A recurring situation in physics is that an observable is a linear functional of an unknown \emph{positive} measure, and the conjugate variable carries an arithmetic structure---it lives on a lattice, on a continuum, on a bounded interval. We show that this alone often decides the physics, and that the verdict flips with the structure. For a locally compact abelian group the L\'evy--Khintchine representation of an infinitely divisible law contains a Gaussian component if and only if the group is not totally disconnected. Hence the principle: \emph{discrete forbids the Gaussian, connected allows it}. We apply it in five places and obtain three qualitatively different kinds of answer. On $\Z$, where the Gaussian is forbidden, the Yang--Mills topological charge distribution is not infinitely divisible---an anomaly, with a model-free inequality $b_2\le-1/12+\langle Q^2\rangle/4$ and a real-time counterpart that puts a floor $\kappa_4/\kappa_2\ge J^2$ under any sphaleron rate. On $\R^d$, where it is allowed and \emph{extremal}, no admissible nonlocal form factor decays faster than a Gaussian. On the lattice $d\Z$, whose dual circle is connected so that the Gaussian survives the reduction, a measured decoherence curve must be a L\'evy exponent, a criterion that is complete, decidable by linear programming, and already sampled by the harmonics of a Talbot--Lau interferometer; three measured points must satisfy $(F_2-F_1)^2\le F_1F_3\le F_1(F_1+2F_2)$, with the walls saturated by a single momentum quantum and by Gaussian noise. On the real line with bounded support, where infinite divisibility is impossible outright, gravitational time-dilation decoherence admits no master equation whenever the internal spectrum is finite. We also report where the machine fails and why: it is blind in semiclassical limits, which closes the holographic R\'enyi line, and four further targets were pre-empted. We state the resulting pre-screen, and record one withdrawn claim in full.
A recurring situation in physics is that an observable is a linear functional of an unknown \emph{positive} measure, and the conjugate variable carries an arithmetic structure---it lives on a lattice, on a continuum, on a bounded interval. We show that this alone often decides the physics, and that the verdict flips with the structure. For a locally compact abelian group the L'evy--Khintchine representation of an infinitely divisible law contains a Gaussian component if and only if the group is not totally disconnected. Hence the principle: \emph{discrete forbids the Gaussian, connected allows it}. We apply it in five places and obtain three qualitatively different kinds of answer. On Z, where the Gaussian is forbidden, the Yang--Mills topological charge distribution is not infinitely divisible---an anomaly, with a model-free inequality b2≤−1/12+⟨Q2⟩/4 and a real-time counterpart that puts a floor κ4/κ2≥J2 under any sphaleron rate. On Rd, where it is allowed and \emph{extremal}, no admissible nonlocal form factor decays faster than a Gaussian. On the lattice dZ, whose dual circle is connected so that the Gaussian survives the reduction, a measured decoherence curve must be a L'evy exponent, a criterion that is complete, decidable by linear programming, and already sampled by the harmonics of a Talbot--Lau interferometer; three measured points must satisfy (F2−F1)2≤F1F3≤F1(F1+2F2), with the walls saturated by a single momentum quantum and by Gaussian noise. On the real line with bounded support, where infinite divisibility is impossible outright, gravitational time-dilation decoherence admits no master equation whenever the internal spectrum is finite. We also report where the machine fails and why: it is blind in semiclassical limits, which closes the holographic R'enyi line, and four further targets were pre-empted. We state the resulting pre-screen, and record one withdrawn claim in full.
Let Lu be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge Q. Averaging eiLuQ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces η(ξ)=D∣ξ∣α (0<α≤2), so coherences decay at Γab=D∣qa−qb∣α. For integer charge differences, the process descends to the wrapped variable Θu=Lumod2π; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when 0<α≤2, while α>2 yields an analytic obstruction. This boundary is exact, verified numerically here.
A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise (α=d/p) and Bochner subordination of Brownian phase by an α/2-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws (α=0.5,1,1.5,2) confirm this law-independence. In continuous time, α-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.
A localization rate F(Δ) is producible by some Markovian translation-covariant completely positive dynamics if and only if it is a continuous negative definite function of the separation---a L\'evy exponent. We close this question for every standard decoherence mechanism, and the two sides close differently. \emph{Compatible, by construction}: every mechanism whose rate is a self-minus-cross pairing with a positive Fourier symbol is a L\'evy exponent with an explicit positive L\'evy measure. Di\'osi--Penrose is one, with measure ∣μ~(k)∣2/k2; so are collisional decoherence, thermal photon scattering, and CSL. \emph{Anomalous}: gravitational time dilation does not couple to position but to internal energy, giving F(Δ)t=−log∣ϕH(gtΔ/ℏc2)∣ with ϕH the characteristic function of the internal energy. This is a L\'evy exponent if and only if that energy distribution is infinitely divisible. An unbounded thermal ladder is, with L\'evy measure cn=qn/n verified to 10−9; \emph{every bounded internal spectrum is not}, so no Markovian translation-covariant completely positive master equation reproduces its separation profile. For a two-level internal system the coefficients alternate, cn=(−1)n+1rn/n with r=p/(1−p); for a thermal ladder truncated at M levels the first negative coefficient sits at n=M exactly, with depth qM. The obstruction is therefore sharp in principle and geometrically soft in practice, and it is invisible in the leading L\'evy coefficients, which reproduce the untruncated answer to six figures.
A gravitationally induced loss of matter coherence is the most-discussed empirical window on the quantum–gravity interface, and a recurring claim is that its shape — most often a short-time Gaussian onset |ρ₀₁(T)| = e^(−½(T/τ_G)²), as in finite-memory variants of the Diósi–Penrose proposal — diagnoses the underlying mechanism (non-Markovianity, state-dependence, or a genuinely quantum-gravitational record process). We prove the opposite, in a sharp and useful form: the gravitational dephasing curve is mechanism-blind, and the structural distinction that would mark a quantum-gravitational record process is not a function of the reduced channel. Three results, each computed at high precision. (i) The Gaussian onset is not a signature: a state-independent, irreversible record commitment ("seal") firing at a ramping hazard λ(t) = a·t reproduces it exactly (|ρ₀₁(T)| = e^(−aT²/2), residual 0), and a constant hazard gives the pure exponential semigroup — the most Markovian channel there is. (ii) Any state-independent irreversible seal is CP-divisible: |ρ₀₁(T)| = e^(−∫₀ᵀλ) is monotone, so it can never produce revivals; revivals occur exactly for Gaussian kernels whose running integral goes negative — the underdamped (still spectrally positive) kernel is the canonical case, with the boundary at ω₀τ = 3.644173671645632… — a non-Markovianity orthogonal to the structural axis. (iii) Operational undecidability from the dephasing curve: a continuous (Ornstein–Uhlenbeck) classical-noise ontology and a matched sparse-seal record ontology produce bit-identical free-induction coherence |ρ₀₁(T)| for all T (the identity ∫λ_OU = χ is sympy-exact; an independent numerical integration gives the machine-zero residual max|C_OU − C_seal| = 7.1×10⁻¹⁰²), so free-induction decay cannot separate them. Multi-time dynamical decoupling (echo/CPMG) can probe reversibility — it distinguishes a refocusable noise from an irreversible commitment — but the genuine quantum-gravitational content, whether the matter's closed-system record process is indivisible in Barandes' sense / unrefinable in the sealed-record sense, is structural (the off-diagonal support of the decoherence functional between sparse commitments) and is invisible to passive reduced-channel measurement, at every order, because a bath dilation is always divisible and a closed unitary system is indivisible yet a trivial channel (an invasive protocol that reconstructs Γ(t) would insert a seal at each conditioning step, and so cannot certify the unintervened process). We make the structural axis precise, exhibit the consistent-histories functor that turns "unrefinable ⟺ indivisible" into a theorem at the kinematic level, and isolate the one open obligation (whether the gravitational record process is sparsely or continuously sealing). The widely used finite-memory Diósi–Penrose-family kernel is, on this analysis, one illustrative CP-divisible member of the undecidable class rather than a discriminating prediction. The practical upshot is a sharpened no-go: a free-induction onset-shape measurement, however precise, does not distinguish gravitational quantum decoherence from classical gravitational noise — the reversibility of the mechanism is probed only by multi-time decoupling, and the structural indivisibility that would mark a genuinely quantum-gravitational record process is reachable by no reduced-channel measurement at all.
Finite Determinacy (FD) was developed from a representational question: whether theuse of real numbers for physical quantities should be read as implying arbitrarily precisephysical determination. Replacing that idealisation by objectively finite determinationleads to bounded shared resolution depth for related records. Close finite-informationand relational antecedents in quantum foundations were identified subsequently and areacknowledged here; they were not the route by which the present framework was developed,and no historical priority is claimed for those principles. This companion paper asks whatdynamical restrictions follow once the resulting finite-record geometry is combined withexplicit finite-rate premises.For self-containment the paper first reproduces the common foundation: bounded jointdepth and split-invariant refinement give exponential resolution and a non-zero two-registerwidth-product floor; a homogeneous, finitely additive area regularity promotes that scalarconstraint to a local area measure; and a primitive tick/readout role asymmetry suppliesorientation, after which the two-member sector is symplectic. The positive symmetric formused for compatible complex structure remains an independent structural input. Dynamicsthen adds a universal pace ceiling rmax, a saturated intra-pair exchange ceiling |dv/du| ≤ L,and an inter-pair causal conversion c∗. In the quantum branch, explicit coherent-recordand record-faithfulness assumptions lead through Weyl–Heisenberg representation theoryto finite-step unitary evolution and the Schrödinger law-form. In the gauge branch, recordgeneratedtransport and holonomy lead, under a minimal local reversible continuum closure,to Yang–Mills curvature-square dynamics. In gravity, locality, reversibility and isotropy givea minimal hyperbolic principal part, while an explicit spatially relational continuum closureleaves two transverse-traceless modes.The proposed contribution is therefore not a new finiteness or relationality principle,nor an exclusive derivation of familiar terminal equations. It is an independently developedfinite-geometric route from the physical interpretation of real-number representation to acommon pre-Hilbert, pre-gauge and pre-metric substrate, followed by sector-specific dynamicalconsistency conditions.
TL;DR — The Core Finding Contrary to the initial expectation that arithmetic structures (e.g., the coprime-to-6 mask or the Möbius mask) generate distinct number-theoretic spectral signatures, this work provides numerical and formally certified evidence that the Non-Ergodic Extended (NEE) phase is entirely governed by the preservation of first-neighbor connectivity (d=1). Suppressing the d=1 hopping channel induces a topological collapse into strong spatial localization (D2≈0.288→0.045, a factor of 6.4), representing a single-particle spatial analogue of Hilbert Space Fragmentation (HSF). Furthermore, the deep arithmetic logic of the Möbius function acts as topological white noise, in compliance with the Möbius Randomness Principle (Sarnak conjecture): since 86.05% of the interaction variance is concentrated in short-range bonds (d≤5), deeper arithmetic channels are swamped by the finite-size GUE noise floor (σGUE∼1/N). Theoretical and Physical Framework Deterministic-stochastic hybrid Hamiltonian. Combines an exact Riemann–von Mangoldt diagonal potential (inverted deterministically via the Lambert WW function) with Power-Law Random Banded Matrix (PRBM) off-diagonal disorder. Sparse hopping constraints. Hopping is filtered by deterministic arithmetic sieves (coprime-to-6 mask Ξ6, square-free Möbius mask Ξμ, Liouville mask Ξλ) and controlled random Bernoulli ensembles. Single-particle Hilbert space fragmentation. Without d=1, adjacent spatial transitions (i→i+1) require higher-order virtual paths (i→i+5→i+1) that suffer severe energy detuning from the fluctuating diagonal potential. The one-dimensional metric graph shatters into decoupled spatial clusters. Coupling variance dominance. The power-law decay (d−1.5 in variance) concentrates 86.05% of the interaction weight in d∈{1,2,3,5}. Deeper arithmetic channels (d≥6) remain buried beneath the finite-size GUE noise floor. Parameter-free operating point. ν=0.75 (exact center of the critical window 0.5<ν<1); local resonance coupling analytically derived as ϵc=π2; Maslov phase 7/8. Mechanized Formal Verification in Lean 4 The algebraic, topological, and connectivity constraints governing the model are certified 100% sorry-free by the Lean 4 kernel (v4.34.1 with Mathlib4), across 13 Level N1 modules (23 theorems, 11 definitions, 34 total verified statements): WheelChannelStructure.lean — Confinement to active chiral sublattices (d≡1,5(mod6)). UnitCellMeasure.lean — Unit cell totient measure (∑d=16Ξ(d)=2 ⟹ φ(6)/6=1/3). ChiralPhaseAbsorption.lean — Invertibility of coprime residues modulo 6. KatoRellichRegime.lean — Self-adjointness domain window (1/2<ν<1). LocalResonanceCoupling.lean — Hybridization resonance condition (ϵc=π2). ChannelDensityContraction.lean — Strict density hierarchy (4/15<1/3<1/2<1). NonErgodicPhaseBound.lean — Non-triviality of the bipartite ceiling (1/4<1). PolyphaseFilterZ6.lean — Complete stopband annihilation for non-coprime channels. FirstNeighborPreservation.lean — Guaranteed base coordination degree (w1=2). DihedralQuantization.lean — Reflection involution and percolation threshold bounds. BipartiteInterferenceBound.lean — Certified upper bound D2≤1/4 under chiral equiprobability. ExpectedDegreeBound.lean — Mean degree lower bound (⟨deg⟩local≥2). FractalDimensionUpperBound.lean — Generalized Jensen inequality for the IPR and universal bound D2≤φ(m)/m. Empirical validation of Lean 4 axioms. Extensive GPU audits up to N=12 000 confirm that the chiral equiprobability axiom (c1=c5=1/2) holds strictly at the ensemble level (⟨W1−W5⟩=−0.000048≈0), with local mass fluctuations decaying monotonically as N→∞. Comprehensive Experimental Validation Matrix Observable Empirical Result Benchmark Physical significance Macroscopic alignment (R2) >0.999999 R2→1 Autonomous Weyl law tracking; zero empirical rescaling Bulk spacing ratio ⟨r⟩ 0.5891±0.0024 GUE 0.5996 / Poisson 0.3863 Intermediate non-ergodic extended statistics SFF ramp exponent γ 0.6086±0.0103 GUE γ=1 Sub-diffusive spectral ramp characteristic of NEE FSS thermodynamic limit D2∞ 0.113±0.003 Anderson D2=0 39σ rejection of Anderson localization Chalker scaling β=2D2 β=0.5549±0.0020 2D2=0.5562±0.0054 0.2σ agreement between transport and fractal support Multifractal curvature ΔD D0−D2=0.730 Ergodic ΔD=0 Strict convexity of Dq for q∈[−3,4]; width Δα=3.726 First-neighbor removal D2=0.0448±0.0009 Preserved: 0.2882 Sixfold collapse into strong localization (HSF) Short-range rescue (d=1) D2=0.2849±0.0007 Unforced: 0.0556 Forcing d=1 restores the NEE phase in random masks Topological pinning (ε=εc) D2no/D2with≈0.17 Healed ratio →1→1 Collapse cannot be overcome at the critical resonance Riemann zeros control (SFF) γzeros=0.94±0.12 GUE γ=1γ=1 Real zeros are ergodic; the NEE ramp is intrinsic to the mask Repository Structure and Reproducibility The codebase is fully vectorized, GPU-accelerated (PyTorch cuSOLVER in complex128), and audited via an automated SHA-256 manifest. The notebook regenerates every figure and table of the manuscript from a fixed random seed (SEED = 42). PRBM_Modular_Lean/ — Standalone Lean 4 (v4.34.1) project containing all 13 verified .lean files and lakefile.lean. Modules 00a–00c — System setup, forensic audit of the 100,000-zero Riemann dataset, and vectorized Hamiltonian engine (complex128). Modules 01–03 — Exact diagonalization at N=10 000; local level statistics P(s) and ⟨r⟩; spectral form factor ensemble averaging (M=100, N=15 000). Modules 04–06 — Channel-density scaling and finite-size scaling; FSS collapse of P(d2,N) across N∈{2000,4000,8000,12000}; AIC/BIC model selection. Modules 07–09 — Anderson transition sweep (ν∈[0.5,3.0]); systematic mask comparison suite; Möbius equivalence and density sweep. Modules 10–12 — Full multifractal spectrum Dq,f(α); microscopic GPU diagonalization at N=12 000; quantum dynamics (MSD, survival probability, Shannon and entanglement entropies). Modules 13–15 — Riemann zeros control SFF; coupling sweep over ε/εc∈[0.2,4.0]; chiral equiprobability audit ⟨Wr⟩ across residue classes. Module 16 — Automated high-resolution vector export (PDF/PNG 300 DPI) and cryptographic manifest generation (MANIFEST_SHA256.txt). Open-Access Dual Licensing Source code (Lean 4, Python, PyTorch): released under the MIT License. Free for academic, educational, and commercial replication. Manuscript, figures, manifests, and data: released under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Declarations Author contributions. Conceptualization, methodology, software, formal analysis, formal verification, investigation, data curation, validation, writing (original draft and revision), visualization, project administration. Competing interests. The author declares no competing interests, financial or otherwise. Funding. No external funding was received for this research. Ethics. Ethical approval was not required, as this research is theoretical and numerical and involves no human participants or animal subjects. Generative AI disclosure. Google Gemini and DeepSeek were used exclusively as auxiliary tools for language editing, adversarial logical review ("red-teaming"), and code debugging. They were not used to produce any original scientific content: all model design, numerical experiments, theoretical derivations, and interpretations are the author's own work. Every AI-assisted output has been reviewed and validated by the author. Scientific Epilogue "In the beginner's mind there are many possibilities, but in the expert's there are few." — Shunyu Suzuki What began as an inquiry into whether number-theoretic structures dictate quantum chaos revealed a simpler, deeper, and more universal physical law: geometry and local connectivity precede arithmetic complexity. The code, formal proofs, and data deposited here are provided to ensure complete auditability, verifiable down to the individual logical inferences of the Lean 4 kernel. Archival snapshot: October 2026 · Certified Level N1 · Built with Lean 4, PyTorch, NumPy, SciPy, and Google Colab. Version: v4 ( submission EPJB-D-26-00793)
Complete positivity and translation covariance force the localization rate F(Δ) of any Markovian open dynamics---environmental decoherence and objective collapse alike---to be a continuous negative definite function of the path separation Δ, i.e.\ a L\'evy exponent. We turn this structural fact into a finite, model-free, and \emph{complete} test on data. Sampling F at commensurate separations Fk≡F(kd) reduces the L\'evy--Khintchine representation to a linear feasibility problem whose dual is a Chebyshev positivity cone; the cone is polyhedral only for two points. Three points already give closed-form walls, (F2−F1)2≤F1F3≤F1(F1+2F2) together with F2≤4F1, saturated respectively by a single momentum quantum and by Gaussian noise---the two extreme rays of the admissible cone. The test detects a 1% super-quadratic admixture from three points, and it is necessary \emph{and} sufficient, so a curve that survives is realized by an actual noise measure. A visibility-versus- separation scan therefore falsifies Markovian translation-covariant evolution with no model and no fitting.