Quantum phenomenology from a classical integer automaton: an emergent Schrödinger sector, a measurable collapse law, exact exchange statistics, CHSH violation without randomness, and a surviving isotropy no-go
A single bit-exactly reversible, integer-valued lattice automaton in which a broad range of quantum phenomenology arises as measurable, checksum-reproducible behaviour of one classical substrate. A second-order Frenkel–Kontorova / sine-Gordon integer automaton supports localized particles with additive mass towers (1 : 2 : 3 and 1 : 4 : 9), emergent Lorentz contraction (fitted c to 0.3 %), and a token-accounted local-time sector with a general-relativity-like time well (ω ∝ n to 0.06 %). Within it, a first-order Visscher–Feistel integer Schrödinger sector is realized in pure integer arithmetic with exact bit-reversibility. The integer/floor-reversible lifting of the Visscher scheme was already noted by Fredkin (1999) and Martin-Delgado (2004); the contribution here is its first computationally tested and validated realization — against exact dispersion, group velocity and envelope reduction — embedded in an interacting substrate. Main results: Emergence bridge. The narrow-band envelope of the classical automaton obeys the Schrödinger equation with an effective mass fixed by the curvature of the exact discrete dispersion (group velocity to ≤0.2 %, curvature to ≤2 %). The dimensionless spreading of two microscopically different substrates collapses onto the single universal law √(1+τ²). Deterministic quantum statistics. Bohmian tracers reproduce double-slit single-object interference (p < 10⁻⁴), with trajectories matching the weak-measurement reconstruction of Kocsis et al. (2011). Bell / CHSH. A value of S = 2.79 from position-valued outcomes with a no-signalling control (0.3 %), via explicitly nonlocal guidance in configuration space. Measurement. A measurable Gaussian collapse law (R² = 0.997); a quantum eraser with literal bit-reversible un-measurement of the entire measurement chain; Pauli exclusion as a bit-conserved invariant; and relaxation to the Born rule with a reversible arrow of time. Surviving the isotropy no-go, and a falsifiable prediction (new in v2). On the live integer engine, velocity anisotropy vanishes as k² in the infrared with first-principles coefficients in both 2D and 3D (agreement ≈ 1 %) — rotational isotropy emerges as an infrared fixed point, meeting the sharpest no-go every lattice candidate faces. From the same structure follows a concrete, quadratically Planck-suppressed dispersion prediction (δγ,2 ≈ 2×10−40 GeV−2, subluminal), lying only ~50× below the tightest ultra-high-energy cosmic-ray bound — the programme's first falsifiable departure from standard QM/GR. Individual phenomena have known precedents — double-slit Bohmian trajectories (Philippidis, Dewdney & Hiley 1979), Born-rule relaxation (Valentini & Westman 2005), the Schrödinger update scheme (Visscher 1991; integer lifting Fredkin 1999, Martin-Delgado 2004) — and are framed accordingly. The contribution is their tested, validated realization on one exactly reproducible integer substrate, several genuinely new exact constructions (tested integer bit-reversible Schrödinger sector; literal bit-exact un-measurement; exchange statistics as a bit-conserved invariant; two-substrate universal collapse; measured lattice isotropy with a Planck-scale dispersion prediction), and a pre-registered, falsification-first methodology (criteria committed to version control before each run; fourteen self-retractions on record). This is an existence result, strengthened by a survived no-go and a falsifiable prediction — not a claim that our universe is this automaton (that step requires a prediction to pass experiment): a fully classical, deterministic, reversible integer computation can carry quantum phenomenology, quantitatively and reproducibly, and this substrate meets a no-go every discrete candidate must survive. Preprint v2 (self-contained; bibliographic audit completed against arXiv/DOI sources). Code, pre-registered gates, per-run reports and figures are included in this archive. Produced in an AI-assisted workflow (Anthropic Claude) under the author's direction; all pre-registration and interpretation is on record in the version history.
A single bit-exactly reversible, integer-valued lattice automaton in which a broad range of quantum phenomenology arises as measurable, checksum-reproducible behaviour of one classical substrate. A second-order Frenkel–Kontorova / sine-Gordon integer automaton supports localized particles with additive mass towers (1 : 2 : 3 and 1 : 4 : 9), emergent Lorentz contraction (fitted c to 0.3 %), and a token-accounted local-time sector with a general-relativity-like time well (ω ∝ n to 0.06 %). Within it, a first-order Visscher–Feistel integer Schrödinger sector is realized in pure integer arithmetic with exact bit-reversibility. The integer/floor-reversible lifting of the Visscher scheme was already noted by Fredkin (1999) and Martin-Delgado (2004); the contribution here is its first computationally tested and validated realization — against exact dispersion, group velocity and envelope reduction — embedded in an interacting substrate. Main results: Emergence bridge. The narrow-band envelope of the classical automaton obeys the Schrödinger equation with an effective mass fixed by the curvature of the exact discrete dispersion (group velocity to ≤0.2 %, curvature to ≤2 %). The dimensionless spreading of two microscopically different substrates collapses onto the single universal law √(1+τ²). Deterministic quantum statistics. Bohmian tracers reproduce double-slit single-object interference (p < 10⁻⁴), with trajectories matching the weak-measurement reconstruction of Kocsis et al. (2011). Bell / CHSH. A value of S = 2.79 from position-valued outcomes with a no-signalling control (0.3 %), via explicitly nonlocal guidance in configuration space. Measurement. A measurable Gaussian collapse law (R² = 0.997); a quantum eraser with literal bit-reversible un-measurement of the entire measurement chain; Pauli exclusion as a bit-conserved invariant; and relaxation to the Born rule with a reversible arrow of time. Surviving the isotropy no-go, and a falsifiable prediction (new in v2). On the live integer engine, velocity anisotropy vanishes as k² in the infrared with first-principles coefficients in both 2D and 3D (agreement ≈ 1 %) — rotational isotropy emerges as an infrared fixed point, meeting the sharpest no-go every lattice candidate faces. From the same structure follows a concrete, quadratically Planck-suppressed dispersion prediction (δγ,2 ≈ 2×10−40 GeV−2, subluminal), lying only ~50× below the tightest ultra-high-energy cosmic-ray bound — the programme's first falsifiable departure from standard QM/GR. Individual phenomena have known precedents — double-slit Bohmian trajectories (Philippidis, Dewdney & Hiley 1979), Born-rule relaxation (Valentini & Westman 2005), the Schrödinger update scheme (Visscher 1991; integer lifting Fredkin 1999, Martin-Delgado 2004) — and are framed accordingly. The contribution is their tested, validated realization on one exactly reproducible integer substrate, several genuinely new exact constructions (tested integer bit-reversible Schrödinger sector; literal bit-exact un-measurement; exchange statistics as a bit-conserved invariant; two-substrate universal collapse; measured lattice isotropy with a Planck-scale dispersion prediction), and a pre-registered, falsification-first methodology (criteria committed to version control before each run; fourteen self-retractions on record). This is an existence result, strengthened by a survived no-go and a falsifiable prediction — not a claim that our universe is this automaton (that step requires a prediction to pass experiment): a fully classical, deterministic, reversible integer computation can carry quantum phenomenology, quantitatively and reproducibly, and this substrate meets a no-go every discrete candidate must survive. Preprint v2 (self-contained; bibliographic audit completed against arXiv/DOI sources). Code, pre-registered gates, per-run reports and figures are included in this archive. Produced in an AI-assisted workflow (Anthropic Claude) under the author's direction; all pre-registration and interpretation is on record in the version history.
In a finite, timeless configuration space where probability is counting and the population of an outcome class is proved to equal N|α|², we construct a refinement chain from local devices—nearest-neighbour beamsplitters and per-cell phase increments, all on the 3,600-value phase lattice of the companion models—and evolve wavepacket classes by literal enumeration. The small-angle brickwork realizes the lattice Hamiltonian H = −2η cos k + V(x): the effective mass m* = 1/(2η) is the inverse hopping rate—mass as resistance to record restructuring, obtained from the move set rather than posited—and the potential is a cell-dependent phase cost per closure step. Counted populations reproduce the Schrödinger spreading law σ²(t) = σ₀² + (t/2m*σ₀)², the group velocity to seven significant figures against the exact device chain, and harmonic oscillation at ω = √(2ηκ), with Σₓ nₓ = N exact at every stage—the counting form of unitarity, tested exactly. Two findings go beyond agreement. First, the counted packet has a literal edge: populations fall to one configuration and then to zero exactly where the Born value crosses 1/N, so a counting world's particle has finite support where the continuum Gaussian merely thins—the rational-probability prediction rendered in position space. Second, the ontology's record-relative definition of classes is load-bearing for dynamics: counting once at the measurement stage (the chain as one composite device) yields residues of O(1/N), measured log–log slope −0.94, whereas imputing an integer census to unrecorded intermediate stages—partitions where no records are—degrades the scaling to slope −0.62 through the amplitude granularity of few-configuration tail classes. A definitional clause changes dynamical accuracy by a factor of order √N. What is not shown is the formal continuum limit: given [CS] the discrete effective dynamics is derived, with the same conditional status as the companion theorems, and the passage to the continuum equation remains the framework's open problem. The cell structure is a named input [CS]—but not a foreign one: its three clauses are the discrete shadow of the framework's own emergent-space program, and we locate each, finding that a preferred partition is what the stability criterion already selects, that adjacency is what the elementary refinement move set already supplies, and that only the graph's large-scale regularity—together with an assumed uniformity of hopping rate, which is a flatness assumption—remains open. Reading the cells as emergent space has an immediate consequence: the construction's maximal group velocity is one instance of the framework's conjectured invariant speed, computed rather than posited, since a single elementary move defines both the refinement step and the spatial adjacency. Relaxing uniformity—a position-dependent hopping rate, hence a position-dependent effective mass—is the discrete form of curvature, and is the natural next model.
This framework preserves the empirical success of existing physical theories while proposing an alternative set of foundational assumptions from which their standard formulations emerge as effective limits. Abstract — Key Innovative Points Discrete Updating FabricThe theory postulates a fundamental discrete fabric whose elementary constituents evolve through local update processes, replacing continuous fields and background spacetime as primary entities. Irreversible Local Dynamics as FundamentalUnlike standard approaches based on unitary or reversible evolution, irreversibility is taken as ontological. Physical law emerges from irreversible local updates rather than from imposed symmetries or variational principles. Emergent Time and SpacetimeTime is not assumed as a fundamental parameter but arises from the counting of update events. Spacetime geometry emerges as a coarse-grained description of update activity across the fabric. Quantum Probability from Microstate CountingQuantum probabilities, including the Born rule, emerge from combinatorial counting of compatible microstates of the fabric, rather than being postulated axiomatically. Gravity as Update-Rate ModulationGravitational phenomena are interpreted as variations in local update rates, leading to effective curvature, time dilation, and relativistic behavior at macroscopic scales. Mass as Resistance to UpdateMass and inertia arise from stable topological structures within the fabric that resist updates, providing a dynamical origin of mass without introducing fundamental mass parameters. Fundamental Dissipation and IrreversibilityDissipation is intrinsic to the fabric dynamics and not an emergent statistical effect. The arrow of time follows directly from the irreversible nature of local updates. Extension Beyond Fundamental PhysicsThe same discrete framework accounts for macroscopic dissipative phenomena—such as friction, stick–slip dynamics, superlubricity, growth, and self-organization—linking fundamental physics and material behavior within a single process-based model. iversality Without Fundamental SymmetryLarge-scale physical laws emerge as dynamical attractors of the updating fabric, arising through elimination of unstable configurations rather than enforcement of fundamental symmetries. Emergent Rules Beyond the Born Rule in the Unified Fabric Framework The same discrete-fabric + local-update + coarse-graining structure generates multiple quantum rules without postulation. 1. Schrödinger Dynamics as an Emergent Long-Wavelength Limit Local deterministic updates on a discrete fabric admit spectral modes.In the long-wavelength limit, their evolution converges to a Schrödinger-type equation. No quantum dynamics is postulated.Only update rules and spectral structure are assumed. 2. Superposition from Coarse-Grained Microstate Counting Macroscopic amplitudes arise as sums over compatible microstates.Linearity follows from additive contributions at the amplitude level. Superposition is therefore emergent.It reflects coarse-graining, not a fundamental axiom. 3. Interference as Cross-Terms in Microstate Combinatorics Probabilities are quadratic because amplitudes sum before squaring.Cross-terms correspond exactly to interference effects. Interference is not assumed.It is a combinatorial consequence of microstate superposition. 4. Tensor Product Structure and the Origin of Entanglement Composite systems correspond to composite regions of the fabric.Global microstates factorize only if compatibility constraints separate. Entanglement arises when constraints couple subsystems.It is structural, not mysterious. 5. Decoherence from Phase Inaccessibility Observers access only coarse-grained variables.Rapidly varying microstate phases average out. Off-diagonal terms are dynamically suppressed.Decoherence emerges without postulation. 6. Measurement Collapse as Constraint Reduction Measurement is an interaction that reduces compatible microstates.The effective state updates accordingly. Collapse is not a new law.It is constraint updating under coarse-graining. 7. The Arrow of Time from Asymmetric State Counting Entropy is defined as the logarithm of compatible microstates.Local updates generically move toward larger compatible sets. Irreversibility is structural.It does not rely on subjective ignorance. 8. Coulomb Friction from Discrete Event Statistics Friction equals dissipated update energy per unit distance.The Coulomb law emerges from counting active interfacial events. The friction coefficient is not fundamental.It decomposes into microscopic event parameters. 9. Superlubricity as the Event-Free Limit Superlubricity occurs when locking events are suppressed.Dissipation is rare, not forbidden. This is a limit regime, not an exception.It confirms friction as emergent. 11. Conservation Laws from Update Invariance If the update rule is invariant under local transformations,conserved quantities emerge automatically. Noether’s theorem becomes emergent.Symmetry is a property of the update, not an axiom. 12. Classical Determinism from Phase Averaging At large scales, phase information becomes inaccessible.Only expectation values survive. Classical trajectories emerge statistically.Determinism is an effective limit. 13. Locality from Finite Update Propagation Updates propagate only through neighboring fabric nodes.Information speed is bounded. Causality is structural.Nonlocality is only apparent at coarse scales. 14. Emergent Speed Limit (c) The maximum update rate defines a universal speed.No signal can exceed it. The speed of light is not fundamental.It is a fabric update bound. 15. Metric Geometry from Update Delays Spatial curvature corresponds to spatially varying update rates.Time dilation is an update slowdown. Geometry is dynamical.The metric is emergent. 16. Gravity from Update Resistance Mass corresponds to structures resisting updates.Nearby updates slow down in response. Gravitational attraction emerges.No force field is postulated. 17. Equivalence Principle as a Structural Identity All stable update-resisting structures slow updates identically.Their response is universal. Inertial and gravitational mass coincide.Equivalence is automatic. 18. Quantization from Discrete Spectrum A finite update lattice admits discrete eigenmodes.Energy levels are quantized. Quantization is not imposed.Continuum is an approximation. 19. Noise Spectra (1/f) from Avalanche Dynamics Update cascades produce scale-free event statistics.Long-range correlations appear. 1/f noise is generic.It signals discrete underlying dynamics. 20. Criticality without Tuning The fabric naturally operates near instability thresholds.Small perturbations trigger cascades. Self-organized criticality emerges.No parameter tuning is required. 21. Thermodynamic Laws as Counting Theorems Temperature measures update activity density.Heat is disordered update flow. Thermodynamics is not independent.It is bookkeeping of updates. 22. Growth and Morphogenesis from Update Bias Directional update biases produce structured growth.Forms emerge without templates. Morphogenesis is dynamical.Shape follows update flow. 23. Fine-structure value and other unknow meaning The fine-structure value (~137) corresponds to the effective degeneracyof coherently contributing update configurations in an elementary coupling process.Proton–electron mass ratio the number of internal update cycles required to stabilize a topological vortex (proton), and a simple oscillatory update mode (electron). Mass measures cumulative resistance to updates. Lepton mass hierarchy Different stability thresholds of the same update pattern. Each mass corresponds to a different coherence regime.→ Families are metastable realizations of one structure. Reduced Planck constant Energy × time associated with one elementary update event. → ħ is the discrete unit of updating. Speed of light Maximum update propagation speed across the fabric. → c is the causal limit of the fabric. Bekenstein–Hawking entropy Number of blocked or saturated update configurations at the horizon. → An event horizon is an informational saturation boundary. Universal critical exponents They arise from the topology of update avalanches,independent of microscopic details. → Criticality is a natural fabric regime. Explanations Rarely Given in Existing Theories 1. Why probabilities are quadratic and not cubic or linear Most theories postulate P=∣ψ∣2P=|\psi|^2P=∣ψ∣2.Here it follows from coherent counting plus phase cancellation. The square is not a choice.It is the minimal stable coarse-graining. 2. Why quantum phases are physically unobservable but indispensable Phases affect interference but not outcomes directly.They label micro-configurations, not macrostates. Phase is structural memory.Measurement erases access to it. 3. Why constants are dimensionless and few Only ratios of update counts survive coarse-graining.Dimensional parameters wash out. Dimensionless constants are structural.They encode geometry, not dynamics. 4. Why “free parameters” cluster around simple numbers Many constants are order-one ratios or small inverses.They reflect combinatorial suppression. Fine tuning is apparent.It is state counting in disguise. 5. Why quantum randomness is stable and reproducible Randomness is not noise.It is deterministic micro-dynamics seen through coarse-graining. Statistics are invariant.Details are inaccessible. 6. Why identical particles are truly identical Microstates differ, macrostates do not.Labels are erased by update symmetry. Indistinguishability is enforced.It is not a postulate. 7. Why spin is quantized and half-integer values exist Spin counts topological winding in update cycles.Only discrete windings are stable. Half-integer spin reflects double-cover topology.No mystery remains. 8. Why gauge free
This pre-print, Paper 7 in the SORT Substrate series, systematically dismantles the mysticism of quantum mechanics by demonstrating that its core postulates are not fundamental laws of nature, but unavoidable mechanical consequences of a discrete, 3D Isotropic Vector Matrix (IVM) undergoing integer-valued torsional phase slips. By refusing to assume any a priori quantum postulates, this framework derives the entirety of observed quantum phenomena from strict geometric necessity: Spin and the Pauli Exclusion Principle: Particle spin is physically instantiated as the topological winding number (W) of a closed defect in the lattice. The spin-statistics theorem emerges naturally from the (−1)W holonomy under 2π rotations within the non-simply-connected SO(3) topology (and its SU(2) double cover). Consequently, Pauli Exclusion is not an arbitrary rule, but a literal geometric impossibility—two identical topological knots cannot occupy the exact same lattice struts simultaneously without fracturing the substrate. Superposition and Born's Rule: Quantum superposition is mathematically reduced to classical multi-mode acoustic standing waves vibrating across the discrete grid. Born's Rule (cn2) is derived independently from both mode energy partitioning and Fermi's Golden Rule applied to resonant phonon exchange, proving that "probability" is merely a measure of physical lattice stress intensity. Entanglement and Bell Inequalities: The mandatory violation of Bell inequalities is shown to be a trivial consequence of the IVM's tensegrity architecture. Entangled particles are not communicating via "spooky action at a distance"; they are topologically tethered nodes within a contiguous, non-local mechanical web, where stress propagates along shared invariant tension lines. The Measurement Problem: Wavefunction collapse and decoherence are re-identified as classical thermal dissipation. Measurement is simply the mechanical grounding of a resonant local defect into the macroscopic heat bath of the surrounding lattice environment. This paper establishes that the "quantum realm" is merely the small-amplitude, high-frequency acoustic limit of a solid-state universe.
Deterministic Quantum Mechanics (DQM) is a research program investigating whether the probabilistic content of quantum mechanics can be understood as an emergent statistical description of an underlying deterministic physical theory. This paper consolidates the program’s cumulative results into a single account.The central proposal is that a complete microscopic state evolves deterministically, that the quantum state is a coarse-grained ensemble descriptor rather than the complete physical state, and that measurement outcomes are selected by which deterministic basin of an enlarged phase space a microscopic trajectory occupies. Under this proposal, Born probabilities are basin measures rather than fundamental stochastic laws.For a qubit, this program is realized exactly: a hidden variable λ on the unit sphere, a preparation-dependent density μ_r(λ) = [r·λ]₊/π, and the deterministic response A(n,λ) = sgn(n·λ) reproduce the qubit Born rule P(±|r,n) = (1 ± r·n)/2 exactly, together with sequential-measurement statistics and the associated expectation values. This preparation measure is itself shown to be the deterministic pushforward of a uniformly distributed microscopic preparation coordinate, removing any appeal to fundamental preparation-level stochasticity.Measurement is then developed as deterministic basin selection within a finite-time Hamiltonian architecture, with a finite-time stability theorem governing when microscopic perturbations cannot dislodge a trajectory from its basin. Entangled correlations are reproduced by an autonomous, rotationally covariant Bell Hamiltonian containing dynamical setting registers and a conserved-momentum phase shear; this mechanism reproduces the singlet correlation E(a,b) = −a·b and the CHSH value 2√2 in its ideal limit, while preserving exactly uniform local marginals at finite setting inertia and yielding an explicit, quantitatively bounded finite-inertia correction. A finite Hamiltonian bath (Model-C) supplies a deterministic amplification and record-retention mechanism; its analytic structure — including an exact completed-square cancellation of the static bath backreaction and an exact finite-bath spectral equation — is derived independently of, and is shown to survive, a documented numerical implementation failure (the C14 program), which is retained in the record as a diagnosed, non-Hamiltonian right-hand side rather than as evidence against the underlying construction.Extending the program beyond the qubit exposes a genuine mathematical obstruction: no deterministic basin defined directly on the qutrit pure-state manifold ℂP² can reproduce an arbitrary rank-one Born probability, because positivity forces any exact basin onto a measure-zero point. This is proved rigorously. The resolution is not the abandonment of determinism but the enlargement of the hidden state beyond the quantum ray itself, to a genuine mechanical phase space such as the cotangent bundle T*SU(3), on which the quantum state re-emerges as a coarse-grained moment and on which contextual, Kochen–Specker-compatible measurement dynamics can in principle be realized.Throughout, the paper enforces a strict evidentiary hierarchy — mathematical construction, analytic theorem, numerical validation, and experimental confirmation are never conflated — and closes with an explicit account of what has been proved, what remains open, and what would be required to render the program experimentally falsifiable.
Objective Reality presents a unified account of physical reality in which quantum possibility and classical actuality are inseparable aspects of one evolving state. The quantum state carries the coherent possibilities—the amplitudes, phases, correlations, and interference structure—while a single history coordinate identifies the configuration that is temporarily actual. Neither component alone constitutes the physical world; objective reality is the evolving relation between them. This relation is expressed by the complete state Aₜ = (ρₜ, Dₜ, Cₜ, Rₜ, uₜ, Ξₜ), advanced through a single reversible bridge law, Aₜ₊Δₜ = BΔₜ[Aₜ]. Unitary quantum evolution generates the probability current. That current determines the admissible transitions between candidate classical sectors. A reversible selector then converts those transition weights, together with a retained auxiliary coordinate, into one actual successor. Decoherence and spectral stability identify durable record-bearing sectors; conserved quantities are carried through explicit accounting variables; and the inverse bridge reconstructs the preceding complete state. The construction proves normalization, Born equivariance, reversible selection, complete-state invertibility, conservation, classical reduction, relativistic integrability under microcausality, and the persistence of exactly one history coordinate. When the ensemble distribution agrees with the quantum sector weights, the bridge transports it into the next Born distribution while each individual realization follows one determinate history. The resulting formalism connects quantum evolution, measurement, wave–particle behavior, emergent classical records, and relativistic state evolution within one mechanics. It also establishes a common foundation for investigating selected-source gravity, emergent spacetime, vacuum energy, black-hole information, and experimentally falsifiable quantum–classical bridge dynamics.
We present the foundational layer of a research program that derives quantum-mechanical, relativistic, and thermodynamic structure from a single ontology: a ledger of sealed records. The ledger is governed by three axioms: R, laws are laws of whole sealed histories; S, no distinction without a record; and C, couplings are fixed by self-consistency under refinement. The program builds on the stochastic-quantum correspondence of Barandes' indivisible processes, replacing configuration trajectories by committed records as the primitive. The main results, each at stated scope and with machine-verified receipts, are as follows. First, the Born layer: record weights compose through square roots, and tame-class reconstruction recovers the Weyl algebra and the Schrodinger representation, via Stone-von Neumann, from record towers, with ladder operators, selection rules, and exact continuum targets approached at machine precision. Second, Lorentz signature: the commitment structure of records forces a Lorentzian (1,d) split with derived orientation classes. The spatial dimension d = 3 is selected by a minimality principle graded as input in the text, not derived. The operational identification lapse = local clock rate is verified to 3.7 x 10^{-8} in curved record geometries as a model-internal consistency receipt, with first 3+1-dimensional cosmological checks presented qualitatively and no cosmological number claimed in this batch. Third, the arrow of time: reflection positivity is a theorem for eventless sectors with finite primitive Markov presentations, the entropy-production functional is the path-measure relative entropy sigma = D(P_AB || P_BA), the classical identity of stochastic thermodynamics in record-pair form, and the presentation hypothesis is sharp. An explicit reversible, valid, finite-rank process, the "record clock", fails reflection positivity, separating record laws from general reversible processes. Fourth, thermality: horizon temperature T = kappa / 2 pi is recovered in exact lattice constructions, presented in full in the companion horizon paper and cited here only as a consistency credential. We state precisely what is axiom, what is theorem, what is named identification, and what is imported mathematics; the full dependency audit is part of the paper. We also delimit the program's frontier: the tame class, the regularity stratum of the emergent continuum, and continuous sector families.