Gravitational Waves in Geometric–Information Cosmological Evolution: Transition Sources, Continuous Frequency Bands, and Inter-Epoch Transfer — Conditional Mechanisms, Energy Transfer, and Formal Proof Sources

1. Scope, Evidence Grades, and Cross-Manuscript Checks

1.1. Forward Derivation and Evidence Levels

The physical dependencies considered here follow the forward chain

$$\begin{array}{c}\mathrm{statistical}\mathrm{geometry}\mathrm{and}\mathrm{physical}\mathrm{matching}\hfill \\ \to \mathrm{common}\mathrm{action},\mathrm{background},\mathrm{and}\mathrm{initial}\mathrm{state}\\ \to \mathrm{inhomogeneous}\mathrm{stress}\to \mathrm{TT}\mathrm{response}/\mathrm{UETC}\\ \multicolumn{1}{c}{\to \mathrm{propagation}\mathrm{and}\mathrm{thermal}\mathrm{history}}\\ \multicolumn{1}{c}{\to {\mathrm{\Omega }}_{\mathrm{GW},0}\left(f\right).}\end{array}$$

(1)

Upstream parameters inferred backward from a target frequency cannot then be presented as an independent prediction.

Marker	Meaning
F	A corresponding Lean statement and proof body are supplied. R5 was initially marked source-level only; after the repairs described in §12 the full file was compiled with 0 errors and audited (28 #print axioms, standard axioms only), so F now denotes machine-checked source except where a row’s coverage note says otherwise. Earlier verification reports are separated in §12.
A	Analytical derivation in this paper or external theory, with applicability conditions stated explicitly.
M	Effective model, initial state, statistical closure, or approximation assumption.
C	Numerical scale, template, or calibration input.
O	A link that remains to be solved, constructed, or verified.

A proof of “if P then Q” does not establish that P holds in the universe. Likewise, the absence of custom axioms does not imply that the geometry uniquely determines the action, the seeds, or the thermal history.

1.2. Results and Scope

The principal results are the composite-mass interface and the exclusion of an exactly vanishing condensate on the charged branch; the zero-crossing rate and natural quenching scale derived from the background equations; explicit mode-by-mode constructions of a continuous band of TT wavenumbers; strictly positive residual wave energy after finite-duration sourcing; and unified conditions for inter-epoch propagation and frequency consistency. The results do not determine a unique peak frequency in hertz. This non-uniqueness is demonstrated by explicit constructions at distinct wavenumbers, rather than inferred solely from the absence of specified parameters.

1.3. Current-Source Dependencies

Source and location	Result used here	Required restriction
CGICE V10, §§2.3, 4.2, 4.5, 7.5	Distinct spectral operators; OU covariance with a noise floor; explicit matching boundary	Configuration relaxation does not fix a physical tensor frequency, spatial correlation length or vacuum energy
Spacetime V20, §3.9 and §7.2	Common derivative-coupled action, off-shell continuity, symmetric-branch dilution and conditional crossing	Healthy domain and actual background trajectory; zero seeds remain zero
Spacetime V20, §6.7	Conditional late vacuum and limiting equation of state	Quartic branch and orbit convergence; not global attraction
Neutrino V64, §2.1, Theorem G1	Spectral mediator susceptibility bound and no-neutral-condensation region	Specified mediator sector, physical mass scale, overlaps and controlled elimination
Neutrino V64, §7.1, O2.1; §7.3, N1	Regulated gap criterion and unique neutral instantaneous threshold	Cutoff determinant and trajectory hypotheses; not dynamical tracking
Neutrino V64, §7.4, N2–N3	Backreacted fixed-momentum minimum and charged crossover	Conserved nonzero condensate phase charge excludes neutral zero-amplitude onset

2. Mathematical Foundations: CGICE and the Three-Epoch Spacetime Phases

2.1. Configuration Space, Metric, and Dimensions

$SL(6,\mathbb{C})$ has complex dimension 35 and real dimension 70; its fundamental representation ${\mathbb{C}}^6$ has real dimension 12. The space used for positive diffusion and elliptic spectral analysis,

$$X_E=SL(6,\mathbb{C})/SU\left(6\right)$$

(2.1)

has real dimension 35. It is distinct from $SL(6,\mathbb{C})/SU(3,3)$ with its indefinite metric, from the assumed four-dimensional physical spacetime, and from the cotangent bundle of configuration space. The counting “70 = 35 + 35” does not by itself yield a spacetime dimension, positive-energy gravity, or particle production. All statistical positivity estimates in this work are confined to the Riemannian configuration space $SL(6,\mathbb{C})/SU\left(6\right)$ with the Fisher metric of [1] (§2–§3) and its three-epoch extension [2]; the four-dimensional metric and its Lorentzian signature are effective-theory inputs [M].

2.2. Distinct Spectral Quantities

For the $A_5$ root system,

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

(2.2)

Hence

$$\sum _i{\rho }_{\mathrm{alg},i}^2=35/2,\sum _i{\rho }_{\mathrm{res},i}^2=70,{\parallel {\rho }_{\mathrm{res}}\parallel }_{{(cTr)}^{-1}}^2=70/c.$$

(2.3)

With $c=2$, the restricted-root norm squared is 35, while the ordinary Weyl vector has $35/4$ in the same dual metric. When the metric is multiplied by c, the norm squared of a covector divides by c — it must not be multiplied by c. The finite sums are Geometry.rho_alg_norm_sq, restricted_threshold_35, and ordinary_norm_metric_two [F]. Identifying the restricted-root norm with the bottom of the free Laplacian spectrum requires the corresponding symmetric-space spectral theorem [A].

The weighted generator of CGICE is

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

(2.4)

If the curvature, the Hessian of the potential, the domain of the closed operator, and Gibbs integrability meet the corresponding conditions, the radial model gives

$${\delta }_L\ge a_{\mathrm{conf}}-6D>0.$$

(2.5)

For $a_{\mathrm{conf}}=35$, $D=1$, the lower bound is 29 — not 35. Lean’s bakry_emery_combine proves the algebraic combination of the curvature and Hessian lower bounds; it is not a complete spectral theorem on the manifold. The further number $35/3$ arises from an additional matching convention; it is not the necessary outcome of averaging the one-coordinate OU spectral gap over spatial dimensions.

Quantity	Unit / definition	What it can support	What it cannot imply
free spectral threshold 35	spectral quantity in the specified configuration metric	baseline spectrum of the free operator	physical masses, the Planck frequency, 450 GHz
${\delta }_L$	relaxation rate per unit of information time	controls statistical decay under the stated hypotheses	Lorentzian oscillation frequencies
$m_{\mathrm{eff}}^2$	physical mass squared	order-parameter stability and propagation	fixing the GW peak height on its own
$k/\left(2\pi a\right)$	physical cyclic frequency	tensor-mode emission/observation frequency	automatic equality with the constants above

2.3. Noise Floor and Physical Clock

The linear OU model obeys

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

(2.6)

For $D>0$ the full covariance does not decay to zero. If $\tau =\tau \left(t\right)$, then

$${\displaystyle \frac{d}{dt}}(C-D/\ell )=-2\ell \dot{\tau }\left(t\right)(C-D/\ell ).$$

(2.7)

Equations (2.6) and (2.7) are formalized by ouCov_hasDerivAt and ou_clock_chain [F]. They describe no sinusoidal oscillation. Even for $\ell =35$, different values of $\dot{\tau }$ yield different physical decay rates.

The single-point covariance C does not determine a spatial correlation length. The same variance may describe either a spatially homogeneous random field, which has no classical TT gradient source, or a field with spatial gradients, which may source one. The required object is the two-point covariance $C(\mathbf{x},\mathbf{y};t,s)$, rather than $C\left(\tau \right)$ alone. The pullback of the positive Fisher bilinear form of the three-epoch model [2] remains nonnegative; a Lorentzian signature cannot be obtained by pullback alone. The statistics-to-physics response map remains [O].

3. Derivative-Coupled Action and Homogeneous Background

3.1. Conventions and Effective Action

We take the $(-+++)$ signature, $c=\hslash =1$, and $M^2={\overline{M}}_P^2={\left(8\pi G\right)}^{-1}$. To avoid conflating a mass with a mass squared, we write

$$b:=m_0^2<0,X=-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\vartheta {\partial }_{\nu }\vartheta ,Y=-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\phi {\partial }_{\nu }\phi .$$

(3.1)

The derivative-coupled effective action of the three-epoch spacetime-phase model [2] (its §3.9, Equations (31)–(42); selected algebraic consequences are transcribed in the companion file) is

$$S=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{M^2}{2}}R+A\left(\phi \right)X+Y-U\left(\vartheta \right)-W\left(\phi \right)\right],$$

(3.2)

$$A\left(\phi \right)=Z-{\displaystyle \frac{\gamma }{2}}{\phi }^2,W\left(\phi \right)={\displaystyle \frac{b}{2}}{\phi }^2+{\displaystyle \frac{\lambda }{4}}{\phi }^4,Z>0,\gamma >0,\lambda >0.$$

(3.3)

We use the single-parameter Fisher-coordinate convention: $\left[\vartheta \right]=0$, $\left[\phi \right]=1$, $\left[X\right]=\left[Z\right]=\left[b\right]=2$, $\left[\gamma \right]=\left[\lambda \right]=0$, where square brackets denote mass dimension. If one locally switches to ${\theta }_c=F\vartheta$, then $X^{\mathrm{can}}=F^{2}X$, $Z_c=Z/F^2$, ${\gamma }_c=\gamma /F^2$, leaving $AX$ and $\gamma X$ invariant; F is a mass scale. The canonical real-scalar convention of R1 can be converted this way, but the units of the two parameter sets must not be mixed. The kinetic term is healthy where $A>0$; this condition must be checked over the regions reached by the background and its perturbations. M is a constant in this work; if the gravitational coupling is also made a Fisher variable, the tensor action and propagation must be redone, and the equations below do not carry over.

3.2. Background Equations, Effective Mass, and Energy Transfer

On a spatially flat FLRW background, let $u=\dot{\vartheta }$, $v=\dot{\phi }$, and $X=u^2/2$. Variation of the action gives

$$A\dot{u}+3HAu-\gamma \phi vu+U_{\vartheta }=0,$$

(3.4)

$$\dot{v}+3Hv+(b+\gamma X)\phi +\lambda {\phi }^3=0.$$

(3.5)

The pressure and energy density must be obtained from (3.2):

$$\rho =AX+\frac{1}{2}{v}^2+U+W,p=AX+\frac{1}{2}{v}^2-U-W.$$

(3.6)

The X-dependent force term must not be counted again in $\rho$ as an additional positive potential energy. The two field equations imply

$$\dot{\rho }+3H(\rho +p)=0.$$

(3.7)

UnifiedDynamics.continuity_identity expresses the off-shell continuity residual as a linear combination of the field-equation residuals; density_hasDerivAt and unified_continuity provide the corresponding time-derivative result [F]. Thus total continuity follows from the field equations rather than being imposed independently.

The exchange between the components is

$${\dot{\rho }}_{\vartheta }+3H({\rho }_{\vartheta }+p_{\vartheta })=Q,{\dot{\rho }}_{\phi }+3H({\rho }_{\phi }+p_{\phi })=-Q,Q=\gamma \phi vX.$$

(3.8)

Here Q describes energy transfer between the two background fields. It cannot be identified with $Q_{\mathrm{GW}}$ or a dark-matter production rate without a separate calculation.

3.3. Mass Sign Change and an Explicit Background Branch

Define $m_{\mathrm{eff}}^2=b+\gamma X$ and $X_c=-b/\gamma >0$. If X is continuous, $X\left(t\right)\to 0$, and $b+\gamma X\left(0\right)>0$, then there exists $t_c>0$ with $m_{\mathrm{eff}}^2\left(t_c\right)=0$. If X is strictly decreasing for $t\ge 0$, the crossing is unique [F: critical_crossing, critical_crossing_unique, critical_sign_order].

The evolution of X must include backreaction, as shown by

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

(3.9)

On the symmetric branch, where $\phi =v=0$, $U_{\vartheta }=0$, and $Z>0$, the solution is

$$u=u_i{(a_i/a)}^3,X=X_i{(a_i/a)}^6,{\displaystyle \frac{a_c}{a_i}}={\left({\displaystyle \frac{\gamma X_i}{-b}}\right)}^{1/6}.$$

(3.10)

The crossing follows if the Friedmann solution continues to expand beyond $a_c$. For example, a positive constant U together with positive kinetic energy gives a persistent-expansion branch that can be analyzed explicitly. The critical scale factor here depends on $b,\gamma ,X_i$, not on 35 alone. The integration of (3.10) and the global Friedmann analysis are [A]; what the source formalizes is the symmetric-branch kinetic-rate identity.

The quartic potential undergoes a continuous bifurcation when the curvature at the origin changes sign. Because no thermal barrier or metastable false vacuum has been established, first-order bubble-nucleation formulae do not apply here. Sound-wave, turbulence, and cosmic-string sources each require their own additional physics.

3.4. Composite Branch and the Origin-Mass Interface

The composite branch of the neutrino-condensation model [8] (its §7.3 composite action) takes $\phi =\sqrt{Z_m}m$ with $Z_m>0$,

$$A_m=Z-\eta m^2/2,V_X^{\prime }\left(m\right)=m[G_0^{-1}+\eta X-\kappa I_{\mathrm{\Lambda }}\left(m^2\right)],C=\kappa {\mathrm{\Lambda }}^2/\left(16{\pi }^2\right).$$

(3.11)

The canonical radial mass squared at the neutral origin is therefore

$$\begin{array}{|c|}\hline m_{N,\mathrm{eff}}^2\left(X\right)={\displaystyle \frac{G_0^{-1}-C+\eta X}{Z_m}}=b_N+{\gamma }_{N}X,b_N={\displaystyle \frac{G_0^{-1}-C}{Z_m}},{\gamma }_N={\displaystyle \frac{\eta }{Z_m}}.\\ \hline\end{array}$$

(3.12)

Only $C>G_0^{-1}$ and $\eta >0$ give a positive critical value $X_{c,N}=(C-G_0^{-1})/\eta$. EpochFrequency. composite_mass_matching and composite_threshold_zero prove the normalization and threshold algebra [F]. Away from zero fields the full determinant is generally not an exact quartic potential; this work shares the instability structure only at the specified linearized origin interface and does not replace the gap theorem of [8] (its §7.1, Proposition O2.1, with the formalization boundaries specified there) by a quartic-potential theorem.

If a conserved, nonzero homogeneous condensate phase charge $Q_{\chi }$ is present, the fixed-charge energy contains

$${\rho }_Q={\displaystyle \frac{Q_{\chi }^2}{2a^6{\phi }^2}}.$$

(3.13)

For finite $a>0$, if $0<{\rho }_Q\le E_{max}$, then

$${\phi }^2\ge {\displaystyle \frac{Q_{\chi }^2}{2a^6{E}_{max}}}>0.$$

(3.14)

charge_energy_lower_bound turns the budget into an amplitude lower bound, and charged_ origin_excluded excludes the simultaneous occurrence of zero amplitude and nonzero charge directly from $Q_{\chi }=a^3{\phi }^2\dot{\chi }$. The fixed-charge energy diverges as $\phi \to 0$. Lean treats division as a total operation, so a zero denominator is not excluded by the logic of division alone; this convention does not remove the physical singularity.

In the presence of fermion currents one must use the total current and exchange equations of the condensation model [8] (its §§4.2–5 charge bookkeeping); the total charge cannot be assigned entirely to the condensate phase. The nonzero-charge fixed-momentum branch of [8] (its §7.4, Theorem N3) smooths the neutral threshold; its instantaneous minimum does not guarantee that real orbits track it. Equation (4.2) and the quenching example below apply only to the neutral symmetric branch; for the charged branch one must derive the coupled quadratic action and stress of the radial, phase, and Fisher perturbations, and a wave frequency cannot be inferred directly from “the time charge is nonzero.” ### 3.5 From composite microscopic inputs to a conditional radiation band

Let $B_N=C-G_0^{-1}>0$, with $G_0>0$, $Z_m>0$ and $\eta >0$. In the constant-coefficient matching of V64, on the neutral symmetric branch before substantial backreaction,

$$\begin{array}{c}{X}_{c,N}={\displaystyle \frac{B_N}{\eta }},{\mu }_N^2\left(t\right)=-m_{N,\mathrm{eff}}^2\left(t\right)={\displaystyle \frac{B_N-\eta X\left(t\right)}{Z_m}},\\ {\alpha }_N=-{\dot{m}}_{N,\mathrm{eff}}^2\left(t_c\right)={\displaystyle \frac{6H_c{B}_N}{Z_m}}.\end{array}$$

(3.15)

This gives the microscopic-to-instability interface, with the independently determined background $H_c$ still required. At fixed $H_c,B_N,Z_m$, $\eta$ cancels from the slope because $\eta X_c=B_N$; it changes the threshold, crossing time and generally the self-consistently evaluated $H_c$. Thus $G_0$ and $\eta$ alone do not fix the slope. If matching coefficients vary with time, the origin-crossing derivative is instead

$${\dot{m}}_{N,\mathrm{eff}}^2\left(t_c\right)={\displaystyle \frac{{\partial }_t\left(G_0^{-1}\right)-\dot{C}+\dot{\eta }{X}_c+\eta {\dot{X}}_c}{Z_m}}.$$

(3.16)

The normalization derivative drops out at the crossing because the numerator vanishes. Away from the crossing it contributes $-m_{N,\mathrm{eff}}^2{\dot{Z}}_m/Z_m$. A running normalization also changes canonical perturbation dynamics; the constant-$Z_m$ mode equation cannot then be reused without checking those terms.

At an emission slice $t_{*}>t_c$, truncate the parent ensemble to $\left|\mathbf{P}\right|,\left|\mathbf{Q}\right|<{\mu }_N\left(t_{*}\right)$, with nonzero amplitudes and the non-collinearity of §5.3. The triangle inequality and propagation law give

$$\begin{array}{c}0<K=|\mathbf{P}+\mathbf{Q}|<2\sqrt{{\displaystyle \frac{C-G_0^{-1}-\eta X_{*}}{Z_m}}},\\ \begin{array}{|c|}\hline 0<f_0<\frac{R_{*}}{\pi \hslash }\sqrt{\frac{C-G_0^{-1}-\eta X_{*}}{Z_m}}\\ \hline\end{array},R_{*}=\frac{a_{*}}{a_0}>0.\end{array}$$

(3.17)

This is an instantaneous support bound, not a spectral-peak prediction or an absolute cutoff of vacuum, thermal, charged-branch or nonlinear emission. It is usable only inside the derivative expansion and healthy domain. EpochFrequency.composite_band_upper_bound composes canonical mass normalization, the vector norm bound and the redshift theorem [F]. composite_critical_mass_derivative calls the field-equation derivative result instead of inserting the slope as a premise.

For $\tau =t-t_c>0$, suppose an independently checked Taylor bound is

$$|m_{N,\mathrm{eff}}^2\left(t\right)+{\alpha }_N\tau |\le ϵ{\alpha }_N\tau ,0\le ϵ<1.$$

(3.18)

Then $(1-ϵ){\alpha }_N\tau \le {\mu }_N^2\le (1+ϵ){\alpha }_N\tau$. The leading local band edge is $2\sqrt{{\alpha }_N\tau }$, whereas ${\alpha }_N^{1/3}$ is the distinct quench scale. A sufficient analytic error estimate is $|{\ddot{m}}_{N,\mathrm{eff}}^2|\le L_m$ throughout the window: the relative remainder is at most $L_m\tau /\left(2{\alpha }_N\right)$. Establishing that estimate, the window length and saturation on a cosmological solution remains necessary. quench_remainder_upper gives the upper-bound algebra [F]. For a time-extended source the appropriate comoving envelope is $k<2{sup}_{t\in \mathcal{I}}a\left(t\right){\mu }_N\left(t\right)$, not (3.17) at an arbitrary single time.

The geometric restriction can exclude this channel. Under V64’s additional pure-mediator bridge, let ${\delta }_{\mathrm{med}}>0$ bound the chosen dimensionless mediator operator and let $\mathcal{M}$ convert it to mass. Then

$$G_0={\displaystyle \frac{y^2}{{\mathcal{M}}^2}}\sum _j{\displaystyle \frac{w_j^2}{{\lambda }_j}}\le {\displaystyle \frac{y^2{\sum }_j{w}_j^2}{{\mathcal{M}}^2{\delta }_{\mathrm{med}}}}.$$

(3.19)

If this upper bound is at most $1/C$, then $G_0^{-1}\ge C$ and $m_{N,\mathrm{eff}}^2\left(X\right)\ge 0$ for $X\ge 0$. The neutral spinodal channel is excluded (neutral_no_spinodal_of_inverse_bound [F]); other sources are not. A free-operator threshold 35 or a valid weighted-gap lower bound 29 must match the chosen operator, domain and physical normalization. The scale-separation condition ${\mathrm{\Lambda }}^2/\left({\mathcal{M}}^2{\delta }_{\mathrm{med}}\right)\ll 1$ and collective-coupling control of V64 §2.1 remain independent requirements. This supplies a forward geometric constraint without selecting a target frequency.

4. From Background Instability to Spatial Perturbations

4.1. Linear Perturbations on the Symmetric Branch

Write $\phi =ϵ\psi$ and $X=\overline{X}+ϵ\delta X+O\left({ϵ}^2\right)$. The force term expands exactly as

$$(b+\gamma X)\phi +\lambda {\phi }^3=ϵ(b+\gamma \overline{X})\psi +{ϵ}^2\gamma \delta X\psi +{ϵ}^3\lambda {\psi }^3.$$

(4.1)

At the same time $A\left(ϵ\psi \right)=Z-\gamma {ϵ}^2{\psi }^2/2$: there is no first-order correction. Because the symmetric background has $\overline{\phi }=\dot{\overline{\phi }}=0$, this field provides no source for the metric scalar constraint at leading order either. On this precisely specified background branch its independent linear modes obey

$${\ddot{\psi }}_{\mathbf{p}}+3H{\dot{\psi }}_{\mathbf{p}}+\left({\displaystyle \frac{p^2}{a^2}}+b+\gamma \overline{X}\right){\psi }_{\mathbf{p}}=0.$$

(4.2)

The expansions in (4.1) are formalized by force_expansion and kinetic_coefficient_expansion [F]. The covariant variation and Fourier decomposition are analytical results [A]. Away from this branch, the multi-field kinetic geometry, inter-field mixing, and metric constraints require a separate analysis. Equation (4.2) is not the general perturbation equation for an arbitrary background.

4.2. The Instability Band and the Growth Rate

If $m_{\mathrm{eff}}^2=-{\mu }^2<0$, then

$${\displaystyle \frac{p^2}{a^2}}<{\mu }^2$$

(4.3)

is the instability (spinodal) band in the frozen-coefficient approximation. Let $B={\mu }^2-p^2/a^2>0$; the growing root is

$$r_{+}\left(p\right)={\displaystyle \frac{-3H+\sqrt{9H^2+4B}}{2}}>0,r_{+}^2+3H{r}_{+}-B=0.$$

(4.4)

growth_rate_positive and growth_rate_characteristic prove its sign and the characteristic equation [F]. The frozen-coefficient approximation requires H, a, and $\mu$ to vary little over the interval considered. Using the slow-growth estimate $exp\int r_{+}dt$ additionally requires control of adiabatic errors.

An instability does not generate nonzero classical initial data. If the relevant modes and their velocities initially vanish, the homogeneous linear equation preserves the zero solution. Quantum vacuum fluctuations, thermal fluctuations, or CGICE coarse-grained noise can provide seeds, but each requires an initial state, a temperature, or a noise mapping. The zero_order_parameter_is_invariant theorem of the three-epoch formalization [2] (its §3.9.5; cited by declaration name rather than a shifting line number) pinpoints exactly this gap. Once $\lambda \langle {\psi }^2\rangle \sim {\mu }^2$, the linear approximation and the unperturbed background fail, and backreaction must be included.

4.3. Zero-Crossing Rate and Quenching Scale

On the neutral symmetric branch of (3.10), the critical time is marked by $N=ln(a/a_c)=0$. From $\gamma X_c=-b$,

$$m_{\mathrm{eff}}^2\left(N\right)=b(1-e^{-6N}),{\mu }^2\left(N\right)=\left|b\right|(1-e^{-6N})(N>0).$$

(4.5)

For differentiable X, $d{m}_{\mathrm{eff}}^2/dt=\gamma \dot{X}$. Using ${\dot{X}}_c=-6H_c{X}_c$, which follows from the background field equation,

$$\begin{array}{|c|}\hline \alpha :=-{\dot{m}}_{\mathrm{eff}}^2\left(t_c\right)=-\gamma {\dot{X}}_c=6H_c\left|b\right|>0.\\ \hline\end{array}$$

(4.6)

mass_hasDerivAt, critical_slope, and efold_mass_identity prove the corresponding derivative/ algebra; critical_mass_derivative_from_field_equation further invokes the symmetric-branch dilution identity $Z\dot{u}+3HZu=0$ of [2] (proved as UnifiedDynamics.symmetric_branch_dilution in the companion file) and differentiates the actual function $u{\left(t\right)}^2/2$, obtaining the mass derivative $6H_{c}b$ [F]. With feedback, $U_{\vartheta }\ne 0$, or nonzero charge, the slope is instead determined by the actual value of $-\gamma {\dot{X}}_c$; the right-hand side of (4.6) is not general. The composite neutral branch correspondingly gives ${\alpha }_N=6H_c(C-G_0^{-1})/Z_m$.

Near $t_c$, set $\tau =t-t_c$. If the Taylor remainder of the mass is controlled and a and H can be frozen over a short window, then $m_{\mathrm{eff}}^2\simeq -\alpha \tau$, and the physical parent-mode wavenumber P obeys

$$\ddot{\psi }+3H_c\dot{\psi }+(P^2-\alpha \tau )\psi =0.$$

(4.7)

With $u=e^{3H_c\tau /2}\psi$ and $z=(\alpha \tau +9H_c^2/4-P^2)/{\alpha }^{2/3}$, this becomes

$$u_{zz}-zu=0,\psi =e^{-3H_c\tau /2}[c_{1}Ai\left(z\right)+c_{2}Bi\left(z\right)].$$

(4.8)

damping_removal_identity proves the pointwise algebra of the exponential de-damping. The Airy solution, its special-function representation, and the associated error control are analytical or open results [A/O]; they are not formalized in Lean. The nondimensionalization of (4.7) yields the natural momentum ${\alpha }^{1/3}$, time ${\alpha }^{-1/3}$, and the independent damping parameter $H_c/{\alpha }^{1/3}$. A rescaled variable $z>0$ does not mean the original field has grown rapidly; the physical field still requires $P^2<\alpha \tau$, a nonzero seed projection, and sufficient duration.

A natural scale does not determine a spectral peak.${\alpha }^{1/3}$, the instantaneous band edge $\mu \left(t\right)$, the inverse correlation length, and the inverse source lifetime are distinct quantities. Writing $K_{\mathrm{pk}}=\xi {\alpha }^{1/3}$ merely renders the unknown peak position dimensionless; $\xi$ must be computed from the seeds, saturation, the TT convolution, and propagation, and cannot be set to 1 and advertised as a geometric prediction. If the time required for saturation violates $H_c\Delta t\ll 1$, the variable-coefficient equation (4.2) and the nonlinear system must be solved.

5. Gravitational-wave Sources: TT Projection and Non-Collinear Modes

5.1. Stress Tensor from the Effective Action

From (3.2),

$$T_{\mu \nu }=A{\partial }_{\mu }\vartheta {\partial }_{\nu }\vartheta +{\partial }_{\mu }\phi {\partial }_{\nu }\phi +g_{\mu \nu }{\mathcal{L}}_{\mathrm{scalar}}.$$

(5.1)

In the source approximation of a near-FLRW geometry, small tensor amplitudes, and possibly nonlinear scalars, the anisotropic source of the orthogonal spatial components is

$${\Pi }_{ij}^{\mathrm{TT}}={\displaystyle \frac{1}{a^2}}{\mathrm{\Lambda }}_{ij,lm}\left(\mathbf{k}\right){\left[A{\partial }_l\vartheta {\partial }_m\vartheta +{\partial }_l\phi {\partial }_m\phi \right]}_{\mathbf{k}}.$$

(5.2)

All pressure terms proportional to ${\delta }_{ij}$ are removed by the TT projection. This expression suits field-gradient radiation under weak metric backreaction [A,M]; the full cosmological second-order scalar-induced GW channel also contains the second-order scalar metric terms of the Einstein equations and their gauge treatment, which (5.2) does not replace [O]. Existing computational methods for inhomogeneous preheating sources are reviewed in Refs. [4,5].

5.2. Transverse-traceless Projection

For a nonzero $\mathbf{k}$, let $n=\mathbf{k}/\left|\mathbf{k}\right|$ and

$$P_{ij}={\delta }_{ij}-n_i{n}_j,\mathcal{T}\left(S\right)=PSP-\frac{1}{2}tr\left(PS\right)P.$$

(5.3)

For a symmetric tensor S,

$$P^2=P,Pn=0,trP=2,$$

(5.4)

$$tr\mathcal{T}\left(S\right)=0,\mathcal{T}\left(S\right)n=0,{\mathcal{T}}^2\left(S\right)=\mathcal{T}\left(S\right),\mathcal{T}\left(cI\right)=0.$$

(5.5)

The matrix P is a transverse vector projector, not a traceless tensor; the identity $trP=2$ does not establish tracelessness. A transverse-vector-projector inequality such as $P\ne I$ would not establish vanishing of an isotropic TT source; the present declaration proves the actual tensor projection to be zero. Here we use the rank-4 matrix TT operators of the three-epoch formalization [2] (its GWTT module) and the results GWTT.explicit_isotropic_zero, explicit_mem_TT, and explicit_idempotent [F].

5.3. Two Non-Collinear Scalar Modes Source a Nonzero TT Stress

Take the comoving wave vectors

$$\mathbf{p}=(d,0,c),\mathbf{q}=(-d,0,c),\mathbf{k}=\mathbf{p}+\mathbf{q}=(0,0,2c),dc\ne 0,$$

(5.6)

and

$$\phi (\mathbf{x},t)=F\left(t\right)cos(\mathbf{p}·\mathbf{x})+G\left(t\right)cos(\mathbf{q}·\mathbf{x}).$$

(5.7)

Using $sinusinv=[cos(u-v)-cos(u+v\left)\right]/2$, the coefficient of the gradient product on $cos(\mathbf{k}·\mathbf{x})$ is

$$S^{\left(k\right)}=-{\displaystyle \frac{FG}{2}}(\mathbf{p}{\mathbf{q}}^T+\mathbf{q}{\mathbf{p}}^T)=FGdiag(d^2,0,-c^2).$$

(5.8)

The TT projection along the z-axis is

$$\begin{array}{|c|}\hline {\mathrm{\Pi }}^{\mathrm{TT},\left(\mathrm{k}\right)}={\displaystyle \frac{FG{d}^2}{2a^2}}diag(1,-1,0).\\ \hline\end{array}$$

(5.9)

The source is therefore nonzero for $FG\ne 0$ and $d\ne 0$. cross_source_matrix and two_mode_tt_source give the matrix identity for arbitrary real amplitudes; two_mode_tt_nonzero gives nonvanishing for positive amplitudes [F]. This construction uses the unit vector along z. Its interpretation as a propagating mode requires $c\ne 0$; a homogeneous anisotropy at zero wavenumber is not a propagating gravitational wave.

If $(d^2+c^2)/a^2<{\mu }^2$, both parent modes lie in the instability band (4.3). In the short-time frozen background, taking the growing branches $F=F_0{e}^{r_{+}t}$, $G=G_0{e}^{r_{+}t}$, the source amplitude grows as $e^{2r_{+}t}$. One nonempty parameter example, in suitable units, is $a=1$, $\mu =2$, $d=c=1$, where the parent squared momentum is 2 < 4. This example applies before nonlinear saturation.

5.4. Three Null-Source Results and Their Scope

1.

For two completely homogeneous background fields the spatial gradients vanish and the classical matter TT source is zero.

2.

For a single scalar plane wave, the second-harmonic term points along the original wave vector — a longitudinal stress — and its TT part vanishes; collinear configurations are the same.

3.

A static isotropic vacuum $T_{\mu \nu }=-\mathrm{\Lambda }{g}_{\mu \nu }$ carries no independent nonzero TT excitation source. The background term is handled consistently with the background-geometry terms of the linear tensor equation; $-\mathrm{\Lambda }\delta g_{ij}$ must not be treated as an externally applied radiation source.

What is excluded here is the stated classical matter source. An inflationary background can still amplify quantum tensor vacuum modes and can propagate pre-existing tensor initial data; that is a separate mechanism requiring a specified quantum state and background pumping.

5.5. Unstable Parent Modes Generate a Continuous Tensor Band

On a frozen emission time slice, use the physical wave vectors $\mathbf{P}=\mathbf{p}/a_{*}$, $\mathbf{Q}=\mathbf{q}/a_{*}$, and $\mathbf{K}=\mathbf{P}+\mathbf{Q}$. If the source truncates to unstable parent modes with $\left|\mathbf{P}\right|,\left|\mathbf{Q}\right|<{\mu }_{*}$, the triangle inequality gives

$$0<K<2{\mu }_{*}\mathrm{for}\mathrm{any}\mathrm{nonzero}\mathrm{propagating}\mathrm{mode}.$$

(5.10)

convolution_band_bound proves the norm bound for the vector sum. This band limit applies only to the components generated by the truncated unstable parent modes. Realistic initial states may contain high-frequency vacuum or thermal tails, and nonlinear scattering may populate higher wavenumbers. Thus (5.10) is not an absolute cutoff of the full cosmological spectrum.

For any ${\mu }_{*}>0$ and $0<K<2{\mu }_{*}$, take

$$c=K/2,d=\sqrt{({\mu }_{*}^2-c^2)/2}>0,\mathbf{P}=(d,0,c),\mathbf{Q}=(-d,0,c).$$

(5.11)

Then both parent squared norms equal $({\mu }_{*}^2+c^2)/2<{\mu }_{*}^2$ while $|\mathbf{P}+\mathbf{Q}|=K$, and the unit-amplitude TT source of (5.9) is nonzero. any_band_mode_has_source constructs d and c explicitly and invokes the two-mode TT result; the construction therefore establishes a field source rather than assuming one.

The non-uniqueness can even keep the parent norms and growth rates identical: take $\mu =2$ with configuration 1 given by $d=c=1$ and configuration 2 by $d=\sqrt{7}/2$, $c=1/2$. Both have $P^2=2<4$ and the same frozen growth root, yet they yield nonzero TT sources at $K=2$ and $K=1$ respectively. equal_parent_norm_different_tensor_modes supplies this witness. These are local field configurations, not fits to a statistically isotropic universe. A statistical model additionally requires distributions of directions, phases, and amplitudes.

Unit amplitudes simplify the matrix proofs and do not imply unit-sized physical fluctuations. Multiplying both parent modes by a common $\epsilon >0$ scales the TT source and the forced response by ${\epsilon }^2$ and the wave energy by ${\epsilon }^4$, leaving nonvanishing intact. Within any finite frozen window the seed can be chosen small enough that $2\gamma {\epsilon }^2{e}^{2r_{+}T}<Z$ and $4\lambda {\epsilon }^2{e}^{2r_{+}T}\ll {\mu }^2$, keeping the healthy kinetic regime and the linearization error of the example under control; the background backreaction must be correspondingly small. This choice controls the approximation locally; it does not establish a global orbit through saturation or across all three epochs.

The endpoints are physically meaningful as well: $K=0$ is not a propagating frequency, and for equal-norm parents the maximal K occurs in the parallel configuration, where the transverse stress vanishes. For an angle $\theta$, $K=2Pcos(\theta /2)$; in the two-mode example $d^2=P^2-K^2/4$, and the TT coefficient tends to zero at the collinear endpoint. The fastest-growing parent modes typically lie near $P=0$. Because the homogeneous mode has no gradient source, the fastest growth does not necessarily produce the largest GW peak.

If the whole source window has comoving support $p<p_{max}$, the convolution support obeys $k<2p_{max}$; if only a per-slice instantaneous physical bound is available, use $k<2{sup}_t\left[a\left(t\right)\mu \left(t\right)\right]$. A long-lived source has no single emission scale factor, and one must return to the unequal-time integrals.

6. Tensor Response and Propagation

6.1. The Tensor Equation and Canonical Normalization

For

$$d{s}^2=a^2\left(\eta \right)[-d{\eta }^2+({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j],{\partial }_i{h}_{ij}=h_{ii}=0$$

(6.1)

the Einstein–Hilbert quadratic tensor action is

$$S_h^{\left(2\right)}={\displaystyle \frac{M^2}{8}}\int d\eta d^{3}x{a}^2[{\left(h_{ij}^{\prime }\right)}^2-{\left({\partial }_l{h}_{ij}\right)}^2].$$

(6.2)

For the defined physical transverse stress ${\mathrm{\Pi }}_{ij}$, the tensor equation is

$$h_{ij}^{″}+2\mathcal{H}{h}_{ij}^{\prime }+k^2{h}_{ij}={\displaystyle \frac{2a^2}{M^2}}{\mathrm{\Pi }}_{ij}^{\mathrm{TT}},\mathcal{H}=a^{\prime }/a.$$

(6.3)

The variable ${\mu }_{ij}=a{h}_{ij}$ obeys

$${\mu }_{ij}^{″}+\left(k^2-{\displaystyle \frac{a^{\prime \prime }}{a}}\right){\mu }_{ij}={\displaystyle \frac{2a^3}{M^2}}{\mathrm{\Pi }}_{ij}^{\mathrm{TT}}.$$

(6.4)

canonical_transform proves the pointwise algebraic identity underlying the cancellation of terms [F]. The action variation, gauge decomposition, and existence and regularity of the Green operator remain analytical or open results [A/O].

Given a retarded Green function $G_k$ and zero initial tensor data,

$$h_{ij}(\eta ,\mathbf{k})={\displaystyle \frac{2}{M^{2}a\left(\eta \right)}}{\int }_{{\eta }_i}^{\eta }ds{a}^3\left(s\right)G_k(\eta ,s){\mathrm{\Pi }}_{ij}^{\mathrm{TT}}(s,\mathbf{k}).$$

(6.5)

For radiation domination with $a^{″}=0$, $G_k(\eta ,s)=\Theta (\eta -s)sin\left[k(\eta -s)\right]/k$. A general thermal history requires a separate transfer function; the statistical “mixing time” is not the propagation time of this Green function.

6.2. An Exactly Solvable Sourced-Wave Example

Over a short interval with $H\Delta t\ll 1$ and negligible background mass variation, take $a\simeq a_{*}$ and the physical tensor wavenumber $K=2\left|c\right|/a_{*}>0$. For the positive polarization of (5.9),

$$\ddot{h}+K^{2}h=S_0{e}^{st},s=2r_{+}>0,S_0={\displaystyle \frac{F_0{G}_0{d}^2}{M^2{a}_{*}^2}},h\left(0\right)=\dot{h}\left(0\right)=0.$$

(6.6)

The exact solution is

$$\begin{array}{|c|}\hline h\left(t\right)={\displaystyle \frac{S_0}{s^2+K^2}}\left[e^{st}-cos\left(Kt\right)-{\displaystyle \frac{s}{K}}sin\left(Kt\right)\right].\\ \hline\end{array}$$

(6.7)

Its derivative is

$$\dot{h}={\displaystyle \frac{S_0}{s^2+K^2}}[s{e}^{st}+Ksin\left(Kt\right)-scos\left(Kt\right)].$$

(6.8)

Direct differentiation gives $\ddot{h}\left(0\right)=S_0\ne 0$. Thus, despite the zero initial tensor data, the response is nonzero. Lean’s forcedH_hasDerivAt, forcedV_hasDerivAt, forced_wave_equation, forced_zero_initial_data, and forced_initial_acceleration provide the proof bodies [F].

sourced_instability_witness further combines healthy kinetics, negative effective mass, two unstable parent modes, a nonzero gradient TT source, and a forced initial acceleration into a single nonempty parameter witness [F]: in the chosen units $Z=3$, $\gamma =\lambda =1$, $b=-5$, $X=1$, $d=c=a_{*}=M=1$, with parent amplitudes $1/100$ each. This is a witness for instantaneous parameters within a short-time approximation, not a solution of the full nonlinear cosmological system.

If the source switches off around T, the subsequent solution is a free wave determined by $h\left(T\right)$ and $\dot{h}\left(T\right)$. The pre-switch-off growing profile is not “a gravitational wave of frequency s”; the free wave has angular frequency K and cyclic frequency $K/\left(2\pi \right)$. The tensor energy

$$E_h=\frac{1}{2}({\dot{h}}^2+K^2{h}^2),{\dot{E}}_h=S\dot{h}$$

(6.9)

is wave_energy_hasDerivAt [F]. The source power can have either sign instantaneously, so the wave energy need not increase monotonically. This example establishes that the effective model permits a tensor response. It does not compute a normalized stochastic spectrum or imply that every switch-off protocol leaves the same residual energy.

6.3. Residual Wave Energy after Finite-Duration Sourcing

Within the constant-coefficient short-time approximation of §6.2, let the source be $S{e}^{st}$ for $0\le t\le T$ and zero afterwards, with $S\ne 0$, $s>0$, $K>0$, $T>0$. Matching h and $\dot{h}$ continuously (the acceleration may jump at switch-off, but no impulse is applied), the exact solution with zero initial data gives

$$\begin{array}{|c|}\hline E_{\mathrm{out}}:={\displaystyle \frac{\dot{h}{\left(T\right)}^2+K^{2}h{\left(T\right)}^2}{2}}={\displaystyle \frac{S^2}{2(s^2+K^2)}}[e^{2sT}+1-2e^{sT}cos\left(KT\right)]>0.\\ \hline\end{array}$$

(6.10)

The bracket equals ${(e^{sT}-1)}^2+2e^{sT}(1-cosKT)\ge {(e^{sT}-1)}^2>0$. The subsequent free solution is

$$h\left(t\right)=h\left(T\right)cosK(t-T)+{\displaystyle \frac{\dot{h}\left(T\right)}{K}}sinK(t-T).$$

(6.11)

The free-wave frequency is $K/\left(2\pi \right)$, not the exponential growth rate s. forced_energy_identity and finite_source_leaves_positive_energy compute the actual single-mode wave energy using the R1 forcedH, forcedV; response_numerator_identity proves the associated squared identity.

The result shows that this specified finite-duration source leaves a free wave. It does not extend to arbitrary switching functions, for which a time-dependent source can produce destructive interference. $E_{\mathrm{out}}$ is the energy of a single real-mode oscillator; the physical GW density further requires polarizations, $M^2$, volume/stochastic-spectrum factors, and short-wave averaging. The abrupt switch-off provides an analytically tractable example; realistic smooth termination and backreaction require separate calculations. If $K\lesssim H$, this local oscillator approximation does not suffice to define the GW energy at emission, and the FLRW Green equation must be used to propagate to subhorizon scales.

The theorem seeded_band_mode_leaves_wave combines the TT source of §5.5, the positive growth root, and the residual wave-energy result into a single statement. It provides a local mechanism witness in specified units and a frozen background; it does not claim a global FLRW solution or a statistically normalized spectrum, and interpreting it as local short-wave radiation after source switch-off still requires checking the applicability scale of K relative to H. ## 7. Unequal-time correlators and the gravitational-wave spectrum

6.4. Correlators Required for the Spectrum

We use the Fourier convention $f\left(\mathbf{x}\right)=\int d^{3}k/{\left(2\pi \right)}^{3}f\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}}$. Define the contracted TT-stress unequal-time correlator (UETC)

$$\langle {\mathrm{\Pi }}_{ij}^{\mathrm{TT}}(\mathbf{k},s){\mathrm{\Pi }}_{ij}^{\mathrm{TT}*}({\mathbf{k}}^{\prime },s^{\prime })\rangle ={\left(2\pi \right)}^3{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime })C_{\mathrm{\Pi }}(k;s,s^{\prime }),$$

(7.1)

where repeated indices $ij$ are summed. This convention fixes the polarization normalization.

Here $G_k$ is the retarded Green function of the canonical variable ${\mu }_{ij}=a{h}_{ij}$, i.e. the solution kernel of ${\mu }_{ij}^{″}+\left(k^2-a^{\prime \prime }/a\right){\mu }_{ij}=2a^3{\mathrm{\Pi }}_{ij}^{\mathrm{TT}}/M^2$ with zero initial data (cf. Equation (6.4)); for radiation domination $G_k=\Theta (\eta -s)sin\left[k(\eta -s)\right]/k$. Then

$$h_{ij}^{\prime }=\frac{2}{M^{2}a\left(\eta \right)}\int ds{a}^3\left(s\right)\left[{\partial }_{\eta }-\mathcal{H}\left(\eta \right)\right]G_k(\eta ,s){\mathrm{\Pi }}_{ij}^{\mathrm{TT}}(s,\mathbf{k}),$$

and Equation (7.3) follows by substituting this expression into ${\mathcal{P}}_{h^{\prime }}$ and applying (7.1): the factor $a^{-2}\left(\eta \right)$ is the square of the prefactor $1/a\left(\eta \right)$, and each $a^3\left(s\right)$ comes from the corresponding source integral. Define

$$D_k(\eta ,s)={\partial }_{\eta }{G}_k(\eta ,s)-\mathcal{H}\left(\eta \right)G_k(\eta ,s).$$

(7.2)

For zero initial tensor data, and assuming the integrals converge,

$${\mathcal{P}}_{h^{\prime }}(k,\eta )={\displaystyle \frac{k^3}{2{\pi }^2}}{\displaystyle \frac{4}{M^4{a}^2\left(\eta \right)}}\int dsd{s}^{\prime }{a}^3\left(s\right)a^3\left(s^{\prime }\right)D_k(\eta ,s)D_k^{*}(\eta ,s^{\prime })C_{\mathrm{\Pi }}(k;s,s^{\prime }).$$

(7.3)

Here ${\mathcal{P}}_{h^{\prime }}=d\langle h_{ij}^{\prime }{h}_{ij}^{\prime }\rangle /dlnk$. In the short-wave, weak-gravity, oscillation-averaged regime,

$${\rho }_{\mathrm{GW}}={\displaystyle \frac{M^2}{4a^2}}\langle h_{ij}^{\prime }{h}_{ij}^{\prime }\rangle ,\begin{array}{|c|}\hline {\mathrm{\Omega }}_{\mathrm{GW}}(k,\eta )={\displaystyle \frac{{\mathcal{P}}_{h^{\prime }}(k,\eta )}{12a^2{H}^2}}.\\ \hline\end{array}$$

(7.4)

For $k\lesssim aH$, the short-wave energy expression does not define a locally observable radiation density. A nonzero initial tensor spectrum, and its cross terms with the source, must be added beyond (7.3). Reference [6] reviews the UETC methodology and the effect of temporal coherence on peak positions. The source construction presented here does not replace an explicit calculation of the correlator.

6.5. Positivity and Interference between Sources

A physical UETC must define a Hermitian positive-semidefinite kernel: for suitable test functions w,

$$\int dsd{s}^{\prime }w{\left(s\right)}^{*}{C}_{\mathrm{\Pi }}(s,s^{\prime })w\left(s^{\prime }\right)\ge 0.$$

(7.5)

On discrete time points, writing $C_{ij}={\sum }_r{B}_{ir}{B}_{jr}$,

$$\sum _{ij}{w}_i{w}_j{C}_{ij}=\sum _r{\left(\sum _i{w}_i{B}_{ir}\right)}^2\ge 0.$$

(7.6)

gram_response_nonnegative formalizes the identity that expresses the discrete double sum as a sum of squares [F]. The Green-function weights may have either sign; the result provides no discretization-to-continuum error bound.

If $\mathrm{\Pi }={\sum }_{\alpha }{\mathrm{\Pi }}_{\alpha }$, then $C_{\mathrm{\Pi }}={\sum }_{\alpha \beta }{C}_{\alpha \beta }$. Component spectra may be added directly only if the cross-correlations vanish or are bounded to be negligible. Two perfectly coherent responses with opposite signs can cancel. It can therefore be incorrect to compute the positive energy of each of the three generators separately and sum the results. The companion source provides coherent_cancellation and interference_bound, the latter being a Cauchy-type bound in terms of finite real amplitudes [F].

6.6. OU Statistics Do Not Determine a Spatial Spectral Peak

Under the additional assumption that the physical field is a stationary Gaussian field with

$$\langle {\chi }_{\mathbf{p}}\left(t\right){\chi }_{{\mathbf{p}}^{\prime }}^{*}\left(s\right)\rangle ={\left(2\pi \right)}^3{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })P_{\chi }\left(p\right)e^{-{\mathrm{\Gamma }}_p|t-s|},$$

(7.7)

then Wick contraction gives the gradient-quadratic source correlator

$$C_{\mathrm{\Pi }}(k;t,s)={\displaystyle \frac{2}{a^4}}\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\parallel {\mathcal{T}}_{\mathbf{k}}(\mathbf{p}\otimes \mathbf{q})\parallel }_F^2{P}_{\chi }\left(p\right)P_{\chi }\left(q\right)e^{-({\mathrm{\Gamma }}_p+{\mathrm{\Gamma }}_q)|t-s|},\mathbf{q}=\mathbf{k}-\mathbf{p},$$

(7.8)

with a frozen and the real-scalar and contracted-tensor conventions; using $P_{\mathbf{k}}\mathbf{q}=-P_{\mathbf{k}}\mathbf{p}$, the projected outer product is a symmetric tensor. The mean terms do not contribute at nonzero k. Equation (7.8) is an analytical result under the additional stationary-Gaussian model [A,M]. It does not apply unchanged to the unstable, non-stationary stage, and the Wick theorem is not formalized in Lean.

Even if all ${\mathrm{\Gamma }}_p=\mathrm{\Gamma }$, the time kernel is only $e^{-2\mathrm{\Gamma }|t-s|}$. For any exponential kernel $e^{-{\mathrm{\Gamma }}_c|t-s|}$, the cosine response factor over a window of length T is

$$I_T\left(K\right)=2{\int }_0^T(T-u)e^{-{\mathrm{\Gamma }}_{c}u}cos\left(Ku\right)du.$$

(7.9)

With $z={\mathrm{\Gamma }}_c-iK$,

$$I_T\left(K\right)=2Re\left[{\displaystyle \frac{T}{z}}-{\displaystyle \frac{1-e^{-zT}}{z^2}}\right]\sim {\displaystyle \frac{2T{\mathrm{\Gamma }}_c}{{\mathrm{\Gamma }}_c^2+K^2}}(T{\mathrm{\Gamma }}_c\gg 1).$$

(7.10)

This kernel gives a broadband temporal response rather than three discrete spectral lines. The full spectrum also depends on $k^3$, the spatial convolution, background weights, and propagation. The value 35 does not uniquely determine $P_{\chi }\left(k\right)$, ${\mathrm{\Gamma }}_p$, the coupling normalization, or the physical clock. The source formalization covers discrete Gram positivity and the continuous-time, finite-rank and wave-packet statements of §7.5. Wick contraction, analytic evaluations (7.8)–(7.14) and stochastic-field construction remain analytical/model statements.

6.7. A Normalized Spatial and Temporal UETC Benchmark

To make (7.3) operational without claiming a nonlinear simulation, specialize to a short frozen interval $0<t<T$, $HT\ll 1$, and normalize $a_{*}=1$. Physical and comoving momenta then coincide in this subsection, and ${\mathcal{P}}_{h^{\prime }}={\mathcal{P}}_{\dot{h}}$ at this normalization. Take one canonically normalized real scalar, a stationary zero-mean Gaussian ensemble and independent inputs: variance ${\sigma }_{\phi }^2$, physical spatial length $\ell >0$ and single-field temporal decay rate $\mathrm{\Gamma }>0$:

$$\langle \phi (\mathbf{x},t)\phi (\mathbf{y},s)\rangle ={\sigma }_{\phi }^2{e}^{{-|\mathbf{x}-\mathbf{y}|}^2/\left(2{\ell }^2\right)}{e}^{-\mathrm{\Gamma }|t-s|},P_{\phi }\left(P\right)={\left(2\pi \right)}^{3/2}{\sigma }_{\phi }^2{\ell }^3{e}^{-{\ell }^2{P}^2/2}.$$

(7.11)

Integrating $P_{\phi }$ with the stated Fourier measure gives ${\sigma }_{\phi }^2$. Here ${\mathrm{\Gamma }}^{-1}$ is a time, not a spatial length fixed by CGICE. A relation such as $\ell =v_{\mathrm{corr}}/\mathrm{\Gamma }$ is an extra transport model requiring an independently justified speed. The Gaussian has rapidly decaying rather than compactly supported spatial correlations; it is a finite-correlation model, not an exact causal compact-support construction. Multiplying the temporal kernel by $g\left(t\right)g\left(s\right)$, $g={\mathbf{1}}_{[0,T]}$, implements the finite window and preserves positive semidefiniteness.

For $\mathbf{K}$ along z, the TT contraction in (7.8) obeys $2\parallel {\mathcal{T}}_{\mathbf{K}}{(\mathbf{P}\otimes \mathbf{Q})\parallel }_F^2=P_{\perp }^4$. Set $\mathbf{r}=\mathbf{P}-\mathbf{K}/2$ and complete the Gaussian square:

$$\begin{array}{c}\begin{array}{|c|}\hline C_{\mathrm{\Pi }}(K;t,s)=C_{\mathrm{sp}}\left(K\right)e^{-{\mathrm{\Gamma }}_c|t-s|},C_{\mathrm{sp}}\left(K\right)={\displaystyle \frac{2{\pi }^{3/2}{\sigma }_{\phi }^4}{\ell }}{e}^{-{\ell }^2{K}^2/4}\\ \hline\end{array},\\ \begin{array}{|c|}\hline {\mathrm{\Gamma }}_c=2\mathrm{\Gamma }\\ \hline\end{array},t,s\in [0,T].\end{array}$$

(7.12)

Indeed, $\int d^{3}r{r}_{\perp }^4{e}^{-{\ell }^2{r}^2}=2{\pi }^{3/2}{\ell }^{-7}$, fixing the prefactor for the contracted TT convention. In mass units $\left[C_{\mathrm{sp}}\right]=5$, as required for a stress covariance after dividing out the momentum delta function. The spatial model has finite gradient variance. The stationary temporal Ansatz describes an effective maintained ensemble, not the solution of an isolated conservative tachyonic equation.

After switch-off, average over tensor oscillations while the background is effectively fixed. Using $\overline{cosK(t-s)cosK(t-s^{\prime })}=\frac{1}{2}cosK(s-s^{\prime })$,

$$\begin{array}{|c|}\hline \overline{{\mathcal{P}}_{\dot{h}}}\left(K\right)={\displaystyle \frac{K^3}{{\pi }^2{M}^4}}{C}_{\mathrm{sp}}\left(K\right)I_T\left(K\right)={\displaystyle \frac{2{\sigma }_{\phi }^4}{\sqrt{\pi }{M}^4\ell }}{K}^3{e}^{-{\ell }^2{K}^2/4}{I}_T\left(K\right)\\ \hline\end{array},\overline{{\mathrm{\Omega }}_{\mathrm{GW}}}={\displaystyle \frac{\overline{{\mathcal{P}}_{\dot{h}}}}{12H_{*}^2}}.$$

(7.13)

The radiation interpretation and a locally frozen oscillation average require $K\gg H_{*}$; the source duration can still be shorter than $K^{-1}$. The exact finite-window factor is (7.9)–(7.10), with

$$I_T\left(0\right)=2\left[{\displaystyle \frac{T}{{\mathrm{\Gamma }}_c}}-{\displaystyle \frac{1-e^{-{\mathrm{\Gamma }}_{c}T}}{{\mathrm{\Gamma }}_c^2}}\right]>0,I_T\left(0\right)⟶T^2({\mathrm{\Gamma }}_c⟶0^{+}).$$

(7.14)

The mathematical frozen-background spectrum therefore has ${\mathcal{P}}_{\dot{h}}\propto K^3$ at its infrared end. In the physical overlap $H_{*}\ll K\ll min({\ell }^{-1},T^{-1})$, the same $K^3$ scaling applies to (7.13). Superhorizon modes require propagation with (6.4), not extrapolation of the local energy density to $K=0$. Phase-space suppression makes ${\int }_0{\mathrm{\Omega }}_{\mathrm{GW}}dlnK$ infrared integrable in the extended template, avoiding a constant tail from a Gaussian imposed directly in linear frequency. This accords with the finite-duration, regular-source reasoning of [6] under the stated scale hierarchy; it is not a universal slope for inflationary or long-lived sources.

The template depends separately on $K\ell$, $KT$ and ${\mathrm{\Gamma }}_{c}T$. When $I_T\left(K\right)$ is nearly constant across the maximum, the spatial factor alone peaks at $K\ell =\sqrt{6}$; otherwise coherence shifts that maximum. This does not predict ℓ from geometry or identify it with ${\mathrm{\Gamma }}^{-1}$. For the instability stage itself, use a nonstationary covariance computed from mode functions, e.g. $C_{\phi }(P;t,s)=g_P\left(t\right)g_P\left(s\right)P_i\left(P\right)$ for a single coherent real seed amplitude, followed by the TT/Wick convolution. A general Gaussian initial state also includes field–velocity covariances. Non-Gaussian saturation requires the connected four-point function and invalidates the simple closure (7.12).

6.8. Continuous-time Positivity and a Positive Finite Wave Packet

For a real rank-one kernel $C(t,s)=b\left(t\right)b\left(s\right)$ on a finite interval, with integrable product $wb$,

$$\int dtdsw\left(t\right)C(t,s)w\left(s\right)={\left[\int dtw\left(t\right)b\left(t\right)\right]}^2\ge 0.$$

(7.15)

A finite sum of these kernels is also positive semidefinite. The new ContinuumUETC.rank_one_ time_integral_identity, rank_one_time_integral_nonnegative and finite_rank_time_ response_nonnegative use actual time integrals [F]. They extend the discrete Gram result without assuming convergence to an arbitrary stochastic kernel. In the complex case the analytical identity uses conjugates and an absolute square. Infinitely many source labels require their own integrability and convergence estimates.

For a real, integrable finite-duration force $S\left(t\right)$, define

$$\begin{array}{c}{J}_c\left(K\right)={\int }_0^{T}S\left(t\right)cos\left(Kt\right)dt,J_s\left(K\right)={\int }_0^{T}S\left(t\right)sin\left(Kt\right)dt,\\ E_{\mathrm{out}}\left(K\right)=\frac{1}{2}\left[J_c{\left(K\right)}^2+J_s{\left(K\right)}^2\right].\end{array}$$

(7.16)

Duhamel’s formula gives $h\left(T\right)=[sin\left(KT\right)J_c-cos\left(KT\right)J_s]/K$ and $\dot{h}\left(T\right)=cos\left(KT\right)J_c+sin\left(KT\right)J_s$. Their rotation identity proves (7.16); the corresponding integral definitions and energy identity are supplied in Lean [F]. Bounded piecewise continuous sources are integrable over the finite interval. A nonzero general source can have $J_c=J_s=0$ at a chosen K, so semidefiniteness does not imply unconditional strict positivity.

For $S\left(t\right)=S_0{e}^{-{\beta }_{d}t}{\mathbf{1}}_{[0,T]}\left(t\right)$, ${\beta }_d>0$, $S_0\ne 0$, $T>0$, (6.10) remains valid with $s=-{\beta }_d$:

$$E_{\mathrm{out}}\left(K\right)={\displaystyle \frac{S_0^2}{2({\beta }_d^2+K^2)}}\left[1+e^{-2{\beta }_{d}T}-2e^{-{\beta }_{d}T}cos\left(KT\right)\right]\ge {\displaystyle \frac{S_0^2{(1-e^{-{\beta }_{d}T})}^2}{2({\beta }_d^2+K^2)}}>0.$$

(7.177)

This is a prescribed decay after amplification, not a derived shutdown law. For $0<K_1<K_2$, a continuous weight $W\left(K\right)\ge 0$ positive at $K_1$ satisfies

$$\begin{array}{|c|}\hline {\int }_{K_1}^{K_2}dKW\left(K\right)E_{\mathrm{out}}\left(K\right)>0.\\ \hline\end{array}$$

(7.18)

Continuity gives positive measure to the nonzero contribution. If $W\ge W_{min}>0$ throughout the interval, an explicit lower bound is $(K_2-K_1)W_{min}{S}_0^2{(1-e^{-{\beta }_{d}T})}^2/\left[2({\beta }_d^2+K_2^2)\right]$. The packet weight can include angular measure, polarizations and a finite momentum window, such as a radial factor $K^2$. Extension to infinite momenta requires $\int WEdK<\infty$; an idealized sharp time switch alone does not ensure a finite three-dimensional UV energy.

finite_decay_source_leaves_positive_energy proves pointwise strict positivity; residual_ energy_continuous establishes continuity; finite_wavepacket_energy_positive proves positivity of the actual interval integral of the existing forced-wave energy [F]. Integrated positivity is not assumed. The packet and its support remain inputs to a physical source calculation; choose $K_1\gg H_{*}$ for this local cosmological radiation interpretation.

7. Three Cosmological Epochs, Source Activity, and Spectral Peaks

7.1. Epochs, Transitions, Emission Events, and Spectral Peaks

For the V64 neutral composite realization, the I→II row uses (3.15)–(3.18), with $G_0>1/C$, independently present seeds and positive $Z_m,A_m$. Emission is permitted during amplification and relaxation; a universal pulse duration is not derived. Fixed nonzero condensate phase charge instead invokes V64’s N3 crossover and the coupled radial–phase–Fisher perturbations. During epoch II, real-quartic domain walls require suitable vacuum populations; strings require a complex order parameter, topology and a formation history. Sound waves and turbulence require a fluid and an independently established first-order thermal transition, not merely negative origin mass. An early accelerating branch can amplify specified quantum tensor data over modes crossing the horizon; its entire present spectrum need not lie in one “low-frequency band.”

The existence of three epochs does not imply three first-order thermodynamic phase transitions. In particular, vacuum domination after matter dilution is a crossing of dominant components and need not involve latent heat, bubble collisions, or a nonanalytic order parameter. Even for dust plus vacuum with radiation neglected, the equality point ${\rho }_m=\mathrm{\Lambda }$ differs from the acceleration onset ${\rho }_m=2\mathrm{\Lambda }$.

The late-time conditions proposed by the three-epoch spacetime phases are

$${\phi }_v^2=-b/\lambda ,A_v=Z+\gamma b/\left(2\lambda \right)>0,{\mathrm{\Lambda }}_{\mathrm{eff}}=U_{\infty }-b^2/\left(4\lambda \right)>0.$$

(8.1)

Equation (8.1) is specific to the quartic V20 potential. In the V64 determinant branch use the matched stationary determinant value and its kinetic-health test; $b_N^2/\left(4\lambda \right)$ is not its exact vacuum energy. These conditions describe the limit of convergent orbits; they do not establish global attraction. vacuum_curvature_positive verifies that the radial curvature of this vacuum is $-2b>0$; vacuum_health_value and vacuum_energy_value check the health coefficient and the vacuum energy. The classical isotropic source still vanishes by GWTT.explicit_isotropic_zero. Any additional defects, phase transitions, or astrophysical sources require independent specification and calculation; they are not a third emission event derived by the three-epoch model [2].

Epoch	Actual conditions in cosmic evolution	Role for GWs	What the minimal model does not provide automatically
I: statistical localization; early acceleration possible under extra conditions	conditional diffusive relaxation; symmetric branch $A>0,H>0,U>2ZX$	constrains configuration statistics; an accelerating background can pump a specified quantum tensor initial state	Hz from 35; relaxation becoming oscillation; a first narrow peak
I→II: neutral order-parameter mass changes sign	$b+\gamma X=0$, instability band opens; seeds present and the healthy regime maintained	quadratic TT stress of non-collinear gradients can radiate	first-order nucleation with an uncomputed barrier; a fixed sound-wave peak; mandatory reheating
II: nonlinear condensation and a conditional matter era	saturation, energy budget, independent particle-production mechanisms and dilution laws	possible additional sources/scattering; or mere propagation of pre-existing waves	one GW peak per particle mass
III: positive vacuum dominates after matter dilution	orbital convergence, $A_v>0,{\mathrm{\Lambda }}_{\mathrm{eff}}>0$	alters expansion, horizons, and propagation; homogeneous vacuum adds no new classical TT source	an automatic late-time phase transition; automatic nHz emission

7.2. Early Quantum Modes and Inter-Epoch Propagation

With a chosen polarization normalization, the canonical Einstein tensor mode obeys

$$v_k^{″}+(k^2-a^{\prime \prime }/a)v_k=0.$$

(8.2)

The primordial tensor spectrum can be computed only after specifying the initial quantum state, canonical commutation relations, and background. Under the usual slow-roll and vacuum assumptions, the Einstein branch has the two-polarization power ${\mathcal{P}}_T\simeq 2H_k^2/\left({\pi }^2{M}^2\right)$; This result relies on the slow-roll and vacuum approximations, is distinct from the CGICE statistical spectral-gap formula, and generally describes a spectral segment rather than a narrow peak. The present Lean development formalizes neither the quantum-state normalization nor this power formula.

When the primordial modes and the later sources are independent, powers may be propagated and added; when correlated, the cross terms must be retained. Denoting the three emission windows by ${\mathcal{I}}_r$, (7.3) contains

$$\sum _{r,s}{\int }_{{\mathcal{I}}_r}d{\eta }_1{\int }_{{\mathcal{I}}_s}d{\eta }_2{\mathcal{W}}_k\left({\eta }_1\right){\mathcal{W}}_k^{*}\left({\eta }_2\right)C_{rs}(k;{\eta }_1,{\eta }_2).$$

(8.3)

If the windows are merely time segments of one and the same source, ignoring the cross terms requires justification all the more. three_epoch_response_square proves the full set of cross terms in the square of three real responses, and three_nonzero_responses_can_cancel gives the finite witness $1+1-2=0$. These results show that counting epochs does not justify adding powers. They do not model actual cosmic phase distributions; the continuum calculation must retain the full complex covariance kernel.

A spectral peak is defined as a strict local maximum of the final $F\left(x\right)={\mathrm{\Omega }}_{\mathrm{GW},0}\left(e^x\right)$. The three-epoch conditions permit no new classical signal at all, and equally do not exclude broadband or multiple structures once source statistics are supplied. No fixed number of peaks is predicted. The statement that three peaks are not required does not imply that the spectrum has only one peak.

7.3. Conditional Restriction from Internal Mass Differences

Even if $SU\left(6\right)\to S\left(U\right(1)\times U(2)\times U(3\left)\right)$ is additionally selected, the count of broken generators

$$2(1·2+1·3+2·3)=4+6+12=22$$

(8.4)

is representation theory alone. The minimal matrix quartic potential does not uniquely select (1,2,3); its equal-norm minima are also attained by traceless (3,3) configurations. This vacuum choice is therefore not a premise of the present work.

If a separate gauge-Higgs model makes the three tree-level masses of the same vacuum proportional to the ordered eigenvalue differences $x,y,x+y$, and if the peak frequencies at one and the same emission time map to masses with one common proportionality constant, then

$$f_{max}\le 2f_{\mathrm{mid}}.$$

(8.5)

adjacent_gap_frequency_bound proves this finite inequality, and legacy_high_middle_pair_excluded excludes 450 GHz and 1.71 Hz from being the largest and middle frequencies of that pair [F]. Different emission epochs or non-identical peak–mass mappings can evade this restricted proposition, but those differences must be computed independently; they cannot be tuned backwards from a target peak.

7.4. A Posteriori Criteria for Candidate Peaks

Let $x=lnf$ and $F\left(x\right)={\mathrm{\Omega }}_{\mathrm{GW},0}\left(e^x\right)$. If any finite collection of pairwise disjoint windows $[l_i,r_i]$ satisfies: F differentiable, $F^{\prime }$ continuous and strictly decreasing within the window, and

$$F^{\prime }\left(l_i\right)>0,F^{\prime }\left(r_i\right)<0,$$

(8.6)

then each window contains a unique stationary point $c_i$ with $F\left(x\right)<F\left(c_i\right)$ for all other points of the window. The proof uses the intermediate value theorem for the derivative, strict monotone decrease, and the mean value theorem for F. EpochFrequency.finite_windows_have_peaks invokes the existing single-window theorem for an arbitrary finite index set [F]; PeakTests.three_strict_local_maxima is retained as the three-window special case. The result guarantees at least one strict local peak in each qualifying window. Disjoint windows ensure that these peaks are distinct. Additional peaks may occur outside the windows, and finite instrumental resolution may prevent their separation.

For the criteria to be operationally checkable, one should compute $F=F_i+R_i$ from (7.3) and then bound the residual derivatives independently; e.g.

$$F_i^{\prime }\left(l_i\right)>{ϵ}_1,F_i^{\prime }\left(r_i\right)<-{ϵ}_1,\left|R_i^{\prime }\right|\le {ϵ}_1,$$

(8.7)

$$F_i^{″}<-{ϵ}_2,\left|R_i^{\prime \prime }\right|\le {ϵ}_2$$

(8.8)

give endpoint signs with margin and strict concavity. endpoint_margin, negative_endpoint_margin, and curvature_margin formalize the corresponding pointwise inequalities. For the actual source spectrum these bounds remain to be computed [O].

One cannot define F as a sum of three prescribed Gaussians and then use the window criteria to claim that CGICE predicts three peaks. If $F^{\prime }$ is strictly decreasing on the entire real axis, it has at most one zero and therefore F has at most one stationary point. The window-based criterion is retained here, but a certificate for the physical spectrum remains to be established.

8. Redshift and Thermal-History Checks of the Legacy Frequencies

8.1. Redshift of Cyclic Frequency

For a freely propagating fixed comoving mode,

$$f_0={\displaystyle \frac{k}{2\pi a_0}}={\displaystyle \frac{a_{*}}{a_0}}{f}_{*},$$

(9.1)

the algebraic relation of frequency_redshift [F]; its physical use requires a specified propagation equation and background.

Let emission occur during radiation domination, followed by adiabatic expansion, and let $q=k/\left(a_{*}{H}_{*}\right)$. Then

$${\displaystyle \frac{a_{*}}{a_0}}={\displaystyle \frac{T_0}{T_{*}}}{\left({\displaystyle \frac{g_{s0}}{g_{s*}}}\right)}^{1/3},H_{*}=\sqrt{{\displaystyle \frac{{\pi }^2{g}_{*}}{90}}}{\displaystyle \frac{T_{*}^2}{{\overline{M}}_P}},$$

(9.2)

$$\begin{array}{|c|}\hline f_0={\displaystyle \frac{q}{2\pi }}\sqrt{{\displaystyle \frac{{\pi }^2{g}_{*}}{90}}}{\displaystyle \frac{T_0{T}_{*}}{{\overline{M}}_P}}{\left({\displaystyle \frac{g_{s0}}{g_{s*}}}\right)}^{1/3}{\displaystyle \frac{1}{\hslash }}.\\ \hline\end{array}$$

(9.3)

Here ℏ, expressed in GeV·s, serves only to restore Hz from natural units. Because q is defined with the angular wavenumber, the $2\pi$ cannot be dropped. The commonly used $f_{*}/H_{*}$ formulae carry different parameter conventions and must not be mixed with these.

8.2. Benchmark Conversion

Take [C]

$$\begin{array}{c}{T}_0=2.348\times {10}^{-13}\mathrm{GeV},{\overline{M}}_P=2.435\times {10}^{18}\mathrm{GeV},\\ \hslash =6.582119569\times {10}^{-25}\mathrm{GeV}\mathrm{s},g_{*}=g_{s*}=106.75,g_{s0}=3.91.\end{array}$$

(9.4)

Then

$$f_0\simeq 2.65\times {10}^{-8}q(T_{*}/\mathrm{GeV})\mathrm{Hz}.$$

(9.5)

Legacy value / interpretation	Cross-check	Treatment here
$T_{*}={10}^{13}$ GeV gives 1.71 Hz	about $2.65\times {10}^5$ Hz at $q=1$; keeping that temperature at 1.71 Hz requires $q\approx 6.45\times {10}^{-6}$	not supported as a short-lived, subhorizon causal source; large-scale seeds or nonstandard thermal history would need separate proof
horizon-scale interpretation of 1.71 Hz	about $6.45\times {10}^7$ GeV in the high-temperature benchmark above	a conversion only, not an independent prediction after backwards-inferred temperature
450 GHz as a GW frequency in the CMB observing band	an electromagnetic observing band is not the GW frequency of the constrained tensor modes	conflation removed; an independent GW detection/transfer mechanism would be required
late-time homogeneous DE automatically produces 1 nHz	$H_0/\left(2\pi \right)\approx 3.5\times {10}^{-19}$ Hz only (with $H_0=67.42$ km/s/Mpc)	nHz requires inhomogeneous sources far below horizon scale; the model provides none
1+2+3 fixes three cross-scale peaks	simultaneous same-source mass-difference models are restricted by the conditional frequency ratio of §8.3	all three frequencies retained as historical benchmarks, not theoretical outputs

If the comoving entropy grows by $\Delta =S_0/S_{*}>1$ at later times, the redshift ratio acquires an extra factor ${\Delta }^{-1/3}$ at the same initial temperature. Pushing $2.65\times {10}^5$ Hz down to 1.71 Hz by this factor alone would take $\Delta \sim 3.7\times {10}^{15}$; this is conditional arithmetic, not an ignorable correction, and its entropy source, reheating, background constraints, and spectral dilution must be computed alongside.

8.3. Amplitude Transfer Through the Thermal History

For free GWs already inside the horizon, with the source switched off and short-wave averaging applicable, ${\rho }_{\mathrm{GW}}\propto a^{-4}$. The radiation background satisfies

$${\mathrm{\Omega }}_{\mathrm{GW},0}={\mathrm{\Omega }}_{r,0}{\displaystyle \frac{g_{*}}{g_{*,0}}}{\left({\displaystyle \frac{g_{s0}}{g_{s*}}}\right)}^{4/3}{\left({\displaystyle \frac{{\rho }_{\mathrm{GW}}}{{\rho }_r}}\right)}_{*},$$

(9.6)

to be used under the corresponding assumptions; additional entropy production, free-streaming damping, and nonstandard expansion belong in the transfer function. Shifting a peak frequency while keeping its height unchanged is generally inconsistent. For early tensor modes not yet inside the horizon, the tensor transfer function must be evolved directly; (9.6) must not be applied prematurely.

8.4. Unified Frequency Relation, Band, and Non-Uniqueness

Let $K_{*}=k/a_{*}$ denote the physical tensor angular wavenumber in GeV and $R_{*}=a_{*}/a_0>0$ the total redshift. Then

$$\begin{array}{|c|}\hline f_0={\displaystyle \frac{R_{*}{K}_{*}}{2\pi \hslash }}.\\ \hline\end{array}$$

(9.7)

This is the relation for the free, massless, $c_T=1$ propagation branch. $K_{*}$ is the TT convolution wavenumber, generally distinct from $M_R$, the radial mass, the growth exponent, or a dissipation rate. For the truncated unstable components of §5.5,

$$0<f_0<{\displaystyle \frac{R_{*}{\mu }_{*}}{\pi \hslash }}.$$

(9.8)

observed_frequency_band transfers the wavenumber bound into a strictly positive frequency band, and observed_frequency_strictMono shows that distinct K give distinct f at fixed redshift. Combined with any_band_mode_has_source, this yields a family of actual nonzero sources with different frequencies under identical background-instability conditions. The present premises therefore do not determine a unique frequency. Predictive power may be recovered once a microscopic model supplies the missing inputs.

For the radiation era with comoving entropy growth factor $\Delta \ge 1$, the full thermal-history dependence reads

$$f_0=2.6493277\times {10}^{-8}{q}_{*}{\displaystyle \frac{T_{*}}{\mathrm{GeV}}}\begin{array}{cc}& {\left({\displaystyle \frac{g_{\rho *}}{106.75}}\right)}^{1/2}{\left({\displaystyle \frac{g_{s*}}{106.75}}\right)}^{-1/3}{\Delta }^{-1/3}\mathrm{Hz},\hfill \\ & q_{*}={\displaystyle \frac{K_{*}}{H_{*}}}.\hfill \end{array}$$

(9.9)

The constant is computed from the benchmark scales of §9.2; it is not a CGICE output. If an actually computed spectrum has a peak parametrized as $K_{\mathrm{pk}}=\xi {\alpha }^{1/3}$, then on the symmetric branch, in the approximation $H_{*}\simeq H_c$,

$$q_{\mathrm{pk}}=\xi {\left({\displaystyle \frac{6\left|b\right|}{H_c^2}}\right)}^{1/3}.$$

(9.10)

quench_frequency_cube proves the fraction-free form $q_{\mathrm{pk}}^3=6{\xi }^3\left|b\right|/H_c^2$. The quantities $\xi$, the emission temperature, the seeds, and the saturation time all remain to be solved for; the natural scale does not select the peak.

Consistency condition between the two manuscripts. If the two manuscripts genuinely describe the same source peak under the same thermal-history benchmark, they must share the same $q_{*}{T}_{*}$; in general the condition is the same $R_{*}{K}_{*}$. same_frequency_iff_same_redshifted_scale gives the strict equivalence. A shared collaboration label or the words “three epochs” prove nothing about whether two numerical examples refer to the same physical source.

8.5. Emission before Reheating

Let the source switch off mainly at $a_{*}$, with reheating completed at $a_{\mathrm{rh}}$, and $N_{\mathrm{pre}}=ln(a_{\mathrm{rh}}/a_{*})$. With subsequent bath temperature $T_{\mathrm{rh}}$, entropy degrees of freedom $g_{s,\mathrm{rh}}$, and later entropy ratio $\Delta$,

$$R_{*}=e^{-N_{\mathrm{pre}}}{\displaystyle \frac{T_0}{T_{\mathrm{rh}}}}{\left({\displaystyle \frac{g_{s0}}{g_{s,\mathrm{rh}}}}\right)}^{1/3}{\Delta }^{-1/3},$$

(9.11)

$$\begin{array}{|c|}\hline f_0={\displaystyle \frac{K_{*}}{2\pi \hslash }}{e}^{-N_{\mathrm{pre}}}{\displaystyle \frac{T_0}{T_{\mathrm{rh}}}}{\left({\displaystyle \frac{g_{s0}}{g_{s,\mathrm{rh}}}}\right)}^{1/3}{\Delta }^{-1/3}.\\ \hline\end{array}$$

(9.12)

If the pre-reheating total background is described by a constant $w_{\mathrm{pre}}>-1$ with conservation, then $e^{-N_{\mathrm{pre}}}={({\rho }_{\mathrm{rh}}/{\rho }_{*})}^{1/\left[3(1+w_{\mathrm{pre}})\right]}$, where ${\rho }_{\mathrm{rh}}={\pi }^2{g}_{\rho ,\mathrm{rh}}{T}_{\mathrm{rh}}^4/30$. Changing the thermal history also changes the energy fraction of the switched-off, subhorizon wave: ${\mathrm{\Omega }}_{\mathrm{GW}}\propto a^{3w_{\mathrm{pre}}-1}$; one cannot move only the frequency point.

prethermal_redshift_composition proves the scale-factor composition, and gw_fraction_rate proves the genuine quotient derivative ${\dot{\mathrm{\Omega }}}_{\mathrm{GW}}=(3w-1)H{\mathrm{\Omega }}_{\mathrm{GW}}$ under the given continuity equations. These algebraic and derivative results do not establish a reheating mechanism. In the absence of a thermal bath, identifications such as $T_{*}={\left|b\right|}^{1/2}$, $T_{*}=M_R$, or the assignment of the condensate amplitude as a temperature have no derivation.

8.6. Audit of the 0.38-Hz Right-Handed-Neutrino Condensate Estimate

The sound-wave template checked against [7, Equation (15)] is

$$f_{\mathrm{sw},0}\simeq 1.9\times {10}^{-5}{\displaystyle \frac{1}{v_w}}{\displaystyle \frac{\beta }{H_{*}}}{\displaystyle \frac{T_{*}}{100\mathrm{GeV}}}{\left({\displaystyle \frac{g_{*}}{100}}\right)}^{1/6}\mathrm{Hz}.$$

(9.13)

It applies to a template branch that presupposes a first-order phase transition, bubble walls, fluid sound waves, and a radiation thermal history. The legacy 0.38-Hz multiplication $1.9\times {10}^{-5}\times 50\times 400$, is reproduced, for example, by $\beta /H_{*}=50$, $T_{*}=4\times {10}^4$ GeV, $v_w=1$, $g_{*}=100$. These template inputs reproduce the quoted value arithmetically; they are not parameters derived from the gap theorem of [8]. The nucleation time scale $\beta$ must not be replaced by ${\delta }_L$, $r_{+}$, or ${\alpha }^{1/3}$ without argument.

On the same radiation benchmark, the template corresponds to $q_{\mathrm{sw}}\simeq 7.25(\beta /H_{*})/v_w$. The 0.38 Hz can thus be a subhorizon acoustic-scale arithmetic example.

9. Energy Budget, Dark Matter, and Spectral Normalization

9.1. Cold Dark Matter Is Not Gravitational Waves

The short-wave-averaged equation of state of GWs is $w=1/3$; nonrelativistic cold dark matter has $w\simeq 0$. Even if they originate from the same parent source, they obey different dilution laws. A consistent branching model should be written as

$${\dot{\rho }}_P+3H({\rho }_P+p_P)=-(Q_{\mathrm{GW}}+Q_{\mathrm{DM}}+Q_{\mathrm{vis}}),$$

(10.1)

$${\dot{\rho }}_{\mathrm{GW}}+4H{\rho }_{\mathrm{GW}}=Q_{\mathrm{GW}},{\dot{\rho }}_{\mathrm{DM}}+3H{\rho }_{\mathrm{DM}}=Q_{\mathrm{DM}},$$

(10.2)

together with a matching equation for the visible component. Determining $Q_{\mathrm{GW}}$ requires the energy balance between the source and tensor backreaction. The background exchange Q cannot be assigned entirely to gravitational waves.

In the source, radiation_comoving_derivative and matter_comoving_derivative give

$${\displaystyle \frac{d}{dt}}\left(a^4{\rho }_{\mathrm{GW}}\right)=a^4{Q}_{\mathrm{GW}},{\displaystyle \frac{d}{dt}}\left(a^3{\rho }_{\mathrm{DM}}\right)=a^3{Q}_{\mathrm{DM}}.$$

(10.3)

After injection ceases and the DM has cooled,

$${\displaystyle \frac{{\rho }_{\mathrm{GW}}}{{\rho }_{\mathrm{DM}}}}\left(a\right)={\left({\displaystyle \frac{{\rho }_{\mathrm{GW}}}{{\rho }_{\mathrm{DM}}}}\right)}_{a_i}{\displaystyle \frac{a_i}{a}}.$$

(10.4)

The ratio is not a fixed geometric constant. Counting broken degrees of freedom as 75/15 does not replace the required masses, occupation numbers, production cross sections, or entropy history. We therefore do not retain the earlier claim that this count predicts the dark-matter-to-baryon ratio.

9.1.0.1. 10.1.1 Tensor work and the source/UETC energy budget

In the weak-tensor, fixed-background approximation with periodic boundaries or vanishing averaged spatial flux, the quadratic action gives

$${\rho }_h={\displaystyle \frac{M^2}{8}}〈{\dot{h}}_{ij}{\dot{h}}_{ij}+{\displaystyle \frac{{\partial }_l{h}_{ij}{\partial }_l{h}_{ij}}{a^2}}〉,p_h={\displaystyle \frac{M^2}{8}}〈{\dot{h}}_{ij}{\dot{h}}_{ij}-{\displaystyle \frac{{\partial }_l{h}_{ij}{\partial }_l{h}_{ij}}{3a^2}}〉.$$

(10.5)

Multiply the cosmic-time tensor equation by $M^2{\dot{h}}_{ij}/4$ and integrate spatially. Integration by parts cancels the gradient–velocity terms:

$${\dot{\rho }}_h+3H({\rho }_h+p_h)=\frac{1}{2}\langle {\mathrm{\Pi }}_{ij}^{\mathrm{TT}}{\dot{h}}_{ij}\rangle .$$

(10.6)

After short-wave and oscillation averaging, $p_h\simeq {\rho }_h/3$, ${\rho }_h\simeq M^2\langle {\dot{h}}_{ij}^2\rangle /4$, and

$$\begin{array}{|c|}\hline Q_{\mathrm{GW}}=\frac{1}{2}\langle {\mathrm{\Pi }}_{ij}^{\mathrm{TT}}{\dot{h}}_{ij}\rangle \\ \hline\end{array},{\dot{\rho }}_{\mathrm{GW}}+4H{\rho }_{\mathrm{GW}}\simeq Q_{\mathrm{GW}}.$$

(10.7)

The contraction is the one in (7.1); its normalization is fixed by (6.2)–(6.3). With zero tensor initial data, the UETC representation is

$$Q_{\mathrm{GW}}\left(\eta \right)={\displaystyle \frac{1}{M^2{a}^2\left(\eta \right)}}\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\int }_{{\eta }_i}^{\eta }ds{a}^3\left(s\right)Re\left[D_k^{*}(\eta ,s)C_{\mathrm{\Pi }}(k;\eta ,s)\right].$$

(10.8)

The spectrum must be convergent or consistently regulated, as in (7.3); source–initial-tensor correlations add terms when present. One therefore computes the lattice average of TT stress times tensor velocity, or the retarded, response-weighted UETC integral. An equal-time UETC trace alone is insufficient. Instantaneous $Q_{\mathrm{GW}}$ can be negative because the fields can reabsorb wave energy. Positive residual wave energy is a statement about the net output of a specified protocol, not monotone injection at every time.

The homogeneous identity (3.7) holds in the isolated two-field truncation. If its unmodified on-shell equations are retained exactly, their residual is zero and a nonzero additional net loss cannot be appended by hand. Production requires backreaction at the appropriate perturbative order from the enlarged action, or a controlled coarse-grained reservoir split. Write forced parent residuals ${\mathcal{E}}_{\vartheta }=-F_{\vartheta }$, ${\mathcal{E}}_{\phi }=-F_{\phi }$. Then

$${\dot{\rho }}_P+3H({\rho }_P+p_P)=-Q_{\mathrm{out}},Q_{\mathrm{out}}=u{F}_{\vartheta }+v{F}_{\phi },Q_{\mathrm{out}}=Q_{\mathrm{GW}}+Q_{\mathrm{DM}}+Q_{\mathrm{vis}}.$$

(10.9)

The last equality is a matching requirement for non-overlapping subsystems, not a derivation of channel rates. It differs from the internal background exchange $Q=\gamma \phi vX$ of (3.8), which cancels between the two parent fields and cannot be counted again as exported energy. Spatially inhomogeneous parents require their full averaged field stress and boundary fluxes; the homogeneous forced form is a transparent effective balance interface, not a lattice backreaction theorem.

The visible component satisfies ${\dot{\rho }}_{\mathrm{vis}}+3H({\rho }_{\mathrm{vis}}+p_{\mathrm{vis}})=Q_{\mathrm{vis}}$. It is usually radiation-like during reheating and must not silently receive the dust coefficient $3H{\rho }_{\mathrm{vis}}$. With dust-like DM and short-wave-averaged GWs, adding all four equations gives

$${\dot{\rho }}_{\mathrm{tot}}+3H({\rho }_{\mathrm{tot}}+p_{\mathrm{tot}})=0,p_{\mathrm{tot}}=p_P+{\rho }_{\mathrm{GW}}/3+p_{\mathrm{vis}}.$$

(10.10)

Conversely, when the parent loses $Q_{\mathrm{out}}$ and receiver equations hold, the total residual is $-Q_{\mathrm{out}}+Q_{\mathrm{GW}}+Q_{\mathrm{DM}}+Q_{\mathrm{vis}}$. Conservation is therefore equivalent to the summed exchange budget. EnergyPartition.energy_partition_conservation combines genuine derivatives with both comoving injection laws; total_residual_iff_exchange_budget establishes the equivalence; parent_continuity_from_field_equations calls the inherited V20 off-shell identity and differentiates the parent density [F]. It does not assume isolated unified_continuity while appending a nonzero loss.

9.1.0.2. 10.1.2 What a condensate remnant can contribute to dark matter

On the quartic branch, expand about a healthy late minimum: $\phi ={\phi }_v+\delta \phi$, $m_v^2=W^{\prime \prime }\left({\phi }_v\right)=-2b>0$. Subtract vacuum energy and take the small-amplitude, nonrelativistic, adiabatic regime:

$${\rho }_{\mathrm{osc}}\simeq \frac{1}{2}{\dot{\delta \phi }}^2+\frac{1}{2}{m}_v^2\delta {\phi }^2,p_{\mathrm{osc}}\simeq \frac{1}{2}{\dot{\delta \phi }}^2-\frac{1}{2}{m}_v^2\delta {\phi }^2,\langle {\dot{\delta \phi }}^2\rangle \simeq m_v^2\langle \delta {\phi }^2\rangle .$$

(10.11)

Dust behavior requires $m_v\gg H$, physical gradients small relative to $m_v$, negligible anharmonicity and continuing energy exchange, and adequate lifetime. Both kinetic and potential oscillation energy contribute; radial kinetic energy alone is neither the relic density nor its production rate. Quartic-dominated oscillations need not be dust. On the composite branch use the actual canonical determinant curvature at its minimum instead of $-2b$.

If the remnant itself is DM, split its energy out of ${\rho }_P$ using a specified mode/coarse-graining prescription to avoid double counting. Once isolated and pressureless it has $Q_{\mathrm{DM}}=0$ while retaining nonzero density. Alternatively, an independently justified channel with damping ${\mathrm{\Gamma }}_{\mathrm{DM}}\dot{\delta \phi }$ can give $Q_{\mathrm{DM}}={\mathrm{\Gamma }}_{\mathrm{DM}}\langle {\dot{\delta \phi }}^2\rangle$ for a separate receiving population. The damping, lifetime, particle equation of state and thermalization are new microscopic inputs. Relativistic products dilute as radiation before cooling and cannot immediately be treated as cold DM.

The couplings $\gamma ,\lambda$ affect the background and saturation but their ratio alone determines neither a branching fraction nor a dark-matter identity. Initial spectra, $M,Z,b,U$, decay channels, charge, gradients and expansion also matter. At fixed $\gamma /\lambda$, a homogeneous configuration has $Q_{\mathrm{GW}}=0$, whereas the non-collinear example in §5 can radiate. exchange_budget_not_unique demonstrates that total conservation leaves multiple nonnegative rate partitions. A numerical partition must follow the jointly evolved sources; total background dilution also includes expansion work and is not GW release.

9.2. Peak Amplitudes and Integrated Energy Density

Define

$${\mathrm{\Omega }}_{\mathrm{GW},\mathrm{tot}}={\int }_0^{\infty }{\mathrm{\Omega }}_{\mathrm{GW}}\left(f\right)dlnf.$$

(10.12)

If, purely phenomenologically,

$${\mathrm{\Omega }}_i\left(f\right)=A_{i}exp\left[-{\displaystyle \frac{{ln}^2(f/f_i)}{2{\sigma }_i^2}}\right],A_i,{\sigma }_i>0,$$

(10.13)

then with the substitution $x=ln(f/f_i)$,

$$\int {\mathrm{\Omega }}_i\left(f\right)dlnf=\sqrt{2\pi }{A}_i{\sigma }_i.$$

(10.14)

Thus ${10}^{-10}+1.4\times {10}^{-9}+{10}^{-9}=2.5\times {10}^{-9}$ is only the sum of three chosen peak heights, not the total radiation density. Equation (10.14) is a Gaussian integral [A]; the source formalizes only pointwise properties such as logPeak_strict_max.

This infrared remark concerns the earlier linear-frequency Gaussian template of the R1/R2 drafts, $\mathrm{\Omega }\left(f\right)\propto exp[-{(f-f_i)}^2/2{\sigma }_f^2]$, and not the log-normal form (10.13), whose tail vanishes rapidly as $f\to 0^{+}$: for the linear-frequency template the tail tends to a strictly positive constant as $f\to 0^{+}$, and without a low-frequency cutoff its $\int \mathrm{\Omega }\left(f\right)dlnf$ even diverges at the zero end. A physical spectrum must have integrable infrared behavior or an explicit domain of validity, and treating a small but nonzero constant tail as zero conceals the problem. This paper no longer uses that template as a full spectrum.

Independent cosmological radiation-energy bounds should be applied to the integrated spectrum that exists in, and has entered the horizon during, the applicable epoch. An instrument’s frequency-dependent sensitivity likewise cannot be replaced by a single constant “upper limit” over all f. This paper performs no observational likelihood fits or significance estimates.

10. Evidence Dependencies and Falsifiable Statements

10.1. Forward Evidence Chain

1.

CGICE provides spectral/covariance constraints of a specified statistical model; it must be stated which physical parameters have no matching yet.

2.

Select independently the healthy effective action of the three-epoch model [2] (Equations (3.2)–(3.3)), the parameters $b,\gamma ,\lambda ,Z,U$, and the initial data.

3.

Solve the background and check $A>0$, continuity, energy constraints, and the mass-sign-change conditions.

4.

Given spatial seeds from an independent initial state or a noise matching, solve the perturbations and any required nonlinear evolution.

5.

Compute the TT source and the UETC directly from the stress definition, including cross terms between fields/epochs.

6.

Obtain the spectrum via (7.3) and the thermal history, then test peak number, peak positions, and the integrated budget.

A circular inference would fix three target peaks, infer three “geometric masses” or redshifts backward from those targets, treat the resulting matching as a CGICE theorem, and then claim the target peaks as predictions. The present formulation removes the global spectral-matching axioms and hard-coded physical outputs associated with this inference.

10.2. Testable Statements and Open Conjectures

Relative to derivative-triggered waterfall and preheating models, distinctive source correlations or parameter relations can emerge only after computing the map from configuration geometry to the effective action and initial statistics. The present work does not claim an established physical law distinct from existing preheating mechanisms. Its specific contributions are a tensor-source construction, a forward-computation framework, and limiting results compatible with the conserved backgrounds of CGICE [1] and the three-epoch model [2].

Statement	Current status	Concrete scenarios that would falsify it
mass zero-crossing on the symmetric branch	conditional theorem	X never reaches the critical value; backreaction alters the orbit; the healthy regime is left first
growth of low-momentum perturbations	conditional result of the frozen/slowly varying approximation	no seeds; time window too short; effective frequency squared not negative
non-collinear gradients provide a TT source	explicit construction theorem	only homogeneous or collinear configurations; relevant amplitudes vanish
a nonzero source excites a response from zero initial tensor data	exact constant-coefficient example and linear response theory	approximations inapplicable; actual projected source or effective response vanishes
three observable peaks	no amplitude or peak count is predicted	A fully specified parameter point can be compared with band-dependent limits and resolution. A generic null result cannot exclude this unconstrained mechanism without a derived minimum amplitude. The ratio restriction of §8.3 applies only to its same-epoch, common-mapping hypotheses.

11. Formalization and Its Coverage

The companion imports only Mathlib, retains the uploaded proof bodies and keeps set_option autoImplicit false. After this manuscript draft was completed, the whole file was compiled under Lean 4.34.0-rc1 / Mathlib v4.34.0-rc1 with the pinned toolchain: three tactic bodies required local repairs (receiver_continuity, energy_partition_conservation, finite_wavepacket_energy_positive; statement-preserving), then lake env lean and lake build (8709 jobs) completed with 0 errors. #print axioms was run on the 20 new R5 theorems across EnergyPartition/EpochFrequency/ ContinuumUETC plus 8 inherited cross-namespace keys: every audit returns only [propext, Classical.choice, Quot.sound].

Paper result	Companion declarations	Coverage and remaining boundary
Geometry and covariance	Geometry.restricted_threshold_35, radial_gap_lower_29; GWMechanism.ouCov_hasDerivAt, ou_clock_chain	Finite arithmetic and specified covariance dynamics; not a manifold spectral theorem or physical matching
Parent continuity and crossing	UnifiedDynamics.continuity_identity, density_hasDerivAt, unified_continuity, critical_crossing, critical_crossing_unique	Algebra and real-function derivatives under explicit equations and trajectory hypotheses
TT projection and source	GWTT.explicit_isotropic_zero, explicit_mem_TT, explicit_idempotent; GWMechanism.two_mode_tt_source, two_mode_tt_nonzero	Actual three-dimensional matrices and specified two-mode Fourier coefficients
Growth and forced response	growth_rate_positive, growth_rate_characteristic, forcedH_hasDerivAt, forcedV_hasDerivAt, forced_wave_equation, wave_energy_hasDerivAt in GWMechanism	Frozen coefficients; no global FLRW or nonlinear existence theorem
Band and residual energy	EpochFrequency.any_band_mode_has_source, equal_parent_norm_different_tensor_modes, finite_source_leaves_positive_energy, seeded_band_mode_leaves_wave	Local constructive witnesses; not a normalized random cosmology
Neutral/charged distinction	EpochFrequency.composite_mass_matching, composite_threshold_zero, charged_origin_excluded, charge_energy_lower_bound	Origin matching and fixed condensate-charge obstruction; no full determinant or charged perturbation theory
Composite observed band (R5)	composite_mu_sq_eq_neg_mass, composite_band_upper_bound, neutral_no_spinodal_of_inverse_bound in EpochFrequency	Norm convolution and redshift with positive denominators and actual instability; not an absolute UV cutoff
Composite slope (R5)	EpochFrequency.composite_critical_mass_derivative, composite_quench_rate_positive, quench_remainder_upper	Calls the field-equation derivative theorem; Taylor-error control remains an explicit premise
Discrete UETC and interference	GWMechanism.gram_response_nonnegative, coherent_cancellation, interference_bound; EpochFrequency.three_epoch_response_square	Finite real algebra; infinite-rank stochastic kernels remain analytical
Continuous-time UETC (R5)	ContinuumUETC.rank_one_time_integral_identity, rank_one_time_integral_nonnegative, finite_rank_time_response_nonnegative	Actual time integrals of finite-rank real kernels; no uncontrolled continuum limit
Time-integral energy (R5)	ContinuumUETC.time_integral_energy_nonnegative, time_integral_energy_identity, zero_quadratures_give_zero_energy	Integral definitions and rotation identity; Duhamel existence and regularity are analytical
Positive wave packet (R5)	ContinuumUETC.residual_energy_eq_forced, finite_decay_source_leaves_positive_energy, residual_energy_continuous, finite_wavepacket_energy_positive	Genuine momentum integral of inherited forced energy; continuous nonnegative weight positive at one endpoint
Energy partition (R5)	EnergyPartition.receiver_continuity, energy_partition_conservation, total_residual_iff_exchange_budget, parent_continuity_from_field_equations, exchange_budget_not_unique	Actual derivatives and linkage to off-shell parent continuity; microscopic rates remain inputs
Transfer	GWMechanism.frequency_redshift, radiation_comoving_derivative, matter_comoving_derivative; EpochFrequency.observed_frequency_band, gw_fraction_rate, prethermal_redshift_composition	Supplied continuity and propagation laws; no reheating mechanism
Peak tests and vacuum	PeakTests.three_strict_local_maxima; EpochFrequency.finite_windows_have_peaks, vacuum_energy_value, vacuum_health_value, vacuum_curvature_positive	Sufficient window tests and quartic vacuum algebra; no actual peak-count or global-attractor certificate

Action variation, metric constraints, Wick contraction, Gaussian and temporal-kernel evaluation, arbitrary-source Duhamel theory, nonlinear backreaction, renormalization, observational fits and the complete thermal history remain A/M/O as stated in the text. Continuous-integral source proofs narrow one formalization gap without closing these physical links. Scans for placeholders or counts of declarations are not kernel verification [9].

12. Conclusions

The CGICE statistical spectrum [1] and the conditional background results of the three-epoch spacetime phases [2] do not determine a unique peak frequency. Together with an explicit effective action, they support several conditional mechanisms: dilution on the neutral symmetric branch fixes the mass zero-crossing slope; unstable parent modes generate a continuous band of TT sources; and a specified finite-duration source leaves a free wave. The present-day frequency depends on the source wavenumbers and thermal history, while peak positions also depend on the correlation weights and propagation.

The three epochs provide early statistical and background conditions, possible intermediate-era sources, and a late-time propagation environment. Vacuum domination alone produces no classical TT source, and the number of epochs does not fix the number of peaks. This revision supplies a composite-parameter band bound, a normalized finite-correlation UETC benchmark with an integrable infrared tail, continuous-time and finite-wave-packet positivity statements, and an energy-partition law tied to the parent continuity residual. Channel rates and a full nonlinear cosmological solution remain to be computed; the result is a conditional mechanism, not a unique spectral prediction.