Geometric–Information Relaxation and Derivative-Coupled Three-Stage Cosmology on SL(6,C)

1. Scope and Mathematical Status

The central question is what geometric diffusion and information dissipation actually constrain in a proposed three-stage cosmology. The effective stages considered here are early statistical localization, intermediate matter and dark-matter formation, and late dark-energy domination. A stage name is an interpretation to be tested against a physical response model. It is not a conclusion obtained merely by counting dimensions or assigning a name to a proposition.

General relativity has a dynamical metric. The additional task here is to construct a map from a configuration-space probability law to a physical metric and stress-energy tensor. Neither the selection of SL(6,C) among possible symmetry groups nor this response map is established. The mathematical results below concern specified objects, not a complete derivation of spacetime.

We use one evidence vocabulary throughout:

Label	Meaning
S	A concrete statement with a Lean proof body supplied in the accompanying source; compilation verified locally under the pinned build (see Appendix A)
A	An analytic derivation or an external mathematical result with stated applicability conditions
M	A model definition, constitutive assumption or approximation
C	A chosen calibration or illustrative numerical input
O	An open construction or physical mechanism

The labels distinguish epistemic roles and are not an ordering of confidence. A conditional statement marked S still depends on all its explicit hypotheses. S means a supplied proof body, not a newly checked theorem, and it does not promote an effective-action assumption M into a derived law.

1.1. Physical Motivation and the Meaning of Emergence

The programme asks whether one statistical geometry can organize three otherwise separate questions: why geometric fluctuations become controlled, how persistent massive excitations form, and why a slowly varying energy component can dominate after matter dilutes. We retain this motivation. The common objects are the probability law on geometric configurations, its moments, compact internal charges, and their free-energy balance. They describe distinct aspects of a candidate underlying dynamics; none alone is a physical spacetime metric.

Here “emergence” has three operational meanings: localization is convergence of a probability law at specified noise and tolerance; condensation is the appearance of stable nonzero order-parameter minima and populated massive modes; late domination is an inequality among physical densities on an expanding solution. A viable interpretation must relate these events on one clock and conserve total stress energy through their transitions. Spatial dimensionality and Lorentzian signature are a further construction, not synonyms for these three events.

SL(6,C) is a candidate symmetry, not a uniquely selected group. Its compact real form and rank-five Cartan sector offer concrete representation and fluctuation calculations [1], but neither a three-generation particle spectrum nor a four-dimensional metric follows from these counts. Eldan’s localization method [2] motivates studying concentration; it is not an identification of that stochastic process with the autonomous Langevin process used here. This distinction also prevents transferring Euclidean dimension-dependent bounds to $X_E$ without a theorem.

Three-stage overview. Early statistical localization controls a probability law under the stated curvature, confinement and initial-law hypotheses. The common effective action can concurrently support accelerated expansion if its potential dominates its physical kinetic energy. An intermediate origin instability is triggered when the evolving Fisher kinetic energy crosses a mass threshold; reaching a condensate, reheating and forming stable matter are additional dynamical steps. Late acceleration is possible when the same action approaches a healthy vacuum of positive energy and other components dilute. These regimes share field equations and proper time, but their occurrence and ordering are not automatic. CGICE-1 through CGICE-4 describe concurrent stochastic, moment, internal-transport and dissipative structures, not equations switched on at separate epochs. The link from their stochastic law to the physical fields remains a coarse-graining construction. None of these regimes is a proved change of spacetime dimension or signature.

2. Geometry, Operators and Normalization

2.1. Different Spatial Categories

Object	Real dimension	Role
SL(6,C)	70	Candidate real Lie group of symmetries
C6	12	Fundamental complex representation; a geometric base requires separate construction
$X_E$ = SL(6,C)/SU(6)	35	Complete, noncompact, contractible Riemannian symmetric space
$X_P$ = SL(6,C)/SU(3,3)	35	Separate indefinite-metric quotient
$T*X_E$	70	Canonical cotangent phase space for a separately specified Hamiltonian system

All positive diffusion, elliptic spectral and Gibbs arguments in this paper refer to $X_E$. They do not transfer automatically to $X_P$. The real Lie-algebra Cartan decomposition is

$$\mathfrak{sl}{(6,\mathbb{C})}_{\mathbb{R}}=\mathfrak{su}\left(6\right)\oplus \mathfrak{p}.$$

This is a vector-space decomposition of Lie algebras, not an additive decomposition of groups. A physical four-dimensional Lorentzian section requires independent metric, reality and causality conditions.

2.2. Restricted Roots and Three Different Rates

For $A_5,$ the ordinary Weyl vector and restricted half-sum are

$${\rho }_{alg}=\frac{1}{2}(5,3,1,-1,-3,-5),{\rho }_{res}=2{\rho }_{alg}.$$

Root multiplicity two gives Euclidean squared norms 35/2 and 70. Under the dual of the metric $c·Tr,$ squared norms divide by c. Thus c=2 gives

$$\parallel {\rho }_{res}{\parallel }^2=35,{\parallel {\rho }_{alg}\parallel }^2=35/4.$$

The finite calculation is S (CGICECore.restricted_threshold_35). Identifying the restricted norm with the free Laplacian spectral bottom requires the Riemannian symmetric-space spectral theorem and its metric convention (A).

Let D>0 and define

$$L=D\Delta -\nabla V_{conf}·\nabla ,\mu =Z^{-1}{e}^{-V_{conf}/D}d\mathrm{vol},$$

$${\delta }_L=\underset{\begin{array}{c}f\ne 0\\ \int fd\mu =0\end{array}}{inf}{\displaystyle \frac{{D\int |\nabla f|}^{2}d\mu }{\int f^{2}d\mu }}.$$

The infimum is over an appropriate closed Dirichlet-form domain. The Witten operator is

$$W_V\left(h\right)=-h^2\Delta +{|\nabla V|}^2-h\Delta V.$$

For compatible closed operators, Gibbs normalization and the corresponding unitary transformation, the nonzero gaps obey

$${\delta }_L={\displaystyle \frac{{\delta }_W(2D,V)}{4D}}.$$

A coefficient identity alone does not establish these operator conditions.

The third quantity, $r_{\mathrm{perp}}=35/3$, is an independent matching definition. It is not the gap of a coordinate of a three-dimensional OU process divided by three. As in the companion CGICE source, transverse_rate_of_matching proves only the arithmetic consequence $r_{\mathrm{perp}}=r_{\mathrm{parallel}}/3$; the matching premise itself is a separate model input [M].

Quantity	Value or condition	Interpretation
Free Laplacian bottom	35 at $c=2$	Geometric spectral baseline
Radial diffusion lower bound	$a-6D$, with $a>6D$	Sufficient positive relaxation estimate
Matching rate $r_{\mathrm{perp}}$	$35/3$	M; requires an application-specific mapping
Reference information time	$6\pi /35$	Dimensionless; not fixed to Planck time
Signed RG parameter	$-36/35$	M in a specified beta function
RG fixed-point square	$12{\pi }^2/35$	Conditional algebraic result
$g_{TC}^2$ (D1 coupling square)	$24{\pi }^2/35$	Independent matching convention

At $a=6D$ the displayed sufficient bound loses strict positivity. This does not prove that the true spectral gap vanishes or that a physical phase transition occurs.

2.3. Relation to localization literature

The cited Klartag–Lehec work [3] establishes bounds up to dimension-dependent polylogarithmic factors. It does not provide the exact constants 35 or 29 in this model. Our radial confinement argument instead uses the chosen potential and curvature bound, together with the appropriate Bakry–Émery/Poincaré framework [4]. No general claim about the resolution of KLS is required.

3. CGICE Equations and the Three Effective Stages

3.1. Unified Dynamical Basis

Write $\tau$ for the configuration-process time, reserving t for physical cosmological time. In an explicitly stated Euclidean linearization, CGICE-1 is

$$d{X}_{\tau }=(-\nabla V_{conf}\left(X_{\tau }\right)+F_{\tau }\left(X_{\tau }\right))d\tau +Bd{W}_{\tau },B{B}^T=2\mathsf{D}.$$

A manifold SDE additionally needs a connection or horizontal-lift convention. For F=0 and isotropic diffusion, its adjoint evolution is

$${\partial }_{\tau }p=L^{*}p=D\Delta p+div(p\nabla V_{conf}).$$

CGICE-2 describes moments of this same probability law. For linear drift $-HX,$

$$C^{\prime }=-HC-C{H}^T+2\mathsf{D},$$

$$C\left(\tau \right)=e^{-H\tau }{C}_0{e}^{-H^T\tau }+{\int }_0^{\tau }{e}^{-Hu}2\mathsf{D}{e}^{-H^{T}u}du.$$

This is a Lyapunov covariance equation. A filtering Riccati equation would require another state and a measurement model. For the symmetric, centered one-dimensional quartic process,

$$C^{\prime }=-2aC-2q\mathbb{E}\left[X^4\right]+2D=-2aC-6q{C}^2+2D-2q{\kappa }_4,{\kappa }_4=\mathbb{E}\left[X^4\right]-3C^2.$$

The Riccati reduction requires the additional Gaussian closure ${\kappa }_4=0$. CGICECore.quartic_cov_decompose and gaussian_closure_reduces prove the algebra, not exact Gaussianity of a quartic process.

The distinction matters for the review: quartic_cov_decompose has no Gaussian-closure hypothesis. Only gaussian_closure_reduces assumes it. Section 6.6 retains the exact cumulant correction and supplies error estimates instead of treating the approximation as an exact consequence of CGICE-1. Curvature corrections to Euclidean moment equations require their own estimates.

CGICE-3 is internal adjoint transport on a compact Lie algebra:

$$j^{\prime }=-g[a,j],{\displaystyle \frac{d}{d\tau }}{\displaystyle \frac{1}{2}}\langle j,j\rangle =0.$$

It is not a sourced topological-charge equation or a probability continuity equation. CGICE-4, for fixed reference measure and appropriate regularity, is

$${\displaystyle \frac{d}{d\tau }}\mathrm{KL}(p\parallel \mu )=-D\int p{|\nabla log(p/\mu )|}^{2}d\mathrm{vol}.$$

An independently chosen scalar closure $I^{\prime }=-\gamma (I-E)$ is not a universal identity for KL or individual Brownian paths. Neither equation implies cyclic entropy conservation; in the companion CGICE source, zero_prefix_sums_force_zero proves that a prefix-sum law ${\sum }_{i<n}\Delta S_i=0$ for every n forces $\Delta S_n=0$, excluding nontrivial irreversible cyclic increments.

3.2. Physical Responses During the Three Stages of Cosmic Evolution

Effective stage	Mathematical benchmark	Additional physical construction
Early localization	Radial confinement; conditional spectral relaxation	Statistical-to-physical map, potential-dominated expanding solution, exit
Matter/CDM formation	Kinetic mass crossing; scalar minima; internal Casimir	Seed growth, matrix vacuum selection, reheating, masses and production rates
Late DE domination	Conditional limits of the action-derived stress; OU covariance separately	Healthy positive residual vacuum, dilution, attraction and normalization

A cosmological extension must determine a response and its expansion dynamics. For the principal effective model these are specified by the action in §3.9:

$$T_{geo}^{\mu \nu }={\mathcal{T}}^{\mu \nu }[p,C,j,g_{phys}],H^2={\displaystyle \frac{{\rho }_{tot}}{3{\overline{M}}_P^2}},{\dot{\rho }}_{tot}+3H({\rho }_{tot}+p_{tot})=0.$$

The action defines a candidate response [M]; its field equations imply conservation [A/S in the scalar reduction]. Its microscopic derivation and identification with CGICE stochastic dynamics remain O. A common solution in the healthy kinetic domain, phase criteria and a populated intermediate regime are required before the three stages can be claimed as a dynamical sequence. Internal exchange is calculated in §7.2, not chosen as an external source function.

3.3. Common Proper Time, Units and Transition Data

To compare stages, introduce the physical scale factor $a_{\cos }\left(t\right)>0$, $H={\dot{a}}_{\cos }/a_{\cos }$ and $N=log(a_{\cos }/a_{\mathrm{ref}})$. In formulas involving density as a function of scale factor, the symbol a means $a_{\cos }$; the coefficient a in $V_{conf}$ remains a separate confinement parameter. Set $\hslash =c=k_B=1$ for the physical sector. $\tau$, N and the geometric coordinates in the fixed reference metric are dimensionless; H and $\dot{\tau }$ have inverse-time units. Physical densities and pressures have energy dimension four, ${\overline{M}}_P^2$ has dimension two, and a source $Q_i$ has dimension five. A numerical geometric covariance must therefore be multiplied by a dimensionful response coefficient before it enters Friedmann’s equation.

All principal field equations use one FLRW proper time. The expansion variable obeys the kinematic identity

$$N\left(t\right)=log{\displaystyle \frac{a_{\cos }\left(t\right)}{a_{\mathrm{ref}}}},\dot{N}=H.$$

(1)

There is no imposed constant relation between information time $\tau$ and N in the principal model. Einstein and field equations determine H(t) from initial data in §3.9. If a stochastic localization estimate is evaluated on this solution, $\tau \left(t\right)$ must be supplied by a controlled coarse-graining or lapse prescription; an action for smooth mean fields alone does not determine that prescription. Section 3.5 and Section 3.6 retain earlier transport and covariance-clock constructions as conditional diagnostics, not additional equations imposed on §3.9.

Let $t_{loc}$ denote attainment of a chosen statistical tolerance, $t_c$ an origin-mass crossing, and $t_{DE}$ a specified density crossing. A sequential interpretation needs $t_{loc}<t_c<t_{DE}$ and a nonempty matter-dominated interval. A smooth solution of §3.9 carries continuous energy through $t_c$ without an imposed jump law; condensation need not be completed at that instant. Statistical tolerance, origin instability and density domination are distinct events. Radiation and reheating require additional field or bath dynamics and may occupy intervening intervals.

3.4. Conservation and Preservation of the Expansion Constraint

For a spatially flat effective FLRW sector, use finite total density and pressure, constant positive ${\overline{M}}_P^2$, the action-derived total continuity equation of §3.9, and

$$\dot{H}=-{\displaystyle \frac{{\rho }_{tot}+p_{tot}}{2{\overline{M}}_P^2}}.$$

The two scalar equations give ${\dot{\rho }}_{tot}=-3H({\rho }_{tot}+p_{tot})$ by the off-shell identity (19). Consequently

$${\displaystyle \frac{d}{dt}}\left(3{\overline{M}}_P^2{H}^2-{\rho }_{tot}\right)=0.$$

(2)

Thus the Friedmann constraint, if imposed on initial data, is preserved by these evolution equations. UnifiedDynamics.unified_continuity derives the actual density derivative from both scalar equations, and UnifiedDynamics.action_friedmann_hasDerivAt connects it to the retained Friedmann theorem. CosmologicalBridge.total_continuity remains reusable component algebra; friedmann_constraint_preserved is a real-line constancy theorem. The corresponding interval statement follows analytically on a connected existence interval. No global solution through singularities or kinetic degeneracy is asserted.

For each component,

$${\displaystyle \frac{d}{dt}}\left(a_{\cos }^3{\rho }_i\right)=a_{\cos }^3{Q}_i-3H{a}_{\cos }^3{p}_i.$$

(3)

The product-rule algebra is comoving_pressure_work. A comoving dust transfer model can conserve total mass, but a system containing vacuum pressure cannot omit the pressure-work term. If an idealized instantaneous condensation releases potential energy $U_{-}-U_{+}$ into radiation, energy matching requires ${\rho }_{r,+}-{\rho }_{r,-}=U_{-}-U_{+}$; transition_energy_matching then proves continuity of ${\rho }_r+U$. This matching condition does not compute reheating, latent heat, surface stress or transition duration.

3.5. Wasserstein Dynamics and Spatial Volume Reconstruction

On a fixed complete Riemannian base, with finite second moments and appropriate entropy/domain hypotheses, set $\xi =V_{conf}+D(logp+1)$. The CGICE continuity equation is

$${\partial }_{\tau }p+{div}_{X_E}\left(pv\right)=0,v=-\nabla \xi .$$

Its variational discretization is the JKO minimization

$$p_{k+1}\in \underset{p\in {\mathcal{P}}_2\left(X_E\right)}{\mathrm{argmin}}\left\{\mathcal{F}\left[p\right]+{\displaystyle \frac{W_2^2(p,p_k)}{2\Delta \tau }}\right\}.$$

(4)

Existence and convergence on this noncompact base require coercivity, lower semicontinuity and moment control; [5] supplies the foundational variational formulation, not a ready-made CGICE spacetime theorem. The finite-graph flux-force proof is not a formal proof of this minimization or its continuum limit.

To obtain physical expansion, introduce a physical spatial patch $\Sigma$, a coarse-graining/projection $\mathsf{\Pi }:X_E\to \Sigma$ and an increasing clock $\tau \left(t\right)$ [M/O]. Where differentiation under the pushforward is justified, the projected number density n and physical velocity b obey

$${\partial }_{t}n+{div}_{\Sigma }\left(nb\right)=0,b\left(y\right)=\dot{\tau }\mathbb{E}[d\mathsf{\Pi }\left(v\right)\mid \mathsf{\Pi }\left(X\right)=y]$$

(5)

for a time-independent projection; moving projections and moving volume forms add terms. This is a weak pushforward construction under stated regularity, not an identification of p with mass density. On a flat physical patch assume an isotropic homologous flow $b(t,y)=H\left(t\right)y$; this is an extra spatial symmetry condition. If ${\mathsf{\Phi }}_t$ is its flow and $J_t=detd{\mathsf{\Phi }}_t>0$, Jacobi’s formula gives

$${\dot{J}}_t=\theta J_t,\theta ={div}_{\Sigma }b=3H,N\left(t\right)={\displaystyle \frac{1}{3}}{\int }_{t_0}^t\theta \left(s\right)ds,a_{\cos }\left(t\right)=a_{\mathrm{ref}}{e}^{N\left(t\right)}.$$

(6)

Thus the scale factor is reconstructed from spatial volume transport once that transport has been constructed. For homogeneous n, (5) gives $\dot{n}+3Hn=0$ and $a_{\cos }^{3}n$ is constant. This counts conserved particles, not vacuum energy; (3) still governs pressure work. TransportClock.density_dilution formalizes the scalar compatibility step. Jacobi’s formula, the pushforward and the flow existence theorem remain A/O.

A direct identification can give the wrong sign. A centered isotropic Euclidean Gaussian with covariance $C{I}_3$ has continuity velocity $b=(\dot{C}/\left(2C\right))y$, hence $H=\dot{C}/\left(2C\right)$. During relaxation from above the OU noise floor, $\dot{C}<0$ and $H<0$ (covariance_volume_contracts). Statistical concentration would then correspond to contraction, not inflation. A different response must be specified and tested; optimal transport cannot remove this missing input by itself.

3.6. A Covariance-Defined Clock and Its Regularity

Let $E\left(t\right)=C\left(t\right)-C_{\infty }>0$, $E_{ref}=E\left(t_0\right)>0$, and choose $\beta >0$. Define the following constitutive response [M]:

$$N_E\left(t\right)=\beta log{\displaystyle \frac{E_{ref}}{E\left(t\right)}},a_{\cos }\left(t\right)=a_{\mathrm{ref}}{\left({\displaystyle \frac{E_{ref}}{E\left(t\right)}}\right)}^{\beta }.$$

(7)

This uses a dimensionless covariance ratio. It is an inverse-size response, not the covariance-volume identification just ruled out. Along the linear moment model, let $\lambda \left(t\right)=\dot{\tau }\left(t\right)>0$, a lapse between physical and information time, and $\dot{E}=-2\ell \lambda E$. Differentiating (7) gives

$$H={\dot{N}}_E=-\beta {\displaystyle \frac{\dot{E}}{E}}=2\beta \ell \lambda ,\theta =6\beta \ell \lambda .$$

(8)

The second equality is the compatibility constraint with (6); neither the existence of a projection realizing it nor the lapse is supplied by the OU equation. In this optional diagnostic, constant ℓ gives an integrated clock proportionality with coefficient $\nu ={(2\beta \ell )}^{-1}$ and $m=1/\beta$. It is not imposed on the unified action. This moves the assumption into an explicit covariance-volume response; it does not eliminate it. For the quartic process of §6.6 the covariance error must be propagated through (7)–(8). For varying coefficients, changing equilibrium covariance adds forcing to $\dot{E}$, and (8) must be recomputed. The symbol $\lambda \left(t\right)$ here denotes a lapse, distinct from the quartic coupling in §3.9.

On any interval $I=[t_0,t_1]$, assume E differentiable and positive and $|\dot{E}/E|\le L$. Then the mean-value inequality gives

$$|N_E\left(t\right)-N_E\left(s\right)|\le \beta L|t-s|,s,t\in I.$$

(9)

For the OU model with bounded lapse, $L=2\ell {sup}_I\lambda$ suffices. Alternatively $E\ge \epsilon >0$ and $|\dot{E}|\le M$ give $L=M/\epsilon$. A finite-time zero of E is excluded; there is no unconditional global regularity through it. TransportClock.eFold_hasDerivAt, ou_eFold_hasDerivAt and eFold_interval_lipschitz work with the actual covariance-dependent function, not an arbitrary external N. covariance_tail_hasDerivAt instantiates the older tail-composition theorem with that function. The information-to-proper-time lapse remains independent data.

For $k=2\beta \ell$ constant, ${\ddot{a}}_{\cos }/a_{\cos }=k\dot{\lambda }+k^2{\lambda }^2$. Constant positive lapse forces constant H, so it cannot describe a decelerating matter interval between two accelerating intervals. Friedmann dynamics imposes $\lambda =H/k$ but then the clock relation alone supplies no new evolution equation. A unified model must determine $\lambda$, thermal coefficients and stress response together, rather than treating this identity as an endogenous inflation/exit mechanism.

3.7. Fisher Pullback, Vielbein Data and the Signature Obstruction

To form a spacetime tensor, one needs a family of normalized distributions $p(z;\vartheta )$, not merely one scalar Fisher functional. With a fixed reference measure and differentiable parameter dependence, define

$${\mathcal{I}}_{AB}\left(\vartheta \right)=\int p(z;\vartheta ){\partial }_{A}logp(z;\vartheta ){\partial }_{B}logp(z;\vartheta )d\mathrm{vol}\left(z\right).$$

(10)

For a parameter field $\vartheta :M_4\to \mathcal{P}$, its pullback is

$K_{\mu \nu }={\mathcal{I}}_{AB}{\partial }_{\mu }{\vartheta }^A{\partial }_{\nu }{\vartheta }^B$.

It is a symmetric covariant two-tensor, but

$$K(v,v)=\int p{[v^{\mu }{\partial }_{\mu }{\vartheta }^A{\partial }_{A}logp]}^{2}d\mathrm{vol}\ge 0.$$

(11)

No real pullback of positive Fisher geometry has a timelike negative direction. In particular a rank-four immersion into $X_E$ gives a positive induced metric, not a Lorentzian one. This obstruction is not removed by calling ${\partial }_{\mu }{\vartheta }^A$ a vielbein. A 35-dimensional target, a six-dimensional complex representation and a four-dimensional tangent bundle remain distinct objects.

An explicit conditional physical geometry uses a rank-four coframe ${e^a}_{\mu }$ and the independently selected internal form ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$, so $g_{\mu \nu }={\eta }_{ab}{e^a}_{\mu }{e^b}_{\nu }$. Equivalently a positive lapse, three-metric and shift specify an ADM Lorentzian metric. This supplies a usable geometric setting, but assumes signature and dimension; compactification or a real-section selection must still derive them [O]. Induced-gravity ideas [6] motivate an effective action on such a background; they do not automatically create Lorentzian signature from a positive Fisher matrix.

The Lean construction TensorResponse.pullback uses genuine bilinear forms and linear maps on fibers, proves composition, symmetry, internal-isometry invariance and nonnegativity, and proves positive_pullback_not_timelike. scoreGram_nonnegative gives a finite score approximation. It does not construct a smooth tensor bundle, a statistical family, a rank-four coframe or a global section; these distinctions limit the formal claim precisely.

3.8. Covariant Action and Stress-Energy Response

A precursor to the coupled response of §3.9, recovered when the derivative coupling is absent, is the action on the assumed Lorentzian geometry:

$$S_{eff}={\int }_{M_4}\sqrt{-g}\left[{\displaystyle \frac{{\overline{M}}_P^2}{2}}R-\mathsf{\Lambda }-{\displaystyle \frac{Z}{2}}{g}^{\mu \nu }{\mathcal{I}}_{AB}\left(\vartheta \right)D_{\mu }{\vartheta }^A{D}_{\nu }{\vartheta }^B-U\left(\vartheta \right)+{\mathcal{L}}_{matter}\right]d^{4}x.$$

(12)

Here $Z>0$, the statistical metric is positive on the represented modes, and U, $\mathsf{\Lambda }$, ${\overline{M}}_P^2$ and field content are independent effective inputs. For dimensionless $\vartheta$, Z has mass dimension two and $U,\mathsf{\Lambda }$ have dimension four. The symmetry must act by isometries of $\mathcal{I}$ and preserve U. For a linear representation $D_{\mu }\vartheta ={\partial }_{\mu }\vartheta +{\mathcal{A}}_{\mu }\vartheta$; generally one uses its action vector fields on parameter space. The connection transforms as ${\mathcal{A}}_{\mu }^{\prime }=R{\mathcal{A}}_{\mu }{R}^{-1}-\left({\partial }_{\mu }R\right)R^{-1}$. covariantRate_transforms formalizes the corresponding single-direction fiber law. It is not a global connection construction.

Writing $K_{\mu \nu }={\mathcal{I}}_{AB}{D}_{\mu }{\vartheta }^A{D}_{\nu }{\vartheta }^B$, assume the statistical metric has no additional dependence on the physical metric when $\vartheta$ is fixed. Metric variation then gives

$$T_{\mu \nu }^{\vartheta }=Z{K}_{\mu \nu }-g_{\mu \nu }\left({\displaystyle \frac{Z}{2}}{g}^{\alpha \beta }{K}_{\alpha \beta }+U\right).$$

(13)

scalarStress and scalarStress_symmetric encode this fiberwise tensor algebra; metric variation and the covariant divergence are analytical, not formalized here. For homogeneous fields in proper time, ${\rho }_{\vartheta }=Z{\mathcal{I}}_{AB}{\dot{\vartheta }}^A{\dot{\vartheta }}^B/2+U$ and $p_{\vartheta }=Z{\mathcal{I}}_{AB}{\dot{\vartheta }}^A{\dot{\vartheta }}^B/2-U$. Hence ${\rho }_{\vartheta }+p_{\vartheta }\ge 0$. A constant positive potential with stationary fields can supply $w=-1$; nonzero configuration noise alone does not imply that stress tensor.

For compactly supported variations, diffeomorphism invariance of the total action yields the on-shell Ward identity ${\nabla }_{\mu }{T}_{tot}^{\mu \nu }=0$, including the order parameter, gauge fields and any bath. Off shell, scalar Euler-Lagrange terms contribute ${\mathcal{E}}_A{\nabla }_{\nu }{\vartheta }^A$; exchanging subsystems need not conserve separately. If thermal coefficients are externally prescribed functions of temperature, the bath and its energy exchange must be included. Thus the stochastic dissipative process is not derived merely by appending (12); integrating out specified bath degrees of freedom with a controlled noise/dissipation limit remains O.

Covariance is not arbitrary Lie invariance. A natural tensor response satisfies $\mathcal{T}[{\phi }^{*}g,{\phi }^{*}\vartheta ]={\phi }^{*}\mathcal{T}[g,\vartheta ]$. Its Lie derivative generally does not vanish: even the constant line metric $d{x}^2$ has ${\mathcal{L}}_{x{\partial }_x}g=2g$. Requiring ${\mathcal{L}}_{X}g=0$ is a Killing-symmetry condition, not a test of general covariance. lie_derivative_dilation_counterexample checks the coordinate arithmetic only. Consequently the universal zero-Lie-derivative condition is replaced by tensor naturality and the on-shell conservation requirement.

3.9. Unified Covariant Action and Conditional Autonomous Transitions

3.9.1. Sign, Dimensions and the Healthy Kinetic Domain

We adopt signature $(-,+,+,+)$ on the independently specified $M_4$. For dimensionless statistical parameters ${\vartheta }^A$ and a real scalar order parameter $\varphi$, define

$$X=-\frac{1}{2}{\mathcal{I}}_{AB}{g}^{\mu \nu }{\partial }_{\mu }{\vartheta }^A{\partial }_{\nu }{\vartheta }^B,Y=-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi ,A\left(\varphi \right)=Z-\frac{\gamma }{2}{\varphi }^2,W\left(\varphi \right)=\frac{m_0^2}{2}{\varphi }^2+\frac{\lambda }{4}{\varphi }^4.$$

(14)

For homogeneous fields, $X={\mathcal{I}}_{AB}{\dot{\vartheta }}^A{\dot{\vartheta }}^B/2\ge 0$ and $Y={\dot{\varphi }}^2/2\ge 0$. The proposed physical action [M] is

$$S=\int \sqrt{-g}{d}^{4}x\left[{\displaystyle \frac{{\overline{M}}_P^2}{2}}R+A\left(\varphi \right)X+Y-U_{geo}\left(\vartheta \right)-W\left(\varphi \right)\right].$$

(15)

Take $Z>0$, $\gamma >0$, $m_0^2<0$, $\lambda >0$ and positive-definite $\mathcal{I}$ on the retained parameter modes. We require $A\left(\varphi \right)>0$ along the solution. Crossing $A=0$ makes the kinetic metric degenerate; solutions and stability cannot be extrapolated through it using this effective model. Large $\varphi$ would give a ghost direction unless higher operators change the model. Here $\left[X\right]=2$, $\left[Y\right]=4$, $\left[Z\right]=2$, $\left[\varphi \right]=1$, $\left[m_0^2\right]=2$, $\left[\gamma \right]=\left[\lambda \right]=0$, and $\left[U_{geo}\right]=\left[W\right]=4$, in mass units.

This is a derivative coupling between fields, not a curvature coupling $F(\vartheta ,\varphi )R$. The reduced Planck mass is constant. An actual nonminimal curvature coupling would require new metric equations and extra stress terms. The review’s positive-sign contraction $\mathcal{K}=+{\mathcal{I}}_{AB}{\partial }^{\mu }{\vartheta }^A{\partial }_{\mu }{\vartheta }^B/2$ is negative on a homogeneous solution in this signature; identifying it with positive kinetic energy reverses the intended trigger. Equation (14) corrects that sign.

The geometric CGICE potential $V_{conf}$ is defined on configuration space with its own normalization; it cannot be inserted as a physical energy density by relabeling it. A possible response is $U_{geo}\left(\vartheta \right)={\mu }^4\mathcal{V}\left(\vartheta \right)$ with dimensionless $\mathcal{V}$, obtained, for example, from a specified expectation of $V_{conf}$ over $p(z;\vartheta )$. The scale $\mu$, parameter map, finite expectation and coarse-graining prescription are additional inputs. A steep statistical confinement potential is not a proof of slow roll in this physical potential.

For an SU(6) Hermitian traceless adjoint, replace ${\varphi }^2$ by $Tr{\mathsf{\Phi }}^2$, use $Y=-Tr{\left(D\mathsf{\Phi }\right)}^2/2$, and $W=m_0^{2}Tr{\mathsf{\Phi }}^2/2+{\lambda }_{mat}Tr{\mathsf{\Phi }}^4/4$. A scalar direction $\mathsf{\Phi }=\varphi T$, $Tr{T}^2=1$, gives $\lambda ={\lambda }_{mat}Tr{T}^4$. Holding this direction fixed is a reduction assumption, not a proof that transverse matrix equations vanish. Gauge dynamics requires the gauge kinetic term and Gauss constraint. The proof chain below covers the scalar reduction; §5.3.2 tests the matrix-vacuum claim separately.

3.9.2. Field equations and the stress tensor

Writing $V_{force}(\varphi ,X)=\frac{1}{2}(m_0^2+\gamma X){\varphi }^2+\lambda {\varphi }^4/4$, the identity

$ZX+Y-U_{geo}-V_{force}=AX+Y-U_{geo}-W$

is UnifiedDynamics.lagrangian_rearrangement. Because $V_{force}$ depends on metric-contracted velocities, it is an instantaneous force potential, not a velocity-independent potential energy to add to $ZX$ in $\rho$. Metric variation, holding $\mathcal{I}$ independent of g at fixed fields, gives

$$T_{\mu \nu }=A{\mathcal{I}}_{AB}{\partial }_{\mu }{\vartheta }^A{\partial }_{\nu }{\vartheta }^B+{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +g_{\mu \nu }{\mathcal{L}}_s,{\mathcal{L}}_s=AX+Y-U_{geo}-W,$$

$$\rho =AX+Y+U_{geo}+W,p=AX+Y-U_{geo}-W.$$

(16)

In particular, the derivative-coupling contribution to $\rho$ is $-\gamma X{\varphi }^2/2$, not $+\gamma X{\varphi }^2/2$. The covariant scalar equations are

$$A\left(\square {\vartheta }^A+{{\Gamma }^A}_{BC}\partial {\vartheta }^B·\partial {\vartheta }^C\right)+A_{\varphi }\partial \varphi ·\partial {\vartheta }^A-{\mathcal{I}}^{AB}{\partial }_B{U}_{geo}=0,\square \varphi +A_{\varphi }X-W_{\varphi }=0.$$

(17)

For a homogeneous single-parameter reduction with $\mathcal{I}=1$, put $u=\dot{\vartheta }$, $v=\dot{\varphi }$, $U_{\vartheta }={\partial }_{\vartheta }{U}_{geo}$. These become

$${\mathcal{E}}_{\vartheta }=A\dot{u}+3HAu-\gamma \varphi vu+U_{\vartheta }=0,{\mathcal{E}}_{\varphi }=\dot{v}+3Hv+(m_0^2+\gamma X)\varphi +\lambda {\varphi }^3=0.$$

(18)

For a general Fisher target the first equation uses the target covariant acceleration $D_t{\dot{\vartheta }}^A$ and ${\mathcal{I}}^{AB}{\partial }_B{U}_{geo}$; choosing a single canonical coordinate globally is not assumed. Equations (16)–(18) are analytic variational results. Lean encodes the scalar residuals and their consequences, not the underlying manifold variational calculus.

3.9.3. Conservation derived from the two field equations

Direct differentiation, including $\dot{A}=-\gamma \varphi v$, gives

$$\dot{\rho }=Au\dot{u}-\gamma \varphi vX+v\dot{v}+U_{\vartheta }u+(m_0^2\varphi +\lambda {\varphi }^3)v,\dot{\rho }+3H(\rho +p)=u{\mathcal{E}}_{\vartheta }+v{\mathcal{E}}_{\varphi }.$$

(19)

Thus the on-shell total continuity equation follows from both field equations. No independent $Q_{DE}$ or separate assumption of total continuity is needed. UnifiedDynamics.continuity_identity is the off-shell identity, density_hasDerivAt differentiates actual functions, unified_continuity inserts the equations, and action_friedmann_hasDerivAt connects the result to (2). The derivative hypothesis for $U_{geo}\left(\vartheta \left(t\right)\right)$ is its chain rule $U_{\vartheta }u$; it does not assume U constant during the transition. The corresponding covariant Ward identity follows analytically from the action, with suitable variations and boundary conditions.

The proposed continuity proof in the review used instead $\rho =X+U+V_{force}$, $p=X-U-V_{force}$, $\dot{X}=-6HX$, $\dot{U}=\dot{\varphi }=0$. Its actual residual is

$$\dot{\rho }+3H(\rho +p)=-3\gamma HX{\varphi }^2,$$

(20)

which is nonzero, for instance when all four factors equal one. review_continuity_residual and review_residual_counterexample record this obstruction. Setting $\dot{\varphi }=0$ also does not mean adiabatic tracking of a moving minimum. The corrected chain retains both velocities and their coupling, rather than trying to prove the false cancellation.

3.9.4. One autonomous system and its domain of existence

Together with Einstein evolution and ${\dot{a}}_{\cos }=H{a}_{\cos }$, the physical background is

$$\dot{\vartheta }=u,\dot{u}=-3Hu+{\displaystyle \frac{\gamma \varphi vu-U_{\vartheta }}{A}},\dot{\varphi }=v,\dot{v}=-3Hv-(m_0^2+\gamma u^2/2)\varphi -\lambda {\varphi }^3,$$

$$\dot{H}=-{\displaystyle \frac{A{u}^2+v^2}{2{\overline{M}}_P^2}},{\dot{a}}_{\cos }=H{a}_{\cos },3{\overline{M}}_P^2{H}^2=\rho \mathrm{on}\mathrm{initial}\mathrm{data}.$$

(21)

For $U_{geo}\in C^2$, ${\overline{M}}_P^2>0$, $A>0$, $a_{\cos }>0$, this finite-dimensional vector field is locally Lipschitz and the usual local ODE existence/uniqueness theorem applies [A]. The Friedmann constraint propagates. Extending the solution requires control of the fields, curvature and a positive lower bound for A; the Lean results do not prove existence, a global basin or singularity avoidance. Smooth solutions have continuous total energy at a mass crossing, so no manually matched phase boundary is present in this model. Idealized jumps used in other approximations require separate matching conditions.

The action is conservative as a full field system; Hubble damping exchanges local field energy with expansion work. It does not by itself derive the CGICE diffusion coefficient, friction/noise relation, Gibbs measure or information-time lapse. Obtaining a stochastic CGICE equation from specified eliminated degrees of freedom remains an open-system calculation [O]. A probability-law covariance at equilibrium can be nonzero while the smooth distribution-parameter field has zero time derivative: these are different quantities.

3.9.5. Conditions for the Three Regimes

Define $X_c=-m_0^2/\gamma >0$. The origin mass is positive when $X>X_c$, zero at $X_c$, and negative below it. Acceleration of a scalar-dominated flat solution requires

$$U_{geo}+W>2(AX+Y),\rho +p=2(AX+Y)\ge 0.$$

(22)

On the symmetric branch, both positive origin mass and acceleration can hold if

$X_c<X<U_{geo}/\left(2Z\right)$; this interval is nonempty when $U_{geo}>2Z{X}_c$. The dimensionless values $m_0^2=-1$, $\gamma =Z=1$, $X=2$, $U_{geo}=10$ illustrate simultaneous inequalities after choosing units, not a computed cosmological history. Large statistical variance does not imply large X, and $X>X_c$ does not imply slow roll. No automatic inflation or exit follows.

Moreover the exact kinetic balance is

$$A(\dot{X}+6HX)=2\gamma \varphi vX-U_{\vartheta }u.$$

(23)

Expansion alone therefore does not guarantee monotone X. For $A>0$, $H\ge 0$, a nonpositive right-hand side suffices for $\dot{X}\le 0$. In the invariant special branch $\varphi =v=0$, $U_{\vartheta }=0$, one obtains $\dot{X}=-6HX$ and $X=X_0{e}^{-6(N-N_0)}$. This is a consequence of field equations, not a chosen relation between $\tau$ and N. If $N\to \infty$ and $X_0>X_c$, the branch reaches $N_c-N_0=log(X_0/X_c)/6$. Exactly zero order-parameter data remain exactly zero even after instability; a seed perturbation is needed to leave that branch. The linearized equation is $\delta \ddot{\varphi }+3H\delta \dot{\varphi }+(m_0^2+\gamma X)\delta \varphi =0$. Its growth, backreaction and eventual stabilization must be checked on the actual solution.

More generally, $X\left(t\right)\to 0$, continuity on $t\ge 0$, and $m_0^2+\gamma X\left(0\right)>0$ imply at least one positive-time zero of the mass. Strict decrease makes that zero unique and orders its signs; it must be established independently, e.g. from a strict form of (23). The Lean theorem UnifiedDynamics.critical_crossing uses the stronger whole-line continuity hypothesis for convenience but draws a positive-time conclusion; critical_crossing_unique restricts strict monotonicity to nonnegative times. Neither assumes the existence of a crossing as a premise. Finally, late convergence gives vacuum pressure only with the conditions of §6.7. A proof of these three regimes on one nonempty, globally healthy, matter-producing solution family remains O.

4. Early Statistical Localization

4.1. Radial Confinement

Let $s=r^2$ on $X_E,$ and choose

$$V_{conf}={\displaystyle \frac{a}{2}}s+{\displaystyle \frac{q}{4}}{s}^2,a>0,q\ge 0.$$

Cartan–Hadamard geometry gives Hess $s\ge 2g$. The chain rule gives

$$Hess\left(s^2\right)=2sHesss+2ds\otimes ds,$$

$$Hess{V}_{conf}=\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{qs}{2}}\right)Hesss+{\displaystyle \frac{q}{2}}ds\otimes ds\ge (a+qs)g\ge ag.$$

For r>0, $Hess\left(r^4\right)=2r^2$$Hess\left(r^2\right)+8r^2$ dr⊗dr. Using s avoids the nonsmooth distance coordinate at the origin. PhaseI.radial_hessian_lower supplies the directional quadratic-form arithmetic. The manifold Hessian and comparison theorem remain A, not Lean-defined tensor objects.

A scalar Poisson relation does not replace this tensor estimate. For example, $V(x,y)=2x^2-y^2$ has $\Delta V=2>0$ but $HessV=diag(4,-2)$. Thus a positive trace of the Hessian does not imply positive curvature in every direction.

With $Ric=-6g$ in the stipulated convention,

$$DRic+Hess{V}_{conf}\ge (a-6D)g.$$

Under normalized Gibbs measure, complete diffusion, and the closed form/semigroup hypotheses, the Bakry–Émery argument yields ${\delta }_L\ge a-6D$. At a=35,q=4,D=1 this gives ${\delta }_L\ge 29$; D=2 instead gives 23. PhaseI.poincare_lower_bound converts an explicitly assumed Poincaré inequality for energy and variance functionals into a Rayleigh bound. It does not construct the geometric semigroup.

4.2. Mixing, Clocks and Dimensionality

For $f_0=p_0/\mu -1$ in $L^2\left(\mu \right),$ spectral contraction gives

$$\parallel f_{\tau }{\parallel }_2\le e^{-{\delta }_L\tau }{\parallel f_0\parallel }_2.$$

When $0<\epsilon <\left|\right|f_0{\left|\right|}_2,$ a sufficient time is $\tau \ge (1/29)log\left(\right|\left|f_0\right|{|}_2/\epsilon )$. It depends on the initial state and tolerance. Physical units require $t=t_{*}\tau$ with a separately determined $t_{*}$. We use no dimension-only mixing formula.

At positive noise, concentration does not entail an exact decrease of probability-support dimension. In a Euclidean strongly convex gradient model, synchronous coupling gives ${d|X-Y|}^2/d\tau \le -2a{|X-Y|}^2$. This contraction is not a proof of physical expansion. The illustrative expression $(35/6)log\left(1000001\right)\approx 80.59$ is a volume-based calibration, not the observable e-fold count ∫Hdt; it is not used as a result of this model. Strong convexity alone also does not exclude every possible single-field slow-roll model.

A purely statistical three-interval diagnostic can be defined by $E\left(\tau \right)=E_0{e}^{(}-2\ell \tau ),$ with $E_0>{\theta }_1>{\theta }_2>0$ and $\ell >0$. Its crossing times ${\tau }_i=log(E_0/{\theta }_i)/(2\ell )$ satisfy $0<{\tau }_1<{\tau }_2$. These input thresholds do not identify condensation or DE dominance without the separate physical criteria in §3.2.

4.3. What Localization Contributes to an Early Cosmological Regime

In the reference model $V_{conf}=35r^2/2+r^4$ at $D=1$, confinement can suppress deviations from a normalized Gibbs state despite negative background curvature. The physical motivation is control of geometric fluctuations before a long-lived effective geometry is used. This is compatible with a persistent thermal noise floor and does not require a probability measure to collapse onto a lower-dimensional support.

For a separately justified increasing information-time lapse, the spectral estimate is

$$\parallel f_{\tau \left(t\right)}{\parallel }_2\le e^{-{\delta }_L[\tau \left(t\right)-\tau \left(t_0\right)]}{\parallel f_{\tau \left(t_0\right)}\parallel }_2,\tau \left(t\right)-\tau \left(t_0\right)\ge {\displaystyle \frac{log(\parallel f_{\tau \left(t_0\right)}{\parallel }_2/\epsilon )}{{\delta }_L}}.$$

(24)

The second inequality is a sufficient information-time interval to attain tolerance, when the initial norm exceeds $\epsilon$. It does not set a number of e-folds. A genuine inflationary solution must satisfy $H>0$, ${ϵ}_H=-\dot{H}/H^2<1$, admit a controlled perturbation calculation, and have an exit with an energy budget. From (2) and Raychaudhuri,

$${\ddot{a}}_{\cos }/a_{\cos }=-({\rho }_{tot}+3p_{tot})/\left(6{\overline{M}}_P^2\right).$$

The conditional algebra is CosmologicalBridge.acceleration_from_raychaudhuri. In the unified scalar model, UnifiedDynamics.acceleration_criterion makes it the explicit potential-dominance inequality (22). Neither the dimensional number 35, the lower bound 29, nor the volume calibration 80.59 supplies that inequality. A large X relative to the mass threshold can coexist with a larger potential, but such a hierarchy must be specified and sustained, not inferred from confinement.

For the kinetic trigger, a sufficient localization-before-crossing test is ${\delta }_L[\tau \left(t_c\right)-\tau \left(t_0\right)]>log(\parallel f_{\tau \left(t_0\right)}{\parallel }_2/\epsilon )$, with continuous increasing $\tau$ and a crossing $t_c$ obtained from the field solution. This still requires the stochastic lapse. The special branch in §3.9.5 determines $N_c$ from its dynamical dilution law, but does not fix $\tau$ or ensure exit from acceleration. Temperature is needed for a bath/reheating calculation, not to define the principal trigger.

4.4. Stability of Confinement Under a Controlled Effective Correction

The cumulant-expansion proposal can be made into a quantitative sufficient condition without assuming that convergence implies convexity. Suppose $V=V_{conf}+U$ and a certified uniform bound ${\parallel HessU\parallel }_{op}\le L$ holds. Then

$$DRic+HessV\ge (a-L-6D)g.$$

(25)

Under the same analytic diffusion hypotheses as §4.1, $L<a-6D$ retains a positive gap lower bound. PhaseI.perturbation_gap_lower proves the directional addition of these bounds; it does not certify the bound on U.

For example, in a specified complete normed function space controlling two derivatives uniformly, assume $\parallel {\kappa }_n{\parallel }_{C^2}\le n!A{B}^n$ and $U={\sum }_{n\ge 2}{g}_c^n{\kappa }_n/n!$. For $z=|g_c|B<1$, absolute convergence gives ${\parallel U\parallel }_{C^2}\le A{z}^2/(1-z)$. This yields the sufficient condition $A{z}^2/(1-z)<29$ for the benchmark $a=35,D=1$, provided the norm controls the covariant Hessian. These norm bounds and their uniformity on noncompact $X_E$ have not been established for CGICE. A finite sampling result or an unrelated cluster-expansion citation cannot supply them. The distinction between convergence and a small enough Hessian perturbation is essential.

5. Internal Charge, Transport and Matter Formation

5.1. Positive Casimir and Actual Trajectories

For complex matrices, B(X,Y)=Re tr(XY) is ad-invariant by trace cyclicity (InternalCharge.trace_pairing_ad_invariant). It is indefinite on sl(6,C). For $H=diag(1,-1,0,0,0,0),$ B(H,H)=2 and $B(iH,iH)=-2$.

On the compact real form su(6), use $⟨X,Y⟩=-Re$ tr(XY). Since $X*=-X,$ its quadratic form equals tr(X*X), hence is positive definite. For an actual differentiable compact trajectory,

$${\displaystyle \frac{d}{d\tau }}{\displaystyle \frac{1}{2}}\langle j,j\rangle =-g\langle [a,j],j\rangle =0.$$

The source provides an arbitrary finite-dimensional orthonormal-coordinate version: if $j_i^{\prime }={\sum }_k{A}_{ik}{j}_k$ and $A_{ik}=-A_{ki}$, then ${\sum }_i{j}_i^2$ is constant (InternalCharge.casimir_hasDerivAt, casimir_conserved). It also contains the concrete su(2)/cross-product example from CGICE. Constructing an orthonormal su(6) basis and proving that its adjoint matrix realizes these hypotheses is still A/O. The matrix trace identity is not presented as that missing construction.

5.2. Wasserstein comparison and time

For absolutely continuous source measures with finite second moments and the hypotheses of the relevant transport theorem, the dynamic minimum-kinetic-energy problem has value $W_2^2/2$:

$$\underset{{\partial }_{\tau }\rho +div\left(\rho v\right)=0}{inf}{\int }_0^1\int \frac{1}{2}\rho {\left|v\right|}^{2}d\mathrm{vol}d\tau .$$

For smooth stationary variations using $+\phi ({\partial }_{\tau }\rho +div\left(\rho v\right))$, integration by parts gives $v=\nabla \phi$ and ${\partial }_{\tau }\phi +{|\nabla \phi |}^2/2=0$. Transport.kinetic_completion supplies the associated finite-dimensional square completion. A static optimal map written $T=exp(-\nabla u)$ uses a different potential convention. Benamou–Brenier [7] supplies the dynamic formulation, and McCann [8] supplies a Riemannian polar-factorization result. The cited McCann paper treats compact manifolds; its citation alone does not establish the extension to noncompact $X_E$. The relevant existence, uniqueness and integrability hypotheses there remain explicit analytic requirements.

The actual CGICE diffusion corresponds, under the appropriate assumptions, to a free-energy Wasserstein gradient flow [5], generally not this fixed-endpoint geodesic. The vector field $n=-\nabla u/|\nabla u|$, where $\nabla u\ne 0$, is not a scalar clock. A global physical time requires a Lorentzian metric, a time function with suitable gradient, and causal conditions. Internal advection, optimal transport and condensation remain distinct equations.

5.3. Separate Confinement and Symmetry-Breaking Potentials

Use $V_{conf}$ for the statistical confinement potential, $U_{geo}$ for its separately specified physical response, and $V_{force}(\mathsf{\Phi },X)$ for the derivative-coupled order-parameter force at fixed X. A tachyonic origin cannot obey the same everywhere-positive Hessian lower bound used for statistical confinement. Coupling and backreaction may modify the statistical potential; any continued use of the §4 gap then needs the perturbation control of §4.4.

For the scalar example

$$V_{SB}\left(\varphi \right)={\displaystyle \frac{m^2}{2}}{\varphi }^2+{\displaystyle \frac{\lambda }{4}}{\varphi }^4,m^2<0,\lambda >0,$$

$$V_{SB}\left(\varphi \right)+{\displaystyle \frac{{\left(m^2\right)}^2}{4\lambda }}={\displaystyle \frac{\lambda }{4}}{\left({\varphi }^2+{\displaystyle \frac{m^2}{\lambda }}\right)}^2.$$

The global minima $\pm \sqrt{-m^2/\mathsf{\Lambda }}$ have positive Hessian; the origin has negative Hessian. The VEV123CW scalar results verify this example and its parameterized lower bound. The principal substitution is now $m^2=m_0^2+\gamma X\left(t\right)$, as justified by the scalar field equation. These minima belong to the frozen-X force landscape, not automatically to a minimum of the full velocity-dependent Hamiltonian or a dynamically attained matrix vacuum.

For a compact $\mathrm{SU}\left(6\right)$ adjoint field with eigenvalues $u_i$, $\sum u_i=0$, a gauge-boson mass convention has $m_{ij}^2=g^2{(u_i-u_j)}^2$. An actual one-loop calculation must specify action, gauge, field/ghost spectrum, renormalization scheme and species-dependent subtraction constants; the schematic sum is ${\sum }_s{n}_s{m}_s^4[log(m_s^2/{\mu }^2)-c_s]/\left(64{\pi }^2\right)$. No numerical condensate scale follows without those inputs and a stability calculation.

For multiplicities (1,2,3) of three distinct eigenvalues, the compact stabilizer $S\left(U\right(1\left)\times U\right(2\left)\times U\right(3\left)\right)$ has dimension 13, so 22 generators are broken and 66 massive-vector polarizations occur in a four-dimensional Higgs model. A generic six-distinct-eigenvalue configuration has 30 broken generators and 90 such polarizations. Neither count is the 35 noncompact directions of a different SL(6,C) model.

5.3.1. Kinetic Triggering, Scalar Ordering and the Energy Budget

The field Equation (18) supplies, rather than independently prescribes,

$$m_{eff}^2\left(t\right)=m_0^2+\gamma X\left(t\right),V_{force}(\varphi ,X)=\frac{1}{2}{m}_{eff}^2{\varphi }^2+\frac{\lambda }{4}{\varphi }^4,X_c=-m_0^2/\gamma .$$

(26)

At fixed X above $X_c$, the scalar origin is the unique minimum; below $X_c$ it has negative Hessian and the minima satisfy

$${\varphi }_{\pm }\left(X\right)=\pm \sqrt{-m_{eff}^2/\lambda },V_{force}({\varphi }_{\pm },X)=-{\displaystyle \frac{{\left(m_{eff}^2\right)}^2}{4\lambda }},{\partial }_{\varphi }^2{V}_{force}({\varphi }_{\pm },X)=-2m_{eff}^2>0.$$

(27)

The symbol ${\varphi }_{\pm }$ denotes minima, whereas $v=\dot{\varphi }$ in §3.9 denotes velocity. UnifiedDynamics.mass_sign_criterion, critical_crossing, critical_crossing_unique and broken_scalar_minimum supply the conditional sign-to-crossing-to-minimum chain. The minimum proof reuses ThermalBridge.broken_minimum, whose algebra accepts any negative mass squared; its historical namespace does not make the new trigger thermal. The analytic square root exists for $m_{eff}^2<0$, $\lambda >0$; the Lean minimum statement takes its defining square as a premise.

This is a coupling-triggered instability of an order parameter, conceptually related to the waterfall mechanism in hybrid models [9]. Fisher target geometry can also influence perturbation stability [10], but that literature supplies neither an exact CGICE reduction nor stability for the present coefficient A. The local origin Hessian changes continuously through zero. A sufficiently long negative-mass interval, nonzero fluctuations, backreaction and attraction to a healthy broken branch are needed for actual ordering. During that process $\dot{X}$ need not remain negative, and multiple crossings are possible unless monotonicity is established.

Energy flows according to (19) and §7.2; (27) is not a latent heat to add to radiation by hand. The two-scalar action contains no particles, bath distribution or decay channel from which a radiation abundance can be computed. Such a claim requires additional interactions and a nonequilibrium calculation, as illustrated by reheating analyses [11]. Small oscillations about a quadratic minimum can have an averaged dust-like equation of state only when their frequency is large relative to H and transfers are controlled. Stable cold particles additionally require lifetimes, momenta and production rates. Therefore mass instability, condensation, matter domination and cold-dark-matter abundance remain different steps.

Retained thermal comparison. The previous $m_T^2=c_T(T^2-T_c^2)$ example is still represented by ThermalBridge.massSq_positive_above, massSq_negative_below, symmetric_minimum and critical_origin_hessian. It is an optional bath model, not the main transition clock. Its scalar minimum formulas are obtained from (27) by replacing $m_{eff}^2$ with $m_T^2$. If temperature-dependent free energy F is used, its equilibrium energy is $F-T{\partial }_{T}F$; an evolving bath and its exchange must be included. No thermal coefficients are silently added to (15).

5.3.2. Matrix Vacuum and Particle Interpretation

For a four-dimensional Hermitian traceless adjoint order parameter, the invariant notation is $I_k=tr{\mathsf{\Phi }}^k={\sum }_i{u}_i^k$, $k=2,\dots ,6$. A possible local potential contains

$$V_{tree}=\frac{m^2}{2}{I}_2+{\lambda }_3{I}_3+{\lambda }_{4a}{I}_4+{\lambda }_{4b}{I}_2^2+{\displaystyle \frac{{\lambda }_6}{{\mathsf{\Lambda }}_{EFT}^2}}{I}_6+\dots .$$

(28)

The dimensions are $\left[\mathsf{\Phi }\right]=1$, $\left[{\lambda }_3\right]=1$, and dimensionless quartic and displayed sextic couplings. This is a sample EFT truncation, not the complete degree-six invariant basis. Coleman–Weinberg corrections [12] and finite-temperature terms depend on the actual spectrum, gauge fixing and renormalization prescription. They do not determine a ${10}^{16}$ GeV vacuum merely from five stationarity equations.

There is a stronger obstruction to the review’s automatic $(1,2,3)$ claim. For the minimal frozen-X adjoint potential $V={\sum }_{i=1}^6[m_{eff}^2{u}_i^2/2+{\lambda }_{mat}{u}_i^4/4]$, ${\sum }_i{u}_i=0$, ${\lambda }_{mat}>0$, $m_{eff}^2<0$, square completion gives

$$V={\displaystyle \frac{{\lambda }_{mat}}{4}}{\displaystyle \sum _{i=1}^6}{\left(u_i^2+{\displaystyle \frac{m_{eff}^2}{{\lambda }_{mat}}}\right)}^2-{\displaystyle \frac{6{\left(m_{eff}^2\right)}^2}{4{\lambda }_{mat}}}.$$

(29)

The lower bound is attained by $(r,r,r,-r,-r,-r)$ with $r^2=-m_{eff}^2/{\lambda }_{mat}$. Every unconstrained scalar summand must attain its minimum for equality, so $|u_i|=r$; tracelessness then requires three plus and three minus signs. Thus this particular classical force potential favors (3,3), not three distinct eigenvalues of multiplicities (1,2,3). UnifiedDynamics.six_mode_lower_bound gives the bound and three_three_witness checks an explicit saturating traceless example. The classification of all equality cases is analytic. The kinetic-domain condition $Z-\gamma {\sum }_i{u}_i^2/2>0$ must also hold for the vacuum to be admissible; otherwise even this branch lies outside the effective theory. Additional invariants, loop corrections or field content may alter selection, but must be specified and compared across competing orbits and transverse fluctuations. The earlier (1,2,3) generator count remains conditional on selecting that pattern.

The physical proposal is that stable massive excitations of a selected condensate can populate visible and dark sectors. Casimir conservation constrains internal motion; it neither creates particles nor establishes a collisionless equation of state. A sterile-neutrino or seesaw realization additionally needs fermion representations, Yukawa operators, a mass matrix, production and decay rates, and lifetime bounds. Those constructions are not provided here. No neutrino mass or dark-matter ratio is imported as a new prediction.

5.4. Multiplicity Enumeration and Signatures

Positive, ordered, pairwise-distinct integer multiplicities summing to six necessarily form (1,2,3). This is MultiplicityLabels.canonical_positive_sumSix_nondeg_unique. The sum constraint is representation dimension, not rank; the rank is five. Pairwise distinctness is an input, not a dynamical rejection of (2,2,2).

A Vandermonde measure depends on eigenvalues, not multiplicities. For eigenvalues $\alpha ,\beta ,\gamma$ and multiplicities $n_i$, the cubic trace is $n_1{\alpha }^3+n_2{\beta }^3+n_3{\gamma }^3$ under $\sum n_i{\alpha }_i=0$; it is not generally $\sum n_i^3$. Choosing $(1,2,3)$ and $(-3,0,1)$ gives cubic trace $-24$. Actual vacuum selection requires an action and comparison of gauge-inequivalent stable stationary points.

The labels (3,1),(2,2),(1,3) enumerate positive ordered pairs summing to four. They are internal labels only. Along a continuous nondegenerate real metric path, inertia is locally constant. An ordinary black-hole interior remains locally Lorentzian; exchanging coordinate roles does not produce three time directions. These labels are not the three effective stages of §3.

5.5. Abundance evolution and a conditional positive bound

For x>0, let $Y>Y_{e}q>0$ solve

$$Y^{\prime }=-\lambda (Y^2-Y_{eq}^2),\lambda >0,b=\lambda [1-{(Y_{eq}/Y)}^2].$$

Then $Y^{\prime }<0$, ${(1/Y)}^{\prime }=b$ and $0\le b\le \mathsf{\Lambda }$. The Lean source proves these pointwise results on an actual solution record. To integrate them, require a positive reference $x_f$, local absolute continuity of $1/Y$, and integrability of b on each finite interval. Then

$${\displaystyle \frac{1}{Y\left(x\right)}}={\displaystyle \frac{1}{Y\left(x_f\right)}}+{\int }_{x_f}^{x}b\left(s\right)ds.$$

If the nonnegative cumulative integral is bounded by $B<U_{\infty }$ then

$$Y\left(x\right)\ge {\left({\displaystyle \frac{1}{Y\left(x_f\right)}}+B\right)}^{-1}>0.$$

DMAbundance.positive_limit_from_reciprocal_identity proves positivity of a given limit from the explicit integral identity and bound. The general fundamental-theorem-of-calculus step from tail-integrable coefficients is not asserted as a Lean theorem. TailIntegrable means IntegrableOn f (Ioi x_f); it is never defined by the value of an integral without an integrability requirement.

Under these regularity conditions, finiteness of ∫ b is the exact criterion for a positive limit. Finiteness of $\int \mathsf{\Lambda }$ is sufficient, not necessary. For example, $\mathsf{\Lambda }=1,$$Y=1+e^{(}-x),$$Y_{e}q=sqrt(Y^2-e^{(}-x))$ gives $Y\to 1$ despite divergent $\int \mathsf{\Lambda }$. The source also provides a rational positive-relic solution and a decreasing solution converging to zero. These are mathematical examples; a physical relic density additionally requires mass, cross section, entropy density and Hubble history.

5.6. Conservative Two-State Transfer

Let B and D denote nonnegative comoving dust masses, u,v>0, and N=B+D. Set

$$\left({\displaystyle \genfrac{}{}{0pt}{}{B^{\prime }}{D^{\prime }}}\right)=\left(\begin{array}{cc}-u& v\\ u& -v\end{array}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{B}{D}}\right).$$

The characteristic polynomial is $\mathsf{\Lambda }(\mathsf{\Lambda }+u+v)$. The stationary vector is proportional to (v,u), with DM/baryon ratio u/v. Positivity does not specify a numerical ratio such as 5.47.

The solution is

$$B_{*}={\displaystyle \frac{vN}{u+v}},B\left(\tau \right)=B_{*}+(B_0-B_{*})e^{-(u+v)\tau },D=N-B.$$

For $0\le B_0\le N$ and $\tau \ge 0,$ the exponential interpolation preserves nonnegativity. For N>0 the ratio converges to u/v. The TwoState namespace supplies the differential equations, initial values, conservation, positivity, limits and uniqueness of the scalar reduced problem. These results concern the stated linear system; they are not a derivation of the reaction rates from CGICE.

6. Dissipation and Late-Time Phenomenology

6.1. Free Energy and Equilibrium Restrictions

For fixed $V_{c}onf,D$ and normalized p,

$$\mathcal{F}\left[p\right]=\int V_{conf}p+D\int plogp,\mathcal{F}\left[p\right]-\mathcal{F}\left[\mu \right]=D\mathrm{KL}(p\parallel \mu ).$$

With $\xi =\delta F/\delta p$, $J=-p\nabla \xi$ and zero boundary flux, the square completion and continuity equation yield $F^{\prime }=-\int p{|\nabla \xi |}^2$. The companion finite-graph construction is reproduced with explicit vertices, edges, divergence and differentiable probability trajectories. CGICECore.freeEnergy_dissipation proves a genuine derivative identity, and freeEnergy_nonincrease_deriv proves its nonpositive sign. The source’s finite graph is a model with source-vertex edge mobility; a continuum convergence theorem is not assumed.

For continuous diffusion, a periodic probability law and the KL dissipation identity imply zero integrated dissipation. Positivity, connectedness, regularity and the zero-dissipation characterization then force $p=\mu$. This rules out a nontrivial periodic probability distribution in this fixed equilibrium model. It does not assert that each random path stops, and it does not concern externally driven systems. Monotonicity alone, without a characterization of zero dissipation, is insufficient.

If the volume form varies with time, $\partial \tau p+v_{\tau }p=L*\tau p$, where $v_{\tau }={\partial }_{\tau }log\left({\mathrm{dvol}}_g\tau \right)$. If V varies, $F^{\prime }$ contains the external-power term $\int p{\partial }_{\tau }V$. Cooling or driving therefore requires a separately derived dissipation balance.

6.2. OU Covariance, Uniqueness and Noise Floor

For $dX=-\ell X$$d\tau +sqrt\left(2D\right)dW$ with $\ell >0$,

$$C^{\prime }=-2\ell C+2D,C\left(\tau \right)={\displaystyle \frac{D}{\ell }}+\left(C_0-{\displaystyle \frac{D}{\ell }}\right)e^{-2\ell \tau }.$$

The single definition CGICECore.ouCov is used throughout the proof file. Its initial value, actual derivative, uniqueness, limit and positivity appear as named theorems. For $C_0>0,D\ge 0,\tau \ge 0,$ C stays positive. For D>0 the limit is D/ℓ, not zero. If $E=C-D/\ell ,$ then

$$E^{\prime }=-2\ell E,{\left(E^2\right)}^{\prime }=-4\ell E^2\le 0.$$

Negative E approaches zero from below; it is not a positive energy density. The OU solution applies to the linear model or a controlled approximation, not a global quartic drift without moment closure.

The potential with Hess $V_{c}onf\left(o\right)=35g$ has local drift $\ell =35$ in the same metric/time convention. This $\ell =35$ is a chosen model input [M], not a consequence of the spectral threshold of §2.2: the free Laplacian bottom, the Witten gap and the diffusion relaxation rate are distinct quantities in the companion CGICE analysis, and the drift rate is not fixed by that threshold alone. A separate rate 35/3 requires a separate effective model or a simultaneous transformation of time, noise and drift. At $\tau =4.01,$$e^{(}-2·35·\tau )\approx 1.24033\times {10}^{-122}$.

6.3. Retained Phenomenological Comparisons for the Equation of State

The models in §§6.3–6.3.2 are earlier response comparisons, preserved for their explicit algebra and observational diagnostics. They are not imposed on the derivative-coupled action, whose pressure and late limit are derived in §6.7. Their external Q and covariance-to-density conversion must not be silently identified with the exchange in §7.2; in particular their phantom branch is not a solution of the healthy scalar stress in (16).

Suppose, as an extra response Ansatz, ${\rho }_{D}E={\rho }_{*}+\kappa E\left(\tau \left(t\right)\right)$. The physical continuity equation then implies

$$w_{DE}=-1+{\displaystyle \frac{2\ell \dot{\tau }\kappa E+Q_{DE}}{3H{\rho }_{DE}}}.$$

CosmologicalClosure.eos_from_continuity proves the algebra for $H{\rho }_{D}E\ne 0$. The stress-energy interpretation, clock and expansion solution remain additional inputs. Entropy or covariance decay alone cannot determine w.

For the different, separately conserved model

$$\rho \left(a\right)={\rho }_v(1-2ϵa^{-3}),{\rho }_v>0,a>0,$$

the identity $p=-\rho -a{\rho }^{\prime }\left(a\right)/3$ gives $p=-{\rho }_v$. On the positive-density domain $a^3>2\epsilon ,$

$$w\left(a\right)=-1-{\displaystyle \frac{2ϵ}{a^3-2ϵ}},w_0=-1-{\displaystyle \frac{2ϵ}{1-2ϵ}},w_a=-{\displaystyle \frac{6ϵ}{{(1-2ϵ)}^2}}=-3w_0(1+w_0).$$

Here $w_a=-w^{\prime }\left(1\right)$ is the local CPL coefficient, not an assertion of an exact CPL function at all a. The sign of $\epsilon$ selects the branch: for $0<\epsilon <1/2,$$w_0<-1$ and $w_a<0$ (phantom); for $\epsilon <0,$$w_0>-1,$$w_a>0,$ and density positivity holds for every a>0. The illustrative $w_0=-0.73$ lies in the $\epsilon <0$ branch $(\epsilon =-27/146,$$w_a=+0.5913),$ with $\rho /{\rho }_v=100/73$ at a=1. ModelAlgebra supplies the elimination, inverse and sign statements. No combination of $-0.73$ with negative $w_a$ is claimed for this density law.

6.3.1. An explicit clock-to-fluid bridge

As an optional comparison, use §3.6’s covariance-volume response with a reference epoch of zero information and expansion coordinates. Its coefficient $\nu ={(2\beta \ell )}^{-1}$ gives $m=2\ell \nu =1/\beta$ in (30). This comparison is not the proper-time identity (1) or an additional constraint on (21):

$$E\left(N\right)=E_0{e}^{-mN},m=2\ell \nu >0,{\rho }_g={\rho }_{*}+{\rho }_E,{\rho }_E=\kappa E_0{e}^{-mN}.$$

(30)

Here ${\rho }_{*}>0$ and $\kappa$ are independent dimensional response inputs: since the geometric covariance $E_0$ is dimensionless, $\kappa$ carries energy dimension four so that ${\rho }_E=\kappa E_0$ is an energy density. ${\rho }_E$ is not the probability density p or the covariance itself. The sign of ${\rho }_E$ is not fixed by (30): $\kappa E_0$ may be positive or negative, and the nonnegative-tail and phantom branches below correspond to opposite signs. For separate conservation, $Q_g=0$, the pressure is fixed:

$$p_g=-{\rho }_{*}+\left({\displaystyle \frac{m}{3}}-1\right){\rho }_E,{\rho }_g+p_g={\displaystyle \frac{m}{3}}{\rho }_E.$$

(31)

Thus a nonnegative decaying excess with $m>0$ gives $w_g\ge -1$ wherever ${\rho }_g>0$, approaching $-1$ as $N\to \infty$. A negative excess cannot be called a positive energy component; it gives an effective phantom contribution and requires an independent stability analysis. Choosing the positive response ${\rho }_E=\kappa E^2$ instead yields $m=4\ell \nu$ and the same conservation algebra, but a different response model. For $Q_g\ne 0$, (31) acquires $Q_g/\left(3H\right)$; if $Q_g$ is also specified through branching, it cannot be ignored while fitting w.

ClockResponse.ou_clock_identity connects the existing OU definition to the N-clock tail. tail_physical_hasDerivAt differentiates it along an actual differentiable $N\left(t\right)$. continuity, enthalpy, no_phantom and tail_limit establish the conditional fluid relations. This construction still assumes the response law, the expanding geometry and the clock; it is not a derivation of them from SL(6,C).

At $m=3$, $p_g=-{\rho }_{*}$ and the response is precisely vacuum plus a dust-like term. With $a_{ref}=1$, identifying ${\rho }_{*}={\rho }_v$ and $\kappa E_0=-2{\rho }_vϵ$ recovers the density in §6.3. density_ansatz_match and positive_tail_epsilon_sign show that a nonnegative tail requires $ϵ\le 0$. The positive-$ϵ$ phantom branch is therefore not the positive-tail realization of the same closure. In the nonnegative $m=3$ branch, background expansion alone is degenerate with a reassignment of dust and vacuum densities; discrimination requires perturbations or another observable.

6.3.2. Domination, Acceleration and a Nonempty Late-Stage Benchmark

Include separately conserved dust ${\rho }_m=M{e}^{-3N}$, $M>0$. Then

$$H^2={\displaystyle \frac{M{e}^{-3N}+{\rho }_{*}+{\rho }_{E0}{e}^{-mN}}{3{\overline{M}}_P^2}},{\displaystyle \frac{\ddot{a}}{a}}>0⟺M{e}^{-3N}+(m-2){\rho }_{E0}{e}^{-mN}<2{\rho }_{*}.$$

(32)

The equivalence follows from (31) and is formalized by ClockResponse.acceleration_criterion; it does not equate acceleration onset with equal densities. For ${\rho }_{E0}\ge 0$ and $m>0$, the two decaying terms vanish as $N\to \infty$, giving late acceleration on the expanding solution. With these specified continuous positive densities, choosing the positive square root for H and integrating $dt/dN=1/H\left(N\right)$ constructs a regular background on each finite N interval; $H\to \sqrt{{\rho }_{*}/\left(3{\overline{M}}_P^2\right)}$ implies an infinite future proper-time interval. This elementary quadrature is A-level, not a Lean solution-existence theorem.

In the dust-like case $m=3$, set $M_{eff}=M+{\rho }_{E0}>0$. The distinct thresholds are

$$N_{acc}=\frac{1}{3}log{\displaystyle \frac{M_{eff}}{2{\rho }_{*}}},N_{eq}=\frac{1}{3}log{\displaystyle \frac{M_{eff}}{{\rho }_{*}}},N_{eq}-N_{acc}=\frac{1}{3}log2>0.$$

(33)

A matter interval after condensation requires $M_{eff}{e}^{-3N_c}>{\rho }_{*}$; a decelerating dust-dominated interval requires the stronger $M_{eff}{e}^{-3N_c}>2{\rho }_{*}$. These are explicit open sets of positive reduced-model data. CosmologicalBridge.dust_vacuum_regime_witness checks the concrete point $M_{eff}=8$, ${\rho }_{*}=1$ at scale factors 1 and 3. It checks inequalities, not a microscopic transition or the initial inflationary stage. The model therefore supplies a conditional, nonempty matter-to-vacuum crossover without claiming a full three-stage universe.

Finally, ${\rho }_{*}$ has not been calculated. An exponentially small excess can coexist with an arbitrarily large constant term. Neither $\ell =35$ nor $\ell =35/3$, and neither a fitted relaxation duration nor the existence of an OU floor, solves the cosmological-constant hierarchy problem.

6.4. Single-Field Kinetic Stability

For $P\left(X\right)=-\kappa X+\mathsf{\Lambda }{X}^2$ with $\kappa ,\mathsf{\Lambda }>0,$ using the k-essence perturbative framework [13],

$$P_X=-\kappa +2\lambda X,K=P_X+2X{P}_{XX}=-\kappa +6\lambda X.$$

At $X=\kappa /\left(6\mathsf{\Lambda }\right),$ K=0 and regular sound-speed propagation is not established. Both positive kinetic coefficient and positive gradient coefficient require $X>\kappa /\left(2\mathsf{\Lambda }\right)$. A positive ratio $P_X/K$ does not remove a ghost if both numerator and denominator are negative. For timelike X>0 and $\rho >0$ in this healthy region, $\rho +p=2X{P}_X>0,$ so the minimal model cannot support healthy phantom behavior. These conditional obstructions do not rule out every multi-field or higher-derivative theory; such alternatives need their own perturbation analysis.

6.5. Cooling, Changing Reference Measures and the Information Arrow

The thermal construction makes it essential to distinguish fixed equilibrium from driving. On a fixed volume form with $D\left(\tau \right)>0$, $V\left(\tau \right)$, normalized instantaneous ${\mu }_{\tau }=Z_{\tau }^{-1}{e}^{-V_{\tau }/D_{\tau }}$, and sufficient regularity, set $K=\mathrm{KL}(p\parallel {\mu }_{\tau })$ and $I=\int p|\nabla log(p/{\mu }_{\tau }){|}^2$. Differentiating both arguments gives

$$K^{\prime }=-DI-\int p{\partial }_{\tau }log{\mu }_{\tau },{\mathcal{F}}_{\tau }\left[p\right]-{\mathcal{F}}_{\tau }\left[{\mu }_{\tau }\right]=D_{\tau }K.$$

(34)

Consequently ${\left(DK\right)}^{\prime }=D^{\prime }K-D^{2}I-D\int p{\partial }_{\tau }log{\mu }_{\tau }$. A cooling schedule is not free equilibrium relaxation. DrivenEntropy.moving_reference_balance proves the product-rule step, and kl_nonincrease states its separate sign requirement; the PDE identity (34) remains analytic. For a moving metric, the volume-form term in §6.1 must also be included.

The no-periodic-distribution argument of §6.1 is the relevant classical fixed-reference obstruction. Watanabe–Oshikawa [14] give a different no-go theorem for specified equilibrium quantum systems; it must not be used as a theorem about every cosmological open system. A proposed oscillatory or time-crystal completion must display driving, energy exchange and perturbative stability.

6.6. Non-Gaussian Correction and Uniform Covariance Control

This subsection concerns the centered, symmetric one-dimensional Euclidean process with drift $-ax-q{x}^3$, $a>0$, $q\ge 0$, $D>0$. Assume a nonexplosive solution, moment integrability sufficient for Itô’s formula and localization/removal of stopping times. These are analytic requirements; the following moment equations are not automatically equations on curved $X_E$.

Let $M_j=\mathbb{E}\left[X^j\right]$, $C=M_2$, ${\kappa }_4=M_4-3C^2$, and let $G\ge 0$ solve the Gaussian-closed Riccati equation. Subtracting the exact covariance equation gives

$$\delta =C-G,{\delta }^{\prime }=-A\left(\tau \right)\delta -2q{\kappa }_4,A=2a+6q(C+G)\ge 2a.$$

(35)

This is QuarticError.exact_difference and restoring_rate_lower. The fourth-moment equation and Jensen’s inequality give

$$M_4^{\prime }=-4a{M}_4-4q{M}_6+12DC,C^2\le M_4,$$

$$12DC\le 2a{C}^2+18D^2/a\le 2a{M}_4+18D^2/a,M_4^{\prime }\le -2a{M}_4+18D^2/a.$$

Consequently, on forward intervals,

$$M_4\left(\tau \right)\le e^{-2a\tau }{M}_4\left(0\right)+{\displaystyle \frac{9D^2}{a^2}}(1-e^{-2a\tau })\le M:=max\{M_4\left(0\right),9D^2/a^2\},\left|{\kappa }_4\left(\tau \right)\right|\le 2M=:K.$$

(36)

The last inequality follows from $-2M_4\le {\kappa }_4\le M_4$. fourth_moment_differential_bound proves the polynomial Young-inequality certificate under explicit moment premises; fourth_moment_bound integrates it on $[0,\tau ]$, and cumulant_abs_bound proves the cumulant estimate. Itô’s formula, Jensen’s inequality for the law, and existence of its moments remain A, not hidden hypotheses asserted globally.

Variation of constants in (35), using $A\ge 2a$, yields the sharper analytical bound

$$\left|\delta \left(\tau \right)\right|\le e^{-2a\tau }\left|\delta \left(0\right)\right|+{\displaystyle \frac{qK}{a}}(1-e^{-2a\tau }).$$

(37)

An independently formalizable squared-error route uses

$${\left({\delta }^2\right)}^{\prime }\le -2a{\delta }^2+{\displaystyle \frac{2q^2{K}^2}{a}},\delta {\left(\tau \right)}^2\le e^{-2a\tau }\delta {\left(0\right)}^2+{\displaystyle \frac{q^2{K}^2}{a^2}}(1-e^{-2a\tau }).$$

(38)

squared_error_differential_bound supplies the differential certificate, scalar_comparison supplies integrating-factor control on an actual finite-interval trajectory, and covariance_error_bound combines them into (38). For solutions existing for all forward times with uniform hypotheses these estimates give $lim\; sup\left|\delta \right|\le qK/a$, a persistent error floor rather than convergence to a Gaussian. Small $qK/a$ in the chosen covariance units is required for accuracy. With a physical density coefficient ${\kappa }_{\rho }$, the corresponding absolute density error is at most $|{\kappa }_{\rho }\left|\right|\delta |$ when the same external clock and response are used. If the clock itself is reconstructed from covariance, its error must also be propagated; on $E,\widehat{E}\ge \epsilon >0$, $|N_E-N_{\widehat{E}}|\le \beta |E-\widehat{E}|/\epsilon$ for a common reference.

There is no reason for ${\kappa }_4$ to vanish at late time for fixed $q>0$: the stationary density is proportional to $exp[-(a{x}^2/2+q{x}^4/4)/D]$, not a Gaussian. More directly, at Gaussian initial moments $M_4=3C^2$, $M_6=15C^3$,

$${\kappa }_4^{\prime }\left(0\right)=-24qC{\left(0\right)}^3<0(q,C\left(0\right)>0).$$

(39)

The polynomial identity and sign have Lean proof bodies. A perturbative stationary diagnostic, obtained by differentiating normalized Gibbs expectations at $q=0^{+}$, is

$C_{\infty }=D/a-3q{D}^2/a^3+O\left(q^2\right)$ and

${\kappa }_{4,\infty }=-6q{D}^3/a^4+O\left(q^2\right)$, with $a,D$ fixed. This local expansion is analytical and distinct from the uniform, nonperturbative bounds (36)-(38); no convergent all-order series is asserted. Decay of relative entropy toward this non-Gaussian equilibrium cannot establish zero fourth cumulant.

6.7. Late Vacuum of the Common Action: Value, Health and Conditional Limits

Consider a solution for which $u\to 0$, $v\to 0$, $\varphi \to {\varphi }_v$, $U_{geo}\left(\vartheta \left(t\right)\right)\to U_{\infty }$, with

$${\varphi }_v^2=-m_0^2/\lambda ,A_v=Z+{\displaystyle \frac{\gamma m_0^2}{2\lambda }}>0,{\mathsf{\Lambda }}_{eff}=U_{\infty }-{\displaystyle \frac{{\left(m_0^2\right)}^2}{4\lambda }}>0.$$

(40)

The first relation supplies the scalar broken vacuum; it does not determine $U_{\infty }$. Square completion gives $W\left({\varphi }_v\right)=-{\left(m_0^2\right)}^2/\left(4\lambda \right)$, and continuity of the polynomial stress yields

$$\rho ⟶{\mathsf{\Lambda }}_{eff},p⟶-{\mathsf{\Lambda }}_{eff},w=p/\rho ⟶-1.$$

(41)

The positive limiting density makes the denominator nonzero eventually. The chain is UnifiedDynamics.vacuum_potential_value, vacuum_density_pressure, density_pressure_limits and late_w_limit. These are limit implications for actual functions, not a statement that every initial condition is attracted to the vacuum. The symbol m in Lean is $m_0^2$, so its square in vacuumDensity means ${\left(m_0^2\right)}^2$, not $m_0^2$ itself.

For a stationary solution one must also have $U_{\vartheta }\left({\vartheta }_{\infty }\right)=0$. With (40), $u=v=0$ satisfies the scalar equations (vacuum_rest_equations), and the expanding flat Einstein branch has $H_{\infty }=\sqrt{{\mathsf{\Lambda }}_{eff}/\left(3{\overline{M}}_P^2\right)}$. Around an isolated scalar vacuum with $U_{\vartheta \vartheta }>0$, the two canonical squared masses are $U_{\vartheta \vartheta }/A_v$ and $-2m_0^2$, both positive. On this homogeneous de Sitter background the decoupled linear characteristic polynomials are $s^2+3H_{\infty }s+M_i^2$ with roots of negative real part. This is an analytic local linear test; it does not prove global nonlinear attraction, gauge/matrix stability or a basin containing the earlier unstable trajectory. For the general Fisher target, the relevant covariant Hessian and kinetic metric replace the scalar quotient.

The health and positive-energy conditions are nonvacuous: in chosen units $Z=\gamma =\lambda =1$, $m_0^2=-1$, ${\varphi }_v=1$, $U_{\infty }=1$ give $A_v=1/2$ and ${\mathsf{\Lambda }}_{eff}=3/4$ (vacuum_health_positive_witness). Changing only $U_{\infty }$ to zero gives ${\mathsf{\Lambda }}_{eff}=-1/4$ (residual_energy_need_not_be_positive). Thus neither the sign nor the small observed magnitude of vacuum energy follows from symmetry breaking. A constant term in $U_{geo}$ is an independent gravitational input; calling it an OU residue does not solve its normalization or radiative stability.

The OU result $C_{\infty }=D/\ell$ describes variance of the configuration law. It does not equal X, which is quadratic in the time derivative of its parameter field, nor does it specify $U_{\infty }$. A stationary distribution with nonzero variance can have $X=0$. Any map from that variance to (40), and its uncertainty, must be derived from the open-system response described in §3.9.1. The non-Gaussian controls of §6.6 bound the stated statistical approximation; they do not convert it into physical vacuum energy.

If an additional, separately conserved pressureless sector has ${\rho }_m\ge 0$ and $a_{\cos }\to \infty$, then ${\rho }_m\propto a_{\cos }^{-3}\to 0$. For a stationary positive vacuum, vacuum_eventually_dominates supplies ${\rho }_m<{\mathsf{\Lambda }}_{eff}$ eventually. More generally, (41), ${\rho }_m\to 0$ and $p_m\to 0$ imply negative total $\rho +3p$ eventually (UnifiedDynamics.late_accelerating_stress); the Friedmann/Raychaudhuri relation then gives acceleration. Particle-sector dilution, an expanding solution lasting to that limit, and a prior matter-dominated interval are separate conditions. The two-field action alone has not produced that particle sector, so the statement is a conditional route to late domination, not a proved complete cosmic history.

7. Branching, Normalization and Cosmological Conservation

7.1. Joint Instantaneous Constraints

Assume initial comoving DM/baryon ratio r, baryon normalization $B_i>0$ relative to today’s critical density, converted fraction $f\in [0,1),$ DE branch $d\in [0,1],$ and no other sources. BH storage is empty at the end. DM subsequently redshifts as dust; DE produced at a single redshift $z_{*}$ has $w=-1$ thereafter. Put $A={(1+z_{*})}^3$ and denote the injected present DE contribution by $\omega$. Then

$$B_f=B_i(1-f),R={\displaystyle \frac{r+(1-d)f}{1-f}},\omega =A{B}_{i}fd.$$

For $e=\omega /\left(A{B}_i\right),$ the joint solution is

$$f={\displaystyle \frac{R-r+e}{R+1}},d={\displaystyle \frac{e}{f}}.$$

For R>r>0 and e>0, feasibility requires and is ensured by $e\le (R-r)/R$. Indeed $d\le 1$ is $e\le f$; substitution gives $eR\le R-r$. This bound implies e<1<r+1 and therefore 0<f<1. The two constraints determine f and then d uniquely. Branching.joint_constraints and joint_physical_range give an explicit algebraic construction and a sufficient feasible range; necessity and uniqueness follow analytically as above.

For the illustrative choices $r=5,R=5.47,B_i=0.06,A={\left(5.29\right)}^3$ and $\omega =0.682,$Branching.benchmark_joint_solution verifies both constraints and the fraction bounds using exact rationals. $B_i$ and $\omega$ here are chosen inputs, not measured outputs. In particular $B_f$ must still be compared with baryon data. The arithmetic of this instantaneous example does not fit a complete cosmology.

The value 5.47 is an illustrative target, not identified with every observational analysis.

7.2. Internal Exchange Fixed by the Common Action

For the specific subsystem convention

$${\rho }_{\vartheta }=AX+U_{geo},p_{\vartheta }=AX-U_{geo},{\rho }_{\varphi }=Y+W,p_{\varphi }=Y-W,$$

the two Euler–Lagrange equations give

$${\dot{\rho }}_{\vartheta }+3H({\rho }_{\vartheta }+p_{\vartheta })=\gamma \varphi \dot{\varphi }X,{\dot{\rho }}_{\varphi }+3H({\rho }_{\varphi }+p_{\varphi })=-\gamma \varphi \dot{\varphi }X.$$

(42)

These exchange terms are computed from the evolving fields and the chosen action. They are not arbitrary functions used to connect stages, and their direction can change with $\varphi \dot{\varphi }$. UnifiedDynamics.theta_exchange, phi_exchange and exchange_balance encode the rate algebra; unified_continuity is the actual total trajectory theorem. Subsystem bookkeeping depends on where the interaction energy is assigned, but the total stress (16) and its conservation do not. On a smooth solution there is no externally inserted energy jump at $m_{eff}^2=0$.

Equation (42) replaces the former four-component injection prescription in the principal model. It does not say each subsystem source vanishes, nor that an independently defined baryon/BH/CDM network has been derived. Existing Branching.sources_sum_zero and SourceQ results are retained as algebra for optional population models, not as consequences of (15). Their source laws would need a specified microscopic or coarse-grained coupling to enter the enlarged action. The instantaneous constraints of §7.1 and population checks of §7.3 are likewise conditional phenomenology; a factor ${(1+z_{*})}^3$ is not a prediction of the two-field vacuum model. Radiation, reheating and black-hole populations require additional variables and their equations.

7.3. Joint Constraint Completeness and Population Histories

The integrated source now supplies Branching.joint_unique, feasible_necessary and feasible_sufficient_sharp. Under $R>r>0$, $e\ge 0$, $0\le f<1$, $0\le d\le 1$ and the two simultaneous equations, the feasible region is exactly $eR\le R-r$. Within it $f>0$, the solution is unique and equals §7.1. Necessity follows from $e=fd\le f$ and $(R+1)f=R-r+e$; sufficiency also implies $e<1$, so the earlier extra bound $e<r+1$ is redundant. This completes the algebraic evidence chain without claiming a physical rate model.

For continuous injection, let $dL\left(z\right)\ge 0$ denote a dimensionless positive measure of comoving energy loss normalized to today’s critical density, including the DE branch. With $w=-1$ after injection and the other assumptions of §7.1, its present contribution is

$$\Delta {\Omega }_{DE}=\int {(1+z)}^{3}dL\left(z\right),{(1+z_{eff})}^3={\displaystyle \frac{\int {(1+z)}^{3}dL\left(z\right)}{\int dL\left(z\right)}}.$$

(43)

This definition requires a positive finite denominator and finite third moment. It is a moment-equivalent redshift, not the mean redshift or the peak of the star-formation history. Madau–Dickinson [15] provide a historical star-formation baseline; the BH population requires an IMF, formation efficiencies, delay distributions, lifetimes and branching fractions before $dL$ is known. A stellar-population integral is not supplied by a lattice-gauge simulation. The reference $z_{*}=4.29$ remains a chosen single-epoch benchmark.

For the exact input values of §7.1, $e\simeq 0.07678$, $f\simeq 0.08451$, $d\simeq 0.90856$ and $B_f\simeq 0.05493$. These differ substantially from a 5.4

8. RG and Index-to-Coupling Matching

8.1. Coupling Conventions and Applicable Scales

In a specified four-dimensional compact $\mathrm{SU}\left(6\right)$ gauge theory with 15 fundamental Dirac flavors [M] and no other contributions, ${tr}_F\left(T^a{T}^b\right)={\delta }^{ab}/2$ gives $C_A=6$, $T_F=1/2$ and $b_0=12$. It is a field-content assumption [M], not a consequence of target-space dimension alone. Under $T\to \sqrt{2}T$, $C_A$, $T_F$ and this $b_0$ all double; $g\to g/\sqrt{2}$ preserves $gT$. Asymptotic freedom of the perturbative branch does not establish ultraviolet completeness of a noncompact gravitational theory.

With ${M=10}^{1}6$ GeV and $g_0=0.5$ as inputs,

$$g^2\left(\mu \right)={\displaystyle \frac{g_0^2}{1+b_0{g}_0^{2}log(\mu /M)/\left(8{\pi }^2\right)}}.$$

The formal pole is about $3.71388\times {10}^4$ GeV and the formal D1 crossing about $9.81879\times {10}^4$ GeV. Both lie in a strong-coupling extrapolation region; neither computes a condensation temperature. No instanton enhancement exponent from a different gauge group is inserted into this running law.

For the separate specified flow $\beta \left(g\right)=\eta g/2+b{g}^3/\left(8{\pi }^2\right),$$\eta =-36/35,b=12,$ one finds $g_{*}^2=12{\pi }^2/35$ and $\beta `\left(g_{*}\right)=36/35$. It is locally IR-attractive as log $\mu$ decreases. The D1 definition gives $g_T{C}^2=24{\pi }^2/35$. Their ratio two is algebraic — a consequence of the realification and basis-rescaling normalization analyzed in the companion CGICE source — not a mechanism relating two physical couplings. CGICECore.beta_fixedpoint, signed_fixedpoint and gTC_sq_eq_two_gStar_sq express that scope.

8.2. Heat Kernel and Fredholm Problem

For $P=-{\nabla }^2+E,$ the local heat coefficient is $a_2=R/6-E,$ consistent with $e^{(}-tE)=1-tE+\dots$ in flat space. Connection-curvature squares occur at $a_4$ [16]. The ratio 36/35 is a beta-function Ansatz; these heat coefficients do not derive a full FRG equation. Local expansions on a noncompact manifold do not automatically define a finite global heat trace.

Following the Callias-type research route, with an applicable theorem such as [17], a possible index route starts on a specified Riemannian Clifford module, with self-adjoint $D_0,\phi$ and a common operator core, and considers $B=D_0+i\phi$. Formally,

$$B^{*}B=D_0^2+{\mathsf{\Phi }}^2+i[D_0,\mathsf{\Phi }].$$

Domain, closedness, adjoint, ellipticity and a suitable coercivity condition outside a compact set must all be proved. Full anticommutation is not inferred merely from a principal-symbol relation. An additional doubling produces

$$\mathcal{D}=\left(\begin{array}{cc}0& B^{*}\\ B& 0\end{array}\right),\Gamma =\left(\begin{array}{cc}I& 0\\ 0& -I\end{array}\right).$$

One would compute ind B, not a nonzero ordinary index for the whole self-adjoint operator. A 35-dimensional spinor bundle is not supplied with an even-dimensional chiral splitting by notation alone. No boundary class or index-one value is asserted here.

The chosen conditional matching convention is

$$g_{match}^2={\displaystyle \frac{8{\pi }^2}{r_{\perp }|indB|}}.$$

IndexMatching.couplingSq takes an integer index, and unit_index_matching proves substitution of |index|=1 and $r_{p}erp=35/3$. This is a matching calculation, not an index theorem or a derivation of its physical normalization.

8.3. Noncompact Regularization Without an Index Shortcut

For a nonnegative self-adjoint Laplace-type operator P on a specified finite-rank Hermitian bundle over $X_E$, choose smooth cutoffs $0\le {\chi }_R\le 1$, equal to one on $B_R$ and supported in $B_{2R}$. At $t>0$, define the localized heat quantity

$${Tr}_{{\chi }_R}\left(e^{-tP}\right)=Tr\left({\chi }_R{e}^{-tP}{\chi }_R\right)=\int {\chi }_R{\left(x\right)}^{2}tr{K}_t(x,x)d\mathrm{vol}\left(x\right).$$

(44)

The trace formula presupposes the heat-kernel and local trace-class hypotheses. If $tr{K}_t(x,x)\le B_t$ uniformly, it is bounded by $B_t\int {\chi }_R^2$ and is finite for each compact support. As $R\to \infty$ this need not stay finite: homogeneous free kernels on infinite-volume $X_E$ do not acquire a global trace by exhaustion. Alternatively choose a fixed nonnegative integrable weight w and use $Tr\left(w^{1/2}{e}^{-tP}{w}^{1/2}\right)\le B_t\int w$. Without a uniform diagonal bound, require the weighted diagonal itself integrable. NoncompactControl.weighted_sum_bound formalizes only a finite quadrature analogue, not this infinite-dimensional operator theorem.

The local small-t expansion [16] can be integrated against a fixed compact cutoff, retaining cutoff-dependent coefficients. Removing the cutoff or interchanging $R\to \infty$ and $t\downarrow 0$ requires uniform integrable remainders. A Dirichlet heat trace on $B_R$ is another possible scheme, but its boundary coefficients differ from (44). A confining Witten operator may have discrete spectrum under separately checked conditions; this cannot be inferred for the free operator from a regularized trace.

For the Callias route of §8.2, require outside a compact set a quadratic-form bound

$${\mathsf{\Phi }}^2\ge m^{2}I,|\langle u,i[D_0,\mathsf{\Phi }]u\rangle {|\le c\parallel u\parallel }^2,m^2-c>0.$$

(45)

Then ${\parallel Bu\parallel }^2\ge (m^2-c){\parallel u\parallel }^2$ for compactly supported exterior test sections, using $D_0^2\ge 0$. For a first-order Dirac operator, boundedness of the displayed commutator requires the suitable compatibility of the principal symbol and $\mathsf{\Phi }$; it is not assumed for an arbitrary matrix potential. exterior_coercivity proves the quadratic-form arithmetic, not domain compatibility. Elliptic local compactness, closure/adjoint control and the hypotheses of the chosen Callias theorem [17] are still required to conclude Fredholmness. Cutoff localization produces derivative-of-cutoff terms, typically controlled by $\parallel \nabla {\chi }_R{\parallel }_{\infty }^2{\parallel u\parallel }^2$ in a squared-operator estimate; localization_error records the elementary error bookkeeping, not an IMS theorem on this bundle.

A weighted or cutoff supertrace is not automatically an integer and is generally not independent of t: weights obstruct the unweighted cyclic cancellation. Hence neither (44) nor (45) computes an index-one boundary class or licenses the coupling substitution as a derived prediction. This responds to the noncompact-domain concern without replacing a Fredholm problem with finite-volume numerology.

9. Numerical and Observational Scope

9.1. Certified Geometry Versus Sampling

Sampling uses the positive Riemannian geometry $X_E$. A noisy exponential-map step is a Langevin discretization, not by itself a Hamiltonian Monte Carlo algorithm. For smooth one-dimensional restrictions,

$${\displaystyle \frac{f(x+h)-2f\left(x\right)+f(x-h)}{h^2}}=f^{\prime \prime }\left(x\right)+{\displaystyle \frac{h^2}{12}}{f}^{\left(4\right)}\left(x\right)+o\left(h^2\right).$$

For $f\left(x\right)=x^4$ the expression is exactly $12x^2+2h^2$, as proved in NumericalBounds.quartic_central_difference. Finite-step exactness for quadratic functions does not imply general unbiasedness.

A finite minimum of sampled Rayleigh quotients is an upper bound on the minimum over all directions. Finite position sampling likewise does not cover a noncompact space. A global lower certificate requires an analytic tail estimate and a compact-region covering with controlled Hessian variation. Given certified sample lower bound K, covering radius $\epsilon$ and operator-norm Lipschitz constant L, the transferable bound is $K-L\epsilon$. certified_cover_bound proves its scalar inequality, assuming the error bound rather than deriving it from random samples.

9.2. Tensor Sources and Observable Tests

For $k\ne 0$ and a symmetric spatial stress S in three dimensions,

$$P=I-{\displaystyle \frac{k{k}^T}{k^{T}k}},{\mathsf{\Pi }}_k^{TT}\left(S\right)=PSP-{\displaystyle \frac{tr\left(PS\right)}{2}}P.$$

The source defines this explicit operator and provides the transverse projector’s symmetry, idempotence, trace and annihilation of k. It supplies an explicit nonzero TT witness and the vanishing of the projected isotropic tensor cI. The concrete operator also has supplied proof bodies for zero trace, transversality, symmetry on symmetric inputs, and idempotence (GWTT.explicit_mem_TT, explicit_idempotent). These bodies are retained from the supplied baseline, which reports a prior successful build; the current source compiles under the pinned build (0 errors).

Three stage names do not determine three spectral peaks. No numerical peak frequency, amplitude or signal-to-noise ratio is predicted here. An observable GW calculation needs an anisotropic tensor source, its unequal-time correlation, propagation, redshift and detector response. The local EoS relation of §6.3 is instead a definite conditional test of that density law, with model choice and data calibration kept distinct.

9.3. Restored Redshift Diagnostic and Historical Observational Context

A frequency correction can be retained as a dimensional diagnostic. For an adiabatic radiation-era horizon source with angular scale $H_{*}$, define $f_{*}=H_{*}/\left(2\pi \right)$ and

$$f_0={\displaystyle \frac{T_0}{T_{*}}}{\left({\displaystyle \frac{g_{s0}}{g_{s*}}}\right)}^{1/3}{\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{{\pi }^2{g}_{*}}{90}}}{\displaystyle \frac{T_{*}^2}{{\overline{M}}_P}}.$$

(46)

Converting the natural-unit frequency with $\hslash =6.582119569\times {10}^{-25}$ GeV s, and choosing $T_0=2.348\times {10}^{-13}$ GeV, ${\overline{M}}_P=2.435\times {10}^{18}$ GeV, $g_{*}=g_{s*}=106.75$, $g_{s0}=3.91$, gives approximately $2.65\times {10}^{-8}(T_{*}/\mathrm{GeV})$ Hz. Thus ${10}^{13}$ GeV corresponds to about $2.65\times {10}^5$ Hz in this benchmark; 1.71 Hz corresponds to about $6.45\times {10}^7$ GeV. A bubble source can involve a different length scale, such as ${\beta }^{-1}$, so (46) does not compute its spectral peak. No three-peak or detection-significance claim is restored.

The explicit finite-dimensional TT results in §9.2 establish neither homogeneous initial tensor data nor unequal-time correlations.

Planck 2018 [18] and DESI 2024 [19] are retained as dated observational context, not a current combined-data fit. Their inferred parameters depend on model and data choices. The density-law relation $w_a=-3w_0(1+w_0)$ can be tested conditionally, but the numerical target $w_0=-0.73$ is an illustrative choice, not a derived measurement. Likewise, proton lifetime and companion neutrino masses need separate operators and source calculations and are not outputs of this manuscript.

10. Conclusions and Outstanding Constructions

The three-stage cosmological evolution scheme based on SL(6, C) proposed in this paper possesses a common effective physical action.. The Fisher derivative coupling changes the order-parameter mass on a single proper-time trajectory. Both field equations imply total continuity, internal exchange and preservation of the Friedmann constraint. This exclude the fixed information-to-expansion clock and arbitrary DE injection from the principal mechanism. Early acceleration, critical ordering and late vacuum pressure now have explicit inequalities and limiting conditions within the same background equations.

The action also exposes restrictions hidden in the proposed narrative. Healthy propagation requires $A>0$; kinetic energy need not decrease in the presence of backreaction. Continuous decay and an actual positive initial mass imply a crossing, while uniqueness requires monotonicity. The minimal matrix quartic force potential selects (3,3), so (1,2,3) requires additional dynamics. A broken scalar vacuum gives positive DE only if $U_{\infty }$ exceeds the negative condensate contribution. The Lean additions supply concrete proof bodies for these scalar identities and conditional limit/crossing statements; they do not certify tensor variation, full matrix selection, nonlinear attraction or particle production.

The remaining programme is to derive or constrain the effective action and its stochastic open-system limit, select a physical real section, establish a globally healthy solution family with seeded condensation and a populated matter interval, and calculate perturbations and observables with independently fixed parameters. The Fisher-signature obstruction, covariance-error estimates and noncompact-index limitations remain applicable. A unified effective equation set is a concrete advance over separate response prescriptions, but is not yet a first-principles derivation of three spacetime phases. The cosmological constant problem, four-dimensional Lorentzian emergence, inflationary exit and observational preference remain unresolved.

Appendix B.1. formal correspondence

Manuscript relation	Lean evidence	Scope
Stability of confinement (25)	PhaseI.perturbation_gap_lower	Directional arithmetic with a certified perturbation bound as premise
Thermal Hessian changes sign at $T_c$	ThermalBridge.massSq_positive_above, massSq_negative_below, critical_origin_hessian	Polynomial thermal Ansatz
Scalar broken minimum and stability	ThermalBridge.broken_minimum	Scalar minimum with $v^2$ specified; no matrix vacuum selection
OU-to-N clock relation (30)	ClockResponse.ou_clock_identity, tail_physical_hasDerivAt	Exact linear model and differentiable physical clock
Conserved pressure and nonphantom sign (31)	ClockResponse.continuity, enthalpy, no_phantom	Chosen density response with Q=0
Old density Ansatz as m=3 subcase	ClockResponse.density_ansatz_match, positive_tail_epsilon_sign	Algebra and sign correspondence
Late acceleration condition (32)	ClockResponse.acceleration_criterion	Stress-energy criterion; response assumed
Friedmann constraint (2)	CosmologicalBridge.friedmann_constraint_hasDerivAt, friedmann_constraint_preserved	Actual differentiable trajectories; no existence theorem
Pressure work and transition matching	CosmologicalBridge.comoving_pressure_work, transition_energy_matching	Product-rule algebra and explicit matching premise
Joint branching completeness	Branching.joint_unique, feasible_necessary, feasible_sufficient_sharp	Simultaneous normalized instantaneous constraints
Changing-reference entropy (34)	DrivenEntropy.moving_reference_balance, kl_nonincrease	Product derivative and sign condition; PDE step analytic

Appendix B.2. Completeness conditions for the three-stage emergence theorem

A complete result must exhibit one initial-data set and a common solution for the configuration dynamics, order parameter, physical clock, metric and component densities. It must prove localization, a populated stable matter sector and late acceleration at ordered times, verify matching and stability, and compute observables without reusing them as calibration inputs.