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.
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.
On a homogeneous space M = Sp(2n,R)/U(n) carrying a compact group action, a kinematic no-go establishes that no quantity derivable from the manifold, the metric, the base point and the group can label the members of a certain triple. This paper measures in the layer that premise does not reach: the layer in which a clock and a system are separated and one is conditioned on the other. The central result is structural — the triple and the clock are one consequence, not two. A single input, an irreducible of the compact group occurring with multiplicity m, forces at once that the sector is Sym2(Cm), that the commutant is u(m), and that the available clock is the Cartan of su(m) inside it, of dimension m−1. At m = 2: three lines, a four-dimensional commutant, a torus of rank two, a clock unique up to scale, weights {+1, 0,−1}. At m = 1: one line and no clock at all, measured in the same block, as the arena’s own falsifier. The clock is not assumed periodic; its compactness and minimal period are measured, and its overlap function, its three orthogonal readings and its sensitivity to the clock-state phase freedom follow in closed form from covariant-POVM formulae. With a distinguished tangent direction admitted, one exact measurement lifts the degeneracy: the projection of the adjoint square, restricted to the sector, is exactly diagonalwith integer equispaced spectrum. Two limitation theorems then show that admitting the direction buys classes but not one invariant direction, and that no spectral invariant can supply the one remaining bit. A family index is therefore a reading: the value a line takes on a clock derived from the same multiplicity that produced the line, with a residual bit gauge-fixed relative to an orientation the construction does not supply.
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.
We observe that the ingredients a relational construction already requires contain a gauge structure, and that its group is the one the frame cannot see. LetM= Gbig/H be a homogeneous space. Three independently established facts close into a statement when placed together. An operational quantum reference frame is a covariant POVM on a homogeneous space Σ ∼= G/Hx, and the relativization map expressing a system observable relative to it is defined only on the Hx-invariant part of the system algebra. The canonical connection of a reductive homogeneous space is h-valued: the projection of the Maurer–Cartan form onto the isotropy algebra. And the isotropy directions carry zero Fisher–Bures weight, which, that metric being the maximal monotone one, is the first statement in the vocabulary of the geometry. Four consequences. Gauge invariance is not imposed on the relational description; it is the condition under which an observable can be expressed relative to a frame at all. The Cartan decomposition splits what the frame cannot resolve from what it measures. The structure is non-abelian whenever the isotropy is, and for Sp(2n,R)/U(n) the isotropy is U(n). And the size of that isotropy and the multiplicities of its content are fixed by the field content, NB = 2 dimG+nH, a gauge sector carrying multiplicity two because the adjoint is real and a charged scalar one because the charge is complex. We then measure what the gauge algebra’s commutant inside the isotropy retains of it: the double centraliser is the full type-preserving algebra L λ u(dλ), of dimension 82 against the gauge algebra’s 12. The commutant does not remember the gauge group; it remembers the isotypic decomposition.No gauge group is derived here, and nothing is identified with anything physical.
Complex Hilbert space is usually assumed at the start of quantum mechanics. This paper formulates a reverse construction for the one-particle and operator-algebraic setting. Starting from a real Hilbert space carrying an orthogonal representation, the admissible imaginary units are identified with the skew-adjoint square roots of − Id in the Real commutant of the complexified representation. The novelty is not the isolated use of complexification, spectral orientation, or polar decomposition, but their order of dependence: first the symmetry representation defines an intrinsic operator-valued phase space; then an oriented internal time flow selects a distinguished point of that phase space; only afterward are the usual complex one-particle, observable, probability, and Weyl–CCR structures invoked under their standard hypotheses. In the terminology used here, this oriented internal flow is the reversible or inner-generative side of a dual-time distinction. It is not a second observer clock and not an additional Lorentzian coordinate: mathematically it is the strongly continuous orthogonal group U ( t ) = exp ( t A ) on the real Hilbert space. Its Stone-dual variable is energy/frequency λ ; canonical conjugation pairs λ with − λ ; positive energy orients this pair and selects the canonical phase on the dynamic sector through A = J | A | = | A | J . Thus inner-generative time is the source of phase orientation, while irreversible record time, projection, memory, and decoherence enter only as later reduced-dynamics layers. No spectral gap at zero is required, and zero modes remain visible as residual phase data rather than being hidden by the construction. The selected phase is invariant under time-orientation-preserving symmetries and changes sign under time reversal, giving the usual antiunitary action after complex reconstruction. The paper also makes explicit the monoidal limitation: two selected local phases do not make the raw real tensor product the standard composite system; the standard composite is recovered only after imposing the J -balanced, equivalently complex, tensor product rule. Thus the result is a reconstruction of compatible complex kinematics on the selected one-particle/operator-algebraic sector, not a derivation of all quantum foundations from purely real data.
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.
Exact permutation covariance does not itself supply a sparse metric geometry. For permutation-covariant, full-support k-body interaction orbits, the incident interaction norm scales as (k−1N−1)∣JN∣∥h∥; such orbits cannot have uniformly finite range in metrics of diverging diameter. A permutation-invariant Hamiltonian diagonal in the graph basis has ground-state multiplicity at least N!/∣Aut(G)∣ whenever G is a ground graph. We give a quantitative version for approximately symmetric energies, together with fixed-radius connectivity and unitary-isometry obstructions. The trace-distance asymmetry of an arbitrary state also bounds its probability of selecting one labeled graph.
We then construct two exactly solvable finite relational history Hamiltonians. A cyclic clock with a reversible incidence update has a unique ground state; gauge reduction yields a cycle Laplacian with a rank-one pin and the exact gap ΔN,L=E∗(1−cosθL), where tan(LθL/2)=1/sinθL and 0<θL<π/L. In particular, ΔN,L/E∗=π2/(2L2)−2π2/L3+O(L−4), independently of the number of vertices. In an alternative order-free architecture, one clock qubit per unordered pair gives commuting four-local terms, 2N ground states and a size-independent gap E∗(1−1/2). A fixed-weight control penalty selects (qN) ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.