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 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 H̃ₖin 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): H̃ₖin H H + O (). POVM E (d) =|W () |²/‖W‖² d is covariant. = √2 is the temporal origin. Trinity Formula (Section IV): Nₜotal = N + ln (··kₚiv/) 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.
Hillard et al. (Fri,) studied this question.