Phase transport specifies the change of phase along a path, but a predictor must also determine the path from initial data. We study this distinction in a conservative projector–connection model on the physical--observation dual-axis structure. A homogeneous two-mode reduction preserves the field equations, Gauss constraint, and energy, and has a global classical flow. Horizontal transport followed by a fixed analyzer gives a nonvanishing coherent readout on an invariant plane. With known parameters, no active input, and zero Gauss momentum, we derive a closed system of seven inner products for the planar branch. The Jacobian of the phase rate and its first six time derivatives has full rank at an explicit rational state, proving that seven is the minimal smooth predictive dimension on an open dense set. The initial phase is independent of these seven states. Near equilibrium, the linear observable motion reduces to two states; on a separate invariant family, internal motion leaves the phase constant. These results are placed in a protocol-dependent realization theory based on future phase increments. We also derive a background-dependent frequency law and the leading nonlinear phase coefficient. Signed perturbations and half-amplitude contrasts isolate the cubic contribution, while an explicit correction separates it from the cubic term introduced by coherent integration. Dynamic references and finite-window array processing illustrate the distinction between instantaneous alignment and phase prediction. All numerical examples are synthetic; the proposed physical tests require independent calibration of a device realizing the stated action and readout.
Phase transport specifies the change of phase along a path, but a predictor must also determine the path from initial data. We study this distinction in a conservative projector–connection model on the physical--observation dual-axis structure. A homogeneous two-mode reduction preserves the field equations, Gauss constraint, and energy, and has a global classical flow. Horizontal transport followed by a fixed analyzer gives a nonvanishing coherent readout on an invariant plane. With known parameters, no active input, and zero Gauss momentum, we derive a closed system of seven inner products for the planar branch. The Jacobian of the phase rate and its first six time derivatives has full rank at an explicit rational state, proving that seven is the minimal smooth predictive dimension on an open dense set. The initial phase is independent of these seven states. Near equilibrium, the linear observable motion reduces to two states; on a separate invariant family, internal motion leaves the phase constant. These results are placed in a protocol-dependent realization theory based on future phase increments. We also derive a background-dependent frequency law and the leading nonlinear phase coefficient. Signed perturbations and half-amplitude contrasts isolate the cubic contribution, while an explicit correction separates it from the cubic term introduced by coherent integration. Dynamic references and finite-window array processing illustrate the distinction between instantaneous alignment and phase prediction. All numerical examples are synthetic; the proposed physical tests require independent calibration of a device realizing the stated action and readout.
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 quantum phase reference is part of the state from which an observation record emerges. Within the physical-observation dual-axis structure, we describe such a reference by a compact quantum coordinate coupled to a finite-dimensional physical system. A covariant rotor action gives the joint evolution and expresses the readout's phase acceleration through ordered correlations and a covariant force. In a qubit model whose dynamics preserve two adjacent observation momenta, the first harmonic depends on four complex correlations. For fixed nondegenerate parameters, its full complex history requires eight real state variables. For phase increments alone, a common amplitude and phase factor can be removed, leaving three complex correlation ratios. We prove that these six real variables are locally minimal at generic states for every fixed nondegenerate parameter choice. Six suitably timed phase estimates and an initial reference recover the ratios by solving real linear equations; an error bound relates uncertainty in those estimates to the subsequent phase prediction. The role of the retained correlations is illustrated by preparations with equal marginals and equal initial phase, rate, and acceleration, but different future phases. We also describe how the state continues through a zero of the first harmonic and how finite integration windows affect the available information. The resulting theory connects a joint quantum action to phase prediction from finite records, subject to independent physical characterization of the phase inertia and coupling.
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.
Coherence lifetimes reflect both the evolution of a quantum system and the conditions under which its state is reconstructed. We study these contributions in Physical--Observation Dual-Axis (PODA) dynamics. For a coherent-state superposition in a passive thermal cavity, we derive the interference envelope and obtain initial-rate lifetimes of 19.73 and 28.89 ms from published apparatus parameters. These values agree with the scale of the reported decay, although the envelope lifetime and the time extracted from a tomographic fit need not coincide. The observation coordinate is then allowed to evolve under the same coherent action. Projection onto an isolated cavity mode determines the frequency response and its reciprocal optical force; propagation of the initial covariance gives the relative-phase variance, including the readout reference. For coupled quadratic motion with a classical canonical preparation and a fixed readout reference, a fluctuation--response identity determines the phase-coherence curve from the reciprocal susceptibility and preparation energy. A finite set of stable modes gives recurrent or quasiperiodic coherence. A second relation predicts the ratio of two coherence curves when only the initial observation covariance changes. The common thermal attenuation cancels from this ratio, while the induced physical response remains. Together, these relations connect coherence to response and preparation measurements without introducing a fitted phase-decay rate.
This paper studies phase formation, continuous comparison, and cross-state transport when the observation map varies with the state of an observer. Starting from the classical model y = h(x), we prove that a complete representation of a nontrivial family of observation maps requires an independent variable position for the current map. State realization and its canonical quotient then yield an observer O = (SO, ΓO) and an observation map FO : X × SO → Y , establishing a three-layer paradigm of the physical world, the observer, and the observation world. For a selected scalar coherent complex readout, the observation map induces the response g = r ◦ (FO|M). On the nonzero domain B = {m : g(m) = 0}, the observed-phase one-form is Ag=Im(gdg). Amplitude–phase reduction of the full complex-response bundle produces a principal U(1) phase bundle. Given g and the standard phase normalization, we prove the existence and uniqueness of the compatible connection Ωg=prU(1)∗ωU(1)−πph∗Ag, whose horizontal condition reproduces the local phase law. The first-order differential functional equation on lifted path space, c∗Ωg=0, is established as the basic phase-transport equation. It is equivalent to the group-valued differential form, the covariant-derivative form, the continuously lifted real-phase form, and the finite path-transport functional. The equation is uniquely induced by the complex response, is equivalent to horizontal lifting under the unique compatible connection, uniquely determines finite transport and path composition for prescribed initial data, is covariant under local phase-gauge transformations, and is invariant under positive amplitude rescaling. Inverse same-source double-state phase transport follows directly on fixed-physical-state slices. Finally, three representation levels are constructed: the full complex-response level G0, the phase-differential and transport level G1, and the task-quotient observation level G2. Their relative completeness is proved for their respective problem classes, providing a unified mathematical structure from variable observation maps, complex responses, and phase connections to phase transport, maximal invariants, and target recoverability.
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.