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 \( \|\mu_\gamma\|_{TV}\leq e^{2c_\gamma} \), where \( c_\gamma \) 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 \( \ell^2 \); nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on \( L^2 \). In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation \( \|e^{T\mathsf B}v\|_2^2 \). 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.
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 ( |\mu_\gamma|{TV}\leq e^{2c\gamma} ), where ( c_\gamma ) 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 ( \ell^2 ); nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on ( L^2 ). In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation ( |e^{T\mathsf B}v|_2^2 ). 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.
Schrödinger’s cat asks how a coherent microscopic event can leave a single, durable macroscopic record. We construct a finite-apparatus model within the physical–observation dual-axis (PODA) framework, using specified quantum representations and apparatus couplings. A common action supplies the mode interactions used for source preparation and controlled amplification. For the resulting orthogonal two-branch states, entanglement negativity across the record bipartition and trace distance from the incoherent mixture equal half the full recombination visibility. Gaussian pointers determine finite-resolution probabilities and conditional states, while a canonical-reservoir construction prepares the event selector with a bounded error. To describe individual records, we retain the complete wavefunction and let actual pointer and memory coordinates follow its natural probability current. Their joint motion reproduces the statistics of successive readouts under the stated conditions. After branch preparation, a finite sequence of smooth, predetermined readout and writing gates admits an autonomous Hamiltonian with positive apparatus kinetic coefficients. We prove almost-sure global existence and Born equivariance for this apparatus, and derive bounds on terminal record errors and retention failure over a finite interval. This realization permits individual records to coexist with full-state coherence without requiring additional environmental decoherence for record formation. Three proposed qubit–cavity experiments test write–erase recovery, finite-resolution backaction, and single-mode revival. Their count probabilities and calibration requirements lead to quantitative tests of the model and conditional bounds on additional irreversible damping.
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.
Enlarging observation space extends the available event families, but its effects on correlation bounds and dynamical spectra depend on different assumptions. Within the quantum representation of the physical--observation dual-axis structure (PODA), the real Gram geometry of the Born pairing yields the dimension-independent CHSH bound 22 and a two-dimensional saturating configuration. For a fixed d-dimensional maximally entangled state, arbitrary-rank binary projections give an even--odd dependence, whereas rank-one events yield 2+(42−4)/d. These bounds admit an experimental comparison at fixed preparation. We then introduce an independent joint field with a quadratic action and positive gradient energy along observation space. On the full CP1 with its standard projective metric and Hopf connection, we derive the covariant eigenvalues and degeneracies for integer phase weight. In the minimal nonzero sector ∣q∣=1, the mass levels obey Mn2=mX2+[2n(n+2)+1]mS2, where mX is an independent local mass parameter and mS is determined by the lowest observation eigenvalue and the gradient coefficient. The two lowest distinct mass levels uniquely reconstruct both scales, giving mX2/mS2=(7−r)/(r−1) with r=M12/M02∈(1,7]. Higher levels satisfy relations such as 3M22=8M12−5M02 without further parameters. Exactly equal mass spacings occur if and only if mX=mS. These spectral results depend on the specified action, full observation geometry, and phase representation. The scale ratio remains a dynamical parameter, while the rest-energy relation retains the form E0,n=Mnc2.
A finite amplitude dynamics, a configuration-space propagator, and a mechanical action are different mathematical objects. We give a conditional one-dimensional bridge between them while retaining the assumptions that make each identification possible. Common-additive-shift invariance selects a relative two-mode quadratic energy; a supplied first-order kinetic term and action scale turn its graph Laplacian into reciprocal norm-preserving dynamics. The resulting amplitude action vanishes on solutions and therefore cannot simply be identified with the configuration action. A convergent ordered-jump expansion retains diagonal dwell phases, hopping factors, and time-simplex measure. On a supplied spatial lattice, inverse-square scaling gives second-order fixed-band convergence and strong convergence for each fixed square-integrable state, but not all-mode real-time operator-norm convergence. The raw return-kernel density has two distinct mesh-subsequence limits. A declared Gaussian resolution instead gives an explicit uniform lattice-to-continuum bound, separate finite-volume control, and locally uniform removal of resolution at nonzero time. With independently declared mass and action calibration, the limit has the free mechanical action and exact finite-slice normalization. For smooth positive variable couplings, midpoint edges and uniform site measure select a divergence-form operator. Different measures or orderings retain the same classical principal symbol while changing lower-order terms. The Liouville coordinate and amplitude transformation fixes a local correction, endpoint factors, and the internal measure; these must be carried together. Smooth-orbit lattice and state-level time-slicing estimates give controlled comparisons without asserting a raw variable-coefficient kernel limit. The contribution is the organized conditional bridge and its obstructions, not the invention of the standard component methods or a derivation of geometry, the action-scale value, physical couplings, or empirical probability from primitive complementarity.
We calculate the formation and measurement of entanglement from an initially independent two-source preparation in physical--observation dual-axis (PODA) geometry. A common matter--field action determines the linear response kernel, and a Dirac reduction fixes the leading spin coupling. Excitation and return of the field generate an ordered exchange operator that creates a second joint amplitude. The component of state motion normal to the product-state manifold determines the initial loss of local purity. In a finite cavity, a polynomial embedding bounds the difference between the full evolution and an effective exchange description over the long preparation interval, with all physical mode families of the chosen background included. At g=8×10−4, the total-variation error in the joint readout distribution is below 7.56×10−5, and an independent bound establishes source entanglement before measurement. We then derive the effects of local transport, unread observation degrees of freedom, and finite Gaussian pointers on the correlations and conditional states. Exact reduction conditions determine when spin variables alone suffice for later predictions. A rotor model and a worked source-to-detector calculation connect these results to independent calibration. The quantum representation is conditional on stated operational assumptions. The numerical bounds apply to the linear-source Hamiltonian defined here; their extension to a microscopic apparatus requires estimates for the additional orbital and nonlinear field terms.
The same position need not imply the same future: observing an address may discard information about past events, active memory, and records. This work determines the data needed to reconstruct a temporal law and reconcile its spatial, coherent, and record readouts. Admissible extensions of a history define a graded generator of actualization; marked event rates specify its dynamics in physical time. We establish separate criteria for recovering the event count from an address and for the predictive sufficiency of that address. A distinguishing future test yields a sharp lower bound on the error of a purely spatial description. On a finite strongly connected state space, marked cycle probabilities classify strictly positive laws up to local changes of clock. For factorized first-event records, outcome fractions form a complete metric invariant after a common reparametrization of age has been factored out. On a finite irreducible state space, zero configurational entropy production characterizes the unitary continuation of the same temporal form. Its invariant complex sectors realize all finite-dimensional autonomous Hamiltonian dynamics, up to a global phase. Preparation, probabilistic readout, and retention of a sector require separate operational conditions. The zero-support condition for address jumps acquires the precise meaning of preserving the admissible state space, provided that the continuous evolution is also checked. For a phase-pure QND writer, an independently calibrated rate of record discrimination predicts transition suppression and a sufficient event-count budget on a finite interval. The joint pressure of transport and recording distinguishes realizations with identical separate marginals. Marked irreversibility may persist even when the configurational dynamics is reversible. These results define a testable sequence: reconstruct the event law, determine its sufficient readouts, and compare predictions across instruments. Equilibrium event algebras, protected phase constructions, and local clocks for infinite systems extend this sequence under explicitly stated assumptions. This preprint develops the temporal sector of the Theory of Structural Articulation (TSA).