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.
This paper develops the representation-level correspondence between Q electrodynamics (QQED), its finite-coherent operator/Fock realization, and renormalized local QED. The primitive physical object remains the complete Q of Paper II-A; the purpose here is not to introduce a second ontology but to show how the same finite-response mother is represented by response-space operators, Fock sectors, Hamiltonian/TOPT amplitudes and, when the response is unresolved, the ordinary local-QED organization. The exact branch is followed from its cut-plane master response through an operator-valued response lift, cocycle-Fock dynamics, a balanced contractive interaction, coefficientwise optical and TOPT-Euclidean reconstruction, and the complete one-loop self-energy/vacuum-polarization/proper-vertex Ward family. The ultraviolet proofs remain in Paper II-B. The construction is then extended to the charged-Fock domain, relative infrared completion, non-Fock charged asymptotic sectors, dressed one- and multi-Q scattering, asymptotic photon fields and response-resonance sewing.
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate
∥μγ∥TV≤e2cγ, where
cγ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on
ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on
L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation
∥eTBv∥22. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.
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.
With the identity matrix on the stand, the bare trace reports
252 and the originally normalized prepared state reports
1: the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed
S01/dx1 external source with the co-directional readout operator, zero output momentum, double Laplace parameters
z=w=6c(1−i), the same preparation, and an incoming/observing two-ball window of radius
B, we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the
P6 grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight
K and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from
289 fields through auxiliary, Ward and symmetry reductions to a
103-field system, counts matter feedback once, and constructs a rational
55-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank
42; keeping six light matter directions yields a
61-dimensional second-order system whose constant term has rank
56, and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical
g00 source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in
[−1.355200,−1.309214],
[2.140271,2.186257] and
[−1.742600,−1.696614] times
10−20 in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full
289×289 quantum tensor
ΠT, and the finite integral of the mother-Euler amplitude derivative is exactly
[WTH289−ΠT]a. Coincident clocks give
WT=T; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.
We relate positive observable quotients to Page--Wootters (PW) clockdescriptions. Physicalization identifies a quotient by its kernel, andjointly faithful clock reductions recover the same exactness relation.Admissible PW data give a faithful functor into an independently definedquotient-clock category after representative lifts are identified modulophysical null directions. With algebraic lifts, it is full on its essentialimage at the level of retained state-and-clock data. A restricted converse characterizes strictly periodic, globallydeparametrizable realizations with unitary one-slice conditioning:one reference reduction must be unitary and the relative reductions mustform a strongly continuous periodic unitary group. The construction includesthe closed-operator domains required for unbounded generators. Retaining clock-event probabilities gives a separate strictness theorem.Normalizable stationary models with additive constraints and covariantperiodic clocks have state-independent clock marginals. A positivetwo-qubit constraint quotient violates this invariant despite invertiblereductions. This separates it from the additive class; its realization isitself interacting. We then define a broader stationary-conditioning class allowing generalizedcovariant clocks, unrestricted self-adjoint interactions, and separableenvironmental dilations. Its normal quantum-instrument data descendisometrically to positive quotients. Conversely, every normalized amplitudequotient-clock datum has an exact realization: a positive projectorconstraint selects the image of its isometric history map. Standardinstrument dilation extends the result to absolutely continuous normalcompletely positive instruments. For finite-dimensional physical and outputspaces with Sobolev-regular histories, a bounded finite-rank interactionsuffices by standard Hermitian completion and spectral isolation. Hencethis normalized operational sector has no obstruction to realization inthe broad class. For a fixed canonical periodic clock, we then bound the interactionremaining after arbitrary bounded clock-local Hermitian terms are allowed.Translation Fisher information gives a lower bound valid under separableenvironmental dilations. For instruments with orthogonal amplitude columns,a finite-rank construction attains the exact minimum while selecting theprescribed physical kernel. The modulated qubit family yields an explicitstrength--frequency threshold and a finite-event approximation-error bound. An exact Gaussian differential-image quotient coupled to a full oscillatorprovides an additive realization with matching physical metrics,conditioning, reconstruction, and relational probabilities, including finitereading intervals. These results distinguish restricted strictness frombroad representability; they neither identify all microscopic realizationsnor establish equivalence or strict superiority across every QRT and PWformulation.