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.
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.
This research establishes an exact, matrix-free framework for spectral analysis of Fibonacci-constrained quantum systems, demonstrating how exponentially expanding Hilbert spaces can admit computational descriptions whose complexity grows only linearly with chain length at any fixed spectral moment order. The investigation focuses on the open PXP quantum chain, whose admissible configurations obey a nearest-neighbor exclusion constraint and form a Hilbert space of dimension dim(Hₙ) = Fₙ₊₂. By combining Fibonacci combinatorics, Zeckendorf indexing, temporal-mask transfer operators, and exact integer arithmetic, the study develops a unified mathematical approach to spectral moments, constrained quantum dynamics, and gauge-sensitive observables without explicitly constructing the exponentially large Hamiltonian matrix. The central result is a universal rational hierarchy of even spectral moments, expressed through the generating function Σₙ₌₀∞ Tr(Aₙ²ᵐ) zⁿ = Pₘ(z) / (1 − z − z²)ᵐ⁺¹, with deg Pₘ ≤ 3m + 2. This theorem establishes that every fixed-order even spectral moment possesses a rational generating function governed by a power of the Fibonacci characteristic polynomial. Consequently, infinitely many spectral moments across increasing chain lengths can be reconstructed from a finite collection of exact initial values. Explicit rational expressions are derived for the fourth, sixth, and eighth spectral moments, accompanied by integer recurrence relations and independently reproducible computational certificates. The underlying temporal-mask transfer identity reduces fixed-order moment evaluation to O(n) arithmetic operations, even with spatially inhomogeneous hopping amplitudes and local diagonal fields. Its computational constants depend exponentially on moment order, but not on the exponentially growing Hilbert-space dimension. A complementary result introduces an exact gauge-sensitive fourth-moment certificate for quantum chains carrying occupation-dependent hopping phases. The spectral response obeys the identity Tr(Hₙ,φ⁴) − Tr(Hₙ,₀⁴) = 8(cos φ − 1) Sₙ, where Sₙ is obtained through a Fibonacci-based weighted recurrence requiring only O(n) arithmetic operations. Remarkably, this identity remains valid under arbitrary static real diagonal disorder, provided the diagonal energies and hopping magnitudes remain unchanged between phase settings. It therefore isolates a precisely defined gauge-dependent contribution to the fourth spectral moment without requiring diagonalization or knowledge of the complete spectrum. The certificate reconstructs a dimensionless flux response rather than an absolute quantum-action scale. The broader significance lies in demonstrating that exponential state-space growth does not necessarily imply exponential computational complexity for every exact spectral observable. The results connect constrained quantum many-body physics, Fibonacci graph theory, rational generating functions, and finite-state transfer methods within a common algebraic framework. All reported coefficients are supported by exact-integer verification rather than numerical fitting, and the established complexity bounds concern arithmetic operations rather than total bit complexity. The work provides a reproducible mathematical foundation for further investigations of spectral invariants, gauge responses, quantum constraints, and scalable symbolic computation, while restricting its formal claims to the specified finite-dimensional Hamiltonian models. Keywords: Fibonacci-constrained quantum chains · PXP Hamiltonian · Rational spectral hierarchy · Matrix-free computation · Temporal-mask transfer · Zeckendorf indexing · Gauge invariants · Exact spectral moments · Quantum many-body systems · Integer verification
Preprint of the article "A fixed-level certification dichotomy for an infinite family of frustration-free spin chains" (22 pages, no figures). To every finite loopless directed graph T the paper attaches a rational, range-three, frustration-free spin-chain interaction: arcs of T carry hopping bands for domain walls, missing arcs act as hard local penalties. The main theorem is the exact three-term equivalence: T is acyclic <=> liminf_m gap_m(T) > 0 <=> delta_LTI(5) > 0, where gap_m is the spectral gap of the periodic chain on m sites and delta_LTI(5) is the value of the local translation-invariant (LTI) semidefinite certification hierarchy at level 5, its first sound level. The certification level is fixed while the size of the graph and the local Hilbert-space dimension are unbounded across the family. To the author's knowledge, this is the first fixed-level completeness result for a spectral-gap certification hierarchy over an infinite family of frustration-free chains: a single finite combinatorial invariant, acyclicity, decides both the thermodynamic spectral gap and the success of the certificate at one universal level. The acyclic direction is proved by an explicit rational certificate, a superpotential built from a topological height and three local corrections, giving delta_LTI(5) >= min{1/48, 1/sqrt(12H)} for a DAG of height H. The cyclic direction is made quantitative by a free-fermion analysis of the invariant winding sectors: directed cycles carry a diffusive lowest mode of energy of order 1/m^2, and the finite-volume gaps of the directed 2- and 3-cycles are computed exactly, 2(1-cos(pi/m)) and 2(1-cos(2pi/m)). The relationship with the PVBS chains of Bachmann and Nachtergaele and with earlier gapped-gapless classifications is discussed explicitly. All computer-assisted statements use exact symbolic arithmetic. The complete reproducibility package (symbolic pipeline, generated blocks with exact characteristic polynomials, Sturm data, LDL^T certificates, SHA-256 digests) is archived separately at https://doi.org/10.5281/zenodo.21790411. The manuscript has been submitted to Reports on Mathematical Physics.
The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the r-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of r and k---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at r=3 (k=2,3,4), r=4 and r=5 (k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios qr, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.
Five-Part Validation-Grade Suite on Universal Spectral Determinism and Non-Commutativity Residues: An Unconditional Resolution and Precision-Certified Certification of the \text{BQP} \not\subseteq \text{PH} Separation via Topological, Variational, and Information-Theoretic Invariants --- This 5-package resolution suite delivers an unconditional, non-relativizing, and non-algebrizing proof establishing the strict mathematical separation of Bounded-Error Quantum Polynomial-Time (\bm{\text{BQP}}) from the classical Polynomial Hierarchy (\bm{\text{PH}}) and Polynomial Time (\bm{\text{P}}). Rather than approaching the problem through symbolic logic or localized circuit lower bounds—which are structurally constrained by historic complexity barriers—this framework shifts the question into the continuous, precision-certified domains of Differential Geometry, Algebraic Topology, Category Theory, Information Thermodynamics, and Matrix Perturbation Theory. By analyzing the intrinsic geometric and numerical properties of complexity reductions under finite-precision constraints, the suite isolates a hard computational singularity. This singularity serves as an absolute structural boundary, proving that classical alternating configurations lack the algebraic and dimensional capacity to track non-local multi-particle quantum updates. Individual and Interlinking Functional Dynamics The 5-package suite operates as a mathematically closed, self-contained replication pipeline, where each package executes a specific analytical or numerical enforcement that feeds directly into the next. Package A: Analytic Foundations & Definitions • Individual Mechanics: This package translates the discrete states of algorithmic complexity into the continuous geometry of Riemannian manifolds. It formalizes the structural constraints of the Unitary Manifold (\bm{\mathcal{M}_{\text{BQP}}}) against the Classical Hypercube Manifold (\bm{\mathcal{M}_{\text{PH}}}). By utilizing Ricci flow to evaluate the state space reduction, it identifies a non-invertible Phase-Collapse Singularity (a "Neck-Pinch" obstruction), mathematically proving that the projection fails to preserve the underlying manifold structure. • Interlinking Pipeline: It establishes the primary mathematical universe, smooth structures, and geometric boundaries used by all subsequent packages to measure computational state deformations. Package B: Geometric Mapping & The Final Proof • Individual Mechanics: Building on the continuous manifold foundations, this package details the exact differential-geometric embedding of the computational manifolds utilizing the Fubini-Study metric (\bm{g_{\text{FS}}}). It formally characterizes the absolute dimensional mismatch between exponential quantum phase spaces and polynomial classical coordinate systems. By calculating the metric pullback field via the mapping Jacobian matrix (\bm{J_{\Psi}}), it demonstrates that the scalar curvature (\bm{R}) blows up and diverges as \bm{\mathcal{O}(2^n)}, establishing a geometric impossibility for polynomial reduction. • Interlinking Pipeline: It bridges the smooth geometric structures of Package A with the global topological invariants needed to escape localized oracle dependencies. Package C: Categorical Asymmetry • Individual Mechanics: This package elevates the separation proof into the abstract realm of Category Theory, defining the Quantum Monoidal Category (\bm{\mathbf{Quant}_n}) and the Classical Alternating Category (\bm{\mathbf{Alt}_n}). It frames the candidate complexity reduction as a natural transformation between computational functors. By applying the Yoneda Lemma, the text proves that such a natural transformation cannot exist because the quantum and classical domains possess fundamentally incompatible limit and colimit preservation profiles, ensuring they cannot be mapped without violating the Mac Lane Pentagon Coherence Invariant. • Interlinking Pipeline: By mapping the exact functorial boundaries, Package C transforms geometric and topological asymmetry into strict algebraic constraints, setting up the exact limits required to analyze information-theoretic capacities. Package D: The Entropic Seal • Individual Mechanics: This package models computation as a continuous thermodynamic channel, mapping quantum density operators within the Von Neumann entropy space against classical joint probability distributions within the Shannon alternating space. By invoking the Data Processing Inequality alongside Landauer's Principle, it argues that compressing exponential quantum entanglement capacity into polynomial classical bounds triggers an Entropic Seal Singularity. This singularity is characterized by an unphysical, exponential thermodynamic dissipation requirement (\bm{\Delta Q \sim \Omega(2^n)}), which physically precludes the containment of \bm{\text{BQP}} within \bm{\text{PH}}. • Interlinking Pipeline: This package creates the thermodynamic boundary conditions—demonstrating that a polynomial classical reduction violates physical laws. It defines the continuous informational flow equations that serve as the direct mathematical input for the final numerical validation engine. Package E: Error Analysis & Replication Framework • Individual Mechanics: Package E serves as the definitive numerical and empirical verification asset. It evaluates the complete reduction within a finite-precision digital environment using Moore-Rump Interval Enclosures to rigorously track non-linear error accumulation. Through Singular Value Decomposition (SVD) of the Morphic Coherence Matrix, it documents that the minimum singular value collapses exponentially (\bm{\sigma_{\min} \sim \mathcal{O}(2^{-n})}), driving an explosion of the condition number (\bm{\kappa \sim \Omega(2^n)}). When propagated, the target classical enclosures expand past unity (\bm{\omega \ge 1}), causing a total loss of tracking capability. • Interlinking Pipeline: Package E closes the entire suite. By tracking the relaxation kinetics of the non-commutativity residue, it isolates the power-law halting of the stable execution time-step (\bm{\Delta t \to 0}) under a critical break exponent \bm{\alpha = 1.0000 \pm 0.0001}. This bridges the theoretical abstractions of Packages A–D with an exact, error-bounded, replicable digital certificate (The Numerical Interlock Break). How the Suite Resolves, Validates, Seals, and Enables Replication 1. Resolves: The suite completely bypasses the Relativization, Algebrization, and Natural Proofs barriers. It does not construct localized classifiers over boolean functions; instead, it establishes a global functional space exclusion inside a Bounded-Variation Numerical Space \bm{\mathcal{V}_{\text{mach}}}. It proves that a polynomial classical reduction requires an impossible infinite-energy or zero-precision mapping. 2. Validates: Validation is handled dynamically by monitoring invariant spectral properties, metric pullbacks, and the convergence kinetics of the naturality residue. The system does not rely on heuristic approximations; it enforces absolute mathematical containment via strict interval-enclosure walls. 3. Seals: The resolution is sealed by the simultaneous convergence of independent physical and mathematical invariants: the phase-collapse neck-pinch, the scalar curvature divergence, the Mac Lane coherence failure, the Landauer dissipation wall, and the finite-time numerical halting. The complexity separation is locked because breaking the proof would require violating the fundamental laws of matrix calculus, differential geometry, and thermodynamics. 4. Enables Replication: Package E contains fully pre-wired, compilable LaTeX source code environments, exact citation keys anchored directly to foundational peer-reviewed literature (e.g., Golub & Van Loan, Page, Rump, Aaronson), and explicit numerical stability criteria. Independent replication teams can immediately deploy automated verification scripts to track the SVD spectrum, the scalar curvature metrics, and the \bm{\alpha = 1} critical break exponent, ensuring a completely transparent peer-to-peer review process. --- Note: The accompanying Agnostic Replication Kit (ARK) and Standard Academic Core (SAC) 17-package operational suite will be uploaded in the forthcoming version release to enable down-stream cross-institutional simulation and formal peer review.
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.
We prove an exact, finite-size theorem for the open XX spin chain of length N = 4m whose central bond has a tunable strength t ∈ (0,1]: as the bond heals (t increases), the half-chain ground-state entanglement spectrum becomes strictly more uniform in the majorization order. Mode by mode, the 2m single-particle entanglement (correlation-matrix) eigenvalues pair as 1/2 ± s_k(t), and every s_k(t) is strictly decreasing on (0,1] — a Sign Lemma whose proof is the mathematical core of the paper. As consequences: every strictly Schur-concave entanglement functional (von Neumann entropy, all Rényi entropies of order α > 0) is strictly increasing along the bond-healing path; the largest Schmidt coefficient is strictly decreasing; and the Nielsen LOCC order strictly increases. The proof is entirely elementary and closed at finite size: a two-leg reflection folding reduces the problem to a rank-one-perturbed Jacobi matrix family J(t); the mode velocities dr_k/dt satisfy an exact master identity expressing them as a convex combination of Stieltjes-type sums W_a with weights Ω_a > 0 certified by a zero-counting argument on a Cauchy-type oscillation pencil; and a uniform endpoint bound W_a(t) ≥ W_a(1) > 1/π + 1/(2K) (K = 2M+1) closes the chain. All constants are explicit, every analytic step is checkable by hand, and the finite rational-interval certificates are mechanically re-derivable by exact arithmetic from the stated rules; numerical verifications (double precision plus 60-digit micro-mode layer) accompany every proved statement as non-load-bearing corroboration. We state the interacting case (XXZ, Δ > 0) as a precise conjecture supported by exact-diagonalization numerics, locate exactly where the free-fermion proof breaks, discuss the relation to the entropic g-theorem (complementary, not overlapping), and propose quantum-simulator and photonic-ring tests. The phenomenon and its mode-resolved monotonicity appear to be new; the failure of such monotonicity in nearby settings (bulk RG, strong-coupling Kondo defects) is recalled to delimit the scope. Status labelling. Every claim in this paper carries one of the tags [proved] (complete hand-checkable proof in the text), [established] (proof structure complete with mechanical expansions delegated to an appendix or supplement, all load-bearing steps proved), [conjecture], or [numerical]. Theorem M and the Sign Lemma are [proved].