← The Shadow of Existence

P10

The canonical problem of time as a category error

an empirically forced cosmic time, deparametrization, the dissolution of frozen dynamics, and a graviton quantization the de~Sitter horizon closes without a free parameter—the seam where ℏ enters gravity—in Cosmological Relativity

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Introduction: the problem of time, as standardly posed

The canonical quantization of general relativity confronts a difficulty that has resisted resolution for half a century. In the ADM decomposition [ADM1962], spacetime is foliated by spatial hypersurfaces and the Einstein–Hilbert action is recast in Hamiltonian form. The result is a fully constrained system in the sense of Dirac [Dirac1964]: the total Hamiltonian is a sum of the scalar (Hamiltonian) constraint H and the momentum constraint H_i, each weighted by the lapse N and shift Nⁱ, and each vanishing on the physical phase space. The lapse and shift are not dynamical; they are Lagrange multipliers enforcing the constraints. Promoting the scalar constraint to an operator yields the Wheeler–DeWitt equation HΨ= 0 [DeWitt1967]: the physical state is annihilated by the generator that should evolve it. The wavefunction of the universe does not appear to depend on any time parameter at all.

This is the problem of time [Kuchar1992, Isham1993]. Its sharpest statement is that the formalism contains no preferred time variable, that every spatial foliation of a given spacetime is related to every other by a gauge transformation generated by the constraints, and that “evolution” in the coordinate label t is therefore pure gauge—a re-slicing of one fixed four-dimensional object, carrying no physical content. The many proposed responses—identifying an internal time among the canonical variables before quantization, coupling a material clock, treating observables as relational “evolving constants,” or abandoning a fundamental time altogether—each carry well-known costs [Kuchar1992, Isham1993, Rovelli2004, BarbourEnd], to which we return in Section 8.

Our claim is that the difficulty is conceptual before it is technical, and that the conceptual diagnosis is precise rather than vague. The frozen formalism is the canonical expression of a single metaphysical commitment: that the four-dimensional manifold is the object that exists. We will argue, following the ontological foundation of CR [Janzen2025, JanzenCRframework] and the necessity assembled in Section 2, that this commitment is a category error—the confusion of what happens with what exists—and that it is empirically false, the universe having furnished, in the CMB, direct evidence of the very cosmic time the formalism declines to recognize. When that error is declined, the same canonical equations describe a real cosmic time all along.

We stress at the outset the register of this paper. We do not propose a modification of general relativity, a new term in the action, or an additional structure bolted onto the canonical formalism. The equations are those of Einstein and of ADM, unchanged. What changes is the reading. The frozen constraint and a true Hamiltonian generating cosmic-time evolution are, as we will show, the same content under two readings, differing only in whether the manifold of occurrences is granted the kind of existence that belongs to the evolving three-dimensional world. The problem of time is not solved here. It is dissolved.

The paper proceeds as follows. Section 2 earns the premise the argument needs—that a preferred cosmic time is real—on two independent grounds, one conceptual and one empirical. Section 3 reads the unchanged ADM formalism under the two ontologies, and Section 4 isolates why the move is a selection rather than a construction: the bare machinery settles nothing, and everything turns on whether a foliation is distinguished to be selected. Section 5 carries out the deparametrization, structurally and then in flat minisuperspace, marking the boundary between the two. Section 6 lifts it to the closed-S³ layer's propagating sector—the transverse-traceless graviton tower, advanced unitarily in cosmic time—and there the quantization closes: the scale-factor Hamiltonian's deficiency-(1,1) freedom, ordering-independent, is fixed by the de Sitter horizon's own thermal state, without a free parameter, leaving open not the boundary condition but the completion of the interacting tower's definition. Section 7 states the verdict negatively, and Section 8 places the argument against the internal-time, relational, and timeless programmes, where the multiple-choice objection is met not with a mechanism but with a warrant.

The necessity: a cosmic time that is real, not posited

The canonical move of this paper rests on one premise: that there is a preferred cosmic time, and that it is a real feature of the world rather than a convention chosen for convenience. This premise is not assumed. It is forced, on two independent grounds—one conceptual, one empirical—which we summarize here; the full argument is developed in [Janzen2025, JanzenWildGoose, JanzenFortress] and stated as the necessary-and-sufficient augmentation of general relativity in [JanzenModernParallax].

The conceptual ground: occurrence is not existence. To exist is to endure—to persist through an interval. The three-dimensional world exists in this sense, and so do the things in it. An event, by contrast, is something that happens: it occurs at a place and an instant and is gone. A worldline is the continuous succession of events associated with an enduring thing, and four-dimensional spacetime is the manifold of all such events—the totality of what happens. The persistent confusion about time, traceable from Heraclitus and Parmenides through to modern eternalism, is the treatment of occurrences as though they were existents: the reification of the event-manifold into a thing that exists [Janzen2025]. Once spacetime is imagined to exist, every moment along a worldline is imagined to exist on equal footing, the distinction between past, present, and future is declared an illusion, and time is said not to pass. But that conclusion was smuggled into the premise. Granting existence to the manifold of occurrences is not a discovery about time; it is a misuse of the word “exist.” The coherent picture is the one in which an evolving three-dimensional world exists and advances, and events happen in the course of that advance, mapping to a four-dimensional spacetime that represents them without being a further existent of the same kind.

In CR this distinction is structural. What exists is the layer 𝒮_t—a real three-sphere evolving in a real de Sitter substrate [JanzenCRframework, JanzenSlicing]. The four-manifold M is the representational record of the events that happen as the layer advances. The layer's existence and advance are not optional gloss on a fundamentally four-dimensional object; they are the ontology, and M is its shadow.

The empirical ground: the CMB measures the cosmic rest frame. That a preferred cosmic time might be real but undetectable is the situation relativity leaves open: local experiments cannot distinguish a true state of rest, because the relativity of synchrony—a general kinematic consequence of one's chosen rest convention, independent of the light postulate—and the round-trip symmetry of local signalling together hide one's motion. Einstein's identification of simultaneity with frame-relative synchrony was, on this much, a philosophical choice among two empirically equivalent options, not a deduction; treating the local inaccessibility of a cosmic rest frame as evidence of its nonexistence is a modal fallacy [JanzenWildGoose].

The universe, however, did furnish the missing evidence. The cosmological redshift of the CMB is isotropic to high precision, and the primordial temperature anisotropies at the level of one part in 10⁵ have been preserved across the entire history of expansion [Aghanim2020]. Had the expansion rate varied locally with the inhomogeneous matter distribution, the accumulated redshift along different lines of sight would differ by far more than is observed—by more than two orders of magnitude, as we establish quantitatively in the companion paper [JanzenModernParallax]—and the isotropy therefore certifies that cosmological space has expanded uniformly throughout cosmic history. But to speak of a three-dimensional space expanding uniformly through time already presupposes a cosmic time, a cosmic rest frame, and an associated simultaneity. The same data fix our own velocity relative to that frame through the CMB dipole. The cosmic rest frame is thus not a large-scale average or a coordinate convenience but one of the better-measured structures in cosmology. Synchrony is relative; simultaneity is absolute [JanzenWildGoose, JanzenFortress].

The two grounds are one correction. These are not two arguments but two faces of one. The relativity-of-simultaneity premise is precisely what licenses the reification of the block: if there is no objective “now,” the four-dimensional manifold is all there coherently is, and the layer's privileged advance has no referent. Conversely, an empirically real cosmic “now” restores the layer as the existent and demotes the manifold to its record. Declining the block (the conceptual correction) and admitting the cosmic rest frame (the empirical correction) are the same move. The equations of relativity are untouched by it; only the interpretation shifts—from “if two events appear simultaneous they must be simultaneous” to “apparent simultaneity is a local matter with no bearing on the global cosmic simultaneity that cosmology establishes empirically” [JanzenFortress].

This is the premise the canonical argument needs, and it has been earned rather than assumed: there is a preferred cosmic foliation, and it is real.

The canonical formalism, read two ways

We now read the standard canonical formalism under the two ontologies. Nothing in this section alters the equations.

In the ADM form, a spacetime (M,g) is foliated by spacelike hypersurfaces Σ_t with induced metric h_ij and conjugate momentum π^ij, and the line element is

with lapse N and shift Nⁱ. The action becomes S = ∫dt d³x (π^ij h_ij - N H - Nⁱ H_i), and variation of N and Nⁱ enforces the constraints H=0, H_i=0. The total Hamiltonian H = ∫d³x (N H + Nⁱ H_i) is a sum of constraints; on the constraint surface it vanishes.

The frozen reading. Suppose the existing object is the four-dimensional manifold itself. Then a choice of foliation is a choice of how to slice one already-existing thing into a stack of three-dimensional pictures, and a different foliation is a different slicing of the same thing. No slicing is distinguished; the relation between any two is a gauge transformation generated by the constraints; and since the generator of t-evolution is a constraint, it moves a physical state nowhere. Quantum-mechanically, HΨ=0: the state is timeless. This reading is internally consistent. Its content is exactly the content of the premise: if the block is what exists, time does not pass, and the formalism correctly reports that there is nothing for a Hamiltonian to do. The frozen formalism is not a mistake of calculation. It is the faithful canonical image of the block.

The reading CR adopts. Suppose instead, with CR, that what exists is the evolving layer 𝒮_t, and that M is the record of the events that happen as it advances (Section 2). Then a foliation is not a slicing of a pre-existing block; it is a choice of how to coordinate the advance of the existing world, and one such foliation—the cosmic one—tracks that advance rather than cutting across it. On this reading the lapse is not a bookkeeping multiplier but the metrical amount by which the existing layer advances per unit of the time label,

with the shift Nⁱ carrying the foliation's non-orthogonality to the layer. The constraint H=0 is still an equation; but now it is read as the relation that, once solved for the momentum conjugate to the cosmic time, yields the generator of the layer's advance—a true Hamiltonian. The same symbols, the same constraint surface, the same dynamics; a different answer to the single question of what exists. This lapse and shift are the very levels the cosmological development computes on: the lapse is the foliation stacking rate the observable expansion rides—set by the geometry, the density the content that set rate carries, read off the clock rather than sourcing it—and the shift is the synchronization through which that expansion is observed as distance and redshift, with the leaf's bend the matter it carries [JanzenOperator, JanzenCosmogenesis]. That a plasma process rides the one rate or the other is then fixed by what the process is—a process running in the content reads the leaf's rate, a comoving separation read across leaves the stacking rate [JanzenCRcosmology]—not chosen: the same selection of the existent's foliation that turns the frozen constraint into a true Hamiltonian here fixes, one level down, which rate a recombination-era or a nucleosynthesis-era computation must read—and reifying the shift's synchronous slicing as the existent is the block error, met at the level of a rate.

The selection is the whole of the move

It is worth isolating why the foregoing is a recognition rather than a new construction, because this is the crux of the paper.

The ADM machinery propagates canonical data along whatever foliation one supplies. It is indifferent to which foliation that is; it has no preferred time of its own, and in this exact sense the bare formalism contains no resolution of the problem of time and no obstruction to one. Everything turns on what is fed to it. If nothing distinguishes a foliation, the constraint is all there is and the state is frozen. If a foliation is distinguished, the constraint deparametrizes along it and a true Hamiltonian appears. The canonical formalism is the same in both cases; what differs is whether a preferred foliation exists to be selected.

CR supplies exactly that, and supplies it from outside the bare formalism, on the grounds of Section 2: the cosmic foliation is real, forced empirically by the CMB and conceptually by the occurrence/existence distinction. It is moreover the foliation that the rest of the programme independently singles out. The metric-singularity structure of the event horizon determines a unique limiting causal orientation in collapse, generically non-orthogonal to any spacelike slice, and this is the orientation CR reassigns as cosmic time [JanzenBHcausality, JanzenCRframework]; the representational freedom in choosing among Lorentzian metrics on the fixed manifold is a genuine gauge symmetry, organized by the description groupoid [JanzenGroupoid], but the cosmic foliation is not one of its orbits to be quotiented away—it is fixed by the ontology the gauge freedom is defined over.

The consequence is methodological as well as physical. There is no separate “canonical machinery” result to be proved here; the deparametrization that follows is standard once a clock is in hand. The content of this paper is that CR's cosmic time is the physically correct clock, on grounds that are ontological and empirical rather than formal, and that the canonical face of the existing physics is therefore a true Hamiltonian rather than a frozen constraint. The move is the selection. Everything else is reading the textbook on it.

The same selection settles what a mass charge could be. The ADM mass is defined relative to an asymptotic time translation, and exists only where the geometry supplies one—which asymptotically-de Sitter spacetimes do not, there being no global timelike Killing vector, so that no conserved charge is well defined there at all [JanzenSlicing]. On the reading taken here that absence is not a deficiency of the geometry but a misplacement of the question: the time this construction runs on is not recovered from an asymptotic symmetry but selected and measured, so a quantity whose definition waits on an asymptotic Killing vector is waiting on the wrong thing. It is the same move as the paper's own—what the formalism cannot supply from inside, the selection supplies from outside, and the question that presupposed the missing structure lapses with it.

Deparametrization and the true Hamiltonian

We now carry out the reading. The structural statement is independent of any model; the explicit minisuperspace realization that follows is an illustration of it, and we are careful to mark the boundary between the two.

The structural statement. Let the cosmic time of Section 2 be the distinguished foliation parameter, and let p_τ denote the momentum conjugate to it. The scalar constraint, linear in p_τ after the cosmic clock is isolated, takes the deparametrized form

where (q^A,p_A) are the remaining (true, gauge-invariant) degrees of freedom of the layer and H_phys depends on the cosmic time only through them. Equation (3) is the same constraint that, read with no preferred foliation, gives HΨ=0. Read with the cosmic clock selected, it is solved for the generator of advance: classically dq^A/dτ= ∂H_phys/∂p_A, and on quantization

This is a genuine time-dependent Schrödinger equation generating unitary evolution in cosmic time, not the frozen Wheeler–DeWitt constraint. The foliation is fixed—by the ontology, not by a gauge choice—so there is a true time and a true Hamiltonian. The frozen constraint and the true Hamiltonian are (3) read two ways; they are the same content, differing only in whether the manifold is granted existence. Deparametrization here is not a trick that restores a time that was really absent; it is the formal face of a time that was never lost, because the cosmic foliation was real all along.

A zero-duration boundary, and what it does to the evolution. The cosmic time of Section 2 is the real part of the companion framework's complex cosmic time, and that framework's closed contour contains one segment—the lift, joining the collapse's comoving turnaround to the branch point at which a collapse becomes a cosmology—along which the real part does not advance [JanzenCRframework]. The consequence for (4) is immediate and worth stating plainly, because it is unusual: cosmogenesis occupies no cosmic time at all, so the evolution operator across it is U(Δτ=0)= 1. The true Hamiltonian has no interval in which to act there. The geometry nonetheless changes across the segment—the areal radius climbs from the turnaround to zero and the expansion rate is carried from zero to divergent—so the change is not generated by H_phys but is the segment's own analytic content, the contour read at another point of itself. The beginning is therefore not a first instant of the evolution this section constructs; it is a boundary of zero duration within it, and that is why nothing can be lost across it: no evolution acts, so none can be non-unitary.

We mark the limit of the claim. What is established is the classical statement—the foliation parameter does not advance on that segment, so no interval of τ separates the two legs—together with its immediate consequence for the evolution operator. What the deparametrization does not supply is a quantization of the Euclidean segment itself. The state on either side is related by the identity in cosmic time; whether the imaginary-time stretch admits a quantum treatment in its own right, and what that treatment would say about the {q^A,p_A} across it, is not addressed by this deparametrization and is not claimed by it. The gap is named rather than managed, and its shape is definite. The silence is structural, not a matter of refinement—an evolution parametrized by cosmic time cannot describe a segment of zero cosmic duration, so no sharpening of H_phys reaches it. What the deparametrization cannot supply is an account of an imaginary-time stretch across which the classical configuration changes while the foliation parameter does not, and that is the natural point of contact between this construction and Euclidean approaches to the quantum beginning. The classical account of the beginning is complete, and the quantum account is not the deparametrization's to give: the step below gives its kernel, and the framework carries the segment's quantum reading on from there [JanzenCRframework].

The propagator across the lift is Euclidean. The evolution (4) is generated in the framework's complex cosmic time, and on the lift that time runs along the imaginary direction: τ= const+is, so ∂_s=i ∂_τ and (4) becomes ∂_sΨ=- H_physΨ. The kernel carrying the state across is therefore not the unitary U=e^{-i H_physΔτ} but

a Euclidean kernel—real, contracting, and not unitary—with |Δη| the lift's imaginary conformal-time interval. This is consistent with U(Δτ=0)= 1 rather than in tension with it: no cosmic time elapses, and the kernel is not an evolution in cosmic time.

Two things follow, and the first is a check the construction did not have to pass. A mode of frequency ω is damped by e^-ω|Δη| under (5); the framework's independent classical reading of the same segment damps a mode carrying e^ikc_sη by e^-kc_s|Δη| [JanzenCRframework]. With ω=kc_s these are the same expression, term for term. The classical selection rule—frozen content crosses, oscillating content does not—is the Euclidean kernel, read classically. Second, and this is the content the classical reading could not supply: a Euclidean kernel run for a long imaginary interval projects onto the ground state, suppressing each excitation by e^{-(E_n-E₀)|Δη|}. The scope of that statement should be fixed carefully, because the two sectors it separates are not symmetric. The kernel acts on oscillatory content; a frozen, zero-frequency mode is a fixed point of it and passes unchanged. And at the crossing no mode is oscillating: on the contracting leg aH grows without bound as r→0, so the comoving horizon shrinks to zero and every mode exits it and freezes before the branch point [JanzenCRcosmology]. So the projection does not select the frozen content that crosses—it has nothing to act on there. What it acts on is the oscillatory content of modes while they were sub-horizon, which it annihilates; the ground-state projection is the statement that this content, and not the frozen amplitude, is what the segment removes.

The projection is not exact, and where it fails is a scale rather than a caveat. At |Δη|≃1.7×10⁴ Mpc the damping reaches e^-2 only at k∼2×10^-4 Mpc^-1, so modes below that wavenumber are not strongly suppressed and cross with residual excitation. That wavenumber subtends ℓ≈3 on the last-scattering sphere, which is where the framework's independent transmission calculation places the boundary of the filter's reach [JanzenCRframework]. The two routes agree on the scale as well as on the factor.

The obvious objection is that a Euclidean kernel needs a Hamiltonian bounded below, and that Euclidean gravity notoriously lacks one—the conformal factor entering the Einstein–Hilbert action with the opposite sign, so that the Euclidean action can be driven arbitrarily negative. That objection does not reach this construction, and the reason is structural rather than fortunate. The conformal-factor problem arises when the path integral ranges over the conformal factor of the metric. Here it does not. The substrate's scale is α and is fixed—not chosen, but required, as the unique maximally symmetric structure carrying no unforced modulus [JanzenShadowExistence, JanzenSlicing]—so there is no conformal mode to integrate over; and the propagating gravitational degree of freedom of the layer is the transverse-traceless shear of Section 6, which is precisely the sector the conformal mode is absent from. Mode by mode that sector is (13), a harmonic oscillator, whose Hamiltonian is bounded below. The areal radius along the lift is likewise confined, running between the comoving turnaround and the branch point on an interval fixed by the progenitor mass, with the substrate curvature finite throughout [JanzenCRframework, JanzenCircle]: there is no runaway direction for the kernel to diverge along.

And the state the kernel projects onto is not a new posit but one this paper has already selected on independent grounds. Section 6 finds the scale-factor Hamiltonian to have deficiency indices (1,1)—a one-parameter family of candidate quantizations—and closes it, without stipulation, by observing that the boundary at a=0 is the de Sitter horizon, that its surface gravity is κ=1/α, and that the regular Euclidean state at that κ leaves nothing to choose. That is the same condition the kernel (5) enforces}: regularity in the Euclidean continuation. And that regularity is available only because of what the causality companion establishes about the boundary. A regular Euclidean state exists at a horizon only if the Euclidean continuation is smooth there—free of a conical defect, curvature finite. The horizon at a=0 is a metric singularity, at which the spatial measure collapses while the curvature stays finite, and not a curvature singularity [JanzenBHcausality]; had it been the latter there would be no smooth Euclidean section, no regular state to select, and the extension would remain unfixed. The distinction that dissolves the information paradox is the same distinction that makes this quantization unique. The extension-fixing condition and the lift's kernel are one requirement met twice, once at the horizon and once across the beginning.

One approximation is carried and its size is bounded below rather than left open. The kernel's frequency is not constant along the segment—the layer's scale varies with the path parameter—so the projection is adiabatic rather than exact, and the residual at the largest scales quantified above is one face of that. The ceiling on what it can transmit is computed in the adiabatic correction below, and it is small enough that nothing above turns on it. The segment's variational treatment is given below. The relation between the configuration variables {q^A,p_A} on the two sides is described through the kernel's action on the modes, and the segment is exhibited there as a solution of a variational principle in its own right.

The segment rides the marginal congruence (dr/d τ)²+f=1—the γ=1, J=0 case of the effective-potential form (dr/dτ)²=γ²-V_eff with V_eff=f derived in [JanzenThesis, Eq. (4.24)]—which is 12 r²+V= 12 with V(r)= 12 f(r). Continuing τ=is sends r↦-i r' and the Lorentzian action S=∫d τ[12 r²-V] to S=-iS_E with

whose stationary points obey r”=+V'(r)—motion in the inverted potential, the standard signature of a Euclidean solution—with first integral 12 r'²-V=- 12. The lift's closed form r(s)=-(2Mα²)^1/3| sin (3s/2α)|^2/3 satisfies both: the first integral is - 12 identically along the segment, and r” agrees with V'(r) to the precision of the check^P10R11. So the imaginary-time segment is not a formal device but a solution of a variational principle—an instanton connecting the Lorentzian turning point to the branch point.

Its action converges. Near the branch point V≃-M/r while r∝s^2/3, so the integrand of (6) grows as s^-2/3 and is integrable; numerically S_E→1.487 in units α=1 on the forced member. The segment therefore carries a finite weight rather than a divergent one, which is what a saddle-point treatment of the beginning would require of it. Two normalisations are in play, and the relation between them can be settled rather than flagged. The functional (6) carries a unit kinetic term; the gravitational action carries the metric factor and the prefactor implied by (9). They share a solution—the contour's law (dr/d τ)²=2M/r+r²/α² is the Friedmann equation with a=r, the identification 2M↔(8π/3)ρ_d being just M=(4π/3)ρ_d—so the instanton is one trajectory read in two measures, and only its action value differs.

That value follows without further input. On shell the reduced action is S=∫p_a da, and inverting (9) gives p_a=(3/4π)a a; continuing τ=is yields S=-iS_E with

using r'²=f-1 on the segment. The integral converges rapidly and gives S_E^grav=-0.0481 α²/G on the forced member^P10R10.

Its sign is the informative part, and it is not a pathology. A negative Euclidean gravitational action is the Hartle–Hawking sign: the no-boundary de Sitter action is likewise negative, -α²/8G in this convention, yielding an enhanced rather than suppressed weight [HartleHawking1983]. The present value is the same sign and the same order, smaller by a factor of order two, which is what a segment of the contour rather than a full hemisphere should give.

The sign traces to the segment lying on the r lt;0 branch, where p_a lt;0 while r increases. So the beginning's weight is of the no-boundary type, obtained here from a contour the construction already possessed rather than from a boundary condition imposed on the path integral. One register point must be made here, because the vocabulary invites a reading the construction refuses. The segment's parametrisation runs imaginary, and the kernel and action computed above are the kernel and action of that parametrisation; they are not the Euclidean objects of a Wick-rotated spacetime. The geometric-core paper states the guard generally: the construction reaches a real manifold through imaginary instruments, and the continuations it uses—the embedding coordinate, the equatorial seam, the cosmogenesis reassignment—are real analytic continuations on a spacetime that is Lorentzian throughout, not Wick rotations [JanzenGeometricCore]. The de Sitter horizon's Gibbons–Hawking state, by contrast, is a Euclidean continuation, and that paper marks it as distinct in kind from these. So the two must not be conflated: the thermal register in which ℏ is fixed by the period β=2πα is not the register the segment's action belongs to, and dividing that action by that ℏ would be joining two continuations the construction keeps apart. What the action is, is a real integral along a real curve of a real Lorentzian geometry, whose parameter happens to run imaginary over one stretch; what it is not is an exponent awaiting a quantum of action. Its comparison with the no-boundary sign above is a comparison of signs and magnitudes, not an identification of frameworks. The adiabatic correction, and where it fails. The kernel projects exactly only if H_phys is constant along the segment, and it is not: the tensor tower of Section 6 carries frequencies ω_n=μ_n/a, and a varies along the lift, diverging in ω as the branch point is approached. The exact suppression exponent is therefore ∫ω_n ds rather than ω_n|Δη|, and the first question is whether it converges at all. Two different questions are in play here and it is worth keeping them apart, because only one of them is what the word “adiabatic” names.} An adiabatic expansion is controlled by |dω/ds|/ω², and along this segment that quantity is ∝s^-1/3 and diverges at the branch point—so the adiabatic invariant E/ω, which is what an adiabatic approximation conserves, is not conserved through it. The suppression is nonetheless finite, and for a different reason: the exponent ∫ω_n ds converges because ω∝s^-2/3 is integrable, which is a statement about the action integral and not about slow variation. So the correction here is semiclassical rather than adiabatic—the companion cosmology paper names the same object correctly, as a WKB form whose adiabaticity parameter is of order unity [JanzenCRcosmology]^L14. It does: near the branch point |r|∝s^2/3, so the integrand grows only as s^-2/3 and

on the forced member^P10R9. The correction is not small and it runs the conservative way: the constant-frequency estimate using the turnaround value gives 1.4396, so the exact exponent is larger by a factor 2.32 and the suppression is stronger than the naive reading of (5) suggests.

Whether the projection is adiabatic is a separate question with a definite answer. The adiabaticity parameter |dω/ds|/ω² is C/μ_n with C running from 1.72 near the branch point to 0.16 near the turnaround, so the approximation is controlled by the harmonic index alone. With μ_n²=n(n+2), n≥2, this gives 0.61 at n=2, 0.44 at n=3, and 0.16 by n=10. The projection is therefore adiabatic for all but the lowest few harmonics, and degrades to order unity only at n=2 and n=3. That the approximation should fail exactly where the tower is coarsest is expected rather than surprising—there are no modes below n=2 on S³—and it locates the correction rather than leaving it as a caveat. We record, without claiming it, that the harmonic indices at which the treatment loses control are the lowest ones.

What that loss of control costs is bounded, and the bound is one-sided because of the tower's own floor. The suppression falls monotonically with the harmonic index—7.9×10^-5 in amplitude at n=2, 2.4×10^-6 at n=3, and by upwards of an order of magnitude per step thereafter—so the mixing the adiabatic projection neglects can only carry amplitude into more-suppressed modes. And there is no mode below n=2 to carry amplitude the other way: the least-suppressed mode has no source beneath it, so the bound is structural rather than estimated. Taking the whole O(ε) fraction out of n=2 at the worst case leaves that mode more suppressed, not less, and the mode it enriches does not reach the unmixed ceiling. The ceiling on transmitted power is therefore the unmixed n=2 value, 7.9×10^-8—and below 10^-3 even on the naive exponent, which is the weaker estimate. So the residual is located, bounded, and reaches nothing observable^P10R14. The relation between S³ tensor harmonics and observed multipoles is developed in the companion framework [JanzenCRframework], whose perturbation-spectrum treatment carries the source spectrum of a closed S³, discrete by degree, projected to the sky through the flat distance slicing rather than a closed one—the flat/closed decoupling carried by the non-synchrony. That projection is that paper's; this section's own contribution to it is the harmonic-side adiabaticity above, controlled by the harmonic index alone—so the two meet at the lowest harmonics, where that paper's lowest physical mode and this section's loss of control coincide.

An explicit illustration, and its limits. To show the structure is not empty we exhibit it in the simplest cosmological model, flat Friedmann–Lemaître minisuperspace. With scale factor a, conjugate momentum p_a, cosmological constant Λ, and a pressureless (dust) reference fluid serving as the cosmic clock in the manner of Brown and Kucha\v{r} [BrownKuchar1995], the deparametrized gravitational Hamiltonian reduced from the Einstein–Hilbert action (in units G=1, fiducial comoving volume unity) is

equal on shell to the conserved comoving dust energy^P10R1. Hamilton's equation a = ∂H_phys/∂p_a = 4πp_a/3a, together with H_phys= ρ_d on shell, gives

the flat-ΛCDM Friedmann equation. Its solution is the sinh^2/3 law

with the single scale α of the de Sitter substrate; one verifies directly that this a(τ) satisfies (11) identically. This is precisely the expansion history that the companion framework paper derives, by a wholly independent route, as the SdS expansion law of the layered construction [JanzenCRframework]: the deparametrized generator reproduces, in cosmic time, the same flat-ΛCDM evolution that the null-boundary correspondence delivers geometrically. Promoting (9) to an operator on L² of the half-line and applying (4) gives unitary evolution of the cosmological wavefunction in cosmic time.

Two qualifications attach to this illustration. First, it is a minisuperspace toy: a homogeneous truncation with a dust clock, not the full Schwarzschild–de Sitter object of CR, and the operator ordering of p_a²/a and the self-adjoint extension on the half-line are genuine technical questions not settled by the reduction. The illustration establishes that the deparametrized structure is consistent and reproduces the right cosmology; it does not by itself constitute the canonical quantization of the full theory. That next object—the lift to the closed-S³ layer and its propagating degrees of freedom—is carried out in Section 6. Second—and this is a feature rather than a gap—the CR beginning is the branch point, a divergence of the geometry read over the areal chart on a substrate whose curvature is finite there, which the bead's contour passes through rather than ending at [JanzenBHcausality, JanzenCRframework]. The usual motivation for quantizing minisuperspace, to resolve an initial singularity, therefore does not arise. The point of (4) is not singularity avoidance but the demonstration that, on the selected foliation, the canonical generator evolves rather than annihilates.

The closed-S³ lift: the graviton sector on the layer

The minisuperspace illustration of Section 5 truncates the layer to its scale factor alone; it carries no propagating degree of freedom and so cannot exhibit what the true Hamiltonian generates beyond the background expansion. The full layer does carry such a degree of freedom—the transverse-traceless shear of its spatial geometry, the graviton—and the lift of the deparametrization to it is the substantive canonical content. We carry out that lift here. Throughout, the foliation parameter is the same absolute cosmic time of Section 2: the comoving congruence of the layer, which the companion framework paper identifies with the cosmic rest frame the CMB measures and with the S³ family of the null-boundary correspondence [JanzenCRframework]. It is one congruence, carrying the flat-ΛCDM and the closed-S³ slicings as two synchronizations of itself, not two frames.

The layer is best represented by the background geometry on which the cosmology is built—the reassigned de Sitter geometry, distinct (as a Lorentzian metric) from the Schwarzschild–de Sitter cosmology it underlies but sharing its manifold and foliation [JanzenCRframework]. Its closed synchronous slicing is the evolving round three-sphere of radius a(T)=α cosh (T/α) in cosmic time T. A transverse-traceless perturbation of the spatial metric decomposes into the tensor harmonics of S³, h_ij(T,x)=_n φ_n(T) Y⁽ⁿ⁾_ij(x), with the Y⁽ⁿ⁾_ij the standard transverse-traceless rank-two harmonics of the unit three-sphere and Laplace eigenvalues n(n+2)-2, n≥2. The frequency the reduction gives is not that eigenvalue: the second variation of the spatial curvature carries a non-derivative contraction alongside the gradient term, R_ikjlh^kl=-K h_ij pointwise for any traceless symmetric perturbation on a maximally symmetric section, and it displaces the eigenvalue by +2K with no dependence on which harmonic is displaced, so the tower's frequencies are μ_n²=n(n+2)—in the labels m=n+1 below, μ²=m²-1^P10R37. That the displacement is the same at every level is a computed fact and not an extrapolation from the lowest: reducing the action again at a level whose harmonic has components varying in all three directions—so that the gradient terms absent at the floor are present—returns the displaced value and not the eigenvalue, the two differing by exactly two with every step exact, while the same pipeline in the same arithmetic reproduces the floor's own value^P10R38. And the two parametrisations separate the object from its presentation exactly: the difference between them is the same constant at both levels, while the difference between the levels is the same in both parametrisations. The second-order Einstein–Hilbert action in the transverse-traceless sector reduces, mode by mode, to that of a harmonic oscillator with time-dependent mass a³ and frequency μ_n/a,

Read on the absolute foliation exactly as in Section 5, the constraint deparametrizes and (4) becomes

a unitary evolution in cosmic time of a tower of time-dependent oscillators^P10R2, one per tensor harmonic. This is the lift the toy lacked: the abstract gauge-invariant degrees of freedom (q^A,p_A) of Section 5 are here concrete—the graviton modes φ_n—and the deparametrized generator advances them rather than annihilating them. Each mode is a Schrödinger oscillator on L²(R) and is manifestly self-adjoint; the closed topology of the layer enters as the discreteness of the tower, in contrast to the continuous spectra of the flat minisuperspace and planar reductions.

The same tower has a second reading, in the frame of a fundamental observer. Under the null-boundary reassignment the observer's worldline is one of the reassigned null rulings, and the layer's expansion read along it is the areal radius a(τ)=(2Mα²)^1/3 sinh^2/3(3τ/2α) in the observer's proper time τ—the flat-ΛCDM scale factor of the framework paper, the expansion leg of the cosmogenetic bead the framework proves one closed curve [JanzenCRframework]. The cosmic time deparametrized here is the very parameter along that bead: real on the expansion leg the observer inhabits, it is what continues off the real axis, through a bounded contour, along the bead's lap into the antecedent collapse. The tower projected into this frame is (14) with a(T) replaced by a(τ), the discrete index passing to the continuous wavenumber of the observer's locally flat slices. What distinguishes the two readings is the bend: the observer's Friedmann readout is

carrying a pressureless matter term 2M/a³ that is absent from the background geometry's pure-Λ expansion, (a/a)² →α^-2. The matter the observer attributes to the graviton's background is thus the bend of the slicing—a feature of the projection, not of the layer—exactly the reclassification the framework makes for matter content generally [JanzenCRframework, JanzenOperator]. The graviton sector therefore exhibits, in one object, both faces of the layered reading: the discrete unitary tower on the ontological background, and the continuous flat-ΛCDM spectrum a real observer measures, with the matter density appearing only in the latter.

This realizes concretely the loop drawn in Section 7 with the dynamical content. The propagating degree of freedom is the transverse-traceless shear of the evolving layer, advanced by a true Hamiltonian on the absolute foliation—the same structure the companion dynamics paper exhibits for the confined polarized wave [JanzenDynamics], here for the cosmological sector and the full tensor tower. The necessity of the external clock is sharpest here: an internal (area) clock is available at Λ=0 and would serve, but at Λgt;0 it no longer separates the dynamics, and the absolute cosmic time is forced. Of the two background-sector questions, the first is settled in structure. The operator ordering and the half-line self-adjoint extension of the scale factor (Section 5) are one coupled question: in the geodesic coordinate x∝a^3/2 the kinetic operator carries an inverse-square term γ/x² at the origin whose coefficient, across the natural ordering family, attains a maximum γ= 14—strictly below the essential-self-adjointness threshold γ= 34—so the operator is limit-circle at a=0 for every ordering, while the -Λ/8³ term maps to the marginal -x² and is limit-point at a→∞, forcing no condition there. The scale-factor Hamiltonian therefore has deficiency indices (1,1) independently of the ordering [Weyl1910, ReedSimon1975]: a one-parameter family of self-adjoint extensions fixed by a single boundary condition at a=0. The ordering ambiguity cannot render the quantization unique; the residual reduces to the physical choice of that boundary condition. That choice is fixed, for the free scale-factor sector, by the geometry of the boundary itself. It is worth naming what kind of closure that is, because the corpus has a criterion for exactly this and this is its sharpest instance. A one-parameter family of self-adjoint extensions^L10 is a one-parameter family of candidate quantizations—distinct physical predictions indexed by a λ that the formalism does not pin. That is the structure the epistemic reading excludes: a structure carrying an unforced parameter is not a single world but a family, and answers “what is the world?” with a family rather than a world [JanzenShadowExistence]. What follows below is not a stipulation that removes the family but the substrate requiring its own configuration: the boundary at a=0 is the de Sitter horizon, its surface gravity is κ=1/α, and the regular Euclidean state at that κ leaves nothing to choose. The distinction that makes this the criterion's instance and not merely an analogy is worth keeping: λ indexes distinct worlds, so it is an ontological family and inadmissible as it stands, whereas a parameter that merely records an unknown datum of one world—the progenitor's mass, say—is an epistemic gap and not a family at all. The point a=0 is not a generic half-line endpoint but the de Sitter cosmological horizon—the smooth, finite-curvature metric singularity at which the comoving congruence's past null ruling closes, the substrate curvature staying finite while the comoving ruler collapses [JanzenBHcausality, JanzenOperator, JanzenCRframework]. A Killing horizon of surface gravity κ carries a thermal (Hartle–Hawking) state [HartleHawking1976], the unique state regular in the Euclidean continuation and free of a conical defect at the horizon; for the de Sitter cosmological horizon f=1-r²/α² gives κ= 12|f'(α)|=1/α, regularity period β=2π/κ=2πα, and the Gibbons–Hawking temperature T=ℏκ/2πk_B=ℏ/2παk_B [GibbonsHawking1977]. Imposed on the scale-factor wavefunction, Euclidean regularity is exactly the condition that retains the regular indicial branch x^1/2+ν, ν=√γ+ 14 lt;1, and excludes the irregular admixture x^1/2-ν, whose presence is a conical defect—an off-temperature state. This is the Friedrichs extension, and it is a single boundary condition: the deficiency-(1,1) freedom is thereby not merely parametrized but closed, and closed without a free parameter, since the de Sitter horizon fixes one temperature. The physical choice the ordering ambiguity left open is the horizon's own thermal state, and ℏ enters the gravitational sector here, at the seam, scaled by Λ alone—the quantum instance of the substrate's constant ledger, in which the fundamental constants enter as unit gauges over the single scale and the one place a free quantum parameter could sit is closed by the horizon's own thermal state [JanzenGeometricCore]. And that continuation is not a device brought in to close the freedom; it is the substrate's other real form. The global Wick rotation x₀↦ix₀ carries dS₅=SO(5,1)/SO(4,1) to the compact S⁵=SO(6)/SO(5), and the two are the real forms of the one complex group SO(6,C) [JanzenBoundary, JanzenGeometricCore]. So the register in which ℏ enters is the same substrate read on its compact face, and the horizon—where the period is β=2πα—is where the two faces meet: the extension is fixed not by an extra assumption but by the geometry the Lorentzian reading already carries, seen from its other side. This determines the free scale-factor sector; the coupled sector below is where the open structure remains. And what remains there is one of three parts, the other two being settled. The boundary operator is bounded below and its spectrum is computable branch by branch, both shown below; what is not settled is the completion of the ultraviolet definition of the mode sums, the definition itself being delivered below. That the remainder is that part and not another is worth stating, because it bounds what any calculation on the free spectrum can reach: a deformation of the free frequencies is answerable exactly—no multiplicative renormalisation moves the logarithmic coefficient, and the one mass-like displacement that removes it is the curvature offset itself—but whether the coupling acts on the sum that way cannot be settled ahead of the sum's own definition, which is the open item. So the free-spectrum method reaches exactly as far as it reaches, and the remainder is not a finer calculation of the same kind^P10R7. The second background-sector question—the couplings of the tower to the scale-factor sector beyond quadratic order—sharpens this structure rather than threatening it. Deparametrizability is not at risk: the cosmic clock is a Brown–Kucha\v{r} dust fluid (Section 5), whose momentum enters the scalar constraint linearly whatever the graviton content, so the constraint solves for a true Hamiltonian at every order of the coupling—which enters H_phys as interaction terms between a and the modes, not as an obstruction. The free tower above evolves on a(T) as a fixed classical background, which is why each mode is manifestly self-adjoint on L²(R); the coupling question is what happens once the scale factor is itself quantized and back-reacts, and there it acts at the a=0 boundary of the scale-factor half-line. Each mode's kinetic term π_n²/2a³ is, in the geodesic coordinate, the inverse-square operator π_n²/2x² at the origin, so the boundary coefficient is promoted from the c-number γ≤ 14 of the free scale factor to an operator Γ on the tower whose spectrum straddles the 34 threshold. At leading order that spectrum is bounded below by γ and unbounded above—the operator is a sum of squares, displayed below—but the decomposition that follows uses only that both sides of the threshold are occupied, and not that the spectrum has a floor; whether the complete Γ is bounded below is not assumed here ^P10R12, though it does in fact follow: the full inverse-square coefficient is positive on non-degenerate metrics, so Γ=γ+c_n π_n² retains its floor at γ beyond leading order, and the question of a spectrum below - 14—where ν=√Γ+ 14 would leave the reals and no regular branch would exist—has no object. The straddle itself is a computed fact^P10R13: the spectrum does occupy both sides of 34, since Γ=γ+c_n π_n² with c gt;0 has spec Γ=[γ,∞), and γ≤ 14 lt; 34 places spectrum strictly below the threshold while the unboundedness above places spectrum strictly above it—the negation requiring γ≥ 34, three times its own bound. That threshold is a different one from - 14, deciding a different property: 34 separates limit-point from limit-circle and so decides whether a boundary condition must be chosen at all, while - 14 decides whether a regular branch exists to choose. How the straddle falls does not bear on the closure below, which is supplied fibre by fibre and so cannot be broken by the size of the sub-threshold set ^R. As Γ commutes with the radial part, -∂_x²+ Γ/x² decomposes as a direct integral over its spectrum: essentially self-adjoint where Γ≥ 34—the graviton momentum makes the origin limit-point and removes the boundary freedom—and limit-circle where Γlt; 34. The single boundary condition of the free scale factor is thereby replaced, in the coupled theory, by a self-adjoint condition supported on the sub-threshold graviton subspace, its deficiency that subspace rather than a line. At leading order Γ=γ+c_n π_n²; but the cubic and higher self-interactions enter at the same inverse-square order at the origin (π_n²φ_m/a³ in kind)—the potential sector's own cubic being regular there, and so reaching the trace without reaching the boundary coefficient—so the complete boundary coefficient is the singular part of the interacting graviton Hamiltonian, and the completion of the ultraviolet definition of the tower sums is the open frontier—the interacting tower as a defined theory, met here at its boundary face. The spectrum is not open with it: it is computed branch by branch below, and which branch is a question the construction has been shown not to answer internally. Two things can be said about that spectrum without settling the frontier, and both bear on the decomposition. First, it is computable branch by branch: on the natural ordering family the minimum of Γ is 14 under normal ordering and 34 under symmetric ordering with one mode occupied, and the difference between the two is exactly the zero-point quantum of that mode—which is why the threshold sits at 34= 14+ 12, the free boundary coefficient plus one such quantum, rather than at a value the construction chose. Second, the two orderings answer the paper's own question oppositely: under normal ordering the origin remains limit-circle and the boundary freedom survives, while under symmetric ordering it becomes limit-point from the first occupied mode upward. The decomposition below is untouched by the choice, since it uses only that both sides of the threshold are occupied and both orderings occupy both sides; but the physical content is not, and the choice is not one this construction can make internally. Asking which ordering is asking whether the graviton tower's zero-point energy gravitates at the horizon—the cosmological-constant problem in local dress, reached from inside the boundary coefficient rather than imported. That the construction makes no such choice is an exhaustion rather than an omission. The ordering is inert in the bulk—normal and symmetric ordering differ by the c-number zero-point 12ℏω_n, a global phase under the deparametrized evolution—and becomes physical only in the boundary coefficient, so any selector would have to act there; and each candidate the construction offers fails to, for a structural reason: the horizon's thermal state and the seam's characteristic structure act downstream, closing the extension given Γ (the regular branch x^1/2+ν exists for either ordering); the substrate isometry, positivity, the single-scale ledger and covariance under configuration redefinition each respect both orderings; and the deparametrization, which supplies a preferred vacuum and so makes normal ordering available, does not make it mandatory—and, by solving the constraint rather than imposing it, removes the anomaly-freedom lever a Wheeler–DeWitt quantization would have used. What is left is not a residual freedom the quantization failed to close but a single physical datum—whether vacuum energy gravitates—localized precisely as one computable quantity: an epistemic gap of the kind the construction admits, not an ontological family of the kind it excludes^P10R21. The boundary condition on that sub-threshold subspace, however, is supplied by the same principle that fixed the free sector. Thermal (Hartle–Hawking) regularity is Euclidean smoothness at the horizon mode by mode: across the direct integral it imposes the regular branch x^1/2+ν, ν=√ Γ+ 14, on each sub-threshold fibre at the one horizon period β=2πα (the surface gravity κ=1/α belongs to the background horizon, not to the graviton content, and so is common to every fibre), and asks nothing of the limit-point fibres Γ≥ 34. The self-adjoint condition on the sub-threshold subspace is therefore the de Sitter horizon's thermal state at every order of the coupling, exactly as in the free sector; what remains open is accordingly not the boundary condition but the definition of the interacting tower—the spectrum of Γ and the ultraviolet definition of the tower sums—the first of which admits a verdict here, and it is affirmative, while the second is delivered below by the construction's own regulator and only its completion stays open^P10R15. The DeWitt form carries exactly one negative direction and it is the conformal one, which is why the Wheeler–DeWitt equation is hyperbolic; on transverse-traceless momenta that term drops and the form is tr((gπ)²), equal to tr(S²) for the symmetric S=L^TπL with L the Cholesky factor of g, hence positive definite for every non-degenerate spatial metric. So the cubic term's apparent unboundedness is an artefact of truncation: π²(1-λφ+⋯) is the expansion of π²/(1+λφ), whose full coefficient is positive wherever the metric is non-degenerate.^R The truncated form is easily misread, because a truncation of a positive function reads as an operator that has lost its floor: the first-order term -λφπ² sampled alone does go arbitrarily negative, while the resummed coefficient on the same points never falls below γ. The deparametrization that yields a true Hamiltonian is the same move that yields a bounded-below one, both being the removal of the conformal direction—though the positivity speaks only of the interior, the degenerate boundary being what the thermal condition above is for. What remains genuinely open is accordingly the completion of the ultraviolet definition of the tower sums, the definition itself being delivered below by the construction's own regulator—a different thing from a residual freedom in the quantization at the boundary, and not settled by being shared with every interacting field theory. And what remains open is narrow, because the coefficient's response to the coupling is settled even though the coupled sector is not. The logarithmic coefficient is the 1/m term of d(m)μ(m), hence an exact functional of the spectrum, so the deformations that could discharge it can be enumerated without solving the coupling: an overall rescale μ²→(1+ε)(m²-1) returns L=(15/4)√1+ε, which vanishes only where every frequency in the tower does. So no multiplicative renormalisation of the frequencies discharges the log—and that is the class the leading back-reaction lives in, the coupling acting through a and through Γ, whose instantaneous expectation is the same sum rescaled. A mass-like shift μ²→m²-1+δ returns L=-(δ-1)(δ+15)/4, so the one such discharge is δ=1: there μ_n=n+1 and d(m)μ(m)=2m³-8m is a polynomial, the 1/m term not cancelling but not existing—and +1 is exactly the offset that makes these frequencies non-integer, being the net of the Laplacian's -3 and the curvature term's +2. The enumeration is one formula and does not depend on where the offset is put: at μ²=m²+u the coefficient is -u(u+16)/4, vanishing at u=0 and at u=-16 alone.^P10R17 That is a characterisation, and a measurement and a structural remark sharpen it.^R The measurement first. The tower is a discrete sum rather than a momentum integral—the harmonics run from n≥2 with no zero mode and no soft region—so the infrared is regulated by the compactness of the layer and needs nothing. The ultraviolet is untouched by that: with μ_n∼n, ⟨ π_n²⟩∼n in the instantaneous ground state, and a degeneracy growing as n², the shell contribution goes as n³ and the sum diverges quartically—the ordinary zero-point degree of a field in four dimensions. The third of those inputs is a formula and not only a scaling: since S³=SU(2) is parallelizable, a frame-indexed field of frame-spin s is L²(SU(2)) V_s, Peter–Weyl returns level-j totals of 1, 3 and 5 times (2j+1)² for s=0,1,2—the component counts—and the transverse-traceless part is the two extreme summands, exactly two fifths of the symmetric-tracefree total. Organised by eigenvalue, the degeneracy is 2(n-1)(n+3), ten at the floor n=2, so the shell contribution is 2n³ and the leading constant is settled rather than assumed^R. And the constant is the propagating-component count rather than a universal: the scalar tower's counting function grows with leading coefficient 13 against the tensor tower's 23, so the growth is n² per propagating component and not per tensor rank. So “standard” is the right word and it is a number. And the leading term's counterterm is one this framework already carries. The quartic piece of the one-loop vacuum energy is independent of mass and of field—it survives at m=0 and is the same for every species—so it is a constant vacuum energy, and the counterterm a constant vacuum energy requires is a cosmological-constant term. That is the framework's single dimensionful constant, absorbed into the one observed curvature with no bare-Λ-versus-vacuum split [JanzenGeometricCore]; and since ℓ_P is a gauge-combination rather than a second physical length, the cutoff is not smuggling a scale in either.

The subleading term is measured on the same spectrum rather than estimated. Written in the labels m=n+1 that carry μ²=m²-1 and d=2(m²-4), the spectral zeta at the origin is ζ(0)=10 exactly—the binomial expansion terminating there, so the value is an identity and not an asymptotic estimate, and it is a functional of the degeneracy alone, the frequency entering at s=0 as unity whatever its offset—and the zero-point sum's logarithmic coefficient is 15/4, the 1/m term of d μ=2m³-9m+15/4m^-1+ 78m^-3+…, which a hard cutoff carrying no zeta function reproduces. And the scale factor does not enter it: the free tower's frequencies are μ_n/a, so a factors out of the sum identically and the coefficient's independence of the expansion is exact rather than adiabatic—which is also why the coupled sector, and not a time-dependent free one, is where what remains lives^P10R3.

The structural remark is about the terms after it, and it is sharper than the leading case because their counterterms are not cosmological-constant terms. Only the quadratic and logarithmic successors carry the mass, which is to say only they renormalise masses and couplings; in the standard curved-space expansion the first goes with the Ricci scalar and the second with curvature-squared invariants, and a curvature-squared coupling is not an entry in this framework's ledger. The statement about curvature-squared couplings is two claims and not one, and the useful thing is that the criterion separating them does not depend on the operator dimension at all. The framework's dimensionful content is one length, and the gauge length is a combination over it rather than a second physical length; so a coefficient carrying any power of length is admissibly the substrate scale to that power times a pure number, and it introduces a second length only if that pure number is left undetermined. The question at every dimension is therefore the same question—whether each coefficient is a number the construction determines—and this section already holds one instance of it settled, the shear's own coefficient being exactly twice a real scalar's and therefore a count of propagating modes rather than a number chosen to fit. What the dimension ladder decides is how many such numbers there are, which is the renormalisability question and not the single-scale one^P10R41. The reason that is not fatal here is a property of the backgrounds this construction admits rather than of the divergences. On a maximally symmetric geometry of radius α every curvature invariant is a pure power of 1/α² [JanzenGeometricCore], so ∫√g R², ∫√g R_μνR^μν, ∫√g R_μνρσR^μνρσ and ∫√g are not four independent functionals but four multiples of one; and the substrates the construction admits are a one-parameter family, the α-foliation, every descent step above the last changing the scale and nothing else [JanzenGeometricCore, JanzenAlgebroid]. So the counterterm basis is one-dimensional on the admitted family, and a divergence of any degree requires exactly one counterterm where a generic theory requires three.

And that reduction does not in fact need the family, which matters because the family's own flatness does not survive second order in the shear. In four dimensions C²=E₄+2 (R_μνR^μν-R²/3) is an algebraic identity on any metric, with E₄ the Gauss–Bonnet density: the curvature-squared invariants are dependent because the dimension is four and not because the background is symmetric, so no order in the shear can separate them. What the admitted backgrounds then supply is the rest of the step rather than the whole of it: on an exact vacuum-Λ space the Ricci terms are constants and C²=E₄-8Λ²/3 exactly, the Gauss–Bonnet part shifting the on-shell action by a topology-fixed constant and the Ricci remainder being what the field equations trade for the stress tensor. So a change of subtraction point is a cosmological-term renormalisation, reached from an identity rather than from a property a deformation could remove^P10R23. The restriction is the on-shell one and it is the whole restriction: off shell C² is an independent invariant and nothing here bounds it.

What that argument does not reach is stated here rather than left for a reader to find, because it is the whole of what remains. A degeneracy among counterterms is a property of the background class: the functionals are distinguished by their response to varying the background, and on a one-parameter family they cannot be told apart. The first thing to check is therefore which background the tower is defined on, and it is this section's own: the closed synchronous slicing a(T)=α cosh (T/α) of §5, whose Ricci scalar is the constant 12/α². That geometry is exactly de Sitter, so the degeneracy holds on the very background the free tower uses, and the argument above is not an appeal to a nearby object.

Where it genuinely stops is elsewhere, and the section has already named the place. The free tower evolves on a(T) as a fixed classical background; the coupled sector is what happens once the scale factor is itself quantized and back-reacts, and that is precisely the regime the boundary operator Γ belongs to. The natural worry there is that a counterterm basis is a statement about a class of fixed backgrounds, so that once the scale factor is quantized there is no fixed background left to state it on. That worry does not survive being checked, and what replaces it is sharper.

The collapse of the three quadratic invariants is not a consequence of maximal symmetry: in four dimensions their deficit Riem²-2 Ric²+ 13R² is exactly C_μνρσC^μνρσ, every Friedmann geometry is conformally flat for every scale factor, and on such a geometry the Gauss–Bonnet^L17 combination is an exact total derivative, √g (R²-4 Ric²+ Riem²)=24 d(a³/3+ a)/dT.

So ∫√g Ric² and ∫√g Riem² are both fixed by ∫√g R² up to a boundary term whatever a does: no scale factor breaks it, because no scale factor can make an FRW geometry anything but conformally flat^R. And because that is an identity holding pointwise in a(·) rather than an evaluation on a chosen class, it survives superposition and descends to the quantized sector as an operator relation: the coupled sector never required a class of fixed backgrounds, because the statement was never made by evaluating on one. What maximal symmetry did buy is a weaker and different thing, and that half does not survive: on a constant-curvature background terms of different dimension—∫√g, ∫√g R, ∫√g R²—are proportional as well. What that half actually needs is weaker than a constant-curvature background and weaker than the cosh: it is exactly that R be √g-almost-everywhere constant, and two things that are not the cosh keep it so. A cosmological term with radiation gives R=4Λ exactly, radiation being traceless; and the free tower's own bare zero-point energy is exactly radiation-like—its energy is S/a with S free of a, so the density goes as a^-4, the pressure is a third of it, and the trace vanishes identically. So the naive back-reaction of the quantized tower breaks nothing, and both cases serve as null controls that measure the numerical floor rather than assuming it.

What does break it is the logarithm, which is to say the very constant at issue. The spectral Z(s) has a pole at s=-1 of residue 15/4, so the renormalized energy carries a ln a—and a ln a spoils the exact a^-4 that made the bare sum traceless. The trace is then Θ=ρ-3p∝15/4a^-4, independent of the renormalization scale: it is scheme-independent, which is what an anomaly is, and is why the split between the constant and the logarithm being a convention does not reach it. So R acquires an a^-4 piece and is constant if and only if that residue vanishes. The three integrals then span three dimensions rather than two, at a ratio thirty-five decades above the floor the null controls measure, with the departure signed by the identity I₂I₀-I₁²=I₀² Var_√g(R)≥0 and growing as the square of the residue: the logarithm's counterterm is a genuinely new entry, and the one ultraviolet constant becomes observable once the scale factor is quantized^P10R24.

That is the first of a sequence, and saying what the sequence is says what defining the interacting tower consists of. It is the rank of the counterterm basis, one rank per operator dimension, taken on the class the construction admits—which, the scale factor being quantized, is not a class of fixed backgrounds—together with, at each dimension, the data that settles that many numbers. Two entries of that sequence are computed. At dimension four the three quadratic invariants have rank two of three pointwise on a general scale factor, adjoining the Gauss–Bonnet combination leaves it at two—so that combination is already inside the pointwise span and the reduction to one functional is the total-derivative step rather than a second degeneracy—and as functionals on the one-parameter family the rank is one, which the trace anomaly then opens by an exact Gram identity. The vanishing of the Weyl invariant there is an identity in the scale factor rather than an evaluation, and it is read against a control that separates them: the same combination on an anisotropic metric with three different rates does not vanish.

At dimension six there is no degeneracy at all, and that is the finding. Five dimension-six scalars are pointwise independent on a general scale factor—no relation whatever—while on the one-parameter family two of them vanish identically and the rest are pure powers, so the rank there is one. So the collapse at dimension six is entirely the one-parameter evaluation, and at dimension four it was not. At dimension four the collapse rests on an identity holding pointwise in the scale factor, which is why it descends to the quantized sector as an operator relation; at dimension six there is no such identity to descend, so the half of the argument that already fails under quantization is the whole of the collapse there—and the fixed-background reading buys strictly less at the higher dimension, not more. What that count is a fact about is said before it is used: it is read off a chosen five-element subset, so it is a fact about those five—but it is a lower bound on the full basis's rank, and a lower bound is the direction the argument needs, completion being able only to raise it. And off the classical history the count that matters is taken modulo the scale factor's equation of motion, for the reason given below.

And the narrower of the two questions this raises is not closer than the other; it is finished. Whether the coefficient the free spectrum settles survives the coupling is answered by the same counting, and the counting is fixed to an observable rather than to a way of writing the field. The expansion is that of the vacuum energy in the ratio of the gauge length to the scale factor and in no other ratio; a counterterm of operator dimension 2k contributes at an order in that ratio equal to 2k-4 exactly, and reading that backwards, second order admits one operator dimension and no other, a logarithm carrying no power of the scale factor and so riding at the same order. The interaction therefore enters at second order and cannot populate the zeroth at any order, while the free tower populates the zeroth alone—its large-label weight carrying no logarithm of the label, checked against a weight that does carry one so that the test discriminates. And the warrant for that is the vacuum energy being a spectral invariant rather than any count of vertices, which matters because a vertex count is a property of the split: the framework offers two lengths, so a redefinition off the grading is dimensionally admissible and generates vertices carrying none of the gauge length—exhibited on a free oscillator, where a canonical point transformation leaves every spectral quantity fixed and yet reads as a genuinely interacting truncation, and with a control that returns a non-zero shift on a genuine quartic so that the silence on the generated one is a finding. So that coefficient lives at a power no order of the coupling reaches, and it survives unconditionally rather than conditionally, on a warrant that no change of variables can move^P10R41.

The divergence does reach second order in the coupling, and it does so by an identity rather than by a search that failed to find a reason for it to vanish. At that order two channels contribute. Every cubic here is non-resonant—no matrix element is diagonal in the energy—so the generator of the redefinition it induces exists term by term, and the redefinition has a unique quartic completion whose own second-order shift is exactly zero. What survives is therefore the actual quartic's deviation from the completion the cubic itself generates, which is a difference of two expectation values and not a residue of either. And no symmetry, parity or count of modes can remove it: one channel is a sum of non-negative numerators over positive denominators, the whole is even in the cubic coupling, a purely cubic interaction already gives a non-zero value, and a total derivative changes no spectrum. Vanishing requires one coincidence between the two couplings, of codimension one—and that locus is exactly where the interaction is a redefinition and there is nothing to renormalise. The parity escape is closed by this section's own arithmetic, the cubic overlap being a determinant that does not vanish on a traceless shear. The instrument is calibrated in both directions: exactly zero on two unitary conjugations, with both channels individually non-zero and equal, and the textbook value on a genuine pair^P10R44.

That ratio is settled, and it is settled by reading the construction rather than by arguing about it: it is taken from the world. The two objects share no argument—the substrate scale is a curvature radius of the four-geometry and has zero derivative with respect to every quantum and gravitational gauge, while the gauge length is not a length on any space at all but a combination of conversion factors whose dimension is a length by unit-counting and has zero derivative with respect to the curvature—so no relation among the inputs of either can fix their ratio. Calling the second a gauge combination therefore fixes that it introduces no new scale; it fixes neither a value nor a functional form. And the place the quantum constant enters is not the place a value is fixed: the horizon's thermal condition defines a temperature and so arrives with its own new quantity, adding one equation and one unknown, whose net constraint on what the ledger already holds is zero. The separating test is an arithmetic—the ratio has non-vanishing derivative with respect to every measured input, where the shear's coefficient has vanishing derivative with respect to all of them, being a mode count^P10R42.

So the affirmative branch is a second number owed and not a second length introduced, and the single-scale ledger stands either way, an undetermined dimensionless number not being a scale. What it costs instead is an entry in a different ledger this construction also keeps—the count of dimensionless constants spent—which stands at one for these very sums. And a determination is not merely absent but declined, which is the honest form of the statement: because the substrate horizon's entropy is exactly a multiple of the squared ratio, determining the ratio would be determining that entropy as a pure number, and the object required is a state count for that horizon. The construction takes the horizon's temperature and never its entropy, on the stated ground that a temperature carries a dimension where the entropy is a cross-register pure number. This is therefore a single-scale ledger with one number read from the world, and that is stated here rather than left to be inferred.

And the size of what the affirmative branch would add is bounded without computing it, by the geometry rather than by an estimate. The scale factor of the admitted family has a positive minimum—its only stationary point is the turnover, where the second derivative is positive—so the expansion parameter is largest exactly there and nowhere else, and it is bounded above by the product of the curvature with the squared gauge length at every epoch, a number of order 10^-122. So the case in which the scale factor's own powers undo the suppression does not exist, and it does not exist because the background turns over rather than because the epochs happen to cooperate^P10R43. And the new number is separately visible rather than degenerate with the first: the zeroth-order counterterm reaches the trace through the logarithmic term of the trace formula at one power of the scale factor and the second-order term through the other term of the same formula at another, so the two are distinguishable by where they appear.

And then the counterterm it would need collapses, because the substrate background is not merely conformally flat but exactly maximally symmetric. That is computed from the metric's own connection rather than assumed: every component of the curvature is the maximally symmetric form with sectional curvature one third of the cosmological constant. So every component of the Weyl tensor vanishes, and with it both cubic Weyl contractions—the pair that carries pure gravity's own two-loop counterterm—and all eight algebraic invariants of the next dimension are exact rational multiples of the cosmological constant cubed, of rank one among themselves. And the covariant derivative of the curvature vanishes identically here in every component, so the collapse is not confined to the Weyl part: the whole derivative sector at that dimension vanishes as well, and what is left to count is the algebraic part alone. And that argument does not stop at one dimension. Any invariant carrying an explicit covariant derivative contains that vanishing factor, so the derivative sector is identically zero at every dimension; and at operator dimension 2k the algebraic invariants are polynomials of degree k in a curvature that is itself the maximally symmetric form, hence rational multiples of the k-th power of the one input. Verified at the next dimension up on ten invariants, each homogeneous of the right degree and of rank one among themselves, and then for an arbitrary rational combination of them. The scope belongs in the same sentence as the result: that is a rank over the values these invariants take on this background, where the rank sequence of the counterterm basis is a rank over the admitted class, and the two are different objects^P10R45. The control separates that from an evaluation: the same eight on an anisotropic metric are of rank two together with these, with both cubic Weyl contractions non-zero and a non-vanishing curvature gradient, returning the quadratic invariant's own value quoted earlier in this section.

Which settles the coefficient's status without computing its value, and that is the useful half. For an arbitrary rational combination of the basis, the residue has vanishing derivative with respect to every quantum and gravitational gauge and is homogeneous of degree three in the cosmological constant. So whatever the computation returns is a rational number times the one input cubed, and the count of dimensionless constants spent stays at one^P10R44. The scheme dependence of the finite part is shown rather than asserted: a logarithmic mode sum regulated at two cutoffs an octave apart carries the same logarithmic coefficient and finite parts differing by a constant, so the finite part is a convention and the coefficient is not.

And what that subtraction has to absorb is one double sum over pairs of labels, which is fewer sums than the order of the calculation suggests. Because the identity above sets a difference of two quartic operators against each other, a free expectation of either is a sum over two contractions and not over intermediate states: the identity has already performed the sum a third-order reading would have left open. The index set closes from below on its own, the degeneracy vanishing at the label that would otherwise have to be excluded by hand, and the two field variances multiply to the minimum-uncertainty product exactly—which is the check that they were not scaled independently of each other.

That sum diverges, and it diverges two powers above the free tower's own quartic. The action being second order in derivatives supplies exactly three quartic structures, and their large-label weights put the leading behaviour at the sixth power of the label cutoff. So the zeroth-order subtractions cannot reach it, which is the case that matters: a divergence the existing subtractions absorb would have changed nothing, and this one requires a subtraction of its own. Two independent routes agree on where it lands—the mode count in labels and the momentum cutoff carrying the squared gauge length both give an energy density at the next operator dimension, which is where the dimensional bookkeeping had already put it, so the counting and the bookkeeping are not the same argument twice. And the quantity the rate turns on is computed rather than bounded, so the rate is exact. The object the double sum actually reaches is the level-summed overlap, the field variances depending on the level alone—and on a space whose isometry group acts transitively, with each level a single irreducible representation, that sum is a constant tensor by homogeneity and Schur's lemma. The completeness relation is verified exactly in every component on a basis checked orthonormal, and at the one level this section owns the density is a computation rather than a theorem, the frame-constant harmonics summing to exactly the degeneracy. The control separates the identity from the basis: dropping a single element leaves the components disagreeing with the projector and the trace short by one. So the level-summed overlap is the product of the two degeneracies times a pure number—and that product is exactly the degeneracy already counted, so the residual factor has no dependence on either label at all. The fall-off exponent is zero where the reversing threshold was three, checked from both sides, and the sixth power is therefore exact rather than conditional^P10R46.

And it never reaches a four-label overlap at all, which is what makes the rate factorise. The quartic vacuum expectation of a multi-mode free field equals its Wick sum with no departure, and because the two-point function is diagonal that sum is supported on one or two distinct labels and never on three or four—computed from the mode operators rather than assumed. So the pairings differ in how the indices are arranged and not in how many labels they carry, each submitting to the same level-summed identity with a different pure number, so an off-diagonal arrangement moves the coefficient and not the rate. The two-label term being a product of two single-label moments is what lets the rate factorise, and that is derived here rather than assumed^P10R46.

So the subtractions are three and not two, and the third is the one this row was opened over. The quartic and the logarithm sit at zeroth order with dimensionless coefficients; the third sits at second order, at the next operator dimension, with a coefficient that is a rational multiple of the one input. And the same argument gives the whole sequence rather than one entry of it: at every operator dimension the basis takes values of rank one on this background, in rational multiples of the corresponding power of the one input, with the derivative sector absent throughout.

What the section holds is therefore the subtraction's whole form, and the conventions it leaves behind are three, necessary, and countable. They are the finite part of the sixth-order coefficient, the finite part of the cosmological one, and the scale inside the logarithm—and the asymmetry between them is what everything below turns on, the third being the only one that is a scale rather than an additive constant and the only one whose action on the energy density carries a power of the scale factor rather than none. They are necessary rather than listed: leaving any of the three unsubtracted leaves an infinite limit, and subtracting all three leaves exactly the finite part, so the count of conventions is the count of structures that have to be fixed. And the logarithm's own coefficient is not among them: an octave in the subtraction point moves the finite part by exactly that coefficient times a logarithm of two and moves the coefficient not at all, and the same shift acts identically on every member of the family—so a measured difference fixes the shift's action and never a point of it, which is why the coefficient is a computed rational and not a convention.

Two of the three reach nothing observable, and the reason is the collapse above rather than a cancellation. The three move the energy density along three directions, of which the first two are pure constants here because every counterterm's value on this background is a rational multiple of a power of the one input. So the rank is two of three and the kernel is exhibited rather than counted: an explicit combination of the sixth-order and cosmological conventions shifts the energy density by exactly zero, identically in the scale factor. The convention this row added is therefore rank one with the cosmological one—new as a subtraction and not as a number taken from the world. The control puts the degeneracy where it belongs: give the sixth-order counterterm a value that does not collapse and the rank is three with no kernel at all, so the degeneracy is this background's and not the rank test's.

And the one observable this section names is untouched by a grading rather than by the cases tried. The trace of a term carrying an inverse power of the scale factor is that term times four minus the power exactly, so the fourth inverse power is the whole kernel of the trace and the zero is there alone. The anomaly is that coefficient, and it has vanishing derivative in all three conventions: the two additive ones move only the trace's constant part, which is the cosmological term the framework absorbs into the one observed curvature, and the third moves nothing at all, being traceless. Controls at the neighbouring powers return non-zero, so the vanishing is radiation's own property and not the formula's—and that the constant part does move is what makes this a grading and not a silence. And the tower's spectrum is untouched at fixed scale factor, a convention being a multiple of the identity there: every spacing exactly unchanged while the total energy shifts, against a control that is not a multiple of the identity and moves every spacing.

The third convention does reach an observable, and that observable is the one this section already records the mode sums as spending. The logarithmic counterterm's coefficient splits: a divergent part fixed at the anomaly, which is scheme-independent, plus a finite part which is the subtraction point's convention. So the constant is neither downstream of the three nor one of them—it contains exactly one, and that the anomaly is scheme-independent while the constant is a convention are both true of the same object, separated by where each appears. This section's one spent dimensionless constant is that convention, and saying so is what makes the accounting closed: the dependence is on a constant already paid for, and nothing new is owed^P10R47.

And one number makes the third convention and the observable direction together. Set the anomaly to zero and the logarithm, its convention and the fourth-power direction all go at once, the rank falling to one and every convention absorbing into the single constant. So the anomaly buys exactly one observable direction and no more—the two ranks differ by one—which is this section's own shape holding for the conventions and not only for the counterterm: the saving mechanism and the thing it saves us from are one number. That also fixes the reach of an earlier result rather than correcting it: the finding that no observable depends on the subtraction point rests on a one-dimensional counterterm basis, which is the reading on the one-parameter family and is right there, while the anomaly makes the basis two-dimensional once the scale factor is quantized—so that finding does not reach this sector, and the protection here comes from the ledger entry instead.

Every rank above is a rank over the values these objects take on the substrate background, so whether any of it survives the scale factor becoming an operator is a question in its own right—and it is one object rather than several. Every counterterm value on this family is a function of the single operator, and for functions of one operator a combination vanishes as an operator exactly where the function vanishes on the spectrum: the operator-valued basis and the basis of values are the same object, with equal ranks and, normalised, the identical kernel. So a rank descends here with no appeal to linearity at all. The expectation-value reading adds nothing, an expectation of a function of one operator being phase-independent and so seeing the spectral measure alone, against a non-commuting control whose phase derivative does not vanish. And the reading in which the family itself is quantized has no object: the curvature radius is a coefficient of the Hamiltonian rather than an observable of it, so its operator carries a one-point spectrum on which every function of it is a multiple of the identity—which is this section's own superselection result, exhibited as an arithmetic rather than quoted.

On that reading the descent is free at the order this section's back-reaction is semiclassical, and it fails at the next one. Pure radiation leaves the curvature scalar at exactly four times the cosmological constant, so the fourth inverse power there is the logarithm's alone; and along that deformation every invariant of the higher dimension shifts along the same function, so the rank of one descends at first order. At second order the rank is two, the breaking carried by the eighth inverse power and by squared logarithms. The control separates the two: putting a sixth inverse power into a single invariant gives rank two already at first order, so the first-order agreement is the deformation's property and not the test's.

Item by item, and they do not fail together. The rate survives at every order—the weights carry no scale factor, so a state contributes one moment multiplying the sum—and so do both counts, of subtractions and of conventions. What is order-dependent is the per-dimension rank alone. And the kernel survives in number while rotating: with the anomaly present the free direction necessarily carries the logarithm's own convention, in a computed ratio whose classical limit is the kernel above. So the finding that the added subtraction costs no observable survives the descent, with the absorption now into the constant already spent—the same coefficient appearing for the fourth time in this section, which is what a structural quantity does^P10R48.

And the ranks above are the whole sector's rather than a chosen subset's, which is a mechanism and not a larger count. At a maximally symmetric point the gradient of every realizable invariant is a multiple of the identity, so the two independent curvature directions are indistinguishable there; and the realizable cubics are exactly the subspace that condition cuts out, which the invariants already computed span. The control separates the realizable from the merely algebraic: a monomial that violates the condition raises the rank, and it is not an invariant this construction produces.

The semiclassical reading of the scale factor is then not a choice but a consequence, and the order it truncates has a size. The expansion parameter's maximum is the geometry's own: one stationary point, at the turnover, with positive second derivative, so the parameter is largest exactly there and equals the product of the curvature with the squared gauge length—of order 10^-122 on the measured inputs, and its square of order 10^-244. And the truncation follows rather than being imposed: an independent second-order source would move every second-order coefficient while leaving every first-order one untouched, and the coefficients here carry no such amplitude—so the rank the iterated first-order deformation gives bounds the construction's own second order from below rather than being it, and the two are separated below, where the genuine source is computed and the bound turns out to be attained.

At that order the rank is two of three, and the second direction is a single combination of the eighth inverse power with its squared logarithm—and it brings no fourth convention. The weights carry no scale factor, so a deformed geometry enters the mode sum as one multiplying moment and the divergent structures stay the three already named: the accounting survives with a rank change and nothing more^P10R49.

And the coefficient that has now arrived four times in this section is one object rather than four agreeing numbers. It is the residue of the tower's spectral series at the pole the logarithm's subtraction sits on, and the partial sum's logarithm, the trace's fourth inverse power and the curvature scalar's return it as computed objects—all four vanishing together on a control weight built without the term that produces it. So the saving mechanism, the observable direction, the surviving kernel and the anomaly are one quantity seen from four sides, which is what a structural number does and is why this section names it once.

And that pole is one entry of a table the whole scheme turns out to be. The spectral series of the weight carrying the frequency has simple poles at arguments four, two and zero with residues 2, -9 and 15/4; the series of the weight carrying its reciprocal has poles at two and zero with residues 2 and -7. Each pole contributes its residue over its own location to the corresponding power of the cutoff, checked pole by pole; the logarithmic coefficients are the residues at the origin exactly, so the anomaly is an entry of this table rather than a separate object; and the second-order sum's leading coefficient is a product of two residues. So the three subtractions, the anomaly and the value-rank are all statements about the pole data of one series, and the coefficients this section names are polynomials in its residues. The subleading integer powers mix, and that is a property of partial sums and not of the spectrum—the leading weight's own partial sum already carries the second power—which is this section's division between a coefficient and a convention seen from the other side.

And the vertices those coefficients are made of are available in closed form at the level this section owns, which turns the outstanding arithmetic from a calculation into a sum. The lowest harmonics are constant in the frame, so a perturbation there is a left-invariant metric on the three-sphere—and for those the geometry is known to all orders rather than to the order one expands to. The extrinsic curvature gives K_ijK^ij-K²=-6H²+6(β₊²+ β_-²) exactly in the two shear parameters β_±, the anisotropy velocities summing to zero so that volume and shear separate with no cross term and the isotropic limit returns the closed Friedmann Lagrangian; and the three-curvature, a fixed quadratic in the metric's three eigenvalues at unit determinant and equal to six on the round metric, expands as

the cubic exactly the hexagonal invariant and the quartic exactly isotropic, so one number carries it. And the calibration is what identifies this as the section's own level rather than a neighbouring problem: the ratio of the first two coefficients is eight, which is this tower's frequency-squared at the floor, and the degeneracy the same reduction carries is ten, which is the tower's own degeneracy there—two numbers the spectrum fixes, arriving from the metric.

One of the three quartic structures that reduction exhibits is a parameterisation rather than an interaction, which is this section's own redefinition identity arriving from the other side. In the exponential variable the kinetic term is exactly quadratic—no momentum-squared-times-field-squared vertex at any order—where in the linear one the same term carries both a cubic and a quartic, and the map between the two is a point transformation of unit Jacobian. So that structure is the completion the cubic itself generates, and in the variables where it is absent the physical shift is carried by the potential's quartic and the cubic alone.

The shift at this level is then a number rather than a structure. Computed exactly in a two-mode Fock space, it is 4/27(63μ²-200)μ^-4 times ℏ² times the gravitational coupling over the comoving volume and the cube of the scale factor, which at this level's own frequency is 19/27 of that combination and is positive, the quartic beating the cubic. And the threshold belongs in the same sentence as the value, and it is a criterion rather than a number: the shift is positive exactly where twice the quartic's coefficient exceeds the cubic's squared coupling divided by the frequency-squared, which at this level's own vertex numbers puts the turning point at a frequency-squared of 200/63. That figure and the tower-wide criterion derived below are in one convention, and they agree. The criterion is a ratio of two energies, so it is invariant under a change of convention—rescaling the field carries the couplings with it, and matching the two kinetic terms fixes the rescaling with nothing left over—and the criterion below, evaluated on this reduction's own two-mode sector, returns exactly the ratio this reduction banks, with no parameter fitted between them. What separates the two numbers is the object and not the normalisation: two frame-constant modes against a whole level summed over its degeneracy. The level's quartic is not a proportional slice of the sector's, and the factor between their two ratios drifts across the tower—which a convention factor cannot do. Both return the same sign.^P10R67P10R50 The criterion is kept in that undivided form, because dividing it through by the quartic's coefficient reverses it wherever that coefficient is negative—the two forms then disagreeing in sign and not in size, so the divided form carries a positivity condition it does not state. And keeping it undivided is what makes the quartic's coefficient decide whether the coupling matters at all: were that coefficient negative the criterion would fail at every level for any coupling whatever, the squared coupling being a square and every level's frequency-squared at least eight. It is not negative—the passage from the action to the Hamiltonian, written out below, returns it positive at every level—so the criterion is that twice it, times the frequency-squared, exceed the squared coupling, which is satisfied at vanishing coupling and fails at large. The sign therefore waits on the coupling only where the coupling exists, and on half the tower it does not exist. The transverse-traceless multiplet at each level is a pair of conjugate representations of the isometry group, fixed two independent ways—its dimension is the degeneracy already in use, and its Casimir less the isotropy group's is the eigenvalue already in use—and the number of invariants in its triple product is exactly zero at every even level, two at the first two odd ones, and eight at every odd level above them. So the level's own cubic vertex does not exist at any even level, in any basis and under any truncation, the squared coupling vanishing there identically; and with the quartic's coefficient positive the criterion holds at every even level on that coefficient alone. The back-reaction's sign is settled positive on half the tower with no bound, no estimate and no two-point object. Two consequences the selection rule was not fitted to return with it: at the lowest odd level the chiral half carries exactly one cubic invariant, which is the count this construction banked independently for that multiplet, and no level carries an invariant vector at all—which is why the cubic's one-quantum amplitude cancels identically, read above as an arithmetic and here as a selection rule.

At the odd levels the coupling does exist, and it is not an integral at all. A level's harmonics are matrix elements of one representation, so a triple overlap of three of them is a product of two recoupling coefficients, and summing over the degeneracy each factor contributes exactly one: the degeneracy sum of squared same-level triple overlaps is an algebraic identity with no integral and no two-point object anywhere in its statement. A second route returns the same number through the object that looked like the obstruction—the level's kernel is a character, so the double integral is a class function of the relative element and collapses to a single class integral, exactly, the character squared integrating to one always and the character cubed integrating to the number of invariants. So the sum is the cube of the degeneracy, over the volume, times that number, and the vanishing at even levels is this one formula's vanishing case rather than a separate result.

Which locates the obstruction more precisely than a wall in front of the value. The same quantity has a representation carrying no two-point object, and on a group manifold the kernel route closes as well: what the coincidence limit stands in front of is a route and not the object. And a bound that sums the levels by completeness is loose by exactly one power of the label—such a bound being the fourth power of the degeneracy where the object is the third—so the coupling's own growth is of the seventh degree against the target's eighth in the level-summed convention, and the ratio falls like one over the label rather than tending to a constant^P10R60P10R59.

And the coefficients that ratio is built from are written down, the first of them not being recoupling data at all. The vertex's own weight factorises into a pointwise part and a recoupling part, and the pointwise part is the cubic term of the substrate's curvature integrand on transverse-traceless jets—the third-order companion of the second-order identity above. On such jets it is

obtained on twelve jets and confirmed on six drawn independently of them. The basis closes at rank six of nine, and the deficiency is exactly three identities that hold on transverse-traceless jets—one structure vanishing by tracelessness, one relating the derivative-free invariant to the second-derivative structures, and one coinciding pair—so Eq. (17) is a representative of a three-parameter family and is reported as one}, every member returning the same value on every jet. Two facts about its shape are derived rather than assumed: no parity-odd cubic structure can exist at all, three indices on the symbol and two on the derivatives against six on the tensors being eleven, which cannot pair; and at frame-constant h every derivative structure vanishes and the identity collapses to the single cubic invariant the constant multiplet carries.

The recoupling part is then exact, and it is a function of the level rather than a constant. The degeneracy-summed square of a same-level triple overlap is, in each channel, the product of the three right dimensions with one squared 9j symbol—reached by summing over every right label, projection and Clebsch–Gordan coefficient before any 9j is named. Summed over the admissible channels it runs 126/125 at m=3, 444/1715 at m=5, 1441/7875 at m=7, strictly decreasing to the limit 81/640; the variation is of order m^-2 about a finite non-zero value, so it moves the constant and leaves every exponent where it was. And the channel count arrives out of the same structure: the eight assignments of three harmonics to the level's two irreducible parts, six of them failing the triangle rule at m=3 and m=5 and all eight admissible and non-vanishing from m=7 upward—the count read above as a selection rule, recovered here as arithmetic on a different object.

Which derivatives the identity carries is the question the assembly turns on, and the jet variables themselves answer it. They are the covariant gradient and its second covariant derivative, not the frame derivative, and the difference is not a convention: assembled with the frame derivative the level's recoupling sum depends on which member of the three-parameter family is used—120575/6 against 19775/6 at the lowest odd level—while with the covariant derivative it is invariant in all three directions and at every level. And the reason the distinction is easy to miss is exhibited rather than asserted: in the transversality condition the two connection terms cancel identically, one against the trace of the structure constants and the other against the perturbation's own symmetry, so transversality cannot see which derivative it was written with and the cubic vertex can.

The frame this is computed in is derived from the substrate rather than adopted. Of the four combinations of invariant frame and generator sign, exactly one carries the structure constants the construction already banks; it is orthonormal for the unit-radius round metric in these coordinates and returns the banked volume 2π²—two independent ties to the same geometry. On it the frame derivative acts on a harmonic's right index by a constant spin-one generator, in each of the three directions, and the two sides are inequivalent everywhere, so the banked structure constants are what force the choice. Transversality in that frame is one explicit linear condition, and the transverse-traceless content is its kernel: dimension 4j+2 at left spin j, with right Casimir j'(j'+1) for j'=j±2—the two-part multiplet read above as a selection rule, arriving here as the kernel of a map. The amplitude is proportional to a single recoupling symbol on the right labels at every admissible triple of a channel, so each channel's degeneracy sum is one constant with no basis or phase convention entering, and the two mirror channels contribute exactly equally at every level—a structure the assembly did not put in.

The split into a derivative-free part and derivative parts is then not a property of the vertex. On transverse-traceless jets the family's own null direction expresses trh³ in the second-derivative structures, so which part carries what moves with the representative and only the total has a degree in the label. That total is of the seventh degree against the target's eighth, so the ratio falls like one over the label, the label times it settling between 11 and 13 from the second odd level upward where the lowest carries 24—which is this section's first statement about the tower's tail rather than about one level of it.

What the comparison then needs is its own absolute scale, and that is a derivation rather than a convention. Assembled through the per-mode Lagrangian the section already uses, the criterion reduces to a statement in which the gravitational coupling and the scale factor cancel exactly, and its left-hand side is the quartic target identically at every label—a closure no fitted factor could produce. Every ingredient is read off oscillator matrix elements rather than adopted: the second and fourth vacuum moments fixing the scale, the three-quantum amplitude, the polarisation factor checked on random symmetric tensors at three and four modes, and the multi-mode three-quantum norm. The volume enters through the harmonics' normalisation and cancels: the level sums are pointwise sums normalised to the degeneracy, which for harmonics orthonormal over the substrate is the degeneracy over the volume, so a quadratic term—one level sum integrated once—carries no volume, while the quartic, two level sums at a point integrated once, carries one inverse power of it, exactly as the square of a triple product of orthonormal harmonics does. So the criterion is a pure number: the vertex's level-summed square against the quartic target must stay below 18.

Against that threshold the residue is empty, and the sign is positive on the whole tower. The largest of the computed ratios is 175/22 at the lowest odd label, below the threshold by the factor 396/175, and every higher level falls further below it—by factors of about 6.6 at the next, 10.4, 14.0, 17.5 and 21.0 at the ones after. So the quartic term dominates the cubic at every odd level, and at every even level the coupling vanishes identically by the selection rule above: the sign is positive everywhere, by a margin that is narrowest at the lowest odd label and grows with the label. That is the object this section has been carrying since the mode sums were first named, and the answer is uniform.

What a uniform sign then does to the sum itself is the opposite of what its steadiness suggests, and the summand is what settles it. The quartic energy a level contributes is (5m⁶-41m⁴+88m²-16)/ (15(m²-1)) in units of the gravitational coupling times ℏ² over the cube of the scale factor and the substrate's volume—positive at every level and growing as the fourth power of the label—and because the criterion's ratio stays strictly between zero and one everywhere, at most 175/396, the net a level contributes is bounded below by 221/396 of that. So the partial sums over the tower are bounded below by a divergent series, and the mode sum has no finite value. The decay established above is the ratio's and not the summand's, and the ratio's falling to zero makes each level's net closer to the divergent quartic term rather than smaller. And a sign that does not alternate forecloses the one route to a finite value that needs no regulator: conditional convergence is available only to a series whose terms change sign, so uniformity settles the sign and closes that route in the same stroke. This is the first statement this section has about the summand rather than about a ratio of two of them, and what it establishes is that the definition the section owes must come from a regulator.

And the construction's own regulator reaches it, which is settled by exhibiting the continuation rather than by an estimate. The regulator is the one the free tower already uses—the spectral sum over the tower's own frequencies, evaluated by the same terminating expansion and calibrated against that tower's banked value before being applied to anything new. In the regulator's own variable the interacting summand is exact: three polynomial terms and one simple pole, whose pole sits at the label 1 and so outside a tower beginning at 3. So the fourth-power growth is not the obstruction the question anticipated. What is new against the free case is that the summand is a rational function, so its large-label expansion does not terminate— but that whole non-terminating tail is the single pole, which the regulator sums by telescoping rather than by continuing it, and which converges absolutely at the point in question. The continuation therefore exists, each of its four pieces separately finite, and it returns the exact rational - 43.

What it costs is one subtraction more than the free case, and the cost is counted rather than asserted. The subtractions a spectral regularisation spends are its summand's non-negative powers of the label: the free summand carries two, and this one carries three. The extra one is the constant term, which the free summand cannot have at all, being odd in the label to every order. So the interacting sum is defined, and the count of three against two is a fact about that summand: a count is not a charge on the ledger—what a ledger entry answers to is a pole, as the free tower's own does, whose counterterm the section has already identified with the curvature-squared invariants, and which of these subtractions answers to one the next paragraphs settle rather than the count. What is not claimed at this point is the renormalised value: the count is three against two, and the count is the claim. And the extra subtraction is not a separate invariant but a separate power of the label: the whole per-level energy is the gravitational coupling times ℏ² over the cube of the scale factor and the substrate's volume, times a function of the label alone, so all three subtractions sit at one power of the scale factor and it is the label's power that distinguishes them^{P10R68P10R67P10R64P10R62P10R81}.

Which dimension that subtraction sits at is then forced rather than chosen, and three routes fix it at the sixth. The interacting quartic energy is the Planck length over the scale factor, squared, times the free tower's own scale, so it enters at second order—and the section's own order rule, that dimension 2k contributes at order 2k-4, admits one dimension there and no other. The power of the scale factor gives the same answer independently: an energy of this kind sits at a^3-2n, so the free tower's a^-1 is n=2, which is why its counterterm is the curvature-squared one, and this sum's a^-3 is n=3—the rule calibrated on the free case's known dimension before being used on the new one. And the section had already placed it: the density above falls as the sixth inverse power of the scale factor, at the operator dimension the mode count and the dimensional bookkeeping put it, and what was missing was only the link from the summand's constant term to that entry.

It is not a volume term, and that settles which way the ledger reads. A volume term is the cosmological one, which sits six powers of the scale factor away on the other side, so the subtraction is a new entry and not one the ledger already carries. And on the class this sector works on it is not degenerate with the free tower's either, which is computed rather than inherited: degeneracy would require one functional to be a multiple of the other, and a^-1 and a^-3 are distinct monomials once the scale factor is itself quantised. The same test separates the two regimes rather than always returning independence—at fixed background every invariant is a multiple of every other and the two are degenerate, which is this section's own one-dimensional statement, and the asymmetry between the dimensions is the section's own as well: at dimension four the collapse rests on an identity pointwise in the scale factor and so descends to the quantised sector, where at dimension six there is no such identity to descend^P10R69.

The quartic's entry at that dimension is empty, and what settles it is a parity rather than a cancellation. What exposes the ledger is not a count of subtractions—those a cutoff trades for the analytic continuation rather than pays—but a pole, which is why the free tower's own counterterm is tied to a logarithm. The spectral function here has poles only at half-integer argument, the underlying zeta's single pole being reached only through an odd power of the frequency; the free weight carries exactly one such power and meets a pole, while the quartic summand regularised above is a rational function of the frequency squared—even in the frequency—and misses every one. At the physical point all four of its arguments are integers, where the spectral function is regular and each term's residue vanishes separately. So for that summand there is no pole, no logarithm, no scale dependence, and nothing for a counterterm to absorb: its coefficient is zero. And no quartic adds a coefficient there, its two-label part included: the free expectation of a quartic vertex is a sum of products of two coincident two-point functions, each a single sum over the tower, so the double sum over pairs of levels is a product of one-loop sums, which carries their logarithms' product and no logarithm of its own once each factor's divergence is subtracted. That holds for the curvature quartic, whose factors are the degeneracy over the frequency and its derivative-carrying companion, and for the kinetic one: in the exponential variable the kinetic trace carries -1/12 tr [H,H]², which vanishes on the commuting two-mode sector above but not on the tower^P10R80.

And the instrument that returns that zero is calibrated on the case whose answer is known, and shown to discriminate. The same residue formula applied to the free tower's own weight at its own physical point returns that tower's banked logarithmic coefficient exactly, rebuilt from the expansion's coefficients and nothing else; and a hypothetical odd power of the frequency does produce a pole, so the zero is a property of the summand's parity and not of the instrument. A zero coefficient is zero for every scalar at that dimension, so this contribution is well defined without any choice of representative—and that convenience is the quartic's alone, the cubic's non-zero coefficient below requiring one. And the count of three subtractions is a fact about the summand rather than a charge on the ledger, a cutoff's bookkeeping traded for the continuation rather than paid^P10R70.

That parity is not an accident of the assembly it was read off, and its shape is what fixes how far the conclusion reaches. A contribution built from N vertices of total degree P carries the frequency to the power -P/2-(N-1), and with this order's own bookkeeping, P=j+2N at order j, that power is 1-j/2-2N: the entire dependence on the number of vertices is -2N, which is even, so the parity is a function of the order alone and not of how the order is assembled. Four different assemblies of the fourth order—a single sextic, two quartics, a cubic with a quintic, and four cubics—return the same parity, each computed rather than argued, and the rule reproduces the case whose answer is already banked, j=0 giving an odd power and with it the free tower's logarithm. That power alternates with j/2—odd at zeroth order, even at second, odd again at fourth—and the order rule inverts to 2k=j+4 with exactly one dimension admitted per order, so each even order answers to one dimension and the assembly cannot be what decides it. What the counting therefore delivers is not the answer but its location: the total parity is the assembly's power together with the vertex coefficients' own, and since the first is fixed by the order, everything turns on the second, one order at a time. And the question was posable without the higher orders being constructed at all: the assembly's half follows from the vertex counting and the perturbative scaling, not from the value of any higher vertex^P10R71.

The two halves of second order carry opposite parities, and the one that decides the entry is the cubic's. The counting turns on whether a vertex coefficient is even in the frequency, and the quartic's is: the summand regularised above is a function of the squared label alone. The cubic's is odd, and settling it settles the entry for the part computed here—the same-level part, in which all three harmonics share one level. Its recoupling sum closes in two orbits, each internally degenerate and each exact—the extreme pair at 9(m²-1)(9m²-25)(9m²-1)³/65536 and the mixed sextet at (m²-25)(m²-9)(m²-1)²(9m²-1)/65536, both over m³—so the sum is an even polynomial of degree ten divided by m³, and is therefore odd in the label. One form carries every level with no case split: the two-channel levels are not a second regime, (m²-9)(m²-25) vanishing at exactly the two lowest of them.

So the same-level sum meets the pole the quartic misses, and at dimension six that part pays. The assembly's remaining factor is an integer power of the squared frequency, so the parity rides entirely on the recoupling sum; the summand's powers of the label are all odd, the continued sum therefore reaches the underlying zeta at odd integer arguments, and one of those is its single pole. The coefficient of the inverse label is -15405/8192—not zero—giving a logarithm and with it a required counterterm, of residue -15405/32768 over the cubic's own odd levels and -15405/16384 over the tower. And the instrument is discriminating because it recovers the other half's answer as a control rather than citing it: run on the quartic summand, whose powers of the label are even throughout, the same construction returns residue zero. What the two halves therefore exhibit is one selection rule read in its two parities, and second order carries a pole contributed by the cubic and not by the quartic^P10R72. The cubic also joins three different levels, and that sum is derived in closed form: by the same recoupling algebra at three independent levels the sum is a product of linear forms. With P=m₁+m₂+m₃ and Q_i=P-2m_i it is (P²-1)_i(Q_i²-1) [f(P)+_if(Q_i)]/ (2¹⁵_im_i(m_i²-1)), with f(s)=(s²-1)²(s²-9)(s²-25) and each channel one term of the bracket, its own triangle conditions carried by the zeros; it reduces on the diagonal to the two orbit forms above and equals the direct computation inside the triangle, on its boundary and at the lowest level. And that sum is not singly logarithmic. Where one label stays low and the other two grow, the sum over the high pair has a logarithm of its own, its coefficient (m²-4)(608m⁴-3866m²+1233)/2520 over the low label's frequency—that level's degeneracy as a factor, the rest an even polynomial in its label, the shape of a four-derivative correction to its quadratic action—and summed over the low label that logarithm diverges logarithmically again, so the bare sum's partial sums grow as 4979/2240 times the square of the cutoff's logarithm, where the same-level line's grow as -15405/16384 times the logarithm itself. So the logarithm of the three-level sum is not yet a coefficient: it moves with how a cutoff treats the high pair against the low label, and it becomes one once that nested divergence is subtracted by the one-loop counterterm inserted at the low level, whose cubic part is the coefficient just given; and the same-level part is not by itself the coefficient of an invariant (below)^P10R79.

And the parity the pole turns on is proved rather than read off those levels, by a symmetry of the channel list itself. The reflection that sends the label to its negative is the Casimir-preserving involution applied to both of the vertex's labels, and it exchanges them; the channel list is closed under the relabelling that induces, and so is each orbit, the swap carrying the count of one label to three minus itself so that the two orbits are exactly its blocks. The recoupling coefficient is invariant under it and the degeneracy factor is odd—each dimension being the label displaced by two, which the reflection negates—so each channel's term at the reflected label is minus the swapped channel's term, and a sum over an orbit closed under the swap is odd in the label. The result holds with the orbit coefficients left as free symbols: it needs the orbit's closure and not the equality of the channels within it, so the mirror coincidence noticed earlier—two channels contributing exactly equally, which the assembly did not put in—is this reflection, with the equality its stronger half. So the parity holds at every level rather than at the computed ones, and the pole does not rest on a fit^P10R74.

The closed forms themselves, which the same symmetry does not reach, are derived from the recoupling algebra as identities in the level. By the Wigner–Eckart theorem each channel's amplitude is one reduced constant times the 3j symbol on the right labels, so the constant can be read at a single right-label triple; at the extreme one, with the first field's right label at the top of its spin and the second's at the bottom, every left magnetic number the vertex reaches lies within six of the top of its spin, of the bottom, or of zero. Written in a magnetic basis rescaled to where each field sits, the angular-momentum operators, the coupled spin-two states and the left 3j symbol divided by the rescalings are each a ratio of factorials times a polynomial in the level, Racah's sum keeping at most seven terms. So the amplitude is a fixed ratio of factorials times a polynomial, and its degeneracy-summed square a rational function of the level: the two extreme channels return the first form and the six mixed ones the second, exactly, at every odd level from nineteen up, and the levels below, computed directly, agree. The orbit degeneracy is derived with them—the mirror equality holds at every level and not only where it was computed—and the amplitude polynomial is the same for each of the three representatives of the vertex's free family^P10R77. The pole's existence needs only that the remaining factor carry an integer power of the squared frequency; the residue's value rests further on the assembly. What is not claimed at this dimension is the renormalised value, the coefficient being what is computed here and not the constant a scheme assigns it.

That assembly is derived rather than carried, and deriving it moves its coefficient while leaving the form the parity needs. For a totally symmetric cubic on degenerate modes the exact second-order shift carries two invariants rather than one—the squared vertex and the square of its own trace, at weights two and nine—read off oscillator matrix elements, calibrated against the textbook single-mode result before the unknown was touched, verified on a generic two-mode vertex with four free coefficients, and separated by a traceless three-mode control that leaves only the first. And the trace channel is absent here at every level, by a count rather than an assumption: the pair contraction vanishes unless the free label is zero, and the same identity on the remaining indices makes a trace require a full singlet, which no channel at any level is. So the assembly is the squared vertex alone—and because both invariants carry the same power of the frequency, an integer power of the squared frequency, the parity above is untouched and the cubic summand still reaches the pole. The turning-point criterion is untouched as well, its own overall factor cancelling in the ratio of two energies that the criterion is^P10R73.

And a non-zero coefficient asks a question a zero one does not. While the coefficient vanished, the entry was well defined for every scalar at that dimension and no representative had to be chosen; a coefficient that does not vanish is the coefficient of something. On the one-parameter family the five dimension-six scalars collapse—two vanish identically and the rest are pure numbers—so the residue at an instant is one equation on three directions, each of which alone reproduces it after a single rescaling, and it names nothing^P10R73. The datum that separates them is on the class this sector works on: the scale factor is quantised, and on a scale factor that varies the scalars are independent, so the tower's divergence is computed there. Each level's second-order term is the double time integral of the cubic's three propagators, expanded locally in derivatives of the scale factor—the expansion checked against the propagator integrated numerically, its residual falling at each order by the next two powers of the frequency—and the logarithm at each order is the inverse-label term of the level sum: -15405/8192 at the instant, -8313/4096 at two derivatives, 14769/8192 at four, and nothing beyond. Off the classical history the test is covariance up to the scale factor's equation of motion, terms proportional to it being removable by a redefinition of the scale factor. The free tower passes it a dimension lower: the same instrument returns a minimally coupled scalar's covariant 1/72R² exactly, normalisation included, and the transverse-traceless tower—two scalar towers less eight modes per level, the static 15/4 being - 14 from the scalar towers and 4 from the eight—is covariant up to that equation, what the eight add off the history being such a term. The same-level cubic does not pass it: its dimension-six logarithm is the reduction of no covariant combination, the lower invariants times powers of the cosmological constant included, even up to the equation of motion, where a control built from covariant scalars and a multiple of the equation is recovered by the same test. So the same-level coefficient is not by itself the coefficient of an invariant, and the test is put to the complete second order^P10R78. The complete second order does not pass it either. Up to the equation of motion and total derivatives, covariance at this dimension is one linear condition: the covariant scalars and the lower invariants times powers of the cosmological constant span every combination of the jet monomials but one direction, so each functional carries one number, zero exactly when it is covariant. The order's double logarithm—the cubic over all three levels, read from the high pair's logarithm at each low level and checked against the three-level sum itself, with the curvature and kinetic quartics—carries 6023/20160 in the unit where the inverse cube of the scale factor carries one, its three parts - 13·197131/6720, 367/48 and 389/160. And the single logarithm cannot restore it: the subtraction's freedom moves it along the double logarithm's own directions, none of them covariant, so with one condition in all its share is whatever that freedom makes it. So the complete coefficient is not the coefficient of an invariant: the order is computed and its carrier is not covariant from inside, which places the difference in the reduction—the transverse-traceless sector the deparametrization leaves—rather than in the truncation of the expansion^P10R80.

And the two halves settle a question about the regulator itself, which is worth stating because it is a choice the section has been making silently. Continued in the squared frequency, the cubic's inverse cube of the label expands into half-integer powers, so it meets the spectral function's poles at infinitely many arguments, one per order of that expansion, and the residue there is an infinite sum; continued in the label it is a single term. Continued in the squared frequency the quartic's is likewise the single term, its powers being integers there. So the two continuations agree on whether there is a pole and differ in how much work it takes to read its residue, and which one is the right instrument is decided by the summand's parity rather than by the passage's habit^P10R72.

The cubic carries the non-resonance the identity above requires as an arithmetic rather than an assumption—its one-quantum vacuum amplitude cancels identically, the two channels being equal and opposite, so it reaches the vacuum only at three quanta. And the energy density it implies falls as the sixth inverse power of the scale factor, which is the operator dimension the mode count and the dimensional bookkeeping already put it at, arriving here from the vertices themselves^P10R50.

The construction's own second order is reached at this level, and reaching it is what turns the rank there from a bound into an equality. The reduction by itself has no second-order datum: its three eigenvalue logarithms satisfy exactly one relation, so the geometry is exhausted by the scale factor and the two shear parameters, and where the first-order system determines its one unknown uniquely the second-order one returns a family, the free state parameter still in the answer. What supplies the datum is the interacting ground state, which at this level is a finite object. The cubic's first-order correction to the field variance vanishes identically by parity—it reaches only odd states where the variance reaches even ones—while the quartic's is twice its coupling and linear in it, so the interacting variance departs from the free one at first order in the coupling. And it is resolved rather than inferred: the truncated two-mode Hamiltonian's lowest eigenvalue tracks that prediction at three couplings with the residual falling as the next order, and the state's own variance departs from the free value by more than twice its agreement with the prediction. The source's power is then computed rather than chosen—restored to physical units the correction carries a second-order energy density falling as the sixth inverse power of the scale factor, the same operator dimension the vertices already gave.

With that source in place the rank at second order is still two, and the reason is this section's own gradient result. Across every realizable invariant the source's second-order contribution is exactly twice the first-order pattern with the opposite sign—the same direction in invariant space—so it adds a function and not a direction. So the bound the iterated deformation gives is attained: the second-order rank at this level is an equality and nothing is missing from above^P10R51. The scope belongs in the same sentence: the equality is at the level whose vertices are exact and whose state is finite-dimensional, and the state is the perturbative one—a cubic potential being unbounded below, the diagonalisation means what it says exactly where it agrees with the expansion, which is what the three couplings establish.

The quartic's algebraic part is settled at every level, and by the level this section owns. On a symmetric three-by-three matrix the difference between the two algebraic quartic structures—the fourth trace and half the square of the second—factorises with the matrix's own trace as a factor, so it vanishes wherever that trace does. That is an identity in the entries and refers to nothing else: it holds at every point of every field, however the field depends on position. And the tower it applies to is the transverse-traceless one, whose perturbation is traceless at every point and whose tracelessness is linear, so the collapse holds on any superposition, within a level or across levels—which also fixes where the tower starts, the label's floor being where the transverse-traceless degeneracy first becomes positive rather than a convention. The identity does not run backwards: a vanishing trace is sufficient and not necessary, the remaining factor having its own roots, so tracelessness here comes from the construction and not from the identity. With the algebraic quartics spanned by one structure, the frame-constant family sweeps the invariants independently and determines that structure's coefficient outright, at a single rational for the whole tower, validated against the full reduction on two constant directions and exact at every order including the cubic that one of them cannot see.

What is not computed is therefore the quartic's derivative sector, which the frame-constant level cannot see because its harmonics have no gradients—and it now has a size rather than a name. Enumerated at a point over the admissible one-jet of a transverse-traceless field, with the pairs tracelessness and transversality forbid excluded, the full contractions of two derivatives with two fields span eight independent invariants. A level supplies one relation among their integrals, exhibited on the harmonic rather than assumed, and the only other candidate total derivative reduces to that same relation by the identity above—so at most seven survive as integrated invariants, eight being the exact pointwise count and seven an upper bound with one relation exhibited rather than the absence of others proved^P10R52. What a level can determine is far less than either number, and by the same argument that made the level-summed overlap exact. The isometry group acts transitively, so a sum over a complete basis of a level of any bilinear in the harmonics has the same coincidence limit at every point; and that constant tensor is invariant under the stabiliser of the point, which for the three-sphere leaves only the metric and the volume form to build it from. The three bilinears a level defines are accordingly tiny: the undifferentiated pair is one-dimensional and fully determined, being the transverse-traceless projector normalised by the degeneracy; the mixed one is one-dimensional and orientation-odd, forced to be so because the first is constant and the volume form is the only odd invariant available; and the doubly-differentiated one is two-dimensional with one direction fixed by the eigenvalue equation, so it carries a single free number.

Over the whole set of contractions the level-summed values therefore have rank three, and then parity takes one more. A level is the sum of two helicity representations of equal dimension, identically in the label, so the orientation-odd bilinear cancels over a full level and the rank falls to two—of which one is fixed by the level's own labels. So a level carries one unknown number and not seven. The scope belongs in the sentence: that is the rank of the level-summed values, so it bounds what a level-summed measurement can determine and leaves the covariant basis at its eight pointwise structures. And one such reduction is already in hand, the second anchor carried to fourth order returning the derivative sector as exact rationals at second and fourth order against a control that returns exactly zero at the floor at every order, where the gradients are absent.

That one unknown number is not free either, and what closes it is a second eigenvalue the harmonics carry. Computed from the connection in coordinates, the harmonics are eigenstates of the curl as well as of the Laplacian, with the two eigenvalues related so that the curl's is the label itself and its sign is the helicity—checked at a level whose frame components vary and at one whose components are constant, so it is not an artefact of either case. Two relations follow. The odd pairing of a curl with an undifferentiated harmonic cancels over a full level because the two helicities have equal dimension, which derives the parity step above rather than arguing it; and the pairing of two curls is helicity-blind and is an even contraction of the two-derivative sum with two volume forms, so it solves for exactly the number the eigenvalue trace had left free. The fully determined sum satisfies both relations identically, component by component.

So both level-summed coordinates are closed functions of the label alone, with no free constant in either—and both vanish at the label below the tower's floor and not at the floor itself, the degeneracy's own starting point reappearing in the level-summed values rather than being imposed on them. Which settles how far a level-summed route can reach, and the answer is not far: the span is two-dimensional at every level and both of its coordinates are functions of the label, so the coordinate pairs of any number of levels span the same two dimensions and a further level returns a point in that plane. At most two combinations of the covariant coefficients are determinable from levels, however many levels are spent—so the covariant expansion is not the cheaper route but the only one that reaches them, and it is now a route to level-independent numbers whose tower sums are already closed in the label. The scope is again in the sentence: what that argument closes is the label dependence, the coefficients themselves being level-independent numbers that no level-summed measurement separates more than two combinations of—and they are reached instead by the expansion below^P10R54.

The coefficients are closed too, as exact rationals, by a route that takes no measurement at all. In the left-invariant frame the structure constants are constant—computed from the frame fields rather than assumed—so the metric components are the identity, no index is raised, and the quartic coefficient of the curvature functional under an exponential deformation is a universal polynomial in the deformation and its first two derivatives with rational coefficients. So the coefficients are the solution of a linear system over the rationals on exact jets, with no configuration, no integral and no level entering it.

The basis closes rather than opening. Enumerated on transverse-traceless jets, the two-derivative sector admits 372 admissible contractions of pointwise rank 8, the second-derivative sector 672 of rank 4, and the purely algebraic sector 60 of rank 1; their union has rank 12, and the 945 available total derivatives span 5 directions, all of them inside that union. So the integrated space has dimension exactly seven—an equality where the count above gave only an upper bound—and the second-derivative sector is entirely removable by parts, the five divergences taking all four of its directions beyond the two-derivative sector and exactly one of that sector's eight, which is what the drop from eight to seven always was. Dropping transversality alone raises the two-derivative sector's rank from eight to eleven, so the reduction is the constraints and not the enumeration.

And two structural facts came out of the enumeration rather than being put into it. The algebraic quartic is not a separate unknown: the union's rank does not rise when the algebraic sector is adjoined, because the curvature commutator fixes the antisymmetric part of the deformation's second derivative to the deformation itself—so the natural split into one algebraic coefficient and a set of derivative ones is not a split this geometry has. And the integrated space is reached by the two-derivative sector alone.

The seven coefficients are solved on twelve jets and then verified pointwise on every jet not used to solve them, and they carry a single denominator with seven different numerators:

And this extends the cap above rather than escaping it, which is what says why the route had to be this one. The per-configuration route caps too, and lower: the whole frame-constant sector has measurement rank one over any number of configurations, and six transverse-traceless configurations spanning both available levels reach rank four—against seven unknowns. So no route that measures configurations reaches these coefficients at any expense, and what makes the expansion reach them is that it is an identity and never takes a measurement. The scope is in the sentence: what is closed is the coefficients of the quartic two-derivative sector in a named basis^P10R55.

Put through the level reduction, those seven contractions close both orders in the tower's own two variables and in nothing else. Every one of their level-summed values lies in the three invariant structures above, and the combination the seven rationals pick out uses both surviving coordinates with non-zero weight—so it needs the whole two-dimensional level object rather than collapsing onto one of its coordinates, and the weight standing on the orientation-odd structure is non-zero too, so the parity argument that set that structure to zero is load-bearing here and not decorative: without it the level object would carry a third number and the rank above would not be the operative count for this combination. Reading the second order required its own pointwise identity, which the same machinery supplies—the second-order coefficient of the curvature is a fixed combination of the deformation's squared gradient and its square, with the mixed contraction's coefficient exactly zero—solved on jets, verified on six held out, and checked against the frame-constant value and the harmonic's second-order anchor, neither of which it was fitted to. With that, both level sums are closed in the degeneracy and the frequency alone:

And the tower sum factorises, which is the structural fact that makes the assembly a product rather than a double sum. Taking the two factors from different levels, every monomial is of degree one in each level's numbers, because every contraction pairs a harmonic with itself—so the sum over the tower is a product of single sums and not an irreducible double one, and setting the second level equal to the first returns the level object above exactly, the diagonal rather than a different object. Summed unweighted to a cutoff the two orders differ by two powers of it, which is the same gap the free tower's quartic divergence and the order-λ² sum already showed. And the physical weighting is carried, with the gap surviving it untouched. The free vacuum puts one inverse frequency on each propagator, and that factor has no closed-form sum, so the leading terms are bracketed rather than fitted—between the label less one and the label, compared on their squares so that no root is ever compared, both bounds being exactly summable. Both return the same leading power and the same leading coefficient in every sum involved: the weighted second order at the fourth power of the cutoff and the weighted fourth order at the sixth, so the gap is two powers weighted where it was two unweighted, and the weighting moves both ends alike. And a control arrives that was not designed: the weighted second order lands on the same power as the free tower's quartic divergence, read independently and earlier, so the weighting is the tower's own and not one chosen here—with both weighted leading coefficients negative, as every level's value is.

What is left is one passage, and naming which one is the result. Carrying a level sum to the vertex coefficient the criterion above uses takes four steps: the level sum to a per-mode coefficient; the pairing combinatorics, which the level sum has already performed and which must not be applied twice; the passage from the action to the Hamiltonian, which carries a sign; and the mode normalisation with its powers of the scale factor. The third is where a sign can turn, and it is written out here. With the lapse unity, the deformation traceless so that the determinant and hence the measure carry no dependence on it, and the kinetic scalar's second-order part computed rather than assumed—it is a quarter of the squared velocity's trace, and positive—the Legendre transform returns a potential that is the curvature integral with a single power of the scale factor and an overall minus. The sign is then read off a frequency the construction already holds rather than chosen: the resulting equation of motion carries the expansion's own friction and a frequency whose square is the label's, matching the tower's banked value, where the opposite choice would make every level of the tower unstable. So the vertex coefficient is positive at every level, which is the sign this construction's second-order computation already carried, and it is consistent with the negative level-summed coefficient closed above, the passage from the action to the Hamiltonian carrying the sign between them. And the pairing count is spent rather than repeated: the level object is built from three terms, and those are the three pairings of four factors, so applying the combinatorial factor again would treble the answer^{P10R56P10R57P10R58}.

And the same argument says which object the sign is a property of, by showing that the obvious candidate is not one. The level-summed cube of the shear is not invariant under an orthogonal recombination of a level's own basis—exhibited by rotating an orthonormal pair inside its span and watching the sum move—so it is not a property of the level at all, which is sharper than reading the vanishing third order as a property of one direction. The squared coupling that the sign criterion actually uses is built from pairwise contractions and is invariant, so the level-summed sign is a well-posed question and the object that answers it is well posed. Its value is the odd-level coupling computed above: that coupling is a sum over squared triple overlaps, pairwise in each index but with each pairing joining two different integration points, because a vertex is an integral of a product of three harmonics, so it reads as an integral over two points of the level's bitensors where the count used their coincidence limit—and the recoupling identity removes the integral altogether, the degeneracy sum of squared same-level triple overlaps being algebraic^P10R53P10R54.

And the form holds at the order this construction operates at while the rank at the order beyond it is now an equality rather than a bound—so what carries an order attached is everything past the second, whose parameter is the cube of a number of order 10^-122, and the binding scope is accordingly the level rather than the order. The distinction is kept because a definition that holds at one order of an expansion and is silent at the next is a definition with an order attached, however small the next order is. The inference to guard against has just that shape—a result true where it was computed, read as true where it was wanted—and a quantity being negligible is not the same as a scheme being specified.

The shape of that is worth stating plainly, because it is not an accident of this tower. On a fixed background the one constant is unobservable because of a degeneracy—and the degeneracy is switched off by exactly the constant it was hiding. Were there no logarithm, R would stay constant and the counterterm would stay invisible; but then there would be no constant to hide. The saving mechanism and the thing it saves us from are the same number, so the sums cannot be carried into the coupled sector free of charge by that route: the counterterm basis is one-dimensional at fixed background and two-dimensional once the scale factor is quantized. And it does not rest on the substrate scale being indefinite, which is the reading the statement invites and which fails twice over. A superselection rule forbids coherence between sectors and not an ensemble over them, and the three integrals are read off the ensemble linearly, so an ensemble over distinct curvature radii spans three dimensions inside a superselected theory and with no anomaly at all; and the question has no object as put, since α is a coefficient of the physical Hamiltonian rather than an observable of it, so the corresponding operator is a multiple of the identity with a one-point spectrum, and a central operator with one point decomposes nothing. Neither route is the one taken here: the rank is three already at a single curvature radius, which is where the a^-4 anomaly puts it, and the operator the Ricci scalar depends on commutes with nothing in the coupled algebra. Centrality is a relation to an algebra, and the theory's one genuine superselection label is the boundary operator—central in the radial algebra, labelling the boundary condition and not the background scale^P10R25. The back-reaction here is semiclassical and the interacting theory is not built; what higher orders could do is narrow, since restoring the degeneracy would need the total trace exactly constant—a pure cosmological term—and an a^-4 anomaly is not one. And the mode in which the constant is observable is fixed, because no state resolves the curvature it is read off. At this order the excitation trace vanishes on the diagonal of the adiabatic basis, which is the harmonic virial theorem and not an operator identity: with ω_n=μ_n/a the excitation energy is S/a with S free of a at fixed occupation, so the pressure is a third of the density in every stationary state and in every mixture of them. Written in the field operators themselves the trace carries two terms and not none—π_n²/a⁶ and μ_n²φ_n²/a², the second a potential-like term from the tower's own gradient energy, with μ_n²≥8 for every mode so that it is neither zero nor small—and their combination is a^-4(a_n²+a_n²): purely off-diagonal, non-zero on any squeezed state, and untouched by normal ordering, which has no contraction to take. The conclusion the diagonal statement was carrying survives and no longer rests on it. That combination is an inverted oscillator, and an inverted oscillator has no normalizable eigenvector at any real eigenvalue: in the momentum representation the null equation is solved by √p J_±1/4(p²/2μa²), both solutions regular at the origin and neither square-integrable at infinity, the modulus squared falling as 1/p with the envelope exponent holding at -1/2 exactly at every real eigenvalue. So the curvature has no point spectrum, and it has none without any term being absent—which is the stronger warrant, since it does not turn on an absence^P10R32. The spectrum is purely continuous and two-sided: the same operator puts spectrum below 4Λ as well as above it.

And the obstruction is the anomaly coefficient itself, which is this result's shape for the third time and from a third direction. The commutator of the curvature with the momentum conjugate to the scale factor gives Var(R) Var(p_a)≥ ℏ²4⟨R'⟩², saturated to a third of a per cent by narrow admissible states, and the right-hand side vanishes exactly when the residue does—which is the case in which the curvature is constant and there is nothing to resolve. So the same number that makes the counterterm observable is what forbids any state from sharpening the curvature the counterterm is read off, and the constant is observable distributionally rather than as an ordinary sharp-valued observable. That is unconditional rather than weakened: there are no special states to except, the ordinary reading being excluded by a theorem and not by a missing calculation. One consistency there was not put in by hand: the expectation of a^-4 converges at the origin only above a threshold in the boundary index, and the horizon's own thermal condition clears it on either ordering—the boundary condition that closes the deficiency is the same one that makes the curvature's expectation exist^P10R26. What the coupling is genuinely needed for is smaller than that, and it is smaller than the order of the promotion suggests. At higher order the cubic and above put genuine tower operators into the trace, and neither the bound nor the level-set argument turns on their absence. The commutator is an algebraic identity and not a dynamical one: [f(a) T,p_a]= iℏf'(a) T for any tower operator T, commuting or not and ordered any way, since p_a differentiates the scale-factor function alone and the tower factor rides through—so the inequality holds verbatim in form at every order, and its form is blind to the ordering. And pointwise cancellation is excluded by the trace rather than found to be absent: the trace of an energy term h(a) T is (h+ah')T/2π²a³ exactly, so a term c a^-n ln^ma contributes 2π²Θ=c a^-(n+3) [(1-n) ln^ma+m ln^m-1a], and the counterterm's own a^-4 is the member of that family at n=1, m=1. The whole family is closed under the operation rather than annihilated by it, and closure in one direction only is what settles the question. From a term at n=1 the highest power surviving in ∂_aΘ is ln^m-1a with coefficient exactly -4mc, while a term one logarithm higher reaches both ln^ma and ln^m-1a: so the terms chain downward in the logarithm and never upward, the coefficient system over log powers is lower triangular with diagonal -4m, and ∂_aΘ≡0 therefore forces every coefficient to vanish. The leading logarithm has no partner, and that is why nothing cancels the residue at any (n,m)—the pure-power case, where n=1 kills the coefficient outright, is this statement at m=0. What a tuned coefficient would do is relocate the obstruction rather than remove it—at exactly twice the residue a second-logarithm term annihilates the constant coefficient the counterterm sits in and leaves the next logarithm up at sixteen times the residue, with the curvature's variance at that same state eleven decades above the measured floor^P10R28—but that relocation is a fact about the trace formula's closure and not about this theory, because the term it needs does not exist here. The relocation must sit at the power of a the counterterm sits at, since that is what lets it reach the constant coefficient at all. And in the reduced theory the only scales are a and the Planck length, so the vacuum energy is a^-1_j f_j(ℓ_P/a)^j against a polynomial in ln (a/ℓ_P): the power is 1+j, one vertex pair costs ℓ_P², and the interaction therefore enters at j≥2. So the counterterm's own power is populated at j=0 alone—by the free tower, whose large-label weight 2(m²-4)√m²-3 is a pure Laurent series and carries no logarithm of the label, so the pole there is simple and the logarithm is capped at the first power. The second logarithm is therefore unavailable at every order of the coupling and not merely at the lowest, and asking the two families to cancel each other instead has no solution, one coefficient being unable to annihilate two different powers of a at once. The cubic's own sum carries no logarithm either: the three-harmonic overlap on the spatial section obeys an exact multiplet sum rule, weight n₁n₂n₃ on the triples satisfying the triangle inequality with odd perimeter, and every term of the soft expansion has the form j(j²-3)^p—an odd polynomial terminating above the first power when p is an integer, an even-power series otherwise—so the inverse first power, which is what a harmonic number would occupy, has coefficient exactly zero at every order^P10R29. The level-set argument likewise survives on every fibre that exists: wherever the cubic's tower operators are simultaneously diagonal, the curvature is multiplication by 4Λ+_j c_j a^-n_j, whose derivative has no zero while r gt;0, whose level sets clear to a polynomial of degree six and are therefore finite, and a direct sum over such fibres manufactures no point spectrum from parts that have none^P10R27. A tuned coefficient can make the right-hand side vanish, and that is not sharpness: the bound goes vacuous while the curvature's variance stays five decades above the measured floor, and two distinct powers of a agree at isolated points rather than on a set of positive measure. And the correlated case is a question about one operator family rather than about states. Because the trace equation is algebraic in the matter trace—R=4Λ+κΘ carries no derivative of the scale factor—the curvature commutes with it exactly, so the curvature is a direct integral of 4Λ+_kκ_k a^-m_kT_k, one operator on the excitation factor at each scale. A constant eigenvalue branch then forces its own value: every power is negative, so the family tends to 4Λ and a branch constant on a set of positive measure can only be 4Λ itself—leaving the condition that _kκ_k a^-m_kT_k be singular for almost every scale. At finite truncation that is exactly the singularity of the excitation operators' own pencil, a condition carrying neither the scale nor the couplings, because with distinct powers no two monomials of the determinant can meet on the curve the scale traces; and the pencil is non-singular at every truncation computed, in exact arithmetic, for the two non-commuting operators the cubic supplies and for three at the powers where they collide^P10R30. The one place the truncation speaks for itself runs toward the affirmative rather than away from it: at odd truncation a ladder matrix is singular and the coupling system does acquire solutions, but every one of them switches off at least two couplings, and the operators the tower actually carries have purely continuous spectrum and so no null vector at all. And in the tower the criterion is not a limit of determinants at all but an ordinary differential equation—one equation in the momentum representation, whose order is the number of powers of the field the trace carries. Which powers it carries is not a free choice, because the parametrisation of the spatial metric is not one: a cubic term with two powers of momentum is present or absent according to how the shear is written, and only one way of writing it separates the shear from the volume. In the volume-preserving parametrisation the trace of the spatial metric's logarithmic derivative is the expansion alone, so the extrinsic-curvature invariant carries no shear–volume cross term; the volume element is the scale factor's own cube, so the cosmological term carries no shear at all; and both statements are exact rather than leading order. There the momentum-weighted cubic vanishes identically—the cubic term of the shear's kinetic trace is the trace of a commutator—and it vanishes at every level rather than for a single mode, the first kinetic correction being a quartic. The parametrisation linear in the perturbation gives the same term a non-zero value, and that is the whole of the difference between them^P10R36.

So the cubic the construction carries is the spatial curvature's own, and it reaches the trace by an identity rather than a computation. For a constant frame metric the scalar curvature is a closed form in the elementary symmetric functions of the perturbation, and about the round sphere it is 6-2 tr h²-10/3 tr h³ to that order, with the volume element contributing nothing and the first-order term absent because the first curvature perturbation of transverse traceless data vanishes on a maximally symmetric section. And the cube's trace is three times the determinant for any traceless symmetric matrix, so the cubic term is a determinant rather than an integrand. The closed form answers to three checks that share no machinery: a coordinate Ricci scalar at two metrics, its own covariance under rotations of the frame, and the standard anisotropy potential of the closed homogeneous cosmology, which it reproduces exactly through cubic order with both cubic terms included.

At flat order that term does not reach the single-mode problem: for one wave vector transversality is algebraic and makes that vector a null eigenvector of the polarisation, so the determinant—and with it the whole term—vanishes identically across the flat polarisation space, lifting only for two wave vectors together, and what would put a third power of one excitation into the equation sits exactly on that degenerate set. On the closed section it does reach it, and the overlap is a determinant rather than an integral. Transversality there is differential rather than algebraic, and the lowest transverse traceless harmonics—identified by their own eigenvalue and degeneracy as the tower's lowest level, and transverse across the whole five-parameter multiplet at once—have constant components in the left-invariant orthonormal frame. So the mixed components are a similarity transform of those constants, the determinant is theirs pointwise, and the overlap is that determinant times the volume of the section, with no integration performed at all—non-zero, and non-zero for the ordinary reason that a determinant is^P10R34. That the determinant is basis-dependent does not reach any ratio of two such overlaps, and the reason is representation-theoretic rather than computational: the symmetric cube of the five-dimensional multiplet contains the trivial representation exactly once, so every cubic overlap at that level—whether it couples one excitation or three, and whether it comes from the kinetic or the potential side—is a multiple of one symmetric trilinear form, and in the ratio of two of them the form cancels and a pure number remains.

So the operator content is three structures and the null equation is third order, with no first-order term in it. At that order the operator is regular: the third power of the field is a third derivative with a constant coefficient, so there is no singular point anywhere on the line and the whole question is at the two ends; and the cubic needs no symmetrisation of its own, being a power of one symmetric operator and leaving an exact total derivative as its Lagrange difference. The counting at the two ends is a genuinely different problem at odd order—there the two deficiency indices need not agree, and a first-order operator on a half-line is the standing example of one that admits no self-adjoint extension at all—but the mechanism that separates them needs the spectral parameter to change the asymptotic count between the two half-planes, and here it enters neither the characteristic exponent nor the prefactor. At the dominant balance the three branches carry real parts ±√3/2|β|^1/3 and zero, and on the marginal branch the modulus is exactly p^-2/3, which is below the square-integrability threshold by exactly one sixth and is settled by an exact solution rather than an estimate of order. Two of the three are square-integrable at each end, the far end being the same equation under one change of signs, so the indices are equal and a self-adjoint realisation exists—a circle of them^P10R35. And three coefficients, less one rescaling and one overall factor, leave a single dimensionless invariant, a fixed fractional power of the scale factor and strictly monotone in it.

What a count like that is not is a property of the trace operator, and this is the place to say so. A redefinition of the field by its own square moves the two cubic couplings against each other while generating quartics, so a cubic truncation is not closed under it and the ratio of the two couplings takes every value, both signs included. What the truncation does fix is one combination of them, and that combination is non-zero: the cubic is really present, it is the curvature's, and the equation really is of third order, while the ratio is a feature of where the expansion was cut and not a datum about the operator. So the criterion at third order is a question about one real variable. The rescaling identity is exact, so the whole family carries a single dimensionless invariant—a fixed fractional power of the scale factor times the squared frequency—and whether zero is an eigenvalue survives both the rescaling and the positive prefactor, while that power is a diffeomorphism of the half line, so a set of positive measure in the scale factor is one in the invariant.

What it is not is a question about the operator family, and that is what makes it answerable. No self-adjoint realisation of this operator has a domain independent of the invariant: the witness is the marginal branch itself, which the operator annihilates to square-integrable order while the second derivative carries it out of the space, and a self-adjoint domain must contain the deficiency subspaces. On a boundary triple the analytic object is a finite Weyl function rather than the family. Zero is an eigenvalue of a realisation precisely where the determinant of that realisation's parameter minus the Weyl function vanishes—a one-by-one object where the deficiency is one and three-by-three where it is three, hence one scalar equation in one real variable in either case—and the realisation is held fixed as a finite-dimensional datum while the domain is free to move^P10R38. And both halves of that hypothesis hold, for a structural reason rather than a technical one. Transform the family and the third-order term becomes a potential while the multiplication operator becomes the kinetic one: the object is unitarily a second-order Schrödinger operator with a cubic potential, for which the Weyl function is a classical object—and no rescaling absorbs the transformation, because the scaling that normalises the cubic term is forced to the identity. Real-analyticity then follows from the equation's own coefficients: they are of degree one in the invariant and the boundary data do not depend on it, so the fundamental system is entire, and reality of the equation carries the analyticity to the real axis. And that the function is not constant is proved rather than measured: an exact two-solution identity gives its derivative in the invariant as an integral of a square over the half line divided by a square, hence strictly positive at every value—so there is no interval of constancy and no sub-family on which it is constant, and the half that could have failed for a boundary reason fails for no reason at all^P10R39.

The same transformation explains why this route reaches where the other two could not, and the explanation is not that it is cleverer. The obstruction that stopped them is representation-independent—the weighted integral over the whole line diverges in the transformed picture exactly as it did in the original—but the half line is the opposite case, and that is the entire point: there the recessive solution decays faster than any exponential and is dominated exactly, so the same weight is integrable. The half-line solution object exists precisely where the whole-line operator object does not, and no operator family, form domain or parameter-independent domain appears anywhere in the statement. The dependence on the choice of boundary triple splits, and it splits the useful way: non-constancy is independent of that choice unconditionally, a change of triple being a constant Möbius map of non-vanishing determinant, while analyticity on a given set is not, the same map having a pole a choice can move onto the real axis—so the statement is proved for the triple constructed and transferred wherever the denominator does not vanish.

So the criterion closes, and it closes more sharply than the second-order case did: monotonicity gives at most one value of the invariant at which zero is an eigenvalue—a single scale factor rather than a discrete set, and outside the measured window none at all—so the condition cannot hold on a set of positive measure and the curvature has no eigenvector at this order either. Two things bound that, and both are the construction's own. The deficiency count, re-derived here from the endpoint thresholds, comes out at the lower of the two values a scalar second-order expression on a line admits, so the larger case is not reached rather than being disposed of. And the whole statement is the criterion at the cubic truncation, whose count is a property of that truncation and not of the trace operator—which is what the section says two paragraphs above, and is the reason this closure is a closure of the third-order question and not of the row.

And there is a second place the degeneracy ends, at fixed background, which is the shear. The deficit is C²; for an anisotropic shear over an isotropic expansion the invariant is C²=2_i(σ_i+Hσ_i)² at second order in the amplitude^P10R4—so it carries two derivatives more than the shear itself, and for a mode of frequency ω it grows as ω²σ², which is what makes it ultraviolet-sensitive at all; and the propagating content of this tower is the transverse-traceless shear. So the tower's own shear is a calculation and not a question about meaning—entering at second order in the mode amplitude, with the sub-leading heat-kernel coefficients as its instrument. That calculation is run, and its answer is smaller than the question sounds: for a transverse-traceless perturbation the first variation of the Ricci scalar vanishes identically and √g is unchanged, so R² at second order in the amplitude is 2 R times R at that order—pointwise, with no integration by parts—and ∫√g R² is accordingly a multiple of the Einstein–Hilbert term rather than a new structure.

With the Gauss–Bonnet combination contributing no field equation, the three quadratic invariants leave ∫√g R² and ∫√g C², of which only the second is new: the shear therefore costs exactly one new counterterm, and it is the Weyl-squared one^R.

And its coefficient follows from what the tower is, without a second-order spectrum. Each mode is a harmonic oscillator of mass a³ and frequency μ_n/a, which is term for term the reduction of a minimally coupled massless scalar on this background, so the tower is two such degrees of freedom carrying the transverse-traceless spectrum. The C² coefficient is a property of the field content rather than of the background, which is why it is fixed without one: for a single real scalar the entire C² content of the second heat-kernel coefficient [Gilkey1975] sits in 2 Riem²-2 Ric²—independently of the curvature coupling, which is why the answer needs none—and that combination is 3C²-E₄, giving 1/120 in units of (4π)^-2 and therefore 1/60 for the tower^P10R8: exactly twice a real scalar's. The count is of physical modes, and that is the reading on which the number stands: the constraints here are solved rather than gauge-fixed, so no ghost sector enters and no covariant spin-2 tabulation applies. The overall normalisation by which the coefficient enters a divergence is convention-dependent; the ratio is not.

Two things about that entry are settled and belong beside it. First, the two helicities are not identified by any symmetry the layer connectedly carries: on S³= SU(2) the connected isometry group is (SU(2)_L×SU(2)_R)/ Z₂, transverse-traceless rank two carries |j_L-j_R|=2, and each level is (j,j+2)⊕(j+2,j) with 2j+1=n-1—whose dimensions sum to exactly the 2(n-1)(n+3) derived above^P10R5. They are inequivalent irreducibles, and the map exchanging them is the exchange of the two factors, an orientation-reversing isometry—so a parity-odd term is not excluded by the geometry, and whatever excludes it does so through the state. Second, a topological term contributes no field equation, so a coefficient multiplying one is not read off any dynamics; its only remaining home is the horizon entropy, on which this construction is deliberately uncommitted [JanzenGeometricCore]. So whether such a term is an entry in this ledger is not a separate question from that one.

One caveat belongs in the statement rather than after it: the dimension-four scalars also include the parity-odd Pontryagin density, which is not a curvature-squared invariant of the kind counted above and is a total derivative, but which can be non-zero at second order in the metric perturbation and reverses with the handedness—so the zero is a property of the mode chosen and not of the geometry [JanzenDynamics]. Which modes give the zero is a question about the configuration and not a general rule: on a plane-wave pair the density goes with the relative phase of the two channels, vanishing where they are a quarter-cycle apart and largest where they are in step, so a prescription reaching for one handedness may be reaching for the case that vanishes.

And what that scopes, first of all, is this computation. A counterterm answers to a parity-odd divergence of the effective action, which is a property of the theory and the state; it is not a property of a basis mode. The two helicities span the same space in either polarisation basis, and on a parity-symmetric state the two contributions cancel whatever an individual basis element returns. So a mode-level evaluation cannot fix what counterterm the theory needs, in either direction — the caveat bounds its own instrument rather than deferring the question to a choice of polarisation.

Two things would have to be settled before a parity-odd term could be scored at all, and they are independent. Whether the tower is chiral on the corpus's own criterion — which is not that a reflection exists but that no connected action completes it, a condition applied to the radiating sector and not to this one. And whether a total derivative is a ledger entry. The density above contributes no field equation, so it is not a coupling in the sense the constant-counting audit means, and the ledger counts no counterterm coefficient [JanzenGeometricCore].

The propagating sector itself is a clean, unitary, deparametrized evolution—a true Hamiltonian generating the graviton on the layer, in cosmic time.

What the coupled sector establishes is one claim, and it is worth stating as one. Once the scale factor is quantized, the graviton tower's renormalized zero point makes a single ultraviolet constant physically observable, and the construction has no route left by which to hide it again. The constant is carried by a residue of the spectral zeta function that is an exact functional of the free spectrum: no multiplicative renormalisation of the frequencies removes it, and the one mass-like displacement that does is precisely the curvature offset which makes those frequencies non-integer in the first place. The logarithm that residue produces has no partner to cancel against, the coefficient system over logarithmic powers being lower triangular with non-vanishing diagonal, so the family is closed downward and never upward. The interaction cannot reach the power of the scale factor at which the counterterm sits, because in the reduced theory that power is populated at zeroth order alone, by a free tower whose large-label weight is a pure Laurent series, and the cubic's own overlap sum carries no logarithm of the label. And the curvature operator that results has no eigenvector—not because it is a function of the scale factor alone, which it is not, the trace carrying the tower's momentum and its gradient energy as genuine operators, but because the operator it carries is an inverted oscillator, whose point spectrum is empty at every real eigenvalue and stays empty under normal ordering.

And the trace carries a third power of the field as well, from the spatial curvature rather than the kinetic term, so that step is made twice and by different arguments. At quadratic content the operator is an inverted oscillator with empty point spectrum. At third order it is regular, self-adjoint realisations exist, and the couplings run along a single monotone invariant—and there the eigenvector question is settled by the one object whose analyticity does not require a domain to be carried along that invariant, a finite Weyl function, which is real-analytic and strictly monotone. The second closure is the sharper of the two: strict monotonicity allows at most one value of the invariant rather than a discrete set, so the condition fails on a set of positive measure by a wider margin than the first argument gave. Both are scoped by the same thing, and it is stated rather than implied: the third-order statement is the criterion at the cubic truncation, whose deficiency count is a property of that truncation and not of the trace operator. So what remains open is one thing, and it is not a category—and it is the completion of a definition rather than a verdict this construction could fail, the question it reduces to having been settled above in both of its arms. The interacting tower's divergence structure does reach second order in the coupling, and it does so by an identity rather than by a search that failed to find a reason for it to vanish. What follows from that is the definition's incompleteness and not the ledger's refutation: the affirmative branch is a second dimensionless number owed and not a second length introduced, so the single-scale ledger stands either way—an undetermined dimensionless number not being a scale—and that number is already counted, the ledger of dimensionless constants spent standing at one for these very sums, while what the branch adds is bounded by the background's own turnover and the counterterm it would need collapses on an exactly maximally symmetric substrate. So the one open thing is the renormalised value at the paying dimension, which a measurement fixes: the complete second order there is computed—the cubic over all three levels, its nested divergence and both quartics—and it is not covariant up to the scale factor's equation of motion, so the counterterm it needs has a component outside the covariant invariants and the invariant it would belong to is a stated limit of the reduction rather than a part still owed. So the two extension questions this sector meets pair, and the pairing is the clearest thing it has produced about its own shape: the one at the origin of the scale factor is closed, without a free parameter, by the de Sitter horizon's own thermal state; the one at unbounded momentum is open, and it is not a new object standing beside the construction but the object the construction was opened to address.

Dissolution, not solution

The result of Sections 4–5 is best stated negatively. We have added nothing to the formalism. We have not introduced a new time variable into general relativity, nor a matter field whose role is to be a clock, nor a modification of the constraint algebra. We have identified, on external grounds, which foliation is physically real, and read the existing constraint on it. The problem of time, on this account, was never a problem internal to the formalism; it was the formalism faithfully reporting the consequences of an ontological premise—the block—that is both a category error and empirically false.

This is why the appropriate verdict is dissolution rather than solution, and the distinction is not merely rhetorical. A solution would add structure to make a defective formalism work. A dissolution removes a mistaken premise and finds the formalism was working all along. The four epistemic rules that govern such corrections [JanzenFortress, JanzenShadowExistence]—resist the naïve identification of appearance with reality; prefer a reading on which the observed structure is a structural necessity rather than a tuned permission; consolidate phenomena under one principle; treat accumulating patches as a red flag—each favour the CR reading over the frozen one. The cosmic time is a structural necessity of the ontology and an empirical datum, not a tuned addition. The reading consolidates: the correction that turns the frozen constraint into a true Hamiltonian refuses the very reification the completed-horizon inference also rests on, both inferences granting existence to a manifold of occurrences that no observer's present ever contains [JanzenBHcausality, JanzenFortress]. The two share a diagnosis and not a premise. The completed horizon is withheld by general relativity's own causal structure, for every admissible exterior slicing and with no ontology among the hypotheses, so what this reading supplies there is the account of why the error was available rather than the result that removes it. And the shared diagnosis is the cited paper's own rather than a reading imposed on it: the causality paper names the existence–occurrence distinction as the one it turns on—that the horizon occurs without ever existing on a finite exterior slice—and records that it reaches that distinction from standard general relativity alone [JanzenBHcausality]. The one move that would have called for an ontological refusal is not an exterior-time claim at all: a defender of the completed horizon may grant that no finite exterior time contains it and hold instead that it exists on the maximally extended geometry. At that one site the refusal would be weaker than what is already available—the extension's stacked horizon points display the causal ordering of events the metric assigns no separation between, so there is nothing there for an ontology to withhold existence from^P10R75. One ontological correction, one canonical payoff, and one standing error of the literature it explains.

The reading also closes a loop with the dynamical content of the programme. The true Hamiltonian generates the advance of the layer 𝒮_t; the geometry of that advance is what the operator and range papers read directly, where the matter content of a layer is the bend of its spatial cut off the vacuum profile and gravitational radiation is the transverse-traceless shear of the evolving layer, up to the wall at which a symmetric layer can no longer carry the wave and ordinary inhomogeneous evolution begins [JanzenOperator, JanzenRange]. The canonical face (a Hamiltonian that advances the layer) and the geometric face (matter and radiation as features of the advancing layer's intrinsic geometry) are two descriptions of the one evolving three-dimensional world. The cosmogenesis re-expansion is one such advance: the companion cosmogenesis paper [JanzenCosmogenesis] runs the collapse-and-re-expansion thermal history—the layer's advance along cosmic time, generated by this true Hamiltonian rather than frozen—through a nuclear network to the primordial light-element abundances. The companion foundation roots these faces in a single augmentation of general relativity [JanzenModernParallax]; the present paper supplies the canonical one, and with it the recognition that CR's distinctive contribution to quantum gravity is not a new dynamics but an absolute cosmic time—real, forced, and already implicit in the formalism—that turns the constraint into a generator. This dissolution is the canonical face of the substrate's maximal symmetry gathered in the geometric core [JanzenGeometricCore]: the problem of time is identified in the constraint-algebroid construction as the base-dependence of the substrate's coset metric [JanzenAlgebroid], and the selection of the forced foliation is what deparametrizes it away—the structural and the canonical sides of one dissolution.

Relation to existing approaches

The deparametrization carried out in Section 5 is, as a technique, entirely standard, and it is important to be clear about what is and is not new here. Kucha\v{r}'s survey [Kuchar1992] classifies responses to the problem of time into those that locate an internal time before quantization, those that quantize first and seek time afterwards, and those that abandon time outright; Isham's review [Isham1993] maps the same terrain. The construction of Section 5 sits squarely in the first class, and its minisuperspace realization uses a dust clock of exactly the Brown–Kucha\v{r} type [BrownKuchar1995]. We claim no novelty in the mechanism.

What CR adds is not a mechanism but a warrant. The standard and well-known objection to internal-time approaches is the multiple-choice problem: many candidate internal times exist, inequivalent quantizations follow from different choices, and the formalism provides no principle to prefer one [Kuchar1992, Isham1993]. This objection has force precisely when the choice of clock is a matter of formal convenience internal to the theory. It loses that force when the clock is fixed from outside the formalism by ontology and observation. CR's cosmic time is not selected because it makes the equations tractable; it is selected because the occurrence/existence distinction identifies the evolving layer as what exists, and because the CMB measures the corresponding cosmic rest frame (Section 2). The multiple-choice problem is thus not solved within the space of formal clocks but dissolved by the recognition that one clock is physically real and the others are not. This is the same pattern as the rest of the paper: a difficulty that is genuine so long as a mistaken premise is held, and that disappears with the premise.

The timeless and relational programmes—Page and Wootters' conditional probabilities [PageWootters1983], Rovelli's relational quantum mechanics and evolving constants of motion [Rovelli2004], Barbour's denial of fundamental time [BarbourEnd]—take the opposite path with full consistency: they accept that no preferred foliation exists and reconstruct the appearance of time from correlations within a fundamentally timeless structure. We do not claim these programmes are internally flawed. We claim that the premise they share—that there is no objective cosmic now—is the very premise the CMB refutes [JanzenModernParallax], and that once a real cosmic time is admitted, the elaborate reconstruction of temporal appearance from timeless correlations is solving a problem that no longer arises. The choice between the programmes is, in the end, the choice of Section 2: whether to grant the manifold of occurrences the existence that the evidence assigns to the evolving layer.

Conclusion

The canonical problem of time is the block universe in Hamiltonian dress. Read the four-dimensional manifold as the existent, and the constraint correctly reports that nothing evolves; read the evolving three-dimensional world as the existent and its record as the manifold, and the same constraint, deparametrized along the cosmic time that the universe has let us measure, is a true Hamiltonian generating a unitary advance that reproduces the observed flat-ΛCDM expansion. Lifted beyond the background to the layer's transverse-traceless sector, that same Hamiltonian advances the graviton tower of the closed-S³ layer unitarily—the propagating degree of freedom a fundamental observer reads as the flat-ΛCDM graviton, with matter the bend of its slicing rather than an ontological content. The equations never changed. What changed was the answer to a single question—what exists—and that answer is not a matter of taste but one the occurrence/existence distinction and the isotropy of the CMB settle together. The problem of time was the canonical symptom of a category error; read without it, the formalism was describing a real cosmic time all along. Read so, the dissolution is one of a family: the same distinction that turns the frozen constraint into a true Hamiltonian is shown, in the companion framework paper's first synthesis, to dissolve a wider set of general relativity's standing problems together—the problem of time among them, and the unitary evolution established here underwriting the resolution of the black-hole information paradox on the realised, globally hyperbolic spacetime [JanzenCRframework]. That places the canonical face within the framework's one synthesis: the maximally symmetric de Sitter substrate the framework reads at once as general relativity's solution space [JanzenRange], as the discrete CPT and charge structure, and as the gauge sector on its conjugate real form, is read here on its canonical constraint—where the deparametrized generator is the layer's true Hamiltonian, and the quantum of action enters the gravitational sector at the seam, scaled by Λ alone [JanzenGeometricCore].