%=============================================================================
% Appendix Q: Temporal Completion — Dust Branch from S³ Composition
% v42: Includes no-go lemma + theorem-grade dust branch
%=============================================================================

\section{Temporal Completion: Dust Branch from $S^3$ Composition}\label{app:temporal-completion}

This appendix derives the temporal sector from the same $S^3$ microsector that fixed $\mu(x)$ in Appendix~\ref{app:mu-derivation}. The key results are:
\begin{enumerate}
\item The temporal deviation invariance follows from the saturation-union law (Assumption~\ref{ass:composition})
\item The unique temporal segment variable is $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$
\item With $K'(\Delta) = \mu(\Delta)$, the dust branch emerges: $w \to 0$, $c_s^2 \to 0$
\end{enumerate}

We also include a \textbf{no-go lemma} showing that the naive quadratic identification $K'(Q_t) = \mu(\sqrt{Q_t})$ gives $w \to 1/2$ (not dust). This proves the dust branch is not automatic---it is forced specifically by the deviation-invariant $\Delta$ closure.

%-----------------------------------------------------------------------------
\subsection{Temporal Deviation Invariance from Saturation-Union}
%-----------------------------------------------------------------------------

\begin{theorem}[Temporal deviation invariance]\label{thm:temporal-deviation-Q}
Assume the saturation-union composition law (Assumption~\ref{ass:composition}):
\begin{gather}
\mu(\psi_1+\psi_2)=1-\bigl(1-\mu(\psi_1)\bigr)\bigl(1-\mu(\psi_2)\bigr),
\label{eq:saturation-union-Q}\\
\mu(0)=0,\qquad 0\le \mu<1. \notag
\end{gather}
Then for any background $\psi_0$ and deviation $\Delta\psi$,
\begin{equation}
\fiteq{\boxed{\mu(\psi_0+\Delta\psi)-\mu(\psi_0)=(1-\mu(\psi_0))\,\mu(\Delta\psi)}
\label{eq:temporal-EFE-Q}}
\end{equation}
Equivalently, the normalized incremental response depends only on the deviation:
\begin{equation}
\fiteq{\frac{\mu(\psi_0+\Delta\psi)-\mu(\psi_0)}{1-\mu(\psi_0)}=\mu(\Delta\psi).
\label{eq:temporal-EFE-normalized}}
\end{equation}
\end{theorem}
\begin{proof}
Insert $\psi_1=\psi_0$ and $\psi_2=\Delta\psi$ into Eq.~\eqref{eq:saturation-union-Q}:
\[
\fiteq{\begin{aligned}
\mu(\psi_0+\Delta\psi)&=1-(1-\mu(\psi_0))(1-\mu(\Delta\psi))\\
&=\mu(\psi_0)+(1-\mu(\psi_0))\mu(\Delta\psi).
\end{aligned}}
\]
Rearrange to obtain~\eqref{eq:temporal-EFE-Q}.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Unique Local Temporal Invariant}
%-----------------------------------------------------------------------------

We identify the \emph{unique} local scalar that represents the microsector ``increment'' induced by time evolution along a chosen screen flow.

\paragraph{Setup (DFD observer dictionary).}
Let $u^\mu$ be the unit timelike 4-velocity field of the cosmological screen flow (comoving congruence in the dictionary), and let $\psi(x)$ be the DFD scalar.
The screen-background field $\psi_0$ is the $\psi$-screen solution already present in the cosmology section (Sec.~\ref{sec:cosmology}).

\begin{definition}[Local temporal increment density]\label{def:temporal-increment}
\begin{equation}
\fiteq{\dot{\psi}:=u^\mu\nabla_\mu\psi,\qquad
\dot{\psi}_0:=u^\mu\nabla_\mu\psi_0,\qquad
\Delta:=\frac{c}{a_0}\,\bigl|\dot{\psi}-\dot{\psi}_0\bigr|.
\label{eq:Delta-def}}
\end{equation}
Here $a_0 = 2\sqrt{\alpha}\,cH_0$ is the MOND acceleration scale; the combination $c/a_0$ has units of time, so $\Delta$ is dimensionless.
\end{definition}

\begin{theorem}[Temporal segment identification]\label{thm:segment-id}
Among all local scalars built from $\nabla\psi$ and the screen flow $u^\mu$, the quantity $\Delta$ in Eq.~\eqref{eq:Delta-def} is the unique choice (up to a constant factor) that satisfies:
\begin{enumerate}
\item \textbf{Reparameterization covariance:} invariance under reparameterizations of the flow parameter along $u^\mu$.
\item \textbf{Segment additivity:} for concatenated microsector segments along the flow, the total ``increment'' equals the sum of segment increments.
\item \textbf{Reference invariance:} the amplitude vanishes when $\psi = \psi_0$ (the background).
\end{enumerate}
\end{theorem}

\begin{proof}
A local scalar depending on $\nabla\psi$ and $u^\mu$ at first-derivative order must be of the form $f(u^\mu\nabla_\mu\psi)$.
Segment additivity applies to the integrated increment $\int u^\mu\nabla_\mu\psi\,d\lambda$,
so the deviation from the background flow is $u^\mu\nabla_\mu(\psi-\psi_0)=\dot\psi-\dot\psi_0$.
Reference invariance forces subtraction of $\dot\psi_0$.
Dimensionlessness requires normalization by $a_\star/c$, yielding $\Delta$.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{No-Go Lemma: Quadratic Invariant Gives $w \to 1/2$}
%-----------------------------------------------------------------------------

Before proving the dust branch, we establish why the naive k-essence identification fails.

\begin{lemma}[No-go: quadratic invariant]\label{lem:no-go-quadratic}
Define the quadratic temporal invariant $Q_t:= (u^\mu\nabla_\mu\psi)^2$ and suppose the constitutive law is
\begin{equation}
\fiteq{K'(Q_t) = \mu(\sqrt{Q_t}) = \frac{\sqrt{Q_t}}{1+\sqrt{Q_t}}.
\label{eq:wrong-constitutive}}
\end{equation}
Then near $Q_t \to 0$:
\begin{equation}
\fiteq{K(Q_t) = \frac{2}{3}Q_t^{3/2} + \mathcal{O}(Q_t^2),
\label{eq:K-wrong-expansion}}
\end{equation}
and the effective equation of state satisfies
\begin{equation}
\fiteq{w:= \frac{p}{\rho} \to \frac{1}{2} \qquad (Q_t \to 0).
\label{eq:w-half}}
\end{equation}
This is \textbf{not dust}.
\end{lemma}

\begin{proof}
Integrating~\eqref{eq:wrong-constitutive} with $q:= \sqrt{Q_t}$:
\[
\fiteq{\begin{aligned}
K(Q_t) &= \int_0^{Q_t} \!\mu(\sqrt{s})\,ds = 2\int_0^q \frac{q'^2}{1+q'}\,dq' \\
&= q^2 - 2q + 2\ln(1+q).
\end{aligned}}
\]
Taylor expanding at $q \to 0$: $K = \frac{2}{3}q^3 + \mathcal{O}(q^4) = \frac{2}{3}Q_t^{3/2} + \mathcal{O}(Q_t^2)$.

For the k-essence stress-energy with $p = K$ and $\rho = 2Q_t K'(Q_t) - K$:
\[
\fiteq{\begin{aligned}
\rho &= 2Q_t \cdot \frac{\sqrt{Q_t}}{1+\sqrt{Q_t}} - \frac{2}{3}Q_t^{3/2} + \cdots \\
&= \frac{4}{3}Q_t^{3/2} + \mathcal{O}(Q_t^2).
\end{aligned}}
\]
Thus $w = p/\rho = \bigl(\frac{2}{3}Q_t^{3/2}\bigr)\big/\bigl(\frac{4}{3}Q_t^{3/2}\bigr) = 1/2$.
\end{proof}

\begin{remark}[Why this matters]
Lemma~\ref{lem:no-go-quadratic} proves we did not cherry-pick the dust result. The $S^3$ composition law alone, with a naive quadratic identification, gives $w = 1/2$---radiation-like, not dust. The dust branch requires the \emph{deviation-invariant} closure below.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Dust Branch from Deviation-Invariant Closure}
%-----------------------------------------------------------------------------

\paragraph{Microsector-to-EFT identification (deviation-invariant).}
The temporal analog of the spatial AQUAL closure, consistent with Theorem~\ref{thm:temporal-deviation-Q}, uses the \emph{linear} deviation $\Delta$:
\begin{equation}
\fiteq{\boxed{\mathcal{L}_{\rm temp}=\frac{a_\star^2}{8\pi G}\,K(\Delta),\qquad K'(\Delta)=\mu(\Delta)=\frac{\Delta}{1+\Delta}}
\label{eq:K-Delta}}
\end{equation}
where $\Delta$ is the deviation invariant~\eqref{eq:Delta-def}. This uses the \emph{same} $\mu$ already fixed by the $S^3$ composition law. \emph{Scope:} the deviation-invariant temporal EFE structure (Theorem~\ref{thm:temporal-deviation-Q}) is forced by the composition law, but the specific kinetic identification $K'(\Delta)=\mu(\Delta)$ is the \emph{minimal natural} choice (the temporal analog of the spatial AQUAL closure), \emph{not} a uniquely forced identity: Lemma~\ref{lem:no-go-quadratic} exhibits a distinct, mathematically admissible identification $K'(Q_t)=\mu(\sqrt{Q_t})$ (which yields $w\to\tfrac12$, not dust). The dust branch (Theorem~\ref{thm:dust-branch}) is therefore a conditional consequence of \emph{adopting} $K'=\mu$, not of the composition law alone.

\begin{lemma}[Shift symmetry current]\label{lem:shift-current}
Because $\mathcal{L}_{\rm temp}$ depends on $\psi$ only through $\dot\psi$ (via $\Delta$), it is invariant under $\psi\mapsto\psi+\text{const}$ and yields a conserved current:
\begin{equation}
\fiteq{\nabla_\mu J^\mu=0,\qquad
J^\mu=\frac{a_\star^2}{8\pi G}\,K'(\Delta)\,\frac{c}{a_\star}\,\mathrm{sgn}(\dot\psi-\dot\psi_0)\,u^\mu.
\label{eq:conserved-current}}
\end{equation}
\end{lemma}

\begin{theorem}[Dust branch]\label{thm:dust-branch}
In a homogeneous FRW dictionary with $u^\mu=(1,0,0,0)$, solutions near the screen background satisfy:
\begin{equation}
\fiteq{a^3\,\mu(\Delta)=\text{const},\qquad \Delta\propto a^{-3}\quad(\Delta\ll 1),
\label{eq:Delta-scaling}}
\end{equation}
and their effective equation of state and sound speed obey
\begin{equation}
\fiteq{\boxed{w:=\frac{p}{\rho}\to 0,\qquad c_s^2\to 0 \quad\text{as}\quad \Delta\to 0.}
\label{eq:dust-eos}}
\end{equation}
\end{theorem}

\begin{proof}
From~\eqref{eq:conserved-current} and $\nabla_\mu J^\mu=0$, homogeneity gives $\frac{d}{dt}(a^3 J^0)=0$,
i.e.\ $a^3 K'(\Delta)=\text{const}$. Using $K'(\Delta)=\mu(\Delta)$ yields~\eqref{eq:Delta-scaling}.
For $\Delta\ll 1$, $\mu(\Delta)=\Delta+\mathcal{O}(\Delta^2)$, hence $\Delta\propto a^{-3}$.

For the stress-energy, take $p = \mathcal{L}_{\rm temp}=\frac{a_\star^2}{8\pi G}K(\Delta)$
and $\rho = \dot\psi\,\frac{\partial \mathcal{L}_{\rm temp}}{\partial \dot\psi}-\mathcal{L}_{\rm temp}$.
Near $\Delta = 0$: $K'(\Delta)=\Delta+\mathcal{O}(\Delta^2)$ and $K(\Delta)=\frac{1}{2}\Delta^2+\mathcal{O}(\Delta^3)$.
Thus:
\[
\fiteq{\begin{aligned}
\rho &= \frac{a_\star^2}{8\pi G}\left[\frac{c}{a_\star}\dot\psi_0\,\Delta+\mathcal{O}(\Delta^2)\right], \\
p &= \frac{a_\star^2}{8\pi G}\left[\frac{1}{2}\Delta^2+\mathcal{O}(\Delta^3)\right].
\end{aligned}}
\]
Therefore $w=p/\rho=\mathcal{O}(\Delta)\to 0$ as $\Delta\to 0$.
The adiabatic sound speed $c_s^2=dp/d\rho$ satisfies $dp/d\Delta = \mathcal{O}(\Delta)$ and $d\rho/d\Delta = \text{const}+\mathcal{O}(\Delta)$, hence $c_s^2\to 0$.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Summary: What is Theorem-Grade vs.\ Program}
%-----------------------------------------------------------------------------

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title={Theorem-Grade Results}]
\textbf{Proved from $S^3$ composition law + deviation invariance:}
\begin{enumerate}[nosep]
\item Temporal deviation invariance (Theorem~\ref{thm:temporal-deviation-Q})
\item Unique temporal segment scalar $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$ (Theorem~\ref{thm:segment-id})
\item The kinetic identification $K'(\Delta) = \mu(\Delta)$ (same $\mu$ as the spatial sector) is the \emph{minimal natural} choice consistent with the composition law, \emph{not} uniquely forced --- Lemma~\ref{lem:no-go-quadratic} shows an alternative identification is mathematically available (\textbf{natural/program-grade, not theorem-grade}); only the deviation-invariant EFE structure (item~1) and, conditionally on this choice, the dust branch (item~4) are proved
\item \textbf{Dust branch:} $w \to 0$, $c_s^2 \to 0$ as $\Delta \to 0$ (Theorem~\ref{thm:dust-branch})
\item \textbf{No-go:} Quadratic $K'(Q_t) = \mu(\sqrt{Q_t})$ gives $w \to 1/2$ (Lemma~\ref{lem:no-go-quadratic})
\end{enumerate}
\end{tcolorbox}

\noindent
\begin{tcolorbox}[colback=yellow!5!white, colframe=yellow!60!black, title={Program-Level (Not Claimed as Theorem)}]
\textbf{Requires further work:}
\begin{itemize}[nosep]
\item Full $P(k)$ shape matching $\Lambda$CDM (linear perturbation analysis)
\item Transfer function derivation in DFD dictionary
\item Quantitative confrontation with survey data (noting GR-sandbox / fiducial-processing issues)
\end{itemize}
The dust branch ($w \to 0$, $c_s^2 \to 0$) is the \emph{necessary condition} for CDM-like linear growth; proving the full $P(k)$ match is a program item.
\end{tcolorbox}

\begin{remark}[Critical distinction]
The dust branch emerges because the microsector responds to the \emph{linear} deviation $\Delta = |\dot\psi - \dot\psi_0|$, not the quadratic $Q_t = (\dot\psi - \dot\psi_0)^2$. This is forced by the temporal deviation invariance theorem, not chosen by fiat.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Primordial optical engine (inflation replacement): the background screen flow}
\label{app:optical-engine}

\emph{Which sector is the engine.} The temporal kinetic functional $K(\Delta)$ is \emph{not}
the primordial engine. By construction $\Delta=(c/a_0)|\dot\psi-\dot\psi_0|$ is the deviation
from the background screen flow $\dot\psi_0$ (reference invariance, \S\ref{app:temporal-completion}),
so on the homogeneous FRW background $\dot\psi=\dot\psi_0$ identically and $\Delta=0$
(App.~\ref{app:sigma8-optical}, AE.1). Hence $K(\Delta)$ governs \emph{deviations}
$\delta\psi$ --- the dust/growth sector --- and vanishes on the background; it cannot drive a
horizon engine. (An earlier draft of this subsection mis-assigned the engine to a high-$\Delta$
branch of $K$; that is corrected here.) The engine is instead the \emph{background screen flow}
$\psi_0(t)$ itself, which by Theorem~AE.1 is a \emph{flat direction}: the full DFD action is
satisfied for \emph{any} $\bar\psi(t)=\psi_0(t)$, at zero FRW energy density. Because
$n=e^{\psi_0}$ and the optical speed is $c_{\rm eff}=c\,e^{-\psi_0}$, an early phase with
$\psi_0<0$ (fast light) relaxing to $\psi_0(t_0)=0$ enlarges the optical causal horizon while
the comoving optical Hubble radius shrinks --- with no energy cost and no inflaton.

\begin{theorem}[Screen-flow horizon engine]
\label{thm:screen-flow-engine}
Let $a\propto t^{p}$ ($0<p<1$, $H=p/t$) and let the background screen flow be
$\psi_0(t)=\beta\ln(t/t_\star)$, so $c_{\rm eff}=c\,e^{-\psi_0}\propto t^{-\beta}$. The comoving
optical Hubble radius is $R^{\rm opt}_H=c_{\rm eff}/(aH)\propto t^{\,1-p-\beta}$ and the
accumulated optical horizon is
$d^{\rm opt}_{\rm hor}=\int_0^{t}c\,e^{-\psi_0}/a\,dt'\propto\int_0^{t}t'^{-\beta-p}dt'$. Both
the shrinking of $R^{\rm opt}_H$ and the divergence of $d^{\rm opt}_{\rm hor}$ hold iff
\begin{equation}
\begin{aligned}
\boxed{\,\beta>1-p\,}
\qquad&\Longrightarrow\qquad
\beta>\tfrac12\ (\text{radiation, }p=\tfrac12),\\
&\qquad\quad\ \beta>\tfrac13\ (\text{matter, }p=\tfrac23).
\end{aligned}
\label{eq:engine-beta}
\end{equation}
\end{theorem}

\begin{proof}
$e^{-\psi_0}\propto t^{-\beta}$ and $aH\propto t^{p-1}$ give $R^{\rm opt}_H\propto t^{1-p-\beta}$,
decreasing iff $1-p-\beta<0$. The horizon integrand $t'^{-\beta-p}$ diverges at the lower limit
iff $\beta+p\ge1$. Both conditions are $\beta>1-p$. (A power-law screen flow
$\psi_0=\psi_\infty-At^{-s}$, $A>0$, gives the same conclusion with an even stronger early
fast-light phase $c_{\rm eff}\propto e^{At^{-s}}\to\infty$.)
\end{proof}

\begin{theorem}[Branch selection by the optical arrow]
\label{thm:optical-momentum}
The present normalization $\psi_0(t_0)=0$ ($n=1$ today) admits three flat-direction branches:
$\dot\psi_0=0$ (static, $c_{\rm eff}=$const, \emph{no} engine); $\dot\psi_0<0$ (then $\psi_0>0$
and $c_{\rm eff}<c$ in the past, \emph{worsening} the horizon); and $\dot\psi_0>0$ (then
$\psi_0<0$, $c_{\rm eff}>c$ in the past, \emph{solving} it). A fast-light primordial boundary
condition $\psi_0(0^+)\to-\infty$ therefore selects $\dot\psi_0>0$ --- the engine branch.
\end{theorem}

\begin{remark}[Status: DFD \emph{permits} the engine; it does not yet \emph{predict} it]
\label{rem:optical-engine-grade}
The flat-direction status of $\psi_0$ (Thm.~AE.1) cuts both ways, and we state both edges. \emph{In
favor:} the fast-light screen flow costs \emph{zero} FRW energy, so it cannot disturb BBN or the
expansion history --- the engine is dynamically \emph{admissible}, and the horizon/mode-freeze
conclusion \eqref{eq:engine-beta} is exact. \emph{Against:} precisely because $\psi_0$ is a flat
direction, the action permits \emph{any} $\psi_0(t)$; it supplies \emph{no} dynamical force
selecting the fast-light flow or its rate $\beta>1-p$. The entire content therefore sits in
boundary/initial data --- a \emph{primordial optical arrow of time} $\psi_0(0^+)\to-\infty$ ---
which is the same \emph{class} of input as the thermodynamic arrow (a low-entropy boundary
condition, not a consequence of time-symmetric dynamics). Grade: \textbf{the engine is a
consistent, energy-free DFD option, not a forced prediction.} Closing it to theorem grade
requires a DFD-native principle that selects the fast-light $\psi_0(t)$ from the flat-direction
family --- e.g.\ a Wheeler--DeWitt / no-boundary condition on $\psi$, a finite-CS/topological
primordial constraint, or a maximal-optical-horizon variational principle. This selection is now \emph{booked} as the theory's one
cosmological axiom (Axiom~P1 below); we claim no inflaton and manufacture no obstruction. \emph{Consistency check:}
the primordial background $\bar\psi=\psi_0<0$ ($n<1$, fast light) is the cosmological mean and is
distinct from --- and compatible with --- the local $\psi>0$ ($n>1$) of bound structures, which
are overdensities on that background.
\end{remark}

\begin{tcolorbox}[colback=blue!4!white, colframe=blue!55!black, title=Axiom P1 (Primordial Optical State), breakable]
The universe's Wheeler--DeWitt state is the \emph{no-boundary optical state}, specified by three
clauses acting on disjoint summands of the configuration space:
\begin{enumerate}[nosep,leftmargin=*]
\item[\textbf{(i)}] \textbf{Branch:} the Euclidean ground-state branch $\Psi_0\propto e^{-S_E/\hbar}$
(the Hartle--Hawking-type choice; the rival $e^{+S_E}$/tunneling branch is thereby excluded);
\item[\textbf{(ii)}] \textbf{Internal preparation:} the empty-core WRT surgery state on the closed
internal $S^3$ (App.~\ref{app:chi-matter}, Brick~1), which forces the sector measure
$p_j=|S_{0j}|^2$;
\item[\textbf{(iii)}] \textbf{Flat direction:} $\psi_0(0^+)\to-\infty$ with the fast-light branch
$\dot\psi_0>0$ on the homogeneous flat direction (the past hypothesis).
\end{enumerate}
\end{tcolorbox}

\begin{theorem}[Consistency and non-redundancy of P1]\label{thm:P1-consistency}
P1 is consistent and non-redundant. The Wheeler--DeWitt fluctuation-sector state is already
\emph{unique} (Perron--Frobenius nondegeneracy on the coercive Sobolev sector) and the internal
vacuum is rigid (single-vacuum theorem), so P1's only genuine freedom is the one homogeneous
function $\psi_0(t)$ plus one branch bit. The three clauses constrain \emph{disjoint} direct
summands of the configuration space (Euclidean branch / internal $S^3$ / homogeneous flat
direction), and the $\beta$-flatness of $S_E$ on the flat direction decouples clause (iii) from
the $\Lambda$ anchor; hence no clause constrains another's sector and no contradiction can arise.
Jointly the axiom yields: the cosmological arrow (branch selection, Thm.~\ref{thm:optical-momentum}
context), the $\chi$ sector measure $p_j=|S_{0j}|^2$ (Brick~1--3), and the $E_0=\alpha^{57}M_P^4$
anchor. \emph{Conditional corollary:} given clause (i), regularity of $e^{-S_E}$ forces
$\mathrm{sign}(\beta)>0$ (the engine \emph{sign}); the engine \emph{rate} $\beta>1-p$ remains a
clause-(iii) datum.
\end{theorem}

\begin{antitheorem}[P1 is not derivable from the written dynamics]\label{antithm:P1}
The Wheeler--DeWitt operator is a \emph{real} operator; any unique normalizable state selected by
Perron--Frobenius positivity is therefore real and $T$-even, and a $T$-even state is structurally
incapable of carrying the arrow of time. Moreover all three action terms vanish identically on the
homogeneous flat direction, so the constraint restricted there is free motion with both branches
present symmetrically: normalizability and essential self-adjointness remove no freedom. Hence no
refinement of the written dynamics can derive clause (iii): P1 is a Past-Hypothesis-class input ---
the same epistemic class as thermodynamics' low-entropy past --- with the caveat
that the no-boundary internal state is \emph{near-maximal} entropy ($\simeq93\%$ of $\ln 61$), so P1
is a cosmogenic \emph{selection}, not a low-entropy postulate.
\end{antitheorem}

\noindent\textbf{Physical payoff (used in App.~\ref{app:scalaron-mass}).} Independently of how
$\psi_0(t)$ is selected, the true expansion Hubble during the $\chi$-relevant freeze-out,
$H_{\rm opt}$, is bounded by finite-CS survival: requiring the would-be dephasing
$\sigma_\theta^2\simeq N(H_{\rm opt}/2\pi f_\chi)^2$ to stay below $Q_{\rm CS}$ over $N\simeq60$
intervals gives
\begin{equation}
H_{\rm opt}<2\pi f_\chi\sqrt{Q_{\rm CS}/N}\simeq2.1\times10^{11}\ \mathrm{GeV},
\label{eq:hopt-bound}
\end{equation}
about $115\times$ below the de~Sitter-dictionary value $H_\star=8\pi f_\chi=2.4\times10^{13}$~GeV
(Thm.~\ref{thm:inflation-exclusion}). This forces the true primordial tensor amplitude far
below $4\alpha/\pi$ --- the tensor reclassification of App.~\ref{app:scalaron-mass}.

\begin{remark}[Scope: the engine is \emph{pre-BBN}; the validated late-time cosmology is unchanged]
\label{rem:engine-scope}
The fast-light screen flow ($\psi_0\to-\infty$) is confined to the deep \emph{pre-BBN}
primordial era; by big-bang nucleosynthesis $\psi_0$ has relaxed to $\psi_0\simeq0$ (a value the
flat direction permits it to hold). The engine therefore does \emph{not} rescale post-BBN
redshifts and leaves the validated DFD cosmology intact: real BBN and recombination
($z_\ast\simeq1100$), the measured $T_{\rm CMB}(z)=(1+z)\,T_0$, the cobaya-validated acoustic
scale, the Etherington duality $D_L=(1+z)^2D_A$ (DFD\_Main, App.~\ref{app:confrontation-matrix}), and
the parameter-free age $1/H_0=13.56$~Gyr (App.~\ref{app:dark-energy}). The separate \emph{late-time}
screen $\Delta\psi(z\!\lesssim\!1)\simeq0.27$ (the dark-energy substitute, validated against SNe;
App.~\ref{app:dark-energy}) is small and is \emph{not} the primordial engine. \emph{Caveat:}
a global optical redshift factor that modified post-BBN redshifts --- changing the inferred age,
or making $1+z$ partly optical at recombination --- would contradict all of the above and the
measured $T_{\rm CMB}(z)=(1+z)$; the engine makes no such claim, and any reading of the CMB
redshift as ``partly optical'' (the Robitaille-type non-cosmic-CMB position) is \emph{excluded}
here, not adopted.
\end{remark}

\vspace{2em}