Born probabilities determine the distribution of individual records, but leave their temporal relations open. In the U(1) M-event model a transported circle phase carries path memory; fixed transports on the same fiber nevertheless commute. We extend the event selector to a nontrivial compact connected finite-dimensional Lie group within the physical–observation dual-axis structure. One action governs the selector, its conjugate charge and their coupled motion on the two-axis base. A Haar initial selector, independent of the other initial data and prescribed controls, preserves the Born probabilities at every observation. Non-Abelian transport changes the relations between records. For balanced hemisphere readout on the full SU(2) group, we derive the exact law Pmis = α/π, relating the mismatch probability to the transport’s conjugacy angle. In a U(1) × SU(2) realization, a central charge sustains the original M response, while a periodic potential preserves its orbit and phase integral and makes finite deviations exactly harmonic. A classical pointer action gives hemisphere readout and a two-period SU(2) return for the stated apparatus preparation. With the faithful fixed-axis coupling, four blocks of 25 periods produce α25 = 1.012373567 . . .. If readout retains the selector after either outcome, the third supercycle differs from the initial record with probability 96.6746%, and the sixth agrees with it with probability 93.3491%. Independent response and matrix measurements fix these values before the event sequence is observed. Scans of axis angle and block length test the relation between noncommuting transport and event memory.
Born probabilities determine the distribution of individual records, but leave their temporal relations open. In the U(1) M-event model a transported circle phase carries path memory; fixed transports on the same fiber nevertheless commute. We extend the event selector to a nontrivial compact connected finite-dimensional Lie group within the physical–observation dual-axis structure. One action governs the selector, its conjugate charge and their coupled motion on the two-axis base. A Haar initial selector, independent of the other initial data and prescribed controls, preserves the Born probabilities at every observation. Non-Abelian transport changes the relations between records. For balanced hemisphere readout on the full SU(2) group, we derive the exact law Pmis = α/π, relating the mismatch probability to the transport’s conjugacy angle. In a U(1) × SU(2) realization, a central charge sustains the original M response, while a periodic potential preserves its orbit and phase integral and makes finite deviations exactly harmonic. A classical pointer action gives hemisphere readout and a two-period SU(2) return for the stated apparatus preparation. With the faithful fixed-axis coupling, four blocks of 25 periods produce α25 = 1.012373567 . . .. If readout retains the selector after either outcome, the third supercycle differs from the initial record with probability 96.6746%, and the sixth agrees with it with probability 93.3491%. Independent response and matrix measurements fix these values before the event sequence is observed. Scans of axis angle and block length test the relation between noncommuting transport and event memory.
The Born rule assigns a probability law to a physical state and a complete observation state, but a probability law is not the event realized in one trial. Building on the physical–observation dual-axis state space, we construct an event layer that separates these two mathematical objects. Every measurable finite Born kernel admits a deterministic circle lift whose event cells have the required normalized Haar volumes. On a principal U(1) selector bundle, cell membership is gauge invariant and Haar reduction returns the Born trace law exactly. A gauge-invariant action yields both selector transport and an observation-axis equation in which mixed curvature is the cross-axis force. The Fisher metric of the Born response supplies its minimal kinetic geometry, while a least-interface principle fixes connected binary event cells. Curvature determines relative transport but cannot select an absolute event without an initial selector; pure principal transport preserves Haar equilibrium but cannot create it. Bell violation further requires the deterministic event map to be contextual on the implemented joint setting. The event law leads to a concrete test of single-event continuity. The physical control is an azimuthal cycle and the observation coordinate is calibrated independently from the binary Born response. The benchmark states three additional closures explicitly: a reduced spherical completion with rotationally invariant unit-Chern curvature, minimal free response dynamics, and a selector-transparent returning instrument. The fixed dimensionless protocol X = 0.100 and ωS/Ω = 4 has a unique balanced periodic response orbit and predicts the protocol constant κM = 0.03080966337. No event record enters this calculation. A returning measure-and-prepare instrument then predicts a triangular mismatch recurrence: 0.980702% after one cycle, 99.9684% after 102 cycles, and 0.0632% after 204 cycles, although every single-cycle marginal remains balanced. The memoryless realization of the same instrument gives 50% mismatch at every nonzero lag. Separate response, phase-calibration, and event ensembles prevent the predicted orbit from being programmed into the test, while causal-break and selector-scrambling controls bound ordinary apparatus memory. This numerical “M-event fingerprint” converts the century-old question of objective single-event continuity into a preregisterable, falsifiable serial-correlation experiment. It is a protocol-specific prediction, not a new natural constant; a null result falsifies the stated minimal realization rather than every possible deterministic completion.
A closed observation path need not return the event selector to its initial position. The displacement left by transport can relate successive records even when their individual probabilities are unchanged. A nonzero charge conjugate to the selector also acts back on the observation coordinate, so the path and its transport must be determined together. For a driven U(1) × SU(2) realization of the PODA (physical–observation dual-axis) action, charge return requires the initial traceless charge to commute with the period holonomy. The coupled equations then admit a variational form whose geometric term is the holonomy angle. Unit-frequency compensation excludes nearby charged periodic motion, whereas nonresonant compensation yields a unique local branch with linear neutral stability of the response and charge at fixed charge length. Four-block symmetry reduces this branch to one scalar consistency equation. The angle susceptibility is α ′ (0) = −0.2490598046 . . .. On a finite continuation interval, we prove that exactly one charge gives W3 = −I and W6 = I; its calculated value is ρ∗ = −0.1377922246 . . .. Passive readings that retain the motion and selector therefore give an exact reversal after three cycles and return after six, while independent Haar preparation preserves the Born marginals. Calibration of the response and charge fixes these correlations before the event records are compared. Bounds on angle errors and finite samples specify the precision with which the prediction can be tested.
Born equilibrium and temporal event correlations are two properties of the same dynamical ensemble. We connect them in a reservoir-coupled realization of physical–observation dual-axis transport. A stable affine response with phase-covariant Gaussian kicks admits an exact phase coordinate that isolates the persistent innovation. Under response controllability, its variance D is positive exactly when every initial joint distribution converges to the unique response–Haar equilibrium. The convergence holds after conditioning on the current response. An explicit finitetime bound transfers to every prescribed Born event partition, and the optimal preparation-uniform exponential rate is D/2. Correlated kicks with D = 0 retain a rotating invariant even when both response and recorded phase fluctuate. The same Fourier multiplier governs the relaxation of a prepared phase harmonic and the decay of its matching two-time M-event fingerprint. This identity fixes their rate relation within one channel and leads to a two-stage protocol: strong mixing prepares equilibrium, followed by quieter fingerprint readout. Variable-width event cells reveal harmonics invisible to balanced counts. A complete numerical preparation–readout sequence and finite-time bounds for perturbed instruments connect these results to measurable populations and correlations, with the original M-event cycle phase unchanged.
Can a measurement cycle stabilize an observation response without erasing the event memory carried through it? We address this question in the classical variational event layer of a physical– observation dual-axis model. An explicit canonical record coupling preserves the selector angle but disturbs its conjugate charge, which also controls the subsequent response. We prove an obstruction to informative readout that preserves the complete selector pair on an open phase-space domain, and derive a resolution–backaction bound for smooth binary event cells. The free benchmark orbit is hyperbolic, with Floquet multipliers 2.408265755267 and 0.415236565073. A meridional reset generated by a response potential and selective dissipation adds no selector phase. At fixed selector charge and with pair-transparent readout, its isotropic contraction factor stabilizes the response precisely when a < 0.415236565073. The attracting response circle retains a neutral phase direction; a convergent asymptotic-phase correction reduces the nearby dynamics to the original rotation. For independent Gaussian cycle noise, an exact Fourier law determines recurrence broadening. Unrestored martingale charge kicks instead produce cubic accumulated phase variance in the linear response regime. The resulting theory separates response stabilization, instantaneous phase preservation, and conservation of the coupling that carries memory into the next event. It supplies calculable readout and reset mechanisms while preserving the original ideal M-event phase.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
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.