Abstract We present DLST Ver.11.0.0, a compact recurrence-classification framework for exact physical histories, relational time, and relativistic closure. Ver.11 explicitly corrects the earlier interpretation of DLST: finite signal speed, microscopic length, compactness, finite entropy, finite Hilbert dimension, or a discrete evolution label do not by themselves establish a fundamental discrete clock or exact recurrence. The central objects are instead the physical identity space, the stabilizer of an individual state, and the common global kernel of the evolution. This separates state-specific recurrence, universal recurrence, gauge stationarity, and observational indistinguishability. For autonomous discrete quantum evolution, we prove that a common exact projective return for all pure-state rays and density operators exists if and only if the one-step projective update [U] is a torsion element of PU(H), equivalently U^N = exp(i phi) I for some finite N. Membership in a finite projective unitary subgroup is therefore a sufficient special case. Finite-volume exact Clifford dynamics on finitely many prime-dimensional qudits provides an explicit conditional realization: the projective Clifford group is finite, so an exact Clifford quantum cellular automaton has a nonzero common recurrence period. This bridge does not require a finite quantum-state cardinality. Its scope is exact and finite-volume; infinite-lattice Clifford QCA or arbitrarily small irrational non-Clifford phases need not recur globally. Complementary positive criteria are given by finite same-law closure and spectral commensurability. For unitary evolution, exact recurrence of an occupied state requires its active eigenphase or energy differences to lie on a common arithmetic lattice. These results clarify why finite-dimensional Hilbert space alone is insufficient: exact recurrence is controlled by projective order and spectral arithmetic rather than vector-space dimension. The relativistic obstruction side is strengthened. In 1+d-dimensional Poincare-covariant quantum dynamics with d >= 2, if the scalar/projective translation kernel contains any nonzero translation—timelike, spacelike, or null—Lorentz covariance and additivity force the entire translation subgroup into the kernel. Hence a nontrivial translation representation cannot possess a nonzero universal spacetime period. A related connected simple-group obstruction excludes noncentral universal projective periods under the stated representation hypotheses. For relativistic scattering sectors, Haag-Ruelle wave-operator equivalence transfers the asymptotic energy spectral measure to the interacting sector. A nonzero non-atomic energy component is incompatible with the affine spectral lattice required for exact projective return, yielding an exact nonreturn criterion without assuming that every state belongs to the scattering sector. Independently, any strictly monotone extensional physical witness along the complete admitted history forbids exact recurrence. Ver.11 also proves a finite-window nonidentifiability result: for any finite observation horizon and tolerance, one can construct finite-dimensional Hamiltonians whose propagators remain arbitrarily close throughout the observed interval while belonging to different global recurrence classes. Finite measurement precision therefore cannot replace an exact structural bridge from observed dynamics to all-history recurrence. Related audits show that finite Hilbert dimension, Type II_1 finite-entropy language, gauge stationarity, and cosmological labels are not exact recurrence classifiers by themselves. Finite-dimensional completeness is also incompatible with a nontrivial exact continuous unitary representation of the standard connected de Sitter group under the stated assumptions. The resulting classification contains both positive and negative bridges. Positive exact routes include finite same-law closure, projective torsion, finite-volume exact Clifford QCA, and affine spectral lattices. Obstruction and nonreturn routes include Poincare symmetry, non-atomic scattering spectra, and strict extensional monotonicity. Gauge equivalence, compactness, finite observational windows, and coarse finiteness statements remain nonclassifying without additional physical input. The principal methodological change is therefore a shift from presumed temporal discreteness to fail-closed actuality promotion: a recurrence or nonrecurrence claim is promoted only after the physical identity relation, complete evolution law, and exact all-history bridge have been independently specified. DLST Ver.11.0.0 does not establish that the actual universe is periodic, aperiodic, fundamentally discrete in time, a finite Clifford automaton, or governed by a universal microscopic tick. C4 Ver.1.0.0 and OUROBOROS Ver.2.0.0 remain separate downstream research programs rather than premises of DLST. Deterministic reproducibility checks accompany the release for the finite-projective, Clifford-group, spectral, permutation, and finite-window constructions. These computational audits verify explicit finite examples and implementation details; the analytic theorems remain the scientific basis of the classification. DLST Ver.11.0.0 thus provides a compact framework connecting exact recurrence, dynamical stabilizers, projective unitary torsion, Clifford quantum cellular automata, Poincare symmetry, Haag-Ruelle scattering, operator-algebraic finiteness, de Sitter representation theory, and relational physical time under an explicit claim boundary.
Abstract We present DLST Ver.11.0.0, a compact recurrence-classification framework for exact physical histories, relational time, and relativistic closure. Ver.11 explicitly corrects the earlier interpretation of DLST: finite signal speed, microscopic length, compactness, finite entropy, finite Hilbert dimension, or a discrete evolution label do not by themselves establish a fundamental discrete clock or exact recurrence. The central objects are instead the physical identity space, the stabilizer of an individual state, and the common global kernel of the evolution. This separates state-specific recurrence, universal recurrence, gauge stationarity, and observational indistinguishability. For autonomous discrete quantum evolution, we prove that a common exact projective return for all pure-state rays and density operators exists if and only if the one-step projective update [U] is a torsion element of PU(H), equivalently U^N = exp(i phi) I for some finite N. Membership in a finite projective unitary subgroup is therefore a sufficient special case. Finite-volume exact Clifford dynamics on finitely many prime-dimensional qudits provides an explicit conditional realization: the projective Clifford group is finite, so an exact Clifford quantum cellular automaton has a nonzero common recurrence period. This bridge does not require a finite quantum-state cardinality. Its scope is exact and finite-volume; infinite-lattice Clifford QCA or arbitrarily small irrational non-Clifford phases need not recur globally. Complementary positive criteria are given by finite same-law closure and spectral commensurability. For unitary evolution, exact recurrence of an occupied state requires its active eigenphase or energy differences to lie on a common arithmetic lattice. These results clarify why finite-dimensional Hilbert space alone is insufficient: exact recurrence is controlled by projective order and spectral arithmetic rather than vector-space dimension. The relativistic obstruction side is strengthened. In 1+d-dimensional Poincare-covariant quantum dynamics with d >= 2, if the scalar/projective translation kernel contains any nonzero translation—timelike, spacelike, or null—Lorentz covariance and additivity force the entire translation subgroup into the kernel. Hence a nontrivial translation representation cannot possess a nonzero universal spacetime period. A related connected simple-group obstruction excludes noncentral universal projective periods under the stated representation hypotheses. For relativistic scattering sectors, Haag-Ruelle wave-operator equivalence transfers the asymptotic energy spectral measure to the interacting sector. A nonzero non-atomic energy component is incompatible with the affine spectral lattice required for exact projective return, yielding an exact nonreturn criterion without assuming that every state belongs to the scattering sector. Independently, any strictly monotone extensional physical witness along the complete admitted history forbids exact recurrence. Ver.11 also proves a finite-window nonidentifiability result: for any finite observation horizon and tolerance, one can construct finite-dimensional Hamiltonians whose propagators remain arbitrarily close throughout the observed interval while belonging to different global recurrence classes. Finite measurement precision therefore cannot replace an exact structural bridge from observed dynamics to all-history recurrence. Related audits show that finite Hilbert dimension, Type II_1 finite-entropy language, gauge stationarity, and cosmological labels are not exact recurrence classifiers by themselves. Finite-dimensional completeness is also incompatible with a nontrivial exact continuous unitary representation of the standard connected de Sitter group under the stated assumptions. The resulting classification contains both positive and negative bridges. Positive exact routes include finite same-law closure, projective torsion, finite-volume exact Clifford QCA, and affine spectral lattices. Obstruction and nonreturn routes include Poincare symmetry, non-atomic scattering spectra, and strict extensional monotonicity. Gauge equivalence, compactness, finite observational windows, and coarse finiteness statements remain nonclassifying without additional physical input. The principal methodological change is therefore a shift from presumed temporal discreteness to fail-closed actuality promotion: a recurrence or nonrecurrence claim is promoted only after the physical identity relation, complete evolution law, and exact all-history bridge have been independently specified. DLST Ver.11.0.0 does not establish that the actual universe is periodic, aperiodic, fundamentally discrete in time, a finite Clifford automaton, or governed by a universal microscopic tick. C4 Ver.1.0.0 and OUROBOROS Ver.2.0.0 remain separate downstream research programs rather than premises of DLST. Deterministic reproducibility checks accompany the release for the finite-projective, Clifford-group, spectral, permutation, and finite-window constructions. These computational audits verify explicit finite examples and implementation details; the analytic theorems remain the scientific basis of the classification. DLST Ver.11.0.0 thus provides a compact framework connecting exact recurrence, dynamical stabilizers, projective unitary torsion, Clifford quantum cellular automata, Poincare symmetry, Haag-Ruelle scattering, operator-algebraic finiteness, de Sitter representation theory, and relational physical time under an explicit claim boundary.
Abstract We investigate the logical and mathematical conditions under which finite information capacity can support a finite physical state space and exact recurrence in autonomous dynamics. The analysis distinguishes four notions that are often conflated: finite entropy capacity, finite Hilbert-space dimension, finite complete-state cardinality, and uniform nonzero state separation. Finite von Neumann entropy and finite Hilbert-space dimension do not imply a finite state set; a qubit provides an explicit counterexample because its entropy is bounded and its Hilbert space is finite-dimensional while its pure-state space forms a continuous Bloch sphere. Under admissible mixing, however, finite entropy bounds the number N of perfectly distinguishable states according to N <= exp(S_max), establishing a finite distinguishability capacity without implying finite total state cardinality. We then characterize the missing geometric condition precisely. For any subset of a compact metric state space, finite cardinality is equivalent to the existence of a uniform nonzero minimum separation between distinct states. Thus uniform state separation is not an independent weaker substitute for finite state cardinality within compact state geometry, but its metric equivalent. For density operators on a d-dimensional Hilbert space equipped with trace distance, a delta-separated complete-state set satisfies the explicit packing bound |S| <= (1 + 2/delta)^(d^2 - 1). When such a finite complete-state representation is combined with bijective autonomous dynamics, the finite-state recurrence result established in DLST Ver.7.0.0 implies an exact finite period P such that F^P = id. The paper further separates exact finite recurrence from finite-dimensional quantum unitarity. An irrational-eigenphase unitary provides a counterexample showing that finite-dimensional unitary evolution need not possess finite global order even modulo global phase. Approximate quantum recurrence therefore remains mathematically distinct from exact recurrence generated by a permutation of a finite state set. We also formulate exact recurrence on finite dynamically closed quotient spaces, distinguishing operational or equivalence-class recurrence from recurrence of complete ontic states. These results clarify the roles of entropy bounds, causal-diamond arguments, compact quantum state geometry, state distinguishability, uniform discreteness, and reversible finite dynamics in DLST. Entropy bounds and finite-dimensional quantum mechanics motivate finite-information descriptions but do not by themselves establish finite complete-state cardinality. The remaining physical question is whether fundamental physics independently supplies a finite complete-state space, a physically meaningful uniform state separation, or a finite dynamically closed quotient. DLST Ver.8.0.0 therefore identifies the precise mathematical boundary between finite information capacity and exact recurrence while explicitly isolating the additional physical structure required to cross it.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.
Abstract We investigate the limits of reconstructing relativistic particle-mass information from finite causal-source records, with particular emphasis on stochastic event histories, spectral identifiability, and classical–quantum jump dynamics. Izanagi Core Ver.3.0.0 distinguishes information determined by a finite Markov source from additional structures required to define quantum time evolution, local field interactions, stable particle excitations, and physical mass ratios. Three interconnected problems are examined: the nonuniqueness of source-compatible quantum spectral realizations, the preservation of classical event histories under quantum operations, and the existence of stationary states in source-driven bosonic field models. For finite irreducible reversible continuous-time Markov sources, we analyze stationary occupation measures, relative event clocks, typed transition histories, and spectral representations. A specified Markov generator determines its own relaxation spectrum, but that spectrum does not uniquely determine a physically admissible quantum Hamiltonian or relativistic particle masses. Explicit constructions demonstrate that compatible quantum representations can possess different internal spectral-gap ratios while reproducing the prescribed classical source information. The canonical Hilbert-space representation of a stationary Markov record further distinguishes stochastic time translation from positive-energy quantum dynamics: the nontrivial spectrum of the record-shift generator is symmetric about zero. This obstruction applies to the specified representation and does not exclude alternative quantum reconstructions. We formulate classical–quantum jump instruments in which transitions of a classical source induce completely positive operations on an independently specified quantum system. Exact preservation of the complete typed source-history law, including transition identities and waiting-time distributions, is characterized by the scalar-effect condition J_ji*(I) = q_ij I on every directed transition, when preservation is required independently of the quantum state. For source-conditioned transition intensities whose total discrepancies are uniformly bounded by δ, a coupling construction yields the finite-time total-variation estimate TV ≤ 1 − exp(−δT). The estimate is sharp in an explicit comparison model, while control over finite observation intervals does not imply uniform agreement over arbitrarily long times. To examine locality and quantum measurement structure, we construct conditional interactions with externally supplied bosonic free fields using future-supported Weyl displacement operators. The analysis distinguishes relativistic locality, overlap of localized operations, commutation of quantum state channels, and phase-sensitive coherent control. Weyl displacement channels can commute even when their implementing unitary operators differ under an exchange of order by a nontrivial central phase. Such phases can be detected only when an additional coherent ordering interface is available. These constructions clarify the distinction between preserving a classical event record and reconstructing the full operational content of an interacting quantum field. Two stationary-state results identify further limitations of one-way classical–quantum coupling. First, a product state consisting of the stationary source distribution and a designated pure free-field vacuum is stationary precisely when every positive-rate jump operation preserves the corresponding vacuum projector, J_ji(P) = q_ij P, under the stated autonomous-source dynamics. Nontrivial Weyl interactions can violate this condition without changing the classical source-history statistics. Second, for a stationary source driving a bosonic Fock field through equally weighted positive and negative Weyl displacements, we obtain an exact energy-balance relation, EH_f − EH_f = Γ_Q t. Here Γ_Q is the nonnegative transition-rate-weighted displacement energy. When Γ_Q > 0, the specified dynamics admit no normal joint stationary state with finite expected free-field energy. This heating obstruction depends on unbiased displacement kicks and nondissipative free-field evolution between transitions. It is not a universal consequence of classical-source autonomy: dissipative dynamics can balance energy injection and permit finite-energy stationary states. The results are placed in the context of reversible Markov processes, quantum spectral theory, GNS representations, canonical commutation relations, generalized free fields, completely positive quantum instruments, quantum dynamical semigroups, and classical–quantum hybrid dynamics. Particular attention is given to the distinction between established mathematical structures and their conditional application to source-to-matter reconstruction. The stationary-state and heating criteria describe a specified class of one-way quantum jump models; they do not constitute a general decoherence–diffusion theorem or a universal prohibition on stationary classical–quantum coupling. The principal unresolved issue is the physical generation of matter from causal-source dynamics. In the constructions studied here, the Markov generator, spacetime geometry, quantum carrier, interaction operators, and measurement interfaces are supplied as independent inputs rather than derived from the Minimal Realization Source Law (MRSL). Accordingly, the results do not establish a dynamically generated Lorentz-covariant matter sector, persistent physical identity between source events and particle excitations, stable isolated particle poles in the coupled relativistic theory, or a uniquely identifiable physical particle-mass ratio. Izanagi Core Ver.3.0.0 therefore provides a mathematically explicit account of the distinction between source-history preservation, quantum dynamical consistency, and physical mass identification, while leaving the source-generated origin and quantitative determination of relativistic particle masses open.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.