An Aether-Scalar-Tensor Structure for the Dark Sector from Dual-4DD Breathing
从 Dual-4DD 呼吸导出暗扇区的 Aether-Scalar-Tensor 结构

1. Introduction

Dark matter, dark energy, and MOND phenomenology — the three pieces of the dark sector — are treated independently in the standard programme: cold dark matter is a particle, dark energy is a constant or an extra scalar field, and MOND is an empirical acceleration relation. Each carries its own free parameters and fits its own observations.

This paper places all three in a single structure, and introduces no dark-matter particle. The structure comes from SAE's dual-4DD picture: two four-dimensional timelike sectors with nearly antiparallel time orientation, each carrying a breathing clock τ₁, τ₂. Their symmetric combination τ_breath = (τ₁+τ₂)/2 acts as a khronometric aether; their antisymmetric combination C = τ₁−τ₂ acts as a DBI scalar. From these, the khronon supplies geometric dust that serves as dark matter, DBI-C supplies dynamical dark energy and MOND, and a 4-form field strength supplies the cosmological constant; the conservation laws of all three reduce to a single remainder-conservation axiom.

This paper follows the SAE cosmology series: Cosmo I gives the cosmological constant Λ = 2(ω₂²−ω₁²)/c², Cosmo II/III give the MOND acceleration a₀ and the geometric origin of its 3/2 power, Cosmo IV treats the effective gravitational constant, and Cosmo V proposes the C field as dynamical dark energy. Here we assemble these into a single modified-gravity action of the aether-scalar-tensor type — narrowing one free function of that class into two terms with structural origins — completing three sub-goals (the khronon's Mukohyama dust, the tensor speed c_T = 1, and the cosmological constant Λ₁), and revising a₀ from an earlier vacuum scale to a horizon scale.

The value of this structure lies not in its ability to fit existing data — there is no shortage of models that fit dark-sector data — but in its constraints on functional form, and in the falsifiable predictions those constraints entail. SAE's prior restricts the form of the action, not merely its parameters: the DBI kinetic term follows from the Nambu-Goto action of a brane rather than being chosen freely; the 3/2 power of the spatial term follows from the p = d = 3 theorem in three dimensions rather than being fit; the conservation laws follow from an axiom rather than being inserted as initial conditions; and c_T = 1 follows from the universality of c, confirmed by GW170817. It is these constraints that let the framework issue hard predictions the standard cold-dark-matter-plus-cosmological-constant model (ΛCDM) cannot — predictions that pin the framework down: an intrinsic dark-energy equation of state w_int ≥ −1 (dark energy structurally does not cross the phantom divide), and a redshift scaling of a₀ (a₀ grows with the horizon, differing from constant-a₀ by more than a factor of four at z ≳ 3). Both can be adjudicated by current and near-term observation.

This is a deliberate trade-off. A model with enough freedom fits almost any data, and fitting everything is equivalent to excluding nothing: it is precise but unfalsifiable, and therefore cannot be adjudicated or renewed. A framework whose structure is constrained by a prior, by contrast, has no room for post-hoc parameter adjustment and can only make hard predictions — much as general relativity, with almost no free parameters, made predictions like light bending and gravitational waves that were computed before they were measured. The two hard predictions of this paper are of this kind: they can be falsified, and that is the point; if falsified, the falsification itself indicates where the prior must be revised.

The remaining quantities — the dust abundance Ω_d, several couplings and khronometric parameters, the absolute value of the a₀ coefficient — are set by observation or boundary conditions, just as the gravitational constant in general relativity or the mass spectrum in the standard model. Throughout, we label prior-derived structural constraints, posterior-set values, and unresolved open problems separately.

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2. The dual-4DD breathing structure

We summarize what is needed; full derivations are in the SAE foundations and in Cosmo I–V.

Two four-dimensional timelike sectors (4DD) coexist with nearly antiparallel time orientation and a slight symmetry breaking, parametrized as $$\varepsilon = \frac{\omega_2-\omega_1}{\omega_2+\omega_1} \approx 0.0127,$$ where ω₁, ω₂ are the frequencies of the sectors' breathing clocks τ₁, τ₂. The hard parameters are taken as T₁ = 20 Gyr, T₂ = 19.5 Gyr (SAE inputs, not re-estimated here).

The key decomposition combines the two clocks into $$\tau_{\rm breath} = \frac{\tau_1+\tau_2}{2}\ (\text{symmetric}),\qquad C = \tau_1-\tau_2\ (\text{antisymmetric}).$$ The symmetric combination τ_breath defines the normal to the causal foliation and acts as a khronometric aether; the antisymmetric combination C is a dynamical scalar. The two are not independent fields but two combinations of the same pair (τ₁, τ₂) — a point that matters in §6 when we discuss coupling.

Two established structural results (see the T2 working record) are used here. First, C is non-compact: the speed of light bounds the misalignment rate Ċ itself, and this bound is unreachable within a 4DD (it is a relativistic ceiling, not a channel to a fifth dimension), so C accumulates along the timelike direction without winding, and shift symmetry C → C + const forces U(C) = const. Second, there is a single light cone: the two 4DD share one c, and opposite time orientation means they correspond to the two directions of one cone rather than to two metrics. This structurally forbids the dangerous bimetric/disformal couplings once envisaged in Cosmo V; the "two-metric reconstruction" is thereby fixed as an aether-scalar-tensor reconstruction, by structure rather than by choice, and this is also a prerequisite for passing GW170817.

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3. The action and its commitment layering

3.1 Template and SAE's narrowing

We take the aether-scalar-tensor action of Skordis–Złośnik (2021) as a template: $$S_G = \int d^4x\, \frac{\sqrt{-g}}{2\kappa}\Big[R - 2\Lambda - \tfrac{K_B}{2}F_{\mu\nu}F^{\mu\nu} + 2(2{-}K_B)J^\mu\nabla_\mu\phi - (2{-}K_B)\mathcal{Y} - \mathcal{F}(\mathcal{Y},Q) - \lambda(A^\mu A_\mu{+}1)\Big] + S_m[g],$$ with $Q = A^\mu\nabla_\mu\phi$, $\mathcal{Y} = (g^{\mu\nu}{+}A^\mu A^\nu)\nabla_\mu\phi\nabla_\nu\phi$, $J_\mu = A^\nu\nabla_\nu A_\mu$, $F_{\mu\nu} = \nabla_\mu A_\nu {-} \nabla_\nu A_\mu$. Matter couples to a single metric g (satisfying the equivalence principle), and the aether kinetic term is of Maxwell form.

AeST leaves $\mathcal{F}(\mathcal{Y},Q)$ a free function. SAE's narrowing fixes it at both ends: $$\boxed{\ \mathcal{F}_{\rm SAE}(\mathcal{Y},Q) = 2T\big[\sqrt{1-Q^2/c^2}-1\big] + M_Y^4\, y^{3/2}\ }$$ The time part is of DBI (tachyon) form, derived from the Nambu-Goto action of a brane (§4.2); the 3/2 power of the spatial part follows from the p = d = 3 theorem (§4.2, §5). This is SAE's core structural gain over AeST: a free function reduced to two terms with structural origins.

We take the scalar φ = C (the antisymmetric combination) and the aether $A_\mu = -N\nabla_\mu\tau_{\rm breath}$ (hypersurface-orthogonal, i.e. khronometric), with $Q_c = c$ as a prior (from the c-ceiling of §4.2).

A sign convention should be fixed to avoid mismatch with the $w_C$ derivation below. Here $\mathcal{F}_{\rm SAE}$ follows the AeST template convention, in which the action contains $-\mathcal{F}$; the DBI subterm therefore contributes $-2T[\sqrt{1-Q^2/c^2}-1]$ to the action, its physical Lagrangian has the form $-\sqrt{1-Q^2/c^2}$ (the brane derivation of §4.2), and the constant $+2T$ is absorbable — into $\Lambda_{\rm 4form}$ or by a zero-point subtraction. The factor of 2 in $2T$ comes from the template's $(2-K_B)$ normalization of the $\mathcal{Y}$ term, matching the DBI term to the $\mathcal{Y}$ term in the weak-field expansion. The $w_C = q^2 - 1$ of §4.2 uses the physical DBI stress-energy tensor, not the sign of $\mathcal{F}$ itself.

3.2 Consistency of the spatial scale: α is not a free constant

The MOND acceleration a₀ is precisely the normalization scale of the quasi-static $\mathcal{Y}^{3/2}$ term. If the coefficient of that term were a constant α, then a₀ would be constant, contradicting the a₀ ∝ H(z) of §5. We therefore do not write the spatial term with a constant coefficient $\alpha\mathcal{Y}^{3/2}$, but write the horizon scale into the non-dimensionalization: $$y \equiv \frac{\mathcal{Y}}{a_0(\tau)^2},\qquad a_0(\tau) = \frac{c^2}{2\pi R_A(\tau)},$$ where R_A(τ) is the apparent horizon (§5). The power 3/2 is fixed by p = d = 3, and the scale a₀(τ) is supplied by the horizon sector. The MOND normalization of §3 and the horizon a₀ of §5 are then the same a₀(τ), and the two are consistent.

This writing requires one remark at the level of the variation, or it would introduce higher derivatives. Since $a_0(\tau) \propto R_A^{-1} \propto H$ and $H = \dot a/a$ contains a time derivative of the metric, varying $a_0(\tau)$ together with the metric in the local field equations ($\delta S/\delta g_{\mu\nu}$) would produce Ostrogradsky-type higher-derivative (ghost) terms. Our position on this is consistent with the physical origin of a₀ (§5): $a_0(\tau)$ is the information read-out noise floor set by the global causal horizon, and in the local (galactic-scale) variation it is treated as an external evolving background that does not participate in the local metric variation — the back-reaction of local matter is too small to shift the horizon noise floor, so no higher-derivative variation term arises. Correspondingly, the spatial term containing $a_0(\tau)$ is not a complete covariant local Lagrangian but an effective term normalized by the horizon read-out on an FLRW background; its full covariantization (expressing $R_A^{-1}$ as a foliation-invariant local geometric scalar — the aether expansion $\theta = \nabla_\mu A^\mu = 3H$, spatial curvature, or another, e.g. $a_0(\tau) \simeq c\theta/6\pi$ in the flat approximation) is left as an open problem.

3.3 Separating the baseline action from the coupling branch

The baseline action and an optional interaction branch are listed separately. The baseline action contains no khronon–C mixing term: $$S_{\rm base} = S_{\rm grav}[g,A] + S_{\rm DBI\text{-}C}[C,A] + S_{Y}[C,A] + S_{\rm 4form} + S_m[g].$$ There is a further term $$S_{\rm mix} = g\int d^4x\,\sqrt{-g}\,\theta Q,\qquad \theta = \nabla_\mu A^\mu,$$ which is not part of the baseline, but an open interaction branch (see §6, HP2, layer 2); its coupling g is a free Wilson coefficient, not a number fixed by the dual-4DD structure. This separation is necessary to keep "intrinsic w ≥ −1" (a baseline result) distinct from "effective crossing" (a g-dependent optional branch).

3.4 Commitment layering

Labeling each term's status is required by SAE's honesty. The table below both guards against overstating the derivation of structural constraints as a full numerical derivation, and acknowledges the prior constraints that are genuinely present — constraints that ΛCDM/AeST do not have.

Term	SAE status	Free coefficient	Observational role
$R - 2\Lambda_{\rm 4form}$	AeST template + prior 4-form form	Λ₁ value posterior	constant part of dark energy
DBI time term $\sqrt{1-Q^2/c^2}$	prior (brane-derived)	tension T posterior	dark energy, w_int ≥ −1
spatial term $y^{3/2}$	power prior (p = d = 3)	scale = horizon a₀(τ)	MOND / a₀
khronon aether (HSO)	prior (breathing foliation normal)	α, λ posterior	dark matter (Mukohyama dust)
$-\tfrac{K_B}{2}F^2$	AeST template	K_B posterior	aether kinetic (c_T = 1)
$S_{\rm mix} = g\theta Q$	open interaction branch (non-baseline)	g free Wilson	HP2 effective crossing (undetermined)

The prior-derived items (single metric, c_T = 1, DBI form, y^{3/2} power, the conservation laws) are structural constraints, not values; the posterior items (Λ₁ value, T, K_B, α, λ) are on the same footing as G measured in general relativity or the nineteen parameters measured in the standard model, and do not detract from this being a framework with structural constraints.

Three interfaces should be made explicit. C is a dynamical scalar (with its own equation of motion, contributing ρ_C(z)), not a pure background quantity. HSO = khronometric is a choice SAE fixes, and it is precisely the interface of the version tension between §4.1 (dust) and §4.4 (c_T = 1). Relative to AeST, SAE fixes the two terms of $\mathcal{F}$ and $Q_c = c$, and leaves K_B, T, α, λ undetermined.

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4. The three objects

4.1 Khronon → dark matter (Mukohyama integration-constant dust)

The HSO aether reduces the gravitational sector, in the infrared, to projectable Hořava–Lifshitz gravity. Consider its vacuum FRW dynamics (flat, unit lapse). Varying the scale factor gives the acceleration equation $$3(1-3\lambda)(2a\ddot a+\dot a^2) + 6\Lambda a^2 = 0,$$ which has a first integral (sign check: $\tfrac{d}{dt}[3(1-3\lambda)a\dot a^2+2\Lambda a^3] = \dot a \times$ the acceleration equation, conserved exactly): $$3(1-3\lambda)a\dot a^2 + 2\Lambda a^3 = \mathcal{C}\quad\Longrightarrow\quad H^2 = \frac{\mathcal{C}/a^3}{3(1-3\lambda)} - \Lambda_{\rm eff}.$$ Even in vacuum (no matter of any kind), a term $\rho_d = \mathcal{C}/a^3$ appears — a pressureless dust. This is precisely Mukohyama's (2009) "dark matter as an integration constant": the Hamiltonian constraint of the projectable version is global (a spatial integral) rather than local, and this looser constraint lets dust emerge as an integration constant, whose classical solution mimics general relativity plus cold dark matter. The dust need not be promoted to an independent dynamical field; it emerges only when the equations are solved — and is therefore geometric and particle-free.

Epistemology. The scaling $\rho_d \propto a^{-3}$ is a structural consequence of the first integral, one instance of the remainder-conservation axiom (prior); the abundance Ω_d is the integration constant, set by conditions in the very early universe (posterior). This agrees, independently, with the earlier result from the brane shift charge (conservation prior, value posterior). The posterior status of the abundance is on the same footing as Ω_c determined by observation in ΛCDM, or the dust fraction as a free initial value in AeST; what is distinctive to SAE is the geometric (rather than particle) ontology of the dust and the axiomatic origin of its conservation. This dust term is precisely the $A_m/a^3$ already used in Cosmo III. The effective gravitational coupling is $G_{\rm eff} \propto 1/(1-3\lambda)$. Positivity of the dust (ρ_d > 0 and G_eff > 0) requires $(1-3\lambda) > 0$, i.e. λ < 1/3, which is off the general-relativity limit λ = 1; the observational viability of this regime belongs to the khronometric α, λ parameter-space open problem listed below.

Version tension. Mukohyama dust arises in the projectable Hořava theory (lapse depending on time only, global constraint), and the projectable version has a known problem — an extra scalar graviton mode that is a ghost or strongly coupled at low energy (Blas–Pujolàs–Sibiryakov). The c_T = 1 / healthy structure of §4.4 belongs to the non-projectable (khronometric) version. These are different Hořava versions; the literature (e.g. MNRAS 2023) notes that realizing cold-dark-matter-like behaviour in the non-projectable version is not automatic ("if one can"). Whether one can simultaneously have Mukohyama dust (projectable) and the healthy khronometric structure (non-projectable) is therefore an unresolved structural problem, not a settled fact; the stability of the dark-matter sub-goal's foundation depends on it. §7 and the open-item list mark this as a foundational open problem.

4.2 DBI-C → dark energy and MOND

Substituting the decomposition into $Q = A^\mu\nabla_\mu C$, the cross terms cancel and $$Q \propto |\nabla\tau_1|^2 - |\nabla\tau_2|^2 \propto \omega_2^2-\omega_1^2 \propto \Lambda,$$ so the time kinetic energy of C is of order Λ (small). In the DBI form, the equation of state $w_C = q^2 - 1$ ($q = Q/c$) is slightly above −1 — C is dark energy, consistent with Cosmo V.

The brane derivation of the DBI form (prior, not analogy). The worldvolume action of C is of Nambu-Goto type, $\mathcal{L} = -T\sqrt{-\det(g+\partial C\partial C)/\det g} = -T\sqrt{1+g^{\mu\nu}\partial_\mu C\partial_\nu C}$; the determinant of the induced metric gives $\det h/\det g = 1-\dot C^2/c^2$, so the time part is $-T\sqrt{1-Q^2/c^2}$. This is not a chosen function but a consequence of brane geometry; the dual-4DD has an embedded brane structure (time arrows reversed but not exactly 180°, with a symmetry breaking). $Q_c = c$ is a prior.

The structural bound (hard). Since $w_C = q^2 - 1 \ge -1$ ($q^2 \ge 0$), together with a positive 4-form Λ₁ (w = −1), the total dark energy satisfies $w_{\rm int} \ge -1$. This is the standard non-phantom property of tachyon/DBI, a hard structural theorem rather than a fit — see §6, HP2.

MOND (the spatial part). The spatial term $y^{3/2}$ gives the AQUAL structure in the quasi-static, deep-MOND limit; the 3/2 power follows from the p = d = 3 theorem in three dimensions (3/2 = 3DD ÷ 2), and the normalization scale a₀ is supplied by the horizon sector (§5). The baryonic Tully–Fisher relation $v^4 = GM_b a_0$ is a first integral of the nonlinear Poisson equation and is exact.

4.3 4-form → the cosmological constant

The explicit Λ in the action comes from a 4-form gauge field. In four dimensions the field strength is $F_{\mu\nu\rho\sigma} = f\cdot\varepsilon_{\mu\nu\rho\sigma}$, its equation of motion makes f constant, and its stress-energy $T_{\mu\nu} = -\tfrac12 f^2 g_{\mu\nu}$ is proportional to the metric — a cosmological constant $\Lambda_{\rm 4form} = (\kappa/2)f^2$.

The dual-4DD gives two 4-form sectors, with field strengths f₁, f₂, so $\Lambda_1 \propto f_1^2$ and $\Lambda_2 \propto f_2^2$ (opposite sign). Hence $$\boxed{\ \Lambda_1+\Lambda_2 = 0\ }$$ is the remainder-conservation axiom acting on the vacuum (4-form) sector — the same axiom as the khronon dust conservation, and as antimatter B₁+B₂ = 0 and zero energy E₁+E₂ = 0. The observed Λ is the single-sector Λ₁, whose prior form (Cosmo I, dual-face reciprocity) is $\Lambda_1 = 2(\omega_2^2-\omega_1^2)/c^2$ (the "2" being the dual-face factor); its value is posterior (from the measured T₁, T₂), and differs from one breath to the next. In the dual-frame picture (Cosmo V): Λ_total = 0 in the geometric frame, while Λ₁ > 0 in the Jordan frame we inhabit, the two related by the conformal factor A²(C).

Consilience. Both the Λ₁ of the 4-form and the dynamical dark energy of DBI-C are ∝ (ω₂²−ω₁²) — the same breathing-asymmetry scale; together they compose the observed dynamical dark energy.

4.4 The tensor speed c_T = 1

In khronometric gravity the linearized spin-2 tensor perturbation mode has speed $c_t^2 = 1/(1-\beta)$, β being one of the khronometric couplings, so $c_T = 1 \iff \beta = 0$. (SAE's ontology holds that gravity is not a force and that there is no independent graviton particle; here $c_T$ is the propagation speed of the linearized tensor mode, not the speed of a graviton particle.)

In SAE, β = 0 follows from structure rather than tuning. The DD Breakthrough theorem holds that c is defined at the birth of the one-dimensional-time sector, that all DD share one spacetime, and that c is universal; taken seriously, this implies that all propagating modes — including the tensor mode — propagate at the same c on this one shared spacetime, so $c_t = 1$, i.e. β = 0. This is the same universality-of-c axiom as STEP 0 (Ċ bounded by c) and STEP 1 (single light cone) of §2. GW170817/GRB170817A is consistent with this and constrains deviations strongly (|β| < 10⁻¹⁵) — noting that the observation confirms the closeness of the gravitational-wave and light speeds, not the SAE universality-of-c derivation itself. This meets c_T = 1 by a different route than AeST's Maxwell-form $F^2$, but in SAE it is structurally forced rather than a choice of form.

Parameter constraints. c_T = 1 only fixes β. The other two khronometric parameters α, λ are independently constrained (post-Newtonian parameters, pulsars, big-bang nucleosynthesis, absence of Cherenkov radiation); the scalar-mode speed c_s is barely constrained by gravitational waves and is a known soft spot of khronometric gravity. Full parameter viability (α, λ within the intersection of these constraints) is an independent check inherited from khronometric gravity and does not hold automatically.

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5. The horizon origin of a₀

5.1 From vacuum scale to horizon scale

Earlier (Cosmo II/III), a₀ = (π/2)c(ω₂−ω₁) was a vacuum scale and constant. But intermediate-redshift data disfavour this: MUSE-DARK III (Ciocan et al. 2026) and MIGHTEE-HI (Vărăşteanu et al. 2025) show a₀ increasing with redshift at z ≲ 1.5 (tentatively, ~2.4σ). This directly contradicts the a₀ decline given by Cosmo III's η(t) factor — a closed-FRW artifact, of the same kind as the C-dust misassignment. The a₀ decline given earlier by Cosmo III's η(t) had the opposite sign to these data; here we revise it to a horizon scaling, and η(t)'s high-redshift decline is not retained. The old form a₀ = (π/2)c(ω₂−ω₁) is reduced to a z = 0 normalization, whose present-matching gives ω₂−ω₁ ≈ H₀/π²; the corresponding update to Cosmo III's text awaits z > 3 data.

5.2 The mechanism: a₀ as the information read-out noise floor of the horizon

The revised picture is SAE-native, and promotes the numerical coincidence a₀ ∼ cH₀ from a coincidence to a structure. SAE's ontology of gravity (the four-force prequel) reads gravity as information read-out: a system's gravitational response corresponds to the information it can read within the causal horizon. What a galactic core feels is the noise floor projected inward as the cosmic horizon continually flushes causal information with expansion. Inverting through an Unruh-type relation, this noise floor corresponds to an acceleration $$a_0 \sim \frac{c^2}{R_A} \sim cH(z),$$ with $R_A$ the apparent horizon. When the Newtonian acceleration from baryons $g_N > cH$, the baryons' "command" overrides the noise floor (the Newtonian regime); when $g_N \le cH$, it is drowned by the noise floor of the expanding horizon (the MOND regime). MOND is then neither dark matter nor a vacuum aether, but the geometric noise floor projected inward as the horizon consumes causal information. This mechanism accounts for the coincidence a₀ ∼ cH₀ (a₀ ∼ cH/2π, with the status of the coefficient discussed in §5.3; today H = H₀) and for the direction of the a₀(z) increase.

5.3 The geometric coefficient and its three-layer status

The numerical coefficient is understood in three layers, whose observational and derivational statuses differ.

Write $a_0 = c^2/(2\pi R_A)$, with $2\pi R_A$ the circumference of the horizon's great circle. Both the radius $R_A$ and the circumference $2\pi R_A$ are lengths, and $c^2/R_A = cH$ (coefficient 1) and $c^2/(2\pi R_A) = cH/(2\pi)$ (coefficient 1/2π) are both dimensionally valid; so 2π is not forced by dimensions, but is the result of the geometric choice to take the circumference rather than the radius. Its justification rests on a "read-out must close" closure-loop principle, which is not yet axiomatically proven (§7 open problem).

The three layers of a₀ are then:

Layer one — the scaling direction (hard prediction). a₀ grows with the horizon ($\propto R_A^{-1}$). This is a falsifiable hard prediction (§6, HP1). The slope cancels systematic error in the ratio and is therefore measurable.

Layer two — the absolute value of the coefficient (not observationally distinguishable). The absolute value of $\chi \equiv a_0/(cH_0)$ is limited by a systematic floor. The initial-mass-function uncertainty in the mass-to-light ratio is about 20% and is systematic (it does not reduce as $\sqrt N$ with sample size), and the Hubble tension contributes about 8%. To distinguish 1/(2π) = 0.159 from 1/6 = 0.167 requires χ to about 5%, which a pure a₀(z) survey route cannot reach persistently (barring independent breakthroughs in both the initial mass function and the Hubble tension). The absolute value of the coefficient therefore cannot serve as an observationally falsifiable hard prediction; the predictive power of a₀ is in the scaling (slope), not the coefficient (absolute value).

Layer three — the value 1/(2π) of the coefficient (motivated preference, not a completed derivation). Although observation cannot distinguish them, 1/(2π) and 1/6 are not symmetric a priori. 1/(2π) has structural support (great-circle normalization of the horizon), while 1/6 and other alternatives have no corresponding structural motivation. We therefore prefer 1/(2π) among the candidates — a legitimate prior preference, much as in physics θ_QCD ≈ 0 is preferred by strong-CP symmetry, or O(1) group-theory factors are preferred by geometry, even when observation cannot yet distinguish them. This differs in kind from "a numerical coincidence that happens to fall within the error band but has no derivational origin": 1/(2π) is the simplest candidate and carries a geometric derivation chain. But it remains a motivated preference rather than a completed derivation — the closure-loop question "why the circumference rather than the radius" has not been axiomatically proven, just as θ = 0, before the Peccei–Quinn mechanism gave it a dynamical explanation, was a strongly motivated preference rather than a proven prediction.

5.4 Distinguishing SAE from Verlinde emergent gravity

On the absolute value of the coefficient, SAE and Verlinde emergent gravity cannot be distinguished (if both give 1/2π the values coincide, and the coefficient is unmeasurable). SAE is distinguished by its mechanism (information read-out, more ontological than an entropic-force phenomenology), by its horizon choice (SAE takes the apparent horizon $R_A = c/\sqrt{H^2+kc^2/a^2}$, which differs from other horizon definitions under curvature or at high redshift — structurally distinct and, in principle, measurable), and by the scaling direction. Moreover, the great-circle normalization that gives 2π is a geometric choice on a purely causal manifold with no thermodynamic-temperature residue, so its 2π does not arise from an "erase-ℏ" kind of fitting; but, as noted in §5.3, the step "take the circumference rather than the radius" itself awaits proof by the closure-loop principle, so what distinguishes SAE from Verlinde here is the character of the source of 2π (causal geometry rather than thermodynamics), not a completed derivation of the coefficient.

Thus the horizon-sector form of a₀ is $$\boxed{\ a_0(z) = \frac{c^2}{2\pi R_A(z)} \xrightarrow{\ \text{flat}\ } \frac{cH(z)}{2\pi}\ }.$$

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6. Predictions

6.0 The parameter-locking protocol

All predictions in this section use only independent inputs, to preclude the circularity of "tuning parameters against the data to be tested": T₁, T₂ are SAE inputs (not re-estimated here); Ω_d, Ω_Λ are constrained by the cosmic microwave background; a₀(0) is anchored by the local radial-acceleration relation / SPARC; the coupling g is not used in the baseline. In particular, the DESI (w₀, wₐ) is not used to fix parameters; the a₀(z) of MUSE/MIGHTEE is used only for an a posteriori test of the scaling. Lock first, then compute.

We restrict the hard predictions to no more than three, to avoid dilution. Below we give two hard predictions and one via-negativa expectation.

HP1 · The horizon scaling of a₀

$$\frac{a_0(z)}{a_0(0)} = \frac{R_A(0)}{R_A(z)} \simeq \frac{H(z)}{H_0}\ (k\simeq0).$$ Numerically: z = 1 → 1.79, z = 3 → 4.57, z = 4 → 6.33. Compared with the constant-a₀ branch, this differs by more than a factor of four at z ≳ 3. Because a₀(0) is anchored by the local radial-acceleration relation, the falsifiable content of HP1 is the shape of the redshift ratio and slope, not the absolute normalization. Absolute-calibration errors (overall mass-to-light ratio, Hubble tension) partly cancel in the ratio; but redshift-dependent systematics do not cancel automatically — stellar-population evolution, gas content, disk thickness, non-circular motion, selection function, feedback, and resolution can all vary with redshift. The strength of HP1 is therefore that it gives a definite scaling shape (whose difference from constant-a₀ is far larger than these systematics), not a claim that systematics have vanished. This slope test forms a key asymmetry with the unmeasurable absolute coefficient (§5.3): the predictive power is in the shape of the slope.

The primary adjudicator is the high-redshift radial-acceleration relation / baryonic Tully–Fisher relation, especially at z > 3.

Relation to current data. Current tentative data (the linear-in-z fit of MUSE-DARK III, which itself reports "faster than H(z)") are not necessarily on the side of HP1 — their growth may already be somewhat steeper than ∝H. So what HP1 predicts is the specific form "exactly ∝H": if high-redshift data confirm "faster than H," then the exact ∝H is falsified (while the mechanism "a₀ grows with the horizon" survives — only its coefficient-fixed ∝H does not). In practice we first test γ = 1 in $a_0 \propto H^\gamma$. This makes HP1 cleanly falsifiable while not presuming that current data are already on its side. z > 3 is the decisive test.

HP2 · Intrinsic dark energy is non-phantom

There are three layers. The first is a parameter-free structural bound; the second and third depend on a free coupling and have a different status from the first.

Layer one — a genuinely hard bet (parameter-free structural theorem). A positive DBI-C plus a positive Λ₁ gives $$w_{\rm int}(z) \ge -1.$$ SAE's intrinsic dark energy structurally does not cross the phantom divide. This is a hard prediction that a non-parametric reconstruction would falsify, not a soft statement, and it is not to be downgraded together with the softer layers below. The falsifier: if a reconstruction independent of the CPL parametrization and independent of a dark–dark interaction assumption robustly gives intrinsic w < −1, the positive-sign DBI-C structure of SAE is falsified.

On DESI: its CPL best fit (w₀, wₐ) ≈ (−0.75, −0.86) extrapolates to w < −1 at high redshift (apparent phantom crossing). We treat this apparent crossing as an effective reconstruction, not an intrinsic target. The reason: low-order extrapolation fails repeatedly in cosmology (the deceleration parameter, constant-w, CPL alike), because cosmological quantities are dynamical solutions rather than low-order polynomials, and w(z) must be doubly differentiated from integral distance quantities and is sensitive to the parametrized shape; the CPL high-redshift crossing may come partly from the parametrization tail extrapolated into a data-sparse region, and cannot be equated directly with a crossing in the intrinsic field's p/ρ. Moreover, DESI measures a reconstructed effective expansion history, not the microscopic p/ρ of any field; if the dark sector has an interaction, intrinsic w ≥ −1 and effective w_eff < −1 can coexist. This judgment is itself contested: some non-parametric reconstructions also suggest crossing; we take the side "the apparent crossing is partly a parametrization artifact," and place the burden of adjudication on a reconstruction independent of CPL. To be clear, we do not claim to have falsified DESI — DESI's departure from ΛCDM is real, and SAE is consistent with it; what we distinguish is only two different objects, "the w of the effective reconstruction" and "the w of the intrinsic field," the former of which may cross while the latter (in this framework) does not.

Layer two — soft (the effective-crossing branch, one parameter, non-baseline). The kinetic mixing θQ of khronon and C (which preserves shift symmetry, since Q is invariant under C → C + const, so U(C) = const is preserved) gives a source $$S_\theta = -3gQ(\dot H + 3H^2).$$ When an observer reconstructs assuming "matter conserved independently," the effective equation of state $w_{\rm eff} = w_C - S_\theta/(3H\rho_C)$ can cross −1 while the intrinsic $w_C$ does not — consistent with the post-DESI literature on "effective crossing via a dark interaction." But the coupling g is free: the field redefinition (τ₁, τ₂) → (τ, C) does not fix it (the redefinition generates ∇τ·∇C, not θQ; θQ is an allowed rather than a forced operator), and the dual-exchange symmetry (θQ is odd under τ₁ ↔ τ₂) only constrains g ∝ ε, not its magnitude or sign. Effective crossing further requires $gQ < 0$, and the sign is likewise undetermined. This mixing is of kinetic-braiding type, and its true w_eff must be obtained by a full variation (the fluid-decomposition formula is only diagnostic). It should be emphasized that g is not a backdoor parameter free to fit arbitrary data: any nonzero θQ coupling must leave a distinctive imprint at the cosmological perturbation level — on the growth rate $f\sigma_8$ and on the integrated Sachs–Wolfe effect of the microwave background (the source must reach 10–30% of $H\rho_C$, not a small correction). Therefore, if future observation sees an effective crossing only in the background $H(z)$ but finds no perturbation feature corresponding to this operator in the perturbation data, this branch is likewise falsified. So this is an SAE-permissible one-parameter dark-sector extension whose one parameter is itself independently constrained at the perturbation level, not a free escape route for post-hoc fitting.

Layer three — the commitment boundary (open). If a foundational two-clock (τ₁, τ₂) action is written down and fixes g and its sign, the effective crossing of layer two can be promoted to a prediction. This is a larger structural task, left for the future, and does not block the layer-one bet of the mainline.

Remainder-conservation confirmation. The exchange source among khronon dust, C, and geometry is the field-theoretic version of remainder conservation ($\nabla_\mu T^{\mu\nu}_{\rm total} = 0$, a reallocation of accounts rather than creation of energy from nothing); what is conserved is the total dark/geometric sector.

HP3 · A particle-free dark sector (via negativa)

Direct detection of conventional dark-matter particles (WIMPs, axions, etc.) continues to yield null results, consistent with this framework. We list this as a "via-negativa expectation" rather than a strong positive prediction (a null result cannot prove "there is none"); true discrimination still lies in a₀(z), the external-field effect, structure growth / lensing, and the full microwave-background spectrum.

A demoted note on deceleration turnaround. Late-time weakening of dark energy means the acceleration slows earlier than in ΛCDM (DESI's cross-statistics already hint at "slowing acceleration at low redshift," in the same direction); but its specific turnaround redshift depends on the free g of HP2, layer two, and is therefore only directional, with the specific value awaiting the fixing of g. A Roman + Euclid reconstruction of q(z) tests its direction.

Structural statements (not falsifiable hard predictions, listed for reference). Λ_total = 0; c_T = 1 (confirmed by GW170817); the geometric turnaround is unobservable; ξ > 0 would falsify the causal-reinforcement prior. These are structural statements, not among the three hard predictions above.

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7. Remainders

SAE's honesty requires stating the remainders openly, without passing them off as predictions, while preserving remainder conservation.

• Dust abundance Ω_d: posterior (set by the microwave background); $\rho_d \propto a^{-3}$ is remainder conservation, and the value is an integration constant (Mukohyama). Same footing as Ω_c in ΛCDM, or the dust fraction in AeST.
• a₀ coefficient 1/(2π): its absolute value is unmeasurable (persistent systematic floor, not a hard prediction); its value 1/(2π) is a motivated preference (supported by great-circle geometry, better-reasoned than 1/6, the θ = 0 analogy), not a completed derivation (the closure-loop principle is open), and not a numerical coincidence fitted post-hoc. Three layers in §5.3.
• khronon–C coupling g: a free Wilson coefficient; not fixed by the field redefinition, with dual-exchange constraining only g ∝ ε and the magnitude and sign undetermined.
• Λ₁ value: form prior, value posterior. K_B, α, λ: posterior (inherited khronometric parameter space).
• Closure-loop principle: the great-circle normalization rests on the "read-out must close" motivation and is not yet axiomatically proven (open).
• Hořava version tension: khronon dust (§4.1, Mukohyama) is cleanest in the projectable version (with its ghost/strong-coupling problem of the extra scalar mode), while c_T = 1 / healthy (§4.4) is in the non-projectable (khronometric) version; realizing integration-constant dust in the khronometric version is an open structural task, not an automatic inheritance — the stability of the dark-matter sub-goal's foundation depends on it.
• Full Boltzmann (CLASS/CAMB full spectrum): marked as a prerequisite for quantitative work, not an optional engineering task. Without a full-spectrum computation one can only say "structurally corresponds to the AeST microwave-background lesson," not "passes the microwave-background test."

The relations above — B₁+B₂ = 0, E₁+E₂ = 0, Λ₁+Λ₂ = 0, ρ_d ∝ a⁻³, khronon ↔ C ↔ geometry exchange — are instances of one remainder-conservation axiom.

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8. Comparison with, and discrimination from, existing frameworks

Relative to AeST. AeST obtains its phenomenological flexibility from a free function plus background initial values; SAE reduces part of that freedom to structural consequences of the brane-DBI, p = d = 3, and the khronon integration constant, and is narrower in function space, hence more easily falsified. This does not classify SAE as a subset of AeST (SAE makes hard predictions AeST does not), but a structural narrowing. c_T = 1 follows from the universality of c.

Discriminating SAE from Verlinde. Not in the a₀ coefficient (unmeasurable), but in the mechanism (information read-out), the horizon choice ($R_A$), and the a₀ scaling.

Discriminating SAE from ΛCDM plus particle dark matter. The a₀ scaling (z > 3), intrinsic w ≥ −1, and no dark-matter particles (WIMP null, via negativa). The external-field effect (e.g. the dynamics of dark-matter-deficient galaxies) is in principle a further discriminant, whose operational quantitative criterion is left for future work.

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9. Conclusion

We have derived, from dual-4DD breathing, an aether-scalar-tensor structure that places the three dark-sector effects in a single structure, with no particles: the khronon gives geometric dust, DBI-C gives intrinsically non-phantom dynamical dark energy and MOND, and a 4-form gives the cosmological constant; all conservation laws reduce to one remainder-conservation axiom; and the tensor speed c_T = 1 follows from the universality of c and is confirmed by GW170817. We have revised a₀ from a vacuum scale to a horizon scale, and give two hard predictions that later data can adjudicate: the horizon scaling of a₀ (adjudicated by the high-redshift radial-acceleration relation) and the non-phantom bound on intrinsic dark energy (adjudicated by non-parametric reconstruction).

We have stated all remainders openly: the dust abundance, the coupling g, and the khronometric parameters are posterior; the absolute value of the a₀ coefficient is unmeasurable, and its value 1/2π is a motivated preference rather than a completed derivation; whether the projectable and non-projectable versions can coexist is a foundational open problem. Where the framework has revised earlier intermediate conclusions — the C-dust misassignment, the a₀ decline given by η(t), the undetermined dark-energy coupling g — the revisions are noted in the corresponding sections.

The path taken here trades structural constraint for falsifiability. The two hard predictions place the framework where it can be adjudicated — which is the value of the approach: a theory whose structure is constrained by a prior has no room for post-hoc parameter adjustment and can only be tested by hard predictions. If one of them is falsified, the falsification itself indicates where the prior should be revised — something a model with enough freedom to fit everything cannot provide.

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Acknowledgments

The derivations, computations, and internal critique of this paper used a four-AI collaborative method: Claude, ChatGPT, Gemini, and Grok took the roles of conceptual coordination, derivation and review, geometry and topology verification, and adversarial critique, with an additional independent review line for cross-checking. The method was used for drafting, cross-validation, and the surfacing and correction of errors (a methodological record accompanies the prequel); its operation, including several intermediate conclusions that were progressively revised or abandoned, is noted in the corresponding sections.

These AI systems are tools and methods used in this research and do not constitute authors. The entire content of this paper — including its prior assumptions, structural choices, predictions, and everything marked as remainder or open — is the sole responsibility of the author.

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Appendix A · Open-item list

Open item	Nature	If resolved
coexistence of the projectable (dust) and non-projectable (c_T = 1) versions	foundational	secures the foundation of the dark-matter sub-goal
a foundational two-clock (τ₁, τ₂) action	structural	fixes g → promotes HP2's effective crossing to a prediction; or fixes the a₀ coefficient
a₀(τ) full covariantization	structural	expresses $R_A^{-1}$ as a foliation-invariant local geometric scalar (θ, curvature), making the MOND term a complete local Lagrangian rather than an FLRW effective term
axiomatic proof of the closure-loop (great-circle) principle	theory	promotes the a₀ coefficient from "motivated preference" to "structurally motivated choice" (still unmeasurable)
the perturbation level of khronon–C mixing (S₈ / growth / lensing)	computation (viability-critical)	the viability of the θQ branch
khronometric α, λ parameter space (post-Newtonian / pulsar / nucleosynthesis)	inherited check	full viability
full Boltzmann (CLASS/CAMB full spectrum)	engineering	quantitative comparison with the microwave background
the C-field tracking corridor (carried over from Cosmo V)	numerical	U(C) entering the corridor naturally

References

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1. 引言

暗物质、暗能量、与 MOND 现象学——暗扇区的三块——在标准处理中彼此独立:冷暗物质是一种粒子,暗能量是一个常数或一个额外的标量场,MOND 是一条经验的加速度关系。三者各有其自由参数,各自拟合各自的观测。

本文从单一结构把三者放进同一框架,且不引入任何暗物质粒子。这一结构来自 Self-as-an-End(SAE)的 dual-4DD 图景:两个时间取向近反平行的四维时向扇区,各携一台呼吸钟 τ₁、τ₂;其对称组合 τ_breath=(τ₁+τ₂)/2 充当 khronometric aether,反对称组合 C=τ₁−τ₂ 充当一个 DBI 标量。由此,khronon 给出几何尘埃并充当暗物质,DBI-C 给出动态暗能量与 MOND,一个 4-form 场强给出宇宙学常数;三类效应的守恒律同归一条余项守恒公理。

本文承接 SAE 宇宙学系列:Cosmo I 给出宇宙学常数 Λ=2(ω₂²−ω₁²)/c²,Cosmo II/III 给出 MOND 加速度 a₀ 及其 3/2 幂的几何起源,Cosmo IV 讨论有效引力常数,Cosmo V 提出 C 场为动态暗能量。这里把它们装进一个 aether-scalar-tensor 型的作用量结构——将该类理论的一个自由函数收窄为有结构来源的两项——完成三个子目标(khronon 的 Mukohyama 尘埃、张量速度 c_T=1、宇宙学常数 Λ₁),并将 a₀ 从早前的真空标度修正为视界标度。

这一结构的价值不在于它能拟合已有数据(拟合暗扇区数据的模型已经很多),而在于它对函数形式的约束,以及由此产生的可证伪预言。SAE 的先验限制了作用量的形式而非仅其参数:DBI 型的动能项由膜的 Nambu-Goto 作用量给出而非任意选取,空间部分的 3/2 幂由 p=d=3 定理在三维给出而非自由拟合,守恒律由公理给出而非塞入初始条件,张量速度 c_T=1 由 c-普适给出并经 GW170817 确认。正是这些约束,使本框架给出标准冷暗物质加宇宙学常数(ΛCDM)所不能给出的、可被观测判决的硬预言:内禀暗能量方程态 w_int≥−1(暗能量在结构上不穿越 phantom),以及 a₀ 的红移 scaling(a₀ 随视界增大,在 z≳3 与常数-a₀ 相差逾四倍)。这两条预言可被现有与临近的观测判决。

这是一种有意的取舍。一个自由度足够多的模型能拟合几乎任何数据,而能拟合一切等价于不能排除任何东西:它精密,却不可证伪,因而无法被判决、也无从更新。相较之下,一个结构被先验约束的框架没有事后调参的回旋余地,只能做硬预言——正如广义相对论几乎没有自由参数、却做出了光线偏折与引力波这样先算后测的预言。本文所给的两条硬预言正属此类:它们可能被证伪,而这正是其价值所在;若被证伪,证伪本身即指明先验应在何处修正。

其余的量——尘埃丰度 Ω_d、若干耦合与 khronometric 参数、a₀ 系数的绝对值——由观测或边界条件设定,一如广义相对论中的引力常数或标准模型中的质量谱。本文在全文将先验导出的结构约束、后验设定的数值、与尚未闭合的开放问题逐一分层标出。

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2. Dual-4DD 呼吸结构

我们简述所需的结构;完整推导见 SAE 基础篇与 Cosmo I–V。

两个 4-维时向扇区(4DD)以近乎反平行的时间取向共存,其间存在轻微的对称破缺,参数化为 $$\varepsilon = \frac{\omega_2-\omega_1}{\omega_2+\omega_1} \approx 0.0127,$$ 其中 ω₁、ω₂ 为两扇区呼吸钟 τ₁、τ₂ 的频率。硬参数取 T₁=20 Gyr、T₂=19.5 Gyr(SAE 输入,本文不重估)。

关键分解在于把两台钟组合为 $$\tau_{\rm breath} = \frac{\tau_1+\tau_2}{2}\ (\text{对称}),\qquad C = \tau_1-\tau_2\ (\text{反对称}).$$ 对称组合 τ_breath 定义因果叶状结构的法向,充当 khronometric aether;反对称组合 C 是一个动态标量。二者并非独立场,而是同一对 (τ₁,τ₂) 的两个组合——这一点在 §6 讨论耦合时至关重要。

两条已确立的结构性结论(详见 T2 工作记录)在此引用:

其一(C 非紧):光速 c 约束的是错位速率 Ċ 本身,而这一约束在 4DD 内不可达(相对论天花板,非通往第五维的通道)。故 C 沿类时方向累积而不缠绕,是非紧的;移位对称 C→C+const 强制势 U(C)=const。

其二(单光锥):两个 4DD 共享同一个 c,时间取向相反意味着它们对应同一个光锥的两个方向,而非两个度规。这在结构上禁止了危险的 bimetric/disformal 耦合——早前 Cosmo V 所设想的"双度规重构"由此被判定为 aether-scalar-tensor 重构,是结构必然而非选择。这也是通过 GW170817 的先决条件。

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3. 作用量结构与承诺分层

3.1 模板与 SAE 的收窄

我们以 Skordis–Złośnik(2021)的 aether-scalar-tensor 作用量为模板: $$S_G = \int d^4x\, \frac{\sqrt{-g}}{2\kappa}\Big[R - 2\Lambda - \tfrac{K_B}{2}F_{\mu\nu}F^{\mu\nu} + 2(2{-}K_B)J^\mu\nabla_\mu\phi - (2{-}K_B)\mathcal{Y} - \mathcal{F}(\mathcal{Y},Q) - \lambda(A^\mu A_\mu{+}1)\Big] + S_m[g],$$ 其中 $Q=A^\mu\nabla_\mu\phi$,$\mathcal{Y}=(g^{\mu\nu}{+}A^\mu A^\nu)\nabla_\mu\phi\nabla_\nu\phi$,$J_\mu=A^\nu\nabla_\nu A_\mu$,$F_{\mu\nu}=\nabla_\mu A_\nu{-}\nabla_\nu A_\mu$。物质仅耦合到单一度规 g(满足等效原理),aether 动能取 Maxwell 形。

AeST 将 $\mathcal{F}(\mathcal{Y},Q)$ 留作自由函数。SAE 的收窄在于两端固定它: $$\boxed{\ \mathcal{F}_{\rm SAE}(\mathcal{Y},Q) = 2T\big[\sqrt{1-Q^2/c^2}-1\big] + M_Y^4\, y^{3/2}\ }$$ 时间部分是 DBI(tachyon)形,由膜的 Nambu-Goto 作用量导出(§4.2);空间部分的 3/2 幂由 p=d=3 定理给出(§4.2、§5)。这是 SAE 相对 AeST 的核心结构增益:一个自由函数被降为两项有结构来源的项。

符号约定须钉清,以免与后文的 $w_C$ 推导错位。此处 $\mathcal{F}_{\rm SAE}$ 采用 AeST 模板的约定,即作用量中出现的是 $-\mathcal{F}$;故 DBI 子项在作用量中的贡献为 $-2T[\sqrt{1-Q^2/c^2}-1]$,其物理拉氏量的形状为 $-\sqrt{1-Q^2/c^2}$(§4.2 的膜导出),常数项 $+2T$ 为可吸收常数,并入 $\Lambda_{\rm 4form}$ 或以零点规范扣除。系数 $2T$ 中的因子 2 来自模板中 $\mathcal{Y}$ 项前的 $(2-K_B)$ 归一约定,使 DBI 项与 $\mathcal{Y}$ 项在弱场展开中衔接。下文 §4.2 给出的 $w_C=q^2-1$ 使用的是物理 DBI 应力-能量张量,而非 $\mathcal{F}$ 的符号本身。

标量取 φ=C(反对称组合);aether 取 $A_\mu=-N\nabla_\mu\tau_{\rm breath}$(hypersurface-orthogonal,即 khronometric);$Q_c=c$ 为先验(源自 §4.2 的 c-天花板)。

3.2 空间标度的自洽:α 不是自由常数

MOND 加速度 a₀ 正是准静态 𝒴^{3/2} 项的归一化标度。若该项的系数是一个常数 α,则 a₀ 为常数,与 §5 得出的 a₀∝H(z) 不符。因此空间项不写成常数系数的 $\alpha\mathcal{Y}^{3/2}$,而在无量纲化时写入视界标度: $$y \equiv \frac{\mathcal{Y}}{a_0(\tau)^2},\qquad a_0(\tau) = \frac{c^2}{2\pi R_A(\tau)},$$ 其中 R_A(τ) 是表观视界(§5)。幂次 3/2 由 p=d=3 定,标度 a₀(τ) 由视界扇区给。如此,§3 的 MOND 归一化与 §5 的视界 a₀ 是同一个 a₀(τ),二者一致。

这一写法须附一项变分层面的说明,否则会引入高阶导数。由于 $a_0(\tau)\propto R_A^{-1}\propto H$,而 $H=\dot a/a$ 含度规的时间导数,若在局域场方程中对 $a_0(\tau)$ 一并变分($\delta S/\delta g_{\mu\nu}$),将产生 Ostrogradsky 型高阶导数(鬼)。本文对此的立场与 a₀ 的物理起源(§5)一致:$a_0(\tau)$ 是由全局因果视界决定的信息读出底噪,在局域(星系尺度)变分中作为不参与局域度规变分的外部演化背景处理——局域物质的反作用不足以撼动视界底噪,故不引入高阶导数变分项。相应地,含 $a_0(\tau)$ 的空间项在此不是完整的协变局域拉氏量,而是 FLRW 背景上经视界读出归一后的有效项;其完整协变化(把 $R_A^{-1}$ 表达为 aether 展开 $\theta=\nabla_\mu A^\mu=3H$、空间曲率或其它 foliation 不变的局域几何标量,如平直近似下 $a_0(\tau)\simeq c\theta/6\pi$)留作开放项。

3.3 baseline 作用量与耦合分支的分离

baseline 作用量与一个可选的相互作用分支在此分开列出。baseline 作用量不含 khronon–C 混合项: $$S_{\rm base} = S_{\rm grav}[g,A] + S_{\rm DBI\text{-}C}[C,A] + S_{Y}[C,A] + S_{\rm 4form} + S_m[g].$$ 另有一项 $$S_{\rm mix} = g\int d^4x\,\sqrt{-g}\,\theta Q,\qquad \theta=\nabla_\mu A^\mu,$$ 它不是 baseline 的一部分,而是一个开放的相互作用分支(见 §6 HP2 第二层);其耦合 g 是一个自由 Wilson 系数,而非被 dual-4DD 结构固定的数。这一分离对于把"内禀 w≥−1"(baseline 的硬结果)与"有效穿越"(依赖 g 的可选分支)分清,是必要的。

3.4 承诺分层

SAE 的诚实要求逐项标清每一项的地位。下表既防"把结构约束的导出夸大为全量数值推导",也认下先验真有的约束——那些约束是 ΛCDM/AeST 所没有的。

项	SAE 地位	自由系数	观测角色
$R-2\Lambda_{\rm 4form}$	AeST 模板 + 4-form 形式先验	Λ₁ 值后验	暗能量常数部分
DBI 时间项 $\sqrt{1-Q^2/c^2}$	先验(膜导出)	张力 T 后验	暗能量,w_int≥−1
空间项 $y^{3/2}$	幂次先验(p=d=3)	标度=视界 a₀(τ)	MOND / a₀
khronon aether(HSO)	先验(呼吸叶状法向)	α,λ 后验	暗物质(Mukohyama 尘埃)
$-\tfrac{K_B}{2}F^2$	AeST 模板	K_B 后验	aether 动能(c_T=1)
$S_{\rm mix}=g\theta Q$	开放相互作用分支(非 baseline)	g 自由 Wilson	HP2 有效穿越(未定)

先验导出者(单度规、c_T=1、DBI 形、y^{3/2} 幂次、各守恒律)是结构约束,不是数值;后验者(Λ₁ 值、T、K_B、α、λ)与广义相对论中 G 靠测量、标准模型中十九个参数靠测量同类,并不因此损及"这是一个有结构约束的理论"这一事实。

三点须予明确:C 是动态标量(有自身运动方程、贡献 ρ_C(z)),非纯背景量;HSO=khronometric 是本框架所固定的选择,它同时是 §4.1(尘埃)与 §4.4(c_T=1)之间版本张力的接口;相对 AeST,本框架固定了 𝓕 的两项与 Q_c=c,而 K_B、T、α、λ 未定。

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4. 三对象分工

4.1 khronon → 暗物质(Mukohyama 积分常数尘埃)

HSO aether 使引力扇区在红外化约为 projectable Hořava–Lifshitz 引力。考虑其真空 FRW(平直、单位 lapse)动力学。变分标度因子,得加速度方程 $$3(1-3\lambda)(2a\ddot a+\dot a^2) + 6\Lambda a^2 = 0,$$ 它有一个首积分(符号验证:$\tfrac{d}{dt}[3(1-3\lambda)a\dot a^2+2\Lambda a^3]=\dot a\times$加速度方程,精确守恒): $$3(1-3\lambda)a\dot a^2 + 2\Lambda a^3 = \mathcal{C}\quad\Longrightarrow\quad H^2 = \frac{\mathcal{C}/a^3}{3(1-3\lambda)} - \Lambda_{\rm eff}.$$ 即便在真空中(无任何物质),也冒出一项 $\rho_d = \mathcal{C}/a^3$——一个无压尘埃。这正是 Mukohyama(2009)的"dark matter as integration constant":projectable 版本的 Hamiltonian 约束是全局(空间积分)而非局域的,这一更松的约束让尘埃作为积分常数涌现,其经典解 mimic 广义相对论加冷暗物质。尘埃无须提升为独立动力学场,只在解方程时涌现——因而是几何的、无粒子的。

认识论:标度 $\rho_d\propto a^{-3}$ 是首积分的结构结果,是余项守恒公理的一个实例(先验);而丰度 Ω_d 是积分常数,由极早期宇宙条件设定(后验)。这与更早从膜移位荷得到的结论(守恒先验、值后验)独立地一致。丰度的后验地位与 ΛCDM 中 Ω_c 由观测确定、AeST 中尘埃份额为自由初值相同;SAE 独有的是尘埃的几何(而非粒子)本体与守恒的公理来源。此尘埃项即对应 Cosmo III 中已使用的 $A_m/a^3$。有效引力耦合 $G_{\rm eff}\propto 1/(1-3\lambda)$。尘埃的正号(ρ_d>0 且 G_eff>0)要求 $(1-3\lambda)>0$ 即 λ<1/3,这偏离广义相对论极限 λ=1;此 regime 的观测相容性属下文所列的 khronometric α、λ 参数空间开放项。

版本张力:Mukohyama 尘埃出自 projectable Hořava(lapse 只依赖时间、全局约束),而 projectable 版本有已知问题——一个额外的标量引力子模,在低能是鬼或强耦合(Blas–Pujolàs–Sibiryakov)。§4.4 的 c_T=1/健康结构则属于 non-projectable(khronometric)版本。二者是不同的 Hořava 版本;文献(如 MNRAS 2023)指出在 non-projectable 版本中实现类冷暗物质行为并非自动("若能够")。因此,能否同时拥有 Mukohyama 尘埃(projectable)与 khronometric 的健康结构(non-projectable),是一项尚未解决的结构问题而非既成事实;暗物质子目标的地基是否稳固,取决于此。§7 与开放清单将其列为地基性开放项。

4.2 DBI-C → 暗能量与 MOND

将分解代入 $Q=A^\mu\nabla_\mu C$,交叉项抵消,得 $$Q \propto |\nabla\tau_1|^2 - |\nabla\tau_2|^2 \propto \omega_2^2-\omega_1^2 \propto \Lambda,$$ 即 C 的时间动能是 Λ 标度(小)。代入 DBI,方程态 $w_C=q^2-1$($q=Q/c$)略高于 −1——C 是暗能量,与 Cosmo V 一致。

DBI 形的膜导出(先验,非类比):C 的世界体作用量为 Nambu-Goto 型 $\mathcal{L}=-T\sqrt{-\det(g+\partial C\partial C)/\det g}=-T\sqrt{1+g^{\mu\nu}\partial_\mu C\partial_\nu C}$;诱导度规的行列式给出 $\det h/\det g=1-\dot C^2/c^2$,故时间部分为 $-T\sqrt{1-Q^2/c^2}$。这不是选取的函数,而是膜几何的后果;dual-4DD 具有嵌入膜结构(时间箭头反向但非严格 180°、有对称破缺)。$Q_c=c$ 为先验。

结构界(硬):由于 $w_C=q^2-1\ge-1$($q^2\ge0$),再加正的 4-form Λ₁(w=−1),总暗能量满足 $w_{\rm int}\ge-1$。这是标准的 tachyon/DBI 无 phantom 性质,是一个硬结构定理,不是拟合——见 §6 HP2。

MOND(空间部分):空间项 $y^{3/2}$ 在准静态、深-MOND 极限给出 AQUAL 结构;3/2 幂由 p=d=3 定理在三维给出(3/2=3DD÷2),归一标度 a₀ 由视界扇区给(§5)。巴里昂 Tully–Fisher 关系 $v^4=GM_b a_0$ 是非线性 Poisson 方程的首积分,为精确律。

4.3 4-form → 宇宙学常数

作用量中的显式 Λ 来自一个 4-form 规范场。四维中场强 $F_{\mu\nu\rho\sigma}=f\cdot\varepsilon_{\mu\nu\rho\sigma}$,运动方程使 f 为常数;其应力-能量 $T_{\mu\nu}=-\tfrac12 f^2 g_{\mu\nu}$ 正比于度规,即一个宇宙学常数 $\Lambda_{\rm 4form}=(\kappa/2)f^2$。

dual-4DD 给出两个 4-form 扇区,场强 f₁、f₂,故 $\Lambda_1\propto f_1^2$、$\Lambda_2\propto f_2^2$(反号)。由此 $$\boxed{\ \Lambda_1+\Lambda_2=0\ }$$ 是余项守恒公理作用于真空(4-form)扇区的实例——与 khronon 尘埃守恒、以及反物质 B₁+B₂=0、零能量 E₁+E₂=0 是同一条公理。观测到的 Λ 是单扇区的 Λ₁,其先验形式(Cosmo I,dual-face 互反性)为 $\Lambda_1=2(\omega_2^2-\omega_1^2)/c^2$("2"为 dual-face 因子);值后验(由测得的 T₁、T₂),且逐次呼吸不同。双-frame(Cosmo V)图景:几何 frame 中 Λ_total=0,而我们所居的 Jordan frame 中 Λ₁>0,经共形因子 A²(C) 相联。

consilience:4-form 的 Λ₁ 与 DBI-C 的动态暗能量都 ∝(ω₂²−ω₁²),是同一呼吸不对称标度;二者合成观测到的动态暗能量。

4.4 张量速度 c_T=1

在 khronometric 引力中,线性化的自旋-2 张量扰动模速度为 $c_t^2=1/(1-\beta)$,β 为 khronometric 耦合之一;故 $c_T=1\iff\beta=0$。(SAE 的引力本体主张引力非力、无独立引力子粒子;此处 $c_T$ 指线性化张量扰动模的传播速度,非某种引力子粒子的速度。)

在 SAE 中,β=0 由结构给出而非调参。DD Breakthrough 定理主张 c 在一维时诞生时定义、各 DD 共享同一时空、c 普适;认真对待这一点,则所有传播模式——包括张量扰动模——都在这一个共享时空上以同一个 c 传播,故 $c_t=1$,即 β=0。这与 §2 的 STEP 0(Ċ 被 c 约束)与 STEP 1(单光锥)是同一条 c-普适公理。GW170817/GRB170817A 与此相容,并对偏离给出强约束(|β|<10⁻¹⁵)——须注意该观测确认的是引力波与光的速度极接近,而非确认 SAE 的 c-普适推导本身。这与 AeST 用 Maxwell 形 $F^2$ 达成 c_T=1 殊途同归,但在 SAE 中是结构强制而非形式选择。

参数约束:c_T=1 只锁定 β。另两个 khronometric 参数 α、λ 受独立约束(后牛顿参数、脉冲星、大爆炸核合成、无 Cherenkov 辐射);标量模速度 c_s 几乎不受引力波约束,是 khronometric 的已知软肋。完整的参数可行性(α、λ 落在这些约束的交集)是继承自 khronometric 的独立检验,不自动成立。

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5. a₀ 的视界起源

5.1 从真空标度到视界标度

早前(Cosmo II/III)a₀=(π/2)c(ω₂−ω₁) 是真空标度、为常数。但中等红移的观测数据对此不利:MUSE-DARK III(Ciocan 等,2026)与 MIGHTEE-HI(Vărăşteanu 等,2025)在 z≲1.5 显示 a₀ 随红移增大(tentative,~2.4σ)。这直接否定了 Cosmo III 由闭合-FRW 的 η(t) 因子给出的 a₀ 下降——那是一次闭合-FRW 的几何 artifact,与 C-尘埃误置同类。

早前 Cosmo III 的 a₀ 红移下降由闭合-FRW 的 η(t) 因子给出,其方向与上述数据相反;本文将其修正为视界 scaling,η(t) 的高红移下降不再保留。旧式 a₀=(π/2)c(ω₂−ω₁) 降为 z=0 归一化,其 present-matching 给出 ω₂−ω₁≈H₀/π²;Cosmo III 正文的相应更新待 z>3 数据后进行。

5.2 机制:a₀ 是视界的信息读出底噪

修正后的图景是 SAE-native 的,并把 a₀∼cH₀ 的数值巧合从巧合升为结构。SAE 的引力本体(四力前置篇)将引力理解为信息读取:一个系统的引力响应对应于它在因果视界内可读取的信息。星系核所感受的,是宇宙视界随膨胀不断刷新因果信息所投射的底噪(background information flush noise)。以 Unruh 型关系逆推,这一底噪对应一个加速度 $$a_0 \sim \frac{c^2}{R_A} \sim cH(z),$$ 其中 $R_A$ 为表观视界。当巴里昂产生的牛顿加速度 $g_N>cH$ 时,巴里昂的"命令"盖过底噪(牛顿区);当 $g_N\le cH$ 时,被视界膨胀的底噪淹没(MOND 区)。于是 MOND 不是暗物质、不是真空气场,而是视界吞噬因果信息时向内投射的几何底噪。此机制解释了 a₀∼cH₀ 的巧合(a₀∼cH/2π,系数的地位见 §5.3;今天 H=H₀)与 a₀(z) 增长的方向。

5.3 系数几何与三层认识论

数值系数分三层理解,三层的观测地位与推导地位各不相同。

写 $a_0=c^2/(2\pi R_A)$,其中 $2\pi R_A$ 是视界大圆(great circle)的周长。半径 $R_A$ 与周长 $2\pi R_A$ 都是长度,$c^2/R_A=cH$(系数 1)与 $c^2/(2\pi R_A)=cH/(2\pi)$(系数 1/2π)量纲皆合法;故 2π 不是量纲强制的结果,而是"取周长而非半径"这一几何选择的结果。其正当性依赖于一条"信息读出必须闭合"的闭合环原则,而该原则尚未公理化证明(§7 开放项)。

由此,a₀ 的三层地位为:

第一层——scaling 方向(硬预言):a₀ 随视界增大($\propto R_A^{-1}$)。此为可证伪的硬预言,见 §6 HP1。斜率在比值中抵消系统误差,故可测。

第二层——系数的绝对值(观测不可分辨):$\chi\equiv a_0/(cH_0)$ 的绝对值受系统地板限制。质光比的初始质量函数不确定性约为 20% 且是系统性的(不随样本数以 $\sqrt N$ 缩减),哈勃张力另贡献约 8%。要区分 1/(2π)=0.159 与 1/6=0.167 需把 χ 定到 5% 以内,而系统地板使这在纯 a₀(z) 巡天路线上持久地不可达(除非初始质量函数与哈勃张力两个独立领域各自突破)。故系数的绝对值不能作为观测可证伪的硬预言;a₀ 的预言力在 scaling(斜率),不在系数(绝对值)。

第三层——系数的取值 1/(2π)(有据偏好,非完成推导):尽管观测不能分辨,1/(2π) 与 1/6 在先验上并不对称。1/(2π) 有结构支持(视界大圆归一),而 1/6 等替代值没有对应的结构动机。因此在候选中偏好 1/(2π)——这是一个合法的先验偏好,正如物理学中 θ_QCD≈0 由强 CP 对称性偏好其值、群论 O(1) 因子由几何偏好其值,即便观测尚不能分辨。这与"落在误差带内、但无推导来源的数值巧合"性质不同:1/(2π) 是候选中最简且带有几何推导链的一个。但它仍是有据的偏好而非完成的推导——闭合环"为何取周长而非半径"尚未获公理化证明,恰如 θ=0 在 Peccei–Quinn 机制给出动力学解释之前,是有强动机的偏好而非已证的预言。

5.4 与 Verlinde 涌现引力的区分

在系数的绝对值上,SAE 与 Verlinde 涌现引力区分不开(若二者皆给 1/2π 则数值重合,且系数不可测)。SAE 与之相区分之处在于:机制(信息读取,较之熵力唯象更本体化)、视界选择(SAE 取表观视界 $R_A=c/\sqrt{H^2+kc^2/a^2}$,在曲率或高红移下不同视界定义相异,结构上不同且可测)、以及 scaling 方向。此外,给出 2π 的大圆归一是纯因果流形的几何选择、不含热力学温度残迹,故其 2π 不来自"抹去 ℏ"式的凑数;但如 §5.3 所述,"取周长而非半径"这一步本身尚待闭合环原则证明,故此点区分 SAE 与 Verlinde 的是 2π 的来源性质(因果几何而非热力学),而非一个已完成的系数推导。

综上,视界扇区的 a₀ 形式为 $$\boxed{\ a_0(z) = \frac{c^2}{2\pi R_A(z)} \xrightarrow{\ \text{平直}\ } \frac{cH(z)}{2\pi}\ }.$$

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6. 预言

6.0 锁参协议

本节所有预言只用独立输入,以杜绝"拿待验数据反配参数"的循环:T₁、T₂ 为 SAE 输入(本文不重估);Ω_d、Ω_Λ 由宇宙微波背景约束;a₀(0) 由本地径向加速度关系/SPARC 锚定;耦合 g 不用于 baseline。特别地,DESI 的 (w₀,wₐ) 不用于定参;MUSE/MIGHTEE 的 a₀(z) 只用于后验检验 scaling。先锁死,再计算。

以下给出两条硬预言与一条 via negativa 预期。

HP1 · a₀ 的视界 scaling

$$\frac{a_0(z)}{a_0(0)} = \frac{R_A(0)}{R_A(z)} \simeq \frac{H(z)}{H_0}\ (k\simeq0).$$ 数值上:z=1 → 1.79,z=3 → 4.57,z=4 → 6.33。与常数-a₀ 分支相比,在 z≳3 相差逾四倍。因 a₀(0) 由本地径向加速度关系锚定,HP1 的可证伪内容是红移比值与斜率的形状,而非绝对归一。绝对标定误差(总体质光比、哈勃张力)在比值中部分抵消;但红移相关的系统误差不会自动抵消——星族演化、气体含量、盘厚、非圆运动、选择函数、反馈与分辨率都可能随红移变化。因此 HP1 的强处在于给出明确的 scaling 形状(与常数-a₀ 的差异远大于这些系统误差),而非声称系统误差已消失。这一斜率检验与不可测的系数绝对值(§5.3)构成关键不对称:预言力在斜率的形状。

主裁判:高红移径向加速度关系/巴里昂 Tully–Fisher,尤其 z>3。

与现有数据的关系:现有 tentative 数据(MUSE-DARK III 的线性于 z 拟合、并自陈"快于 H(z)")未必落在 HP1 一侧——其增长可能已略比 ∝H 更陡。故 HP1 所预言的是"恰好 ∝H"这一具体形式:若高红移数据坐实"快于 H",则精确的 ∝H 被证伪(而"a₀ 随视界增"的机制仍存活,只是其 ∝H 的具体形式不)。检验上先测 $a_0\propto H^\gamma$ 的 γ=1。这使 HP1 明确可证伪,同时不预设现有数据已在其侧。z>3 为决定性检验。

HP2 · 内禀暗能量非 phantom

分三层。第一层是无参数的结构界;第二、三层依赖一个自由耦合,其地位与第一层不同。

第一层——真硬对赌(无参数结构定理):正的 DBI-C 加正的 Λ₁ 给出 $$w_{\rm int}(z) \ge -1.$$ SAE 的内禀暗能量结构上不穿越 phantom。这是一个会被非参数重构证伪的硬预言,不是软话,不与下面的软层一起降调。证伪者:若独立于 CPL 参数化、独立于暗-暗相互作用假设的非参数重构稳健地给出内禀 w<−1,则 SAE 的正号 DBI-C 结构被证伪。

关于 DESI:其 (w₀,wₐ)≈(−0.75,−0.86) 的 CPL 最佳拟合在高红移外推出 w<−1(表观 phantom 穿越)。本文把这一表观穿越当作有效重构、而非内禀靶标。理由是:低阶参数化的外推在宇宙学中屡屡失准(减速参数、常数-w、CPL 皆然),因宇宙学量是动力学解而非低阶多项式,而 w(z) 尚须对积分型的距离量作两次微分,对参数化形状敏感;CPL 的高红移穿越可能部分来自参数化尾部与数据稀疏区的外推,不能直接等同于内禀场的 p/ρ 穿越。此外,DESI 测的是重构的有效膨胀史,而非某场的微观 p/ρ;若暗扇区存在相互作用,内禀 w≥−1 与有效 w_eff<−1 可以并存。此判断有其争议:部分非参数重构亦暗示穿越;本文取"表观穿越部分为参数化产物"一侧,并把判决的举证责任置于独立于 CPL 的非参数重构。须澄清:本文并不主张已证伪 DESI——DESI 对 ΛCDM 的偏离是真实的,SAE 与之相容;本文所区分的,仅是"有效重构的 w"与"内禀场的 w"两个不同对象,前者可穿越而后者(在本框架中)不穿越。

第二层——软(有效穿越分支,单参数,非 baseline):khronon–C 的动能混合 θQ(保移位对称,因 Q 在 C→C+const 下不变,故 U(C)=const 保持)给出源项 $$S_\theta = -3gQ(\dot H + 3H^2).$$ 观测者按"物质独立守恒"重构时,有效方程态 $w_{\rm eff}=w_C - S_\theta/(3H\rho_C)$ 可穿越 −1 而内禀 $w_C$ 不穿越——与后-DESI 文献"经暗相互作用得有效穿越"相容。但耦合 g 是自由的:变量变换 (τ₁,τ₂)→(τ,C) 不固定它(变换生成的是 ∇τ·∇C 而非 θQ;θQ 是被容许而非被强制的算符);dual-exchange 对称(θQ 在 τ₁↔τ₂ 下为奇)只约束 g∝ε,不固定其大小与符号。而有效穿越还要求 $gQ<0$,符号亦未定。此外该混合项是动能编织(kinetic-braiding)型,其真实 w_eff 须经完整变分(流体分解公式仅为诊断)。须强调 g 并非可自由拟合任意数据的后门参数:任何非零的 θQ 耦合都必然在宇宙学微扰层留下独特印记——结构增长率 $f\sigma_8$、微波背景的积分 Sachs–Wolfe 效应(源须达 $H\rho_C$ 的 10–30%,非小修正)。因此,若未来观测仅在背景 $H(z)$ 上看到有效穿越、却在扰动数据中找不到与此算符对应的微扰特征,则本分支同样被证伪。故此为 SAE-可容许的单参数暗扇区扩展、且其单参数亦受扰动层独立约束,而非用以事后拟合的自由退路。

第三层——承诺边界(开放):若未来写出基本层的 (τ₁,τ₂) 双钟作用量并由之固定 g 及其符号,则第二层的有效穿越可升为预言。这是更大的结构任务,留待将来,不阻塞主线的第一层对赌。

余项守恒确认:khronon 尘埃↔C↔几何的交换源是余项守恒的场论版($\nabla_\mu T^{\mu\nu}_{\rm total}=0$,是重新分账而非凭空造能量);所守恒的是总暗/几何扇区。

HP3 · 无粒子暗扇区(via negativa)

常规暗物质粒子(WIMP、轴子等)的直接探测持续零结果,与本框架相容。我们把此列为"via negativa 预期"而非强阳性预言(零结果不能证"无");真正的分辨仍在 a₀(z)、外场效应、结构增长/透镜、以及完整的微波背景谱。

降为附注的减速回转:暗能量晚期减弱意味着加速比 ΛCDM 更早减慢(DESI 的交叉统计已暗示"低红移加速减慢",方向一致);但其具体回转红移依赖 HP2 第二层的自由 g,故仅为方向性,具体值待 g 固定。Roman + Euclid 的 q(z) 重构验其方向。

结构性陈述(非可证伪硬预言,备查):Λ_total=0;c_T=1(GW170817 已确认);几何回转不可观测;ξ>0 即证伪因果加强先验。这些是结构陈述,不列入上述三条硬预言。

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7. 余项

SAE 的诚实要求把余项摆明、不冒充预言,同时守住余项守恒。

• 尘埃丰度 Ω_d:后验(由微波背景定);$\rho_d\propto a^{-3}$ 是余项守恒,值是积分常数(Mukohyama)。与 ΛCDM 的 Ω_c、AeST 的尘埃份额同地位。
• a₀ 系数 1/(2π):其绝对值观测测不动(系统地板持久,不作硬预言);其取值 1/(2π) 是有据偏好(大圆几何支持,较之 1/6 有理由,θ=0 类比),非完成推导(闭合环原则开放),非 √3 式凑数。三层见 §5.3。
• khronon–C 耦合 g:自由 Wilson 系数;变量变换不固定,dual-exchange 只约束 g∝ε,大小与符号未定。
• Λ₁ 值:形式先验、值后验。K_B、α、λ:后验(继承 khronometric 参数空间)。
• 闭合环原则:大圆归一依赖"信息读出必闭合"的动机,尚未公理化证明(开放)。
• Hořava 版本张力:khronon 尘埃(§4.1,Mukohyama)最干净地在 projectable 版本(带额外标量模的鬼/强耦合问题),而 c_T=1/健康(§4.4)在 non-projectable(khronometric)版本;在 khronometric 版本坐实积分常数尘埃是一项开放的结构任务、非自动继承——暗物质子目标的地基是否稳固取决于此。
• 完整 Boltzmann(CLASS/CAMB 全谱):标为定量化的前置,非可有可无的工程项。没有全谱计算,只能说"结构上对应 AeST 的微波背景经验",不能说"通过了微波背景检验"。

上述关系——B₁+B₂=0、E₁+E₂=0、Λ₁+Λ₂=0、ρ_d∝a⁻³、khronon↔C↔几何交换——是同一条余项守恒公理的多个实例。

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8. 与既有框架的比较与判别

相对 AeST:AeST 以自由函数加背景初值获得现象学灵活性;SAE 将其中一部分自由度降为膜-DBI、p=d=3、khronon 积分常数的结构后果,在函数空间上更窄,从而更易证伪。这不是把 SAE 归为 AeST 的子集(SAE 做出 AeST 做不出的硬预测),而是一次结构化收窄。c_T=1 由 c-普适给出。

判别 SAE 与 Verlinde:不在 a₀ 系数(测不动),而在机制(信息读取)、视界选择($R_A$)、与 a₀ scaling。

判别 SAE 与 ΛCDM 加粒子暗物质:a₀ scaling(z>3)、内禀 w≥−1、无暗物质粒子(WIMP 零结果,via negativa)。外场效应(如暗物质稀缺星系的动力学)在原则上亦为一判别通道,其可操作的定量判据留待后续工作。

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9. 结论

本文从 dual-4DD 呼吸导出一个 aether-scalar-tensor 结构,把暗扇区的三类效应放进同一结构、无粒子:khronon 给几何尘埃,DBI-C 给内禀非 phantom 的动态暗能量与 MOND,4-form 给宇宙学常数;所有守恒律同归一条余项守恒公理;张量速度 c_T=1 由 c-普适给出并经 GW170817 确认。本文将 a₀ 从真空标度修正为视界标度,并给出两条可由后续数据判决的硬预言:a₀ 的视界 scaling(高红移径向加速度关系裁判)与内禀暗能量的非 phantom 界(非参数重构裁判)。

本文诚实地列出全部余项:尘埃丰度、耦合 g、khronometric 参数为后验;a₀ 系数的绝对值观测测不动、其取值 1/2π 为有据偏好而非完成推导;projectable 与 non-projectable 两版本能否兼容,是地基性开放问题。本框架在发展中亦修正了此前的若干中间结论——C 场作为尘埃的误置、η(t) 给出的 a₀ 下降、以及暗能量耦合 g 之未定——相应的修正已在各节说明。

本文所取的是一条以结构约束换取可证伪性的路径。上述两条硬预言会把本框架置于可被判决的位置——这正是其价值:一个结构被先验约束的理论没有事后调参的空间,只能以硬预言接受检验。若其中一条被证伪,证伪本身即指明先验应在何处修正,而这是自由度足以拟合一切的模型所无法提供的。

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致谢

本文的推导、计算与内部批判采用了一套四-AI 协作方法:以 Claude、ChatGPT、Gemini、Grok 分任概念协调、推导与审稿、几何与拓扑核验、以及对抗性批判,另设一条独立复审线作交叉检验。这套方法用于草稿的形成、交叉验证与错误的发现与修正(方法论实录见前置篇);其运作过程,包括若干被逐步修正与放弃的中间结论,已在正文相应各节说明。

上述 AI 系统是本研究所用的工具与方法,不构成作者。本文的全部内容——包括其先验假设、结构选择、预言、以及被标为余项与开放的一切——由作者独立负责。

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附录 A · 开放项清单

开放项	性质	若解决则
projectable(尘埃)与 non-projectable(c_T=1)版本共存	地基性	暗物质子目标地基坐实
基本层 (τ₁,τ₂) 双钟作用量	结构	固定 g → HP2 有效穿越升预言;或定 a₀ 系数
a₀(τ) 空间项的完整协变化	结构	把 $R_A^{-1}$ 表为 foliation 不变的局域几何标量(θ、曲率),使 MOND 项成为完整局域拉氏量而非 FLRW 有效项
闭合环原则(大圆归一)公理证明	理论	a₀ 系数从"有据偏好"升"结构选择有据"(但仍测不动)
khronon–C 混合的扰动层(S₈/增长/透镜)	计算(可行性关键)	θQ 分支的可行性
khronometric α、λ 参数空间(后牛顿/脉冲星/核合成)	继承检验	完整可行性
完整 Boltzmann(CLASS/CAMB 全谱)	工程	微波背景定量对照
C 场追踪走廊(Cosmo V 遗留)	数值	U(C) 自然进入走廊

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