We reconstruct nonrelativistic quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by
20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
Recursive formulations in theoretical physics often generate very large or formally infinite families of alternative histories, even when the internal state space controlling their generation is finite. This work develops a sector-resolved resummation framework based on propagation, interaction, closed-path monodromy or holonomy, and recursive closure (PIHR). Propagation, interaction, and holonomy are treated as primitive dynamical data, whereas recursion is treated as a closure operation acting on a finite return kernel. For systems admitting a unique first-return factorization, an exact resummation theorem is established: all repeated histories are generated by powers of one return operator and summed by its resolvent whenever the history series converges. The framework is mapped onto Dyson resummation, Lippmann–Schwinger scattering, the Bethe–Salpeter equation, Foldy–Lax multiple scattering, Fabry–Pérot interference, double-barrier tunneling, and Aharonov–Bohm transport. A three-arm flux-threaded quantum network then provides a quantitative benchmark. With complete junction mixing, the number of arm-resolved histories grows exponentially with recursive depth, while the exact coherent amplitude remains encoded in a fixed three-dimensional return operator. Through eight internal returns the cumulative path count already exceeds one hundred forty-five million, although the all-orders solution still requires only a three-dimensional resolvent. Using standard linear-systems realization theory, we additionally give a controllability-and-observability criterion under which this three-dimensional representation is minimal among linear realizations of the recurrence-generating function. The resolvent identity and the individual physical models are established results; the proposed contribution is the PIHR factorization, its cross-domain dictionary, and the quantitative separation of path-growth complexity from finite realization dimension.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-
k density matrices: minimal purifications make the fixed-rank stratum the base of a principal
U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at
k=1, at equal weights, and at
k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
Abstract (Initially published 08/16/25 as 2nd article in Engineering Quantum Tunneling in Arbitrary Size Matrices) In this publication, a general theory of finite-state, non-probabilistic quantum mechanics is specified, in which the physical state space is finite at every realizable layer, quantum dynamics is generated by finite Hermitian operators, and spectral realization weights arise from exact finite decompositions rather than from a primitive continuum of physical states. The conventional structures L^2(\{R}^3), -i\hbar\nabla, and continuum integration are replaced, respectively, by finite-dimensional complex state spaces, finite momentum operators, and finite sums over physical channels. Within this construction, position and momentum are assigned to common orthogonal channel projectors and are therefore simultaneously determined on each realized finite channel. The central mathematical result is a Conservative Finite Mesh–Gauge Correspondence connecting finite-volume mechanics on arbitrary polyhedral partitions with finite quantum gauge dynamics. For every finite reconstruction scheme, A_s=B,\mathrm{diag}(q),R_s, so that conservation follows structurally from the incidence identity \mathbf 1^{T}B=0 \quad\Longrightarrow\quad \mathbf 1^{T}A_s=0. This remains exact for extended higher-order stencils, including third-order QUICK. The unique decomposition A_s=D+S+K, with S^T=S and K^T=-K, yields the Hermitian quantum generator H_A=D+S+iK, thereby retaining both symmetric coupling and directed transport, with the latter encoded as gauge phase rather than discarded by symmetrization. The construction extends blockwise to arbitrary finite internal dimension d, with links U_{ab}\in U(d), local gauge covariance, antisymmetric realization currents, holonomy conjugacy, and Wilson invariants. Deterministic verification is supplied on triangular, quadrilateral, and mixed polygonal meshes for upwind, central, and QUICK schemes, with exact structural divergence and Hermiticity, and with non-Abelian gauge tests spanning U(1) through U(8). The resulting framework replaces continuum quantum kinematics with finite physical state space, finite operators, finite relational transport, and finite spectral resolution, while preserving exact unitary evolution, conservation, gauge structure, and finite quantum dynamics on arbitrary coupling geometries. Note: The goal has been to better equip science to tackle problems previously thought to require specific hardware. I believe this has accomplished that goal, in the larger sense. I am optimistic to see the software that will be built from such beginning points. May quantum logic develop, earnestly, and may approximation give way. Dr. Charles Tibedo
We formulate a finite four-sector path architecture in which elementary generators are labelled as propagation (P), interaction (I), holonomy (H), or recursive closure (R). The mathematical starting point is a directed generating system together with its discrete groupoid completion and associated path *-structure. The standard left regular construction on the Hilbert space of admissible reduced paths then represents composable arrows by partial isometries and groupoid inversion by the Hilbert-space adjoint. We state this result explicitly to fix the mathematical setting, not as a new groupoid-representation theorem. The architecture-specific content begins with the PIHR sector decomposition: sector-resolved weighted operators are defined and combined into a self-adjoint architectural operator. We prove that a weight-, holonomy-, and sector-preserving architectural isomorphism induces unitary equivalence of the complete sector-resolved operator family and therefore equality of the corresponding spectra. The central demonstration is a U(1)-twisted cyclic architecture. Its discrete architectural Laplacian has the exact eigenvalues determined by the total loop holonomy, while its continuum limit reproduces the familiar flux-dependent spectrum of a particle on a ring. The example isolates a precise mathematical point: global holonomy can alter spectral data even when local connectivity is unchanged. No derivation of quantum mechanics from the four sectors is claimed. The framework is instead proposed as a mathematically explicit classification scheme for separating local transport, local coupling, global holonomy, and compositional depth, and as a basis for testing stronger uniqueness or reconstruction conjectures in restricted classes of systems.
This paper develops the first mathematical layer of Variable-Signature Quantum Theoryand Obstruction Theory in the finite-dimensional deterministic setting. The core object is asignature-parameterized family inside a fixed ambient complex vector space K ∼=CN, withparameter space Θ, Hermitian forms Gθ, representing matrices Bθ, physical-sector projectionsPθ, and selected sector-preserving observable algebras Aθ. The Paper-I projected physicalsector is not the positive cone of Gθ, but the selected positive sector HIphys,θ = (Eθ,Gθ|Eθ).On each regular stratum Θα, projected positive sectors form finite-rank Hilbert bundles. Thepaper distinguishes ambient comparability from smooth fixed-Hilbert triviality and introducesfirst-layer coarse obstructions to sector representation, transition extension, and conditionalselected-algebra transport. The algebra obstruction is evaluated only after a smooth globalsector representation exists. On regular strata, the paper recovers the ordinary fixed-Hilbertform of finite-dimensional sector kinematics, numerical Born state-effect pairings under sectorrepresentation, and selected algebra transport when the relevant algebra obstruction alsovanishes. This first paper proves mathematical nontriviality criteria for sector geometry; itdoes not claim empirical deviation from standard quantum mechanics or a full operationalreconstruction of measurement theory.