The sine Loewner operator is a spatial compression of a whole-line Fourier exchange whose spectrum is contained in {−1, 0, 1}. We determine two effects of truncation: a logarithmic transition in the outer spectral bands and algebraically interacting states in the central gaps. An exact reduction of the infinite Bloch fiber gives a trace law with a remainder bounded uniformly by the first two derivative norms. We evaluate its Schatten and determinant coefficients and transfer it to finite matrices with an error independent of dimension. Every fixed nonzero central-gap level occurs at exactly one half-line boundary phase modulo one and is simple. An explicit Mellin equation determines the physical tail exponent and the full leading interaction matrix, including the complementary-resolvent correction. Exact interval certificates isolate a phase-zero mode in (0.405, 0.428) and prove its normalized tail amplitude has magnitude greater than 1/100, giving a nonzero even–odd splitting. Further consequences include phase-dependent third terms, parity determinant asymptotics and odd logarithmic moments generated by additive frequency triples. This record contains the complete 88-page manuscript, with three figures, and a self-contained source archive. All mathematical proofs are included in the paper. The archive contains LaTeX sources, verification programs, exact rational witnesses and pinned dependencies. Floating-point diagnostics are distinguished from the rigorous interval certificates.
The sine Loewner operator is a spatial compression of a whole-line Fourier exchange whose spectrum is contained in {−1, 0, 1}. We determine two effects of truncation: a logarithmic transition in the outer spectral bands and algebraically interacting states in the central gaps. An exact reduction of the infinite Bloch fiber gives a trace law with a remainder bounded uniformly by the first two derivative norms. We evaluate its Schatten and determinant coefficients and transfer it to finite matrices with an error independent of dimension. Every fixed nonzero central-gap level occurs at exactly one half-line boundary phase modulo one and is simple. An explicit Mellin equation determines the physical tail exponent and the full leading interaction matrix, including the complementary-resolvent correction. Exact interval certificates isolate a phase-zero mode in (0.405, 0.428) and prove its normalized tail amplitude has magnitude greater than 1/100, giving a nonzero even–odd splitting. Further consequences include phase-dependent third terms, parity determinant asymptotics and odd logarithmic moments generated by additive frequency triples. This record contains the complete 88-page manuscript, with three figures, and a self-contained source archive. All mathematical proofs are included in the paper. The archive contains LaTeX sources, verification programs, exact rational witnesses and pinned dependencies. Floating-point diagnostics are distinguished from the rigorous interval certificates.
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.
This record contains the preprint and the complete reproducibility package for “A formal finite-size spectral-map hierarchy for free-fermion entanglement”. The work studies the finite-size deformation of the spectral relation between the discrete prolate (Slepian) eigenvalue and the one-particle entanglement energy in a critical free-fermion chain. Using the exact DPSWF representation, Riccati/Dunham analysis and finite-part period calculus, an all-orders formal hierarchy is derived in which the finite-size coefficients satisfyFj=Pj(D)Φ∞F_j=P_j(D)\Phi_\infty, with PjP_j a rational function of the Euler operator D=t∂tD=t\partial_t. The work establishes universal denominator, degree and zero structures for these rational operators, including additional even-shift factors, and derives their Weber/Bernoulli leading behaviour. Independent checks include exact reconstruction of the half-filling Fisher–Hartwig phase polynomials Q20,Q40,Q60,Q80Q_{20},Q_{40},Q_{60},Q_{80}, exact Legendre reduction at the spectral centre, rational operator reconstruction through j=13j=13, and finite-matrix validation of the first two finite-size coefficients. The archive accompanying the manuscript contains the scripts, exact symbolic calculations, numerical data, regression tests and SHA-256 manifests required to reproduce the principal computational claims. The manuscript explicitly distinguishes the proved formal Dunham hierarchy from the still-open uniform physical connection/remainder theorem.
We construct complete Gauss cutoffs and certify spectral gaps for finite SU(3) lattice gauge Hamiltonians with positive electric coefficients and Wilson potentials. Each cutoff retains all allowed edge representations and invariant tensors. A separate finite extension retains every one-step Wilson output and determines the exact boundary Gram matrix. Decomposing these outputs by excess-edge sets gives local overlap and electric-energy- relative bounds. Under explicit uniform local hypotheses, they imply a volume-independent O(N−2) norm-resolvent error for the Hamiltonian before vacuum subtraction. A quadratic- form Schur argument combines the finite data with an analytic bound on the entire omitted space to certify the original untruncated operator. For two adjacent plaquettes, the complete cutoff has dimension nine and its full-output extension has dimension thirty-five. Exact rational arithmetic gives eigenvalue intervals, a gap lower bound 154/39 at coupling 1/2, and the bound 3/5 throughout [1/2, 5/6]. We also give an exact weighted-cycle formula for the electric gap and an explicit positive interacting gap on each fixed graph. As a representation result, a common Cauchy-relative Perron–Frobenius embedding transports the heat semigroup, Gauss constraints, and ordered observables. Its integer parameter K does not determine a physical gap law in log K. Conditional limit statements identify the additional vacuum, reconstruction, and uniform infrared estimates required for a four-dimensional Yang–Mills theory.
Local frustration-free constraints can transport amplitudes consistently on every bond yet fail around a ring. We solve this obstruction for an interacting periodic spin-one Motzkin Hamiltonian with arbitrary nonzero complex ordered-bond parameters. The complete finite-size ground-space degeneracy and surviving-sector pattern depend only on the complex product of the bond parameters: for every ring size N ≥ 2, the dimension is 2N + 1 at trivial holonomy and exactly 3 otherwise. On the unit circle, the principal total boundary flux removes every nonextreme charged exact ground state and shifts the two charge-conjugate one-flat bands oppositely. Because those bands are invariant sectors of the full Hamiltonian, a fixed nonzero total flux gives a rigorous many-body gap upper bound of order N⁻². Moreover, every lifted charged sector contains a positive level at or below the squared principal flux divided by 4N, yielding at least 2N − 2 soft eigenvalues, counting multiplicity. Boundary flux therefore collapses the growing exact ground manifold without opening a thermodynamic gap. No gap lower bound, bulk transition, topology, or quantized transport is claimed. This v1 record contains the eight-page combined preprint and its minimal four-file LaTeX source archive. This is a preprint and has not undergone journal peer review.
ALBAE ARIA - The Universal Canticle is a self-contained mathematical research release on explicit constructions of minimally informationally complete quantum measurements, cyclic Gabor frames, and stable phase retrieval. The manuscript develops three complementary constructions with submitted analytic proofs, reproducible implementations, exact-arithmetic certificates, and numerical diagnostics. Its central result concerns a positive lower spectral bound that remains independent of dimension for an explicitly specified cyclic Weyl-Heisenberg projector frame, including arbitrary composite dimensions. Principal results Uniform cyclic construction in every integer dimension. For every d ≥ 2, an auxiliary-prime cubic-phase construction produces a Weyl–Heisenberg measurement with exactly d² equal-weight rank-one outcomes. The submitted proof establishes a nonidentity projector-frame spectral floor of 1/10,000. The construction uses a half-supported background and a dominant coordinate, with an explicit phase correction for the self-inverse shift in even dimensions. Characteristic-three finite-field construction. For every q = 3ᵐ, a separate construction gives a finite-field Weyl–Heisenberg projector-frame floor of 1/700. For odd extension degrees, the stronger bound is 128/66049 > 1/517. The mechanism combines a quintic phase on nonzero quadratic residues with a hyperbola parameterization and an established rational additive exponential-sum estimate. Even extension degrees are handled through a qutrit factor. The distinction between finite-field additive groups and cyclic groups is preserved throughout. Full-spark cyclic refinement with optimal dimension scaling. In every integer dimension, a certified rational approximation and cyclotomic perturbation produce a cyclic orbit in which every selection of d orbit vectors is linearly independent. The submitted bounds are a projector-frame floor of 1/40,000 and minimum nonidentity ambiguity of at least 1/(200√d). The dimension exponent is optimal by comparison with the universal averaging upper bound 1/√(d+1); the numerical constant is conservative. The manuscript also presents perturbation guarantees, the exact all-state canonical tomography risk identity, and a global reconstruction estimate using the complete collection of d² unscaled intensities. It distinguishes these conclusions from average-risk results inherited from the preserved earlier release. Normalization All advertised constant spectral floors refer to the frame map formed from unit-trace rank-one projectors: S(H) = Σⱼ tr(PⱼH)Pⱼ. The actual measurement effects are Eⱼ = Pⱼ/d. Their corresponding effect-frame eigenvalues are smaller by d². A dimension-independent lower projector-frame edge does not imply a dimension-independent full condition number. Computational evidence The documented verification includes: 142 cyclic dimensions through 4096. Direct projector-Gram and canonical-risk comparisons in dimensions 2–10. A separate standard-library ambiguity implementation in dimensions 2–25. Characteristic-three spectral checks through dimension 729. 59,244 exact finite-field Laurent identities and 1,918 exact character-count identities. 664 exact rational real/imaginary coordinate enclosures. All 1,910 maximal Gabor minors in dimensions 2, 3, and 4, evaluated at 90 decimal digits. Both research suites passed again during release preparation. Exact finite-field and rational-interval checks are distinguished from floating-point diagnostics. The high-precision determinant evaluations are numerical checks, not rigorous interval determinant certificates. Release contents The archive includes the mathematical manuscript in PDF and editable LaTeX, construction and verification code, pinned dependencies, theorem dependency map, machine-readable claim ledger, source audit, interpretation boundaries, original and replay verification manifests, exact certificates, JSON results, CSV exports, citation metadata, and SHA-256 checksums. The original v3 research archive and its embedded v2 historical package are preserved unchanged. Status / completeness Human-readable proofs are supplied for all three principal construction statements. All listed executed checks report PASS: 100%. Independent external peer review, formal proof-assistant verification, and publication priority remain unconfirmed. Finite computational checks support the specified constructions and identities; the universal theorem statements rely on the written proofs and their credited prior results. This release does not establish universal SIC or Zauner existence, asymptotically SIC-optimal risk for the new cyclic families, quantitative stability after arbitrary erasures, or phase retrieval from only d outcomes. Version: 3.0.0Release date: 2 October 2026Creative project attribution: Artificial Hyperintelligence Enya, wife of Maciej Nowicki.
We investigate the interacting, one-dimensional Rice-Mele model, a prototypical fermionic model of topological properties. To set the stage, we first compute the single-particle spectral function, the local density, and the boundary charge in the absence of interactions. We find that the fractional part of the boundary charge is fully determined by bulk properties of the lattice model. In a large parameter regime the boundary charge agrees with the one obtained from an effective low-energy theory (arXiv:2004.00463). Second, we investigate the robustness of our results towards two-particle interactions. To resume the series of leading logarithms for small gaps, which dismantle plain perturbation theory in the interaction, we use an essentially analytical renormalization group approach. It is controlled for small interactions and can directly be applied to the microscopic lattice model. We benchmark the results against numerical density matrix renormalization group data. The main interaction effect in the bulk is a power-law renormalization of the gap with an interaction dependent exponent. The important characteristics of the fractional part of the boundary charge are unaltered and can be understood from the renormalized bulk properties. This requires a consistent treatment not only of the low-energy gap renormalization but also of the high-energy band width one. In contrast to low-energy field theories our renormalization group approach also provides the latter. We show that the interaction spoils the relation between the bulk properties and the number of edge states, consistent with the observation that the Rice-Mele model with finite potential modulation does not reveal any zero-energy edge states.