Local information can be lost rapidly while a collective mode survives for a long time. We study this distinction on a physical–observation joint state space. The second variation of relative entropy under a local record map gives a positive Fisher-loss operator; the least loss over normalized noninvariant scores is the spectral gap of the associated posterior-resampling process. The local strength and the compatibility of surviving directions enter separately. A fixed-width interacting ladder has a positive gap at every fixed finite coupling, as follows from an explicit block comparison, although strong correlation makes the bound small. We then reverse the construction. For a given irreducible real Hamiltonian with nonpositive off-diagonal entries, its ground amplitude determines pair-record probabilities and energy-valued loss weights, reproducing the physical excitation gap exactly. A positive trial amplitude gives an error interval controlled by the oscillation of its local energy, without requiring the exact ground state. When signs remain, the energy form is a difference of positive information forms and an explicit cancellation constant controls the gap. Sparse diagonalization of open transverse-field chains up to 65 536 configurations agrees with both the reconstructed information spectra and an independent free-fermion calculation. The critical law n∆n/J → π, the gapped regime, and the ordered finite-chain splitting are distinguished. The result is an explicit connection for specified channels and Hamiltonians, with the requirements of positivity, locality, and energy normalization kept separate.
Local information can be lost rapidly while a collective mode survives for a long time. We study this distinction on a physical–observation joint state space. The second variation of relative entropy under a local record map gives a positive Fisher-loss operator; the least loss over normalized noninvariant scores is the spectral gap of the associated posterior-resampling process. The local strength and the compatibility of surviving directions enter separately. A fixed-width interacting ladder has a positive gap at every fixed finite coupling, as follows from an explicit block comparison, although strong correlation makes the bound small. We then reverse the construction. For a given irreducible real Hamiltonian with nonpositive off-diagonal entries, its ground amplitude determines pair-record probabilities and energy-valued loss weights, reproducing the physical excitation gap exactly. A positive trial amplitude gives an error interval controlled by the oscillation of its local energy, without requiring the exact ground state. When signs remain, the energy form is a difference of positive information forms and an explicit cancellation constant controls the gap. Sparse diagonalization of open transverse-field chains up to 65 536 configurations agrees with both the reconstructed information spectra and an independent free-fermion calculation. The critical law n∆n/J → π, the gapped regime, and the ordered finite-chain splitting are distinguished. The result is an explicit connection for specified channels and Hamiltonians, with the requirements of positivity, locality, and energy normalization kept separate.
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.
Generalized trace dynamics (GTD) seeks both quantum theory as a statistical approximation to matrix dynamics and classical spacetime through collective nonunitary localization. We consolidate the subsequent reconstruction tests of Project Unification, using the available IX-D--IX-O technical record and its foundational manuscripts. Closure here means a precise account of what follows, what fails, and what further microscopic data are needed; it does not mean successful completion of unification. In the finite external-marker realization, source-free Berezin normalization fails, and regular normalized marker observables have vanishing external quotient. This obstruction survives coefficientwise non-Gaussian integration under explicit hypotheses. Positivity on the unchanged finite exterior star algebra removes its nilpotent ideal; a different physical product is therefore indispensable for a nontrivial positive fermionic reconstruction of that route. A postulated first-order kinetic extension admits a local marked constraint lift and nonzero transport curvature, but its tested ensemble loses configuration mass to infinity and retains regulator-dependent shape statistics. Its canonical comparison has no stationary normalized trace-class density, while possessing nonzero response and source-free entangling capacity in a specified partition. Exact exceptional-channel projectors, a bound state in a separately declared current Hamiltonian, and a stable but nonbinding scalar interface remain useful conditional results. Finally, identical reduced density dynamics can admit both nonlocalizing and Born-completing trajectories. Neither an interaction kernel, a covariance scaling exponent, nor a spectral-action expansion selects physical records by itself. We give proofs and falsification controls, distinguish these scoped obstructions from a no-go theorem for GTD, and formulate a bounded next programme: positive statistical reconstruction, physical STM observable assignment, conditional record probabilities, and only then dynamical Lorentzian-leaf and parameter matching.
We introduce double localization, a framework for proving spectral gaps of quantum Gibbs samplers through two complementary operations: geometric localization, which selects updates supported in small spatial regions, and interface localization, which focuses on a model-defined subspace of observables while remaining geometrically global. Our abstract gap theorem combines a global bound on this subspace, called the interface, with local quantum gap estimates under quantitative coupling and geometric assembly conditions. The resulting explicit lower bound controls the gap of the full dynamics without requiring the interface to be invariant under the generator. This separation allows global estimates, including classical comparison, to be combined with local control of the remaining quantum directions.
We apply the framework to Hamiltonians built from the vertex terms of Kitaev's finite-group quantum double construction, without plaquette terms, on finite simple three-regular graphs of girth at least six. For the specified local Davies dynamics, we prove an unconditional spectral-gap lower bound by a positive constant independent of graph size for every fixed nontrivial finite group and each fixed inverse temperature β with 0≤βJ<log(5/3), where J>0 is the coupling strength. For the smallest non-Abelian group S3, double localization yields a positive lower bound uniform in both graph size and the entire interval 0≤βJ≤log3, providing a guarantee that does not follow directly from existing results. This demonstrates how model-specific finite calculations can extend spectral-gap guarantees for the full quantum dynamics.
This paper addresses why a matter record formed through an autonomous rank-opening transition can persist without requiring fundamental information loss. It first proves a finite-unitary recurrence no-go theorem: in a finite-dimensional closed system with a time-independent self-adjoint Hamiltonian, the boundary survival probability cannot permanently converge to zero because arbitrarily accurate late-time recurrences occur. A consistent locking mechanism therefore requires an infinite or effectively continuous spectral sector.Using a self-adjoint Friedrichs-type completion, the boundary survival amplitude is represented as the Fourier transform of its spectral measure. A purely absolutely continuous measure with an � density provides an exact sufficient condition for boundary decay through the Riemann–Lebesgue lemma. Atomic spectral weight is identified as the precise owner of persistent time-averaged boundary memory, while singular-continuous measures are treated separately through the more general Rajchman condition. Sobolev regularity of the spectral density yields explicit power-law bounds on the residual boundary probability.The construction is extended to a fifteen-dimensional boundary space through a matrix-valued spectral measure, producing a sufficient criterion for the simultaneous spectral locking of all fifteen chiral channels. A Friedrichs resolvent analysis identifies real poles, bound states, spectral gaps, and embedded dark states as exact obstructions to complete escape. The completed matter record is further protected by a positive singular-value gap in its synthesis operator.Global evolution remains unitary, exterior fermion parity is preserved, and total information is conserved; only the local overlap with the original boundary sector vanishes. Cosmological expansion is proposed as a model-level source of continuous spectral dispersion, but the paper does not claim that the physical universe realizes the required spectral density or that observed particle lifetimes, masses, and abundances have been derived.
This paper develops physical-resource bounds for operationally recoverable decoherent histories, extending a branch-record framework to include constraints on record energy, formation, observer access, measurement, memory, time, and reset. Decoherent-histories methods identify time-extended alternatives whose diagonal weights may support a probabilistic interpretation, while environmental-record approaches ask which alternatives a specified observer can recover. For uniform branch-record packing, an energy-constrained maximum-entropy argument combined with Holevo and Fano bounds gives a finite-error converse. For a finite Hamiltonian spectrum, the zero-error average-cost and strict expected-energy packing numbers coincide and equal the largest NNN whose NNN lowest levels have mean energy no greater than the declared ceiling. Finite physical mean energy alone nevertheless cannot bound retrospective cardinality: an explicit one-time unitary tagging construction realizes arbitrarily many increasingly rare, orthogonal, high-energy records at bounded mean energy. Solvable thermal record models separate decoherence, record distinguishability, internal energy, formation dynamics, and measurement-compatible consistency. A full controlled copy is exactly scalar- and measurement-refined-consistent under energy-basis registration even when the thermal record is unreadable. A partial-copy benchmark admits an exact Hamming-layer reduction of the direct refined-gate optimization. The thermodynamic conclusions are path dependent. Writing x=βmemΔx=\beta_{\rm mem}\Deltax=βmemΔ, the cross-checked L=10L=10L=10 benchmark has a strict direct final-internal-energy optimum near x=2.15x=2.15x=2.15, with 17 copies per event, in a fixed-memory-temperature gap scan. At fixed gap, the temperature-scan optimum instead occurs near x=2.60x=2.60x=2.60, at the upper edge of the 16-copy feasibility interval. In the weak-record limit, a finite interaction-energy infimum exists only along the fixed-memory-temperature, shrinking-gap path; at fixed gap and increasing temperature, the physical interaction energy diverges. A proof-of-principle experimental roadmap uses controlled record rotations, symmetric decision-only registration, recovery testing, and ancilla-assisted estimation of cross-branch amplitudes. The resulting bounds are vector- and protocol-dependent and constrain operationally recoverable histories. This record includes a self-contained reproducibility archive.
How much of a quantum system’s total correlation is retained in local measurement records? This theoretical study addresses that question for finite-dimensional multipartite systems, with particular emphasis on survival measurements and the onset of correlations. An exact information balance separates recorded correlations from contributions lost through local dephasing, unresolved conditional information, and correlations remaining within measurement sectors. Established Petz recovery theory characterizes when local measurements preserve the complete total correlation. For sufficiently regular trajectories starting from product states, the analysis distinguishes two short-time regimes. At rank-deficient preparations, binary survival-or-departure records capture the complete leading logarithmic correlation term when simultaneous departures from the initial local supports occur with a positive first-order rate. At full-rank preparations, a quadratic expansion uniform over finite-outcome product measurements identifies the optimal leading coefficient accessible to local records, including time-dependent measurement choices. Continuity bounds control the effects of finite state errors. Explicit examples distinguish classical joint jumps from coherent entangling dynamics, while counterexamples establish limitations on reconstructing total correlation from survival records alone. The results clarify when simple local detectors capture the leading correlation signal and when additional measurement resolution or state information is required. The framework concerns product measurements on a single copy and does not require Markovian dynamics.