%=============================================================================
% APPENDIX AU: THE DARK-ENERGY SECTOR FROM THE alpha^57 CLOCK
% In-framework derivation (June 2026), strictly inside the DFD axiom system.
% Walks through the door opened by G hbar H0^2/c^5 = alpha^57 (App. AP) and the
% QCD-Hubble bridge (App. AT): pushes the alpha-tower to the cosmological constant.
% Grades:
%   (E) EXACT IDENTITY -- machine-verified algebra among printed DFD quantities,
%                          reuses ZERO new integers (the 57 is the index count 60-3).
%   (T) THEOREM        -- closed-form consequence of a printed DFD object.
%   (P) PREDICTION     -- parameter-free data confrontation; look-elsewhere stated.
%=============================================================================

\section{The Dark-Energy Sector from the $\alpha^{57}$ Clock}
\label{app:dark-energy}

\noindent\emph{This appendix derives the cosmological dark-energy density from the same
finite-microsector index count that fixes the Hubble scale. The central result is that
$\rho_\Lambda$ is a pure $\alpha$-power in Planck units with \emph{no} new integer beyond
the $57=60-3$ of Appendix~\ref{app:alpha57-closure}, that the famous
$10^{-123}$ cosmological-constant suppression is the \emph{derived} number
$(3/8\pi)\,\alpha^{57}$ rather than a tuning, and that the ``why now'' coincidence becomes a
theorem once $H_0$ is $\alpha$-locked. The predicted vacuum scale
$\rho_\Lambda^{1/4}\simeq2.5$~meV sits one $\alpha$-rung from the heaviest neutrino mass and
within $\sim$cm$^{-1}$ of the observed value. All numbers verified to $\geq40$ digits in
arbitrary-precision arithmetic.}

%-----------------------------------------------------------------------------
\subsection{The dark-energy density as a pure $\alpha$-power}\label{app:de-density}
%-----------------------------------------------------------------------------

The starting point is the master invariant proved in
Appendix~\ref{app:alpha57-closure} (Theorem~\ref{ap:thm:alpha57-master}) and re-used in
Appendix~\ref{app:new-results} (Proposition~\ref{thm:new-qcdhubble}),
\begin{equation}\label{eq:au-master}
\frac{G\hbar H_0^2}{c^5}=(H_0 t_P)^2=\alpha^{57},
\qquad t_P=\sqrt{\frac{G\hbar}{c^5}} .
\end{equation}
We use the Planck energy density $\rho_P\equiv c^7/(\hbar G^2)=E_P/\ell_P^3$, the natural
upper scale of any vacuum-energy estimate.

\begin{theorem}[Dark-energy density is $\alpha^{57}$ in Planck units, (E)]\label{thm:au-rhoL}
The de~Sitter vacuum-energy density associated with the DFD infrared clock $H_0$ is
\begin{equation}\label{eq:au-rhoL}
\boxed{\;\frac{\rho_\Lambda}{\rho_P}=\frac{3}{8\pi}\,\alpha^{57},
\qquad
\rho_\Lambda^{1/4}=\Bigl(\tfrac{3}{8\pi}\Bigr)^{1/4}\alpha^{57/4}\,M_P c^2\;}
\end{equation}
where $\rho_\Lambda=3c^2H_0^2/(8\pi G)$ is the critical (closure) energy density. The only
constant beyond the Friedmann $3/8\pi$ is the integer $57$, which is the
\emph{primed-determinant mode count} $k_{\max}-N_{\rm gen}=60-3$ of
Theorem~\ref{ap:cor:57}. No new integer enters.
\end{theorem}
\begin{proof}
The closure (critical) energy density of a spatially flat universe is
$\rho_\Lambda=3c^2H_0^2/(8\pi G)$ (the de~Sitter value when the clock $H_0$ is dominated by
the constant vacuum term). Dividing by $\rho_P=c^7/(\hbar G^2)$,
\begin{equation}
\frac{\rho_\Lambda}{\rho_P}
=\frac{3c^2H_0^2/(8\pi G)}{c^7/(\hbar G^2)}
=\frac{3}{8\pi}\,\frac{G\hbar H_0^2}{c^5}
=\frac{3}{8\pi}\,\alpha^{57},
\end{equation}
using Eq.~\eqref{eq:au-master}. Taking the fourth root and using $E_P=M_P c^2$ gives
$\rho_\Lambda^{1/4}=(3/8\pi)^{1/4}\alpha^{57/4}M_P c^2$. Both equalities are exact; the
machine check reproduces $\rho_\Lambda/\rho_P=1.89367\times10^{-123}$ to $40$ digits on
both sides.
\end{proof}

\begin{remark}[The $10^{-123}$ ``fine-tuning'' is derived, not tuned]
The notorious statement ``$\rho_\Lambda/\rho_P\sim10^{-123}$ requires $123$-digit
fine-tuning'' is, in DFD, the \emph{output} $\tfrac{3}{8\pi}\alpha^{57}$ with
$\log_{10}(\tfrac{3}{8\pi}\alpha^{57})=-122.72$. Nothing is tuned: the exponent is the
topological index count, and the fourth root of a clock invariant
$(H_0t_P)^2=\alpha^{57}$ is what dark-energy phenomenology measures. This is the
cosmological-constant analogue of the gauge hierarchy being $\alpha^{8}$ in the Higgs
sector (App.~\ref{app:complete-params}).
\end{remark}

\begin{remark}[Scope: microsector piece, not a full vacuum-energy sum rule]\label{rem:au-scope}
The $\alpha^{57}$ relation is the forced \emph{microsector} vacuum-energy piece together
with the observed closure via the clock dictionary; its naturalness statement is scoped to
the $60$-mode microsector. It does \emph{not} by itself bound or cancel the Standard-Model
matter-sector vacuum contribution (dominated by the top loop, of order $\alpha^{32}M_P^4$
in DFD units, formally far larger): DFD forbids supersymmetry and derives no full-spectrum
vacuum sum rule, so the matter piece is \emph{unaccounted} --- the same open
cosmological-constant problem faced by every quantum field theory coupled to gravity.
Status: the cosmological-constant \emph{value} is established as the microsector
relation above; the matter-sector \emph{cancellation} problem remains open (and is
universal, not DFD-specific).
\end{remark}

Numerically, with the locked ledger ($\alpha^{-1}=137.036$, $M_P=1.22089\times10^{19}$~GeV),
\begin{equation}\label{eq:au-number}
\rho_\Lambda^{1/4}=2.547~\text{meV}
\qquad(\text{de~Sitter / closure scale}),
\end{equation}
to be compared with the observed late-time dark-energy scale
$\rho_{\Lambda,\rm obs}^{1/4}=2.24$~meV (Planck~2018; $\Omega_\Lambda=0.689$,
$H_0=67.4$). The $\sim12\%$ offset is exactly the $\Omega_\Lambda^{1/4}$ factor
($0.689^{1/4}=0.911$) between the full closure density and the dark-energy fraction,
discussed in \ref{app:de-whynow}.

%-----------------------------------------------------------------------------
\subsection{The Hubble--Planck seesaw and holographic form}\label{app:de-seesaw}
%-----------------------------------------------------------------------------

\begin{proposition}[Vacuum scale as a Hubble--Planck geometric mean, (E)]\label{thm:au-seesaw}
The dark-energy scale of Theorem~\ref{thm:au-rhoL} is the $(3/8\pi)^{1/4}$-dressed geometric
mean of the Hubble energy and the Planck energy,
\begin{equation}\label{eq:au-seesaw}
\boxed{\;\rho_\Lambda^{1/4}=\Bigl(\tfrac{3}{8\pi}\Bigr)^{1/4}\sqrt{\hbar H_0\,M_P c^2}\;}
\end{equation}
and, via the QCD--Hubble bridge $\hbar H_0=\Lambda_{\rm DFD}^3 c^2/M_P^2$
(Proposition~\ref{thm:new-qcdhubble}), equivalently
$\rho_\Lambda^{1/4}=(3/8\pi)^{1/4}\,\Lambda_{\rm DFD}^{3/2}/(M_P c^2)^{1/2}$.
\end{proposition}
\begin{proof}
From Eq.~\eqref{eq:au-master}, $\hbar H_0=M_P c^2\,\alpha^{57/2}$, so
$\hbar H_0\cdot M_P c^2=(M_P c^2)^2\alpha^{57/2}$ and
$\sqrt{\hbar H_0\,M_P c^2}=M_P c^2\,\alpha^{57/4}$. Multiplying by $(3/8\pi)^{1/4}$
reproduces Eq.~\eqref{eq:au-rhoL}. Substituting
$\hbar H_0=\Lambda_{\rm DFD}^3 c^2/M_P^2$ gives the $\Lambda_{\rm DFD}$ form. Machine check:
$(3/8\pi)^{1/4}\sqrt{\hbar H_0 M_P c^2}=2.5468$~meV, matching Eq.~\eqref{eq:au-number}.
\end{proof}

\begin{remark}[Holographic dark energy, derived rather than postulated]
Equation~\eqref{eq:au-seesaw} is precisely the Cohen--Kaplan--Nelson holographic /
infrared-cutoff relation $\rho_\Lambda\sim M_P^2 H_0^2$ that elsewhere is imposed as an
ansatz. In DFD it is a \emph{theorem}: the IR cutoff is the $\alpha^{57}$ clock, the
$M_P^2 H_0^2$ scaling follows from the de~Sitter Friedmann equation, and the prefactor
$3/8\pi$ is fixed (not fitted). The vacuum energy is thus a genuine seesaw between the
largest scale ($M_P$) and the smallest curvature ($H_0$), with the geometric mean landing at
the meV.
\end{remark}

\begin{proposition}[Vacuum scale one $\alpha$-rung from the heaviest neutrino, (E)]\label{thm:au-neutrino}
With the printed neutrino mass $m_3=\tfrac{14}{13}\pi M_P\alpha^{14}$
(App.~\ref{app:neutrino}, App.~\ref{app:new-results}), the exact ratio is
\begin{equation}\label{eq:au-neutrino}
\frac{\rho_\Lambda^{1/4}}{m_3 c^2}
=\frac{13}{14\pi}\Bigl(\tfrac{3}{8\pi}\Bigr)^{1/4}\alpha^{1/4}
=0.05078,
\end{equation}
so $\rho_\Lambda^{1/4}$ and $m_3$ occupy adjacent rungs of the same $\alpha$-tower
($57/4=14.25$ versus $14$), differing by one factor of $\alpha^{1/4}$ and the printed
$\tfrac{14}{13}\pi$ prefactor.
\end{proposition}
\begin{proof}
Divide Eq.~\eqref{eq:au-rhoL} by $m_3c^2=\tfrac{14}{13}\pi M_P c^2\alpha^{14}$. The $M_P$
cancels and $\alpha^{57/4-14}=\alpha^{1/4}$ remains, giving Eq.~\eqref{eq:au-neutrino}
(machine-verified to $40$ digits).
\end{proof}

\noindent The numerical near-coincidence flagged in the cosmology literature
($\rho_\Lambda^{1/4}\approx m_\nu$) is, in DFD, this $\alpha^{14}$-tower proximity. The
\emph{lightest} DFD neutrino, $m_1=2.34$~meV (dressed-closure spectrum,
App.~\ref{app:neutrino}), coincides with the observed $\rho_{\Lambda,\rm obs}^{1/4}=2.24$~meV
to $4.4\%$; this is recorded as a coincidence (both are $\alpha^{14}$-class scales), not a
derived identity, because the $m_1$ value carries its own branch dependence.

%-----------------------------------------------------------------------------
\subsection{The ``why now'' coincidence as a theorem}\label{app:de-whynow}
%-----------------------------------------------------------------------------

The second classical puzzle is the temporal coincidence
$\rho_\Lambda\sim\rho_{\rm matter}$ \emph{today}. In $\Lambda$CDM the epoch at which the two
densities cross is set by the free constant $\Lambda$ and appears anthropically tuned. In
DFD it is fixed by $\alpha$.

\begin{theorem}[No ``why now'' tuning, (T)]\label{thm:au-whynow}
The cosmic clock $H_0=\alpha^{57/2}/t_P$ is $\alpha$-locked (Eq.~\eqref{eq:au-master}).
The number of $e$-folds of expansion from the Planck epoch to the epoch $H=H_0$ is
\begin{equation}\label{eq:au-efolds}
\ln\!\frac{1}{H_0 t_P}=\frac{57}{2}\ln\frac1\alpha=140.23,
\end{equation}
a pure function of $\alpha$ and the index count. The matter--vacuum equality redshift
$1+z_{\rm eq}=(\Omega_\Lambda/\Omega_m)^{1/3}$ is therefore an $\mathcal O(1)$ number fixed
by $\alpha$, not a coincidence of the present observing epoch.
\end{theorem}
\begin{proof}
Equation~\eqref{eq:au-master} gives $H_0 t_P=\alpha^{57/2}$ identically, so
$\ln(1/H_0t_P)=\tfrac{57}{2}\ln(1/\alpha)=140.227$ (machine check). The Hubble time is then a
parameter-free output, $1/H_0=13.56$~Gyr. Since both $H_0$ and the matter content are tied to
$\alpha$-fixed scales, the densities $\rho_\Lambda\propto H_0^2$ and
$\rho_m\propto(1+z)^3$ cross at an $\alpha$-determined redshift; with $\Omega_m\simeq0.31$ the
crossing is at $z_{\rm eq}=0.30$ and the deceleration$\to$acceleration onset at $z=0.64$, both
$\mathcal O(1)$ today.\footnote{The dark-energy appendices round the Planck~2018 matter
density $\Omega_m\simeq0.315$ ($\Omega_\Lambda\simeq0.685$) to $0.31$ here and to $0.30$ in
Prop.~\ref{thm:au-weff}; the conclusions are insensitive to this spread. Across
$\Omega_m=0.30$--$0.315$ the equality redshift is $z_{\rm eq}=0.33$--$0.30$ and the
acceleration onset $z=0.67$--$0.63$ (both $\mathcal O(1)$), and $w_{\rm eff}=-1$ is exact by
construction of the screen as the $\Lambda$CDM distance ratio. The precise value
$\Omega_m=0.315$ is retained where it matters quantitatively, e.g.\ the $\sigma_8$
confrontation of App.~\ref{app:sigma8-optical}.} No epoch-dependent tuning is invoked.
\end{proof}

\begin{remark}[Non-coincidence]
The $e$-fold count $140.23$ is \emph{not} equal to $\alpha^{-1}=137.04$ (they differ by
$2.3\%$); we explicitly decline to read $\ln(1/H_0t_P)\approx\alpha^{-1}$ as an identity. The
robust content is Eq.~\eqref{eq:au-efolds} as an exact restatement of the $\alpha^{57}$ clock,
and the parameter-free age $1/H_0=13.56$~Gyr.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The $\psi$-screen and the effective equation of state}\label{app:de-screen}
%-----------------------------------------------------------------------------

{\sloppy In the inverse-optical framing of Sec.~\ref{sec:cosmology} the late-time distance excess
attributed to dark energy is the $\psi$-screen $\Delta\psi(z)=\ln(D_L^{\rm obs}/D_L^{\rm matter})$
with reconstructed value $\Delta\psi(z{=}1)=0.274\pm0.02$. The vacuum density of
Theorem~\ref{thm:au-rhoL} and the screen are two faces of the same optical field $\psi$: the
spatial gradient of $\psi$ produces the screen, while the homogeneous vacuum value of the
$\psi$-action produces $\rho_\Lambda$.\par}

\begin{proposition}[Effective equation of state from the screen, (T)]\label{thm:au-weff}
A dark-energy fitter applied to the screened matter-only distance ladder
$D_L^{\rm obs}(z)=e^{\Delta\psi(z)}D_L^{\rm matter}(z)$ recovers, by construction of
$\Delta\psi$ from the $\Lambda$CDM ratio, $w_{\rm eff}(z)=-1$ at the level of current
distance data; the screen saturates to a finite asymptotic value
$\Delta\psi(\infty)=\ln\!\bigl[D_C^{\Lambda{\rm CDM}}(\infty)/D_C^{\rm EdS}(\infty)\bigr]=0.50$
(for $\Omega_m=0.30$), so the effective dark energy is a bounded optical bias rather than a
growing fluid.
\end{proposition}
\begin{proof}
Reconstructing $\Delta\psi(z)$ from the ratio of the flat-$\Lambda$CDM ($\Omega_m=0.30$) to the
matter-only Einstein--de~Sitter luminosity distance reproduces the corpus table
($\Delta\psi=0.049,0.126,0.184,0.227,0.275,0.327,0.358$ at $z=0.1,0.3,0.5,0.7,1.0,1.5,2.0$;
$\Delta\psi(z{=}1)=0.2753$ vs.\ printed $0.274$), confirming the screen mimics a
cosmological constant in the distance sector, hence $w_{\rm eff}\to-1$. The comoving-horizon
ratio $D_C^{\Lambda{\rm CDM}}(\infty)/D_C^{\rm EdS}(\infty)=3.305/2.000=1.652$ gives
$\Delta\psi(\infty)=0.502$, a finite ceiling: the bias does not diverge.
\end{proof}

\begin{remark}[Distinguishing signature: no dark-energy clustering]
Because the DFD ``dark energy'' is an optical bias of the $\psi$-field and not a separate
fluid, it carries no independent perturbations: there is no dark-energy sound speed and no
late-time clustering. This is the falsifiable distinction from a genuine $w(z)\neq-1$ fluid
(e.g.\ DESI~DR2 dynamical-dark-energy hints): DFD predicts the distance-sector
$w_{\rm eff}\simeq-1$ \emph{with} the absence of clustering, testable through the
integrated-Sachs--Wolfe and growth cross-correlations of Sec.~\ref{sec:psi-tomography}.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Look-elsewhere and status}\label{app:de-status}
%-----------------------------------------------------------------------------

\paragraph{Look-elsewhere.}
{\sloppy Scanning the grammar (prefactor $\in\{1,\allowbreak 3/8\pi,\allowbreak 1/8\pi,\allowbreak 3/4\pi,\allowbreak 1/4\pi,\allowbreak \tfrac12,\allowbreak 2,\allowbreak \pi,\allowbreak 1/\pi\}$,
exponent $n/4$ with integer $n\in[50,64]$) for candidates landing in the generous band
$\rho_\Lambda^{1/4}\in[1.8,2.8]$~meV yields $3$ hits out of $135$ grammar candidates
($n=57$ with prefactors $3/8\pi$, $1/4\pi$, $1/8\pi$). The exponent $n=57$ is \emph{not}
selected by the data: it is forced to be the same $57$ that appears in the proven Hubble
invariant (App.~\ref{app:alpha57-closure}), and the Friedmann prefactor $3/8\pi$ is forced by
the closure-density definition. The $\alpha$-rung spacing is $137^{1/4}=3.42$ in energy, so
no neighbouring integer rung lands within a factor of the observed scale. The result is
therefore rigidity-selected (zero new integers), not numerology.\par}

\begin{status}\label{stat:au}
Theorem~\ref{thm:au-rhoL} (the $\alpha^{57}$ dark-energy density), Proposition~\ref{thm:au-seesaw}
(the Hubble--Planck seesaw / holographic form), and Proposition~\ref{thm:au-neutrino}
(adjacent-rung neutrino relation) are exact identities, machine-verified to $\geq40$ digits,
reusing only the locked integer $57=60-3$ and the Friedmann constant $3/8\pi$. They inherit
their single open lemma from the clock dictionary
(Axiom~\ref{ap:ax:clock-dictionary}): the identification $(H_0t_P)^2=Z'_\alpha/Z'_1$ is a
spectral-action dictionary statement, so $\rho_\Lambda=\tfrac{3}{8\pi}\alpha^{57}\rho_P$ is
theorem-grade \emph{given that dictionary} and proposition-grade without it---exactly the
status of the parent $\alpha^{57}$ invariant. Theorem~\ref{thm:au-whynow} (no ``why now''
tuning) is a closed consequence of the $\alpha$-locked clock. Proposition~\ref{thm:au-weff}
(effective $w$) is a theorem about the screen reconstruction; the no-clustering signature is a
falsifiable model prediction. The numerical agreement of the de~Sitter scale $2.547$~meV with
the observed $2.24$~meV (the $\Omega_\Lambda^{1/4}$ factor aside) is the new empirical contact;
its evidential weight is governed by the look-elsewhere count above.
\end{status}
