Here's a description for the Zenodo record. It mirrors the abstract but is written for someone scanning a repository page rather than reading a journal. Third paper in a series on a phase-gradient relational clock observable in cobalt–corrin active-space models. Earlier work defined a discrete observable, τ_pre, on a two-qubit clock marginal of a Jordan–Wigner register, and reported that using τ_pre as the time axis yields a lower Schrödinger residual than an external evolution parameter s, validated against a randomized-monotonic null. In those studies the states whose ordering τ_pre was tested against were themselves generated by evolution in s, leaving the relational reading circular. This paper removes the external trajectory. A Page–Wootters history state satisfying the constraint (H_C ⊗ 1 + 1 ⊗ H_S)|Ψ⟩⟩ = 0 is constructed over a 10-qubit CASCI(5,5) Hamiltonian for a 26-atom cobalt–corrin fragment, and the system states are recovered only by conditioning on clock readings. Main results: Conditional states converge to the externally evolved reference, with mean infidelity falling from 1.9 × 10⁻² at clock dimension d = 64 to 1.0 × 10⁻⁶ at d = 16384. All reported quantities converge as the finite-difference step Δt → 0; the finite-difference residual agrees with the analytic-derivative residual to within 1.6% at the coarsest step used. In that limit, τ_pre lowers the mean residual from 8.1162 to 6.8715 against a null of 0.7114 ± 0.0820 (z = 5.73). The standardized effect is stable across all 45 clock-qubit pairs (z = 5.60–5.87), under orbital reordering (z = 5.66), across a fiftyfold change in step size, and across the null's step-floor parameter, whereas the unstandardized ratio varies with these conventions. A spectrum-preserving control, mixing eigenvectors toward random orthogonal ones at fixed eigenvalues, shows the clock marginal decohering monotonically toward the maximally mixed value; the apparent scrambling threshold is shown to follow the purity cut in the definition of τ_pre rather than any property of the system. The paper states explicitly what is not established: that τ_pre is a physical time parameter, that the external parameter is eliminated rather than relocated into the choice of clock Hamiltonian, that anything is specific to vitamin B12, or that τ_pre is experimentally accessible. Three quantities that might naturally be reported as results are deliberately not, with reasons given. Data and code available on request.
Here's a description for the Zenodo record. It mirrors the abstract but is written for someone scanning a repository page rather than reading a journal. Third paper in a series on a phase-gradient relational clock observable in cobalt–corrin active-space models. Earlier work defined a discrete observable, τ_pre, on a two-qubit clock marginal of a Jordan–Wigner register, and reported that using τ_pre as the time axis yields a lower Schrödinger residual than an external evolution parameter s, validated against a randomized-monotonic null. In those studies the states whose ordering τ_pre was tested against were themselves generated by evolution in s, leaving the relational reading circular. This paper removes the external trajectory. A Page–Wootters history state satisfying the constraint (H_C ⊗ 1 + 1 ⊗ H_S)|Ψ⟩⟩ = 0 is constructed over a 10-qubit CASCI(5,5) Hamiltonian for a 26-atom cobalt–corrin fragment, and the system states are recovered only by conditioning on clock readings. Main results: Conditional states converge to the externally evolved reference, with mean infidelity falling from 1.9 × 10⁻² at clock dimension d = 64 to 1.0 × 10⁻⁶ at d = 16384. All reported quantities converge as the finite-difference step Δt → 0; the finite-difference residual agrees with the analytic-derivative residual to within 1.6% at the coarsest step used. In that limit, τ_pre lowers the mean residual from 8.1162 to 6.8715 against a null of 0.7114 ± 0.0820 (z = 5.73). The standardized effect is stable across all 45 clock-qubit pairs (z = 5.60–5.87), under orbital reordering (z = 5.66), across a fiftyfold change in step size, and across the null's step-floor parameter, whereas the unstandardized ratio varies with these conventions. A spectrum-preserving control, mixing eigenvectors toward random orthogonal ones at fixed eigenvalues, shows the clock marginal decohering monotonically toward the maximally mixed value; the apparent scrambling threshold is shown to follow the purity cut in the definition of τ_pre rather than any property of the system. The paper states explicitly what is not established: that τ_pre is a physical time parameter, that the external parameter is eliminated rather than relocated into the choice of clock Hamiltonian, that anything is specific to vitamin B12, or that τ_pre is experimentally accessible. Three quantities that might naturally be reported as results are deliberately not, with reasons given. Data and code available on request.
I. Introduction The problem of time in quantum gravity is the following. General relativity is a diffeomorphisminvariant theory: there is no preferred time variable, and the Hamiltonian constraint HΨ = 0 impliesthat all physical states are ‘frozen’ in the sense that they carry no manifest time dependence [Dirac1964; DeWitt 1967]. Yet quantum mechanics requires time as a background parameter againstwhich evolution is measured. Reconciling these two facts — the timelessness of diffeomorphisminvariant quantum gravity and the temporal structure required for quantum measurement theory—is the problem of time [Isham 1993; Kuchăř 1992]. The problem is not merely philosophical: ithas concrete consequences for the interpretation of quantum cosmological perturbations, includingthose that seed the CMB anisotropies we observe today.One of the most elegant proposed resolutions is the Page–Wootters (PW) mechanism [Page & Wootters 1983]: time is not a background parameter but a relational quantity, encoded in correlationsbetween a ‘clock’ subsystem and the rest of the universe. Physical time is the reading of the clock,and quantum evolution is the conditional state of the system given a particular clock reading. Thisidea has been placed on rigorous mathematical footing by Höhn, Smith & Lock (HSL) [2021], whoproved the equivalence of three apparently distinct quantization procedures — the Dirac constraintquantization, the Page–Wootters mechanism, and the deparametrized Heisenberg picture — undera precise set of conditions. The HSL result reduces the problem of time, for any system satisfyingthose conditions, to the problem of identifying a clock with the right algebraic properties.In Loop Quantum Cosmology (LQC), relational time has been implemented using material degreesof freedom: a minimally-coupled massless scalar field [Ashtekar, Pawlowski & Singh 2006], a GFTcondensate [Gielen & Sindoni 2016], or dust fields [Brown & Kuchăř 1995; Pawłowski 2014]. Allof these are material clocks: dynamical degrees of freedom that must be added to, or identified1within, the matter content of the model. No worked example exists in which the clock arises fromthe algebraic structure of the kinematic Hilbert space itself — as an invariant of the theory’s gaugegroup rather than an additional field. Such an algebraic clock would be conceptually preferable: itrequires no extension of the Hilbert space, its back-reaction on geometry is controlled, and it is notsubject to the multiple-choice ambiguity [Kuchăř 1992] that afflicts material clock choices when thesystem has multiple degrees of freedom.This paper provides the first such example. The LQC bounce, treated via the Pöschl–Teller (PT)effective potential established in the companion series [Hillard 2026a–d], admits a spectral threshold = √2 that is fixed by the zero of the SU(2) Casimir polynomial W() = ³(²−2)/[C·sinh(/2)],where C = 3.60329… is the Hillard constant [Hillard 2026a, Paper 8]. This zero is a representationtheoretic invariant of the kinematic Hilbert space: it exists prior to any dynamical input, beforeany clock is chosen. We show that instantiates the full HSL relational clock construction. TheClock Decomposition Theorem (Section III) establishes the tensor product factorization H̃_kin H H and constructs the covariant POVM on H. The Temporal Trinity Formula (Section IV)is identified as the HSL clock reading in all three quantization pictures, with explicit clock-changemap T (Appendix A). The result holds exactly: is not a perturbative approximation to a clockbut a genuine algebraic invariant.A relational clock that is an algebraic invariant makes a concrete, falsifiable prediction. The signchange multipole * — the CMB multipole at which the LQC bounce imprint changes sign in thepower spectrum [Hillard 2026a] — is precisely the PW temporal origin in Fourier space: * =/_B. When GFT interactions of strength are included, the PT potential is deformed to U_ =−(2−)sech²(), which remains within the exactly solvable PT family (Section VI). The thresholdshifts to () = √(2−), giving the exact non-perturbative prediction:()/ = √(1 − /2) − 1 −/4 + O(²).* (1)*The sign is negative: GFT interactions push * to lower multipoles. The detection threshold forLiteBIRD is _crit 4/*, placing the prediction within reach of the near-term CMB survey (SectionV). This is the first quantitative prediction from a relational clock interpretation of the LQC bounce.A word on the relationship between this paper and the companion series [Hillard 2026a–d]. ThePWinterpretation of the bounce established here depends on W() only through its functional form:W() ³(²−2). Any function with that polynomial zero structure has a scattering threshold at =√2, and any such function in the Hamiltonian constraint generates the relational clock constructionof Sections III–IV. The validity of the LQG derivation of W in [Hillard 2026a] is therefore not aprerequisite for the results of this paper. The companion series and the present paper are mutuallyreinforcing rather than circularly dependent: [Hillard 2026a] provides the physical motivation andprecise form of W; this paper provides the relational interpretation that explains why has thevalue it has.The paper is organised as follows. Section II reviews the relevant background on LQC, the PTeffective potential, and the HSL formalism. Section III establishes the Clock Decomposition Theorem: spectral factorization (Proposition 1), covariant POVM, and temporal origin. Section IVidentifies the Temporal Trinity Formula in all three HSL pictures (Lemma 1, Proposition 2). Section V derives the physical inner product by group averaging and the full observational prediction(Table 2). Section VI proves that the intertwiner deformation is shape-invariant and promotes theprediction to an exact formula (Proposition 3). Appendix A constructs the clock-change map Texplicitly (Proposition A.1, Corollary A.1).22Main results at a glanceClock Decomposition Theorem (Section III): H̃_kin H H + O(). POVM E(d) =|W()|²/‖W‖² d is covariant. = √2 is the temporal origin.Trinity Formula (Section IV): N_total = N + ln(··k_piv/) is a Dirac observable, clockchange invariant. T acts as identity on (Appendix A).Exact prediction (Sections V–VI): ()/ = √(1−/2) − 1 −/4. LiteBIRD detection threshold:_crit 4/*.First algebraic PW clock in LQC (Section V.4): is an SU(2) Casimir invariant, not amaterial d.o.f. Back-reaction is O(); no Hilbert space extension needed.
This project explores a fully timeless formulation of quantum theory motivated by the Page–Wootters mechanism and canonical quantum gravity. The central assumption is that the universe is described by a single global quantum state ∣Ψ⟩ satisfying a Wheeler–DeWitt–type Hamiltonian constraint, with time appearing only as an internal, relational variable defined by “clock” subsystems. I formulate a simple multi–clock framework, introduce a clock neutrality conjecture, and test it in an explicit finite-dimensional toy model. The model consists of two three-tick internal clocks and a qubit system, with joint step relations that encode a static global state over a 3×3 grid of clock readings. Imposing multi-clock consistency selects a constrained family of states in which the two clock evolutions commute on the support of ∣Ψ⟩. Within the decoherent histories framework, I then show that for histories expressible with either clock, the probabilities for relational events agree whenever the two clocks implement the same underlying physical evolution. In this sense the toy model realises a discrete version of clock neutrality: decoherent-histories probabilities are independent of which internal clock is used as the parameter. The construction does not yet yield new empirical predictions, but it isolates a precise structural condition that any timeless, deterministic description of the universe would need to satisfy in order to be compatible with clock-independent low-energy physics.
This note records a consistency result for the Spectral Quantum Gravity (SQG) framework: that the relational, timeless construction underlying SQG’s emergent time reproduces ordinary Schrödinger quantum dynamics, with departures bound to the framework’s own condensate scale rather than free. In SQG the phase fiber of the substrate carries a conserved U(1) charge Q (bounded below; the conjugate angle compact), and the Osterwalder–Schrader reconstruction supplies a self-adjoint matter Hamiltonian. Conditioning the timeless ground state on the phase-fiber clock, in the Page–Wootters sense, is shown analytically to yield the Schrödinger equation i ∂_θ ψ = (H − E) ψ exactly. This is confirmed numerically to machine precision in a finite model carrying SQG’s two defining features — a generator bounded below and a compact, integer-spectrum clock. The compact, bounded-below clock leaves the equation itself intact and imprints only clock-level fingerprints: a spectral window (modes above the constraint energy cannot be clocked), a finite time resolution of order 1/⟨Q⟩ (the phase states form a covariant POVM; Pauli’s theorem forbids a sharp conjugate time), and a KMS/thermal correction of order T ∝ 1/v from the compact Euclidean direction. All three are governed by the same condensate scale and vanish in the macroscopic-condensate limit, where textbook Schrödinger evolution is recovered. Scope and grade. This is a supplementary, lowest-grade consistency/closure note, conditional on the open decompactification of the phase fiber discussed in the main SQG work. It is explicitly not a derivation of quantum mechanics: the reflection-positive quantum substrate and the self-adjoint generator are inputs, as throughout SQG. The formalism is standard and borrowed — Page–Wootters relational dynamics, covariant (POVM) clocks for generators bounded below, and the KMS structure of a compact Euclidean direction — here applied to SQG’s specific clock. What is computed is the mechanism in a finite model, not the full field-theoretic dynamics of the substrate. The single SQG-specific statement is that any departure from ordinary quantum mechanics is tied to SQG’s own, unobservably small scale. A Python script reproducing all numerical values is included.
In timeless formulations of quantum theory, the Page-Wootters proposal (PW) recovers time and dynamics as relative clock-and-world states. By interpreting the total timeless system as a clock entangled with the world, the states having a definite clock reading appear to recover the temporal states of the world as relative states, and dynamics emerges. But an entangled state admits infinitely many decompositions as a superposition of product states, each decomposition corresponding to a different but equally valid time operator of the same clock system. Different valid choices of the time operator for the same clock system lead to different physical histories, where the world states in a history are superpositions of the world states from another history. Fixing one operator as "the" clock time would import the temporal meaning that the Page-Wootters proposal is meant to explain in the first place. Therefore, the interpretation of PW that the world state is passively conditioned on the clock state cannot hold. The timeless state is resolved into time-dependent states by the intrinsic pointer observables of the world, without having to outsource the role of the time operator to a separate clock. Fortunately, the formalism of PW remains valid, provided that it is interpreted in terms of intrinsic pointer observables of the world itself.
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.
Version 3 (2026-08). One correction, chair-directed after an independent text-only rebuild: the setting's conventions dictionary is regenerated from the code of record. The pointer packet is an ENERGY-basis Gaussian launched at tick zero (v1/v2 described it as a tick-space packet of tick-width sigma_n; the temporal width is in fact the inverse, NC/(4 pi sigma_n) ticks); the tick-state sign convention, the second geometry's sigma_n = 3, the twin/control definitions, and the post-selection probabilities are now printed; a standing text-only rebuild gate (fresh implementation from the setting paragraph alone, matching the exact conditioned state at 3.5e-14) guards the dictionary. Every published number stands (the independent rebuild reproduced the NC = 24 physics cold, the lane's first exterior verification, recorded in the data-and-code paragraph); the dictionary was broken. The methods law is minted program-wide: a geometry prints its dictionary, generated from the code, never transcribed. Version 2 (2026-08). Changes from v1: the clock-width profile is completed end to end: v1's spike-plus-plateau is the interior of a measured disturbance VALLEY with an optimal clock width (minimum at sigma_n* = 2.5; the broad side rises 93x by sigma_n = 24, tracked by the value law against exact evolution at ratios 0.985 and 0.995), so the least-disturbing clock is neither time-sharp nor energy-sharp; a pre-registered identification of the plateau with the sharp-clock limit was killed by a factor of 160 and is ledgered; the open problems are re-posed at the valley's optimum. When a finite quantum clock gates a register's interactions and the run is conditioned on the clock's final reading, the register's conditioned dynamics is derived at two orders and machine-verified against exact evolution. First order: the drive is the complex Aharonov-Albert-Vaidman weak value of the gating window projector (real part = Hermitian envelope, imaginary part = post-selection reweighting); a twin evolved under this envelope removes the entire first-order mismatch a mean-envelope control retains (state-metric deficits g^1.1 vs g^2.1, factor 56 at g = 0.1). Second order: the deficit is governed by the connected two-time window correlator under post-selection, resolved along the interaction picture; the resulting parameter-free value law matches measured deficits at ratios 0.92-1.05 across the clock-width grid and at a second geometry (1.03-1.07). The when/what trade-off of a participating clock is promoted from confounded excess to derived law: the disturbance is the connected fluctuation of the clock's timing answer, its clock-width profile is a predicted spike-plus-plateau (not a power law), and two earlier exponent claims are superseded with the reconciliation recorded. Two honest derivation failures (equal-time invisibility resurrected by the interaction picture; the kernel belonging inside the integral) are ledgered as the steps that located the law.