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.
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.
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.
A detector records an event through a succession of physical changes, from its initial response to the recovery of its working state. We construct a complete finite cycle on the local physical--observation phase space of PODA. An incoming carrier traverses fixed interaction regions, with its arrival, speed, and energy transfer governed by an autonomous Hamiltonian bounded below. Six canonical gates write a smooth event profile, amplify the pointer separation, export the record, reverse the working displacement, and exchange the remaining pointer state with a prepared blank. Their exact reference map identifies the selector and mechanical recoil left by record export. The retained core returns on the event-cell plateaus; narrow transition strips contribute a bounded probability of failure and a recoil that depends on the resolution. With a positive free Hamiltonian present throughout the passage, energy conservation and a finite-time perturbation estimate establish faithful writing, finite retention, and return within prescribed tolerances. An explicit binary realization admits nonzero preparation errors, and independent trajectory integrations check the equations and the endpoint map. These results complete the single cycle for the specified local model and incoming family. The calculation follows the archive, carrier, and waste through the output section, making explicit the resources consumed by reset. Preparation of fresh blanks, replenishment of those resources, and storage beyond the proven holding interval remain physical requirements of continued operation.
This theoretical and computational preprint integrates standard QND measurement and posterior-martingale results with finite-confidence decisions and physically retained observer records. Under a declared quantum instrument and its standard outcome probabilities, the posterior is a bounded martingale on an increasing observation filtration. A precise sufficient identification condition yields asymptotic terminal weights; a binary Gaussian benchmark gives exact mean stopping times, finite-threshold latch probabilities, and a controlled population-transfer bias bound. Version 2.3 is the first public release. It distinguishes accumulated observations from lossy current memory; assumed Born/instrument probabilities from derived consistency results; expected entropy from pathwise ambiguity; continuous confidence endpoints from discrete overshoot and unresolved runs; and Kramers first escape from endpoint readout error. It also clarifies the conditional D18/D19 persistence-selector interface and the separate D1092 finite-resource ledger, with direct attribution to established QND convergence and filtering literature. The companion archive includes the editable manuscript, PDF, equation sources, a fixed validation protocol, executable independent checks, numerical results, a revision ledger, and unchanged historical inputs. Historical simulation summaries are explicitly distinguished from new executions; some older executable sources were unavailable. The new audit includes exact finite-kernel and stopped-posterior checks, Gaussian likelihood-mixture calculations, five neutral and four population-transfer boundary-value cases, and retention/accounting controls. All examples are analytic or synthetic. No laboratory measurements, independent Born-rule derivation, measurement-problem solution, universal physical selector, or new empirical anchor are claimed.