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 $$K_T = L^{-1/2} H L^{-1/2},$$ a strongly commuting clock operator $\widehat L$ is equivalent to a single scalar time rescaling only when it is spectrally scalar on the nonzero-energy support of $\widehat H$. A minimal gauge-invariant realization is built from the residual one-form $$\alpha = d\Phi - A,$$ 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 $\kappa_i$ denotes the logarithmic clock response at vertex $i$ and $D$ is the graph incidence matrix, then pairwise residuals satisfy $$r = D\kappa.$$ 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 $B_{\partial}$. 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.
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.
Version v1.0. This work develops a minimal effective framework for residual-sourced operational-time rescaling in a U(1) phase sector. The gauge-invariant residual one-form α=dΦ−A is used to define a positive phenomenological clock response Lα=L0exp(−λ∣α∣h2),dT=Lαds. In the scalar-lapse sector, the generator with respect to operational time is KT=Lα−1Hs. Consequently, generator variance and pure-state quantum Fisher information scale as Lα−2, while the Bures line element of the same physical trajectory remains invariant under the reparameterization. The scalar lapse therefore changes information rate per unit operational time without, by itself, producing an intrinsic change in the quantum-state geometry. Independently of this kinematic construction, the paper considers an effective residual gradient flow α˙=−∇admA on a fixed Hilbert configuration space. This yields the Lyapunov identity dTdA[α(T)]=−∥∇admA∥H2≤0 and a monotone residual-dissipation functional. The dissipative law is an additional effective assumption and is not derived from the lapse law or from generator rescaling. The manuscript also separates the covariant gauge-invariant core from the foliation-dependent positive spatial norm entering the operational lapse, clarifies the distinction between Fisher-information-rate rescaling and intrinsic Bures geometry, and records the correction required when the background geometry is time dependent. The model is not proposed as a microscopic or fundamental theory of time, irreversibility, decoherence, or quantum measurement. Its purpose is structural: to isolate a common residual variable that can enter clock normalization, generator rescaling, and an independently assumed effective relaxation while keeping their logical dependencies explicit. The paper identifies several open problems required for a more predictive theory, including a microscopic derivation of the clock-response law, a controlled derivation of dissipative mobility, a covariant dynamical clock or foliation sector, quantitative cross-channel observables, and independent empirical tests.
**Version v0.5 — Journal-Polish Update** This manuscript develops a conservative extension framework for Schrödinger dynamics in which an observed quantum system is embedded into an enlarged Hilbert space equipped with an auxiliary clock-information fiber. The fiber is not introduced as a macroscopic hidden spatial dimension, nor as a local hidden-variable register. Instead, it is treated as a quantum auxiliary sector carrying phase memory, clock/reference correlations, and correlation records that may appear as information loss or dephasing after partial tracing. The central contribution of this version is a residual-identifiability framework. Rather than claiming that unitary dilation itself is new, the paper asks whether a calibrated observed system can exhibit structured residuals after independent-bath subtraction and common-bath controls. The main diagnostic object is a log-residual statistic, Rχ(t;σref)=logCobs(t;σref)−logCind(t;σref), together with reference-covariance indices such as Iχ(t;σref)=∂σrefRχ(t;σref), and covariance/correlation-based diagnostics over controlled reference settings. The framework emphasizes that a nonzero residual alone is not sufficient evidence for an admissible clock-information fiber. A candidate residual must be non-factorizing, reproducible, reference-covariant, and not absorbed by independent local bath models, common classical noise, or generic common quantum bath controls. In particular, the admissible-fiber hypothesis is treated as a residual diagnostic condition, not as a proof of a new hidden variable, superluminal channel, or macroscopic extra dimension. The paper proves the elementary consistency statements for the enlarged dynamics: total unitarity under a self-adjoint total Hamiltonian, total energy conservation for time-independent generators, subsystem entropy growth under partial trace, and no-signalling compatibility under local trace-preserving operations. It also presents minimal toy models for phase-memory dephasing, shared-fiber cross-kernels, compact protected fiber sectors, and synthetic residual-fitting demonstrations. The synthetic fitting layer uses a representative model of the form $$C_{\rm obs}(t;\sigma_{\rm ref})=C_{\rm ind}(t)\exp[-a(\sigma_{\rm ref})t^2],\qquada(\sigma_{\rm ref})=a_0+a_1\sigma_{\rm ref}^2,$$ so that Rχ(t;σref)=−a(σref)t2. This example is intended as an identifiability demonstration: after calibrated independent-bath subtraction, a candidate clock-information fiber signature should appear as a reproducible residual with controlled reference covariance, rather than as an arbitrary fitting term. The experimental outlook focuses on Ramsey/spin-echo baselines, dynamical-decoupling response, superconducting qubit and Josephson platforms, tunneling phase-delay diagnostics, and reference-modulation protocols. The manuscript also formulates a common-bath control envelope, ∣Rχ(t;σref)−Rctrl(t;σref)∣≤δR(t;σref), to express the requirement that a proposed fiber residual should not be absorbed by generic common-bath control models within statistical uncertainty. ### Main updates in v0.5 - Shortened journal-style title and abstract.- Removal of version-history language from the abstract.- Strengthened common-bath control criterion.- Addition of a formal common-bath control envelope.- Improved diagnostic separation between independent local baths, common classical noise, generic common quantum baths, and admissible clock-information fibers.- Algorithm-style synthetic residual-identifiability test.- Synthetic residual-fitting demonstration using reference-covariant dephasing.- Improved table formatting and manuscript presentation for journal-style readability. ### Scope and non-claims This work should be read as a structured framework for residual diagnostics in extended Schrödinger dynamics. It is not a hidden-variable theory, not a superluminal-signalling model, not a derivation of measurement collapse, and not a completed microscopic theory of mass, charge, or quantum gravity.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
SCQG Paper 4 of 28 This paper explains how different physical ways of measuring and ordering time relate to one another. Proper time is defined by the spacetime metric along a path. A material clock produces a reading through its own dynamics and calibration. A detector writes a timestamp, and exchanged signals allow separate clocks to be synchronized. A temporal bridge is a map connecting such coordinates under stated physical conditions. Semantic time counts or orders the interactions through which a physical distinction becomes usable, compared, or maintained. Forward actionability is the resolution of uncertainty into a distinction that a later operation can use. Local grounding depends on forward action making that resolution available to a local interaction; measurement realizes the grounding by writing a registration. Access to a readable record, inference meeting a specified accuracy test, and completed durable maintenance have separate first-success times. The paper derives causal access conditions, timing bounds for finite detector windows, synchronization errors, and relations between physical clock time and interaction counts. It defines agreement between event orders and entrainment: agreement maintained through physical coupling over a stated interval. It also distinguishes the accumulated interaction count from the records that remain readable. Applications include quantum clocks, accelerated observers, black-hole observations, cosmological modes, delayed-choice experiments, and reciprocal cosmological coordinates. Each connection states the physical systems, operations, units, calibration, and conditions under which it holds. Commitment Time, Metric Time, and Physical Temporal Interfaces Proper Time, Clocks, Synchronization, Addressability, and Durable Commitment This work develops a unified physical architecture for the many different quantities that physics calls “time.” Modern theories routinely move between coordinate time, proper time, clock readings, detector timestamps, synchronization conventions, thermodynamic arrows, quantum relational clocks, record formation, inference, and historical reconstruction. These quantities are often written with the same symbol t even though they arise from different physical systems and perform different roles. The central claim of this paper is that temporal quantities are strongly typed. A proper-time interval is generated by a metric and a worldline. A clock reading is generated by a material clock, a readout instrument, and a calibration. A detector timestamp is generated by switching, response, record formation, and estimation. A shared operational time is generated by signal exchange and a maintained clock ledger. Semantic time is generated by physically realized interactions that create addressable distinctions. Thermodynamic time is generated by routed physical exchange. Modular time is generated by an algebra–state pair. Lifecycle time is generated by a branch-history coordinate. These temporal objects can be compared, synchronized, or mapped into one another only through explicit physical bridges. Numerical agreement does not make them the same object. The paper collects the principal temporal coordinates into the owner-complete family 𝔗ₑₓₜ(Ξ)= (t, τγ, qC, τ̂C, Tₒₚ, Iᴰ, t̂ₑ, τˢᵉᵐₚᵣₒₚ, Tᵣₑc, Tₐddr, Tcmp, Tinf, Tlock, τedge, smod, u). Here: t is a coordinate parameter; τγ is metric proper time along a worldline γ; qC is the raw output of a physical clock C; τ̂C is a calibrated estimate derived from that output; Tₒₚ is synchronized operational time constructed through signal exchange and a maintained timing ledger; Iᴰ is the detector’s physical operating support; t̂ₑ is an estimated timestamp assigned to an event e; τˢᵉᵐₚᵣₒₚ counts physically realized semantic interaction steps; Tᵣₑc is the first time an extant carrier becomes readable as a record; Tₐddr is the first time the record becomes physically addressable; Tcmp is the first realized comparison time; Tinf is an inference stopping time; Tlock is the time at which a commitment enters a durable shared ledger; τedge is a coordinate generated by routed thermodynamic export; smod is a modular-flow parameter; u is a lifecycle gauge. The complete package Ξ includes the event, worldline or interaction process, clock subsystem, detector, ledger, route data, calibration data, and any thermodynamic, modular, horizon, or lifecycle structure used by the selected branch. A temporal bridge is represented as a partial typed map: ℬA→B : Dom(ℬA→B) ⊆ 𝕋A × ℝA→B → 𝕋B. Every bridge must identify its source type, target type, carrier, causal support, units, domain, calibration, hypotheses, thresholds, uncertainty, exchange law, and failure surface. When the bridge fails, the source temporal quantity remains meaningful; the target comparison simply remains undefined. This framework replaces the assumption of one universal temporal variable with a network of physically realized temporal interfaces. Relativistic addressability Relativistic causal support is necessary for an event to become available at a later anchor: x ∉ J⁺(e) ⇒ qᵛᵃˡₑ,ₓ = 0. An event outside the causal future J⁺(e) cannot be physically delivered to x. Causal accessibility alone is not sufficient. A record becomes addressable only when a surviving carrier reaches the anchor, enters the detector’s operating support, passes the decoder and validation gates, and is written into an actionable register. The paper therefore distinguishes: causal reach,physical delivery,readability,addressability,comparison,inference,and durable commitment. These stages may occur at different times, and some may never occur. Proper time and physical clocks Proper time is generated by metric geometry along a worldline. A material clock is a dynamical system that evolves along that worldline. Its displayed value is a physical measurement outcome, not proper time itself. A clock reading becomes an estimate of proper time only through a calibration map. Periodic clocks also require cycle provenance: the phase shown on a dial does not by itself identify how many complete cycles have elapsed. The paper proves that proper time and clock readout remain type-distinct even when the clock is highly accurate. Their numerical agreement is produced by a calibrated physical bridge. This distinction becomes decisive in accelerated motion, gravitational redshift, horizon-limited observation, finite detector windows, and quantum clocks. Synchronization and clock ledgers Synchronization is not a property that two distant clocks possess automatically. It is produced through signal exchange, a propagation model, calibration, and a maintained ledger of timing relations. Two-way timing protocols contain route-dependent biases whenever the outward and return propagation conditions differ. A synchronization convention can remove some ambiguity only after the propagation model and admissible equivalence class have been declared. Operational time is therefore a constructed relational product. It belongs to the clock network and its synchronization procedure, not to either clock in isolation. The paper separates static order agreement from dynamical entrainment. Two clocks can preserve the same event ordering without matching rates, phases, or physical mechanisms. Semantic time and interaction anchors Semantic time records the order in which physical distinctions become available for downstream action. Its basic increment occurs at a realized interaction anchor: emission, scattering, absorption, detection, amplification, decoding, comparison, or record formation. Free propagation between anchors carries causal support but does not create an intermediate semantic tick. For a free null carrier, ds² = 0, and the carrier accumulates no proper time. The same free segment also produces no intermediate semantic-interaction increment: Δτˢᵉᵐₚᵣₒₚ = 0. The timelike source and receiver can accumulate proper time, operate clocks, and create records. The freely propagating null carrier connects those interactions without generating a chain of local commitment events between them. This gives a precise temporal account of photons, communication channels, interferometers, and delayed-choice experiments. Finite measurement duration Measurement is not instantaneous. Every physical detector has an operating window, switching dynamics, response latency, finite bandwidth, resolution limits, and a nonzero probability of failure. A timestamp is generated only after the detector response has been transformed into a record and processed by an estimator. The achievable precision depends on the measurement duration, signal energy, distinguishability, detector dynamics, and calibration. The paper keeps three limits separate: the quantum speed limit governing how quickly states can become distinguishable; the detector’s finite response and operating support; the statistical precision of the timestamp estimator. Records, inference, and durable commitment A registration can exist before it becomes readable. A readable record can exist before it becomes semantically addressable. An addressable record can exist before comparison. An inference can be reached before the result is locked into a durable shared ledger. The corresponding stopping times satisfy no universal equality: Tᵣₑc ≠ Tₐddr ≠ Tcmp ≠ Tinf ≠ Tlock. Their order depends on the realized physical architecture. A durable commitment requires more than a transient signal. It requires a surviving carrier, decoding, validation, admission, maintenance, and sufficient resistance to noise and decay. The paper distinguishes live semantic activity from durable recorded stock. A system may be processing and transmitting information without having written a stable commitment. Conversely, a durable record may survive after the process that created it has ended. Quantum clocks and relational dynamics
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 Page–Wootters construction and the three equivalent pictures built on it all require a bipartition of a kinematical space into a clock and a system, with commuting Hamiltonians and no interaction term. In the literature that bipartition is input, and its non-interaction clause is acknowledged as an idealisation. We observe that a reductive dual pair supplies it as output: two subgroups of a symplectic group that are each other’s full centraliser act on the oscillator representation with a multiplicity-free joint decomposition, so the two factors commute by theorem rather than by assumption, the tensor structure is canonical rather than chosen, and the sector label is one both factors must agree on. Of six requirements of the relational construction, four are met exactly, one is reduced to a condition on the pairing bijection, and one — the covariant POVM that makes a clock — is not supplied at all. Three consequences. The statement that the clock–system tensor product does not survive to the physical Hilbert space becomes: the constraint selects the diagonal of the joint decomposition. Whether commuting subalgebras form an exact dual pair is finite and checkable, and in a related arena it was checked and answered negatively — a gauge algebra of dimension 12 with a double centraliser of dimension 82 — so exactness must be measured, not assumed. And the oscillator representation is a representation of the double cover, whose covering sign no observable of the pair can see. Finally, an obstruction of the form “one universal clock cannot supply a per-mode cost” is shown to move from universal to uniform without declaring it; a clock universal as an operator but non-uniform in its weights is a third option the standard enumeration does not contain. We name it and do not claim it works. Every object here is an algebraic object on a symplectic arena.