A Cosmological Sector for the E₈ × _ωE₈ Octonionic Unification: Emergence at the Electroweak Epoch, Infrared Gravity in Place of Dark Matter, and Large-Scale Cosmic Flows

1. Introduction: the Missing Sector, and the Observational Moment

The $E_8\times {}_{\omega }E_8$ unification programme [1,2,3,4,5,6,7,8,9,10,11] now fixes, with varying epistemic strength, the gauge content of the standard model coupled to pre-gravitation, the quantisation of electric charge ($Q=N/3$ from the $\mathrm{Cl}\left(6\right)$ number operator), hypercharge without right-sector generators ($Y=Q-T_L^3$), the mass ratios of charged fermions from the exceptional Jordan algebra, a single vector-like colour with strong-$CP$ protection, gravity as the geometric face of the broken $\mathrm{SU}{\left(2\right)}_R$ symmetry, and a new unbroken abelian gauge symmetry: dark electromagnetism $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$, the remnant of $\mathrm{SU}{\left(2\right)}_R\times \mathrm{U}{\left(1\right)}_{Y\mathrm{dem}}$, whose charge is the square root of mass [8,12]. What the programme conspicuously lacks is a cosmology: an account of the expansion history, of the origin and growth of structure, of galaxy rotation curves and cluster dynamics, and of the largest-scale properties of the observed universe. This paper constructs that sector.

Two structural features of the theory constrain the construction from the outset, and both are unusual. First, there is a desert: no new dynamics between the Planck scale and the electroweak scale. Second — and more radically — in this theory classical spacetime itself is not available above the electroweak scale. The symmetry breaking that gives the standard-model bosons their masses is, in this programme, the same event as the left–right symmetry breaking that separates the two four-dimensional spacetime leaves from the six-dimensional split-signature arena, and the same event as the quantum-to-classical transition of the underlying matrix (trace) dynamics [7,8,9,11]. Above the electroweak scale there is no classical metric, no light cone, and no cosmic time; there is only the pre-geometric phase. Consequently the seeds of cosmic structure cannot be laid before the electroweak epoch, because “before” is not, in the classical sense, defined. Any cosmology for this theory must generate its initial conditions at the emergence event itself.

We take this constraint seriously and quantitatively (Sec. 3). It forces a choice between two logical possibilities: either structure is seeded causally after emergence — which we show fails by ten orders of magnitude in length scale, and is in any case excluded by the phase coherence of the acoustic peaks — or the seeds are laid down at emergence, inheriting the acausal correlations of the pre-geometric phase. The second option is available precisely because the pre-geometric phase of trace dynamics is non-local: matrix degrees of freedom are not separated by light cones, so correlations at emergence are not bounded by any horizon. The horizon, flatness, and relic problems are then not solved but dissolved, and no inflaton is required. What replaces the inflationary origin of perturbations is the spontaneous-localisation structure of the quantum-to-classical transition [15,16,17], operating at the emergence event.

The dark sector of the cosmology is constructed in Sec. 5 and Appendices A–F (subsuming, in corrected form, the author’s unpublished gravi–DEM notes), and it supersedes, on one decisive point, the earlier proposal of [12]. In [12] the MOND-like galactic phenomenology was attributed to the $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ vector force itself; but a healthy abelian vector gives repulsion between like charges, and in the late universe only the $+s$ dark-charge sector is populated. The vector $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ is therefore retired as the mediator of the missing-mass phenomenology: it remains in the theory as a genuine unbroken gauge symmetry whose gauged charge, by the anomaly no-go of [8], is carried by a dark sector while the visible fermions are dem-neutral, so that on ordinary matter it produces no long-range force at all (the massless dark photon decouples by a field redefinition; Sec. 5.5), its conceptual role being the early-universe dark-charge separation. In its place, MOND is derived from gravity itself: an infrared functional $S_{\mathrm{IR}}[g,T]$ of the metric and the vacuum’s preferred foliation, added to the Einstein–Hilbert term that the $\mathrm{SU}{\left(2\right)}_R$ Plebanski/MacDowell–Mansouri sector produces [7]. Its static asymptotics are fixed (up to one $\mathcal{O}\left(1\right)$ constant $\xi$) by de Sitter scale invariance of the $\mathrm{SU}{\left(2\right)}_R$ infrared vacuum (Appendices A–C). The single new scale is

$$a_0={\displaystyle \frac{c^2}{\xi {\mathcal{l}}_{\mathrm{dS}}}},$$

(1)

with ${\mathcal{l}}_{\mathrm{dS}}$ the de Sitter radius dynamically selected in the infrared; general relativity holds for accelerations above $a_0$ and MOND below it. The proposed completion contains one foliation scalar and no additional vector or tensor fields; no slip and $c_T=c$ are conditional linear results pending completion of the khronon sector. Dark energy is then not an input: the same infrared vacuum carries ${\mathrm{\Lambda }}_{\mathrm{eff}}=3c^2/{\mathcal{l}}_{\mathrm{dS}}^2=3{\xi }^2{a}_0^2/c^2$ (Appendix D). Matching the observed $a_0$ and $\mathrm{\Lambda }$ gives $\xi \simeq 4.55$, and an $E_6\times E_6$ vacuum-projection argument derives $\xi =\sqrt{2}{\left|S\right|}^{3/2}/\left|D\right|\simeq 3.85$ from the Jordan invariants of the electron-family vacuum — some $18\%$ above the empirical central value, within the $\sim 20\%$ systematic uncertainty of the empirical $a_0$ (Sec. 5.2, Appendix E): the century’s two “dark” coincidences — that rotation curves flatten at $a_0\sim c{H}_0$, and that the vacuum energy is of order the critical density — collapse into the single statement that both trace the one infrared length ${\mathcal{l}}_{\mathrm{dS}}$, with the proportionality constant supplied by the exceptional Jordan algebra. The remaining missing mass — clusters and the acoustic peaks — is closed by particle content the theory has carried since before any cosmological consideration: the three sterile right-handed neutrinos [6], as entropy-diluted thermal relics (Sec. 4).

The construction is also constrained — more accurately, is liberated — by the present observational situation, which the standard $\mathrm{\Lambda }$CDM model faces with growing discomfort. The matched dipole in the number counts of distant radio sources and quasars exceeds the kinematic expectation from the CMB dipole by a factor $\sim 2$, at $4.9\sigma$ [41,42,43] — an excess that has since survived clustering corrections [52] and an independent simulation-based reanalysis of the scanning-law systematics [53], the latter flagging an unresolved photometric-error caveat. Type Ia supernova compilations show a dipolar modulation of the deceleration parameter aligned with the local bulk flow, falling off with redshift [44,45,46]; and the most recent analysis of the Pantheon+ catalogue by Sah, Rameez and Sarkar, correcting for progenitor-age–dependent standardisation, finds that the monopole of the deceleration parameter shifts to positive values — a decelerating universe — while the local dipole survives [47]. These supernova claims are contested: reanalyses with standard light-curve standardisation and peculiar-velocity treatment find the acceleration robust above $4\sigma$ even when a dipole is allowed [48], cosmographic isotropy tests on Pantheon+ find no significant departure from isotropy [49], and the DES five-year sample yields a strongly accelerating fit within the standard framework [50]; the decelerating monopole of [47] rests on the progenitor-age standardisation, itself awaiting independent scrutiny. Meanwhile, from the opposite flank, the DESI DR2 baryon acoustic oscillations combined with CMB and supernova compilations prefer an evolving dark-energy equation of state over a cosmological constant at $\sim 3$–$4\sigma$, the significance depending on the supernova sample and below discovery threshold [51]. Bulk flows measured from CosmicFlows-4 are larger, and coherent to larger scales ($\gtrsim 200h^{-1}$Mpc), than $\mathrm{\Lambda }$CDM predicts [54]. The local luminosity density is anomalously low out to $\sim 300$ Mpc [58,59]. Structures such as the Huge-LQG and the Giant Arc have been reported on Gpc scales [60,61,62], and whether the survey volume exhibits any approach to homogeneity remains contested [63,64,67,68,71]. A cautionary coda to this literature arrived while this paper was in revision: Sylos Labini and Galoppo reported, from a pairwise-direction statistic applied to the DESI DR1 bright-galaxy and LRG samples, coherent anisotropic structures extending to Gpc scales in excess of $\mathrm{\Lambda }$CDM mocks at $>3\sigma$ [69]; Sawala has, however, identified a conflation of luminosity with comoving distances in that analysis — a spurious radial stretch of $(1+z)/h\simeq 1.5$–$1.8$ across the samples — and shows that at the correct comoving calibration, compared against mocks carrying galaxy bias and redshift-space distortions, the excess disappears and the most prominent reported structure reduces to the known Sloan Great Wall [70]. Pending the authors’ response, we treat that detection as unconfirmed and build nothing on it; we record the episode both because it calibrates the standard of care Gpc-scale claims require, and because the corrected null constrains this paper’s own proposal for enhanced large-scale power (Sec. 6). None of this refutes the cosmological principle, and we will not pretend it does. Isotropy at last scattering is secure at the ${10}^{-5}$ level. Some late-time probes report excess large-scale power or aligned dipoles, but their significances, covariances and systematics remain disputed; several modern clustering analyses recover an approach to homogeneity on surveyed scales. The record motivates a quantitative test, not a verdict of a preferred cosmological direction.

The central large-scale result is narrower. Below the acceleration threshold $a_0$, the infrared closure can enhance low-k density and velocity power. Rotationally invariant growth nevertheless preserves statistical isotropy, so it raises the variance of realisation-level bulk flows and dipoles but cannot select an ensemble-level axis. Whether the observed flow amplitudes, cross-probe alignments or homogeneity scale follow is a calibrated-pipeline question (Sec. 6); a direction fixed independently of the random density realisation would require the separate order parameter discussed in Sec. 7. On the background monopole, the theory takes a sharp stand: the relation (1) anchors $a_0$ to a de Sitter infrared and requires genuine late-time acceleration. If the decelerating monopole of [47] survives independent scrutiny of the progenitor-age systematics, the infrared anchor of $a_0$ fails and the galactic sector loses its cosmological origin. A third exposure is that ${\mathrm{\Lambda }}_{\mathrm{eff}}$ is constant, $w=-1$ exactly, so the current DESI-led preference for evolving dark energy [51] presses where the theory cannot bend (Sec. 7.3).

The paper is organised as follows. Section 2 summarises the inputs. Section 3 constructs the initial conditions at the emergence event. Section 4 follows the thermal history, computes the dark radiation, and fixes the collisionless sector. Section 5 presents the infrared gravitational sector and the galactic and cluster phenomenology. Section 6 treats the growth of structure. Section 7 treats the late-time universe. Section 8 collects predictions and falsifiers; Sec. 9 closes with the ledger. The relativistic gravi–DEM construction on which Sec. 5 and Sec. 6 rest is given in Appendices A–F. Claims are tagged [D] (derived here or in cited work), [P] (programme-level working hypothesis), [C] (conditional), [O] (open), [SPE] (speculative), [REQ] (consistency requirement), as in [8].

2. Inputs from the Programme

(i)

Pre-geometric phase and emergence.

The fundamental description is Adler’s trace dynamics [15], generalised to $E_8\times {}_{\omega }E_8$-valued degrees of freedom on a 16-dimensional split-bioctonionic space [2,9,10]. There is no classical spacetime in this phase; a Connes-type modular time provides the ordering parameter [11]. The emergence of classical spacetime is a spontaneous-localisation–driven quantum-to-classical transition [11,15,16], and it coincides with the electroweak symmetry breaking and with the left–right symmetry breaking [8]. The six-dimensional split-signature $(3,3)$ arena resolves into two four-dimensional Lorentzian leaves of opposite signature sharing a $(1,1)$ interface: our leaf, whose geometry (the broken $\mathrm{SU}{\left(2\right)}_R$ sector) is general relativity, and a second leaf whose geometry is the weak interaction [7]. [P] (the gravi-weak identification), [D] (the two-leaf reduction, given the constructions of [7]).

(ii)

The desert.

No new dynamics between the Planck and electroweak scales; the emergent couplings are fixed at the transition, where the fine-structure constant and the mass ratios are determined algebraically [4,5]. [P].

(iii)

Matter content.

One generation is a $\mathrm{Cl}\left(6\right)$ module; electric charge is $Q=N/3$; three generations arise from the flavour structure of the exceptional Jordan algebra, whose eigenvalues fix square-root mass ratios [3,4]. The fermion content includes exactly three sterile right-handed neutrinos ${\nu }_R$ [6]; there is no stable weak-scale relic, and no particle cold dark matter anywhere in the spectrum. Colour is single and vector-like [8]. [D] within the programme.

(iv)

The gravitational and dark sector (gravi–DEM).

As constructed in this paper (Sec. 5, Appendices A–F), superseding [12] on the mediator question:

(a)

Gravity. The $\mathrm{SU}{\left(2\right)}_R$ connection with Plebanski/MacDowell–Mansouri simplicity constraints yields the Einstein–Hilbert action for the emergent metric [7] (Appendix A). GR holds in the high-acceleration regime. [P]/[D]

(b)

Infrared functional. The deep infrared of the $\mathrm{SU}{\left(2\right)}_R$ sector realises de Sitter kinematics with radius ${\mathcal{l}}_{\mathrm{dS}}$; imposing dS scale invariance fixes the static asymptotics of an addition

$$S_{\mathrm{IR}}[g,T]={\displaystyle \frac{a_0^2}{16\pi G}}\int d^{4}x\sqrt{-g}\mathcal{F}\left({\displaystyle \frac{I[g,T]}{a_0^2}}\right),I[g,T]=a_{\mu }{a}^{\mu },$$

(2)

with $a_{\mu }$ the acceleration covector of the infrared foliation (Appendix C) and $a_0=c^2/\left(\xi {\mathcal{l}}_{\mathrm{dS}}\right)$, $\xi =\mathcal{O}\left(1\right)$. The proposed completion contains the foliation scalar but no additional vector or tensor fields. It does not modify the homogeneous background (where $a_{\mu }=0$), and in the static weak-field limit yields the Bekenstein–Milgrom (AQUAL) equation [27] (Appendix A). [P]/[C] (see Sec. 5.3 for its open points)

(c)

Dark electromagnetism. The unbroken $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ vector is standard and healthy, but the anomaly no-go of the companion strong-$CP$ analysis [8] fixes what it couples to: on the visible fermions no anomaly-free $\mathrm{U}\left(1\right)$ can carry the $\sqrt{m}$ pattern (every family-universal anomaly-free abelian charge on $\mathrm{SM}+{\nu }_R$ lies in $\mathrm{span}\{Q,B-L\}$, which $\sqrt{m}$ does not), so the gauged $Q_{\mathrm{dem}}$ carries the parity-mirror of electric charge on a dark sector and the visible fermions are dem-neutral; $\sqrt{m}$ is then a spectral label of the Jordan mass operator, not a gauged charge of ordinary matter. Two consequences follow for the phenomenology. (1) Because $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ is unbroken (massless), a massless kinetically-mixed dark photon is removed from the visible sector by a field redefinition and has no direct signature on ordinary matter by itself [39,40]: there is no $\sqrt{m}$-patterned composition-dependent fifth force to detect or to suppress, and the earlier Eötvös/MICROSCOPE framing of [12] does not apply. (2) The dark charged matter of $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$, if it forms a cosmological relic, appears as a millicharged species with the standard dark-photon/millicharge phenomenology [40], the only visible portal being the kinetic mixing $\epsilon F_{\mathrm{dem}}{F}_Y$. In the late universe only $+s$ dark charges are populated, so the residual like-sign repulsion within the dark sector is negligible; the vector’s roles are structural (the $E_6\times E_6$ charge bookkeeping) and early-cosmological (dark-charge separation at emergence). [D] (the no-go and the massless-decoupling)/[P] (the dark-sector realisation and $\epsilon$)

(d)

Dark energy. The dS infrared vacuum carries ${\mathrm{\Lambda }}_{\mathrm{eff}}=3c^2/{\mathcal{l}}_{\mathrm{dS}}^2=3{\xi }^2{a}_0^2/c^2$; no explicit cosmological constant is introduced (Appendix D). [P]

Remark 1 

(the interpolating function). The dS scale-invariance argument fixes only the two asymptotic limits of $\mathcal{F}$ ($\mathcal{F}\sim \frac{2}{3}{y}^{3/2}$ at small y, deep MOND; $\mathcal{F}\sim y$ at large y, GR; Appendix B). The exact interpolation adopted throughout is the parameter-free

$$\mu \left(x\right)={\displaystyle \frac{x}{1+x}},\mathcal{F}\left(y\right)=y-2\sqrt{y}+2ln(1+\sqrt{y}),$$

(3)

the “simple” function known to fit galaxy data well [28]; care is needed with the superficially natural alternative $\mathcal{F}=y+\frac{2}{3}{y}^{3/2}$ ($\mu =1+x$), whose regimes are inverted — it gives Newton at low and an enhanced force at high acceleration — and which is therefore not an admissible completion of the same asymptotics. The linear-perturbation closure of Sec. 6 is built from (3). [D] (consistency)

3. Initial Conditions from the Emergence Event

3.1. The Pre-Geometric Phase Is Not an Era

In $\mathrm{\Lambda }$CDM with inflation the chain of epochs is: quantum gravity, inflation, reheating, radiation era, BBN, recombination. In the present theory the chain is shorter and its first link is not an epoch. The pre-geometric phase is a matrix dynamics with no metric, no light cones, and no FRW scale factor; its “time” is the modular parameter of the equilibrium state [11]. It is meaningless to ask what the universe was doing “between the Planck time and the electroweak time”: the desert is not a period of classical expansion but the absence of any scale — and that scalelessness is the origin of the near scale invariance of the primordial spectrum (Sec. 3.3).

Classical cosmology begins at the emergence event, parametrised by the temperature $T_{*}$ of the radiation bath it deposits on our leaf; the identification of emergence with electroweak symmetry breaking fixes $T_{*}\simeq T_{\mathrm{EW}}\sim 100$ GeV. [P]. Below $T_{*}$ the theory is the standard model plus the dark sector, so BBN is standard in its reaction mechanism, but not identical in output: the predicted dark radiation changes the expansion rate as quantified in Sec. 4.1. [REQ]

3.2. the Scale Problem: Why Causal Seeding after Emergence Fails

The comoving Hubble radius at emergence is minute. With $H_{*}^2=(8{\pi }^3{g}_{*}/90)T_{*}^4/M_{\mathrm{Pl}}^2$, $g_{*}=106.75$, $T_{*}=100$ GeV,

$$H_{*}^{-1}\simeq 1.4\mathrm{cm},{\displaystyle \frac{a_0}{a_{*}}}\simeq 1.3\times {10}^{15},{\lambda }_{H,\ast }^{\mathrm{com}}\simeq 6\times {10}^{-4}\mathrm{pc}.$$

(4)

Every cosmologically relevant scale is enormously super-horizon at emergence: an 8 Mpc mode exceeds the emergence horizon by $e^{23}$, and the present comoving horizon ($\sim 14$ Gpc) exceeds it by $e^{31}$. [D]. Hence:

No causal process operating at or after the emergence event can seed structure on cosmological scales. The seeds must be inherited from the pre-geometric phase, whose correlations are not bounded by light cones.

The pre-geometric phase can supply them: in trace dynamics the matrix degrees of freedom are not spatially organised, and the equilibrium state carries correlations among the emergent positions of all degrees of freedom with no causal restriction [11,15]. What inflation achieves by stretching a causal patch, emergence achieves by never having been causal. [C] (conditional on the trace-dynamics equilibrium construction, problems (O1)–(O3) of [11]).

The same dissolution disposes of the classic puzzles: the horizon problem (uniformity is inherited from a single pre-geometric equilibrium state) [C]; the flatness problem (the leaves inherit the flat split metric of the bioctonionic background; robustness against localisation fluctuations remains to be shown) [C]/[O]; the relic problem (no classical epoch between $M_{\mathrm{Pl}}$ and $T_{\mathrm{EW}}$, hence no GUT relics) [D]. The entropy problem, honestly, is relocated rather than solved: the emergence event must deposit $\sim {10}^{40}$ Hubble volumes of 100 GeV plasma — entropy $\sim {10}^{88}$ within our present horizon — and this is the size of the underlying matrix ensemble, an input. [O]

3.3. the Seed Mechanism: Spontaneous Localisation at the Transition

The proposal, natural within the programme and with precedent in the collapse literature [16,17]:

The primordial density perturbations are the stochastic residue of the localisation events that create classical spacetime at the electroweak transition. [C]

Four properties are required, and we assess each.

(a)

Adiabaticity and phase coherence. [REQ]

The acoustic peak structure, and decisively the TE anticorrelation at $\mathcal{l}\sim 100$–200, demonstrate that the perturbations at last scattering were adiabatic, super-horizon, and phase coherent; causal “active” seeding is excluded. The localisation residue must therefore behave as a passive, super-horizon curvature perturbation, laid down once, coherently on all scales, at the (modular-time–sharp) transition, and the localisation noise must switch off once classicality is established — consistent with trace dynamics, where localisation rates scale with mass and the post-emergence universe is dominated by microscopic, effectively unitary, degrees of freedom. Plausible; not yet a theorem. [C]/[O]

(b)

Near scale invariance. [C]

Between the Planck scale (which sets the localisation dynamics) and the electroweak scale (which triggers the transition) the theory has no intermediate scale, so the correlations of the pre-geometric equilibrium state are scale-free over the entire range of emergent wavelengths, and the residue inherits a nearly scale-invariant spectrum — in the spirit of Hollands and Wald’s observation that scale invariance needs scale-free initial conditions, not inflation specifically [18]. The observed red tilt $n_s\simeq 0.965$ [73] must come from the weak scale dependence introduced by the transition itself ($T_{\mathrm{EW}}/M_{\mathrm{Pl}}$ being the only small parameter); its sign and magnitude are an open computation. [O]

(c)

Amplitude. [O]

The measured amplitude corresponds to curvature perturbations $\sim 4.6\times {10}^{-5}$. We record, without weight, that the programme’s derived fine-structure constant [5] gives ${\alpha }^2\simeq 5.3\times {10}^{-5}$. [SPE]. Absent a derivation, the amplitude is the one number the cosmology takes from observation, as inflation takes it via the potential normalisation. [O]

(d)

Gaussianity.

Residues summed over $\sim {10}^{88}$ localisation events are Gaussian by the central limit theorem; small local-type non-Gaussianity from the nonlinearity of the transition is expected, unquantified. [C]/[O]

3.4. Baryogenesis and Dark-Charge Separation at Emergence

The emergence event is simultaneously the P- and $CP$-structured left–right breaking: the order parameter is parity-odd [8], the CKM phase is fixed geometrically [13,14], and the transition is far from equilibrium by construction. Two logically distinct realisations of the baryon asymmetry must, however, be separated. (a) Born asymmetric: the localisation event deposits an emergent state with $B\ne 0$ — an initial condition, to which Sakharov’s theorem (which presumes symmetric initial conditions) does not apply; the question is then relocated into the pre-geometric ensemble, alongside the entropy of Sec. 3.2. (b) Dynamical: generated during the transition itself; but since emergence coincides with electroweak breaking, the sphalerons are at best marginally in equilibrium and freeze out immediately — the standard electroweak-baryogenesis window is closing exactly at $T_{*}$ — so realisation (b) requires B-violation in the transition dynamics, not the sphaleron. The theory’s P and $CP$ structure equips either route; deciding between them, and computing ${\eta }_B$ (or exhibiting it as an initial condition), is open. [O]. The same event must populate the $+s$ sector of the dark charge — the dark-charge separation of Sec. 2(iv) — plausibly by the same mechanism that separates matter from antimatter, since $s=\pm \sqrt{m/\kappa }$ is odd under the relevant conjugation; a joint derivation of ${\eta }_B$ and the $+s$ excess is a well-posed target. [O]

4. Thermal History, Dark Radiation, and the Sterile-Neutrino Sector

4.1. the Dark Thermal State, and a Two-Epoch Dark-Radiation Fingerprint

The dark photon of $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ and the three ${\nu }_R$ are far too weakly coupled to the visible plasma to thermalise by scattering (Sec. 5.5); their thermal population is instead a statement about the emergence event itself: the transition deposits a common equilibrium state across all sectors at $T_{*}$ — it is the quantum-to-classical transition of one matrix ensemble, not a coupling-mediated equilibration. [P]. Their temperature is subsequently diluted by visible entropy releases: $T_{\mathrm{dark}}/T_{\gamma }={(3.91/106.75)}^{1/3}\simeq 0.33$. Two consequences follow, at different epochs:

• At nucleosynthesis ($T\sim$ MeV) the dark photon and the three ${\nu }_R$ (then ultra-relativistic) contribute

  $$\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq \underset{{\gamma }^{\prime }:0.054}{\underbrace{\frac{8}{7}{\left(T_{\mathrm{dark}}/T_{\nu }\right)}^4}}+\underset{3{\nu }_R:0.14}{\underbrace{3{\left(T_{\mathrm{dark}}/T_{\nu }\right)}^4}}\simeq 0.19.$$

  (5)
• At recombination the ${\nu }_R$ have mass $\simeq 40$ eV (Sec. 4.2). They have long been non-relativistic ($m/T_{\mathrm{dark}}\simeq 500$ at $z_{\mathrm{rec}}$): they count as matter, not radiation, and only the dark photon remains,

  $$\Delta N_{\mathrm{eff}}^{\mathrm{CMB}}\simeq 0.054.$$

  (6)

[D] (given the thermal-emergence hypothesis). Both values are consistent with current bounds ($N_{\mathrm{eff}}^{\mathrm{CMB}}=2.99\pm 0.17$ [73]; BBN determinations at $\sigma \sim 0.2$–$0.3$). The BBN value gives the approximate helium shift $\Delta Y_p\simeq 0.013\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq +0.0025$. The prediction is the pattern: more dark radiation at BBN than at recombination, with the transition at $1+z_{\mathrm{nr}}\simeq 1.6\times {10}^5$, plus a percent-level shift of matter–radiation equality from the ${\nu }_R$’s relativistic tail. This two-epoch fingerprint is unique to a diluted eV–scale sterile sector and distinguishes it from both a massless dark fluid (equal $\Delta N_{\mathrm{eff}}$ at both epochs) and from no dark radiation. Stage-4-class sensitivity ($\sigma \left(N_{\mathrm{eff}}\right)\simeq 0.03$; the original CMB-S4 construction project is no longer funded [74,75]) together with improved $Y_p$ and D/H would probe both numbers. One caution: at $\sigma \left(N_{\mathrm{eff}}\right)\simeq 0.03$ the CMB-epoch value alone sits only $1.8\sigma$ from zero, so a robust $\Delta N_{\mathrm{eff}}^{\mathrm{CMB}}=0$ at that precision is tension, not refutation — the falsifying measurement is a joint BBN–CMB analysis excluding the ordered pair $(0.19,0.054)$ at a stated threshold. Conversely, if the dark sector is born cold rather than thermal, both signals vanish, so $\Delta N_{\mathrm{eff}}$ directly probes the thermal character of the emergence event. [D]/[REQ]

4.2. the Right-Handed Neutrinos Close the Matter Budget

A collisionless clustering component is not optional. With baryons alone (${\omega }_m={\omega }_b\simeq 0.022$), matter–radiation equality falls at $z_{\mathrm{eq}}\simeq 5\times {10}^2$ and the turnover moves to $k_{\mathrm{eq}}\simeq 1.6\times {10}^{-3}{\mathrm{Mpc}}^{-1}$, far from the observed $P\left(k\right)$; pre-recombination baryons oscillate with the photons instead of growing; the gravitational-potential decay during radiation domination boosts the first acoustic peak and suppresses the third, contrary to Planck; and the CMB lensing amplitude is underproduced. None of this is repaired by $S_{\mathrm{IR}}$, which is inert on the homogeneous background and gated off at recombination (Sec. 6.4), nor by ${\mathrm{\Lambda }}_{\mathrm{eff}}$. [D] (standard physics). The theory must therefore supply a component that clusters like CDM on acoustic and cluster scales — and it does, without new particle content.

The theory contains exactly three ${\nu }_R$ [6], sterile under the visible gauge group, with dark charge zero (the $\sqrt{m}\to 0$ entry of the trace split) but gravitating. With the diluted temperature above, their relic number density per species is $n_{{\nu }_R}/n_{\nu }\simeq 0.10$, and

$${\mathrm{\Omega }}_{{\nu }_R}{h}^2\simeq {\displaystyle \frac{{\sum }_i{m}_{{\nu }_R,i}}{94\mathrm{eV}}}\times 0.10⟹{\mathrm{\Omega }}_{{\nu }_R}{h}^2=0.12⟺\mathrm{tr}{M}_R\simeq 113\mathrm{eV}.$$

(7)

[D] (arithmetic; the trace is an open programme parameter, fixed here by ${\mathrm{\Omega }}_m$, [O]). The quasi-degenerate point is $m_{{\nu }_R}\simeq 38$ eV per species, rounded below to $\sim 40$ eV. It reproduces the entire measured matter density, ${\mathrm{\Omega }}_m{h}^2\simeq 0.14={\mathrm{\Omega }}_b{h}^2+{\mathrm{\Omega }}_{{\nu }_R}{h}^2$, with ${\mathrm{\Omega }}_b{h}^2=0.0224$ from BBN. The kinematics is decisive and welcome:

• the ${\nu }_R$ become non-relativistic at $1+z_{\mathrm{nr}}\simeq 1.6\times {10}^5$, well before matter–radiation equality, so the background expansion history is that of $\mathrm{\Lambda }$CDM up to the percent-level early-radiation shift noted above; [D]
• free streaming begins at the comoving horizon at $z_{\mathrm{nr}}$, ${\lambda }_{\mathrm{fs}}\left(z_{\mathrm{nr}}\right)\simeq 2.5$ Mpc, but continues to accumulate as the relics coast: ${\lambda }_{\mathrm{fs}}\simeq 14$ Mpc by recombination and $\simeq 17$ Mpc today. Above the accumulated scale they cluster approximately like CDM; below it the linear transfer function is progressively suppressed, a graded cut rather than a step. Their late-time thermal velocities still permit nonlinear infall on cluster scales. [D]

Two particle-physics conditions are load-bearing. (i) Negligible active–sterile mixing: any appreciable mixing would add Dodelson–Widrow-type oscillation production to the diluted thermal abundance and activate decay constraints; the abundance bookkeeping above therefore adopts the quarantined active–sterile bridge of the book’s fermion-sector analysis. (ii) Mass-sector consistency: that same combined analysis permits $\mathrm{tr}{M}_R\simeq 113$ eV, near the quasi-degenerate $m_{{\nu }_R}\simeq 40$ eV point, only on its direct-mass (Fork-A) branch, which also stakes inverted active-neutrino ordering, $\sum m_{\nu }=0.0997$ eV and $m_{\beta \beta }=18.1$ meV. These are adopted inputs from the programme’s left–right symmetric neutrino construction [6], not results of the cosmological calculation. Every sterile-sector conclusion below is conditional on this combined package. [C]/[REQ]. The result is a scale-dependent dark sector fixed by particle content: linear modes above the accumulated $\sim 14$–17 Mpc free-streaming length see an approximately CDM-like component; linear galactic-scale power is strongly suppressed, while the relics can still fall nonlinearly into cluster potentials. Galactic missing-mass phenomenology is carried by the infrared gravitational sector. This is the hybrid that MOND phenomenology has long required empirically — Angus’s 11 eV thermal sterile neutrino [35] — here forced, with the mass rescaled to $\sim 40$ eV by the entropy dilution, by a particle content fixed years before any cosmological consideration [6]. Phase-space (Tremaine–Gunn) bounds for cluster cores require $m\gtrsim 2$ eV and are satisfied with more than an order of magnitude to spare. [D]

We emphasise what this is not: it is not a reintroduction of particle cold dark matter. The ${\nu }_R$ are hot-born, free-stream out of galaxies, cannot form galactic halos or cusps, and are invisible to every direct-detection and collider search for weak-scale dark matter. The prediction that all such searches continue to return null results is unconditional. [D]

Relation to earlier MOND–sterile hybrids.

The combination MOND + eV-scale sterile neutrino is not new, and priority must be assigned accurately. Angus [35] showed that an 11 eV thermal sterile neutrino reconciles MOND with clusters and the CMB, and the $\nu$HDM cosmology built on it by Haslbauer, Banik and Kroupa [59] used MOND’s enhanced long-wavelength growth to explain the KBC void and the Hubble tension — a genuine precedent for the large-scale phenomenology of Sec. 6 and Sec. 7. The present construction differs in four respects. (i) Nothing is chosen. In the hybrid literature MOND is postulated, $a_0$ is fitted, and the sterile mass and abundance are selected to fit; here MOND descends from the $\mathrm{SU}{\left(2\right)}_R$ infrared functional, $a_0$ is anchored to $\mathrm{\Lambda }$ with its coefficient computed from Jordan invariants, the three ${\nu }_R$ were fixed by the fermion construction [6] years before any cosmological use, their abundance follows from the emergence-thermal dilution, and the mass is then pinned by ${\mathrm{\Omega }}_m$. (ii) The dilution rescues the hybrid. A fully thermalised eV-scale sterile contributes $\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq 1$, now excluded at high significance [73]; the diluted relic gives the allowed — and distinctive — two-epoch pattern of Sec. 4.1. (iii) A proposed relativistic completion with one khronon and no additional vector or tensor fields; no slip and $c_T=c$ are conditional linear results pending the W1 completion (Appendices A–C), where the earlier hybrids had no relativistic parent. (iv) The initial-conditions sector of Sec. 3 has no counterpart in any MOND cosmology. [D] (the comparisons)/[C] (the constructions compared to)

4.3. the CMB Anisotropies

At last scattering the model contains: baryons and photons (standard), three active neutrinos (standard), the dark photon ($\Delta N_{\mathrm{eff}}\simeq 0.05$), and a clustering component ${\mathrm{\Omega }}_{{\nu }_R}\simeq 0.25$ that on acoustic scales is indistinguishable from CDM; the primordial spectrum is adiabatic, coherent, and nearly scale-invariant (Sec. 3.3); and the infrared functional is gated off on the acoustic scales at that epoch (Sec. 6.4). The acoustic peak structure — including the third-peak height that defeats baryon-only MOND universes [36] — is therefore expected at the $\mathrm{\Lambda }$CDM level, with calculable deviations: the dark radiation of Sec. 4.1, and the graded free-streaming suppression of the ${\nu }_R$ transfer function below the accumulated $\simeq 14$ Mpc recombination scale. The quantitative statement requires a Boltzmann computation with the diluted ${\nu }_R$ distribution function and the $S_{\mathrm{IR}}$ closure included; until it is done the claim “the CMB is fit” carries the grade [C], resting on the proximity of the setup to the computed 11 eV-sterile scenario [35]. [C]/[O]

5. Infrared Gravity: Galaxies, Clusters, and the $a_0$–$\mathrm{\Lambda }$ Relation

5.1. the Static Limit: AQUAL from the Infrared Functional

In the static weak-field limit the variation of $S_{\mathrm{IR}}$ (2) yields the Bekenstein–Milgrom equation [27] (Appendix A; the Einstein–Hilbert ${|\nabla \mathrm{\Phi }|}^2$ piece is replaced, not double-counted, by the aquadratic functional)

$$\nabla ·\left[\mu \left({\displaystyle \frac{|\nabla \mathrm{\Phi }|}{a_0}}\right)\nabla \mathrm{\Phi }\right]=4\pi G\rho ,\mu \left(x\right)={\displaystyle \frac{x}{1+x}},$$

(8)

The interpolation is specified in Remark 1. In spherical symmetry, the algebraic QUMOND-form map is

$$g=\nu \left({\displaystyle \frac{g_N}{a_0}}\right)g_N,\nu \left(y\right)={\displaystyle \frac{1}{2}}\left(1+\sqrt{1+\frac{4}{y}}\right),$$

(9)

so $g\to g_N$ for $g_N\gg a_0$ (GR) and $g\to \sqrt{g_N{a}_0}$ for $g_N\ll a_0$ (deep MOND [26]). Immediately: flat rotation curves; the baryonic Tully–Fisher relation $v^4=G{M}_b{a}_0$ with small intrinsic scatter [29]; and the radial-acceleration relation with the “simple” function, which fits the SPARC data well [28,30]. The static weak-field construction has no gravitational slip, $\mathrm{\Psi }=\mathrm{\Phi }$: lensing follows the same potential that governs dynamics. This is conditional on the khronon completion not reintroducing slip, and is a clean, falsifiable difference from relativistic-MOND constructions carrying extra dynamical fields [37]. [C]/[O]

Two Solar-System remarks keep this honest. First, the magnitudes: $g_N/a_0\simeq 5\times {10}^7$ at Earth’s orbit and $5\times {10}^5$ at Saturn, so the MOND regime is nowhere near. Second, the tail: the simple function approaches GR only as $\mu \simeq 1-a_0/g_N$, leaving fractional anomalies $\sim 2\times {10}^{-6}$ at Saturn and $\sim 2\times {10}^{-5}$ at Neptune, a range probed by planetary ephemerides; combined Solar-System and rotation-curve analyses already disfavour interpolating functions with such slowly decaying tails [31]. Since the dS scale-invariance argument fixes only the two asymptotic limits of $\mathcal{F}$, not the interpolation, an exponentially suppressed high-acceleration tail remains available; but this must be stated as a constraint the derivation of $\mathcal{F}$ has to meet, not waved away. [REQ]/[O]. The corresponding robustness statement also deserves stating: every galaxy, cluster, and cosmological result in this paper probes $x\lesssim$ a few, and is therefore insensitive to the high-acceleration tail — replacing (3) by an ephemeris-safe variant with a faster approach to GR changes nothing else in the paper. [D]

Because matter couples minimally to $g_{\mu \nu }$, the weak equivalence principle is exact at this order: the MOND response is universal, composition-independent, and MICROSCOPE-safe by construction [38]. This repairs the sharpest phenomenological liability of the earlier vector-mediated proposal [12], in which the dark charge ${\sum }_i\sqrt{m_i}$ would have imprinted composition dependence on the fifth force. Under the anomaly no-go of [8] there is, moreover, no residual composition-dependent force to worry about: the visible fermions are dem-neutral, and the unbroken (massless) $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ decouples from ordinary matter by a field redefinition, leaving no $\sqrt{m}$-patterned long-range force on visible matter at all (Sec. 5.5). [D]

5.2. the $a_0$–$\mathrm{\Lambda }$ Relation

The infrared vacuum fixes both the MOND scale and the effective cosmological constant from the single length ${\mathcal{l}}_{\mathrm{dS}}$ (Appendix D):

$$a_0={\displaystyle \frac{c^2}{\xi {\mathcal{l}}_{\mathrm{dS}}}},{\mathrm{\Lambda }}_{\mathrm{eff}}={\displaystyle \frac{3c^2}{{\mathcal{l}}_{\mathrm{dS}}^2}}={\displaystyle \frac{3{\xi }^2{a}_0^2}{c^2}}.$$

(10)

With the Planck value $\mathrm{\Lambda }=1.106\times {10}^{-52}{\mathrm{m}}^{-2}$: ${\mathcal{l}}_{\mathrm{dS}}=\sqrt{3/\mathrm{\Lambda }}\simeq 1.65\times {10}^{26}$ m, and matching $a_0=1.2\times {10}^{-10}$ m ${\mathrm{s}}^{-2}$ (central value; the empirical determination carries a $\sim 20\%$ systematic [28,30]) gives

$${\xi }_{\mathrm{obs}}\simeq 4.55.$$

(11)

[D] (arithmetic). This number is not left as an unexplained $\mathcal{O}\left(1\right)$ input: an $E_6\times E_6$ vacuum-projection argument (Appendix E) fixes it in terms of the Jordan invariants $S,D$ of the exceptional order parameter,

$$a_0={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{\left|D\right|}{{\left|S\right|}^{3/2}}}{c}^2\sqrt{{\displaystyle \frac{\mathrm{\Lambda }}{3}}}⟺{\xi }_{E_6}={\displaystyle \frac{\sqrt{2}{\left|S\right|}^{3/2}}{\left|D\right|}},$$

(12)

and with the electron-family (Dirac-set) vacuum invariants of the mass-ratio programme ($S=-\frac{7}{6}$, $D=-\frac{25}{54}$ [4]) this gives ${\mathrm{\Theta }}_{E_6}\equiv \left|D\right|/{\left|S\right|}^{3/2}=0.3674$, hence $a_0^{E_6}=1.42\times {10}^{-10}$ m ${\mathrm{s}}^{-2}$ and ${\xi }_{E_6}=3.85$, against ${\xi }_{\mathrm{obs}}\simeq 4.55$: some $18\%$ above the empirical central value, inside the $\sim 20\%$ systematic of $a_0$. [C] (on the assumptions of Appendix E). Correction. An earlier version of this paper evaluated (12) with the pair $(S,D)=(-\frac{7}{6},-\frac{25}{64})$ and reported ${\xi }_{E_6}=4.562$, a few-per-mille central-value match. That determinant is a misreading, inconsistent with its own S: for the electron-family matrix with charge centre $q=\frac{1}{3}$ and vanishing coassociative cubic invariant, the slice arithmetic of [4] gives $\mathrm{\Sigma }=3q^2-S=\frac{3}{2}$ and $D=q^3-q\mathrm{\Sigma }=-\frac{25}{54}$ exactly. The few-per-mille match is therefore withdrawn; what survives is an order-unity derivation of the coefficient from the exceptional Jordan algebra, accurate at the $\sim 18\%$ level. Three cautions fix the epistemic weight of this agreement. (i) At the present $\sim 20\%$ systematic precision of $a_0$, all three normalisations catalogued in Appendix E — ${\mathrm{\Theta }}_{E_6}/\sqrt{2}=0.260$, $1/2\pi =0.159$, $1/6=0.167$, in units of $a_{\mathrm{\Lambda }}=c^2\sqrt{\mathrm{\Lambda }/3}$ — straddle the empirical ratio $0.220$ from above and below, and none is discriminated: the agreement is not yet discriminating, and the claim we stake is structural ($a_0$ and $\mathrm{\Lambda }$ from one length, the coefficient from Jordan invariants), not one of achieved precision. (ii) The factor $1/\sqrt{2}$ enters through the equal-weight projection assumption (iv) of Appendix E — a modelling choice, not a theorem. (iii) The selection of the electron-family vacuum among the validated Jordan matrices is open [O], and — sharper after the correction — so is the selection of its Dirac-set invariants: the validated Majorana-set lepton pair $(S,D)=(-\frac{1}{24},-\frac{19}{216})$ gives ${\mathrm{\Theta }}_{E_6}=10.3$ and no viable $a_0$. Falsifiability requires tabulating the ${\mathrm{\Theta }}_{E_6}$ of the up- and down-family vacua alongside (Appendix E). The content of (10)–(12) is that the two great “dark” coincidences — $a_0\sim c{H}_0$ and ${\rho }_{\mathrm{\Lambda }}\sim {\rho }_{\mathrm{crit}}$ — are one statement: both observables measure ${\mathcal{l}}_{\mathrm{dS}}$, with the proportionality constant supplied by the exceptional Jordan algebra. Since ${\mathcal{l}}_{\mathrm{dS}}$ is fixed in the infrared, $a_0$ is epoch-independent: the RAR normalisation does not run with redshift (the effective-distance bookkeeping of Appendix A freezes $R_H$ at its present value in the deep-MOND sector). This cleanly distinguishes the theory from proposals with $a_0\left(z\right)\propto H\left(z\right)$, and is testable with high-z rotation curves. [C] (the freezing itself is a prescription whose derivation from the $\mathrm{SU}{\left(2\right)}_R$ vacuum is open, [O])

5.3. What Remains Open in the Infrared Functional

Rigour requires isolating what the construction of Appendices A–E establishes and what it does not.

• Covariance and the preferred foliation. The acceleration covector is made covariant by a foliation scalar $T\left(x\right)$: $u_{\mu }=-{𝜕}_{\mu }T/\sqrt{-{(𝜕T)}^2}$, $a_{\mu }=u^{\nu }{\nabla }_{\nu }{u}_{\mu }={\nabla }_{\mu }lnN$ (Appendix C). The $\mathrm{SU}{\left(2\right)}_R$ infrared vacuum selects this timelike direction — boosts are spontaneously broken to rotations, and the foliation coincides with the CMB rest frame — so the preferred frame is the order parameter of the vacuum, not an ad hoc insertion; it also protects the early universe (Sec. 6.4). At linear order $a_{\mu }$ is unperturbed by transverse-traceless modes, so the provisional tensor equation has $c_T=c$. This statement, together with the no-slip result, is conditional on the W1 khronon completion. What remains is to derive the foliation dynamics from the $\mathrm{SU}{\left(2\right)}_R$ vacuum and check PPN ${\alpha }_1,{\alpha }_2$, gravitational Cherenkov emission, stability and the completed tensor sector. [C]/[O]
• Derivation of $\mathcal{F}$, ξ, and the frozen $R_H$. The scale-invariance argument fixes the asymptotics of $\mathcal{F}$ but not the interpolation (Remark 1; the ephemeris constraint above may force a faster tail); the coefficient $\xi$ is now derived from $E_6$ invariants under the stated projection assumptions (Appendix E), but the selection of the electron-family vacuum, and the frozen-$R_H$ prescription, still want a derivation from the trace-dynamics infrared. [C]/[O]
• The external-field effect. AQUAL-type nonlinearity implies environmental dependence (the external-field effect), empirically supported in galaxies; the foliation completion inherits it (a mild, derived violation of the strong — not weak — equivalence principle, Appendix C). Its cosmological analogue — how the background acceleration state gates the local $\mu$ — must be made precise, since it controls the onset of the linear-regime enhancement (Sec. 6). [O]

5.4. Clusters

With (9), cluster hydrostatic masses are boosted in the outskirts but under-closed overall — the classic factor-$\sim 2$ MOND shortfall [34,35]. In this theory the residue is not optional and not adjustable: it is the ${\nu }_R$ component of Sec. 4.2. Its linear transfer is suppressed below the accumulated $\sim 14$–17 Mpc free-streaming length, but its late-time velocity dispersion permits nonlinear infall into cluster potentials. The working rich-cluster picture is therefore gas + stars + a $\sim 40$ eV ${\nu }_R$ atmosphere concentrated on ∼ Mpc scales, with the MOND map applied to the total Newtonian field, as set out in Appendix F. Because $\mathrm{\Psi }=\mathrm{\Phi }$, lensing masses equal dynamical masses including the “phantom” MOND contribution; in mergers (Bullet-type) the collisionless ${\nu }_R$ component produces galaxy-aligned lensing peaks displaced from the shocked gas. Quantitative fits to relaxed clusters and to merger lensing maps, with the particle mass fixed at $\simeq 40$ eV by (7), are a no-free-parameter test; the ${\nu }_R$ atmospheres additionally carry a universal phase-space signature (a common m and $T_{\mathrm{dark}}$), distinguishing them from cold halos. [C]/[O]

5.5. The Residual Role Of $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$

The anomaly no-go of the companion strong-$CP$ analysis [8] settles what $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ can and cannot do, and it removes — rather than merely bounds — the experimental liabilities the earlier vector-mediated picture [12] carried. On the visible fermions no anomaly-free abelian charge realises the $\sqrt{m}$ pattern: every family-universal anomaly-free $\mathrm{U}\left(1\right)$ on the $\mathrm{SM}+{\nu }_R$ content lies in $\mathrm{span}\{Q,B-L\}$, and the $\sqrt{m}$ values sit outside it under any sign or chirality assignment. The visible fermions are therefore dem-neutral; the gauged $Q_{\mathrm{dem}}=T_R^3+Y_{\mathrm{dem}}$ carries the parity-mirror of electric charge on a dark sector, and the visible $\sqrt{m}$ is a spectral label of the Jordan mass operator, not a gauged charge of ordinary matter.

Two phenomenological statements follow, and they reverse the earlier reading. First, the equivalence-principle channel is empty, not merely quiet. Since $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ is unbroken and its photon massless, a massless kinetically-mixed dark photon can be removed from the visible sector by a field redefinition and produces no force on ordinary matter by itself [39,40]. There is no $\sum \sqrt{m}$-patterned composition-dependent force to detect: the MICROSCOPE-based bound ${\alpha }_{\mathrm{dem}}\lesssim {10}^{-12}$ quoted for the vector-mediated proposal constrains a coupling the theory now sets to zero, and is not a constraint the present construction must satisfy. Correspondingly, a future detection of a $\sum \sqrt{m}$-patterned long-range force on ordinary matter would falsify this framework, not confirm it — the opposite of the earlier claim — since the theorem forbids visible matter from carrying that charge.

Second, the one genuine visible portal is kinetic mixing. If the dark charged matter of $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ survives as a cosmological relic, kinetic mixing $\frac{1}{2}\epsilon F_{\mathrm{dem}}^{\mu \nu }{F}_{\mu \nu }^Y$ endows it with a small ordinary electric charge $\sim \epsilon g_{\mathrm{dem}}$, i.e. it behaves as a millicharged species with the standard, much-studied dark-photon phenomenology (CMB spectral distortions, $\Delta N_{\mathrm{eff}}$, direct-detection millicharge bounds) [40]. This is a hypercharge-patterned coupling of magnitude $\epsilon$, not a $\sqrt{m}$-patterned one, and $\epsilon$ is undetermined here — a target for the dark-sector construction, not a number this paper predicts. Whether such a relic exists at all depends on the dark-sector content that realises the mirror-canonical charge, which is likewise not fixed here.

The vector’s indispensable roles are untouched by all of this, because they are dark-sector-internal: the $E_6\times E_6$ charge bookkeeping, and the dark-charge ($+s$) separation at emergence (Sec. 3.3). Finally, the feebleness that these considerations imply for any visible coupling is the same feebleness that prevents the dark sector from thermalising by scattering, which is why its thermal population must be attributed to the emergence event itself (Sec. 4.1), not to equilibration with the visible plasma. [D] (the no-go and massless decoupling)/[P] ($\epsilon$ and the dark relic)/[O] (the dark-sector content)

6. Growth of Structure after Last Scattering

6.1. the Linear-Regime Closure

In Newtonian gauge with $\mathrm{\Psi }=\mathrm{\Phi }$, the quasistatic scalar sector closes with the QUMOND-form map (9) applied to the perturbation field (the static-limit closure of Appendix D, built on the interpolating function (3), so that the enhancement switches on once accelerations drop below $a_0$):

$$\begin{array}{cccc}\hfill -k^2\mathrm{\Phi }& =4\pi G{a}^2{\overline{\rho }}_m\nu \left(y_N\right){\delta }_m,\hfill & \hfill y_N& \equiv {\displaystyle \frac{g_N(k,a)}{a_0}},\hfill \\ \hfill g_N(k,a)& \simeq {\displaystyle \frac{4\pi Ga{\overline{\rho }}_m{\Delta }_m(k,a)}{k}},\hfill & \hfill {\Delta }_m^2& \equiv {\displaystyle \frac{k^3{P}_m\left(k\right)}{2{\pi }^2}}.\hfill \end{array}$$

(13)

where $g_N$ is the rms peculiar Newtonian acceleration carried by modes near k (note the dimensionless band amplitude ${\Delta }_m$, not the Fourier coefficient, enters). Two caveats define the closure’s status. The AQUAL nonlinearity does not strictly linearise, and (13) is the standard mean-field treatment [32,33]. More importantly, the nonlinearity responds to the total local field, so the gate at wavenumber k is in truth the environmental acceleration — the total rms field smoothed on scale $\sim 1/k$, which at late times is dominated by smaller scales — and (13) is therefore an upper bound on the enhancement at k; this external-field gating is the same physics as the curl-field corrections below. The growth equation is

$$\ddot{\delta }+2H\dot{\delta }=4\pi G{\overline{\rho }}_m\nu \left(y_N\right)\delta .$$

(14)

[C] (on the quasistatic mean-field closure; valid for $k\gg aH$).

Two structural features follow. (i) The enhancement is acceleration-gated: $\nu >1$ precisely where $g_N<a_0$. Peculiar accelerations on linear scales today are $g_N\sim {10}^{-13}$–${10}^{-12}$ m s${}^{-2}\ll a_0$: the entire linear regime sits below the threshold, and the largest scales, carrying the smallest accelerations, sit deepest. (ii) Self-regulation: as $\delta$ grows, $g_N$ grows, $\nu \to 1$, and growth reverts to Newtonian — the enhancement shuts itself off as $g_N\to a_0$. [D] (given (13))

A third feature is a warning, and we state it as such. Taken at face value, the mean-field closure with $\nu \simeq \sqrt{a_0/g_N}\sim 30$ on linear scales would overproduce late-time power and flows — an unregulated deep-MOND acceleration sustained for a Hubble time corresponds to velocities $\sim a_0{t}_0\sim 5\times {10}^4$ km ${\mathrm{s}}^{-1}$, two orders above the observed few hundred. The classic MOND structure-formation analyses found the same overshoot tendency and identified the mitigations [32,33]: the curl (solenoidal) field of the exact AQUAL equation, absent in the mean-field map, and the external-field gating by neighbouring structure both reduce the effective enhancement, plausibly by an order of magnitude. The observed bulk-flow amplitudes therefore calibrate the effective cosmological gating — a consistency requirement on the pipeline, not a free success — and the same pipeline must confront existing $E_G$/redshift-space-distortion constraints, which probe exactly this $\nu$ and the absence of slip. [REQ]/[O]

6.2. the deep-MOND Growth Law

In the deep regime $\nu \simeq {(a_0/g_N)}^{1/2}$, so the source in (14) scales as $\sqrt{\Delta }$, and the equation is genuinely nonlinear even for small amplitudes. In the matter era the growing solution is a power law: with $4\pi G\overline{\rho }\nu \Delta ={\left(4\pi G\overline{\rho }{a}_{0}k/a\right)}^{1/2}{\Delta }^{1/2}$ one finds

$$\delta \propto t^{4/3}\propto a^2\left(\mathrm{deep}\mathrm{MOND},\mathrm{EdS}\right),$$

(15)

[D] (given the closure): the formal deep-MOND attractor doubles the growth exponent. If a mode remained on this branch from $z=1090$ to the present, its amplitude would acquire an extra factor $\sim 1100$ relative to standard $\delta \propto a$ growth. This is an excluded upper-bound history, not a viable prediction: Eq. (19) shows that even a $20\%$ integrated enhancement requires an average exponent $\overline{p}\lesssim 1.026$, while an unregulated $p=2$ phase can begin only at $z\lesssim 0.2$. Likewise, the often-quoted estimate that a baryonic ${\delta }_{\mathrm{rec}}\sim {10}^{-5}$ mode would reach nonlinearity near $z\sim 2.5$ is only the unregulated toy solution. Whether the regulated, scale- and environment-dependent history produces earlier structure than $\mathrm{\Lambda }$CDM — and whether it can account for the JWST high-redshift candidates [79] without overproducing low-redshift power — is an output of the calibrated pipeline. [C]/[REQ]/[O]

6.3. the Small-Scale frontier: Lyman- $\alpha$, Dwarfs, and Reionization

We now state plainly what may be the model’s decisive vulnerability. After recombination neither component carries unsuppressed primordial power on small scales: the accumulated ${\nu }_R$ free-streaming length is ${\lambda }_{\mathrm{fs}}\simeq 14$ Mpc comoving at recombination, while baryon power is Silk-damped below $\sim 10$–30 Mpc. The surviving primordial power is therefore confined to a narrow window of roughly 15–30 Mpc, narrower than an estimate based only on the $2.5$ Mpc horizon at $z_{\mathrm{nr}}$. Everything below $\sim 15$ Mpc — galaxies, the Lyman-$\alpha$ forest and the sources of reionization — must be regenerated top-down by growth, collapse and fragmentation of the window modes. If, purely as an upper-bound timing exercise, a window mode with ${\delta }_{\mathrm{rec}}\sim {10}^{-3}$ followed $a^2$, it would reach nonlinearity at $1+z\sim 30$. The growth-history constraint (19) rules out applying that unregulated law globally, so neither a $z\sim 10$–30 collapse epoch nor timely fragmentation follows without the calibrated nonlinear calculation. [C]/[REQ]/[O]

The named tests, in increasing order of severity: dwarf-galaxy abundance and satellite counts (top-down fragmentation must reproduce the low-mass end); reionization timing ($\tau$, and the dark-ages/cosmic-dawn 21-cm signal — the direct probe of the recombination-onset MOND era of Sec. 6.4); and, sharpest, the Lyman-α forest at $z\simeq 2$–5, which in $\mathrm{\Lambda }$CDM requires quasi-linear power at $0.1$–1 Mpc already at those redshifts and which excludes thermal warm dark matter far colder than a 40 eV relic whenever the relic is required to carry primordial small-scale power. The present model evades the WDM bound only if the regenerated (non-primordial) small-scale field at $z\sim 3$ mimics the observed flux power — a statement no analytic estimate can secure, requiring the pipeline of Sec. 6.6 extended into the quasi-nonlinear regime, with hydrodynamics. We record this as the model’s sharpest open confrontation: the cascade must bridge roughly two decades, from the 15–30 Mpc window to the sub-Mpc forest; if it underproduces the $z\sim 3$ flux power, the model dies here. [REQ]/[O]

6.4. Why the Primary CMB Is Protected

With the proposed covariant completion $a^{\mu }=u^{\nu }{\nabla }_{\nu }{u}^{\mu }$ (Sec. 5.3), the acceleration that gates $\mathcal{F}$ is that of the matter frame itself. Before decoupling the baryon–photon fluid undergoes driven acoustic oscillations whose accelerations on the peak scales exceed $a_0$ by orders of magnitude: the functional sits on its GR branch, and the mean-field closure leaves the primary anisotropies approximately standard. After decoupling the baryons fall ballistically; their accelerations are the tiny peculiar-gravity values of Sec. 6, far below $a_0$, and the enhancement can switch on. Thus recombination is the proposed onset epoch of the cosmological MOND regime, conditional on the matter-frame gating and its khronon completion; it is not yet a theorem of a completed action. Its direct observational probe is the dark-ages/cosmic-dawn 21-cm signal (Sec. 6.3 and Sec. 8). Near-horizon modes, where the quasistatic closure fails, require the full pipeline. [C]/[O]

6.5. from Isotropic Growth to Realisation-Level Flows and Dipoles

The first result is a no-go statement. Let statistically isotropic initial perturbations obey

$$〈{\delta }_i\left(\mathbf{k}\right){\delta }_i^{*}\left({\mathbf{k}}^{\prime }\right)〉={\left(2\pi \right)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P_i\left(k\right),\delta (\mathbf{k},a)=D(k,a){\delta }_i\left(\mathbf{k}\right).$$

(16)

Any rotationally invariant, even scale-dependent, growth factor gives $P(k,a)=D^2(k,a)P_i\left(k\right)$, which remains statistically isotropic. A dipole vector therefore has $\langle \mathbf{d}\rangle =0$ and $\langle d_i{d}_j\rangle ={\sigma }_d^2{\delta }_{ij}/3$. Infrared growth can increase ${\sigma }_d$; it cannot by itself select an intrinsic cosmological axis. The priority for enhanced MOND large-scale growth belongs to Sanders and Nusser [32,33]; the calculation below asks what that enhancement can and cannot imply for the late-time observables.

Bulk-flow amplitude and the growth-history constraint.

For linear irrotational motion,

$$\mathbf{v}(\mathbf{k},a)=iaHf(k,a){\displaystyle \frac{\mathbf{k}}{k^2}}\delta (\mathbf{k},a),{\sigma }_V^2\left(R\right)={\displaystyle \frac{{\left(aH\right)}^2}{2{\pi }^2}}{\int }_0^{\infty }dkP(k,a)f^2(k,a)W^2\left(kR\right),$$

(17)

where $f=dlnD/dlna$ and ${\sigma }_V$ is the rms magnitude of the top-hat bulk-flow vector. A controlled diagnostic uses a Planck-normalised smooth transfer function with ${\mathrm{\Omega }}_m=0.315$, $h=0.674$, ${\sigma }_8=0.811$ and $n_s=0.965$. At $R=173h^{-1}$ Mpc it gives

(18)

The last two values bracket the reported $428\pm 108$ km ${\mathrm{s}}^{-1}$ CosmicFlows-4 estimator [54,55]. Enhanced infrared growth can therefore reach the observed scale, but this is a feasibility result, not a prediction: the survey window, mode-dependent $fD$ and estimator covariance remain to be propagated, and alternative CosmicFlows-4 reconstructions reduce the tension [56].

The same estimate exposes a stricter consistency condition. Uninterrupted $\delta \propto a^2$ growth from recombination would enhance the density amplitude by an extra $a_0/a_{\mathrm{rec}}\simeq 1090$ relative to $\delta \propto a$. Requiring this enhancement to remain below $20\%$ gives

$$\overline{p}\lesssim 1+{\displaystyle \frac{ln1.2}{ln1090}}\simeq 1.026;p=2\mathrm{can}\mathrm{operate}\mathrm{unregulated}\mathrm{only}\mathrm{from}z\lesssim 0.2.$$

(19)

A present $f_{\mathrm{eff}}\simeq 1.5$–2 is viable only if the modification is strongly time-, scale- or environment-dependent, or is reduced by the AQUAL curl field. The calibrated pipeline must fit $D(k,a)$ and $f(k,a)$ simultaneously; matching a flow amplitude alone is not a success. [C]/[REQ]/[O]

Low-redshift supernovae.

A coherent flow ${\mathbf{V}}_B$ produces, to leading order,

$${\displaystyle \frac{\delta H}{H}}(z,\widehat{\mathbf{n}})\simeq {\displaystyle \frac{{\mathbf{V}}_B·\widehat{\mathbf{n}}}{cz}}.$$

(20)

For $V_B=400$ km ${\mathrm{s}}^{-1}$ and $H_0=73$ km ${\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, the maximal modulation is $5.8\%$, $2.7\%$, $1.3\%$ and $0.9\%$ at $z=0.023,0.05,0.10$ and $0.15$, corresponding to $\Delta H\simeq 4.2,2.0,1.0$ and $0.65$ km ${\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. This has the right order and the required $1/z$ decay for part of the reported Pantheon+ signal [46], but it does not predict a $q_0$ dipole, which requires survey mocks and a joint light-curve/velocity analysis. A KBC-like local underdensity can bias very-low-z distance-ladder estimates [58,59]; neither such an underdensity nor an $8\%$ global $H_0$ shift follows generically from isotropic growth.

Correlated dipoles without an intrinsic axis.

Different observables can have correlated random directions. Write

$${\mathbf{d}}_X=\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\delta }_i\left(\mathbf{k}\right){\mathbf{K}}_X\left(\mathbf{k}\right),r_{XY}={\displaystyle \frac{C_1^{XY}}{\sqrt{C_1^{XX}{C}_1^{YY}}}},C_1^{XY}\propto \int dk{k}^2{P}_i\left(k\right){\Delta }_1^X\left(k\right){\Delta }_1^Y\left(k\right).$$

(21)

Overlapping transfer kernels can make $r_{XY}$ large. For correlated Gaussian three-vectors, the probability of two directions lying within ${30}^{\circ }$ is $6.7\%$ for $r=0$, about $28\%$ for $r=0.6$, and about $71\%$ for $r=0.9$. The kinematic CMB dipole, local bulk flow and low-z supernova dipole are therefore not independent alignment tests. The high-redshift number-count dipole is more demanding:

$${\mathbf{d}}_{\mathrm{obs}}={\mathbf{d}}_{\mathrm{kin}}+{\mathbf{d}}_{\mathrm{clust}}+{\mathbf{d}}_{\mathrm{sys}}.$$

(22)

Explaining an excess requires the clustering term to be sufficiently large and aligned after survey masks, redshift distributions and systematics are included. The calibrated pipeline must calculate the cross-covariances, not merely note similar directions [42,43].

The separate intrinsic-axis conjecture.

If a remnant of the $(1,1)$ interface direction survives at cosmological scales, the vacuum order parameter could break rotational invariance and fix a direction independently of the random density realisation. The induced number-count dipole, domain structure and correlation with the CMB low-ℓ axes are uncomputed [SPE]/[O]. This order parameter is not required for correlated random dipoles generated by shared modes; it is required for an intrinsic axis. The conclusion is therefore deliberately narrow: scale-dependent infrared growth enhances the variance of realisation-level bulk flows and dipoles. Whether it produces the reported amplitudes and cross-probe alignment is open pending the calibrated pipeline; whether there is an intrinsic cosmological axis is a separate open question.

6.6. Late-Time CMB Observables

The candidate differences from $\mathrm{\Lambda }$CDM are late-time ISW and low-L CMB-lensing changes from the modified potential history, together with the dark radiation of Sec. 4.1; their signs and amplitudes require the completed calculation. The implementation path is concrete: replace the Poisson closure in a Boltzmann code by the mean-field relation (13), include the completed khronon sector and the diluted ${\nu }_R$ distribution function, and propagate all observables jointly. [O]

7. The Late-Time Universe: Dipole, Deceleration, and Dark Energy

7.1. the Observational Situation, Stated without Spin

(i) The quasar dipole exceeds the kinematic expectation at $4.9\sigma$; jointly with radio counts, above $5\sigma$ [42,43]; the excess survives clustering corrections [52] and simulation-based treatment of the scanning-law systematics [53]. (ii) Pantheon+ supernovae show a dipole in $q_0$ aligned with the local bulk flow, decaying with redshift [46]; after progenitor-age corrections the monopole $q_0$ shifts positive [47]; against this, standard-standardisation reanalyses find the acceleration robust at $>4\sigma$ with a dipole allowed [48,49,50], while a 2026 forward analysis finds that apparent low-z $H_0$ dipoles are better described by local flows or systematics than by anisotropic expansion [57]. (iii) The Watkins et al. CosmicFlows-4 estimator gives a bulk flow above $\mathrm{\Lambda }$CDM expectations on $\gtrsim 150h^{-1}$Mpc scales [54], but estimator tests and Wiener-filter reconstructions reduce the significance [55,56]. (iv) The homogeneity scale is contested [63,64,67,68], with Gpc-scale structures reported [60,61,62]. (v) The CMB is isotropic to ${10}^{-5}$ apart from modest low-ℓ anomalies [72]. (vi) DESI DR2 baryon acoustic oscillations, jointly with CMB and supernova compilations, prefer $w_0{w}_a$ evolution over a constant $\mathrm{\Lambda }$ at $\sim 3$–$4\sigma$, sample-dependent and below discovery threshold [51]. Each item has critics; it is the pattern that this theory addresses.

7.2. Dipole Amplitudes and Directions: Conditional Outputs

The tilted-frame literature [45,76,77,78] supplies useful kinematics, but a tilt is not an automatic output of the present dynamics. The derived result is that enhanced low-k $fD$ raises bulk-flow variance. If our realisation contains a coherent flow, Eq. (20) then gives a redshift-decaying low-z $H_0$ dipole. The amplitude, the $q_0$ profile, the high-redshift number-count dipole and their cross-covariances are open until the survey-window calculation is performed. Similar measured directions can arise from overlapping kernels and are not, by themselves, evidence for an intrinsic axis.

A local underdensity or outflow can bias very-low-z distance-ladder estimates upward [58,59], but isotropic infrared growth does not require the observer to occupy such a region and does not predict an $8\%$ global shift. If the two-leaf interface leaves a cosmological order parameter, it could fix an axis independently of the density realisation [7]; this remains a separate speculative calculation. [D]/[C]/[O]/[SPE]

7.3. The Monopole: Contested — the Theory Needs the Acceleration

Here the theory takes a falsifiable stand against part of the Sarkar-collaboration reading. The relation (10) anchors the observed $a_0$ to a de Sitter infrared with ${\mathrm{\Omega }}_{\mathrm{\Lambda },\mathrm{eff}}\simeq 0.7$: rotation curves and dark energy are one phenomenon, and the background must accelerate at late times. If the decelerating monopole of [47] survives independent scrutiny — in particular of the progenitor-age standardisation, an astrophysical systematic logically separate from the dipole, and against the standard-analysis literature in which the acceleration stands at high significance [48,49,50] — then ${\mathcal{l}}_{\mathrm{dS}}\gg c/H_0$, whence $a_0=c^2/\left(\xi {\mathcal{l}}_{\mathrm{dS}}\right)\ll {10}^{-10}$ m ${\mathrm{s}}^{-2}$, destroying the infrared anchor of the galactic sector. Conversely the theory permits a genuinely accelerating background together with a coherent local flow that biases low-z monopole inferences and produces a redshift-decaying dipole. Whether the flow is large enough, and whether the age-standardisation correction and a flow-induced modulation are cleanly separable, are questions for a joint data analysis; the theory’s requirement is only that monopole acceleration survive. Distinguishing the two readings requires CMB-frame Hubble diagrams at $z\gtrsim 0.3$ with full covariance, BAO+chronometer $H\left(z\right)$ (standardisation-free), and the redshift profile of the q-dipole.

The anchor also fixes the equation of state. ${\mathrm{\Lambda }}_{\mathrm{eff}}=3c^2/{\mathcal{l}}_{\mathrm{dS}}^2$ with a frozen infrared vacuum is a constant: $w=-1$ exactly, epoch-independent, with no mechanism for $w\left(z\right)$. The theory is therefore exposed on a second, independent front: the DESI DR2 preference for evolving $w_0{w}_a$ dark energy, at $\sim 3$–$4\sigma$ jointly with CMB and supernova data [51], presses against precisely this commitment, and a systematics-vetted $5\sigma$ demonstration that $w\ne -1$ — in particular a confirmed phantom crossing — would refute the constant-${\mathrm{\Lambda }}_{\mathrm{eff}}$ identity as decisively as a decelerating monopole would. Low-redshift flow and anisotropy systematics must be propagated in any joint inference, but the present framework does not predict their amplitude or direction. Its commitments are monopole acceleration and an equation of state constant at $w=-1$. [C]/[REQ]

7.4. Dark Energy without a Cosmological-Constant Problem?

Honesty requires noting what (10) does and does not achieve. It does not solve the old cosmological-constant problem (why the visible-sector vacuum energy does not gravitate at $M_{\mathrm{Pl}}^4$); that question is relocated into the trace-dynamics vacuum, where the programme’s position — emergent gravity does not respond to the pre-geometric zero-point — remains to be made precise. [O]. What it does achieve is the coincidence half: the infrared selects one length ${\mathcal{l}}_{\mathrm{dS}}$, and both ${\mathrm{\Lambda }}_{\mathrm{eff}}$ and $a_0$ are set by it, so their observed conspiracy ($a_0\simeq c^2\sqrt{\mathrm{\Lambda }/3}/\xi$ with ${\xi }_{\mathrm{obs}}\simeq 4.55$) is structural. Why ${\mathcal{l}}_{\mathrm{dS}}\sim {10}^{26}$ m — equivalently why the dS vacuum sits $\sim 60$ orders below Planck — is the residual hierarchy question, plausibly the same question as the smallness of $T_{\mathrm{EW}}/M_{\mathrm{Pl}}$ and the size of the matrix ensemble (Sec. 3). [O]

A candidate mechanism for the relocated half deserves record here, because it rests on structure the programme already owns rather than on new ingredients. [SPE]/[O]. The equations of motion of the underlying spacetime–matter atom are those of a conservative Bateman dual system [2,11]: a cross-kinetic pair — one ordinary and one ghost-signed matrix oscillator at the single non-Planck frequency ${\nu }_0$ the theory owns — whose self-adjoint Hamiltonian is, in the pairing representation, $\hslash {\nu }_0(a^{†}a-b^{†}b)$: the zero-point energies of the pair cancel identically, with no $\frac{1}{2}$ surviving, as in the Feshbach–Tikochinsky quantisation of the damped oscillator and its field-theoretic development [20,21]. The cancellation is exact because the two members are structurally degenerate — they share the one frequency — which is precisely the degeneracy broken supersymmetry fails to deliver, and it survives every internal symmetry breaking that leaves the pair intact; the asymmetry the theory requires for collapse lives in the anti-self-adjoint fermionic sector, which is separately established to annihilate the combined Fock vacuum [19]. If the companion statement holds for the self-adjoint part — with the ghost representation derived from equipartition of the Adler–Millard charge rather than chosen, the principled formulation available in trace dynamics because zero-point structure is there a property of the emergent equilibrium — then the full trace Hamiltonian annihilates the aikyon vacuum, and the standing assumption above (emergent gravity does not respond to the pre-geometric zero-point) becomes a theorem. What this does not yet control is the vacuum energy generated at the emergent level: the cosmological (Seeley–DeWitt $a_0$) term of the spectral action, and the condensate shifts at the electroweak and QCD transitions. A naive doubling with a relative minus sign cancels the Einstein term along with the vacuum term; the known escape — one metric coupled to the difference of the two sectors’ stress tensors, the energy-parity route [22,23,24], cf. the CPT-mirror construction of [25] — ordinarily costs a ghost catastrophe, the divergent phase space for gravitational production of positive–negative pairs. This theory has the one structural escape generic energy-parity models lack: the partner is the time-reversed anti-spacetime leaf, sharing no light cone with ours, so the instability diagram has no arena. The escape obliges two derivations in return: why the partner’s vacuum piece enters the emergent gravitational source (it must, or nothing cancels) while its excitations do not — a selective transparency that must come from the localisation map, not assumption — and a condensate ledger showing the partner runs the electroweak and QCD transitions in time-reversed lockstep (the partner must be the mirrored copy of the visible fields, not the dark sector of Sec. 4, whose different content and temperature would leave an uncancelled residue of order ${\left(200\mathrm{MeV}\right)}^4$). Four computations therefore decide the mechanism: the self-adjoint vacuum expectation with the representation fixed by the equilibrium ensemble; the $a_0$-versus-$a_2$ discrimination in the spectral action; the condensate ledger; and the stability argument. Until the second is done this is a mechanism, not a solution, and we do not claim it as one. If all four are delivered, vacuum energy does not gravitate, ${\mathrm{\Lambda }}_{\mathrm{eff}}$ is sourced solely by ${\mathcal{l}}_{\mathrm{dS}}$, and the cosmological-constant problem contracts to the derivation of one length. [SPE]/[O]

7.5. The Homogeneity Scale: an Open Output

The correlation dimension is determined by the monopole power, not by the mere existence of scale-dependent growth:

$$D_2\left(R\right)=3+{\displaystyle \frac{dln[1+\overline{\xi }\left(R\right)]}{dlnR}},\overline{\xi }\left(R\right)={\displaystyle \frac{1}{2{\pi }^2}}{\int }_0^{\infty }dk{k}^{2}P\left(k\right)W_{\mathrm{TH}}\left(kR\right).$$

(23)

A Planck-normalised diagnostic illustrates the size of the effect. Multiplying the density amplitude of all modes $k<0.01h{\mathrm{Mpc}}^{-1}$ by $B=1,2,4,8$ moves the scale defined by $|D_2-3|<0.03$ only from approximately 77 to 78, 79 and $83h^{-1}$ Mpc. Keeping $|D_2\left(300h^{-1}\mathrm{Mpc}\right)-3|>0.03$ requires $B\simeq 13$, or roughly 170 times more low-k power. Modest infrared enhancement therefore does not generically postpone homogeneity beyond the survey volume.

BOSS galaxies give a transition near $63.3\pm 0.7h^{-1}$ Mpc and DR16 quasars also approach homogeneity on surveyed scales [65,66]; the interpretation of individual Gpc-scale structures remains debated [60,61,62,68]. The claimed DESI anisotropy of [69] is not used as support because the comoving recalibration of [70] removes the excess. The homogeneity scale is an open quantitative output of the calibrated pipeline, not a present prediction of delayed convergence. [REQ]/[O]

8. Predictions and Falsifiers

1.

Monopole acceleration. The background accelerates (${\mathrm{\Omega }}_{\mathrm{\Lambda },\mathrm{eff}}\simeq 0.7$); definitive confirmation of a decelerating monopole destroys the $a_0$–$\mathrm{\Lambda }$ anchor and with it the infrared sector. Sharpest single falsifier. [REQ]

2.

No cold dark matter. The clustering component is the hot-born, laboratory-sterile ${\nu }_R$; the spectrum contains no cold relic. Any confirmed direct, indirect, or collider detection of a cold halo particle falsifies the model outright. [D]

3.

Large-scale flows and dipoles. Enhanced low-k $fD$ raises bulk-flow variance, and a coherent flow produces a redshift-decaying low-z $H_0$ dipole. The flow amplitude, $q_0$ profile, quasar number-count dipole, cross-covariances and homogeneity scale are open outputs of the calibrated survey-window pipeline. An axis fixed independently of the density realisation is not a consequence of isotropic growth and belongs to the separate order-parameter conjecture. [D]/[C]/[O]/[SPE]

4.

Two-epoch dark radiation. $\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq 0.19$ against $\Delta N_{\mathrm{eff}}^{\mathrm{CMB}}\simeq 0.05$ — the diluted-sterile fingerprint. The falsifier is a joint BBN–CMB exclusion of the ordered pair $(0.19,0.054)$; the CMB epoch alone, at Stage-4-class $\sigma \simeq 0.03$, discriminates from zero only at $1.8\sigma$. BBN–CMB concordance at $\Delta N_{\mathrm{eff}}=0$ with that power falsifies the thermal-emergence hypothesis altogether. [D]/[REQ]

5.

Sterile-neutrino sector. $\mathrm{tr}{M}_R\simeq 113$ eV is fixed by ${\mathrm{\Omega }}_m$ (near $m_{{\nu }_R}\simeq 40$ eV for a degenerate trio), conditional on the adopted Fork-A mass package. The comoving free-streaming length grows $2.5\to 14\to 17$ Mpc from $z_{\mathrm{nr}}$ to recombination to today, producing graded small-scale transfer suppression; nonlinear cluster infall and merger lensing remain quantitative tests. The package also stakes inverted active ordering, $\sum m_{\nu }=0.0997$ eV and $m_{\beta \beta }=18.1$ meV. [D]/[C]/[REQ]

6.

Galaxy scales. BTFR/RAR with the “simple” $\mu$ and epoch-independent $a_0$ (no $a_0\left(z\right)\propto H\left(z\right)$ drift) — testable with high-z rotation curves; no gravitational slip ($\mathrm{\Psi }=\mathrm{\Phi }$) in the provisional linear completion, conditional on the W1 khronon sector; external-field effect present. Planetary ephemerides already pressure the simple function’s high-acceleration tail (Sec. 5): the derived $\mathcal{F}$ must have a faster approach to GR. [C]/[REQ]

7.

The $a_0$–$\mathrm{\Lambda }$–$E_6$ relation. $a_0=({\mathrm{\Theta }}_{E_6}/\sqrt{2})c^2\sqrt{\mathrm{\Lambda }/3}$ with ${\mathrm{\Theta }}_{E_6}$ fixed by the electron-family Jordan invariants (Appendix E): no secular drift of the RAR normalisation in a $\mathrm{\Lambda }$-dominated era; consistency of $a_0$ across galaxy families (vacuum universality); and, if the two-$E_6$ alignment is imperfect, $1/\sqrt{2}\to cos\vartheta$ — so precision $a_0$ measurements probe the inter-sector alignment. At the current $\sim 20\%$ precision of $a_0$ all three normalisations of Appendix E survive; an improved absolute calibration of $a_0$ is what discriminates. [C]

8.

Small-scale structure. The Lyman-$\alpha$ forest at $z\simeq 2$–5, dwarf counts, and reionization timing test the top-down regeneration of sub-Mpc power (Sec. 6.3); underproduction of the $z\sim 3$ forest flux power falsifies the model. The sharpest open confrontation. [REQ]/[O]

9.

Equivalence principle. Exact for the MOND sector by minimal matter coupling. The visible fermions are dem-neutral (anomaly no-go, [8]) and the unbroken $\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ photon is massless, so it decouples from ordinary matter by a field redefinition: the theory predicts no $\sum \sqrt{m}$-patterned composition-dependent force. Detection of such a force would therefore falsify the framework (it is forbidden, not predicted). The only visible dark-EM signature is a millicharged dark relic through kinetic mixing $\epsilon$, with standard dark-photon phenomenology; $\epsilon$ is not fixed here. [D]/[O]

10.

Early structure. Whether regulated infrared growth regenerates the required sub-Mpc power, advances massive-galaxy formation relative to $\mathrm{\Lambda }$CDM at fixed ${\mathcal{A}}_s$, and produces a viable 21-cm history is an open output. Recombination is the conditional onset epoch of the enhancement (Sec. 6.4). [C]/[REQ]/[O]

11.

Phase-coherent adiabatic peaks. Required of the emergence mechanism (Sec. 3.3); detected active-seeding admixture would indicate residual localisation noise. [REQ]

12.

Late-time CMB. Enhanced low-L lensing and ISW relative to $\mathrm{\Lambda }$CDM, and consistency with $E_G$/RSD data, computable once the calibrated pipeline (Sec. 6) is run. [O]

9. The Ledger, and Outlook

Three computations dominate the outlook. First, the derivation of $S_{\mathrm{IR}}$ — the full interpolation $\mathcal{F}$ (with its ephemeris-safe tail), the khronon dynamics of the preferred foliation, the frozen-$R_H$ prescription, and the vacuum selection behind the $E_6$ coefficient of Appendix E — from the $\mathrm{SU}{\left(2\right)}_R$ trace-dynamics vacuum, together with the preferred-frame consistency checks. Second, the calibrated Boltzmann pipeline: the closure (13) with curl/external-field corrections in place of the Poisson equation, the diluted ${\nu }_R$ distribution functions, marched through recombination to the CMB spectra, $S_8$, $E_G$, ISW, and the simultaneous $D(k,a)$ and $f(k,a)$ history; bulk-flow and supernova survey windows; the quasar clustering, kinematic and systematic dipoles and their cross-covariance; the homogeneity statistic; and, extended with hydrodynamics into the quasi-nonlinear regime, the Lyman-$\alpha$ flux power on which the model’s survival most directly depends (Sec. 6.3). Third, the joint derivation of ${\eta }_B$ and the $+s$ dark-charge excess at the transition. The first is a theorist’s calculation; the second is engineering on existing codes; the third is the programme’s own quantum-to-classical machinery applied to its natural target.

Table 1. Epistemic status of the cosmological claims.

Statement	Grade
Classical spacetime, EWSB, and L–R breaking are one emergence event at $T_{*}\sim 100$ GeV	[P]
Thermal history below $T_{*}$ standard in mechanism; output shifted by the predicted dark radiation, including $\Delta Y_p\simeq +0.0025$	[D]/[REQ]
All cosmological scales super-horizon at emergence; causal post-emergence seeding excluded	[D]
Pre-geometric (acausal) correlations supply the super-horizon seeds	[C] (on [11])
Horizon, flatness, relic problems dissolved; no inflaton	[C]
Entropy ($\sim {10}^{88}$) relocated to the matrix ensemble; unexplained	[O]
Seeds = localisation residue; adiabatic, coherent, Gaussian	[C]/[O]
Near scale invariance from the scaleless desert; value of $n_s-1$	[C] / [O]
Amplitude $4.6\times {10}^{-5}$ (the ${\alpha }^2$ coincidence)	[O] ([SPE])
Dark sector born thermal at emergence (not coupling-thermalised)	[P]
$\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ retired as MOND mediator (like signs repel); visible fermions dem-neutral (anomaly no-go, [8]), massless dark photon decouples from visible matter, portal is kinetic mixing $\epsilon$	[D]/[P]
MOND from $S_{\mathrm{IR}}[g,T]$, with the vacuum foliation and static asymptotics fixed by dS scale invariance of the $\mathrm{SU}{\left(2\right)}_R$ vacuum	[P]/[C]
Interpolating function (3): asymptotics dS-fixed; exact interpolation chosen	[C]
Ephemeris pressure on the simple-$\mu$ tail; faster approach to GR required of the derived $\mathcal{F}$	[REQ]
$a_0=c^2/\left(\xi {\mathcal{l}}_{\mathrm{dS}}\right)$, ${\mathrm{\Lambda }}_{\mathrm{eff}}=3{\xi }^2{a}_0^2/c^2$; ${\xi }_{\mathrm{obs}}\simeq 4.55$ from data	[D] (arith.)/[P]
${\xi }_{E_6}=\sqrt{2}{\left|S\right|}^{3/2}/\left|D\right|=3.85$ from electron-family Dirac-set Jordan invariants (corrected determinant $D=-\frac{25}{54}$; the few-per-mille match of v1 withdrawn); $18\%$ above ${\xi }_{\mathrm{obs}}$, inside the $\sim 20\%$ systematic of $a_0$	[C] (App. E)/[O] (vacuum selection)
$a_0$ epoch-independent (frozen $R_H$); derivation of the freezing	[C] / [O]
Foliation-scalar covariantisation of $a_{\mu }$ (khronon selected by the $\mathrm{SU}{\left(2\right)}_R$ vacuum); WEP exact, SEP mildly broken; PPN ${\alpha }_1,{\alpha }_2$/Cherenkov checks	[C]/[O]
Tensor GW speed $c_T=c$ at linear order; completed khronon sector and stability outstanding	[C]/[O]
No gravitational slip ($\mathrm{\Psi }=\mathrm{\Phi }$) in the provisional linear completion; lensing = dynamics	[C]/[O]
Weak equivalence principle exact in the MOND sector	[D]
Two-epoch dark radiation: $\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq 0.19$, $\Delta N_{\mathrm{eff}}^{\mathrm{CMB}}\simeq 0.05$	[D] (given thermal emergence)
${\mathrm{\Omega }}_m$ closed by three diluted ${\nu }_R$: $\mathrm{tr}{M}_R\simeq 113$ eV, near 40 eV each; ${\lambda }_{\mathrm{fs}}:2.5\to 14\to 17$ Mpc from $z_{\mathrm{nr}}$ to recombination to today; conditional on the combined Fork-A package	[D]/[C]
CMB peaks reproduced (diluted-sterile hybrid; $S_{\mathrm{IR}}$ gated off pre-recombination)	[C]/[O] (Boltzmann)
Recombination as the onset of the MOND era (conditional acceleration gating through the matter frame and completed khronon sector)	[C]/[O]
Deep-MOND linear growth $\delta \propto a^2$ in the mean-field closure; regulation and full history outstanding	[C]/[O]
Overshoot control: curl-field/external-field gating; flows calibrate the pipeline; $E_G$/RSD	[REQ]/[O]
Sub-Mpc structure regenerated top-down; Lyman-$\alpha$ forest, dwarf counts, reionization (Sec. 6.3)	[REQ]/[O]
Rotationally invariant $D(k,a)$ preserves statistical isotropy; enhanced low-k $fD$ raises bulk-flow variance	[D]/[C]
Flow and SN-dipole amplitudes, quasar cross-correlation and homogeneity scale	[O]
Background monopole must accelerate ($a_0$–$\mathrm{\Lambda }$ anchor); vs. [47]	[REQ]
Interface order parameter fixing an axis independently of the density realisation	[SPE]/[O]
Zero-point ledger of the Bateman pair (vacuum energy does not gravitate): exact pair cancellation at the atomic level; four named computations decide the transfer to the emergent level (Sec. 7)	[SPE]/[O]
Baryogenesis and dark-charge ($+s$) separation at the transition	[O]

Table 1. Epistemic status of the cosmological claims.

Statement	Grade
Classical spacetime, EWSB, and L–R breaking are one emergence event at $T_{*}\sim 100$ GeV	[P]
Thermal history below $T_{*}$ standard in mechanism; output shifted by the predicted dark radiation, including $\Delta Y_p\simeq +0.0025$	[D]/[REQ]
All cosmological scales super-horizon at emergence; causal post-emergence seeding excluded	[D]
Pre-geometric (acausal) correlations supply the super-horizon seeds	[C] (on [11])
Horizon, flatness, relic problems dissolved; no inflaton	[C]
Entropy ($\sim {10}^{88}$) relocated to the matrix ensemble; unexplained	[O]
Seeds = localisation residue; adiabatic, coherent, Gaussian	[C]/[O]
Near scale invariance from the scaleless desert; value of $n_s-1$	[C] / [O]
Amplitude $4.6\times {10}^{-5}$ (the ${\alpha }^2$ coincidence)	[O] ([SPE])
Dark sector born thermal at emergence (not coupling-thermalised)	[P]
$\mathrm{U}{\left(1\right)}_{\mathrm{dem}}$ retired as MOND mediator (like signs repel); visible fermions dem-neutral (anomaly no-go, [8]), massless dark photon decouples from visible matter, portal is kinetic mixing $\epsilon$	[D]/[P]
MOND from $S_{\mathrm{IR}}[g,T]$, with the vacuum foliation and static asymptotics fixed by dS scale invariance of the $\mathrm{SU}{\left(2\right)}_R$ vacuum	[P]/[C]
Interpolating function (3): asymptotics dS-fixed; exact interpolation chosen	[C]
Ephemeris pressure on the simple-$\mu$ tail; faster approach to GR required of the derived $\mathcal{F}$	[REQ]
$a_0=c^2/\left(\xi {\mathcal{l}}_{\mathrm{dS}}\right)$, ${\mathrm{\Lambda }}_{\mathrm{eff}}=3{\xi }^2{a}_0^2/c^2$; ${\xi }_{\mathrm{obs}}\simeq 4.55$ from data	[D] (arith.)/[P]
${\xi }_{E_6}=\sqrt{2}{\left|S\right|}^{3/2}/\left|D\right|=3.85$ from electron-family Dirac-set Jordan invariants (corrected determinant $D=-\frac{25}{54}$; the few-per-mille match of v1 withdrawn); $18\%$ above ${\xi }_{\mathrm{obs}}$, inside the $\sim 20\%$ systematic of $a_0$	[C] (App. E)/[O] (vacuum selection)
$a_0$ epoch-independent (frozen $R_H$); derivation of the freezing	[C] / [O]
Foliation-scalar covariantisation of $a_{\mu }$ (khronon selected by the $\mathrm{SU}{\left(2\right)}_R$ vacuum); WEP exact, SEP mildly broken; PPN ${\alpha }_1,{\alpha }_2$/Cherenkov checks	[C]/[O]
Tensor GW speed $c_T=c$ at linear order; completed khronon sector and stability outstanding	[C]/[O]
No gravitational slip ($\mathrm{\Psi }=\mathrm{\Phi }$) in the provisional linear completion; lensing = dynamics	[C]/[O]
Weak equivalence principle exact in the MOND sector	[D]
Two-epoch dark radiation: $\Delta N_{\mathrm{eff}}^{\mathrm{BBN}}\simeq 0.19$, $\Delta N_{\mathrm{eff}}^{\mathrm{CMB}}\simeq 0.05$	[D] (given thermal emergence)
${\mathrm{\Omega }}_m$ closed by three diluted ${\nu }_R$: $\mathrm{tr}{M}_R\simeq 113$ eV, near 40 eV each; ${\lambda }_{\mathrm{fs}}:2.5\to 14\to 17$ Mpc from $z_{\mathrm{nr}}$ to recombination to today; conditional on the combined Fork-A package	[D]/[C]
CMB peaks reproduced (diluted-sterile hybrid; $S_{\mathrm{IR}}$ gated off pre-recombination)	[C]/[O] (Boltzmann)
Recombination as the onset of the MOND era (conditional acceleration gating through the matter frame and completed khronon sector)	[C]/[O]
Deep-MOND linear growth $\delta \propto a^2$ in the mean-field closure; regulation and full history outstanding	[C]/[O]
Overshoot control: curl-field/external-field gating; flows calibrate the pipeline; $E_G$/RSD	[REQ]/[O]
Sub-Mpc structure regenerated top-down; Lyman-$\alpha$ forest, dwarf counts, reionization (Sec. 6.3)	[REQ]/[O]
Rotationally invariant $D(k,a)$ preserves statistical isotropy; enhanced low-k $fD$ raises bulk-flow variance	[D]/[C]
Flow and SN-dipole amplitudes, quasar cross-correlation and homogeneity scale	[O]
Background monopole must accelerate ($a_0$–$\mathrm{\Lambda }$ anchor); vs. [47]	[REQ]
Interface order parameter fixing an axis independently of the density realisation	[SPE]/[O]
Zero-point ledger of the Bateman pair (vacuum energy does not gravitate): exact pair cancellation at the atomic level; four named computations decide the transfer to the emergent level (Sec. 7)	[SPE]/[O]
Baryogenesis and dark-charge ($+s$) separation at the transition	[O]

Nothing here was retrofitted to the anomalies: the desert, the emergence at the electroweak scale, the $\mathrm{SU}{\left(2\right)}_R$ origin of gravity and its de Sitter infrared, and the three right-handed neutrinos were fixed by particle-physics considerations in the cited works before the cosmological questions were posed. The resulting cosmology gives MOND-like galaxies, sterile-supported clusters and no cold halo particle; its rotationally invariant infrared growth can enhance realisation-level flows without predicting an intrinsic axis. The observed amplitudes and alignments remain tests, not successes already earned. The framework simultaneously stakes its life on a genuinely accelerating background and on regenerating the small-scale power probed by the Lyman-$\alpha$ forest.