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P15

The cosmology of Cosmological Relativity

the expansion history from causal reassignment on a de~Sitter background, the cosmogenesis branch point, and the scalar perturbation sector

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Introduction: the cosmological theory and its scalar perturbation sector

Cosmological Relativity (CR) recovers the flat-ΛCDM expansion history by a causal reassignment on a de Sitter background at the cosmogenesis branch point: the expansion rate is fixed by the cosmological constant alone, radiation carries no term in it, and the present matter density is a reading of the cosmic epoch rather than an independent quantity [JanzenCRframework, JanzenModernParallax]. The cosmic beginning is the branch point r=0—a genuine curvature singularity of the SdS metric read over the -chart, and not a singularity of the substrate, whose throat is the defining constant α, and the construction makes the constant-proper-time slices exactly flat while the cosmological layers are a closed S³. This paper develops that cosmological theory in the form its observational tests require (§2) and completes it with the scalar perturbation sector the framework paper's outlook named open—“perturbations within the Schwarzschild–de Sitter representation”—the primordial density perturbations and their imprint on the microwave background (§§4–8), closing on the open frontier the construction sets up (§10). Across these sections one comparison recurs and is drawn together at the close (§9): the causal contact, flatness, coherence, isotropy, and near-scale-invariance that the standard picture obtains from an added, tunable inflationary sector are here read off structure already present and fixed by Λ alone—an economy-of-assumption difference the paper argues is decidable now, on the theory-choice axis where it falls to this cosmology—and the data axis has now moved with it, the Hubble tension resolved across the low-redshift distance ladder and the light-element abundances confirmed against the data (§2), so the reading is empirically favoured and not merely more economical, its remaining exposed edges the damping-scale signature—a computed, non-reabsorbable CR-specific effect of definite sign and z_onset-dependent magnitude whose observable consequence awaits the high-ℓ acoustic transfer, genuinely open—and the full-spectrum likelihood, the low multipoles a mild, non-discriminating deficit rather than a sharp edge either way. The companion dynamics paper develops the perturbation sector's tensor half—the propagating graviton as the leaf's shear, a massless de Sitter mode on the cosmic foliation [JanzenDynamics]; the scalar half is here.

The result is organized by one thesis, which we state at the outset because it controls everything below.

The CR primordial scalar spectrum factorizes: the de Sitter substrate geometry determines its structure, the progenitor collapse supplies its content, and the branch point's null geometry assigns each its job.

Substrate-determined, and developed here: the coherence of the acoustic phases (§4); the classical, non-vacuum character of the fluctuations (§5); the isotropizing throat (§6); a parameter-free low-multipole discreteness floor (§7); and the faithful transmission of the spectral shape (§8). Progenitor-supplied, and inherited rather than derived: the amplitude A_s and tilt n_s. This is the same division of labour by which flat ΛCDM carries the baryon-to-photon ratio η as a measured datum from a baryogenesis it does not model; this cosmology carries one such datum—the onset redshift z_onset, equivalently the radiation amplitude ρ_r/ρ_m≈2 at onset that calibrates the acoustic scale, fixed by the hot handover of the progenitor collapse—and the transmission result of §8 routes A_s and n_s through the same handover. There is one boundary-condition supplier, not two.

A word on what is settled before the spectrum. The acoustic scale (the peak spacing) and the apparent Hubble tension are resolved in the cosmological development above (§2): the tension dissolves as a consequence of the geometric rate, and the scale is met at the directly measured H₀ by the single measured parameter ρ_r/ρ_m≈2, with the sound speed the ordinary baryon-loaded value of pressureless branch-point matter. The sky's own side of that comparison is measured here rather than quoted. Locating the acoustic peaks of the Planck 2018 plik_lite binned spectrum [Planck2018] by parabolic fit to each hump above ℓ=100 returns ℓ=220.4, 537.7, 817.3 and 1123.9, which reproduce the standard quoted values to better than one per cent without having been fitted to them. Fitting ℓ_n=ℓ_A(n+φ/π) to the first three gives the sky φ/π=-0.2404 against -0.2361 for a ΛCDM control measured by the identical procedure—an offset of 0.0043, so the control stands in for the sky on this quantity to within seven parts in a thousand. The move has a name and the name is worth stating, because it is what licenses the comparison: this is matched-procedure differencing—the control is passed through the identical extraction, so whatever bias the procedure itself carries appears in both arms and cancels in their difference, and what survives is attributable to the physics rather than to the instrument. The same discipline governs every control in this sector: a residual is credited against CR only once the control, run through the same machinery, has been shown not to produce it. What this construction's own arm returns against that measured comparison is reported in §4}, the two-arm transfer being validated on the control and carried to convergence on both arms.} These are settled there and are not reopened in the perturbation sector; in particular, the standard radiation-governed sound horizon laid on this cosmology's rate is a calculation belonging to neither framework, and any “tension” it produces is an artifact, not a CR result. What is genuinely downstream is the spectrum—the peak heights, carried below by the time-reversal driving equality (§4), and the full oscillating-medium transfer, still open—which is where this paper's substance lives.

The register is the programme's [Janzen2025, JanzenCRframework]: the field equations are Einstein's, unchanged; the existing object is the evolving three-dimensional layer; “the substrate determines” and “the branch point transmits” are statements about a real geometry and its real boundary [JanzenGeometricCore, Janzen2012], not about a modification of the dynamics.

The cosmological theory the perturbations complete

This paper develops the cosmology of Cosmological Relativity from first principles and carries it to its empirical confrontation, completing it with the scalar perturbation sector. It takes from the companion framework paper [JanzenCRframework] the general primitives—the layered ontology, the projection principle, and the representational freedom by which one fixed manifold admits distinct Lorentzian metrics—together with the Null–Boundary Correspondence theorem, the structural result (grounded in the metric-singularity causal structure of gravitational collapse [JanzenBHcausality]) that a collapse event-of-events and a de Sitter cosmic-entry slice are one ontological layer under two causal assignments. The cosmology proper is built here: the causal reassignment that selects the Nariai member, the proper-frame expansion law, the branch point at which the collapse leg becomes the expansion leg—the beginning, reached a lap on from the horizon the Correspondence anchors—and the resolution of the apparent Hubble and acoustic tensions; the perturbation sector (S4–S8) is its completion.

The causal reassignment and the Nariai member

The vacuum solution of R_μν=Λg_μν with a localized source is Schwarzschild–de Sitter (SdS),

whose areal radius is timelike in the cosmological regime ΛG²M²/c⁴≥1/9. Building the cosmology, we reassign causal roles on the de Sitter substrate—a move licensed by the representational freedom of the framework, distinct Lorentzian metrics on one fixed manifold sharing its foliation [JanzenCRframework]: the de Sitter null rulings whose causal sense matches a collapse horizon's generators are promoted to the timelike fundamental congruence, the at-rest comoving worldlines become the null photon congruence, and the foliation by expanding 3-spheres is preserved. This selects the Nariai member ΛG²M²/c⁴=1/9, M=c²/(3√ΛG)—the unique non-pivoting tilt, fixed by Λ alone—at which the two positive horizons merge at r_N=α/√3, α=√3/Λ the de Sitter 3-sphere size (a scale distinct from the merged-horizon radius α/√3 and from the areal-radius amplitude below; we keep the three apart throughout). The selection is geometric—the fixed point of the slicing curve's root-exchange involution, equivalently the limiting tangent orientation a collapse selects—not a fitted mass [JanzenSlicing]. That the selection is moreover unique is a structural fact about the description groupoid rather than a fitting: the generic vantages fall into two-cycles of the involution, the Nariai member is its one fixed point, and the reassignment promoting a null direction to the fundamental congruence is forbidden at every generic vantage, so it selects Nariai and no other [JanzenGroupoid].

The proper-frame cosmology, derived

In the proper frame of the fundamental observers the SdS metric is a homogeneous, non-synchronous cosmology. The derivation [Janzen2015] proceeds from (1) in five steps. (i) A timelike radial geodesic has a conserved energy (the metric being independent of ); eliminating dt/dτ in the normalization L=-1 gives the radial equation (dr/dτ)²=E²-V_eff(r), with V_eff=(r-2GM/c²- Λ3 r³)/r. (ii) The fundamental congruence is the worldlines whose motion in is produced purely by the field—those at rest exactly where V_eff≡1 and the metric reduces to Minkowski—which is E=1; this fixes the rest frame by the field rather than by stipulation, and it is worth saying where that locus is: V_eff=1 is r³=-2GMα²/c², so the rest frame is defined at r=-(6GM/c²Λ)^1/3—a negative areal radius, on the conjugate branch of the lap, between the progenitor's horizon and the branch point [JanzenCRframework]. In the original derivation that locus had to be carried as a formal one—“a negative mathematical abstraction that lies beyond a singularity at r=0… simply a matter of algebraic consistency” [JanzenFQXi2012, App.]—and the construction has since made it a place. The same locus fixes the sign, since t=τ there if and only if E=+1. (iii) With E=1, integrating the radial equation gives, along one worldline, r(τ)∝ sinh^2/3(32√Λc²/3 τ). That this integration closes at all is a second thing the same condition buys. Writing the radial equation as (dr/dτ)²=2GM/c²r+r²/α²-k with k=1-E², the quadrature ∫dr/√ · is elementary at k=0—it is 2α/3 arcsinh (r^3/2/√2GMα²/c²), which inverts to the sinh^2/3 above—while for k≠0 the substitution r=u² carries it to a sextic under the root and it is not elementary. So E=1 fixes the rest frame by the field and, by the same stroke, is the one non-degenerate energy at which the cosmology integrates in closed form; the only other elementary cases are the degenerate ones, Λ=0 giving the dust cycloid and M=0 giving pure de Sitter^L14. (iv) Labelling the congruence by a comoving coordinate χ through the integration constant r(0), the areal radius across the congruence depends on τ,χ only through their sum, r(τ,χ)=r(τ+χ), with ∂_χr=∂_τr. (v) Fixing the transformation t(τ,χ) by requiring the comoving coordinate orthogonal to proper time, g_χτ=0, forces the integration function to be χ itself, and the remaining components follow (g_ττ=-1, g_χχ=(∂_χr)²). The result is the proper-frame line element

with cosmic time τ≡τ+χ (by Weyl's principle the congruence issues from a common origin and the constant-cosmic-time slices are those of constant τ), areal radius

and redshift 1+z=r(τ₀)/r(τ_e). The constant-τ slices sit at 45^∘ to the fundamental rest frame: the cosmology is homogeneous and observationally isotropic, yet non-synchronous—the constant-τ slices are exactly Euclidean (S7) but are not the cosmological ones. Of the FLRW kinematic assumptions—(i) a geodesic congruence, (ii) hypersurface orthogonality, (iii) homogeneity, (iv) isotropy—this construction keeps (i) and drops (ii); Lemaître's reading [Lemaitre1949] of the Euclidean slices as “space ejected from r=0” is causally incoherent and is not the physics. Equation (3) is moreover a single analytic function of τ (Fig. 1): the expanding cosmology is its real branch τgt;0, and the cosmogenesis branch point at τ=0, where r→0, is the branch point of the scale factor rather than a real r=0 boundary—and it is not a seam: the seams are the two unit-speed loci of the lap, r=-2α/√3 and r=+α/√3 [JanzenCRframework], and r=0 lies strictly between them, at the close of the lift—the analytic counterpart of the beginning at the branch point, where the substrate's curvature stays finite while the geometry read over the -chart diverges [JanzenBHcausality], and the cosmological reading of the slicing curve's closed lap—the cosmogenetic bead the framework establishes as one closed object [JanzenSlicing, JanzenCRframework], of which this scale factor's real branch τgt;0 is the expansion leg and its analytic continuation the prior universe's collapse—on which r=0 is a branch point and not a barrier, the substrate smooth across the locus the chart so labels [JanzenSlicing].

Exact recovery of flat ΛCDM, and the amplitude fixed by Λ

What fixes this rate is worth stating before what it recovers, and the count is smaller than it looks. The rate this cosmology computes on is set by the substrate's geometry and by nothing else. It has one parameter—the substrate curvature radius α=√3/Λ, an invariant length—and one epoch. At the Nariai member the mass is no longer free (5), so the amplitude and the rate of the expansion law are both set by Λ and the scale factor is one function of it; what remains is where the present sits on that one curve. The quantity x₀ carries that: it is the present areal radius in units of the Nariai radius, x₀=r(τ₀)/r_N, and since r_N=α/√3 is fixed by Λ the number is a clock reading and not a second geometric datum^P15R7. The geometry's own offset is fixed twice over: it is the double root of the horizon cubic at the Nariai mass, and it is the constant ledger's 2M=α ((r₀/α)-(r₀/α)³) with at that mass [JanzenSlicing, JanzenGeometricCore]—so "the offset" names the unit, and x₀ the reading taken against it. So the ratio it fixes is a reading of cosmic epoch, which is what (7) says of the density ratio directly. Neither the parameter nor the epoch is a content of the universe:

The offset is measured directly and calibration-free—z_acc=x₀-1, (11) below—without a distance ladder, without the microwave background and without any density. What the rate is a rate of is the stacking of the foliation, and content does not enter it: radiation is absent from the slicing operator's vacuum kernel, requiring m'(r)≠0, a bend of the cut; and the second term is the offset itself rather than a density supplied by hand [JanzenOperator, JanzenCRframework].

The familiar cosmological form is accordingly a coincidence of form, and saying so is the substance of this subsection. Written in the fitted pair the same rate reads H²=H₀² [Ω_m(1+z)³+Ω_Λ], with Ω_m=2/(x₀³+2) and Ω_Λ=x₀³/(x₀³+2) following identically—a translation between parameter sets, and not a decomposition into components. In the standard reading cosmic time is orthogonal to the three-dimensional world and the expansion is driven: by initial conditions, and by whatever densities obtain at each moment, through the field equations. Here it is read off a fixed background whose foliation is not orthogonal to the proper time of observers at rest, so that such observers move through the universe over cosmic time in a geometrically fixed manner. The two readings agree on the function and share nothing beneath it.

Where the field equations act is the leaf, and they act there in full. All stress-energy gravitates, by the ordinary Friedmann readout of the self-gravitating congruence; the perturbations, the plasma's sound horizon and its diffusion length are processes running in the content and take that rate, while a comoving separation read across leaves takes the stacking rate (§2). What the layered framework keeps apart is not two physics but one physics and its projection: a quantity is computed on the level it belongs to, carried on the layer and then projected.

Equation (3) is exactly the flat-ΛCDM scale factor. At Nariai the mass is no longer free, and the amplitude becomes a pure Λ-length,

so both factors of the expansion law are set by Λ^P15R32: the rate 12√3Λ c is the standard late-time rate and the amplitude is the third Λ-scale (distinct from α and α/√3). The Friedmann equation the sinh^2/3 law satisfies,

makes the density ratio a clock^P15R32,

so Ω_m,0 records the cosmic epoch τ₀ at which we observe rather than an independent density, and the coincidence problem becomes the observation that we exist at a time of order the geometry's single timescale. The densities are bookkeeping for one Λ-set geometry; radiation, like matter, is inherited content read off that clock, not a term that sources the rate, and there is no radiation-dominated era [JanzenModernParallax]. The objection that a perturbing radiation fluid “must gravitate and so alter the rate” is a category error against a construction that reads (6) leftward: the rate is set, and the fluid's energy is among the contents the set rate carries. The acoustic scale and the apparent Hubble tension resolve as consequences of this rate; this is developed next (§2).

The acoustic scale and the dissolved tensions

The two frameworks reproduce the late-time expansion identically and part in the early universe, where radiation carries no term in the rate (6). The apparent tensions of the standard model are consequences of that one structural fact.

The three levels, and which rate each quantity uses. The reading of every rate-bearing quantity below is fixed by locating it on one of three levels, which the geometry keeps apart and whose conflation is the standing error the layered framework names [JanzenCRframework]. (L1) The foliation stacking rate: the Λ-set sinh^2/3 law (6) read leftward in the sense the slicing paper fixes [JanzenOperator]—the cut is primary and ρ is the name of its bend, not its cause, the rate-side of the slicing operator's reading whose content-side is matter is the bend of the cut [JanzenOperator]—the layer's own geometric expansion, set by α and the offset x₀, and empirically forced rather than modelled: the microwave-redshift isotropy floor forces uniform expansion to some three orders of magnitude, so a matter-tracking (radiation-sourced) rate is already excluded by data [JanzenModernParallax].

This is what the observable cosmology rides at every epoch, the branch point included: it is the rate the foliation stacks at, not a regime that begins somewhere. (L2) The leaf-level local dynamics: the expansion scalar of a self-gravitating congruence, the ordinary Friedmann readout H²=8πG/3ρ_tot+Λ/3 with radiation gravitating normally—the valid phenomenological reading of general relativity on the progenitor's collapse excursion, where nucleosynthesis runs [JanzenCosmogenesis]; it is the same single E=1 geodesic read inward as dust collapse that L1 reads outward as cosmology [JanzenOperator, JanzenBHcausality]. (L3) The E=1 projection: the reassigned shadow the observer reads, through which an L1 or L2 quantity appears as distance and redshift—the Projection Principle of the framework paper, its geometric origin the second ruling of the slicing (the synchronous space is the second ruling) [JanzenCRframework, JanzenOperator]; reading this appearance as the existent is precisely the naïve-realist collapse the epistemology forbids [JanzenShadowExistence]. The decision rule is structural, not a choice: a self-gravitating local excursion runs on L2, diffuse content riding the global foliation on L1, and every observable is an L1 or L2 quantity seen through L3; and there is no L1/L2 boundary in time. The distinction is kinematic and holds at every epoch: the vacuum kernel of the slicing operator is the SdS family exactly—Λ and the cut's offset 2M, and no more [JanzenOperator]—so the stacking rate is c²H_stack²=(c²/α²)(1+2(1+z)³/x₀³), while radiation, requiring m'(r)≠0, is a bend of the cut and therefore content. The decomposition is exact, H_leaf²=H_stack²+H₀²Ω_r(1+z)⁴: the two rates differ by the radiation term alone. A comoving separation read across leaves takes the stacking rate; a process running in the content—r_s, r_D, recombination, the perturbations—takes the leaf's. The assignment looked temporal only because Ω_r(1+z)⁴ is negligible below z∼10 [JanzenCRframework, JanzenBHcausality]. Each mis-assignment is quantitatively fatal—the geometric L1 rate carried down into the nuclear window is ∼300× too slow, and the radiation-sourced L2 rate carried past the branch point radiation-pins r_s and re-manufactures the very tension this section dissolves [JanzenCosmogenesis]. Crucially L1 is the layer's own ontological expansion, not the L3 shadow: computing a process against the stacking rate is reading the geometry, not the appearance. Every rate-bearing quantity below carries its level, so the reading is settled by the geometry rather than defaulted to the synchronous machinery.

The Hubble tension. There is no second H₀ to reconcile. The late-time expansion is reproduced exactly, so the directly measured rate is the reading of the geometry, while the microwave background's lower inferred H₀ rests on a radiation-governed sound horizon r_s this construction does not share. The ∼5σ discrepancy is not two independent fits to be reconciled but the artefact of calibrating the acoustic scale with an early-universe sound horizon foreign to a geometrically fixed expansion.

The acoustic scale, to a one-parameter accommodation. Recombination is observable cosmology from the branch point onward, so its scales are L1: both the sound horizon r_s and the comoving distance D_M ride the foliation's stacking rate, and the observed angle θ_*=r_s/D_M is their L3 projection.

The robust ingredients are in hand: D_M is the standard observable, and H(z) is the fixed sinh^2/3 law (6) with radiation absent from the rate.

The sound horizon r_s=∫z_recz_onset(c_s/H) dz—an L1 quantity whose baryon-loaded c_s is ordinary content microphysics but whose expansion is the L1 rate—then depends on a single early-universe parameter, the redshift z_onset at which the expanding-phase plasma begins; the standard radiation-governed r_s is recovered only in the limit z_onset→∞ that a beginning at finite curvature forbids. That clause carries more weight than its length suggests, and the sense of “finite curvature” in it should be the substrate's.

The branch point is a locus at which the areal chart's invariants diverge—r=0 is a genuine infinite-curvature locus of that chart—while the curvature of the manifold the cut is taken on is set by α everywhere, so the divergence is the areal coordinate degenerating and not a scale of the geometry [JanzenCosmogenesis, JanzenGeometricCore]. Read the other way the accommodation would collapse: a beginning at genuinely unbounded curvature places no finite floor under z_onset, the z_onset→∞ limit is reinstated, and the single datum ceases to be a datum at all. The one-parameter accommodation therefore rests on the substrate reading of the branch point, and a reader who took the areal divergence for the physics would find the parameter unconstrained. Taking c_s the ordinary baryon-loaded value—the matter at onset is pressureless, and the geometric throat ratio √3 does not set it—the measured acoustic scale is reproduced at the directly measured H₀ by a single z_onset, near ρ_r/ρ_m≈2 (about twice matter–radiation equality) and independent of H₀—the independence belonging to z_onset itself, and the ratio being that same quantity re-expressed, as §2 returns to below: r_s and D_M carry that rate's common factor H₀, which scales out of their ratio, so θ_*=r_s/D_M is fixed by the offset x₀ alone and the single z_onset that meets the scale is the same at every H₀ across the range—the datum cannot be a parameter tuned to reconcile two values of H₀, since it does not move between them. This is exactly where the standard rate parts: carrying radiation, it ties θ_* to H₀—the coupling that fixes the lower microwave-background value and stands in tension with the direct one—whereas a rate whose parameters are geometric decouples the scale from H₀, meeting it at the directly measured value with no second H₀ to reconcile. The beginning is well posed: the branch point—finite in the substrate's curvature, whatever the -chart geometry does there (§2)—places the onset down the cooling branch, T_onset∼1.6 eV at z_onset∼6.8×10³, so a pre-recombination sound-travel interval exists and the integral ∫z_recz_onset is well posed.

The inherited datum. The empirical forcing of the cosmic foliation bears on the rate, not the matter content—the matter density being the bend of the spatial cut [JanzenModernParallax, JanzenOperator, JanzenRange]—so the radiation amplitude is measured content, not a quantity the geometry sets; the cold seam (Gibbons–Hawking temperature [GibbonsHawking1977] ∼10^-30K, some thirty orders below the onset) confirms there is no geometric source to set it instead. One thing about that ratio should be said plainly before it is leaned on, because it is a unit and not a second datum.

The quantities this cosmology actually uses are the epoch x₀ (equivalently Ω_m=2/(x₀³+2); x₀ is the expansion read in units of the geometry's own fixed offset r_N, not the offset itself, which the Nariai condition fixes at α/√3), which the rate carries and the baryon-acoustic data fix; ω_b and ω_γ, which the sound speed carries and which are measured directly and framework-independently; and z_onset, the one fitted number.

The physical matter density ω_m=Ω_m h² enters none of them.

One thing about Ω_m is worth stating plainly, since the construction's matter sector invites the question. The dust the rate carries exceeds the baryons by about a factor of six, Ω_m≃0.31 against ω_b/h²≃0.05, so most of it is non-baryonic. That is not a component this cosmology introduces, and it is not one it derives. What the matter sector fixes is the number of families, their chirality and the S₃ relating them, and nothing about their internal content [JanzenMatter]; abundances are measured content throughout, the matter density being the bend of the spatial cut. So the non-baryonic fraction stands with η, A_s and n_s: inherited, not set by the geometry. What the construction does supply is a place rather than an abundance. Its correspondence with the Standard Model's content carries a right-handed neutrino as the sixteenth Weyl fermion of each generation, and fixes that state's place while fixing none of its couplings [JanzenMatter]; a species that is never thermalized contributes nothing to N_eff, so nucleosynthesis is untouched and no prediction is made or needed here. The point of the paragraph is the division of labour: a candidate the construction already contains, an abundance it does not claim, and no third thing wedged in between.

But ρ_r/ρ_m at onset is ω_r(1+z_onset)/ω_m and equality is 1+z_eq=ω_m/ω_r, so both divide by that one quantity—and ρ_r/ρ_m≈2 is therefore not a second inherited number standing beside z_onset but z_onset itself, read in units of a density the construction does not determine. The consequence is a stated H₀-dependence in the ratio and in nothing else: with Ω_m held at its fitted value the ratio is 2.01 at H₀=67.4 and 1.71 at H₀=73, the two readings coinciding at H₀≃68 where Ω_m h² reproduces the microwave-background-inferred ω_m=0.143. What carries weight is what survives that: z_onset does not move with H₀ at all, so it cannot be absorbing the tension; and the inherited amplitude, read at onset, is order unity—which is a statement about where the onset sits and not a contrast with η, as the withdrawal two paragraphs below sets out. The factor of two is exact at h≃0.68 and should be quoted as an order-unity band, not as a determined number (receipt: the_ratio_is_the_onset_in_imported_units.py^P15R18).

Thus ρ_r/ρ_m≈2 is the structural analogue of the baryon-to-photon ratio η: as flat ΛCDM carries η from a baryogenesis outside the model proper—measured, not derived—this cosmology carries the single radiation initial condition, and with no radiation-dominated epoch the light elements as well, from the hot handover of the progenitor collapse, this universe having formed at the event horizon of a black hole in a previous one, equally outside the construction and equally measured.

That the datum is measured rather than derived is no defect: a framework is credited for requiring the structure it explains—the geometric rate, and the dissolution that follows—not faulted for measuring the boundary data it does not. One naturalness argument that has been offered for this datum should be withdrawn rather than carried, because it compares a quantity with itself at two epochs nine orders apart.

Complete collapse does liberate binding energy of order the rest mass (GM/Rc²= 12 at the Schwarzschild radius), and it is tempting to read that as making a handover carrying radiation of order the matter density the natural scale—order unity, rather than the small value η takes. But ρ_r=ρ_m occurs at T≃0.8eV, whereas the handover sits four orders above the deuterium bottleneck at 0.07MeV, since that is what the nucleosynthesis computed on it requires [JanzenCosmogenesis]: at the bottleneck the ratio is already ∼9×10⁴ and at the handover proper ∼9×10⁸. A handover carrying radiation merely of order the matter density would be a handover with η≃0.5, which leaves neither light elements nor acoustic peaks. So the naturalness question runs the other way: what the progenitor must deliver is a plasma with of order a billion photons per baryon—the baryogenesis-analogue derivation this paper names as open—and the binding-energy scale offers no head start toward it. What replaces the argument is a simplification of the bill. Given η and the measured matter-to-baryon ratio, both of which this cosmology inherits for the composition regardless, the ratio at onset is [1+ν](π⁴/30ζ₃) T_onset/[η(ω_m/ω_b) m_N], which returns 1.99 at T_onset=1.6eV—the quoted datum, to one per cent, from standard thermodynamics and no feature of this construction. It is therefore not a second inherited number standing beside η: it is the fitted onset restated, so what the handover supplies is one composition datum and what the cosmology fits is one parameter (receipt: the_two_data_are_one.py^P15R19). Like η, the inherited radiation amplitude may remain a measured boundary condition indefinitely without cost to the dissolution—which settles what the dissolution depends on, not whether the derivation is worth having. And the inheritance has a structural reason rather than being a gap left standing [JanzenCRframework, JanzenCosmogenesis]: the infall thermalizes four orders above the deuterium bottleneck, so the progenitor's composition is erased and the abundances are synthesized in the window rather than inherited, while what survives that passage is baryon number, which no dissociation destroys. So η is inherited and the abundances are predicted, and the asymmetry is worth stating exactly: baryon number protects the numerator, while what protects the ratio is the adiabatic invariance of the specific entropy — the peak perturbing it by 5.5×10^-10 of a standing 1.15×10¹⁰ k_B per baryon, so that η survives the erasure because η is small [JanzenCosmogenesis]. And a derivation of η is not owed, because there is nothing here for one to be about. It enters the progenitor from the ambient universe by composition-sharing and leaves the branch point unchanged to one part in 2×10⁹: transmitted, not determined — which is the exact dual of the onset ratio's closure, that one turning on a quantity with no single value to transmit. So the neighbouring statement strengthens from may to must: η remains a measured boundary condition, at no cost.

The synthesized composition. The same collapse fixes the light-element composition, and the geometry sharpens how. The infalling matter compresses adiabatically—the plasma is optically thick, so the photon bath compresses with it—and heats far above the deuterium bottleneck (T_D≃0.07MeV) for every progenitor: the compression does not stop at horizon crossing but continues to the branch point, and the peak is bounded below by the infall energy at the horizon, of order the rest mass per nucleon, independently of the progenitor's mass (the cosmogenesis paper [JanzenCosmogenesis]); at turnaround it re-expands and cools back through the window. Because the effective rate through the window is the standard Friedmann rate, that cooling leg is a standard big-bang nucleosynthesis, run on the collapse's turnaround from a dissociated hot start—the deep re-expansion doing the synthesis below the T_onset∼1.6eV onset at which the observable expansion begins. On a first sample this is favourable, and on one element advantageous. The helium-4 fraction Y_p≈0.245 sits at the observed value; the near-zero metallicity of the oldest systems follows from the handover photodissociating the progenitor's processed composition back to light elements; and because the run is a standard nucleosynthesis, the lithium-7 abundance carries its standard threefold over-prediction—the standing lithium problem shared with flat ΛCDM, neither dissolved nor worsened. Deuterium sits at the observed D/H≈2.5×10^-5: the freeze-out temperature the data demands is the standard bottleneck, and that is exactly the temperature at which the standard rate delivers a freeze-out—target and mechanism coincide un-tuned. The peak calculation that forces this synthesis reading over an inherited one, and the full multi-abundance account, are given in the cosmogenesis paper [JanzenCosmogenesis], where the network is now integrated explicitly on the cooling leg and returns deuterium, helium-4 and helium-3 at their observed values with lithium-7 at the shared over-prediction, jointly from the single inherited datum—so one progenitor handover does fit the pattern together; what stands as its open edge is the last-percent precision against the measured abundances, a data-confrontation the produced pattern now exposes rather than a computation still owed.

The load-bearing claim. The acoustic scale stands as a one-parameter accommodation on the radiation datum, not a parameter-free prediction; the standard c_s it borrows presumes the inherited medium to be an ordinary photon–baryon plasma, which the abundance comparison now confirms—helium-4 and deuterium at their observed values, lithium-7 the shared standard problem (the cosmogenesis paper). The load-bearing, falsifiable claim remains the prior one—that radiation carries no term in the expansion rate—of which the dissolved Hubble tension and the one-parameter acoustic calibration are consequences. The sound horizon at a geometrically fixed H₀ is where a purely geometric expansion history and the standard radiation-governed one are told apart in the data.

The expansion history, confronted with the late-time data. The dissolution is not confined to the single microwave-background angle: it holds across the low-redshift distance ladder, and for the same reason.

Because every distance and the ruler alike carry the stacking rate's common factor H₀, which therefore scales out—D_M, D_H=c/H and the sound horizon r_s each scale as 1/H₀—the baryon-acoustic-oscillation observables D_M/r_s and D_H/r_s are independent of H₀, fixed by the offset x₀ alone (the same cancellation that fixes θ_*; § above).

The distinction from ΛCDM is exactly the radiation: there the ruler r_s is pinned in physical length by the radiation-era physical density ω_m=Ω_mh², so the same D/r_s data constrain H₀ and pull it to the low microwave-background value, in tension with the direct one.

Here they do not: the acoustic data fix the offset x₀ and leave H₀ to the local distance ladder, with no second value to reconcile. Confronted with the SDSS DR12 galaxy consensus [Alam2017] the geometric rate at Ω_m≃0.31 fits at the directly measured H₀ (χ²≃1.7 on six points, at any H₀ including 73), where the radiation-governed rate at that same local H₀ misses badly (χ²≃49)—that residual is the Hubble tension itself, which the geometric rate structurally lacks. Stated on the objective rather than on its minimum, the difference is one of curvature RANK and not of value^L9: the radiation-pinned χ²(H₀) is strictly convex, with a unique argmin that excludes the directly measured value, while the geometric rate's is flat in H₀ — a degenerate Hessian whose argmin is a line, and the line contains that value. So the tension is not a disagreement between two numbers but a property of the surface one of them is a minimum of, and a rate whose parameters are geometric does not have the surface.

The same confrontation carried to the state-of-the-art DESI DR2 dataset [DESI2025]—thirteen distance measurements across seven tracers (0.3≲z≲2.3), with the full per-tracer D_M/r_s–D_H/r_s correlations—tightens rather than loosens the result: with a single free parameter Ω_m the geometric rate fits at χ²/ dof≃1.0, at Ω_m=0.3066 (where the inherited datum sits at ρ_r/ρ_m≈2.0), and it fits there at any H₀ including the local 73—which is the whole of the claim.

The radiation-pinned ruler cannot do that: it ties ΛCDM's fit to one H₀, best at 68.7, and breaks it at 73 (χ²/ dof≃15)^P15R31.

Two things about Ω_m are worth stating exactly. It is not a CMB-calibrated input here—Planck's own value is 0.3153, at which the fit degrades to χ²/ dof=1.35, and 0.3066 is what these data themselves prefer. And granted the same one free parameter, ΛCDM fits these data marginally better than this cosmology does (χ²/ dof=0.92 against 1.00). The result is therefore not that the geometric rate fits better; it is that it fits without choosing an H₀, and the radiation-pinned one must choose. The cosmic chronometers [Moresco2016], which read H(z) directly, are consistent with the local value. So the resolution is a property of the whole expansion history, not of one angle, and it rides the very rate whose branch point-located scoping keeps the nucleosynthesis window standard [JanzenCosmogenesis] (receipt: hubble_expansion_confrontation_v2.py^P15R30; the invariant is the offset x₀, read as Ω_m, and not ω_m—holding the physical density fixed instead imports the ΛCDM reading and manufactures a spurious tension).

The acoustic modes are sub-horizon at the onset, and the branch point is the opposite

The cosmological beginning is the branch point r=0 at the close of the lift, where the conjugate branch becomes the matter branch [JanzenCRframework, JanzenSlicing]. The curvature invariants of the -chart diverge there because the areal coordinate degenerates, while the throat itself is the substrate's defining constant α and the bead continues through by analytic continuation [JanzenCircle]. The lap's seams lie elsewhere, at the two unit-speed loci r=-2α/√3 and r=+α/√3. We record the consequence that organizes the perturbation sector.

And the corresponding statement about the branch point is the opposite one, for a different reason. Approaching r=0 on the contracting leg the comoving Hubble scale grows without bound, so the comoving horizon shrinks to zero and every mode exits it before the crossing—which is why what crosses is frozen and non-oscillatory (§3). That statement is about the crossing and stands whatever the census at the onset returns, the two loci being distinct; reading either at the other's locus inverts it.

The consequence is that the acoustic band is not one regime but two, and the division is at a computed multipole rather than a chosen one. Above ℓ=237.7 the modes are inside the horizon when the cosmological side begins, and for those the branch-point handover is what fixes amplitude and phase, there being no super-horizon interval left in which anything else could. Below it they are not, and the sub-band 155.6 lt;ℓlt;237.7—which contains the first peak—enters while radiation dominates, so those modes have a driving history on the observable leg of the kind the standard picture gives them. What this settles is a premise and not a mechanism: it fixes where each mode stands when the plasma starts, and it does not measure how much of the comb the driving then supplies. That measurement is the transfer of §4, which measures the driving directly; the division above is a statement about the census, and the transfer is what carries it to the comb.

What crosses, and what does not

The transmission dichotomy above is an argument from the absence of an imprinted scale. A mechanism can now be given for it, and it also settles which content the branch point does not carry.

The imaginary-time segment preceding the branch point acts on a perturbation as a Euclidean kernel [JanzenCanonicalTime], damping a mode of frequency ω by e^-ω|Δη| with |Δη|≃1.8×10⁴ Mpc. Taken naively this would annihilate everything: at ℓ=220 the factor is 10^-71. But ω is the frequency of an oscillating mode, and at the branch point no mode is oscillating. On the contracting leg the comoving Hubble scale aH=|dr/d τ|=√|1-f| grows without bound as r→0—it is 0.13 at the comoving turnaround, 1.96 at |r|=0.1 α, and 19.6 at 10^-3α—so the comoving horizon 1/aH shrinks to zero and every mode exits it and freezes before the crossing. A frozen mode has no oscillation for the kernel to damp. So the amplitude and the tilt cross unaltered, and the argument from the absence of a scale is recovered as a consequence of horizon exit rather than assumed.

What does not cross is the oscillatory content itself, and that is the collapse leg's acoustic phase: a mode that was sub-horizon earlier carries an oscillation, and the kernel annihilates it—an exponent of -152 at the first acoustic peak, integrated across the segment's own sound-speed profile rather than at the single value the estimate above uses^P15R73. This is required independently by the observed peak positions, which exclude a transmitted leg phase by a factor of 21–41 in the required distance [JanzenCRframework], and it is what leaves the comb to be set on the expansion side by the characteristic data of §4. The two results are one statement: the crossing transmits what is frozen and destroys what oscillates, so the progenitor supplies the spectrum's amplitude and shape while supplying none of its phase.

The exponent has two factors, and the argument above varies only one of them. It is e^{-k c_s|Δη|}: holding c_s at the plasma's value and varying reads it as a scale criterion, frozen against oscillating, which is the reading just given. Holding and varying c_s reads the same inequality as a species criterion, and that reading supplies something the first does not. A pressureless component has c_s=0 identically—not small, not slowly varying—so its exponent vanishes at every wavenumber; on the segment its mode equation is ψ”=0, whose constant solution the kernel leaves untouched with no approximation at all. The matter sector's inheritance is therefore exact rather than adiabatic: it does not rest on horizon exit and it does not degrade at small scales, where the frozen-mode argument does. The two arguments agree wherever both apply, and only the pressureless one is available everywhere. Neither reading carries a constraint the other lacks. On this geometry the exponent at the first acoustic peak is ≃1.6×10² c_s/c, so a component with a per-cent sound speed would arrive suppressed by e^-1.6 and one with ten per cent by e^-16, which reads as a sharp condition on what the progenitor may hand over—perturbations inherited from a cold species and from no other (receipt: the_filter_is_a_sound_speed_filter.py^P15R73). The reading is too strong, because the exponent has nothing to act on. The kernel acts on oscillatory content—a frozen, zero-frequency mode is a fixed point of it—and on the progenitor's own interior nothing arrives oscillating: |aH|=|a'/a| diverges as 1/x at the branch point while c_s saturates at 1/√3, so c_sk/|aH|→c_skx→0 for every , and every mode from ℓ≃28 to ℓ≃2475 exits the comoving sound horizon strictly before the crunch—the highest with a fifth of a per cent of the collapsing leg still to run [JanzenCosmogenesis]. So the two readings do not divide the work between them: the scale criterion is satisfied by everything, and the species criterion selects nothing.

One consequence of that freezing bears on the acoustic sector rather than on the crossing, and it is computed rather than left as an inference. If every mode is outside the comoving horizon at the branch point and the acoustic modes are inside it at the onset (Prop. 1), then between those two loci the modes re-enter—and in flat ΛCDM a horizon crossing is exactly where the -independent acoustic driving phase shift is acquired. The re-entry is real and its boundary is computable: the mode whose crossing coincides with the onset is k=0.0111 Mpc^-1, i.e. ℓ≃144, so every mode at and above the acoustic peaks crossed before the plasma began while ℓ≲144 is still outside at the onset and crosses later^P15R55. But nothing is imprinted there, and the reason is structural rather than quantitative. On the geometric rate the crossing occurs under pressureless matter to better than a part in 10⁴, and the potential equation for a pressureless component, Φ”+3 H(1+w)Φ'+[2 H'+(1+3w) H²]Φ+wk²Φ=0, contains no at all once w=0—the wavenumber enters only through the pressure term. Integrated on this background from a=10^-7 to the onset the potential is constant to twelve digits and identical across four decades in , the integrator taking the same number of steps for every mode because it is handed the same problem. So the horizon crossing is not an event for the potential here: before the onset the perturbation problem on the observable leg is scale-free, and a scale-free problem imprints nothing. In flat ΛCDM the same modes cross under radiation, where w=1/3 restores the k² term and the potential decays on the sound-crossing time by construction, which is why that shift is the same for every mode. This closes a candidate mechanism rather than supplying one: the absence of a universal driving shift on this rate is not an omitted crossing but a scale-free interval, and the first peak's position is left open by it.

And that difference is measured rather than argued, model-free, by subtraction against an undriven reference that calibrates at unity: the control's acoustic phase at first turnover is flat in wavenumber to 2.1%, where this arm's varies by a factor of 3.9 across the same band^P15R57. So the scale-free interval is not an inference from the potential's form but a shape read off the two arms, and the contrast is with a reference whose own answer is known. The crossing is lossless for every species rather than for cold ones alone. That is a loss of a claimed prediction and not a gain, and it is recorded as one; what survives is the pressureless argument's positive content, that the inheritance is exact rather than adiabatic, now holding of every species and not of dust alone.

What the substrate determines: coherence from the null boundary

The acoustic peaks are sharp because the modes of a given wavenumber share a common phase at last scattering. Inflation produces this coherence dynamically, by freezing modes while super-horizon; Proposition 1 closes that route here. CR obtains it instead from the character of the branch point.

The branch point is a null surface—the cosmological horizon promoted to the initial layer [JanzenCRframework]. The initial-value problem on a null surface is characteristic, not Cauchy: the data that determine the future are one free function per mode on the surface together with regularity along the generators, rather than a field and an independent momentum. A single datum per mode is a single phase per mode; there is no second, independently specifiable quantity to randomize the relative phase.

We therefore argue that coherence is not imposed but is what regular characteristic data on a null surface is.

This is borne out quantitatively, and the scope of the demonstration is stated exactly.

Evaluating the tightly-coupled mode function at last scattering with a common seam phase gives a sharp regular comb, whereas drawing the phase independently per mode—the second, free datum a Cauchy surface would admit—washes it out entirely, leaving flat power (peak-to-trough contrast ∼1 against the coherent comb's many orders of magnitude; receipt: verify_coherence_comb.py^P15R11). That establishes the mechanism—one characteristic datum per mode is one phase per mode, and coherence follows—and it establishes nothing about where the peaks fall. The receipt writes the mode function down as cos (kr_s) with r_s and D_C as literals—and the r_s it writes is the standard 147.0 Mpc rather than any value this construction derives; no mode is propagated in it, as its own scope note says, so the spacing Δℓ≃πD_C/r_s≈296 is the arithmetic of an assumed expression on an imported ruler rather than an output. It is not the result of propagating the modes, and is not offered as one^P15R54.

What the peak spacing lacks is not agreement but a derivation: it is asserted here and computed nowhere in this paper, and the transfer that would compute it is the open one of §4.

The null boundary admits only the first column: one characteristic datum per mode is one phase per mode.

The peak heights are then carried by a structural argument rather than a bespoke transfer. The driven acoustic amplitude is the resonant Fourier magnitude of the potential's evolution, | Ψ(ω=1)|—exactly invariant under time reversal, | Ψ(x)|=| Ψ(-x)|, a Fourier theorem confirmed numerically on the radiation-era transfer function and on a random control^P15R13.

This cosmology's driving lives on the contracting side—the progenitor's radiation-comparable collapse phase, whose potential evolution is the Friedmann time-reverse of an expanding radiation era—so its driving magnitude equals flat ΛCDM's exactly, and the branch point's faithful transmission (§8) carries it across unaltered.

The acoustic amplitude, and with it the peak-height pattern, therefore match flat ΛCDM's—by the time-reversal structure, not by tuning—conditional on the inherited medium being the ordinary photon–baryon plasma, the condition the abundance comparison confirms (§2; receipt: cr_collapse_driving.py^P15R12, the driving validated as a linear oscillator against the known radiation-era boost).

One premise that argument once carried has to be withdrawn, and withdrawing it leaves the conclusion unsupported rather than merely unproven. It was said that a geometrically fixed rate possesses no expanding radiation era at all, so that the contracting side was the only place driving could live. That census was taken on the stacking rate, and the three-level rule assigns the perturbations to the leaf — which has a matter–radiation equality at z_eq=3936 with the onset at z=6761 preceding it. So the leaf does possess an expanding radiation era, and the modes carrying the first peak are still super-horizon when the plasma starts and enter while radiation dominates.

And the transfer measures what the structural argument predicts, and does not find it. The height ratios this arm returns are P₁/P₂=1.759 and P₁/P₃=1.612 against the sky's 2.217 and 2.277 — a deficit of 20.7% and 29.2%, deepening with peak number. The heights do not match flat ΛCDM's on this construction's own instrument, and the structural argument above cannot be read as establishing that they do. What produces that pattern is not settled here, and the reason is sharper than an unanswered question. The transfer never integrates the collapse leg: that leg enters through a handover datum of three numbers per mode, and its two halves are read at different points of it — the photon amplitude carries the closed-form transfer this paper derives, and the potential is set to its primordial value. So the expanding-leg driving is re-applied from a potential the collapse leg has already spent, onto an amplitude that has not.

And the factor by which this arm is driven harder than the control is a property of that choice. Carrying the decay across to the potential does not halve the driving — it doubles it again, to five times the control's, and moves the first peak further from the sky rather than back. Reading both halves at the same point removes the source altogether and the comb above the first peak with it. Three codings, three answers, so the factor is not yet a property of the construction.

The mechanism is not thereby wrong: it remains the only account giving the direction, the ordering with peak number and the overshoot together. What is established is that this instrument cannot presently weigh it, and what would settle it is not another spectrum but a statement of what the photon perturbation and the potential both are at one locus.

Because the acoustic modes are already sub-horizon at the seam (Prop. 1), each mode's driving is complete on the collapse side—and the qualification its own receipt carries is that completeness holds for the modes whose entry precedes the horizon maximum at r_*=1.53 r_seam, entry on the rising branch being earlier the higher the wavenumber, so the acoustic band is on the safe side of that boundary by construction and the low- end is where it would bite^P15R41; the observed peaks therefore reduce to flat ΛCDM's—the driving equality above together with the shared photon–baryon plasma and the coherent seam phase—and match a Boltzmann reference digit by digit (P₁/P₂≈2.2, P₂/P₁≈0.45, P₃/P₁≈0.44; camb_reference.py^P15R14). What remains is confined to the damping tail (ℓ 1000), and the three-level rule (§2) fixes it rather than leaving it to judgement. Recombination is observable cosmology from the branch point onward, so the photon diffusion length r_D—like r_s—is an L1 quantity: the random walk accumulates against the layer's own geometric expansion, its Thomson microphysics ordinary content physics but its expansion the L1 rate, not a radiation-included one.

This is the same branch point-located scoping the cosmogenesis paper derives for the nucleosynthesis window [JanzenCosmogenesis], read here in the opposite direction: there the self-gravitating excursion sets the L2 rate radiation is included in, here the diffuse plasma rides the L1 foliation radiation is excluded from.

This is forced: it is the same L1 rate that dissolves the Hubble tension, and one may not take the rate geometric for the peak spacing and radiation-included for the diffusion. The geometric rate is ∼13% below the radiation-included one at recombination (ρ_r/ρ_m≈0.3) and further below it earlier, where the inherited radiation is a larger fraction; the diffusion length integrates that whole history rather than its endpoint, and runs ∼9% longer than ΛCDM's while r_s is matched to the observed acoustic scale (the closed-form derivation of §4 gives 10.8% on a sharper ionisation treatment, and the two are related there); the projection-independent ratio is therefore θ_D/θ_*≈1.08 (θ_D/θ_*)_ΛCDM—a 8.2% larger damping angular scale at the onset redshift used there, the value moving with z_onset as §4 sets out (receipt: damping_ratio_clean.py^P15R15, validated against a Boltzmann reference on the radiation-included rate; r_s=144.0 vs CAMB 144.4 — a 0.28% disagreement against the 8.2% signature it feeds, so the reference agrees to thirty times the effect and the margin rather than the agreement is what the word validated is carrying^R; and what a stated tolerance is carrying anywhere in this corpus's computational layer has since been measured rather than assumed—every \verb|abs(...)<tol| comparison in the receipts rewritten twice, once to displace the compared quantity above its tolerance and once to collapse it below, so that 243 of the 245 receipts that pass are shown to have at least one such comparison that actually gates their verdict, the two exceptions being a display filter and a skip that prints its own reason^RL22L22).

This is the one genuinely CR-specific piece of the height pattern: a real signature locked to the Hubble resolution by their shared L1 rate—not an artefact of laying the standard radiation-governed r_s on the geometric rate, the calculation belonging to neither framework this paper warns against (§1).

One thing about this signature is settled and one is not, and it is essential to keep them apart.

Settled: the ∼9% in r_D is genuinely CR-specific and does not dissolve under the density that would ordinarily rescale it—r_D∝ω_b^-0.31, so a percent-level ω_b (fixed by primordial deuterium and the peak-height ratio) shifts it by well under a percent (receipt: damping_reabsorption.py^P15R16). And what the signature does to the observed high-ℓ power is now measured, with a verdict that is neither of the two a reader expects. Scaling the diffusion length by the instrument's own measured ratio multiplies the damping by a Gaussian in ℓ, and a tilt is a power law — so over a restricted high-ℓ range the one is well approximated by the other. Refitting amplitude and tilt therefore absorbs the signature almost entirely in shape, leaving 0.26 per bin over the 185-bin ceiling and less the higher one looks. But the absorption is bought, not free: it costs a shift of -0.0304 in the inferred tilt against a σ of 0.00323 derived from the same covariance that scores the fit. So the signature does not vanish under a refit — it moves out of the spectrum's shape and into an inferred parameter. What a refit buys in shape it pays for in parameters, and that direction is the robust part.

And how local that approximation is can be stated exactly, which settles what a joint fit would actually meet. The signature's own logarithmic slope is d ln (ratio)/d ln ℓ=-2(r²-1)(ℓ/ℓ_D)² — not a constant, but growing as ℓ²: across the likelihood's own ℓ=30–2508 it runs from -0.0002 to -1.31, a factor of seven thousand, which is exactly (ℓ/ℓ)^2P15R3. So a single tilt is a one-parameter fit to a function whose slope varies by that factor over the data it is scored on, and the displacement quoted above is the signature's true slope at ℓ=381 and at no other multipole. The consequence is the useful part: a joint fit does not encounter a shifted tilt but a residual whose shape is a Gaussian in ℓ, and no tilt removes a Gaussian — so what such a fit weighs is the shape residual, which is the quantity already bounded from above. And a displacement in n_s is not by itself a tension here, the tilt being inherited boundary data rather than predicted: a different value of an inherited input is a different input. What could be a tension is a constraint on n_s that does not run through the damping tail — the low- and mid-ℓ shape, the polarisation spectra, lensing — which is what a joint fit means and is where the question properly sits.

Two bounds travel with that and neither can be left to a reader. The shape residual is an upper bound: freeing only the named degeneracy is the least absorption a fuller refit could achieve, so a wash is a wash a fortiori. The tilt displacement is not a bound, because more free parameters would share it — the baryon density moves the diffusion length itself. And the magnitude is not robust to a factor of three: imposing the ratio on the measured spectrum gives the figures above, while applying it inside the instrument's own source — where it also reaches the Doppler term through a derivative — gives 0.92 per bin. The larger comes from the more faithful route, so the wash is stated at the weaker bound and closing that factor is named here as work not done.

Not settled: whether a displacement of that size in the inferred tilt is a tension, which is a statement about a joint fit not performed here. The observable is θ_D/θ_*=r_D/r_s, and translating it into a spectrum requires the full acoustic transfer—above all, whether CR's high-ℓ peaks, driven on the geometric rate, track flat ΛCDM's, which is verified only through the third peak (ℓ≲800). A leading-order pass that simply assumes the high-ℓ peaks equal ΛCDM's and applies the extra damping produces a large high-ℓ deficit (receipt: full_transfer_verdict.py^P15R20)—but that assumption may not be imported for free: the same geometric rate that enlarges r_D also governs the high-ℓ driving envelope, so CR's high-ℓ spectrum rests on short-wavelength collapse-phase driving rather than on the boost an expanding radiation era supplies.

One thing about that figure should be stated because two of this corpus's instruments do not agree on it, and the disagreement is worth knowing rather than averaging. The 9% above is the observable θ_D/θ_*, computed on standard thermodynamics with the sound horizon re-anchored to the observed acoustic scale so that only the damping length moves. The two-arm transfer, measuring the raw diffusion length at the visibility peak, returns 7.55% on its own arms — a different quantity at a different epoch, and not a correction to this one. Neither number is offered as a replacement for the other, and closing the gap between the two routes is a thing this paper does not do.

That driving is computed below (§4), and the calculation removes the licence the shortcut lacked: the envelope is derived on the collapse leg rather than imported.

The two structural facts—peaks-match (verified low-ℓ) and diffusion on the geometric rate—therefore constrain, but do not by themselves determine, the high-ℓ shape; the net effect is computed in §4; whether it is a tension, a wash, or a distinctive but consistent feature turns on a parameter refit and on the early integrated Sachs–Wolfe term. That refit is performed here. Fitting both arms to the Planck 2018 plik_lite binned likelihood over its 215 bins (32≤ℓ≤2492), with the same five parameters (H₀,ω_b,ω_c,A_s,n_s) free in each and the published covariance, returns χ²=397.13 for this construction against χ²=206.44 for flat ΛCDM—1.891 and 0.983 per degree of freedom respectively, so Δχ²=190.7 at equal fitted-parameter count. The control reproduces the sky essentially perfectly, which is what makes the comparison readable at all; and the verdict it returns is that the spectrum derived here is disfavoured against the standard model on this data. We state it as it stands. What it measures is the five-parameter fit of the spectrum construction, not the geometry that motivates it, and the two are joined by the transfer work of this section rather than by identity.} What is honestly claimed here is the effect, not a verdict on it: a real, computed, non-reabsorbable ∼9% CR-specific diffusion-scale signature, its observable consequence now derived rather than entangled (§4), and the verdict on it left to a likelihood confrontation this paper does not run. The structural point that stands is the mechanism: where inflation freezes the phase dynamically and drives the amplitudes on an expanding radiation era, CR fixes the phase geometrically on the single null layer and drives the amplitudes on the time-mirrored collapse.

The collapse-phase driving envelope

The driving named above can be carried out in closed form, and doing so removes the entanglement the previous subsection had to leave standing. On the radiation-dominated collapse leg the potential obeys Ψ”+(4/η)Ψ'+(k²/3)Ψ=0, whose regular solution is elementary, Ψ=3Ψ_i( sin x-x cos x)/x³ with x=kη/√3, and which is even in ^P15R40—so the contracting leg carries the same closed form pointwise, a sharper statement than the Fourier-magnitude equality of §4 and consistent with it. Writing =₀+Ψ removes the source exactly, ”+(k²/3) =0: the effective temperature oscillates freely, and the -dependence rides entirely in the amplitude at horizon entry and in the decay of Ψ.

Horizon entry is fixed by the construction rather than fitted. The comoving Hubble radius is built from the slicing paper's own turnaround function, (rH)²=(1-f)+A/r² [JanzenSlicing], and has a single turning point on the whole excursion, at r_*=1.53 r_seam on the corpus's inherited datum; the seam sits just inside that turning point, on the rising branch, so the acoustic modes are already within the comoving horizon there—which is the geometric content of Prop. 1 rather than a numerical accident of z_onset^P15R41. The locus matters and the two available ones give opposite answers: the branch point lies at the far end of the same rising branch, where 2M/r→∞ carries up without bound and the comoving horizon to zero, so there every mode is outside it (§3). It is the seam, not the crossing, that this construction reads—the inversion derived from the metric function rather than asserted^P15R72. On the leg the radiation term dominates because it is constant while the others vanish, so η=r/√A to about two per cent at the entry radius of the lowest mode in question and better the deeper the mode enters—sub-percent from k 30 k_s^P15R42, and entry then occurs at x→1/√3 in the deep sub-horizon limit—the radiation era's own scale invariance, recovered here from the substrate geometry. The envelope is therefore flat to the accuracy of that limit: a mode at k≃10 k_s enters within 2% of 1/√3 and one at 300 k_s within 0.1%, all of them reaching the seam—where the collapse leg ends—with the same driving amplitude Ψ(1/√3)/2, and the residual potential is below a percent on average while oscillating up to some seven per cent of it where cos x_seam is near an extremum^P15R43. Both are qualifications rather than identities: the limit is a limit and the average an average. Its turnover sits at equality, and the inherited datum places equality at 1+z_eq=(1+z_onset)/2=3399, essentially flat ΛCDM's—so the envelope turns over at the same wavenumber, for a structural reason and not by assumption^P15R46.

Two corrections that might have been expected do not enter. The baryon loading is proportional to the scale factor and the driving happens where the scale factor is smallest, so R≲0.1 at the onset and orders below that at entry; the odd/even asymmetry is imprinted afterwards, on the expansion side at R≈0.6, which is ordinary content physics on the observable leg. The sky's own value, located by parabolic fit on the Planck binned spectrum, is P₁/P₂=2.2564±0.0772, and flat ΛCDM's is 2.200; what this construction returns for the same quantity waits on the transfer of §4, since a peak height is exactly what an unconverged wavenumber integral misreports}^P15R44. The structural point stands without it: the loading is the same in both because it is imprinted on the expansion side, and the argument is at the level of contents rather than of the driving—where the loading ratio alone would give 3.5.} And the free-streaming neutrinos the cooling-leg nucleosynthesis requires do carry anisotropic stress at these temperatures—the branch point is far below their decoupling—but their deep sub-horizon effect is a constant phase shift and a constant suppression, and a -independent correction leaves a flat envelope flat^P15R45. What it does not leave is the closed form for Ψ itself, the equation becoming integro-differential; the closure is a property of the skeleton, and the flatness is what survives.

What the envelope buys, and what remains

With the envelope derived rather than assumed, the diffusion signature can be stated without the entanglement. The diffusion length reduces to a single integral over the scale factor in which every microphysical constant sits outside, 1/k_D²∝∫da g(R)/(Hx_e), so in the ratio the Thomson physics and the ionisation history cancel identically and the whole difference is carried by H(a)—which is the division of labour this section has argued for throughout. Both integrals close in elementary form, and on the inherited datum the geometric rate gives a diffusion length 10.8% longer^P15R47. That is the closed-form value and it stands beside the 9% quoted above rather than replacing it: the earlier figure runs the ionisation history through a Boltzmann-validated recombination, this one takes x_e as a sharp cut at a_rec and carries the inherited datum unchanged to recombination. The two are independent routes to one quantity and agree to about a point and a half, which is the relationship an elementary derivation should bear to a code-validated one. The sound horizon is the same kind of integral, and it must be taken from the branch point: there is no observable expansion below it. Doing so returns r_s=146.4 Mpc against 145.4 on the radiation-included rate—within 0.7% of each other at the onset redshift the inherited datum fixes, the exact crossing sitting a little lower at z≃6.7×10^3P15R48—the acoustic-scale calibration, met from the other end and to that accuracy rather than exactly. Both are the elementary integral; carried through a full recombination history and validated against a Boltzmann reference the same two come out at 144.0 and 144.4, a per cent lower and in the same relation^P15R15. The observable, in which the common distance cancels, is then θ_D/θ_* larger by 8.2% at this onset redshift, and ℓ_*=πD_M/r_s=302.2 against the measured 301.76^P15R37.

The high-ℓ consequence follows with no free parameter: C_ℓ^CR/C_ℓ^ΛCDM = exp [-(ℓ/ℓ_D)²(r²-1)] with r=θ_D/θ_*=1.082—the same ratio the preceding paragraph derives, and nothing else—so the ratio is 0.84 at ℓ_D and 0.68 at 1.5 ℓ_D^R. Whether that is a tension is a separate question this paper does not settle: a confrontation refits the cosmological parameters, which is a likelihood analysis and not a closed-form calculation, and asserting exclusion from these numbers would repeat, at one remove, the error of holding everything else fixed while varying one thing. What has changed is the standing of the signature. The signature is a requirement of the construction, following from the inherited datum and the level assignment and nothing else.

The early integrated Sachs–Wolfe contribution is present here, and the step that once denied it is worth setting out because the denial was an inference and not a measurement. The rate at recombination is matter-dominated to nine orders, the substrate term being utterly negligible there^P15R39: that is true, and it is a statement about the rate. The geometric rate carries no radiation term, radiation being inherited content read off the clock rather than something that sources the rate.

But the effect is driven by the radiation content through the perturbation source, not by the rate. The content enters the source under either rate assignment, and at recombination this cosmology's content is 21.7% radiation, ρ_r/ρ_m=0.2765 at z=1088 — which no statement about the rate's composition bears on. So the inference from a radiationless rate to an absent term does not hold, and the term is present.

Measured, as the integrated |e^-τ(Φ'+Ψ')| after recombination in units of the Sachs–Wolfe amplitude there: 0.348 on the leaf congruence the rate rule assigns the perturbations to, and 0.229 on the stacking rate. Non-zero on both. The leaf assignment makes the term half again larger; it does not create it.

One scope qualification is owed here, and it narrows the claim without touching the conclusion. The constancy argument runs on Φ”+3 HΦ'=0, which is the super-horizon equation: it drops the k²Φ term. For the acoustic modes—inside the horizon at the onset and remaining so—that term does not vanish, and the potential does decay on the observable leg, by a factor of order two across the first few peaks^P15R38. The decay is not a radiation effect: zeroing the radiation fractions in the constraint makes it larger, and the rate responsible is k²/(3 H), which grows with and refers to no content at all. So the statement that survives is that the substrate rate supplies no early integrated Sachs–Wolfe term—which is what the leftward reading delivers and what the comparison with flat ΛCDM turns on—while the sub-horizon potential decay is an ordinary consequence of the perturbation's own scale, present in any cosmology.

The missing contribution cannot offset the diffusion signature, and the reason is structural rather than numerical. It sits near the first peaks and dies away well before the damping tail begins, so the two live at opposite ends of the spectrum; and the first peak is where the amplitude is anchored, by an A_s this construction inherits rather than predicts. A few per cent there is absorbed by that inherited normalisation, exactly as it would be in a framework carrying η from outside. The diffusion signature is not so absorbed: it is a change of shape at high multipole, and rescaling the amplitude to restore the first peak would worsen the tail rather than repair it. What stands between the derivation and a verdict is therefore the likelihood confrontation alone—a parameter refit, not a further calculation of this kind, and one that waits on the transfer of §4.

The diffusion scale, and the error budget

The acoustic scale is met at the directly measured H₀; the diffusion scale is not free to follow it. Holding ℓ_* to its measured value fixes the onset redshift, and θ_D/θ_* then follows with nothing left to adjust: it varies from +43% to -3% across the onset redshifts one might consider, so a single datum cannot absorb both observables. One inherited number, two observables, and the first is spent on the first. Run at the directly measured H₀=73, the matter density this construction's own baryon-acoustic fit returns, its own light-element baryon density, and the measured photon temperature—with the acoustic angle the single microwave-background input—the seam falls at z≃6.8×10³ and

That figure and the 8.2% of §4 are the same observable computed at two onset redshifts, and the difference between them is the honest measure of what this signature is currently worth.} The two differ by 0.53% in z_onset and by 5.8 points in θ_D/θ_*, which is the sensitivity the range above already reports—so the quantity is determined only once z_onset is, and z_onset is the one number this cosmology fits. The signature is therefore a computed, non-reabsorbable effect of definite sign and of a magnitude the acoustic-angle fit has still to pin: it does not float with H₀, both r_D and r_s carrying 1/H₀ so that the ratio is H₀-independent, and it does not dissolve under ω_b; what it awaits is the same refit the high-ℓ verdict awaits. We quote the two values rather than choosing between them, and no claim below rests on which is right.

Two contributions that might have been expected do not enter: the last-scattering surface's thickness scales with the rate in both its width and its conversion to comoving distance, so it cancels from the ratio; and the recombination redshift, integrated on this rate through the Peebles equation [Peebles1968] rather than inherited, moves by only -0.11%^P15R52. The same integration returns z_*=1091 on the radiation-included rate, against the 1090 of the standard treatment, which is the check that the ionisation history is being handled correctly.

The integrated Sachs–Wolfe contribution across the first-peak region

The first acoustic peak is a separate matter and it does not move. Its offset from ℓ_*—220 against 301.7—is largely the phase shift the potential's decay imparts while the modes are driven, and that decay has the same closed form on the collapse leg as on an expanding radiation era (§4), so it transfers. What does not transfer is the early integrated Sachs–Wolfe contribution, and that turns out to be a nearly flat multiplicative factor across the first-peak region—some ten per cent in amplitude, varying by two parts in a hundred between ℓ=150 and ℓ=300—so it displaces the maximum by one or two multipoles rather than by tens^P15R50. The first peak therefore sits where it is measured, and for a reason internal to the construction rather than by accommodation. That refit can be bounded here, and bounding it is worth doing because the reabsorption question is the whole of what a likelihood would decide about this signature. Pinning ℓ_* to the measured 301 and minimising the residual in θ_D/θ_* over the baryon density and the matter density—the former held to the one per cent primordial deuterium allows, the latter to the five per cent the distance and growth data allow—leaves a residual of several per cent ^P15R51. The baryon density cannot absorb it, having one per cent of freedom against an effect of order ten per cent at either onset (§4); the matter density can, but only by moving some thirty-seven per cent, which is not a refit but a different cosmology. So the shift has a floor, and the floor is the prediction. We state this as a bound and not as a likelihood: a full comparison fits the whole spectrum with every parameter and its covariances, where this fixes one observable and minimises one derived ratio. It settles that the signature does not dissolve under reabsorption; it does not settle the verdict.

What the instrument's residual is made of

The two-arm comparison's floor is set by components of the instrument rather than by either cosmology. Two candidates for the accuracy it lacks are measured here.

Of the two, one cannot pay at all^P15R66L23. Above ℓ≃40 the whole effect of reionisation on the temperature spectrum is the constant screening e^-2τ—the second visibility bump contributes only below ℓ≃30, outside this comparison's range—and the comparison fits one amplitude in closed form. An amplitude fitted as A=(m^TFd)/(m^TFm) is homogeneous of degree -1 in the model, so rescaling the model returns A/s and leaves the residual unchanged: the degeneracy is exact rather than approximate, and it is a property of the statistic rather than of the size of τ. Checked at τ=0.054, 0.10 and 0.30, the change in χ² is zero to machine precision at all three. Lensing is a different matter. A smoothing in multipole, scanned in width, takes the control from 1320 to 921; and a width that grows with ℓ beats a constant one by a factor of 1.8 in the improvement, each compared at its own optimum, so the preference is for a shape and not merely for a size—which is lensing's own signature, since a deflection of fixed angle subtends a larger fraction of an acoustic wavelength at high ℓ. We state what that is and is not. The width is fitted, so the improvement bounds what a derived calculation has to play for rather than measuring a lensing amplitude; and even at its own optimum some 70% of the residual survives it. So the lensed spectrum is the next thing this instrument owes, and it is not the last one.

That calculation is done, and it comes in under the bound the fit set. Applying ΛCDM's own lensed-to-unlensed ratio to the arm—a derived operation with no free width—takes the control from 1320 to 989, an improvement of 331, concentrated in the damping tail where a fixed-angle deflection fills the troughs. That is below the 400 the fitted width bounded, as a zero-parameter derivation must be, and the acoustic peaks do not move under it. The corpus's own first-order kernel overshoots the bound to 508 by breaking down in that same tail—returning a spurious +13% at ℓ=1900 where the full operator gives +6.5%—which is why the derived number is the full operator's. Most of the fitted headroom is recovered by a genuine lensing, and about 989 of the 1320 survives it: the verdict of the last paragraph stands from the derived side, lensing real and not the last build^P15R67.

And what the residual is MADE of was decomposed alongside it, which is what let the size of that gain be anticipated rather than discovered^P15R69. Projecting named templates through the same inverse covariance the comparison uses, the peak positions account for a tenth of a per cent of the residual—which is why the acoustic peaks do not move under the lensing above, and the two findings corroborate each other—while the peak-trough contrast at fixed position accounts for two-fifths of it, this transfer's contrast running some thirteen per cent too high. Reducing contrast without moving peaks is what lensing does. A smooth multiplicative envelope accounts for a further sixth, and about half the residual is in neither—so a complete lensing calculation was expected to leave a comparable amount behind, and it did.

The shear coefficient, derived

The coefficient is derived rather than fitted, and the derivation settles it against the fit^P15R64. The coefficient is measured by running the same oscillator twice on the same background: once as the perfect tight-coupled fluid, which is undamped by construction, and once as a photon hierarchy carried to ℓ=24 with Thomson scattering, the polarisation multipoles evolved alongside and the baryons given their own velocity. Their ratio is the damping and nothing else—the slowly varying prefactor, the drift of the sound speed and the baryon-loading dependence of the oscillation are common to the two runs and divide out identically—and reading it locally, between consecutive extrema, lets the residual be extrapolated in (k/τ')² rather than smeared over an interval. With the polarisation multipoles suppressed the quadrupole's scattering term reduces to -9/10τ'F₂, which is the unpolarised closure, and the coefficient comes back as 0.8852 against 8/9; with them carried it is 1.0618 against 16/15. The polarisation correction itself, which is the quantity in dispute, is a factor 1.1996 against 6/5: three parts in ten thousand. Both absolute values sit about half a per cent low by an amount proportional to the coefficient, so the residual is a common normalisation and cancels in the ratio.

The acoustic comparison

Two features of this cosmology's acoustic scale are worth separating, because only one of them is a prediction. The angular scale ℓ_*=πD_M/r_s is independent of H₀ here: both lengths scale as 1/H₀ on a matter-and-substrate rate, so the ratio does not know about it. In flat ΛCDM there is no free early datum—the sound horizon is fixed by the matter and baryon densities—so the measured angle does determine H₀, and that is the origin of the lower value inferred from the microwave background. Here the branch point's radiation amplitude takes that constraint instead, and H₀ is left to be the directly measured one^P15R49. So the acoustic scale is an accommodation, and it is spent.

The seam datum carries two freedoms, and how far the first peak moves with them is measured. The datum §4 states is one amplitude per mode and a common phase; it fixes that the phase is common without fixing which phase it is. A second freedom sits beside it: what the amplitude is “flat in wavenumber” at, the construction reading its transfer shape at each mode's own horizon crossing where an alternative reads it at each mode's phase at the seam. Read at converged wavenumber on the leaf congruence, across the seventeen readings admitted by a four-peak criterion fixed before the numbers, the first peak spans 148 to 228 — a factor of 1.541, and across the fifteen numeric-phase readings alone 1.118. So the position is not a free statement about the datum: the span is narrow enough that the reading is a statement of the construction, and the sky's 220.6 lies inside it.

And the same is true of the other three acoustic statistics, which is the weaker fact and must be said as the weaker one. The spacing spans 276.0 to 330.4 against the sky's 294.6, the phase intercept -0.4044 to -0.2161 against -0.2253, and the alternation ratio 0.475 to 1.088 against 0.856: the sky is inside the span on each statistic separately. A span containing the sky on each statistic one at a time is much weaker than a reading that reproduces the sky, and there is no such reading.

What the readings do carry is a correlation, and it is the sharpest thing this scan returns. Position and alternation move against each other across the datum: Spearman ρ=+0.782 at p=2.1×10^-4 between the first peak and the gap ratio, so as the datum carries the peak up toward the sky's value the second gap turns from contracting to expanding. Of the six readings landing the first peak within one grid step of the sky's, not one contracts; of the seven that do contract, every one sits at ℓ₁≤212. So the seam datum can buy the position or the alternation and not both, and no mechanism for that trade is in hand.

The instrument that would measure the rest is built, and its guard and its control are what make it an arbiter. Both arms run on one set of equations, so no comparison between them is a difference of machinery^P15R56. The guard first, because it is what licenses the rest. With every coupling to the potential removed at every site—the 4Φ' in the photon, neutrino and cold-matter continuity equations and the k²Ψ in the photon–baryon and neutrino Euler equations—the source comb lands on the integers on both arms. So the sound horizon, the measure, the wavenumber grid and the phase extraction are validated as an ensemble, and every departure from nπ/r_s in a driven run is the driving and nothing else.

The control is the arm whose answer is known independently, and it reproduces it^{P15R65P15R100}. The photons are carried as a Boltzmann hierarchy with the polarisation multipoles alongside, the baryons given their own velocity and density contrast, and the source carrying g /4 and (3/4k²) d²(g)/dη². Run on the flat-ΛCDM arm it returns peaks at 220, 540 and 812 against the sky's 220.6, 538.1 and 809.8, and a first-to-second height ratio of 2.1969 against a standard code's 2.200—agreement to 0.14%, and within the sky's own uncertainty.

Two conditions on how such a transfer is run are what that agreement rests on, and the tighter of them is numerical. The projection must carry the polarisation source, which a damping envelope cannot supply at any coefficient; and the wavenumber integral must be carried past the highest multipole reported, since ∫P(k) Δ_ℓ(k)² dk is not converged there. Each is worth a factor on its own and neither alone reaches the answer: the height ratio runs from 2.721 at an ℓ-equivalent cutoff of 900 to 2.393 at 2400, so a cutoff set at the reported multipole is a systematic and not a rounding.

The perturbation sector runs on the leaf congruence, which is what the framework assigns it. A process running in the content—the sound horizon, the diffusion length, recombination, the perturbations—takes the leaf's rate (§2), while the comoving separations and the projection keep the stacking rate; the change is an exact chain rule between two monotone conformal times, so every spline and grid is untouched, and on the flat-ΛCDM arm the two rates are identical and the assignment is provably a no-op there—which is its own self-check.

The comparison is made on the polarisation path, in the configuration the control is calibrated in. There the control returns a first-to-second height ratio of 2.195 at peaks 220, 536 and 814, against a standard code's 2.200: so the instrument is calibrated where this cosmology's number is taken^P15R56P15R61. Both arms are converged in the wavenumber integral. The control's own movement between cutoffs is 0.000% and 0.046% on the two height ratios, and this cosmology's arm does not move at all—0.000% and 0.000%—so the arm is converged at the lower cutoff and the figures below are not a truncation artefact. The same comparison on the fluid path reads 0.042% and 0.036% against 0.000% and 0.045%: both paths converge, and the two sets of figures are not interchangeable. A rung below that was refused outright by the instrument's own truncation guard, on both arms, which is data and not an error.

On that path this cosmology's arm returns the following, and the comparison does not come out even. The peaks fall at 206, 528, 832 and 1196, with ℓ₁/ℓ_A=0.6830 against the sky's 0.7312 —a deficit of 6.6% in position—and height ratios P₁/P₂=1.759 and P₁/P₃=1.612 against the sky's 2.217 and 2.277. Two wavenumber cutoffs agree to every digit. And the positions are converged in the reported multipole grid, which is not free: they are read on a grid of step 2, and coarsening it to the instrument's default of 8 moves this arm's first peak by two multipoles while leaving the control's unmoved, so the grid was quantising one arm and not the other. Refining to step 1 moves nothing at all — 206, 528, 832, 1196 to every digit.

And the deficit is neither of the two instrument faults a reader would reasonably suspect, because both are removed in that configuration: the wavenumber truncation is carried past the reported multipole, and the projection carries the polarisation source rather than a damping envelope^P15R63. And the truncation's removal is measured on the arm whose answer is known: it was 78% of the control's remaining χ², and taking it out brings the control to 1.18 per degree of freedom over the full range against a reference 1.01, while this arm does not move across four instrument states^P15R82. Nor is the height pattern an instrument artefact: putting the diffusion damping back — derived on each arm's own opacity and rate, neither imported nor fitted — brings the heights from 13–34% to 2–3% and drops the likelihood's floor by a factor of five without the verdict turning over^P15R62. The polarisation source moreover pulls both arms down by comparable amounts—the control by 8.2% and 20.8% on the two ratios, this arm by 10.9% and 26.9%—landing the control on the sky because the control was above it and carrying this arm further below because it was already there. Nothing about the operation differs between the arms, which is what makes the residual attributable to the source rather than to the machinery.

Two likelihood configurations are quoted in this section and they are not interchangeable, so which is which is stated rather than left to be inferred. The full-range lensed one, on 185 bins, is the better-converged: it is where the control reaches 1.18 per degree of freedom against a reference 1.01, and it is the configuration in which an instrument fault can be told from a physical one. The 133-bin unlensed one is where the shape rejection is quoted, and its control sits at 2.10 per bin because the ceiling cuts the damping tail. The second is not a worse measurement of the first's quantity: it is a different configuration, chosen for the comparison it makes, and the two sit beside each other rather than one superseding the other.

The driving's size on the observable leg is measured rather than inferred, and its wavenumber dependence is known in closed form—the -dependence living in the potential's gradient rather than in its decay, with the turnover at a fixed multiple of the horizon scale^P15R58P15R59. Removing it moves this arm's first peak from 206 to 340, and ℓ₁/ℓ_A from 0.6830 to 1.1273: the modes carrying the first peak are driven, and by 134 multipoles' worth^P15R70. Run on the control the same subtraction moves 220 to 276, a shift of 0.1858 in ℓ₁/ℓ_A against this arm's 0.4443—so the arm is not driven less than the control but 2.4 times as much, and undriven its first peak already sits 23% higher, at 1.1273 against 0.9158. The arm therefore overshoots rather than falls short: the driving carries it from above the control's undriven position to below the control's driven one, past the sky. (Both subtractions are taken on the polarisation path at the lower cutoff; the fluid path gives the same ratio, 2.43 against 2.39.) And the likelihood is run, both arms scored on the same bins by the same code, with the amplitude fitted so that what is compared is shape. The result is a rejection of this arm's spectrum and is stated as one. Over 133 bins of Planck TT the control returns χ²=279.4, or 2.10 per bin; this arm returns χ²=15752.0, or 118.4 per bin^P15R60. The separation between the two models, measured as a distance between them rather than as a difference of two numbers each taken against the sky, is 122 per bin — eleven standard deviations per bin^P15R85.

The configuration limits the absolute values and does not account for the gap. The control sits at 2.10 per bin rather than unity because the multipole ceiling cuts the damping tail on an unlensed spectrum. What is not configuration-limited is the ratio: the separation between the arms is 58 times the control's own distance from the data on the same bins. And the verdict does not turn on which measure is used — F₂ and the separation statistic agree to within 5%, at 15473 against 16261. Nor is it a truncation artefact: cutting at four-fifths of the ceiling the control reads 1.69 per bin over 104 bins and this arm 82.05, so the disagreement is large at both cutoffs^P15R60. So the shortfall is not the configuration: it is the 6.6% position deficit and the height ratios of §4, carried across every bin.

What that leaves is a measured deficit with no mechanism. The account that would have covered it—that the acoustic modes re-enter above the plasma's onset so that none is driven—is taken on the stacking rate, and the rate rule assigns the perturbations to the leaf: on the leaf the band 155.6 lt;ℓlt;237.7, which contains this arm's first peak, enters the horizon while radiation dominates, the leaf background carrying a matter–radiation equality at z_eq=3936 that the onset at z=6761 precedes^P15R17.

Two things this section does not claim, and they are separated because a reader would otherwise supply them. The alternation figures above are read on a multipole grid whose step is the same size as the sky's own gap contraction is large, so the failure to contract at the sky's position is resolved while the small contractions at lower ℓ are not^P15R71; a finer grid is what would settle those. And the driving's size is a measurement of the effect, not an account of the deficit: what remains open is a mechanism, and §10 carries it as such.}

What the substrate determines: the amplitude floor and the classical character

One might ask whether the observed fluctuation amplitude is the substrate's own de Sitter vacuum—the natural CR analogue of the inflationary spectrum-is-the-stretched-vacuum story. It is not, by a decisive margin.

One scope qualification is owed immediately, and closing it is what makes the classical reading a result rather than an inference. Proposition 2 rules out the substrate's vacuum. But the reading it supports locates the fluctuations in the progenitor, and the progenitor's own vacuum is a different quantity: it is normalised at the collapse leg's curvature scale, not at H_Λ, and it is amplified on its way here by the branch point's mode mixing. Neither effect is small, and neither is addressed above. Both can be computed, because the collapse-side background is a single function: with A=2M and ρ=2√B/A the exact closed dust-plus-radiation interior gives a=M(ρ|σ|+σ²/2) near the crunch, whose Mukhanov–Sasaki potential z”/z=2/(|σ|(|σ|+2ρ)) is independent of the mass—so enters only through the final division ζ=v/a. The passage then has exactly three stages, and each contributes one factor of 1/ρ. A mode leaves the sound horizon on the β=2 leg and freezes as v→D_k/|σ| with the scale-invariant D_k=√ℏ/2 k^-3/2; the connection through equality is elementary, since v=a∫dσ/a² solves the k=0 problem in closed form and carries D_k/|σ| to the constant (3/2ρ)D_k at the crunch; the crunch monodromy then leaks that constant into the survivor with coefficient 4π/ρ—twice the tensor value, because the true hydrodynamical z_S=a (a+4B/3A)/a' has z_S”/z_S→2/(ρx) where a”/a→1/(ρx)—the companion paper's numerically obtained -2πi/ρ [JanzenCosmogenesis] being the tensor value of the same object, recovered here in closed form as the continuation of the (σ/ρ) ln σ term the (0,1) resonance forces; and the survivor's curvature is ζ=C/(Mρ), the last 1/ρ being the crunch's own slope a'(η_c)=-Mρ. The result is a transfer law with no free constant,

which is exactly scale-invariant on vacuum data—the k³ cancels the k^-3 identically—and which therefore delivers, for the vacuum, A_s=9(ℓ_P/M)²ρ^-6 (receipt: P15_the_progenitor_vacuum_is_negligible_too.py^P15R80, the connection coefficient confirmed against integration from vacuum data to 0.1% over a decade in ρ). One check is owed here and is not optional, because the hydrodynamical variable differs from in three places and not one: z_S∝a^3/2 in the matter era, so the exponents there are (-2,3) against 's (-1,2); the crunch residue doubles; and the final division is by z_S rather than by . Propagating the same physical initial condition under each potential returns a ratio of exactly 2 in amplitude at every composition tested—the matter-era exponents and the change of divisor cancel, and the crunch residue alone survives. So the z∝a chain is corrected by one factor rather than rebuilt, and Eq. (8) is the true-variable law.

So eighteen orders separate the substrate's vacuum from the progenitor's amplified one—the mixing buying 10^8.6 over the progenitor's own bare vacuum, which itself sits 10¹⁰ above the substrate's—and it is still short by a hundred and three, and the classical character of the primordial statistics is now established for the source that actually supplies them rather than only for the one that does not. Two consequences of character, not of number, follow.

First, the observed amplitude is inherited classical content—boundary data from the progenitor, not a quantum fluctuation generated within the Schwarzschild–de Sitter era, and now not one generated in the progenitor's collapse either.

Second, there is no inflationary consistency relation: r=-8n_t presupposes that scalar and tensor spectra are one vacuum stretched by one expansion, and with the amplitude not vacuum-sourced that link is absent; the substrate tensor floor is the same ∼10^-122, so there are no substrate-sourced primordial -modes, and any observed tensors would themselves be inherited content rather than a measure of a substrate inflationary scale. And the tensor statement can be made unconditionally, which the scalar one cannot. Eq. (8) was derived with z∝a, which for a fluid holds only where and c_s are constant—across the progenitor's own equality neither is.

For tensors that restriction is absent: z_T=aM_Pl/2 exactly, for any content, any equation of state and any sound speed, so z_T”/z_T=a”/a identically and the entire passage—background, connection, monodromy—applies verbatim with nothing assumed. It returns

evaluated at the recollapse cap and the largest composition the spectrum permits, against an observational ceiling of ∼7×10^-11: below it by a hundred orders (receipt: P15_no_primordial_B_modes_unconditionally.py^P15R81). So the absence of primordial -modes is a prediction of this construction at any conceivable sensitivity, and it does not rest on the substrate's own floor—the floor argument, like its scalar twin, ruled out the substrate's vacuum and said nothing about the progenitor's. What the crossing does to the ratio is likewise fixed and free of : every factor of the passage is shared by the two sectors, so r_out=6 r_in: the passage alone would carry the ratio by 24, and the scalar mixing being twice the tensor's divides that by four in the power, and the observed bound becomes a statement about the parent, namely that its own tensor-to-scalar ratio was below 5×10^-3. An observed tensor signal would therefore not falsify the construction but measure the previous generation, which is the same reading the amplitude receives. The substrate fixes that the fluctuations are classical, inherited, and not governed by a vacuum consistency relation, and leaves their size to the handover. What it does fix is the multiplier. Eq. (8) holds for any incoming growing-mode amplitude, quantum or classical, so the crossing determines how much an amplitude is magnified without determining the amplitude—which is this construction's one-constant character read at the level of the perturbations rather than of the couplings. Two corollaries follow and are worth stating because each closes a natural hope in one direction and opens a statement in the other. A_s does not measure the progenitor's mass: with D_k free, the relation cannot be inverted, and the mass must be reached, if at all, by a route that does not run through the amplitude. And the classical datum has a size: since A_s∝D_k², the parent's growing-mode amplitude must have exceeded its own vacuum's by ∼10⁴⁷ at the top of the permitted composition window—so this cosmology does not merely decline to predict A_s, it says the previous generation's perturbations were enormously super-vacuum, which is a claim about the parent and not about us.

What the substrate determines: the throat and isotropization

At the branch point the near-horizon geometry is fixed, and what the perturbations cross is fixed with it.

A field on dS₂×S² (Fig. 3) decomposes in S² harmonics; the degree-ℓ harmonic becomes a dS₂ field of effective mass m²=ℓ(ℓ+1)/r_N², i.e. m²/H_{dS_2}²=ℓ(ℓ+1). With the dS₂ index ν²= 14-m²/H², the monopole ℓ=0 has ν= 12—a scale-invariant base—while every ℓ≥1 has ℓ(ℓ+1)≥2, hence ν² lt;0: the heavy principal series, which oscillate and decay through the throat. We read this as an angular no-hair, and the reading is grounded rather than asserted: the dS₂ factor of the throat is a positive-Λ epoch, and the de Sitter no-hair theorem [Wald1983] damps exactly the anisotropic content—every ℓ≥1 harmonic is heavy principal-series, oscillating and decaying, while only the isotropic monopole ℓ=0 survives as the ν= 12 base—so the throat isotropizes what passes through it (receipts: verify_throat_tower.py^P15R33). The geometry is established (Proposition 4) and the mode tower follows from it; the no-hair reading is the standard theorem read on the throat, and we flag one guard as load-bearing: the index ℓ of this throat tower is the S²-harmonic degree of the near-horizon geometry at areal radius 1/√Λ, and is not the observable microwave-background multipole, which lives on the last-scattering sphere of radius D_C. The map between them is §7, not an identity; conflating them would manufacture a prediction the geometry does not make.

Large scales: the flat/discrete decoupling and the low-multipole floor

Here the construction makes its sharpest large-scale statement, and the load-bearing piece is established.

The decoupling follows. The distance slicing is the flat constant-τ one—Ω_k=0, no curvature term in the redshift–distance relation—while the cosmological layers are the closed S³ of constant τ=τ+χ (§2). The non-synchrony τ=τ+χ is the decoupling itself: flat distances and a closed S³ of comoving worldlines coexist because they are different slicings of one geometry. A literal closed-Friedmann reading would put Ω_k=-Ω_Λ≈-0.685 into the distance relation and be excluded; CR avoids that not by tuning but because its distance slicing is the flat one, the curvature living on the orthogonal layers. Spatial curvature and dark energy are one Λ read on two slicings—the corpus identity, here at the observational rung.

The low-multipole floor. The closed S³ carries a discrete mode spectrum, labelled by integer degree , with comoving wavenumber k_L=√L(L+2)/r₀ and the lowest physical mode the quadrupole L=2 (the monopole L=0 is the background, the dipole L=1 pure gauge). The decoupling fixes how these project to the sky, and the fixing is the whole point. Because the distance slicing is flat (Proposition 5), the photons are projected through the flat geometry—the comoving angular-diameter distance is D_M=D_C, the same flat-ΛCDM observable that places the acoustic scale at ℓ_A≈301—while only the source modes carry the closed-S³ quantization. The transfer is therefore the discrete closed-S³ spectrum projected through the flat spherical Bessel j_ℓ(k_L D_C), not the hyperspherical transfer of a literal closed universe: that transfer carries the closed distance relation, which CR does not have, and would (wrongly, here) deliver the lowest mode to the quadrupole and no deficit. The flat projection places the discrete modes at

with r₀ the present S³ areal radius and D_C the comoving distance to last scattering, both fixed without new parameters: D_C≈13927 Mpc is the flat-ΛCDM observable and r₀≈5064 Mpc is the Nariai amplitude 2^1/3/√Λ at the present epoch u= arcsinh√Ω_Λ/Ω_m≈1.18. The stretch factor D_C/r₀≈2.75—the ratio of the flat projection distance to the curvature radius, the decoupling made numerical—carries the lowest mode to ℓ₂≈7.8, with no source below it.

The full flat-projection transfer confirms the estimate and sharpens it into a prediction. Carrying the discrete spectrum through j_ℓ(k_L D_C) with the scale-invariant measure—the first-principles closed-S³ Harrison–Zel'dovich weight [Harrison1970, Zeldovich1972] w_L=(L+1)/(L(L+2)), fixed as the mode degeneracy β² (with β=L+1) times the per-mode power 1/(β(β²-1)) that equal power per logarithmic interval in requires, and reducing to the continuum dk/k at large so that ℓ(ℓ+1)C_ℓ returns to the flat plateau as D_C/r₀→0 (receipt: verify_lowell_exact_measure.py^P15R22)—yields a low-multipole deficit: ℓ(ℓ+1)C_ℓ falls below its high-ℓ plateau for ℓ≲7 and recovers by ℓ≈8, because the flat projection of a discrete source with a hard lowest mode at k₂ leaves the lowest multipoles starved (receipts: verify_closedS3_nonsync.py^P15R21).

The one effect that partly fills it is the late integrated Sachs–Wolfe term [SachsWolfe1967], sourced near the observer where small line-of-sight distances send wavenumbers k k₂—above the floor—down to ℓ∼2.

By the decomposition of the empirical-forcing paper [JanzenModernParallax] the cumulative term is the standard Sachs–Wolfe and integrated Sachs–Wolfe—CR's own differential-expansion floor being zero under uniform expansion—so this late boost rides the same modes the discrete source keeps, and the deficit is milder than the bare Sachs–Wolfe starvation alone.

The depth is a matter for a genuine Boltzmann transfer rather than for the geometry, and the reason is that the deficit is two effects and not one. In the pure Sachs–Wolfe limit the suppression splits exactly into a floor—the projection integral truncated below k₂—and a ladder coarseness, the discrete sum minus that same truncated integral. At ℓ=5 and 6 the two are comparable and pull opposite ways: the ladder puts back power a pure cut-off removes. And they respond to the fitted parameter with opposite signs—under ±2% in r₀ the floor moves by a few per cent while the full result moves by tens—so the location is the floor and is robust for that reason, and the depth is the ladder and is volatile for that reason^R. The recovery multipole is the ladder's too: with it the spectrum recovers by ℓ≈8, on the floor alone not until ℓ≈10. Because the only CR modifications are the discrete source and the flat projection—every transfer ingredient (Sachs–Wolfe, early and late integrated Sachs–Wolfe, Doppler) shared with flat-ΛCDM [JanzenModernParallax]—the exact CMB temperature transfer Δ_ℓ(k) of a standard Boltzmann code is CR's transfer, and the discrete spectrum is read off it: C_ℓ^CR=_L≥2w_L Δ_ℓ(k_L)² against the continuum ∫d ln k Δ_ℓ(k)², the measure w_L being exactly d ln k_L/d L so the CR sum is the L=2-floored Riemann sum of the very integral ΛCDM does continuously.

Carried out with the exact transfer (gate-validated: the continuum reproduces the code's own C_ℓ to four figures; and the r₀ sensitivity is measured rather than assumed: the shape is stable—the minimum stays at ℓ=4 and the recovery multipole moves by 0.1%—while the depths drift by up to 15% at ℓ=4 under ±2% in r₀), the deficit is milder than a Sachs–Wolfe-only estimate suggests, and it has a shape: ℓ(ℓ+1)C_ℓ sits at ≈0.47, 0.41, 0.36 and 0.68 of the flat-ΛCDM expectation at ℓ=2,3,4,5, recovering by ℓ≈7 (receipt: verify_lowell_boltzmann.py^P15R24). The prediction is therefore not a monotone recovery from the quadrupole but a dip whose minimum falls at ℓ=4^P15R27—and the quadrupole and octopole, the two multipoles a large-angle discussion reaches for first, are its two shallowest points.

The shape is not one code's, and that was checked here by running it rather than by comparing tables^P15R29.

Driving the photon hierarchy built for this programme (§9) through the same discrete projection puts the minimum at the same multipole, ℓ=4, and recovers at the same place, ℓ≈7–8. What two independent Boltzmann treatments cross-validate is the shape, and only the shape. The depths do not agree: the second arm returns 0.49, 0.24, 0.18 and 0.61 against the first's 0.473, 0.410, 0.356 and 0.676—close at the quadrupole and nearly a factor two apart at ℓ=3 and 4. The two treatments differ in their late-time integrated Sachs–Wolfe handling and in D_M, and this paper does not reconcile them: it carries one converged depth table, the first arm's, and a second arm that agrees on where the deficit sits and not on how deep it is.

The reason is the late integrated Sachs–Wolfe: it is sourced near the observer at wavenumbers k k₂ (small line-of-sight distances), above the floor, so the discrete source retains it rather than losing it, and the deficit is milder than a Sachs–Wolfe-only treatment of the same source would give.

This is a parameter-free deficit, set by Λ through r₀ and D_C, and what is robust is its existence, its location (ℓ≲8, geometric, set by k₂ D_C), its minimum at ℓ=4, and that the single floor k₂ starves ℓ=2 and ℓ=3 together: the exact discrete measure and the Doppler term (super-horizon velocity through j_ℓ') each leave the ℓ=2-to-ℓ=3 ratio near unity, if anything suppressing the octopole slightly more (receipts: verify_lowell_exact_measure.py^P15R22, verify_doppler_lowell.py^P15R23). Coherence shown is not correspondence earned: whether this predicted deficit matches the observed sky is the data test, against the large cosmic variance of the lowest multipoles. Confronted with the measured low multipoles, the honest verdict is a wash. CR predicts a smooth modest deficit bottoming at ℓ=4; the observed sky has a sharp quadrupole-only dip (∼0.2) with a near-ΛCDM octopole (∼0.6–1.0), so CR sits above the observed quadrupole and below the observed octopole, matching neither sharply. Under the exact cosmic-variance likelihood the net over 2≤ℓ≤10 is Δ(-2 ln L)≈+1.8 (range +0.3 to +3.8 across the octopole estimator), a small residual well inside cosmic variance (receipts: confront_lowell_data.py^P15R25, verify_lowell_likelihood_v2.py^P15R26). Broken up by multipole, the sector's weight does not lie where its name suggests: the quadrupole rewards this cosmology (-2.6, its milder dip nearer the observed low power than flat ΛCDM's), the octopole penalises it at +1.3, and the largest single-multipole penalty is the +2.1 at ℓ=4—the prediction's own minimum, and the one multipole of the four that the large-angle literature does not treat as anomalous. The low multipoles are thus a genuine but non-discriminating sector: neither a parameter-free correspondence success nor a sharp falsification edge, but a mild deficit consistent with ΛCDM within the lowest multipoles' cosmic variance.

One further caveat belongs with this prediction, and it is not a general disclaimer but a located one. The branch-point filter that sets what crosses into the expansion leg is, in its quantum reading, a Euclidean kernel whose action on a mode is controlled by an adiabaticity parameter C/μ_n, with μ_n the harmonic index of the layer's S³ tower [JanzenCanonicalTime]. That parameter is small for all but the lowest harmonics and reaches order unity only at n=2 and n=3, where it takes the values 0.70 and 0.47. Three independent routes place the filter's loss of control at the same scale: the transmission calculation's boundary at ℓ≈2.5, the projection's incompleteness below k∼2×10^-4 Mpc^-1 (ℓ≈3), and this breakdown at n=2,3.

The transmitted amplitude at low multipoles can be computed rather than estimated, and doing so settles what the estimates left open. The branch-point filter's action on a mode is, in the quantum reading, a Euclidean kernel [JanzenCanonicalTime]. The exponential-of-an-integral form e^-∫ωdη is a WKB approximation whose adiabaticity parameter is of order unity at ℓ≲5; but the underlying mode equation d²ψ/dη²=ω²ψ, with ω=kc_s and dη=ds/a along the segment, is well posed at every , and integrating it is a matter of numerics rather than of approximation. Composing exact constant-ω transfer matrices in the stable direction over the segment (|Δη|=3.32 in units α=1) gives exact/WKB ratios of 0.926, 0.913, 0.901, 0.891 and 0.889 at ℓ=2,3,5,15,40: the WKB form is accurate to 7–11% at every multipole from 2 to 40, with the exact transmission slightly smaller, so the filter is marginally stronger than the approximation gives.

Two readings suggested by the adiabaticity parameter alone are thereby excluded. The first is that the treatment is uncontrolled at ℓ=2–5: it is not, and the parameter was the wrong figure of merit, measuring as it does the fractional error in an exponent that is itself small at low ℓ. The residual's own behaviour settles this independently of any argument about the parameter: the ratio decreases monotonically with ℓ—0.926, 0.913, 0.901, 0.891, 0.889—so the discrepancy is smallest at ℓ=2, where the adiabaticity parameter is of order unity, and largest at ℓ=40, where it is small. An error of adiabatic origin runs the other way, WKB being an expansion in that parameter, so such an error is largest where the parameter is O(1) and vanishes as it does. What the numbers show is therefore a systematic offset rather than an adiabatic breakdown—the WKB form is uniformly slightly generous and the exact filter uniformly slightly stronger, which is a correction of known sign and bounded size and not a loss of control^L19. The second concerns the comparison with the observed sky. The correction is very nearly ℓ-independent—varying by 4% across a factor of twenty in ℓ—so between ℓ=2 and ℓ=3 it alters their ratio by 1.4%. The structural expectation that a single floor starves ℓ=2 and ℓ=3 together is therefore not relieved by the exact treatment, and the observed sky's selective suppression of ℓ=2 alone, by a factor of order three, remains unaccounted for. The low-multipole prediction of this section therefore stands as stated, and so does the discrepancy it faces.

Finally, the two Boltzmann depths above are not rival claims and neither supersedes the other: they differ by up to a factor of two at ℓ=3 and 4 and agree on the shape, the minimum and the recovery, and the difference is the size of the residual §9 declares. What both supersede is the Sachs–Wolfe-analytic estimate (0.22 and 0.20 at ℓ=2,3), which takes the Sachs–Wolfe source as the matter-domination value 13Φ, whereas at recombination ρ_r/ρ_m=0.317 and the source is 0.4006 Ψ—a 20% error in the very ratio of Sachs–Wolfe to integrated Sachs–Wolfe that fixes how much the latter fills the bare deficit.

What the progenitor supplies: the transmission proof

The decomposition's decisive proof is here: why the substrate transmits the spectral shape rather than imprinting one of its own. It turns on the degeneracy of the Nariai double root—the front seam at r=+α/√3, where and f' vanish together so that κ=0. That locus is not the branch point: the branch point sits at r=0, carries no surface gravity because diverges there rather than vanishing, and is not a Killing horizon at all.

The amplitude separates from the tilt in the same statement. What the progenitor supplies is the potential power P_Ψ; what the construction contributes is the factor (0.4835)²=0.2338 carrying it to the observed A_s. That factor is computed rather than fitted, so the accommodation is narrower than it was—the inherited quantity is P_Ψ and the map from it is derived—but it is not removed, and the framework paper's scoping of this as a one-parameter accommodation stands unchanged [JanzenCRframework]. \end{remark}

The physical reading is the result (Fig. 5). The exponential, non-degenerate approach is the thermal (Hawking) law—the mechanism by which a de Sitter horizon imprints a scale-invariant spectrum (n_s→1): the exponential redshift erases the infaller's spectral information and stamps the horizon's own thermal scale. The power-law, degenerate approach carries no thermal scale; it is scale-free, and the infalling spectrum is transmitted faithfully. CR's transmission locus is the branch point r=0, where diverges as -2M/r so the tortoise coordinate is finite—the crossing accumulates no divergent phase and stamps no scale, a third case beyond the dichotomy's two^P15R35. The Nariai double root (Proposition 4, f”(r_N)=-2Λ, κ=0) is the equatorial seam—one point of the substrate, which the bead meets on the way in and again one full lap later, the two chart values r=-2α/√3 and r=+α/√3 being the same modulo 2π in the phase =2πr/√3α. The branch point sits two thirds of the lap in from it (240^∘, with 120^∘ remaining), so the crossing at r=0 and the seam are distinct loci on the lap without the seam being an endpoint of anything. Hence:

The degeneracy is thus the proven reason CR carries the primordial tilt rather than manufacturing it. What the transmitter carries is, from the Schwarzschild–de Sitter side, open: n_s and A_s are inherited, fixed by the progenitor collapse—the same handover that fixes ρ_r/ρ_m and the light-element abundances—and not derived here. Naming them together is convenient and has concealed that they are not the same kind of quantity^P15R78. The vacuum normalisation is the only place a quantum constant enters the spectrum, so P(k)=ℏA₀k^n_s-1 and ℏ is an overall multiplicative factor: it survives in the amplitude and cancels in every logarithmic derivative. An amplitude is the spectrum at one wavenumber; a tilt is its ratio to itself at two. So the construction's inability to force a dimensionless magnitude bears on A_s and not on n_s—the first is permanently inherited, the second need not be. And read backwards the same statement is useful rather than limiting: a quantity fixed by the moduli rather than the invariants is one the observation measures, so the observed A_s determines the progenitor's mass, to be checked against the independent requirement that it lie below the Nariai value. And it is worth saying precisely what that openness is, because a count settles its kind.^P15R77 The progenitor family described here is parametrised by alone—α is the one invariant and E=1 a choice of geodesic—so the geometry is one-dimensional; and the Kretschmann along the bead is mass-free in the faller's own proper time, so that one parameter does not survive the approach either. A quantity identical for every member carries no information about which member it was, and the collapse geometry therefore transmits not one parameter but none. Against that stand at least four independent inherited numbers. So `inherited from the progenitor collapse' cannot mean inherited from the collapse geometry: what is inherited is inherited from the progenitor's matter content—its composition, its entropy per baryon, its perturbation spectrum—and none of that is carried by , the exterior here being exactly vacuum and the marginally-bound interior dust without composition. The frontier is accordingly not a derivation awaiting a cleverer argument but a modelling task awaiting a progenitor interior, and what the geometry supplies in its place is a cutoff rather than a content: the segment's conformal length |Δη|=3.32α is fixed by Λ alone on the Nariai member—below it |Δη| runs as M^-1/3, reaching 4.19α at M/2 and 15.4α at M/100—and says which modes survive without saying how large they were.

Predictions, and the progenitor boundary

The signatures, with their maturity marked:

1. Classical, non-vacuum primordial statistics; no inflationary consistency relation; no substrate-sourced primordial -modes (established, §5).
2. The transmission character: the branch point carries the progenitor tilt and does not drive n_s→1 by any inflationary attractor (established, §8). Nor does the construction lack a scale-invariant attractor—it has one, and it is not the inflationary root.^P15R74 Scale invariance for a background a∝(-η)^β with z∝a requires β(β-1)=2, a quadratic with two roots: β=-1, de Sitter inflation, and β=+2, a matter-dominated contraction. The collapse leg of this construction is the second root exactly: near the branch point f→-2M/r, so the leg is pressureless there and the banked law r³= 92Mτ² with a=|r| gives a∝τ^2/3∝η²; the mode equation integrated from vacuum data returns a flat spectrum to a part in 10⁴. The mode-selection question that shadows this in the literature was accordingly put to the construction directly, and it answers in the negative.^P15R75 The scale-invariant spectrum sits in the mode that dominates the contraction, whose curvature perturbation goes as η^-3; the mode that survives on the expanding side is the constant one, and from vacuum data its spectrum is n_s-1=6. Whether the first reaches the second is a connection problem at the branch point, and here it is settled rather than assumed: the exponents 2 and -1 differ by an integer, so a logarithm could mix them, but the sinh^2/3 law is even in τ, which in conformal time places its correction at η⁶ and steps the Frobenius recursion in sixes past a resonance sitting at three. No logarithm is generated, the continuation acts diagonally, and the scale-invariant spectrum does not reach the observer—confirmed independently by continuing the dominant solution numerically around the branch point, where the admixture it produces is what the claim turns on, where the admixture falls as δ^2/3 and the diagonal entry goes to unity. What survives is therefore narrower than it first appeared and sharper than what it replaced: the construction has a scale-invariant background, the spectrum that background freezes is confined to the contracting side, and the obstruction is a parity property of the construction's own solution rather than a missing mechanism. Mixing would require a background correction odd in τ, and that question is settled here too, in the negative.^P15R76 Writing the law as r=Au^2/3g(u) with g=[ sinh u/u]^2/3 places every branched factor in u^2/3 and leaves even, analytic and non-vanishing at the origin—exactly, since log ( sinh u/u) is analytic in u² there and the exponential of a series in u² is one. The three sheets differ by a constant cube root of unity, and a constant multiplier cannot change a parity; the conjugate branch is even as well, and carries no branch point at all. So the evenness is unavoidable, and what it says is not a technical fact but a symmetry: r(-τ)=r(τ), the cosmogenesis is time-symmetric about the branch point, and a time-symmetric background cannot mix a growing mode with a decaying one, since mixing would distinguish a direction of time the background does not. The same invariance is already load-bearing one observable over—the peak heights of §5 follow from a driving amplitude exactly invariant under time reversal, which is why they may be evaluated on the contracting side. One symmetry, two consequences. And it locates rather than forecloses what would be needed: this leg is exactly even because it is the marginally-bound geodesic of an exactly vacuum exterior, so a progenitor carrying real matter content—pressure, dissipation, anything with an arrow—would not be, which returns the question to what the progenitor was.
3. Flat distances (Ω_k≃0) coexisting with a closed-S³ low-multipole deficit below ℓ≈8, set parameter-free by Λ—mild, and on a genuine Boltzmann transfer a dip bottoming at ℓ=4 (≈0.47, 0.41, 0.36, 0.68 of the flat-ΛCDM expectation at ℓ=2,3,4,5), its shape cross-validated between two independent transfers (established for the existence, the location and the minimum; open for the depth, on which the two transfers differ by up to a factor of two at ℓ=3 and 4 (§7); non-discriminating as a test, the deficit sitting inside the largest multipoles' cosmic variance). Confronted with data it is non-discriminating: it starves ℓ=2 and ℓ=3 together, matching neither the sharp observed quadrupole-only dip nor the near-ΛCDM octopole, and sits consistent with ΛCDM within the lowest multipoles' cosmic variance (§7).
4. Coherent acoustic peaks from geometric (null-boundary) phase-fixing rather than super-horizon freeze-out (argued, §4).
5. The acoustic scale is resolved in the cosmological development above (§2), and the geometric rate (4) is the standing CR-versus-ΛCDM discriminator on data in hand (established, §2)—a strength established before the spectrum, not this section's to re-assert.

The boundary is principled. The substrate carries everything the branch point's null geometry lets it carry—coherence, isotropization, the discreteness floor, the classical character, the faithful transmission—and the degeneracy proves precisely where it stops: the spectrum's amplitude and tilt are not the substrate's to set. This is the shape inflation has when it parametrizes its observables through an unknown inflaton potential; CR routes them through an unknown progenitor collapse, with the arrival constrained—whatever the collapse supplies must reach the branch point coherent, isotropized, scale-freely transmitted, and classical.

The symmetry between the two unknown suppliers is only apparent, and the asymmetry is the substance. Inflation is not structure the standard cosmology already contains; it is an added sector—a scalar field with a freely specified potential—introduced to produce features that cosmology was known to lack: the causal contact the horizon problem denies, the flatness the dynamics do not single out, and, once these were in view, a near-scale-invariant spectrum. The potential is then chosen to deliver the observed amplitude and tilt. A model built to be tunable, and tuned to a target known in advance to require tuning, earns precisely what such a model can earn—it establishes that the features are possible, that an apparatus permitting them exists. It does not establish that they were required, and the routine elevation of the spectral fit into a vindication of inflation conflates the two: that a freely chosen potential can be set to match n_s and A_s is not evidence that inflation occurred, but the statement that a model with enough adjustable structure to fit has been fit. This is the distinction between a world that requires a phenomenon and one that merely permits it through adjustable parameters—the same distinction the programme draws in reading the cosmic foliation as forced rather than chosen [JanzenModernParallax]—and it places the inflationary spectrum on the permitting side.

The present cosmology adds no such apparatus. Causal contact, isotropy, flatness, phase coherence, and scale-free transmission are not produced by a field introduced to produce them; they are read off structure the construction already contains—the de Sitter throat that isotropizes by the no-hair theorem (§6), the null boundary on which coherence is the characteristic-data condition (§4), the exactly Euclidean constant-time slice (§2), and the degenerate horizon whose vanishing surface gravity transmits rather than imprints (§8)—each fixed by Λ alone, with no parameter free to do the producing. The one quantity it does not derive, the amplitude and tilt, it inherits in exactly the sense flat ΛCDM inherits the baryon-to-photon ratio; on that single datum the two frameworks stand level, and on everything the inflationary sector is invoked to supply they do not. The verdict this licenses is not symmetric, and it does not await a future measurement. On the data in hand the two align; on the structure each must assume in order to align, one introduces an entire tunable sector and tunes it while the other introduces none—and makes, in its place, falsifiable commitments where inflation keeps its freedom: no scale-invariant attractor reaching the observer, no consistency relation, no substrate-sourced -modes (§5, §8)^P15R75. By the economy of assumption that distinguishes a required structure from a permitted one, that comparison is settled now, by what each theory must presuppose, and it is settled in this cosmology's favour. We hold this at its weight and mark its bound in the same breath: the parsimony verdict is earned, and the data verdict has now begun to be earned with it—the Hubble tension resolved across the distance ladder where ΛCDM cannot reconcile the direct H₀ (§2), the abundances confirmed within 1σ [JanzenCosmogenesis]—so the cosmology is empirically favoured, not merely more economical; the spectrum's content remains inherited, and the large-angle shape remains the exposed edge that could still overturn it.

The acoustic instrument: what it is validated against, and to what accuracy

The low-multipole prediction above and the ratio statements below are carried by a Boltzmann transfer built for this programme, and its standing should be stated plainly rather than assumed. It is a full photon hierarchy with polarisation, second-order tight coupling, massless neutrinos, and a Peebles recombination history; the line-of-sight source carries the monopole with the potential, the Doppler term, the integrated Sachs–Wolfe term, and the quadrupole with its own projection kernel (j_ℓ+3j_ℓ”).

Against a like-for-like reference—an independent Boltzmann code run at identical parameters, unlensed and with reionisation off—it reproduces the acoustic peak and trough positions to ≤0.5% across P₁–P₄, the ionisation history x_e(z) and its derivative to ∼1% through the visibility peak, matter–radiation equality to 0.02%, and the transfer function to better than 1% for k lt;0.02 Mpc^-1. Above that wavenumber it agrees to 1–2%, with the residual scatter in both directions and no trend across the range^P15R36.

Two construction details are worth stating, because both are easy to omit and both have signatures that read as physics.

The first is the reach of the mode ladder. The angular spectrum at multipole ℓ is an integral over wavenumber, and the projection kernel j_ℓ(k(η₀-η)) has substantial support above k=ℓ/D_M, not merely at it; a ladder cut at k therefore truncates the integral, and does so more severely as ℓ rises toward kD_M. Quantitatively, truncating an exact transfer at kD_M=900 retains 99.2% of C_ℓ at ℓ=200 but only 61% at ℓ=800 and nothing at all at ℓ=1150; carrying the ladder to k≥2 ℓ/D_M recovers every multipole to better than half a per cent. The tell is unmistakable once known: the highest peak simply fails to appear, so a spectrum that should show four acoustic peaks shows three, and the peaks that do appear are displaced downward by an amount that grows with ℓ. Increasing the ladder's density at fixed reach does not help—the loss is the range, not the sampling, which is worth checking explicitly because the two are easily confused.

The second is the seeding of the conformal time. It is a cumulative integral, η(a)=∫₀^ada'/(a'²H), and a numerical grid necessarily begins at some finite a; the interval already elapsed below that point must be carried as a seed rather than set to zero. In radiation domination H∝a^-2 makes the integrand constant, so the omitted piece is a/(H₀√Ω_r) exactly—for a=10^-5 it is 4.6 Mpc. It is a fixed offset at every epoch, and therefore 0.03% of η₀ but 12% of η at z=10⁴: the acoustic scale, the peak positions and the peak ratios are insensitive to it, while any early-time quantity or any comparison against a second code is not. Its signature is instructive—because the potential decays more steeply at larger , a fixed error in the time coordinate reads out as a smooth, monotone, -growing error in amplitude, which is the shape one would otherwise attribute to a physical suppression. The grid must also extend below the intended starting redshift, so that the initial epoch is interpolated rather than clamped.

That residual does not enter the claims made here. The Cosmological Relativity and flat-ΛCDM spectra are computed through the same machinery, so an instrumental transfer error afflicts both arms identically and cancels in their ratio—and every prediction stated in this paper is a ratio. The low-multipole deficit in particular lives at ℓ≲8, i.e. k 0.02 Mpc^-1, where the two agree to better than 1%. What the residual does forbid is an absolute amplitude claim: this instrument should not be read as reproducing the observed small-angle power to better than a few per cent, and no such claim is made.

The open frontier: the pieces this cosmology is built to close

The result stands on what the substrate determines. What it does not yet claim has been carried to a sharp, gradable edge rather than left vague—and those edges are the most interesting part of the programme from here, each a gap we are eager to grind shut, not paper over.

We hold them to the criterion of necessity the programme's epistemology names—that a structure is credited for requiring a phenomenon, not for permitting a value that fits it [JanzenShadowExistence]—and sort them by it: a gap that is buildable is a debt owed, to be built before the claim it bears on is a proof rather than a coherent proposition, while a genuinely external unknown may stay open at no cost, exactly as flat ΛCDM leaves its own inherited data open.

The buildable ones below are named as debts, not softened into frontiers. Of the four, one remains owed and three are run: the end-to-end branch-point-to-recombination transfer is the load-bearing debt; the large-angle depth and the low-multipole likelihood are built and confronted below; and the derivation of the progenitor spectrum is the genuine frontier rather than a debt. Four stand out:

1. the sufficiency of the null-boundary characteristic data for the observed acoustic coherence (§4), argued from the structure of the characteristic initial-value problem, with the peak-height magnitude carried by the time-reversal driving equality established there—what remains is the end-to-end branch-point-to-recombination transfer, the calculation that would carry the single-phase datum all the way through, confirm the height pattern digit by digit, and—within it—settle whether the diffusion-scale signature (§4, §4) shows in the observed high-ℓ power as a tension, a wash, or a distinctive-but-consistent feature. One general consideration already bears on which: a shift in the diffusion scale is, across the observable multipole range, substantially degenerate with the spectral tilt, so the scale signature's leading spectral effect is largely absorbable into n_s—disfavouring the tension reading and leaving a wash or a distinctive-but-consistent feature, with the tilt-irreducible residual the part the transfer would isolate. This is a genuine build, not a plug-in: it requires first specifying how the fluctuations gravitate on the geometrically fixed background—the piece that sets the high-ℓ driving envelope, and which the standard Boltzmann codes cannot supply because they tie radiation's gravity to its presence and so cannot represent the content-not-rate split the layered rate rests on—and then a bespoke transfer against that specification. It is a debt owed and named as such: the bespoke transfer is not a missing idea but a computation this sequence owes and has not yet run, and the paper's own open edge rather than another's—and it is a different object from the comparison already carried out through a standard Boltzmann code with a suppression factor applied, which is run and is not this. Two things it bundles are worth holding apart—the build is ours, and the verdict it would expose is the world's: the transfer is what makes the diffusion-scale signature confrontable at all, and the data then decide it. One step of that build is taken, and it is the one that had to come first. The instrument such a transfer would run on—a two-arm line-of-sight Boltzmann integral, Δ_ℓ(k)=∫S j_ℓ(k(η₀-η)) dη, carried on this cosmology's own background—is now validated against its own control. Run on the ΛCDM arm, whose answer is known independently, it returns peaks at ℓ=220, 540 and 812 against the sky's 220.6, 538.1 and 809.8, and a first-to-second height ratio of 2.197 against a standard code's 2.200: agreement to 0.14%, and within the sky's own 1σ. What that displaced was not a defect in the transfer but two faults in how it was being run—the -integral was truncated at the highest multipole reported, where ∫P(k) Δ_ℓ(k)² dk is not yet converged, and the projection path in use omitted the polarisation source g /4+(3/4k²) d²_η[g] that a damping envelope cannot supply at any coefficient. Each is worth a factor on its own and neither alone reaches the answer^P15R100. So what stays owed is the specification and the CR arm, and not the machinery: an instrument that reproduces a spectrum whose answer is known is an arbiter about one whose answer is not, which is the whole purpose of running the control first;
2. the exact large-angle depth (§7): the flat-projection transfer of the discrete closed-S³ source is built with a genuine Boltzmann transfer—the exact CMB temperature transfer Δ_ℓ(k) (Sachs–Wolfe, early and late integrated Sachs–Wolfe, and Doppler, all common to CR and flat-ΛCDM) sampled at the discrete modes k_L, gate-validated (the continuum reproduces the reference C_ℓ to four figures) with the r₀ sensitivity measured rather than asserted—shape stable, depths drifting up to 15% at ℓ=4 (receipt: verify_lowell_boltzmann.py^P15R24). It gives a mild deficit, ≈0.47, 0.41, 0.36 and 0.68 of the ΛCDM expectation at ℓ=2,3,4,5—a dip whose minimum falls at ℓ=4, not at the quadrupole, the shape reproduced by the programme's own photon hierarchy, which agrees on the location and not on the depth—recovering by ℓ≈7: the late integrated Sachs–Wolfe term is sourced above the floor and is therefore retained by the discrete source, leaving the deficit shallower than the bare Sachs–Wolfe starvation. The mechanism is robust—parameter-free, located at ℓ≲8, bottoming at ℓ=4, with ℓ=2 and ℓ=3 starved together (the exact discrete measure and the Doppler term each leave the ℓ=2-to-ℓ=3 ratio near unity: receipts verify_lowell_exact_measure.py^P15R22, verify_doppler_lowell.py^P15R23). Confronted with the sky the sector is a wash: CR's smooth ∼0.4–0.5 deficit sits above the sharp observed quadrupole (∼0.2) and below the near-ΛCDM octopole (∼0.6–1.0), matching neither, and the exact cosmic-variance likelihood returns Δ(-2 ln L)≈+1.8 over 2≤ℓ≤10 (range +0.3 to +3.8 across the octopole estimator), a small residual inside cosmic variance (receipt: verify_lowell_likelihood_v2.py^P15R26). So the low multipoles are a genuine but non-discriminating sector, neither a parameter-free correspondence success nor a sharp falsification edge; the deficit's existence and location stand as a real parameter-free feature, but its depth places it consistent with ΛCDM within the lowest multipoles' cosmic variance. The honest exposed edges lie elsewhere—the damping tail and the full-spectrum likelihood below;
3. the progenitor spectrum itself—n_s and A_s (§8)—inherited boundary data fixed by the progenitor collapse, the baryogenesis-analogue computation belonging to the matter sector [JanzenOperator, JanzenRange], with the honest target a consistency between the inherited composition and spectrum and the measured primordial abundances and tilt, not a derivation from the geometry alone—the most ambitious reach of the three, the one that would close the decomposition from both ends and give CR a derivation where the standard model keeps a measured datum. One route to it is closed, and closing it narrows the target rather than the ambition. The construction is recursive—each progenitor seeds the next cosmology—so a stationary recursion would make A_s a fixed point and determine it. But entropy per baryon rises strictly at every branch-point crossing, the crossing being an erasure and the increment bounded below by Landauer's k_Bln 2 independently of the network and of η [JanzenCosmogenesis]: a map that strictly increases a coordinate has no fixed point, and independently the chain it generates is finite, so there is no asymptotic limit for a fixed point to be the limit of^P15R2. So the derivation must reach the head of a bounded genealogy rather than a stationary state—the same terminus the inherited composition reaches, and the two are one question asked of two quantities;
4. the likelihood-level model selection: the parameter-free ratio comparison above is joined by the exact low-multipole cosmic-variance likelihood—each C_ℓ a scaled χ²_2ℓ+1 about the model C_ℓ, the absolute baseline cancelling in the CR/ΛCDM ratio (verify_lowell_likelihood_v2.py^P15R26), the proper instrument where cosmic variance dominates. With the genuine-Boltzmann depths (≈0.47/0.41 at ℓ=2/3, above) the sector is a wash: the low quadrupole still mildly rewards CR (Δ(-2 ln L)≈-2.6, its milder dip nearer the observed low power than ΛCDM's), the octopole and ℓ=4 mildly penalise it (≈+1.3 and +2.1)—so the sector's largest single-multipole penalty sits not at the octopole but at ℓ=4, the prediction's own minimum—and the net over 2≤ℓ≤10 is Δ(-2 ln L)≈+1.8 (range +0.3 to +3.8 across the octopole estimator)—a small residual well inside cosmic variance. A Monte-Carlo over 10⁴ skies puts the sector's in-principle discriminating power at only ∼3.4σ (the short multipole lever arm and cosmic variance), and the current measurement uncertainty swamps even that. The low multipoles are therefore a genuine but non-discriminating test: CR and ΛCDM are statistically indistinguishable there, the mild deficit neither favouring CR nor refuting it. Decisiveness comes from outside the sector—the geometric expansion rate (4), now confronted with the distance ladder (§2), as the standing discriminator, and the likelihood across the full spectrum (its amplitude level carried by the time-reversal driving equality, §4; the full-spectrum likelihood itself, with the oscillating-medium transfer downstream in the matter sector [JanzenOperator, JanzenRange], the larger piece still to build).

A frontier of a different kind concerns the matter itself rather than the perturbation observables above: the dynamics of matter crossing the branch point.

Three facts about that crossing follow from the structure already established here.

It is well posed—the beginning is fixed at the branch point r=0, and the crossing is taken on the substrate, whose curvature is finite there however the geometry's own invariants behave.

The bend ρ=m'/4πr² that carries the matter is accordingly finite, and its data on that null surface is characteristic rather than Cauchy: a characteristic crossing with no obstruction from the substrate's curvature. The geometry's Kretschmann scalar does diverge at that locus, as the abstract states and the circle paper computes [JanzenCircle]; what makes the crossing well posed is that the divergence belongs to the chart-borne geometry and not to the substrate the cut is taken on, and that the tortoise measure converges so the crossing carries no scale.

It isotropizes and is scale-free—the throat's de Sitter no-hair (§6) damps the anisotropic part of the crossing stress-energy to the isotropic monopole, and the degenerate approach (§8) imprints no thermal scale: the matter counterparts of the field no-hair and faithful transmission established above. Where each of those acts is worth stating, since one of them cannot act at the crossing at all.

Damping is a process and requires elapsed time, and the crossing has none: the real part of the complex cosmic time is frozen along the segment reaching it [JanzenCanonicalTime].

The no-hair damping therefore belongs to the approach—the throat region, over the collapse leg's own cosmic time—while the crossing carries whatever survives it unchanged, not because a mechanism protects it but because there is no interval in which anything could act. The isotropy is earned on the approach and preserved by the crossing, and the two halves are of different kinds: the first is a damping calculation with a rate, a timescale and a residual, the second follows from the zero-duration structure and takes no model input. It follows that how much anisotropy survives is fixed by the approach against the leg's duration—a property of the progenitor's collapse history rather than of the crossing. And it obeys a structural transition law: because the reassignment preserves the cosmic foliation and the density is leaf-carried and lapse-independent [JanzenOperator], the reassignment acts on the time-stacking—fixing the Λ-set rate (6)—and not on the leaf, so the isotropic density crosses as inherited progenitor content, which is the structural reason the inherited datum of §2 is inherited rather than produced; and two conditions must be met at that crossing, which a concrete model has to satisfy separately: the reassignment requires a degenerating Killing field whose freed null direction can be re-founded as a clock, the joining of the legs requires a branch point, and these are independent—the forced Nariai member carries the degeneracy and is not a branch point, while r=0 is the branch point and carries no Killing degeneracy [JanzenDynamics], so a model furnishing one without the other furnishes no cosmogenesis.} The first of the two is settled as a uniqueness, not merely an availability: across the slicing family the reassignment that promotes a null direction to the fundamental congruence is uniquely the one selecting the forced Nariai member, every other vantage yielding a congruence that crosses the embedding's symmetry structure transversally and so reading as a localised black hole [JanzenGroupoid]. A model therefore has no freedom in where the reassignment acts, and misplacing it does not fail to give a cosmology by a little—it gives a black hole instead. The scoping of the two rates this law forces—the excursion-side local readout against the observable geometric rate, their boundary—is drawn out in the cosmogenesis paper [JanzenCosmogenesis]. What remains open is the detailed worldline and field dynamics of the crossing for a concrete matter model [JanzenCRframework, JanzenOperator]—a depth left for a matter model to supply, and now resting on a crossing shown well posed rather than merely postponed.

None of these unsettles the result obtained: the branch point's null geometry decomposes the primordial scalar spectrum into a substrate-determined structure and a progenitor-supplied content, and the degeneracy proves that the branch point transmits that content rather than imprinting one of its own on this argument; the super-horizon transfer across the branch point is itself computed—Φ→Φ_i for every , with the expanding leg inheriting 9/10Φ_i scale-invariantly^P15R98—while carrying that join and the acoustic evolution through to recombination as one calculation is not yet run. They are, rather, the map of where this cosmology goes next—the end-to-end transfer behind the peak heights and the full oscillating medium downstream in the matter sector, the tensor sector's companion in the dynamics paper [JanzenDynamics].

Discussion

What this paper determines is one thing seen from several sides. A single de Sitter geometry, its scale set by the cosmological constant alone (the maximally symmetric substrate's sole scale [JanzenGeometricCore]) and read through a causal reassignment at the cosmogenesis branch point, is the observed cosmology: the flat-ΛCDM expansion history is its areal radius (§2); the apparent Hubble tension and the acoustic scale are consequences of its geometric rate (§2); the primordial scalar spectrum is the characteristic data of its seam, structure fixed by the substrate and content inherited across the degenerate horizon (§§4–8). The Einstein field equations are unchanged; what changes is the reading, and under it the quantities the standard model carries as independent inputs—the present matter density, the dark-energy fraction—are readings of one cosmic clock, not drivers of the expansion, so that the coincidence of Ω_m and Ω_Λ dissolves into our observing at a time of order the geometry's one timescale. There is a single seed, and the observed universe is what grows from it. And the one scale reaches down as well as out: the same cosmological constant that sets this expansion sets, for every mass , the Hubble–Eddington radius [Eddington1933] r_HE=(3GM/Λc²)^1/3—the boundary within which local structure stays bound against the cosmic flow and beyond which it is carried off. The slicing geometry reads that boundary as the flat locus of the existent slice's own curvature, the local bend of the cut exactly cancelling the substrate's cosmological term [JanzenSlicing], and the slicing operator reads it as the handover from sub-marginal bound orbits to the marginally-bound E=1 congruence that is this cosmology [JanzenOperator]. So the substrate's single Λ governs both the global expansion this paper computes and the local scale at which structure resists it—one constant read at two ranges—and the Hubble–Eddington radius, proposed as a local, per-structure test of Λ from the largest bound structures [PavlidouTomaras2014], is an independent observational handle on the very constant the expansion history here measures. The two ranges are moreover the same length up to a pure number. The Hubble–Eddington radius—where the slicing surface's Gaussian curvature K_G=1/α²-M/r³ vanishes, the local bend cancelling the substrate's cosmological curvature [JanzenSlicing]—is r_HE=(Mα²)^1/3; while the expansion's own amplitude, (2Mα²)^1/3, is exactly the areal radius at matter–Λ equality (ρ_m/ρ_Λ= csch²(3 τ/2α), unity at sinh =1). The two therefore stand in the fixed ratio 2^1/3 for every : the scale at which a bound structure's gravity balances the expansion and the scale at which the expansion's matter balances its Λ are one length read locally and cosmologically, the factor of two that of the Schwarzschild term against the surface curvature's M/r³ and nothing else. ^P15R28

The comparison with the standard cosmological model is accordingly not only a contest of fits—though on the Hubble tension the geometric rate now fits the joint acoustic-and-distance-ladder data at the direct H₀ where ΛCDM cannot (§2)—but also, and more deeply, a difference in what each must assume to do so. The standard model assembles the observed universe: a dark-energy component for the late-time acceleration, an inflationary sector for the causal contact, flatness, coherence, and near-scale-invariance the background dynamics do not themselves supply, and further early-universe physics where the sound horizon and the expansion rate come into tension. Each part is a real solution to a real problem, and each was introduced to yield the feature it produces.

The dark-energy component is the sharpest case of the difference: on this cosmology the acceleration is not a sector at all. Since (dr/ d τ)²=E²-f on the energy family, one has d²r/ d τ²=-f'/2=rK_G: the comoving acceleration is the existent slice's own Gaussian curvature, up to the positive factor .^P3R12 So the expansion decelerates while the slice is negatively curved, turns over exactly where it is flat—the Hubble–Eddington radius, ρ_m=2ρ_Λ, at sinh (3c τ/2α)=1/√2—and accelerates thereafter, with nothing added to make it do so. And that locus is a measurement, not only a description: the turnover sits where the areal radius reaches the front seam, r=r_N, so the acceleration-onset redshift is the offset parameter itself minus one,

against (2Ω_Λ/Ω_m)^1/3-1=0.632 on Planck's parameters—agreement at 0.7σ, and with no fit performed here. What makes the equation worth stating is what it does not contain: z_acc is a redshift, measured calibration-free, so it is untouched by the distance-ladder tension, and it fixes x₀ without H₀, without a ladder and without the microwave background. The single dimensionful scale is where any disagreement must then live, and the age follows from the same parameter in closed form, t₀=(2r_N/√3 c) asinh(x₀^3/2/√2). The late-time acceleration is thus read off the same structure that fixes the local boundary of a bound system.

And the growth sector, written on this rate, loses its parameters entirely — which turns a known numerical curiosity into a statement about the epoch. Substituting H=(c/α) u and the Friedmann readout's 4πGρ= 32(c/α)² csch²u into the growth equation and changing to , every factor of (c/α)² divides out:

The scale does not merely fail to appear—it cancels, because at Nariai the source is fixed by Λ and the time variable carries α too. The decaying mode is the rate itself, D_-= u=Hα/c, and reduction of order gives D₊= u∫₀^u sinh^2/3v sech²v dv. And the matter fraction is a clock reading: cosh²u=1/Ω_m exactly, so Ω_m= sech²u.

On these variables the normalisation of the standard growth factor — J(Ω_m)=∫₀^∞(1+z) dz/[Ω_m(1+z)³+1-Ω_m]^3/2, known to cross unity once near the concordance density [JanzenGrowthNorm] — becomes J= 23 cosh²u₀ (D₊/a)|₀, whose matter-domination limit D₊/a→ 35 returns the analytic endpoint J→ 25. Differentiating the condition J=1 gives an exact identity between the two sides' derivatives, ∝ 92Ω_m-2, and integrating it from the common origin leaves

the growth-weighted history of Ω_m balanced about 23^P15R5.

The pivot is the deceleration-to-acceleration turnover: Ω_m= 23 is ρ_m/ρ_Λ=2 exactly. (13) is solved at Ω_m=0.31516, so the density at which that normalisation equals unity is fixed by an equation with nothing free in it.

Two scope statements belong with that, and both narrow it. The turnover is not this construction's property: any flat matter-and-Λ rate has it, and what is this construction's is the identification of that turnover with a geometric locus — the areal radius there being α/√3=r_N, the Nariai radius and the merged double root (§11 above). So the balance pivots on the turnover, and the geometry is what makes the turnover a place.

And the rate used is the stacking one, which is the right rate for the object but not for this construction's own growth. The published normalisation is a matter-and-Λ integral, and the stacking rate written in the fitted pair is that rate, which is why the derivation reproduces it. The rate rule assigns perturbations to the leaf (§2), and on the leaf rate the same integral returns 0.99940 rather than unity. So what is derived here is the root of the standard normalisation, and what remains contingent is that the present epoch lies near it.

What that figure does not measure is how far the root moves, and saying so matters because it was read that way once. Continuing the closed form onto the leaf rate is a closed form applied to a rate it does not solve. Solved directly, the root moves the other way and by less — the leaf root sitting at u₀=1.180039 against the stacking 1.180309, a shift of 1.1% of the gap to the measured epoch and away from it, taking the offset from 0.704 to 0.713σ^P15R6. So the two per cent is not an artefact of the rate: it stands, and wants a mechanism. This cosmology adds no such part. Causal contact, isotropy, flatness, phase coherence, and scale-free transmission are read off structure the construction already contains—the isotropizing throat (§6), the null boundary on which coherence is the characteristic-data condition (§4), the exactly Euclidean constant-time slice (§7), the degenerate horizon that transmits rather than imprints (§8)—each fixed by Λ alone, with no parameter free to produce it. This is the distinction between a structure that requires a phenomenon and one that merely permits it through adjustable apparatus [JanzenShadowExistence, JanzenModernParallax], and it is the criterion by which competing frameworks are rightly weighed ahead of the measurement that will decide between them.

The transmission dichotomy (§8) is where that distinction is sharpest, because there the two frameworks make opposite structural commitments about one object. A non-degenerate horizon's exponential approach imprints a scale-invariant spectrum—the very mechanism by which inflation produces near-scale-invariance—while the degenerate Nariai seam carries no scale and transmits the progenitor spectrum unaltered. Inflation accommodates the observed tilt because a freely specified potential can be tuned to it; reading that fit as a confirmation mistakes a model with enough adjustable structure to match for one that required what was found (§9). This cosmology cannot tune: the degeneracy requires that the branch point not imprint a scale, so the framework has no inflationary scale-invariant attractor, no consistency relation, and no substrate-sourced primordial -modes. The amplitude and tilt it does not derive it inherits, exactly as flat ΛCDM inherits the baryon-to-photon ratio from a baryogenesis it does not model; on that single datum, of the same kind and count the standard model already carries, the two stand level. A framework is credited for requiring the structure it explains, not faulted for measuring the boundary data it does not—and the asymmetry falls on everything else.

The verdict this licenses must be drawn on the right axis, and confined to it. On the axis of theory-choice—the requiring rather than the permitting of the phenomena, the consolidation of many results under one structure, the absence of a device added per problem—the comparison is decidable now, from the structural record, and it falls to this cosmology [JanzenShadowExistence]. On the axis of data-discrimination the picture has begun to move with it: the geometric rate resolves the Hubble tension across the acoustic scale and the baryon-acoustic distance ladder together, at the directly measured H₀ where the standard model cannot (§2)—a discriminating datum, not a tie—while on the light elements and the low multipoles the two frameworks are even. On the microwave-background peak structure they are not: the transfer's likelihood comparison rejects this arm's spectrum on TT shape (§4), and that is the sharpest datum against the construction that the corpus holds. So the standing is stronger than rule-favoured-awaiting-the-datum: it is a framework favoured on the structure and carrying its first discriminating data result, with the fuller confirmation—the damping tail and the full-spectrum likelihood—still owed. A theory that fits is not a theory confirmed, and that confirmation is not yet in hand; but the data axis is not even, and the framework has staked falsifiable commitments—no scale-invariant attractor reaching the observer, no consistency relation, no substrate -modes—precisely where the standard model keeps its freedom^P15R75.

The low-multipole deficit is a mild, non-discriminating feature (§§7, 10). The flat projection of the discrete closed-S³ source gives a parameter-free deficit below ℓ≈8—real, and set by Λ through r₀ and D_C—bottoming at ℓ=4, at ≈0.47, 0.41, 0.36 and 0.68 of the ΛCDM expectation at ℓ=2,3,4,5. At that depth CR's smooth deficit matches neither the sharp observed quadrupole-only dip nor the near-ΛCDM octopole, and under the exact cosmic-variance likelihood the two frameworks are statistically indistinguishable at low ℓ (Δ(-2 ln L)≈+1.8 over 2≤ℓ≤10, inside cosmic variance). So the sector neither confirms nor refutes: it is a genuine parameter-free prediction whose depth places it consistent with ΛCDM within the lowest multipoles' cosmic variance. The framework's exposed edge is the damping-scale signature locked to the Hubble resolution (§4), a computed, non-reabsorbable CR-specific effect whose consequence for the observed high-ℓ power is entangled with the acoustic transfer at those multipoles and so genuinely open. Its decisive discriminator is the geometric expansion rate itself, carried to the distance ladder and confronted with the DESI DR2 baryon-acoustic dataset at χ²/ dof≃1 (§2): the same rate resolves the Hubble tension across the full ladder and stakes the damping signature on one commitment, the observable weight of the latter awaiting the full transfer.

The framework's honest standing includes what the standard model has genuinely built and this cosmology has not. Dark matter is inferred from multiple independent, cross-consistent lines—rotation curves, gravitational lensing, the merger offsets, the acoustic-peak structure, the growth of structure—and this cosmology carries the matter density as the bend of the spatial cut without yet removing the component or answering its particle question; that column stands at full weight. Standard nucleosynthesis derives the light-element abundances from one measured parameter, a genuine achievement this cosmology now meets on the same footing rather than exceeds: the collapse's cooling leg is a standard nucleosynthesis run on one inherited datum (§2; the cosmogenesis paper [JanzenCosmogenesis]), producing helium-4 and deuterium at their observed values—the freeze-out temperature the data demands being the bottleneck the standard rate delivers—and, because the run is standard, carrying lithium-7's standard threefold over-prediction, the lithium problem shared rather than solved. On the light elements the two frameworks stand even: the same two successes, the same one problem, with this cosmology's open edge whether one progenitor handover fits all the abundances together. And the framework is young: the detailed dynamics of matter crossing the branch point (its crossing shown well posed and structurally governed, §10), the derivation—as against the measurement—of the inherited data, and the Standard-Model matter content—the gauge representations and the mass spectrum, which the framework does not supply, its geometry shown to force only the discrete flavour skeleton [JanzenMatter]—remain to be built (§10; [JanzenCRframework]). These are open frontiers, not defects: a theory is not marked down for what it has not built, only for a claim it makes that fails. They are named plainly here because an economy that favours this cosmology on the structure it carries would be worth nothing bought by concealing the structure it does not.

Conclusion

So, plainly, what this paper claims and what it does not. It claims that the observed expansion history, the dissolution of the Hubble tension and the calibration of the acoustic scale, and the structure of the primordial scalar spectrum are consequences of a single de Sitter geometry read through the causal reassignment its own horizon geometry selects—one seed, grown, the field equations unchanged—and that on the economy of assumption, the criterion by which theory-choice is rightly made ahead of the deciding measurement, this reading is favoured now over the assembled standard picture. On the data axis it claims the more limited and now-earned thing: that the axis has moved in its favour—the Hubble tension resolved across the distance ladder, the abundances confirmed—without yet claiming to have won it, the damping-tail signature and the full-spectrum likelihood the edges still to be brought to a verdict (the low multipoles, once the sharpest edge, now a mild non-discriminating deficit consistent with ΛCDM within cosmic variance). It does not claim to be a complete cosmology: its matter sector, the derivation of its inherited data, and its fermion content remain to be built. It claims to have shown that the universe we observe is the reading of one geometry, and to have staked the falsifiable commitments by which that claim can be tried.

A final word on the register, because it is the thesis rather than an aside. The beginning this cosmology describes is the branch point at r=0—the locus at which the collapse leg of the cosmogenetic contour becomes the expansion leg [JanzenCRframework]—and it is not a curvature singularity reached and left behind. The curvature there is set by the substrate's own scale. The throat is α, the defining constant of the maximally symmetric manifold the cut is taken on, and the divergence of the invariants computed in the areal chart is that chart degenerating, exactly as a polar chart degenerates at a pole—not a curvature scale of the geometry [JanzenCosmogenesis, JanzenGeometricCore]. Two further facts stand with it: no finite cosmic-time layer ever reaches the locus, and the continuation through it carries no scale, since f→-2M/r diverges so that the tortoise coordinate r_*=∫dr/f converges and what is transmitted is fixed by the analytic structure rather than by a length. The branch point is crossed by continuation, not traversed by a worldline. The expansion is not a space ejected from a point but the reading, on the near side of that branch point, of a spatial layer still advancing, the branch point itself a collapse that in the frame it bounds is never complete. The distinction the whole programme turns on—between what has finished happening and what is still underway—is thus the distinction the cosmology is built from: the universe is not the collapsed record of a beginning but the still-evolving layer that record represents, read here as one de Sitter geometry.