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.
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 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.
Physical--observation dual-axis (PODA) Theory holds that physical reality is realized jointly through physical state and observation state. A change of observation state can therefore change the realized physical fact even when the physical state is held fixed. We develop the phase law of this PODA structure. A coherent readout of the joint observation law gives a smooth scalar complex response g. Wherever this response is nonzero, differentiating its polar decomposition gives the observed phase rate and the one-form Ag=Im(dg/g). A transported phase with the same increments differs from the observed phase by a constant reference factor. We derive its differential equation and unique initial-value solution, and show that the equation is horizontal lifting for the unique response-compatible connection on the trivial principal U(1) phase bundle. We call the resulting equation c∗Ωg=0, the Phase Transport Fundamental Equation (PTFE), the first fundamental equation of observation space in PODA Theory. Its group-valued, covariant-derivative and continuous real-phase forms express one transport law: a path and an initial phase determine a unique horizontal lift. Integration gives finite phase comparison, composition along successive paths, and inverse transport between two observation states of the same physical source. Positive amplitude rescaling leaves the law unchanged; synchronized changes of response and fiber coordinates preserve its covariant form. The induced connection is flat and has trivial circular holonomy, although a continuously unwrapped phase may wind. We distinguish the full response G0, the phase-transport structure G1, and the task quotient G2. At the last level, maximal invariants describe the quotient, and a target can be recovered exactly when it is constant on each fiber of the quotient observation. Phase transport and the information retained after a task reduction thus follow from the same joint observation law.
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.
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.
Whether observation dynamics can leave distinct matter records under identical physical initial conditions and subsequent external forcing depends on three questions: whether observation initial states have a well-defined identity, whether they can be prepared, and whether their evolution satisfies the common field equations. We construct a local realization within the existing five-dimensional response space and quadratic common action of PODA. With spatial and temporal references fixed, primitive response transport yields nonzero mixed curvature and identifies the observation coordinate through distinguishable comparison records; the image of its field data is an invariant subspace of the transverse dynamics. The field restoring force and a nonzero current overlap allow a single matter input to prepare arbitrary sufficiently small joint initial states. An explicit eighth-degree polynomial protocol carries the state initially at rest to two final states with identical matter position, matter velocity, and observation configuration, but opposite observation velocities. The input vanishes continuously at both endpoints, and the net work equals the stored energy. After release, the two-frequency displacement and the observation comparison probability obey a common integral identity. We also derive propagation relations for initial-state mismatch, finite readout precision, and control-waveform errors, and retain the complete calculations for matter sources, the one-loop relativistic response, and indirect tests involving common parameters. The result is an existence theorem for preparation and response in the specified local realization, within the stated quadratic dynamics and external matter-source model.