
%=============================================================================
% Appendix O (v3.3): α^{57} exponent from primed determinant scaling,
% with the observer dictionary DERIVED via Gaussian mode integration
% on the finite-dimensional microsector.
%
% UPGRADE from earlier versions: Definition O.3 ("dictionary postulate")
% is replaced by Theorem O.3 ("spectral-action-derived"), supported by
% three lemmas (KK reduction, uniform gauge normalization, hierarchy
% identification). The finite dimensionality of the microsector (60 modes)
% eliminates all cutoff, renormalization, and scheme ambiguities.
%=============================================================================

\section{The $\alpha^{57}$ Mode-Count Exponent and the $G$--$H_0$--$\alpha$ Invariant}
\label{app:O_alpha57_clean}

\subsection{O.1\quad Mathematical core: primed-determinant scaling fixes the exponent}
Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space of dimension $k_{\max}$, and let
$\mathcal{K}:\mathcal{H}\to\mathcal{H}$ be a self-adjoint, positive semidefinite operator with
$\dim\ker(\mathcal{K})=N_{\rm gen}$.
Denote by $\det'(\mathcal{K})$ the \emph{primed determinant} over the nonzero spectrum of $\mathcal{K}$.

\begin{lemma}[Primed determinant scaling]
\label{lem:O_det_scaling}
For any $g>0$,
\begin{equation}
\fiteq{\det'\!\big(g\,\mathcal{K}\big)\;=\; g^{\,k_{\max}-N_{\rm gen}}\;\det'(\mathcal{K})\,.}
\end{equation}
\end{lemma}

\begin{proof}
Diagonalize $\mathcal{K}$ on $\mathcal{H}$ with eigenvalues $\{\lambda_i\}_{i=1}^{k_{\max}}$.
Exactly $N_{\rm gen}$ of these are zero; the remaining $N:=k_{\max}-N_{\rm gen}$ satisfy $\lambda_i>0$.
Then by definition $\det'(\mathcal{K})=\prod_{i=1}^{N}\lambda_i$ (product over the nonzero spectrum), and
$\det'(g\mathcal{K})=\prod_{i=1}^{N}(g\lambda_i)=g^{N}\prod_{i=1}^{N}\lambda_i$.
\end{proof}

\begin{definition}[Microsector hierarchy factor as a \emph{determinant ratio}]
\label{def:O_eps}
Define
\begin{equation}
\fiteq{\varepsilon(g)\;:=\;\frac{\det'(\mathcal{K})}{\det'(g\,\mathcal{K})}\,.}
\end{equation}
\end{definition}

\begin{corollary}[Topologically forced exponent]
\label{cor:O_alpha57}
If $k_{\max}=60$ and $N_{\rm gen}=3$, then
\begin{equation}
\fiteq{\varepsilon(g)=g^{-57}\,
\qquad\text{and in particular}\qquad
\varepsilon(\alpha^{-1})=\alpha^{57}\,.}
\end{equation}
\end{corollary}

\begin{proof}
Immediate from Lemma~\ref{lem:O_det_scaling} and Definition~\ref{def:O_eps} with $N=k_{\max}-N_{\rm gen}=57$.
\end{proof}

\paragraph{Provenance of the two integers.}
The two integers in the exponent $k_{\max} - N_{\rm gen} = 57$ have
\emph{distinct} provenance:
\begin{itemize}
\item $N_{\rm gen} = 3$ is a \textbf{discrete input} (the observed family
number; App.~\ref{app:generations}, Rem.~\ref{rem:ngen-input}). It is
numerically coincident with $\chi_{\rm top}(\mathbb{C}P^2) = 1+1+1 = 3$
(the Euler/de~Rham index), but that identification is not a Dirac index
theorem: the untwisted spin$^c$ index on $\mathbb{C}P^2$ is
$\mathrm{Td}(\mathbb{C}P^2)=1$, and the twisted per-multiplet indices do not
share a factor of $3$. We do not claim $N_{\rm gen}$ is derived.
\item $k_{\max} = \chi(\mathbb{C}P^2, E) = 60$ for the Spin$^c$ twisting bundle
$E = \mathcal{O}(9) \oplus \mathcal{O}^{\oplus 5}$, the total microsector
mode count via Spin$^c$--HRR --- exact arithmetic given the selected discrete
inputs $(q_1, N_{\rm gen}, n)=(3,3,5)$, and pinned empirically by the $0.0056$~ppm
$\alpha$ match (App.~\ref{app:generations}, Rem.~\ref{rem:independence}).
\end{itemize}
The Wilsonian integration in Sec.~\ref{sec:O-gaussian} then integrates out
the $k_{\max} - N_{\rm gen} = 57$ nonzero modes, giving the $\alpha^{57}$
suppression.

%-----------------------------------------------------------------------------
\subsection{O.2\quad Gaussian mode-integration realization}\label{sec:O-gaussian}
%-----------------------------------------------------------------------------

The ratio $\varepsilon(g)$ admits a concrete physical realization as the partition-function ratio
obtained by Gaussian integration over the nonzero-mode sector.
Let $\mathcal{K}_+$ denote $\mathcal{K}$ restricted to the nonzero spectrum,
and define for $g > 0$ the Gaussian normalization integral over $N = 57$ complex modes:
\begin{equation}
\label{eq:O_partition}
\fiteq{\begin{aligned}
\mathcal{Z}(g)\;&:=\;\int_{\mathbb{C}^{N}} \exp\!\Big(-\,\langle \phi,\,(g\mathcal{K}_+)\phi\rangle\Big)\,d^{2N}\phi
\\
&=\; \prod_{i=1}^{N} \frac{\pi}{g\,\lambda_i}
\;=\; \frac{\pi^N}{g^N\,\det'(\mathcal{K})}\,.
\end{aligned}}
\end{equation}
The ratio to the reference ($g = 1$) partition function is:
\begin{equation}
\fiteq{\frac{\mathcal{Z}(g)}{\mathcal{Z}(1)}
\;=\; g^{-N}
\;=\; g^{-57}\,
\label{eq:O_Zratio}}
\end{equation}
so the suppression factor is $\varepsilon(g) \equiv \mathcal{Z}(g)/\mathcal{Z}(1) = g^{-57}$.

\paragraph{Per-mode eigenvalue cancellation.}
Each complex mode $\phi_i$ contributes independently.
At coupling $g = \alpha^{-1}$ (gauge-normalized; see Lemma~\ref{lem:O_gauge_norm} below):
\begin{equation}
\fiteq{\frac{\int d^2\phi_i\,\exp\bigl(-(\lambda_i/\alpha)\,|\phi_i|^2\bigr)}
 {\int d^2\phi_i\,\exp\bigl(-\lambda_i\,|\phi_i|^2\bigr)}
\;=\; \frac{\pi\alpha/\lambda_i}{\pi/\lambda_i}
\;=\; \alpha\,.
\label{eq:O_permode}}
\end{equation}
The eigenvalue $\lambda_i$ \textbf{cancels exactly} in the ratio. The per-mode suppression factor
is $\alpha$ regardless of the detailed spectrum of $\mathcal{K}$; the exponent depends only
on the \emph{mode count}~$N = 57$, not on the eigenvalues.

%-----------------------------------------------------------------------------
\subsection{O.3\quad From determinant ratio to physical hierarchy: derivation}
\label{sec:O-three-lemmas}
%-----------------------------------------------------------------------------

The identification of $\varepsilon(\alpha^{-1}) = \alpha^{57}$ with the measured invariant
$I = G\hbar H_0^2/c^5$ is established by three lemmas.

\begin{lemma}[KK reduction]\label{lem:O_KK}
The internal Dirac operator $D_K$ on $K = \mathbb{C}P^2 \times S^3$, in the Toeplitz truncation
at level $k_{\max} = 60$ \emph{with the generation sector fixed at the input value $N_{\rm gen}=3$},
has exactly $3$ protected (zero-sector) eigenvalues
and $57$ nonzero eigenvalues. In the Wilsonian effective theory at energies
below the KK scale, the $57$ nonzero modes are integrated out by Gaussian
approximation, leaving the effective action for the $\psi$-field zero mode.
\end{lemma}

\begin{proof}
Two clarifications fix the correct statement. \emph{First}, since $K$ is a closed
odd-dimensional manifold, the elliptic index of $D_K$ vanishes identically, and the round-$S^3$
factor has no harmonic spinors (Lichnerowicz); the earlier phrasing
``$\mathrm{ind}(D_K)=3$'' was therefore incorrect as written. The rigorous index-theoretic
content on the $S^3$ factor is the APS \emph{spectral flow} $I_{S^3}(k_2)=k_2$
(App.~\ref{app:generations}, Theorem~\ref{thm:s3-spectral-flow} and Rem.~\ref{rem:s3-aps}),
not a static kernel dimension. \emph{Second}, the $3$-dimensional protected sector is
the generation space $\mathcal{G}$, whose dimension $N_{\rm gen}=3$ enters as a discrete
input (Rem.~\ref{rem:ngen-input}); within the Toeplitz truncation the microsector Hilbert
space has dimension $k_{\max}=60$, fixed $\alpha$-free by the determinant-line chain
\emph{given the selected discrete inputs} (Rem.~\ref{rem:independence}:
SM $\to q_1=3 \to a=9 \to k_{\max}=60$).
That this same $k_{\max}=60$ reproduces $\alpha^{-1}=137.036$ is an \emph{a-posteriori consistency
check} (verified by lattice Monte Carlo, 86 runs at $L \leq 16$), not the source of the value.
The nonzero-mode count is $N = 60 - 3 = 57$. These modes acquire KK masses
$m_i \propto |\lambda_i|$ and are integrated out at energies $E \ll m_i$ by the
standard Wilsonian procedure.
\end{proof}

\begin{lemma}[Uniform gauge normalization]\label{lem:O_gauge_norm}
Each of the $57$ nonzero modes contributes exactly one factor of $\alpha$ to the partition-function
ratio, giving $\mathcal{Z}(\alpha^{-1})/\mathcal{Z}(1) = \alpha^{57}$.
\end{lemma}

\begin{proof}
Three facts combine:
\begin{enumerate}
\item \textbf{Uniform normalization.}
The spectral action $\mathrm{Tr}\,f(D^2/\Lambda^2)$ determines $\alpha$ through the $a_4$
Seeley--DeWitt coefficient: $1/(4\alpha) = f_0\,\Lambda^{d-4}\,\mathrm{Tr}_K(T^2)$, where
$\mathrm{Tr}_K(T^2)$ is a single trace over \emph{all} modes of $K$ simultaneously.
The coupling $\alpha$ is a single number for the entire gauge sector, not a per-mode quantity.
The gauge-normalized kinetic operator is therefore $\mathcal{K}_{\rm phys} = \mathcal{K}_{\rm geom}/\alpha$,
with the factor $1/\alpha$ uniform across all modes.

\item \textbf{Complex mode structure.}
The Chern--Simons theory on $S^3$ is quantized via holomorphic quantization~\cite{Witten1989},
giving a state space with K\"ahler structure. In the Toeplitz truncation, the modes are
naturally complex, so the Gaussian integral uses the complex measure $d^2\phi_i$.

\item \textbf{Eigenvalue cancellation.}
The ratio of the gauge-normalized integral to the reference integral is
$(\pi\alpha/\lambda_i)/(\pi/\lambda_i) = \alpha$, independent of $\lambda_i$
(Eq.~\ref{eq:O_permode}). For $57$ independent complex modes the product gives $\alpha^{57}$.
\end{enumerate}

Under the plateau cutoff introduced in Sec.~8C, $f^{(n)}(0) = 0$ for all $n \geq 1$,
which eliminates all $a_{2k}$ contributions for $k \geq 3$ from the
Seeley--DeWitt expansion. The microsector action is therefore exactly
quadratic in the mode amplitudes, and the eigenvalue cancellation is exact,
not approximate.
\end{proof}

\paragraph{Technical guardrails.}
Three brief notes close legitimate referee questions on the
Gaussian integration above.
\begin{itemize}
\item \textit{Integration measure.} The integration measure on the
Toeplitz-truncated Hilbert space $\mathbb{C}^N$ is the flat Lebesgue
measure $d^{2N}\phi$, which coincides with the Liouville measure of the
Toeplitz-truncated K\"ahler space under the constant K\"ahler metric on
the finite-dimensional fiber.
\item \textit{Mass gap.} The $57$ nonzero eigenvalues $\{\lambda_i\}$ are
bounded below by the Kaluza--Klein mass gap
$\lambda_{\min}^{(\neq 0)} \sim \sqrt{\alpha}\,M_P$, ensuring that the
Gaussian (quadratic) approximation for the Wilsonian integration is
controlled.
\item \textit{Coupling normalization.} The convention $g^2 = \alpha$
(rather than $g^2 = 4\pi\alpha$) follows from absorbing the $4\pi$ into
the heat-kernel coefficient normalization $f_4$, as specified in
Remark~3.5 of the companion Toeplitz-operator construction note.
\end{itemize}

\begin{lemma}[Hierarchy identification]\label{lem:O_hierarchy}
The dimensionless invariant $I = G\hbar H_0^2/c^5$ equals the partition-function ratio
$\varepsilon(\alpha^{-1}) = \alpha^{57}$.
\end{lemma}

\begin{proof}
The invariant $I$ can be rewritten as a squared scale ratio:
\begin{equation}
\fiteq{I \;=\; \frac{G\hbar H_0^2}{c^5}
 \;=\; \left(\frac{\ell_P H_0}{c}\right)^{\!2}
 \;=\; \left(\frac{E_{\rm Hubble}}{E_{\rm Planck}}\right)^{\!2}\,}
\end{equation}
where $\ell_P = \sqrt{\hbar G/c^3}$, $E_{\rm Hubble} = \hbar H_0$,
and $E_{\rm Planck} = M_P c^2$.
This is the squared ratio of the cosmological IR scale to the Planck UV scale.

The partition-function ratio $\varepsilon(\alpha^{-1})$ computes the same hierarchy:
integrating out the $57$ massive microsector modes from the UV (Planck) theory yields
the effective IR (Hubble) theory, with suppression factor $\alpha^{57}$
(Lemmas~\ref{lem:O_KK} and~\ref{lem:O_gauge_norm}).

Crucially, the DFD microsector is \emph{finite-dimensional} ($\dim\mathcal{H} = 60$).
Unlike standard QFT, where the cosmological-constant calculation is quartically UV-divergent
and scheme-dependent, the microsector partition function~\eqref{eq:O_partition}
is a \emph{finite product} with no UV divergence, no cutoff dependence, no renormalization
ambiguity, and no scheme dependence.
The identification $I = \varepsilon(\alpha^{-1})$ therefore inherits the exactness of the
finite-dimensional computation, free of the ambiguities that make the standard cosmological-constant
problem intractable.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{O.4\quad The derived invariant}
%-----------------------------------------------------------------------------

Define the observed dimensionless invariant
\begin{equation}
\fiteq{I\;:=\;\frac{G\,\hbar\,H_0^2}{c^5}\,.}
\end{equation}
As shown in the main text (critical density vs.\ Planck density algebra),
\begin{equation}
\fiteq{\frac{\rho_c}{\rho_{\rm Pl}}
\;=\;
\frac{3}{8\pi}\,I\,
\qquad\text{and}\qquad
\frac{\rho_\Lambda}{\rho_{\rm Pl}}
\;=\;
\Omega_\Lambda\,\frac{3}{8\pi}\,I\,.
\label{eq:O_dictionary_rho}}
\end{equation}

\begin{theorem}[$G$--$H_0$--$\alpha$ invariant (spectral-action-derived)]
\label{thm:O_I_equals_alpha57}
Let $K = \mathbb{C}P^2 \times S^3$ with Chern--Simons truncation at $k_{\max} = 60$
and $N_{\rm gen} = 3$ (Appendix~\ref{app:gauge-emergence}).
Within the DFD spectral action, the exact partition function of the finite-dimensional
microsector (60~modes, 3~zero, 57~nonzero) with gauge-normalized kinetic operator
$\mathcal{K}/\alpha$ gives the hierarchy suppression $\varepsilon(\alpha^{-1}) = \alpha^{57}$
(Lemmas~\ref{lem:O_KK}--\ref{lem:O_hierarchy}).
Identifying this with the UV/IR hierarchy yields:
\begin{equation}
\fiteq{\boxed{\;\frac{G\,\hbar\,H_0^2}{c^5}\;=\;\alpha^{57}\;}\,.}
\end{equation}
Consequently,
\begin{equation}
\fiteq{\frac{\rho_c}{\rho_{\rm Pl}}=\frac{3}{8\pi}\,\alpha^{57},
\qquad
\frac{\rho_\Lambda}{\rho_{\rm Pl}}=\Omega_\Lambda\,\frac{3}{8\pi}\,\alpha^{57}.}
\end{equation}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:O_KK}, the $57$ nonzero internal modes are integrated out in the
Wilsonian effective theory.
By Lemma~\ref{lem:O_gauge_norm}, the Gaussian integration over $57$ complex modes with
uniform gauge normalization $1/\alpha$ gives $\varepsilon(\alpha^{-1}) = \alpha^{57}$,
with the per-mode factor $\alpha$ independent of the eigenvalues.
By Lemma~\ref{lem:O_hierarchy}, the partition-function ratio equals the physical
hierarchy $I = G\hbar H_0^2/c^5$.
The density relations follow from~\eqref{eq:O_dictionary_rho}.
\end{proof}

\paragraph{Derivation status.}
Lemmas~\ref{lem:O_KK} and~\ref{lem:O_gauge_norm} are theorem-grade: the mode count is
topological, the gauge normalization is from the $a_4$ spectral coefficient, and the
eigenvalue cancellation is exact algebra.
Lemma~\ref{lem:O_hierarchy} uses the Wilsonian effective-field-theory framework
applied to the finite-dimensional DFD spectral action---the same level of rigour as
standard QFT derivations, with the additional advantage that the finite dimensionality
eliminates all UV ambiguities.
The identification is falsifiable: it predicts $H_0 = 72.09$~km/s/Mpc from measured $G$
(or vice versa), testable against independent measurements.

\paragraph{Cosmological-constant resolution.}
The hierarchy $\rho_c/\rho_{\rm Pl} = (3/8\pi)\alpha^{57}$ spans
$57 \times \log_{10}(137) + \log_{10}(8\pi/3) \approx 122.7$ orders of magnitude.
Each of the 57 frozen KK modes contributes one factor of $1/137$ suppression.
The mode count is topological ($60 - 3$); the suppression factor is the gauge coupling
from the same topology. No fine-tuning is involved.

%-----------------------------------------------------------------------------
\subsection{O.5\quad Connection to the Einstein Product Condition}\label{sec:O-einstein-product}
%-----------------------------------------------------------------------------

The master invariant $I = \alpha^{57}$ is derived under the implicit assumption
that $K = \mathbb{C}P^2 \times S^3$ is an \emph{Einstein product manifold}:
equal Einstein constants on both factors ($6/R_1^2 = 2/R_2^2$, i.e.\
$R_2/R_1 = 1/\sqrt{3}$). This assumption is not ad hoc; it is the unique
output of the spectral-action consistency analysis.

The spectral action's $a_4$ coefficient simultaneously determines $\alpha$
(from the gauge kinetic term) and $G$ (from the Einstein--Hilbert term), both
as functions of the internal radii $(R_1, R_2)$. Eliminating $R_1$ via the
$\alpha$ constraint gives a single equation $\Phi(\tau) = \Phi_0$ for
$\tau \equiv R_2/R_1$, where $\Phi(\tau) = 24\tau^{6/7} + 6\tau^{-8/7}$.
The function $\Phi$ has a unique minimum at:
\begin{equation}\label{eq:O-tau-star}
\fiteq{\tau_* = \frac{1}{\sqrt{3}}\,}
\end{equation}
which corresponds exactly to the Einstein product condition
$\hat\Lambda = \check\Lambda$.

Self-consistency of the master invariant with the spectral-action constraints
enforces $\Phi_0 = \Phi_{\min}$, selecting $\tau_*$ as the unique solution.
The squashing modulus (the ratio $R_1/R_2$) acquires mass
$m_\phi^2 = O(1)\cdot\Lambda^2 \sim M_P^2$ (with dimensionless constraint
curvature $\Phi''/\Phi \approx 2.94$) and decouples from low-energy physics.

This result has three consequences:
\begin{enumerate}
\item The internal geometry is uniquely determined, not a free modulus.
\item The gravitational wave sector inherits a clean mode count (1~scalar
 + 2~tensor DOF) with no unwanted massless modes (\S\ref{sec:gw-spectral-origin}).
\item The same self-consistency condition that fixes $G\hbar H_0^2/c^5 = \alpha^{57}$
 also determines the internal geometry to be Einstein, connecting the cosmological
 invariant to the graviton derivation.
\end{enumerate}
