Abstract The temporal parameter t ∈ ℝ carries three surplus properties (negative extension, loop-admitting topology, and reversal symmetry) that no experiment has confirmed. This paper traces this conclusion from the mass-shell relation E² = p²c² + m²c⁴, which functions as a conserved-total resource constraint: total energy is conserved, and any increase in spatial momentum physically necessitates a decrease in the rate of internal evolution. The Pikovski Hamiltonian formalizes this as Ĥ₀/γ, the internal Hamiltonian multiplied by the Lorentz factor. The Page-Wootters framework derives time emergence from entanglement correlations in a globally static state. Smith and Ahmadi (2020) showed that PaW conditioning recovers the Pikovski time dilation; Höhn, Smith and Lock (2021) proved the full three-way equivalence. Combined with the SI definition of a second (9,192,631,770 oscillations of cesium-133, an operational definition grounding time in internal state transitions), these results establish that the physical content of the temporal parameter is a constraint on internal state-transition rates. The temporal parameter t ∈ ℝ serves as a continuous surrogate for a quantity accessed only through physical process that is non-negative and monotonically non-decreasing. A relational observable N_ref replaces t. Following Gisin and Weatherall, three ℝ-domain properties of t are identified as surplus structure. Closed timelike curves fall outside GR's initial value formulation (Choquet-Bruhat-Geroch); parameter reversal in T-symmetry is shown to be surplus via the algebraic CPT theorem (Borchers, Guido-Longo, Mund, Swanson); and the block universe loses its default status. The arrow-of-time problem decomposes into a process-accumulation arrow (a spectral property of the clock operator, formally independent of entropy) and a thermodynamic arrow determined by boundary conditions. The argument extends Einstein's 1905 relativization of simultaneity to the temporal parameter itself.
Abstract The temporal parameter t ∈ ℝ carries three surplus properties (negative extension, loop-admitting topology, and reversal symmetry) that no experiment has confirmed. This paper traces this conclusion from the mass-shell relation E² = p²c² + m²c⁴, which functions as a conserved-total resource constraint: total energy is conserved, and any increase in spatial momentum physically necessitates a decrease in the rate of internal evolution. The Pikovski Hamiltonian formalizes this as Ĥ₀/γ, the internal Hamiltonian multiplied by the Lorentz factor. The Page-Wootters framework derives time emergence from entanglement correlations in a globally static state. Smith and Ahmadi (2020) showed that PaW conditioning recovers the Pikovski time dilation; Höhn, Smith and Lock (2021) proved the full three-way equivalence. Combined with the SI definition of a second (9,192,631,770 oscillations of cesium-133, an operational definition grounding time in internal state transitions), these results establish that the physical content of the temporal parameter is a constraint on internal state-transition rates. The temporal parameter t ∈ ℝ serves as a continuous surrogate for a quantity accessed only through physical process that is non-negative and monotonically non-decreasing. A relational observable N_ref replaces t. Following Gisin and Weatherall, three ℝ-domain properties of t are identified as surplus structure. Closed timelike curves fall outside GR's initial value formulation (Choquet-Bruhat-Geroch); parameter reversal in T-symmetry is shown to be surplus via the algebraic CPT theorem (Borchers, Guido-Longo, Mund, Swanson); and the block universe loses its default status. The arrow-of-time problem decomposes into a process-accumulation arrow (a spectral property of the clock operator, formally independent of entropy) and a thermodynamic arrow determined by boundary conditions. The argument extends Einstein's 1905 relativization of simultaneity to the temporal parameter itself.
The temporal parameter t ∈ ℝ carries three surplus properties (negative extension, loop-admitting topology, and reversal symmetry) that no experiment has confirmed. This paper traces this conclusion from the mass-shell relation E² = p²c² + m²c⁴, which functions as a conserved-total resource constraint. The Pikovski Hamiltonian formalizes this as Ĥ₀/γ, and the Page-Wootters framework derives time emergence from entanglement correlations in a globally static state. Smith and Ahmadi (2020) showed that PaW conditioning recovers Pikovski time dilation; Höhn, Smith and Lock (2021) proved the full three-way equivalence. Combined with the SI definition of a second (grounding time in internal state transitions), these results establish that the physical content of the temporal parameter is a constraint on internal state-transition rates. Following Gisin and Weatherall, three ℝ-domain properties of t are identified as surplus structure per Weatherall's framework: closed timelike curves, parameter reversal in T-symmetry, and the block universe. The arrow-of-time problem decomposes into a process-accumulation arrow (a spectral property of the clock operator) and a thermodynamic arrow determined by boundary conditions. The argument extends Einstein's 1905 relativization of simultaneity to the temporal parameter itself.
Abstract The temporal parameter t ∈ ℝ carries three properties that no experiment has confirmed as features of physical reality: negative extension, loop-admitting topology, and reversal symmetry. We identify these as surplus structure in the sense of Weatherall, applying Gisin's observation on ℝ-modelling to the temporal parameter specifically. Its operational content is N_ref: the accumulated state-transition count of a reference system, non-negative and monotonically non-decreasing. The caesium-133 hyperfine oscillator is one realisation; the framework is characterised axiomatically (N1–N4, Section 4) and is independent of the choice of clock. We establish two theorems. Theorem 1: restricting the Wheeler–DeWitt scalar-field clock to its operationally grounded domain φ ∈ [0, ∞) halves the minisuperspace solution space, excluding independently contracting universes. Theorem 2: within the Page–Wootters framework, the process-accumulation arrow (a spectral property of the clock operator) and the thermodynamic arrow (a boundary condition on the constraint surface) are formally independent. The Bondi k-calculus then derives the full Lorentz transformation — including the relativity of simultaneity — from three operational inputs: transition-count ratios between inertial observers, the relativity principle, and a finite signal speed c. No prior notion of time, metric, or spacetime is assumed; 1/γ is a theorem rather than an input. Closed timelike curves, parameter reversal in T-symmetry, and the block universe as the default interpretation of the formalism are shown to depend on the surplus structure of ℝ and to lack independent empirical support.
Four axioms (N1–N4) define N_ref as the accumulated state-transition count of a physical reference system; the definition is realisation-independent and structurally excludes the Pauli obstruction. The mass-shell constraint yields the Pikovski Hamiltonian Ĥ_phys = Ĥ₀/γ within the expansion regime, and the budget equation (dτ/dt)² + v²/c² = 1 as the exact constraint. A forgetful functor F: Math_t → Phys_N identifies surplus automorphisms of the mathematical temporal category that have no counterpart in the physical category. The computational gauge characterisation (CG1–CG3) shows that the surplus is mathematically necessary (Stone's theorem), informationally determined for KMS states (modular analyticity), and physically vacuous (axiom N1). The signature theorem proves that Minkowski (1,1) is forced by the non-compactness of the boost group, itself a theorem of counting and the relativity principle. The extension to the full (1,3) signature follows from spatial isotropy via Schur's lemma on the irreducible SO(3) representation. A universality proposition establishes that every physical clock satisfying N1–N4 dilates by 1/γ, with the bridging assumption A_bridge naming what the quantum universe contributes to the derivation. These results supply the axiomatic and categorical foundations for the surplus structure programme introduced in Part I; Part III extends the k-operator developed here into the quantum regime.
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.
The nature of time remains a central problem in fundamental physics, spanning quantum mechanics, thermodynamics, and relativity. In this work, we develop an observer-dependent framework in which time is not assumed as a fundamental background parameter, but emerges operationally from restricted access to a globally timeless quantum state. Building on and extending the Page–Wootters mechanism, we introduce a unified relational formula that shows how three traditionally distinct aspects of temporality arise from the same structure: quantum dynamics from conditioning on physical clock states, thermodynamic irreversibility from partial tracing over inaccessible degrees of freedom, and observer-relative time from the locality of the clock subsystem. No modification of unitary quantum mechanics, collapse postulates, or fundamental time flow is required. Analytical arguments and numerical simulations demonstrate that, in the limit of complete information access, all temporal ordering disappears, while effective Schrödinger dynamics and an arrow of time emerge naturally under informational and observational constraints. These results support a central claim: time is not a property of the universe itself, but a property of descriptions generated by physical observers embedded within it.
AbstractFundamental formulations of quantum cosmology, most notably the Wheeler–DeWitt equation (\hat{H}\vert{}\Psi\rangle = 0), describe the Universe as a closed, static, and unitary system. Within the block-universe interpretation of spacetime, the underlying laws of quantum mechanics are often taken to be time-reversal invariant at the fundamental level. This paper proposes a relational framework wherein temporal asymmetry and the apparent irreversibility of physical events are not intrinsic features of the global wave function, but emergent phenomena resulting from quantum entanglement between an internal observer (or record-bearing subsystem) and the rest of the universe.Utilizing the Page–Wootters relational time mechanism alongside quantum decoherence theory and Landauer's principle, we demonstrate that while the global state remains static and time-symmetric under these formalisms, any internal subsystem capable of forming stable memory states naturally experiences a unidirectional arrow of time. Irreversibility is here interpreted as emerging from observer-system correlation, decoherence, and entropy production during record formation rather than a violation of fundamental unitary symmetry.Key Focus Areas & Subject CoverageQuantum Cosmology & Quantum Gravity: Wheeler–DeWitt Equation, ADM Formalism, Problem of TimeRelational Quantum Dynamics: Page–Wootters Mechanism, Conditional State FormalismThermodynamics & Information Theory: Quantum Decoherence, Landauer's Principle, Arrow of Time, Information DispersalFoundations of Physics: Block Universe, Eternalism, Emergent Temporal AsymmetryManuscript StatusStatus: Revised PreprintField: Theoretical Physics / Quantum Foundations & Cosmology