We formulate Quotient Relational Time (QRT) for regular clock sectors ofconstrained systems. Observational equivalence first defines a quotient;constraints and candidate clock reductions must descend to it. Positivephysicalization is separate data. On a finite-rank sector, an orderedsecond-order constraint admits a selected first-order branch precisely whenits operator Riccati defect vanishes. We characterize that branch as aninvariant graph of the companion system and derive the propagation of itsresidual when factorization is approximate. Norm distortion and the actualbranch residual give a state-error bound, while a nested integral of theRiccati defect gives a sufficient error certificate. The resulting qualityquantities $\mathsf N,\mathsf D,\mathsf R$ are invariant under smoothquotient-frame changes and orientation-preserving clock relabellings.Candidate reductions can be compared on a common normalized initial-dataspace; identifying them with the same parent physical states requires afurther compatibility condition. Compact continuous families admit optimalerror bounds. A scalar calculation calibrates the comparison and makes thecertificate explicit. These results separate algebraic descent, branchselection, positivity, and analytic error control. They do not supply apreferred clock, a positive-frequency prescription, or a physical innerproduct for an unspecified constrained model.
We formulate Quotient Relational Time (QRT) for regular clock sectors ofconstrained systems. Observational equivalence first defines a quotient;constraints and candidate clock reductions must descend to it. Positivephysicalization is separate data. On a finite-rank sector, an orderedsecond-order constraint admits a selected first-order branch precisely whenits operator Riccati defect vanishes. We characterize that branch as aninvariant graph of the companion system and derive the propagation of itsresidual when factorization is approximate. Norm distortion and the actualbranch residual give a state-error bound, while a nested integral of theRiccati defect gives a sufficient error certificate. The resulting qualityquantities N,D,R are invariant under smoothquotient-frame changes and orientation-preserving clock relabellings.Candidate reductions can be compared on a common normalized initial-dataspace; identifying them with the same parent physical states requires afurther compatibility condition. Compact continuous families admit optimalerror bounds. A scalar calculation calibrates the comparison and makes thecertificate explicit. These results separate algebraic descent, branchselection, positivity, and analytic error control. They do not supply apreferred clock, a positive-frequency prescription, or a physical innerproduct for an unspecified constrained model.
We relate positive observable quotients to Page--Wootters (PW) clockdescriptions. Physicalization identifies a quotient by its kernel, andjointly faithful clock reductions recover the same exactness relation.Admissible PW data give a faithful functor into an independently definedquotient-clock category after representative lifts are identified modulophysical null directions. With algebraic lifts, it is full on its essentialimage at the level of retained state-and-clock data. A restricted converse characterizes strictly periodic, globallydeparametrizable realizations with unitary one-slice conditioning:one reference reduction must be unitary and the relative reductions mustform a strongly continuous periodic unitary group. The construction includesthe closed-operator domains required for unbounded generators. Retaining clock-event probabilities gives a separate strictness theorem.Normalizable stationary models with additive constraints and covariantperiodic clocks have state-independent clock marginals. A positivetwo-qubit constraint quotient violates this invariant despite invertiblereductions. This separates it from the additive class; its realization isitself interacting. We then define a broader stationary-conditioning class allowing generalizedcovariant clocks, unrestricted self-adjoint interactions, and separableenvironmental dilations. Its normal quantum-instrument data descendisometrically to positive quotients. Conversely, every normalized amplitudequotient-clock datum has an exact realization: a positive projectorconstraint selects the image of its isometric history map. Standardinstrument dilation extends the result to absolutely continuous normalcompletely positive instruments. For finite-dimensional physical and outputspaces with Sobolev-regular histories, a bounded finite-rank interactionsuffices by standard Hermitian completion and spectral isolation. Hencethis normalized operational sector has no obstruction to realization inthe broad class. For a fixed canonical periodic clock, we then bound the interactionremaining after arbitrary bounded clock-local Hermitian terms are allowed.Translation Fisher information gives a lower bound valid under separableenvironmental dilations. For instruments with orthogonal amplitude columns,a finite-rank construction attains the exact minimum while selecting theprescribed physical kernel. The modulated qubit family yields an explicitstrength--frequency threshold and a finite-event approximation-error bound. An exact Gaussian differential-image quotient coupled to a full oscillatorprovides an additive realization with matching physical metrics,conditioning, reconstruction, and relational probabilities, including finitereading intervals. These results distinguish restricted strictness frombroad representability; they neither identify all microscopic realizationsnor establish equivalence or strict superiority across every QRT and PWformulation.
An instantaneous relational quotient can lose observable directions at asingular clock reading even when a completed state description remainsreversible. We study this distinction through finite-dimensional stratifiedclock atlases. Surjective specialization maps identify the persistentquotient and their kernels identify the missing directions. Their fiberproduct is the canonical crossing relation; a deterministic crossing requiresadditional memory and shear data. We characterize reversible and unitarycrossings and give a composition law through finitely many singular strata,with explicit boundary-extension hypotheses and frame covariance. A positive 4→3→4 model exhibits reversible transport after the lostdirection is retained as memory. It cannot be represented by a clock atlaswhose every instantaneous quotient is an invertible faithful reduction of onefixed physical Hilbert space. This obstruction concerns faithful reducedPage--Wootters perspectives; it does not exclude clock-neutral theories ornonfaithful reductions with auxiliary memory. A separately derived hard-edgeLaurent quotient supplies a complementary analytic example. Its rank dropsfrom four to three, an adapted extension identifies the vanishing line, andits coefficient connection has a logarithmic solution and nontrivialrank-one unipotent monodromy. Locally transported positive metrics exist, butno positive-definite Hermitian form is invariant under that monodromy. Thehard-edge analysis supplies analytic specialization and memory data, not aderived physical crossing law or gravitational clock realization.
Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.
This paper asks a basic structural question: when does a state-dependent clock represent physical structure rather than a mere reparameterization of time? The starting point is deliberately restrictive. If a dynamical vector field is multiplied everywhere by a strictly positive scalar lapse, then the oriented state-space trajectories are unchanged; only their parametrization changes. A state-dependent clock rate therefore does not become physical by declaration. This positive-lapse reparameterization result is used as a no-go gate for subsequent clock models. The paper then identifies three distinct structures that can survive this test: Relational nonuniversality: Different clock sectors can respond differently to the same residual datum, so normalized clock ratios retain observable differential information even when a universal common mode cancels. Covariant lapse-response stress: When the lapse-dependent sector is embedded in a generally covariant action, a nonzero Hilbert-tensor contribution provides a sufficient tensor witness that the lapse dependence is not removable by coordinate-time relabeling alone. Operator nonscalarity: For KT=L−1/2HL−1/2, a strongly commuting clock operator L is equivalent to a single scalar time rescaling only when it is spectrally scalar on the nonzero-energy support of H. A minimal gauge-invariant realization is built from the residual one-form α=dΦ−A, which sources a positive operational lapse and a positive residual generator-cost form. Under the stated admissibility, density, boundary, and coercivity hypotheses, the zero-cost sector is the aligned vacuum almost everywhere. The same framework yields a lapse-dependent Hilbert stress response and a regulated operator realization near clock-degenerate sectors. For a network of clock comparisons, the paper also derives an exact relational structure. If κi denotes the logarithmic clock response at vertex i and D is the graph incidence matrix, then pairwise residuals satisfy r=Dκ. On a connected graph, the global common mode is unidentifiable, while the cycle-space component provides an exact noiseless compatibility test for the scalar vertex-clock model. Loss of a positive lower bound is treated separately as a boundary flag B∂. It signals that the global positive-lapse equivalence argument has left its domain of applicability, but it is not taken by itself as evidence of physical inequivalence. The resulting framework is intended as a theorem-controlled physicality criterion for residual clock–energy models: a physical clock is not identified merely by state dependence, but by relational, covariant-stress, or operator structure that survives the reparameterization test.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0. Its spectrum is λa,k=1−cos((2πk−θa)/L), where eiθa∈spec(M). Thus the exact history sector is isomorphic to Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step U, the minimal number of clock events is the projective order of U. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZL on a rank-r history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
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).