the median over 500 cases is 3.23×10⁻¹⁵, the 90th percentile 8.1×10⁻¹⁵, the maximum 2.2×10⁻¹³. 2. §6.2’s “Stratonovich simulation” label does not fit the reported figure. The number itself is right — the theory value 19.44 and the closed form are exact and confirmed — but a direct Stratonovich–Heun integral of the locked case carries a large O(dt) bias, because the phase-difference modes relax at rate ~K = 30: dt = 0.002 gives 14.94 and dt = 0.001 gives 17.16, so reaching the reported 0.15% would need dt ≈ 10⁻⁵, far beyond what “N = 6, D = 0.25, K = 30, 400 replicas” implies. The exact Itō form of the same quantity, D⁻¹(⟨ΣFi²⟩ + D⟨∇·F⟩), carries no stochastic-integral bias and gives 19.4651 at dt = 0.001, converging to theory as dt falls. So the estimator, not the result, is what the label misnames. Separately, the ω list is never given, so the printed simulated triple is not reproducible as such; the printed theory triple is internally consistent (24.42 − 19.44 = 4.98 exactly) and is reproduced exactly from a reconstructed ω with mean 0.9 and population variance 0.2075. The note’s “var” is unambiguously the population variance — the identity fails for the sample variance. 3. §1’s “the C60 spectral values 1, 4−φ and φ” is loose on the third. 1 is an eigenvalue of A with multiplicity 9; 4−φ is an eigenvalue of L = 3I − A with multiplicity 5, since 3 − 1/φ = 4 − φ; but φ is an eigenvalue of neither (nearest level 0.0565 away in A, 0.1796 in L). Only −φ is, with multiplicity 5. The note says so itself two paragraphs later, so this is internal looseness of wording rather than a numerical error, and the product identity (4−φ)·φ = 3φ−1 = 3.854101966249685 is exact. Two results are worth naming because they are stronger than stated. Aut(A) ≅ Ih is confirmed structurally, not merely by order: order 120, 31 involutions, centre of order 2 and no element of order 4, which pins A₅ × C₂ and rules out S₅ (25 involutions, trivial centre). And in §6.1 the garbled line is ΔSsys = −TC₁, the total correlation: the system’s entropy falls by exactly the correlation that forms, verified symbolically, with C/8 = H(p₁) exactly — which is why the printed closed form for (E/S)op holds. Ψop was rebuilt from ledger quantities alone, with no A, B or C in it, and agrees with the printed form to 2.5×10⁻¹⁷⁵. A scope and status note on the coherence dissipation law. Separates what the C60 adjacency forces from what it forbids, and states the open-system form that survives after the bare coefficient was retired in the record. The law in every surviving form is posited, not forced, and the paper says so wherever it matters. Includes the two-ledger theorem under stated premises, a no-go result limited to coherence functionals built from the adjacency, and a verification script.
Reproducibility note and two wording corrections (28 October 2026). The note’s appendix promises check_dissipation_law.py, which version 1 did not carry. It has been written and run: 109 checks, 109 pass, with 53 derived from the C60 adjacency alone, 45 from the stated premises, 6 internal-consistency, 4 Monte-Carlo and 1 using a reconstructed input. The MEASURED set is empty, and that is itself a check: the note’s own flow-position paragraph says it compares no kernel number with measured data, and that was confirmed rather than assumed. Definitions are tagged as such and are never counted as derivations. Every entry of Table 1 and every number in §3, §4, §5.3, §6.1 and §6.2 reproduces. 1. §3.1’s “fails by at least 19%” is a property of one draw, not a bound. On the setup described, the smallest control deviation is 40.6%, so the floor holds there comfortably. But across 500 random cases (25 draws × 5 β × 4 t) the minimum falls to 10.1%, with 0.6% of cases below 19%. The substantive content is far stronger than the stated floor and should be given instead: every one of the 500 control cases exceeds its own KMS error by a factor of 3.5×10¹³ or more. Suggested wording: “fails by factors of 10¹³ or more above the machine-precision residual.” Relatedly, §3.1’s “relative error 3×10⁻¹⁵” is the typical value, not an attainable bound: the median over 500 cases is 3.23×10⁻¹⁵, the 90th percentile 8.1×10⁻¹⁵, the maximum 2.2×10⁻¹³. 2. §6.2’s “Stratonovich simulation” label does not fit the reported figure. The number itself is right — the theory value 19.44 and the closed form are exact and confirmed — but a direct Stratonovich–Heun integral of the locked case carries a large O(dt) bias, because the phase-difference modes relax at rate ~K = 30: dt = 0.002 gives 14.94 and dt = 0.001 gives 17.16, so reaching the reported 0.15% would need dt ≈ 10⁻⁵, far beyond what “N = 6, D = 0.25, K = 30, 400 replicas” implies. The exact Itō form of the same quantity, D⁻¹(⟨ΣFi²⟩ + D⟨∇·F⟩), carries no stochastic-integral bias and gives 19.4651 at dt = 0.001, converging to theory as dt falls. So the estimator, not the result, is what the label misnames. Separately, the ω list is never given, so the printed simulated triple is not reproducible as such; the printed theory triple is internally consistent (24.42 − 19.44 = 4.98 exactly) and is reproduced exactly from a reconstructed ω with mean 0.9 and population variance 0.2075. The note’s “var” is unambiguously the population variance — the identity fails for the sample variance. 3. §1’s “the C60 spectral values 1, 4−φ and φ” is loose on the third. 1 is an eigenvalue of A with multiplicity 9; 4−φ is an eigenvalue of L = 3I − A with multiplicity 5, since 3 − 1/φ = 4 − φ; but φ is an eigenvalue of neither (nearest level 0.0565 away in A, 0.1796 in L). Only −φ is, with multiplicity 5. The note says so itself two paragraphs later, so this is internal looseness of wording rather than a numerical error, and the product identity (4−φ)·φ = 3φ−1 = 3.854101966249685 is exact. Two results are worth naming because they are stronger than stated. Aut(A) ≅ Ih is confirmed structurally, not merely by order: order 120, 31 involutions, centre of order 2 and no element of order 4, which pins A₅ × C₂ and rules out S₅ (25 involutions, trivial centre). And in §6.1 the garbled line is ΔSsys = −TC₁, the total correlation: the system’s entropy falls by exactly the correlation that forms, verified symbolically, with C/8 = H(p₁) exactly — which is why the printed closed form for (E/S)op holds. Ψop was rebuilt from ledger quantities alone, with no A, B or C in it, and agrees with the printed form to 2.5×10⁻¹⁷⁵. A scope and status note on the coherence dissipation law. Separates what the C60 adjacency forces from what it forbids, and states the open-system form that survives after the bare coefficient was retired in the record. The law in every surviving form is posited, not forced, and the paper says so wherever it matters. Includes the two-ledger theorem under stated premises, a no-go result limited to coherence functionals built from the adjacency, and a verification script.
The complete fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise decomposes into five channels. Closed forms were previously available for two of them. This note evaluates the remaining three for the unitarily invariant ensemble, so that the whole coefficient is explicit for every bulk dimension d_1, every geometric dimension d_2 and every spectrum. The delta1-delta3 channel, reported in earlier work as measured but not evaluated, is A_13 = -(5/3) D (Xi - 1), with D = (d_1 d_2)^2 the total Hilbert-space dimension and Xi the Kubo-Mori spectral functional of the second-order theory. The proof uses only the two operator identities sum_a A_a^2 = D and sum_a A_a M A_a = Tr(M), and is independent of the code. The channel is negative for every spectrum, and not merely because Xi > 1: since the logarithmic mean is dominated by the arithmetic one, Xi >= d_1^2 with equality exactly at a degenerate spectrum, so A_13 <= -(5/3) D (d_1^2 - 1). The quartic commutator channel C is obtained in closed form as a contraction of the divided-difference kernel l_3 with two copies of the mode-sum tensor N, and is proved independent of d_2. The mixed channel B splits as B_1 + B_2. The remainder B_2 is the one that resisted: it is obtained here in closed form for every d_1, every d_2 and every spectrum, B_2 = 3 Xi - chi/2 - 2 + T, with chi = sum_u 1/p_u and T = sum_uvs l_2(u,v,s) p_u, where l_2 is the second confluent divided difference of the logarithm. T is a third spectral functional, and it is shown not to be a function of the other two: the best affine fit to {Xi, chi, 1} over forty interior spectra leaves worst residuals 2.86, 8.34 and 13.80 at d_1 = 3, 4, 5, against values of abs T reaching 82, 297 and 542. T is nevertheless not an alien object. In the variables m_k(t) = sum_u (p_u + t)^(-k) and h(t) = sum_u p_u/(p_u + t), the three functionals are Xi = integral h m_1 dt, chi = integral m_2 dt and T = -integral h m_1^2 dt, so T is the next member of one family. T and chi moreover scale alike, both as p^(-1), so no counting of degrees separates them; only the mode sum does. The derivation has two halves. The double mode sum Q_uvs = sum_ab (J_a)_uv (J_b)_vs (K_ab + K_ba)_su closes on the bulk spectrum alone, by four completeness identities: two that the preceding paper already prints, and two that carry an uncontracted mode operator and are what close the terms of K carrying A_a A_b Phi. Twenty-four cases are involved, and they are tabulated in the derivation note deposited in this record. The resulting divided-difference contractions then collapse by two identities proved from the integral representation of the Kubo-Mori kernels, F_4 = -d_1^2/2 and F_3 = d_1^2/2 - Xi. Three statements that could previously only be checked numerically follow as corollaries: B_2 is independent of d_2 and of the code; at a degenerate spectrum B_2 = -(1/2)(d_1^2 - 1)(d_1^2 - 4); and at d_1 = 2 it reduces to 2 (Xi - chi), that collapse being an accident of the dimension, since T there equals -(3/2) chi - w_12. Consequently the assembled coefficient is explicit at every configuration, its d_2-independent constant collapsing to 2 Xi - chi/2 - 1 + T + C. At d_1 = 2, d_2 = 2 and p = (0.7, 0.3) it reproduces a direct sampling measurement to 0.13 standard errors, at d_1 = 3 with a degenerate spectrum to 0.45, and the formula for A_13 reproduces retrodictively, to 0.55 standard errors, a measurement published before this derivation existed. Every closed form is certified against exact mode sums on an explicit orthonormal Hermitian basis of the full space, link by link rather than only end to end, and the formula for B_2 is tested on held-out spectra it never saw. An erratum to the preceding paper is also recorded. As printed, its third-order mode object L_abc = R([A_a,[A_b,[A_c,rho]]]) is anti-Hermitian, which would force A_13 to vanish identically for every ensemble, every code and every spectrum, contradicting the nonzero delta1-delta3 word measured in earlier work. The third-order deformation carries i^3, and restoring that factor removes the contradiction. That paper's own numerics evidently carried the factor, because it reports nonzero values of A_13 on two families that are not stabilizer codes; no value reported in any of the preceding papers is affected, and what requires correction is one definition as set. Two limitations are stated. Everything here is for the unitarily invariant ensemble, and none of the operator arguments survives for a structured mode family such as a Pauli basis on a stabilizer code. And a discrepancy with a published measurement at d_1 = 3, d_2 = 12 is located at a definite configuration but not adjudicated: the coefficient gives -9492 at the degenerate endpoint there, while the estimator used in this note cannot be carried to d_2 = 12 at the coupling used throughout, the Taylor control eps times the norm of W reaching 1.44. This note is posted here for the first time. It supplies the three missing channels of the five-channel decomposition of A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), corrects one printed definition in it, and evaluates in closed form the delta1-delta3 word measured in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The degeneracy measurement it compares against is reported in Breakdown and Persistence of the Reconstruction-Error Hierarchy in Approximate Holographic Codes (DOI 10.13140/RG.2.2.19676.14723). The derivation note carrying the twenty-four case tables is deposited in this record as B2_derivation_note_2026-10-27.pdf. The scripts reproducing every number in the note are deposited as closed_forms_scripts.zip. They require Python 3.10+ and NumPy; no other dependencies, CPU only.
A pure encoded state of an approximate quantum code is perturbed by exp(i eps W) with W drawn from a centered Gaussian ensemble, and D(eps) denotes the relative-entropy disturbance of a reduced state. This paper derives the complete fourth-order coefficient of the mean disturbance in closed form, and identifies what that coefficient is made of. The derivation has two stages, and they assume different things. The first assumes nothing about the noise beyond that it is centered and Gaussian: the coefficient splits into five channels, one disconnected drift term and four connected ones, each a finite sum over pairs of Karhunen-Loeve modes of the covariance contracted with exact confluent divided differences of the logarithm. For any concrete noise model the whole quartic disturbance is then computable deterministically, with no sampling. The second stage adds one hypothesis, that the mode family is complete and orthonormal, and evaluates every channel. The device throughout is to substitute the integral representation of the divided differences before performing the index sums, which turns each Kronecker delta into a coincidence of resolvent labels, and then to integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] of resolvent traces. No coefficient is fitted and no index contraction survives. What survives is four scalar functions of the Schmidt spectrum: Xi, chi, S_w and T. The assembled coefficient is a function of d_1, d_2 and those four alone, and four is the right number. The five functions 1, Xi, chi, T, S_w are proved linearly independent for every d_1 >= 3, analytically: a degeneration lemma confines the kernel to two dimensions and an expansion about the degenerate spectrum closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2, where the two relations that do make the set reducible are exhibited. The four are not a list that happens to close. Each is a contraction of the Kubo-Mori metric and the Amari-Chentsov skewness tensor on two operator directions, the identity and the maximal-coherence operator of the Schmidt eigenbasis; and those directions are not chosen but read off the mode two-point function, which the Gaussian ensemble fixes identically in the spectrum and for every d_1. The two are Kubo-Mori orthogonal, so chi is the squared classical Fisher length of the spectrum and S_w - chi the squared coherent length, bounded below sharply by d_1^2 (d_1 - 1). Every channel is a rational-weighted sum of metric and skewness contractions, with d_2 the only number entering from outside the geometry, and the skewness tensor enters the whole coefficient exactly once: at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term. The paper is explicit about scope. A dedicated section separates what is proved about the ensemble alone, which is everything, from what the code supplies, which is the Schmidt spectrum and the reading of log d_2 as an area term, and from where the gap lies. On a family of Pauli strings satisfying the Knill-Laflamme conditions on a stabilizer code the first-order mode objects vanish identically, so three of the five channels are not small but exactly zero, and the entire Amari-Chentsov contribution goes with them: the mechanism is a property of the ensemble, not of approximate codes in general. The step from the coefficient to a physical reconstruction-error floor requires fixing the backreaction at a chosen fraction of its Page value, a modelling decision that this paper does not make. The limitations section states four further points, including one archived measurement at d_1 = 3, d_2 = 12 which is located at a definite configuration, quantified, and left unadjudicated with the two reasons given. Every closed form is certified. The checks run from the defining index sums, from the raw five-channel sum built from the mode objects on an explicit orthonormal basis, from the geometric identifications, and, beyond the invariant ensemble, against direct simulation on structured families and stabilizer codes. Two of the latter are retrodictions: coefficients measured and archived earlier in this programme, before the present theory existed, are reproduced to +0.4 and 0.0 standard errors, with a previously unexplained connected excess given exactly by the connected-channel formula. The appendix evaluates the eighteen connected contractions on which the connected channel rests, and the deposited code derives all eighteen of them from the Wick rule and then checks symbolically that they sum to the closed form. This paper supersedes and absorbs three earlier preprints of this programme: 10.5281/zenodo.23644208, 10.5281/zenodo.23684111 and 10.5281/zenodo.23726789, together with the contraction appendix of 10.5281/zenodo.23621356. Those records remain available and are not withdrawn; readers should cite the present paper for any of their results, which are proved here in full and without reliance on unrefereed material. Version 3 corrects the presentation of the appendix table of eighteen connected contractions. Up to version 2 the nine word pairs were listed in five grouped rows, on the ground that the relations T_3 = T_1^dagger and conjugation with u <-> v relate the members of each row; those relations preserve a pair's d_2-degree but not its value, and for three pairs the grouped entry is therefore not the pair's own. The table is now listed pair by pair, the collected sum before the kernel contraction is displayed, and the deposited script derives all eighteen entries from the Wick rule rather than taking them as given. No result of the paper changes: the two matchings of a word pair are added together, and sum_uv w_uv p_u = sum_uv w_uv p_v by the symmetry of w, so the collected sum and everything downstream of it are unaffected. Version 3 also corrects the wording of the proof of the entropy-kernel lemma. The first-order term of the expansion of Tr f(sigma) is Tr[Delta log sigma_0] + Tr Delta, and it is removed by the second, linear term of the relative entropy rather than by Tr Delta = 0 as the earlier versions said; the conclusion of the lemma is unchanged and is verified in the deposited code. A fifth script, an independent review pass over the statements the other four do not cover, is added to the archive. The verification scripts that generate every entry of the certification table, together with the derivation and symbolic verification of the appendix table, are included in the attached archive under the MIT licence.
For a region of a free-fermion ground state the entanglement Hamiltonian K_A = log[(1 - C_A) C_A^-1] splits into a part proportional to the identity and a trace-free remainder. The identity part shifts every level rigidly, so every observable built from level differences is blind to it. This paper is about that part. It is not a bulk quantity. The block carries one ladder per entangling cut, the trace is the difference of the two cut-phase charges, and on the exactly dimerised chain with a single cut carrying the whole charge it has the closed form 2 artanh(v / sqrt(v^2 + t^2)), derived in one line and measured to 10^-11. In exact arithmetic the trace is a definite number: zero to thirty-three digits when the two cut phases agree, and 0.786153199019314523 when they do not. The apparent divergence reported in earlier work was double-precision truncation of a finite sum, and is retired here. Along the two lines through the Rice-Mele plane where the half-line entanglement Hamiltonian is known in closed form, it is proportional to the Bisognano-Wichmann deformation T = (1/2){X, H} with no additive constant, and the ladder spacing and phase are fixed by the literature. Between those lines the commuting operator survives, measured at 10^-13 against controls that fail at 1.5 and 29, but the proportionality does not: the level map acquires an intercept b that is zero to 3 x 10^-9 on both lines and of order 10^-1 in the interior, and is identical whether the region has one entangling cut or two. That intercept is the measured failure of exact Bisognano-Wichmann proportionality. At first order in the staggering v the intercept is derived rather than fitted. The symbol's determinant is det(z - P) = z(z - 1) independently of momentum, so the factorisation is canonical everywhere; its logarithm is affine in an involution R = H/E with R^2 = I; the abelian Szego-type boundary functional is identically even in v and therefore cannot carry an odd intercept; and the first-order piece follows from the chiral zero mode of the dimerised chain, whose amplitudes are (-r)^j C(2j,j) 4^-j in closed form. The result is b/v = 2(E' - K')/(1 - r^2) + K', K' = K(1 - r^2), E' = E(1 - r^2), r = t1/t2, whose expansion in the modulus u is, in exact rationals, h(u)/(pi/8) = 1 - (5/8) u^2 - (9/64) u^4 - (65/1024) u^6 - (595/16384) u^8 - ... So pi/8, -5/8 and -9/64 are consequences rather than recognitions, and the natural elliptic candidate [9E - K]/16 - pi/8 is excluded by 7.7 percent on the first coefficient it does not get to choose. The closed form agrees with direct measurement extrapolated to v -> 0 by two to three orders better than the fitted series it replaces; a previously reported u^6 coefficient of -0.0648 is corrected here to -65/1024 and identified as truncation. The trace-free half is treated in parallel. A Clausius relation with a single temperature holds exactly when that half is proportional to the region's own Hamiltonian. At a ground state it never is, and the measured temperature is zero, which is correct rather than a failure. In a thermal state the same instrument returns 1/beta to five significant figures, and the boundary of the region where it does so collapses onto two dimensionless ratios. Two further results are included. The offset vanishes when the two cuts sit at the same phase in the unit cell, without requiring chirality, time reversal, inversion or cleanliness, and a mirror-symmetrised disordered chain gives a larger offset than its unsymmetrised control, which rules reflection out as the mechanism. And in two dimensions the offset is exactly additive along the boundary for a transverse coupling proportional to the identity in sublattice space, for a reason short enough to state, with a derived crossing past which additivity fails and a sigma_z coupling that breaks it smoothly. Every claim carries one of four grades: [T] proven here, [V] computationally verified with stated controls and a stated failure mode, [C] imported from the cited literature, [H] interpretation. The two exact lines, the commuting deformation, the boundary-locality of the entanglement spectrum, and the first law of entanglement entropy are not this work's and are used as given. Sixteen retired claims, including two withdrawn from earlier versions of this manuscript, are listed with the measurement that killed each. Files in this deposit: the_offset_belongs_to_the_boundary_v8_submission.pdf — 35 pages, gate output verbatim, script listings supplied separately. the_offset_belongs_to_the_boundary_v8_complete.pdf — 59 pages, identical body with all scripts printed inline. offset_v8_scripts.zip — 23 files. Every number quoted in the paper is printed by one of them; the two gate files reproduce their stored output byte for byte. KEYWORDS entanglement Hamiltonian; entanglement spectrum; modular Hamiltonian; Bisognano-Wichmann; Rice-Mele model; SSH model; free fermions; boundary charge; cut-phase charge; corner transfer matrix; Toeplitz operators; Wiener-Hopf factorisation; Szego-Widom asymptotics; algebraic Heun operator; complete elliptic integrals; entanglement first law; area law; Landau-Widom counting function ADDITIONAL NOTES Supersedes earlier versions of this manuscript. Two claims present in versions 1 to 3 are withdrawn and recorded in the kill list: that the incommensurate offset has no value, and that the offset is one quantum. Both were double-precision truncation of a finite sum. The u^6 coefficient of the shape function reported in the companion results ledger is likewise superseded by the derived value. Computational verification, gating and drafting assistance by an AI collaborator (Claude, Anthropic). Every gate is independently re-runnable from the ancillary scripts.