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 $4\to3\to4$ 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.
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.
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.
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.
This paper develops a constructive local-to-global extension of the Complementary Relational Principle (CRP) and Complementary Relational Dynamics (CRD). It investigates how local two-input relations can be assembled into omnidirectional networks with consistent comparison, observation, reduction, and dynamical evolution. Directional states are indexed by the circle S¹, the sphere S², or the frame-rotation group SO(3). These angular parameter spaces are distinguished from physical spatial dimensions. Positive two-direction kernels provide a common relational representation, while cocycle consistency and connection-based transport specify how locally represented states are compared. The mathematical constructions include positivity-preserving finite directional sampling, gauge-invariant transported relations, reference-routed kernels in the presence of curvature, spherical-harmonic reduction, and quantitative memory corrections under weak anisotropic coupling. Exact block elimination produces a joint direction–time memory kernel, and a contraction argument constructs a causal nonlinear memory functional under stated hypotheses. Two physically distinct applications are developed: visibility-weighted multiview reconstruction with error and covariance bounds, and coherent interference through quadratic detector readouts. An observation-fiber criterion characterizes autonomous reduced dynamics, while a two-state error bound quantifies the predictive consequences of discarded information. A specified kernel–connection feedback model supplies a globally well-posed, positivity- and trace-preserving dynamical core. The paper also distinguishes ordinary 360-degree complex-phase return from spinorial 720-degree frame return. The latter is constructed through an associated SU(2) spinor bundle and becomes observable as a relative sign against a coherent reference; it is not inferred from the imaginary unit alone. These results establish a modular relation–observation–promotion architecture without identifying stereopsis, interference, gauge transport, and fluid-mode reduction as the same physical law. The paper is a companion to “Complementary Relational Dynamics and Three-Dimensional Closure” and preserves its conditional Hodge-degree argument.
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.
This study asks which structures can be derived from a single finite relational system to which no prior physical meaning, coordinates, or time are assigned. A selector-free relational lift and its minimal self-sustaining orbit generate an intrinsic separator order and grading, providing a directed order axis without an external clock. Boolean-diamond incidence defines transverse locality, and five-K4 incidence produces an exact S3 minimal persistent complex whose canonical continuation yields a four-dimensional associated realization. An explicit finite witness gives a nonzero commutator between longitudinal incidence and transverse locality, ruling out a simple product of fixed space with independent time. A complementary stable-quotient readout contains a refinement branch that closes as a one-dimensional metric-graph continuum. Combined with the S2 fiber of the standard S4 representation, this gives a regular spatial sector locally of the form (0,1) x S2(R). The S2 Laplace spectrum provides a branch-independent nonzero rest gap for every l >= 1; under the stated internal second-order wave representation this gap is read as a rest-mass parameter. Gravity is represented by an exact finite source-to-field map using the Moore-Penrose Green operator, and the regular three-dimensional branch yields the local 1/r^2 acceleration exponent in the weak-static regime. The same relational development also supplies three distinct internal Hilbert representations. Conditional on the full orthogonal-projector event structure and standard noncontextual probability consistency, Gleason's theorem fixes the Born form in the spatial representation. The incidence representation is noncommutative and yields the Robertson uncertainty relation, while procyclic reaction gives exact discrete unitary dynamics by Haar-measure invariance. No canonical identification of these Hilbert spaces, unique continuous Hamiltonian, measurement-update rule, spin-1/2 structure, or Bose/Fermi statistics is claimed. Complete derivations and branch audits are provided in a Supplementary Formula Atlas.
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.