We establish a deterministic discrete-to-continuum theorem for free-information fields whose values are finite collections of quantum-register density operators. The local state space is the compact convex configuration space introduced in the companion finite theory, embedded in a finite-dimensional Hilbert space with the Hilbert–Schmidt metric. A mesh field carries two contributions: a finite-range quadratic register-mismatch energy and a local free-information density consisting of an energetic term, the relational-entropy hierarchy, and an optional log-determinant faithfulness barrier. Under explicit assumptions on cell quadrature, finite interaction range, directional second moments, discrete Poincaré compactness, and divergence consistency, the discrete Dirichlet forms Mosco-converge to an anisotropic continuum form. We then prove equicoercivity, the full Gamma-liminf inequality, a recovery theorem that respects the density-matrix constraints and the barrier, convergence of minima and almost minimizers, and mesh universality: any two admissible mesh sequences with the same limiting second-moment tensor have the same continuum functional. Natural Neumann and imposed Dirichlet boundary conditions are separated at the form level. In the convex regime the convergence strengthens to Mosco convergence of the complete functionals and therefore to convergence of resolvents and variational semigroups. We also give sharp failure statements: without a convergent directional tensor there need not be a unique full-sequence limit; the local continuum theorem does not solve the overlapping quantum marginal problem; and static Gamma-convergence alone does not derive a physical open-system dynamics. No random-substrate, Lorentzian, gravitational, gauge, particle-physics, cosmological, or empirical claim is made.
We establish a deterministic discrete-to-continuum theorem for free-information fields whose values are finite collections of quantum-register density operators. The local state space is the compact convex configuration space introduced in the companion finite theory, embedded in a finite-dimensional Hilbert space with the Hilbert–Schmidt metric. A mesh field carries two contributions: a finite-range quadratic register-mismatch energy and a local free-information density consisting of an energetic term, the relational-entropy hierarchy, and an optional log-determinant faithfulness barrier. Under explicit assumptions on cell quadrature, finite interaction range, directional second moments, discrete Poincaré compactness, and divergence consistency, the discrete Dirichlet forms Mosco-converge to an anisotropic continuum form. We then prove equicoercivity, the full Gamma-liminf inequality, a recovery theorem that respects the density-matrix constraints and the barrier, convergence of minima and almost minimizers, and mesh universality: any two admissible mesh sequences with the same limiting second-moment tensor have the same continuum functional. Natural Neumann and imposed Dirichlet boundary conditions are separated at the form level. In the convex regime the convergence strengthens to Mosco convergence of the complete functionals and therefore to convergence of resolvents and variational semigroups. We also give sharp failure statements: without a convergent directional tensor there need not be a unique full-sequence limit; the local continuum theorem does not solve the overlapping quantum marginal problem; and static Gamma-convergence alone does not derive a physical open-system dynamics. No random-substrate, Lorentzian, gravitational, gauge, particle-physics, cosmological, or empirical claim is made.
We develop a self-contained finite-dimensional theory of relational-information functionals on hypergraphs whose hyperedges carry quantum registers. The central object is a weighted hierarchy of total correlations evaluated after local product compression. We prove an exact data-processing inequality under compression-compatible block-local product channels and obtain the corresponding monotonicity along local product GKSL semigroups. For the variational layer we separate the continuous base functional on the compact configuration space from its faithful-state dynamics: an extended-valued log-determinant regularization yields interior minimizers, and every smooth objective with compact sublevel sets generates a unique global Riemannian gradient flow whose omega-limit set is critical; isolated critical points give single-point convergence and nondegenerate minima are exponentially stable. We then study the maximum-entropy decomposition of total correlation. Every maximum-entropy level is locally continuous on faithful states with an explicit modulus, invariant under local unitaries, and nondecreasing under local unital channels, while the sum of the connected-information layers is nonincreasing under all local product channels. Individual layers, however, are not data-processing monotones: a known three-qubit counterexample creates irreducible three-party correlation by a local random-unitary channel. The paper therefore gives a rigorous finite-dimensional foundation and an explicit claim boundary; it does not assume global consistency of overlapping hyperedge states, a continuum limit, or a physical realization of the variational flow.
We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
Abstract This paper develops a discrete information-theoretic framework in which physical systems are modeled on a graph or lattice of locally interacting space cells. Its principal contribution is an integrated architecture separating exact consequences of locality and unitarity, continuum-limit derivations, model-family results, toy-model mechanisms, and unresolved physical mappings. Matter-like identity is treated as propagated relational information rather than persistence of the same occupied cells, and candidate mechanisms are audited for causal locality, conservation or ledger closure, and false reconstruction. The quantum branch assumes finite-dimensional local Hilbert spaces and local unitary or completely positive trace-preserving (CPTP) updates. Under those assumptions, norm preservation, finite causal cones, and reduced-state no-signaling follow directly; spinor quantum walks admit a Dirac-like long-wavelength limit and a secondary nonrelativistic Schrödinger envelope. A representative reciprocal q–χ lattice action shows how compact local dynamics can support continuity, spectral response, action-derived scattering, and internal symmetry/charge, while later extensions supply a provisional local gauge/Gauss/holonomy structure. Two geometric realizations remain deliberately distinct: distortable cells and scalar extrinsic-χ compression. Toy models provide mechanism and failure-boundary evidence for relational identity, reconstruction, scattering, photon-like bookkeeping, and topology/charge-current response. The framework does not derive the Standard Model, quantum field theory, or general relativity; it defines a falsifiable program for testing whether compact local information-processing laws can recover required low-energy structures without hidden nonlocal repair or post-hoc tuning.
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.
Abstract Finite-dimensional local Hilbert spaces are widely used as microscopic regulators and have also been proposed as a fundamental information-capacity property of space-time. A finite local dimension, however, is not equivalent to a spatial lattice spacing, and neither is equivalent to a coarse-graining scale. We formulate a regulator-resolved framework that keeps these three notions separate and defines finite-information deviations operationally as the failure of microscopic and effective time evolution to commute with a specified coarse-graining channel. The resulting deviation is a state- and observable-dependent quantity in a common effective state space, rather than a formal subtraction of generators acting on different domains. We derive a decomposition into capacity, spatial-discretization, and coarse-graining contributions; a finite-time capacity bound controlled by discarded state weight and leakage through the cutoff boundary; observable error bounds; and a sharp continuity bound for subsystem entropies. We further show that block entropy and averaged one-cell entropy differ by a relative-entropy measure of total correlation and therefore cannot be interchanged. An information-geometric expansion of neighboring reduced states yields a nonlinear sigma-model effective action on the manifold of coarse-grained states. A single entropy field follows only under an additional one-dimensional closure condition. Finally, we test the framework in an exactly evolved periodic Klein--Gordon chain with independently variable lattice spacing and local number-state cutoff. Spectral peaks are extracted from dynamical matrix elements rather than imposed. Ground-state, dispersion, reduced-state, and entropy errors converge rapidly with local capacity, while the remaining lattice--continuum discrepancy persists at fixed spacing. The results provide a controlled mathematical and numerical foundation for finite-information models without claiming a unique microscopic theory of quantum gravity.
This report develops the mathematical spine of a discrete information-theoretic space-cell framework. A finite region of the substrate is represented by a tensor product of local Hilbert spaces over a graph or cell complex. Closed dynamics are modeled by finite-depth circuits of bounded-range unitary gates; open-system, measurement-inclusive, or coarse-grained dynamics are represented by local completely positive trace-preserving maps. Under these assumptions, norm or trace preservation is exact, local observables acquire support only inside finite update cones, and unconditional local statistics are invariant under distant trace-preserving operations. A one-dimensional symmetric split-step quantum walk is used as an explicit baseline. Selecting a quasienergy branch of its finite-step update operator defines an effective Hamiltonian. In the long-wavelength, small-step regime, the leading generator has Dirac form, with the cell spacing and update interval fixing an effective propagation speed and the local component-mixing angle fixing a mass scale. The Schrödinger equation is then obtained as a secondary positive-energy, low-momentum envelope approximation after factoring out the rest-energy phase. This hierarchy prevents the discrete update, Dirac continuum regime, and nonrelativistic Schrödinger regime from being treated as equivalent claims. The report also proves reduced-state no-signaling for local trace-preserving operations, states the corresponding probability-level result for local POVMs, and shows how Bell/CHSH correlations can coexist with no controllable superluminal communication. Entanglement is interpreted as distributed nonfactorization generated by local interaction history, not as measurement-time signaling. The report closes with explicit proof obligations: Born-rule and measurement derivation, interacting relativistic fields, spin-statistics, gauge structure, curved geometry, exact or experimentally bounded Lorentz recovery, local conservation beyond global norm, and the causal safety of χ or deformation variables.