Bostelmann, Cadamuro and Minz noted that they presented no proof of convergence of finite-dimensional modular data to the continuum, and that this requires further work; the closest constructive framework, the operator-algebraic scaling-limit programme, likewise leaves modular-data convergence untreated. For half-line free fermions we prove such a theorem for compact-local wavelet schemes, and one delimiting it. Positively, for gauge-invariant quasi-free CAR states, strong exhaustion, strong embedded covariance convergence, and endpoint faithfulness imply norm convergence of local state restrictions and strong-resolvent convergence, after faithful block-completion, of the log-odds modular generators; Trotter–Kato gives local modular-automorphism convergence. We verify these inputs for admissible schemes, including the pinned Daubechies embedding along a dyadic refinement sequence and, under a multiresolution-exhaustion hypothesis, the fixed-order Cohen–Daubechies–Vial boundary model; in the pinned model, local state restrictions on smooth windows away from the endpoint converge at rate O(a^{s}), 0<s ≤ p fixed, those reaching it at rate O(a^{1/2}), the exact order of the driving defect for p ≥ 2 with nonzero endpoint value. Negatively, the locality of this convergence is sharp within a subclass (compactly supported orthonormal-translate interior generator, boundary-free pullback modulo compacts, alias-free, and satisfying the hypotheses of the abstract theorem) for the raw (un-renormalized, infrared-unmodified) fixed-cutoff comparison: a non-exact passband at an interior Fermi-window point makes both Powers–Størmer–Araki square-root defects noncompact, so that comparison supplies no same-folium global normality, whereas exact passband with trace-class boundary error does. Compact support forbids exact passband: the locality yielding local modular convergence obstructs the raw global comparison.
Bostelmann, Cadamuro and Minz noted that they presented no proof of convergence of finite-dimensional modular data to the continuum, and that this requires further work; the closest constructive framework, the operator-algebraic scaling-limit programme, likewise leaves modular-data convergence untreated. For half-line free fermions we prove such a theorem for compact-local wavelet schemes, and one delimiting it. Positively, for gauge-invariant quasi-free CAR states, strong exhaustion, strong embedded covariance convergence, and endpoint faithfulness imply norm convergence of local state restrictions and strong-resolvent convergence, after faithful block-completion, of the log-odds modular generators; Trotter–Kato gives local modular-automorphism convergence. We verify these inputs for admissible schemes, including the pinned Daubechies embedding along a dyadic refinement sequence and, under a multiresolution-exhaustion hypothesis, the fixed-order Cohen–Daubechies–Vial boundary model; in the pinned model, local state restrictions on smooth windows away from the endpoint converge at rate O(a^{s}), 0<s ≤ p fixed, those reaching it at rate O(a^{1/2}), the exact order of the driving defect for p ≥ 2 with nonzero endpoint value. Negatively, the locality of this convergence is sharp within a subclass (compactly supported orthonormal-translate interior generator, boundary-free pullback modulo compacts, alias-free, and satisfying the hypotheses of the abstract theorem) for the raw (un-renormalized, infrared-unmodified) fixed-cutoff comparison: a non-exact passband at an interior Fermi-window point makes both Powers–Størmer–Araki square-root defects noncompact, so that comparison supplies no same-folium global normality, whereas exact passband with trace-class boundary error does. Compact support forbids exact passband: the locality yielding local modular convergence obstructs the raw global comparison.
We develop a spacetime-free reconstruction from a faithful state on a structured family of observable algebras. Compatible half-sided modular inclusions generate positive translations and, after finite homogeneous assembly, a pointed Lorentz cone. Each faithful standard split inclusion determines a canonical Petz--Gram transfer, and we isolate a faithful modular boundary-code condition with three clauses: compact discarded response, a uniquely dominant displacement sector, and complete-order recovery of product defects. Under this condition, overlap transport, and the assembly hypotheses, we prove strict tail contraction, completely bounded two-boundary factorization, exact coherently aligned M2(C) embeddings, selection of three response dimensions, and complex multiplication up to the opposite algebra. Exact finite-dimensional and type-III counterexamples show that none of the three local code clauses follows from faithfulness, split or modular nuclearity, positivity, KMS symmetry, or covariance alone. A genuinely interacting spin-1 AKLT/Haldane chain realizes the logical edge algebra without microscopic qubits, with closed-form transfer bands and completely bounded encoding distortion 1+O(3^{-N}). Relative modular two-jets define the response fields (C,u,Q); their Gaussian covariance dynamics are Lie--Poisson with modular-spectrum Casimirs, and pseudodifferential commutators yield the mass-independent Q^{ij} deformation anchor. Spatial locality, branch smoothness, and two-derivative closure remain explicit conditional inputs.
This record freezes the analytical and computer-assisted successor protocol to the HLV Golden Gauge Forge finite-family study. Previous work established an exact golden 6→3 cut-and-project carrier fingerprint, a nested bounded-locality exhaustion, maximum vertex degree 12, maximum edge–face incidence 10, volume-independent positive HLV coefficient bounds, and a strictly positive physical-sector gap for every fixed finite connected complex satisfying the declared hypotheses. A preregistered exact Z3 finite-family study successfully constructed the controlled carrier sequence but did not support the frozen positive volume-uniform-gap candidate. This negative result motivates a locality-based analytical programme rather than further unrestricted enlargement of finite matrices. The primary system is a fixed-lattice Z3 gauge Hamiltonian on a declared class of finite oriented two-complexes with uniformly bounded vertex degree, face incidence, interaction range, and positive electric and magnetic coefficients. The protocol freezes a sequential theorem programme consisting of: an explicit volume-uniform gap for the electric product-state backbone; derivation of system-size-independent locality and interaction constants; a strong-coupling perturbative stability theorem with a volume-independent spectral gap; exponential clustering of spatially separated gauge-invariant observables; a van-Hove/Følner theorem for the canonical HLV origin-connected exhaustion; and, only if these gates are closed, construction of a fixed-cutoff thermodynamic-limit state. The computer-assisted component freezes an eight-radius geometry audit, graph-boundary layers of thickness one to three, radius-two rooted patch-frequency diagnostics, exact local norm and interaction-overlap calculations, a global exact-diagonalization cap at cycle rank 10, independent numerical verification, and complete deterministic replay. HLV carrier specificity is tested against matched controls consisting of uniform weights on the same complex, 50 exact histogram-preserving type-stratified weight permutations, 12 deterministic internal-window phason shifts, and three rational Fibonacci-slope periodic approximants. The protocol explicitly forbids replacing a local perturbation estimate by the global magnetic operator norm, which grows with system size and cannot establish a bulk spectral gap. It also keeps the thermodynamic limit, representation-cutoff removal, and continuum lattice-spacing limit strictly separate. A positive result under this protocol would establish, at most, a fixed-Z3, fixed-lattice strong-coupling theorem for a class of aperiodic bounded-locality gauge complexes and a separate finite test of whether the HLV geometric assignment produces effects not reproduced by matched controls. It would not establish full SU(3) dynamics, representation-cutoff removal, a continuum Yang–Mills theory, or the Yang–Mills existence and mass-gap problem.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.
This preprint develops the theory of spectral closure systems: an operator-algebraic framework for local access to a global compatibility object through typed, lossy closures, organized as a calculus of admissible sections of access maps. Truth is extendability: a local datum is globally valid exactly when it extends along the inverse system, with finite-stage obstruction certificates by compactness. Recovery of local charts is canonical exactly on the sufficiency sector (Petz recovery, conditioning); off it, every totalization is impotent or pays an exact relative-entropy price. Time is the Radon–Nikodym cocycle of the failure of a nonsingular dynamics to preserve belief, realized as the modular flow of its crossed-product envelope; a nontrivial clock forces a noncommutative envelope, with Murray–von Neumann–Connes type read off the ratio set and the flow of weights as envelope invariant. In the quantum perimeter, point-valued global truth is empty (Kochen–Specker), state-valued compatibility survives (Gleason, Christensen–Yeadon), Bell violations measure the classical gluing defect (Fine), and stable classical recordability is characterized as the commutative, modular-invariant, asymptotically sufficient chart sector; on the reflection-positive sector the record process reconstructs a positive-generator dynamics. Globality is relative but closure is terminal: stage sets collapse inward under directed exhaustion, the envelope tower closes outward at the second step by Takesaki duality, unforced finite-entropy residuals decay at a rate uniform in the modular cocycle (a complete logarithmic Sobolev inequality by transference from the circle), and the forcing cost balances exactly in dilation against system–environment correlation. Further results include an independence theorem for the underlying skeleton, a two-tier classification theorem with provable Foreman–Rudolph–Weiss wildness inside its fibers, an envelope universality theorem (the temporal envelopes are exactly the injective von Neumann algebras carrying a Cartan subalgebra), and a sector-split Orlicz envelope for relative entropy beyond bounded densities. Every external input is a named theorem-import with matched hypotheses; the interpretive vocabulary is proved to be a conservative definitional extension of the formal core.
A non-invertible topological line fixes global defect data but does not select a reduced density matrix (RDM) in a spatially twisted sector. For the Kramers-Wannier defect of the critical Ising chain, we choose the equal-weight incoherent mixture of the two charge-sector ground states and reduce it to the ordinary full spin algebra of a complete non-wrapping prefix anchored at the defect endpoint. The finite-size RDM is the convex average of two charge-sector RDMs rather than a sign-zero Gaussian proxy, and an exact descended antiunitary enforces many-body Kramers pairing without determining the entropy. At fixed prefix, the RDMs converge in trace norm to a common Gaussian state as the circumference grows. Convergence is uniform over all ground-doublet preparations, including coherent pure states. The subsequent large-prefix entropy limit follows from the odd-even Toeplitz/Fisher-Hartwig endpoint, yielding half the natural logarithm of two above the homogeneous chain. The matched spin-flip defect has zero excess. The exact joint energy-modified-translation character of the same twisted Hamiltonian records the finite-size roots and their multiplicities, retains chirality in the marked scaling limit, and, using the standard Ising character identities, resolves the four Virasoro towers of the Ising duality-twisted sector.
We develop Interval Algebraic Quantum Field Theory (IAQFT), a finite-precision operational formulation of algebraic quantum field theory (AQFT). Physical experiments provide only finitely many observations at finite resolution and therefore determine regions of compatible states rather than uniquely specified point states. IAQFT represents this information by quantum parcels---non-empty convex weak∗ open regions of state space---with observational parcels determined by finitely many finite-precision expectation constraints. Exact AQFT states remain indispensable mathematical objects and are recovered under appropriate parcel refinement. We establish reduction to exact states, measurement update and no-signalling, and formulate locality through compatible parcel nets. In finite dimensions an exact Jacobian formula for invertible Lüders updates gives a volume-contraction criterion. Spacelike vacuum correlations and strict CHSH violations persist throughout sufficiently small observational parcels. Reeh--Schlieder cyclicity becomes finite-precision local reachability, while parcel equivalence of vacuum refinements implies vacuum GNS equivalence and hence does not evade Haag-type obstructions. Tomita--Takesaki modular automorphisms act on parcels, measurement update is modularly covariant, and modular conjugation induces a duality between normal parcels of an algebra and its commutant. State-dependent modular correlations are continuous under predual-norm variation of faithful normal states; shrinking parcels with vanishing KMS defect recover a KMS state. Bisognano--Wichmann yields a parcel formulation of the Unruh effect, recovering the exact KMS relation and Unruh temperature under refinement. Finally, limiting projection faces and their conjugation actions recover the Murray--von Neumann factor classification.