%=============================================================================
\section{A Topological Link Between $H_0$ and $M_P$}\label{sec:G-derivation}
%=============================================================================

The preceding sections treated $M_P$ (equivalently $G$) as an input parameter. Here we present a dimensionless constraint linking $G$, $\hbar$, $H_0$, $c$, and $\alpha$, such that \textit{given one scale measurement, all others follow from topology}.

%-----------------------------------------------------------------------------
\subsection{The Dimensionless Invariant}
%-----------------------------------------------------------------------------

The primary claim is a purely dimensionless relation, now derived to theorem status via Gaussian mode integration on the finite-dimensional microsector (Appendix~\ref{app:O_alpha57_clean}):

\begin{proposition}[Topological Invariant --- Spectral-Action-Derived]\label{prop:invariant}
DFD predicts the following dimensionless constraint:
\begin{equation}
\fiteq{\boxed{\frac{G\hbar H_0^2}{c^5} = \alpha^{k_{\max} - N_{\rm gen}} = \alpha^{57}}
\label{eq:invariant}}
\end{equation}
where $k_{\max} = 60$ (Spin$^c$ index from Lemma~\ref{lem:E_bundle_kmax60}), $N_{\rm gen} = 3$ (generation count), and $\alpha$ is the fine-structure constant.

\textbf{Theorem-grade status (Appendix~\ref{app:O_alpha57_clean}):}
\begin{itemize}
\item The exponent $57 = k_{\max} - N_{\rm gen}$ is forced by primed-determinant scaling on the finite Toeplitz state space (Lemma~\ref{lem:O_det_scaling}, Corollary~\ref{cor:O_alpha57}).
\item The identification with the observed invariant $I = G\hbar H_0^2/c^5$ is derived via Gaussian mode integration on the finite-dimensional microsector (Lemmas~\ref{lem:O_KK}--\ref{lem:O_hierarchy}, Theorem~\ref{thm:O_I_equals_alpha57}).
\end{itemize}
\end{proposition}

This formulation has several advantages:
\begin{itemize}
\item \textbf{Dimensionless}: No unit conventions or hidden factors
\item \textbf{Symmetric}: Predicts $G$ from $H_0$ \textit{or} $H_0$ from $G$
\item \textbf{Falsifiable}: A single testable constraint
\end{itemize}

\paragraph{Bidirectionality.}
Given $(\alpha, \hbar, c)$ and a measured $G$, the invariant predicts $H_0$. Equivalently, given $H_0$ it predicts $G$. Neither is privileged as ``input''---the constraint is symmetric. This prevents any accusation that one quantity was ``chosen'' to match the other.

\paragraph{Error propagation.}
Taking logarithms and differentiating:
\begin{equation}
\fiteq{\frac{\delta G}{G} = -2\frac{\delta H_0}{H_0}
\label{eq:error-prop}}
\end{equation}
The precision of any $G$ prediction is limited by $H_0$ uncertainty. With current $H_0$ uncertainties of $\sim$1--2\%, the constraint tests $G$ at the $\sim$2--4\% level.

\paragraph{Equivalent form (Planck mass).}
Defining $M_P = \sqrt{\hbar c/G}$, the invariant becomes:
\begin{equation}
\fiteq{M_P = \alpha^{-(k_{\max} - N_{\rm gen})/2} \times \frac{\hbar H_0}{c^2} = \alpha^{-28.5} \times \frac{\hbar H_0}{c^2}
\label{eq:MP-master}}
\end{equation}

\paragraph{Numerical verification.}
Using CODATA values for $G$, $\hbar$, $c$, $\alpha$:
\begin{equation}
\fiteq{\begin{aligned}
\text{LHS: } \frac{G\hbar H_0^2}{c^5} &= 1.587 \times 10^{-122} \quad (\text{at } H_0 = 72.1~\text{km/s/Mpc}) \\
\text{RHS: } \alpha^{57} &= 1.586 \times 10^{-122}
\end{aligned}}
\end{equation}
Agreement to 0.03\% on a quantity spanning 122 orders of magnitude.

%-----------------------------------------------------------------------------
\subsection{Implication for the Cosmological Constant Problem}
%-----------------------------------------------------------------------------

The cosmological constant problem asks: why is $\rho_\Lambda / \rho_{\rm Planck} \approx 10^{-123}$? This is often called ``the worst fine-tuning in physics'' because naive quantum field theory predicts $\rho_\Lambda \sim \rho_{\rm Planck}$.

If Eq.~\eqref{eq:invariant} holds, the ratio is \textit{topologically constrained}:

\begin{proposition}[Cosmological Constant Scaling]\label{prop:CC-scaling}
The critical density satisfies:
\begin{equation}
\fiteq{\frac{\rho_c}{\rho_{\rm Planck}} = \frac{3}{8\pi} \times \frac{G\hbar H_0^2}{c^5} = \frac{3}{8\pi}\alpha^{57} \approx 1.9 \times 10^{-123}
\label{eq:CC-solution}}
\end{equation}
With $\Omega_\Lambda \approx 0.7$: $\rho_\Lambda/\rho_{\rm Planck} \approx 1.3 \times 10^{-123}$.
\end{proposition}

\noindent\emph{Scope.} This identity fixes the \emph{exponent} of the vacuum scale; the $O(1)$
coefficient enters through the clock-dictionary normalization (Rem.~\ref{rem:clock-credit}), and the
statement is microsector-scoped --- the Standard-Model matter-sector vacuum energy is not part of
this identity and remains unaccounted (Rem.~\ref{rem:au-scope}).

\begin{proof}[Derivation]
The critical density is $\rho_c = 3H_0^2/(8\pi G)$. The Planck density is $\rho_{\rm Planck} = c^5/(\hbar G^2)$. Thus:
\begin{equation}
\fiteq{\frac{\rho_c}{\rho_{\rm Planck}} = \frac{3H_0^2}{8\pi G} \times \frac{\hbar G^2}{c^5} = \frac{3}{8\pi} \times \frac{G\hbar H_0^2}{c^5}}
\end{equation}
Substituting Eq.~\eqref{eq:invariant} gives the result.
\end{proof}

The exponent $57 = k_{\max} - N_{\rm gen} = 60 - 3$ traces to topology:
\begin{itemize}
\item $k_{\max} = 60$: the Spin$^c$ index $\chi(\mathbb{C}P^2, E)$ for twist bundle $E = \mathcal{O}(9) \oplus \mathcal{O}^{\oplus 5}$ (Lemma~\ref{lem:E_bundle_kmax60})
\item $N_{\rm gen} = 3$: the generation count (discrete input; the flux-product expression is a bookkeeping identity --- App.~F)
\end{itemize}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Cosmological Constant: Spectral-Action-Derived Resolution (Appendix~\ref{app:O_alpha57_clean})]
The ``fine-tuning'' of $10^{-123}$ is now derived via Gaussian mode integration:
\begin{equation}
\fiteq{\frac{\rho_c}{\rho_{\rm Planck}} = \frac{3}{8\pi}\alpha^{k_{\max} - N_{\rm gen}} = \frac{3}{8\pi}\alpha^{57} \approx 10^{-123}}
\end{equation}
The exponent 57 is topologically forced by primed-determinant scaling (Corollary~\ref{cor:O_alpha57}). The identification with the physical hierarchy is derived via Gaussian mode integration over the 57 nonzero KK modes (Lemmas~\ref{lem:O_KK}--\ref{lem:O_hierarchy}).
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Testable Consequence: The Hubble Constant}
%-----------------------------------------------------------------------------

Interpreted as an $H_0$ prediction from $(G, \alpha)$, the invariant Eq.~\eqref{eq:invariant} yields:
\begin{equation}
\fiteq{H_0 = \sqrt{\frac{\alpha^{57} c^5}{G\hbar}} = \frac{\alpha^{28.5}}{t_P}
\label{eq:H0-prediction}}
\end{equation}
where $t_P = \sqrt{\hbar G/c^5}$ is the Planck time.

Using CODATA values for $G$, $\hbar$, $c$, $\alpha$:
\begin{equation}
\fiteq{\boxed{H_0^{\rm DFD} = 72.09~\text{km/s/Mpc}}
\label{eq:H0-DFD-value}}
\end{equation}

This is a \textbf{zero-parameter prediction}---the value follows entirely from the microsector derivation of $\alpha$ and the topological exponent $57 = k_{\max} - N_{\rm gen}$.

\paragraph{Comparison with observations.}
Recent JWST observations provide high-precision tests of this prediction. Two major collaborations have released results:

\begin{table}[h]
\centering
\caption{Hubble constant: DFD prediction vs.\ observations.}\label{tab:H0-comparison}
\begin{ruledtabular}
\scriptsize
\fittab{%
\begin{tabular}{lcccc}
\textbf{Source} & $H_0$ & \textbf{Uncert.} & \textbf{$\Delta/\sigma$} & \textbf{Ref.} \\
\hline
\textbf{DFD prediction} & \textbf{72.09} & (theory) & --- & This work \\
\hline
\multicolumn{5}{c}{\textit{Local distance ladder (JWST)}} \\
SH0ES JWST combined & 72.6 & $\pm 2.0$ & $-0.3\sigma$ & \cite{Riess2024JWST} \\
SH0ES JWST Cepheids & 73.4 & $\pm 2.1$ & $-0.6\sigma$ & \cite{Riess2024JWST} \\
SH0ES JWST TRGB & 72.1 & $\pm 2.2$ & $0.0\sigma$ & \cite{Riess2024JWST} \\
SH0ES JWST JAGB & 72.2 & $\pm 2.2$ & $-0.05\sigma$ & \cite{Riess2024JWST} \\
CCHP TRGB (HST+JWST) & 70.4 & $\pm 1.9$ & $+0.9\sigma$ & \cite{Freedman2024JWST} \\
CCHP JAGB (JWST) & 67.8 & $\pm 2.7$ & $+1.6\sigma$ & \cite{Freedman2024JWST} \\
\hline
\multicolumn{5}{c}{\textit{CMB-inferred (model-dependent)}} \\
Planck $\Lambda$CDM & 67.4 & $\pm 0.5$ & $+9.4\sigma$ & \cite{Planck2018VI} \\
\end{tabular}%
}
\end{ruledtabular}
\vspace{1mm}
{\scriptsize Units: km/s/Mpc. $\Delta/\sigma \equiv (H_0^{\rm DFD} - H_0^{\rm obs})/\sigma_{\rm obs}$.}
\end{table}

\paragraph{Assessment.}
The DFD prediction $H_0 = 72.09$ km/s/Mpc lies near recent JWST distance-ladder estimates ($\sim$72--73 km/s/Mpc from SH0ES) but above some TRGB/JAGB-based determinations ($\sim$68--70 km/s/Mpc from CCHP). The two JWST teams obtain systematically different results, with the disagreement not yet resolved~\cite{Riess2024JWST,Freedman2024JWST}.

Key observations:
\begin{itemize}
\item The DFD prediction is consistent with all SH0ES JWST measurements within $1\sigma$
\item CCHP results lie $1$--$2\sigma$ below the DFD prediction
\item The Planck CMB-inferred value disagrees at $9.4\sigma$
\end{itemize}

\paragraph{The Hubble tension in DFD.}
The ``Hubble tension''---the $\sim$5 km/s/Mpc discrepancy between local and CMB-inferred values---has a natural interpretation in DFD:
\begin{itemize}
\item \textbf{Local measurements} (Cepheids, SNe Ia) measure actual photon propagation through the $\psi$-field, yielding $H_0 \approx 72$--73 km/s/Mpc
\item \textbf{CMB inference} uses $\Lambda$CDM to extrapolate from $z \sim 1100$, but this model does not account for the $\psi$-screen optical bias (Section~\ref{sec:psi-tomography})
\end{itemize}
The CMB is observed through an accumulated $\Delta\psi \approx 0.30$ (from $\psi$-tomography), which biases distance inferences in the standard framework. The ``tension'' is not a measurement error but a \textbf{model error} in $\Lambda$CDM.

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=The G-H$_0$ Link: Sharp Prediction]
\textbf{Prediction:} $H_0 = 72.09$ km/s/Mpc (zero free parameters)\\[2pt]
\textbf{Status:} Consistent with SH0ES JWST ($<1\sigma$); above CCHP TRGB/JAGB ($1$--$2\sigma$); incompatible with Planck $\Lambda$CDM ($9.4\sigma$)\\[2pt]
\textbf{Interpretation:} The Hubble tension reflects the $\psi$-screen optical bias ignored by $\Lambda$CDM\\[2pt]
\textbf{Test:} As JWST completes its full Cepheid sample ($\sim$2025--2026), the prediction becomes testable at sub-percent precision
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Cosmological Evolution of $G$}
%-----------------------------------------------------------------------------

If the topological constraint Eq.~\eqref{eq:invariant} holds at all times, then as $H(t)$ evolves, so must $G(t)$:
\begin{equation}
\fiteq{G(t) = \frac{\alpha^{57} c^5}{\hbar H(t)^2}
\label{eq:G-evolution}}
\end{equation}

As the universe expands and $H$ decreases, $G$ \textit{increases}. Differentiating
Eq.~\eqref{eq:G-evolution} with $\alpha,\hbar,c$ frozen gives the photon-frame rate
\begin{equation}
\fiteq{\dot G/G = -2\,\dot H/H = +3 H_0\,\Omega_m \approx +6.6\times10^{-11}\,\mathrm{yr}^{-1}}
\end{equation}
(matter-dominated, $\Omega_m\approx0.30$, $H_0=72.1$~km/s/Mpc), which exceeds the
lunar-laser-ranging and binary-pulsar bound $|\dot G/G|\lesssim10^{-13}\,\mathrm{yr}^{-1}$
by a factor $\sim10^3$.

\paragraph{Why haven't we detected varying $G$?}
Measurements of $G$ (lunar laser ranging, binary pulsars) use \textit{atomic} references. In DFD, atomic-frame measurements give:
\begin{equation}
\fiteq{G_{\rm atomic} = G_{\rm photon} \times e^{2\psi_{\rm cosmic}}}
\end{equation}
Reconciliation with the bound requires the atomic-frame screen to evolve as
$2\dot\psi_{\rm cosmic} = -3 H_0\,\Omega_m$, finely cancelling the photon-frame
drift. \emph{This rate is not supplied by the present construction:} the FRW
flat-direction theorem (Thm.~\ref{thm:aext-FRW}, App.~AE) makes the homogeneous
cosmological $\dot\psi$ non-dynamical ($\dot\psi=\dot\psi_0$ identically, a flat
direction of the action), so $e^{2\psi_{\rm cosmic}}$ cannot vary in time as
required. We therefore flag the \emph{all-epoch} promotion of the $G$--$H_0$
invariant as a \emph{conditional self-consistency falsifier}: absent a dynamical
cosmological $\dot\psi$ (currently excluded), the predicted
$+6.6\times10^{-11}\,\mathrm{yr}^{-1}$ photon-frame drift stands in $\sim\!10^3\times$
tension with lunar-laser-ranging. The cavity-atom comparison below probes the
photon-vs-atomic \emph{difference} directly.

This is precisely what the cavity-atom LPI test (Section~\ref{sec:cavity}) can detect: the \textit{difference} between photon-frame and atomic-frame measurements of gravitational coupling.

\paragraph{Connection to early universe.}
At the CMB epoch ($z \sim 1100$), $H(z)/H_0 \sim 33000$. In the photon frame:
\begin{equation}
\fiteq{\frac{G(z=1100)}{G_0} = \left(\frac{H_0}{H(z)}\right)^2 \sim 10^{-9}}
\end{equation}
Gravity was vastly \textit{weaker} in the early universe (photon frame). This may affect interpretation of BBN and CMB constraints on $G$.

\paragraph{Companion note: epoch-consistency and the galactic transition scale.}
The present subsection promotes the topological closure $G\hbar H_0^2/c^5=\alpha^{57}$ from a present-epoch statement (at $z=0$, Sec.~\ref{sec:G-derivation}) to an all-epoch statement $G(t)\hbar H(t)^2/c^5=\alpha^{57}$ (Eq.~\eqref{eq:G-evolution}). A companion note~\cite{Alcock2026AstarZ} observes that the galactic transition acceleration $a_*=2\sqrt{\alpha}\,cH_0$ (Appendix~\ref{app:mu-derivation}, Theorem~\ref{thm:a-star-final}) admits the same epoch-consistency promotion as a program-grade result, $a_*(z)=2\sqrt{\alpha}\,cH(z)$. The argument is conditional: it requires that the Appendix~N derivation, which uses the cosmic horizon-scale integration of the $\psi$ field equation at $z=0$, carries through at earlier epochs with $H_0$ replaced by $H(z)$. The result is not a theorem of the local field equation; it is a consequence of applying the same epoch-consistency rule used here to the Appendix~N functional. High-redshift galactic kinematics (JWST, DESI) discriminate. DFD's \emph{forced} background is the $\Lambda$-carrying optical-time Hamiltonian~\eqref{eq:opt-hamiltonian} (with the frozen $\Lambda=\alpha^{57}M_P^4$ term), giving the $\Lambda$CDM-shape expansion $H^2/H_0^2=\Omega_m(1+z)^3+\Omega_\Lambda$ and hence the \emph{unique} DFD value $a_*(z{=}1)/a_*(0)=H(1)/H_0=\sqrt{0.315\cdot 8+0.685}=1.79$. The matter-only ($\Omega_m{=}1$) curve that would give $\approx 2.83$ is \emph{not} a rival DFD prediction: it is the $\Lambda$CDM-comparison baseline---the denominator of the optical $\psi$-screen distance ratio $\Delta\psi=\ln[D_L^{\rm obs}/D_L^{\rm matter}]$---a reconstruction yardstick, not a DFD expansion history (setting $\Lambda=0$ would contradict~\eqref{eq:opt-hamiltonian}). Caveat: $\alpha^{57}$ fixes the $\Lambda$ \emph{scale}, not the $\Omega_\Lambda=0.685$ split, which inherits the Planck $\Omega_m=0.315$ normalization. A frozen $a_*$ independent of $z$ would falsify the epoch-promotion.

%-----------------------------------------------------------------------------
\subsection{The homogeneous optical-time Hamiltonian (background expansion)}
\label{sec:optical-time-friedmann}
%-----------------------------------------------------------------------------

Theorem~\ref{thm:aext-FRW} establishes that the homogeneous mode of the
\emph{static} field action is a flat direction: with the source written as the
density contrast $\rho-\bar\rho$, a homogeneous configuration carries no net
source and the Euler--Lagrange equation places no constraint on $\bar\psi(t)$.
This is the correct statement for the \emph{perturbation} (galactic/AQUAL)
sector, and it is the reason the cosmic expansion rate does not follow from the
spatial field equation. It does \emph{not} imply that DFD lacks an expansion
law: the background rate lives in a separate, global \emph{energy} sector, which
we now write down.

\paragraph{The optical-time bridge.}
The DFD optical redshift relation $1+z=e^{\psi/2}$ together with the
kinematic definition $1+z=1/a(t)$ gives $\psi(t)=-2\ln a(t)$, hence the
\emph{exact} identification of the Hubble rate with the rate of the homogeneous
optical field,
\begin{equation}
  \boxed{\;H \;=\; -\tfrac12\,\dot\psi\;}\,,
  \qquad e^{\psi}=a^{-2}.
  \label{eq:H-psidot}
\end{equation}

\paragraph{The Hamiltonian constraint.}
A comoving dust shell of physical radius $R(t)=a(t)\chi$ encloses a fixed mass
$M=\tfrac{4\pi}{3}\rho R^3$; in the weak-field (Newtonian-potential) limit its
first integral of motion is $\tfrac12\dot R^2-GM/R=E$, with $2E/R^2\equiv-kc^2/a^2$.
Adding the vacuum term $\Lambda c^2/3$ (with $\Lambda=\alpha^{57}M_P^4$ fixed by
the clock-dictionary closure $G\hbar H^2/c^5=\alpha^{57}$) and rewriting through
\eqref{eq:H-psidot} gives the \textbf{homogeneous optical-time Hamiltonian
constraint}
\begin{equation}
  \boxed{\;
  \mathcal{H}_{\rm opt}\;\equiv\;
  \tfrac14\,\dot\psi^{2}
  \;-\;\frac{8\pi G}{3}\,\rho
  \;+\;k c^2 e^{\psi}
  \;-\;\frac{\Lambda c^2}{3}
  \;=\;0\;}
  \label{eq:opt-hamiltonian}
\end{equation}
paired with the continuity equation, which in optical-time variables reads
\begin{equation}
  \boxed{\;\dot\rho \;=\; \tfrac32\,\dot\psi\!\left(\rho+\frac{p}{c^2}\right)\;}
  \qquad\Longleftrightarrow\qquad
  \dot\rho+3H\!\left(\rho+\frac{p}{c^2}\right)=0 .
  \label{eq:opt-continuity}
\end{equation}
Equation~\eqref{eq:opt-hamiltonian} is the optical-time form of the Friedmann
energy constraint; via \eqref{eq:H-psidot} it is identically
$H^2=\tfrac{8\pi G}{3}\rho-kc^2/a^2+\tfrac{\Lambda c^2}{3}$. No Einstein tensor
is invoked: the background rate is the kinetic energy of the homogeneous optical
field $\psi(t)$ balanced against matter density, curvature, and vacuum energy.

\begin{theorem}[Self-consistency of the optical-time sector]
\label{thm:opt-selfconsistent}
The constraint \eqref{eq:opt-hamiltonian} and continuity
\eqref{eq:opt-continuity}, with the bridge \eqref{eq:H-psidot}, reproduce the
\emph{second} (acceleration) Friedmann equation with the correct relativistic
source,
\begin{equation}
  \frac{\ddot a}{a}\;=\;-\tfrac12\ddot\psi+\tfrac14\dot\psi^{2}
  \;=\;-\frac{4\pi G}{3}\!\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda c^2}{3}.
  \label{eq:opt-acceleration}
\end{equation}
\end{theorem}
\begin{proof}
Differentiate \eqref{eq:opt-hamiltonian} in $t$, substitute
\eqref{eq:opt-continuity} for $\dot\rho$, eliminate $kc^2e^{\psi}$ using
\eqref{eq:opt-hamiltonian}, and insert $\ddot a/a=\dot H+H^2=-\tfrac12\ddot\psi
+\tfrac14\dot\psi^2$ from \eqref{eq:H-psidot}. The result is
\eqref{eq:opt-acceleration} identically (symbolically verified, residual $0$).
The active gravitational mass $\rho+3p/c^2$ \emph{emerges} from the constraint
algebra although the shell motivation was pressureless ($p=0$): the sector is
the full Friedmann system, not its matter special case.
\end{proof}

\paragraph{Consequence for the MOND acceleration scale.}
Because the background rate is now an intrinsic field quantity, the optical
transition scale is set by the rate of the homogeneous optical field itself,
\begin{equation}
  \boxed{\;a_*(z)\;=\;2\sqrt{\alpha}\,c\,H(z)\;=\;\sqrt{\alpha}\,c\,|\dot\psi|\;}.
  \label{eq:a0-psidot}
\end{equation}
The redshift dependence of $a_*$ is therefore not an imported clock but the
evolution of $\psi(t)$ governed by \eqref{eq:opt-hamiltonian}: the MOND knee
tracks the kinetic rate of the cosmic optical field, and the sign
$\partial_z a_*>0$ follows from $|\dot\psi|=2H$ rising with $z$.

\paragraph{Scope (two-channel).}
The genuinely DFD-native content is the bridge \eqref{eq:H-psidot} and the
identity \eqref{eq:a0-psidot}: the expansion rate and the MOND scale are
expressed through the single optical field $\psi$ with no additional clock.
The constraint \eqref{eq:opt-hamiltonian} is \emph{posited} as the background
Hamiltonian, motivated by the Newtonian-limit shell first integral; it is a
separate sector from the AQUAL perturbation equation and is not derived from the
static action. Its coefficient $8\pi G/3$ is the standard one, so the sector
\emph{reproduces} the Friedmann background (a consistency statement, not a new
expansion prediction); the curvature constant $k$ is an integration constant
fixed separately. This resolves the apparent tension with
Theorem~\ref{thm:aext-FRW}: that theorem governs the AQUAL action's homogeneous
mode (contrast source), while \eqref{eq:opt-hamiltonian} is the global energy
constraint (total source) --- the standard background/perturbation split,
written in DFD's optical variables.

%-----------------------------------------------------------------------------
\subsection{The Parameter Structure}
%-----------------------------------------------------------------------------

If Eq.~\eqref{eq:invariant} holds, DFD has the following structure:

\begin{table}[h]
\centering
\caption{DFD input/output structure.}\label{tab:parameter-count}
\begin{ruledtabular}
\scriptsize
\fittab{%
\begin{tabular}{lll}
\textbf{Category} & \textbf{Quantity} & \textbf{Source} \\
\hline
\multirow{3}{*}{Topological} & $k_{\max} = 60$ & $\chi(\mathbb{C}P^2, E)$ \\
 & $N_{\rm gen} = 3$ & Input (App.~F) \\
 & $\alpha^{-1} = 137$ & CS quant. \\
\hline
Observational & $H_0$ or $G$ & Measured \\
\hline
\multirow{5}{*}{Derived} & $G$ or $H_0$ & Eq.~\eqref{eq:invariant} \\
 & $v = 246$ GeV & $M_P \alpha^8 \sqrt{2\pi}$ \\
 & $\rho_c/\rho_{\rm Pl}$ & Eq.~\eqref{eq:CC-solution} \\
 & All masses & $\alpha$-hierarchy \\
 & All mixings & $\mathbb{C}P^2$ geom. \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

The resulting one-anchor tower of scales is displayed as the
$\alpha$-exponent ladder of Fig.~\ref{fig:alpha-ladder}.

\input{fig_alpha_ladder}

\paragraph{Parameter counting.}
DFD introduces \textbf{no continuous fit parameters}. The discrete topological sector is uniquely determined by Standard Model structure:
\begin{itemize}
\item Hypercharge integrality admits $q_1\in\{3,6\}$; $q_1=3$ selected (Lemma~\ref{lem:q1}, Rem.~\ref{rem:q1-honest})
\item Minimal integer-charge lift gives $\mathcal{O}(9) = L_Y^{\otimes 3}$
\item Five hypercharged chiral multiplet types fix $n = 5$
\item Within $E = \mathcal{O}(a) \oplus \mathcal{O}^{\oplus n}$, minimal-padding uniquely selects $(a,n) = (9,5)$ with $k_{\max} = 60$
\end{itemize}

One scale measurement ($H_0$ or equivalently $G$) determines all dimensionful quantities via the invariant $G\hbar H_0^2/c^5 = \alpha^{57}$.

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Zero Continuous Parameters --- Spectral-Action-Derived (Appendix~\ref{app:O_alpha57_clean})]
DFD introduces no continuous fit parameters. Once the discrete topological sector is fixed by Standard Model structure ($k_{\max} = 60$, $N_{\rm gen} = 3$), the exponent in the dimensionless invariant
\begin{equation}
\fiteq{\frac{G\hbar H_0^2}{c^5} = \alpha^{57}}
\end{equation}
is topologically forced by primed-determinant scaling (Corollary~\ref{cor:O_alpha57}). Gaussian mode integration over the finite-dimensional microsector derives the identification with the physical hierarchy (Theorem~\ref{thm:O_I_equals_alpha57}). One scale measurement ($H_0$ or $G$) then fixes all dimensionful quantities.
\end{tcolorbox}

