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% SECTION 12: COSMOLOGICAL IMPLICATIONS
% Part VII: Cosmology and Open Problems
% v2.9+: ψ-Tomography (ψ-Screen) inverse-first cosmology module
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\section{Cosmological Implications}\label{sec:cosmology}

DFD cosmology is treated as an \emph{inverse optical problem}: infer the line-of-sight optical bias field directly from data, and only then interpret what standard cosmology would call ``expansion history,'' ``dark energy,'' and ``dark matter.'' In this framing, GR/$\Lambda$CDM enters \emph{only} as an observer dictionary (how distances/angles are commonly reported), not as ontology.

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% ψ-TOMOGRAPHY (ψ-SCREEN) COSMOLOGY MODULE
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\subsection{$\psi$-Tomography ($\psi$-Screen) Cosmology Module}\label{sec:psi-tomography}

\paragraph{Non-negotiable premise.}
The primary reconstructed object is the ``$\psi$-screen'' on the past light cone:
\begin{equation}
\fiteq{\Delta\psi(z,\hat n)\;\equiv\;\psi_{\rm em}(z,\hat n)\;-\;\psi_{\rm obs},
\qquad \text{dimensionless.}
\label{eq:psitomo-def-dpsi}}
\end{equation}
All GR/$\Lambda$CDM quantities used in this section (e.g.\ $D_L^{\rm dict},D_A^{\rm obs}$) are \emph{reporting-layer} variables that serve as a convenient dictionary for published datasets.

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\subsubsection{DFD postulates and sign conventions}

DFD is formulated on flat $\mathbb{R}^3$ with a scalar field $\psi$ and refractive index $n=e^\psi$.
The one-way light speed is
\begin{equation}
\fiteq{c_1(\psi)=c\,e^{-\psi},}
\end{equation}
and the (nonrelativistic) acceleration of matter is
\begin{equation}
\fiteq{\bm a \;=\;\frac{c^2}{2}\,\nabla\psi.
\label{eq:psitomo-accel}}
\end{equation}
We adopt the gauge choice $\psi_{\rm obs}\equiv 0$, so that $\Delta\psi=\psi_{\rm em}$ in this gauge. With this convention:
\begin{itemize}
\item $\Delta\psi>0$ means $\psi$ (hence $n$) was higher at emission than locally (slower $c_1$ at emission).
\item $\Delta\psi<0$ means $\psi$ was lower at emission than locally (faster $c_1$ at emission).
\end{itemize}

\paragraph{Endpoint vs.\ observable screen.}
Equation~\eqref{eq:psitomo-def-dpsi} is an endpoint definition.
Operationally, each dataset reconstructs an \emph{observable} screen $\Delta\psi_{\rm obs}$ defined by the log-multiplicative bias required by the DFD optical relations below.
When needed, one may represent $\Delta\psi_{\rm obs}$ as a weighted line-of-sight functional
\begin{equation}
\fiteq{\Delta\psi_{\rm obs}(z,\hat n)
=\int_{0}^{\chi(z)}\!d\chi\;W_{\rm obs}(\chi;z)\,\delta\psi(\chi,\hat n),
\label{eq:psitomo-los-functional}}
\end{equation}
where $\chi$ is a dictionary comoving-distance coordinate and $W_{\rm obs}$ is a dataset-specific kernel.
The inverse program reconstructs $\Delta\psi_{\rm obs}$ \emph{directly from data} without assuming a particular $W_{\rm obs}$.

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\subsubsection{Forward model: three primary DFD optical relations}

The module is built around three primary DFD optical relations.

\paragraph{(1) Luminosity-distance bias (SNe Ia).}
Let $D_L^{\rm dict}(z,\hat n)$ be the baseline luminosity distance as typically reported under the observer dictionary.
DFD maps this to an optically biased luminosity distance:
\begin{equation}
\fiteq{D_L^{\rm DFD}(z,\hat n)=D_L^{\rm dict}(z,\hat n)\,e^{\Delta\psi(z,\hat n)}.
\label{eq:psitomo-DL-bias}}
\end{equation}
Equivalently, $\ln D_L^{\rm DFD}=\ln D_L^{\rm dict}+\Delta\psi$.

\paragraph{(1b) Angular-diameter-distance bias.}
Both $D_L$ and $D_A$ are computed from null geodesics of the same optical metric $\tilde g_{\mu\nu}$. The $\psi$-field therefore screens both distances equally:
\begin{equation}
\fiteq{D_A^{\rm DFD}(z,\hat n)=D_A^{\rm dict}(z,\hat n)\,e^{\Delta\psi(z,\hat n)}.
\label{eq:psitomo-DA-bias}}
\end{equation}

\paragraph{(2) Distance duality (Etherington reciprocity).}\label{sec:psitomo-duality}
DFD's optical metric $d\tilde s^2 = -c^2\,dt^2/n^2 + d\mathbf{x}^2$ with $n = e^\psi > 0$ is a smooth, non-degenerate Lorentzian metric. All three conditions of Etherington's reciprocity theorem\cite{Etherington1933,Ellis2007} are satisfied: (i)~photons propagate on null geodesics of a Lorentzian metric; (ii)~geodesics are locally unique ($\psi$ is $C^{1,\alpha}$ away from sources, Appendix~\ref{app:wellposedness}); (iii)~photon number is conserved (no absorption/emission mechanism). Therefore
\begin{equation}
\fiteq{D_L(z,\hat n) \;=\; (1+z)^2\,D_A(z,\hat n).
\label{eq:psitomo-duality}}
\end{equation}
This holds exactly; no $e^{\Delta\psi}$ factor appears. The common screening factor from Eqs.~\eqref{eq:psitomo-DL-bias} and \eqref{eq:psitomo-DA-bias} cancels in the ratio $D_L/D_A$.

\paragraph{Notation.} We distinguish $\Delta\psi_{\rm screen}(z,\hat n)$ (the distance bias relative to the dictionary baseline, measured by Estimators~A and~C below; can be large, $\approx 0.27$ at $z=1$) from $\Delta\psi_{\rm dual}(z,\hat n) \equiv \ln[D_L/(1+z)^2 D_A] = 0$ (the DDR violation parameter, measured by Estimator~B; identically zero). Where the subscript is omitted, $\Delta\psi$ refers to $\Delta\psi_{\rm screen}$.

\paragraph{(3) CMB acoustic-scale screen (angular anisotropy).}
Let $\ell_1(\hat n)$ denote the locally inferred first acoustic peak location from patchwise CMB power spectra.
DFD posits the angular screen mapping
\begin{equation}
\fiteq{\ell_1(\hat n)=\ell_{\rm true}\,e^{-\Delta\psi(\hat n)},
\label{eq:psitomo-cmb-ell1}}
\end{equation}
where $\ell_{\rm true}$ is a sky-independent constant that cancels out of the normalized anisotropy reconstruction below.

\paragraph{Sign of the $\ell_1$ mapping.}
The sign deserves explicit comment. The distance relations Eqs.~\eqref{eq:psitomo-DL-bias}--\eqref{eq:psitomo-DA-bias} give $D_A^{\rm DFD} \propto e^{+\Delta\psi}$: objects appear \emph{farther}, which naively pushes $\ell_1 \propto D_A/r_s$ \emph{higher}. But that scaling applies to the sky-averaged monopole, which is absorbed into $\ell_{\rm true}$. Equation~\eqref{eq:psitomo-cmb-ell1} describes the \emph{direction-dependent} anisotropy: a foreground sightline with $\Delta\psi(\hat n) > 0$ acts as a convergent screen that \emph{magnifies} the angular scale of CMB features in that patch. Larger apparent angular scale maps to \emph{lower} $\ell_1$, hence the negative exponent. This is the same sign as standard weak-lensing magnification of the CMB, where convergence $\kappa > 0$ shifts power to lower $\ell$.

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\subsubsection{Two independent screen estimators and one consistency check}\label{sec:psitomo-estimators}

\paragraph{Estimator A: SNe Ia alone (and its degeneracy).}
From Eq.~\eqref{eq:psitomo-DL-bias}, an operational estimator on each SN sightline is
\begin{equation}
\fiteq{\widehat{\Delta\psi}_{\rm SN}(z_i,\hat n_i)
\;\equiv\;
\ln D_L^{\rm obs}(z_i,\hat n_i) - \ln D_L^{\rm dict}(z_i) - \mathcal{M},
\label{eq:psitomo-est-sn}}
\end{equation}
where $\mathcal{M}$ is an unknown constant absorbing absolute magnitude / distance-ladder calibration.
SNe alone cannot fix an additive constant in $\Delta\psi$ (monopole), because $\Delta\psi\!\to\!\Delta\psi+{\rm const}$ can be absorbed into $\mathcal{M}$.
A robust SN-only product is therefore the anisotropy field
\begin{equation}
\fiteq{\widehat{\delta\psi}_{\rm SN}(z,\hat n)\;\equiv\;\widehat{\Delta\psi}_{\rm SN}(z,\hat n)-\big\langle \widehat{\Delta\psi}_{\rm SN}(z,\hat n)\big\rangle_{\hat n}.
\label{eq:psitomo-est-sn-aniso}}
\end{equation}

\paragraph{Estimator B: SNe + BAO / strong lensing (duality consistency check).}
Etherington's reciprocity~\eqref{eq:psitomo-duality} implies that the observable ratio
\begin{equation}
\fiteq{\boxed{
\widehat{\Delta\psi}_{\rm dual}(z,\hat n)
\;\equiv\;\ln\!\left(\frac{D_L^{\rm obs}(z,\hat n)}{(1+z)^2\,D_A^{\rm obs}(z,\hat n)}\right)
\;=\; 0.
}
\label{eq:psitomo-est-dual}}
\end{equation}
This is not an independent measurement of $\Delta\psi_{\rm screen}$; it is a \emph{consistency check} that the optical metric satisfies Etherington's conditions. Observational confirmation ($\widehat{\Delta\psi}_{\rm dual} = 0.01 \pm 0.02$)\cite{Holanda2010,Liang2011} validates the metric structure.

\paragraph{Estimator C: CMB peak anisotropy (screen at last scattering).}
From Eq.~\eqref{eq:psitomo-cmb-ell1}, define the normalized estimator:
\begin{equation}
\fiteq{\boxed{
\widehat{\Delta\psi}_{\rm CMB}(\hat n)
= -\ln\!\left(\frac{\ell_1(\hat n)}{\langle\ell_1\rangle}\right),
}
\label{eq:psitomo-est-cmb}}
\end{equation}
which fixes the additive constant by construction ($\langle\widehat{\Delta\psi}_{\rm CMB}\rangle=0$). This isolates \emph{angular} structure in the screen at last scattering.

\paragraph{How to obtain $\ell_1(\hat n)$ without $\Lambda$CDM priors.}
Choose a sky patching scheme; estimate local pseudo-$C_\ell$ spectra per patch (beam/mask corrected);
fit a \emph{local peak locator} template around the first peak (only a smooth peaked function is required);
take the maximizing multipole as $\ell_1$ for that patch.

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\subsubsection{Theorem-level internal closure of the reconstructed screen}
\label{sec:psitomo-closure-theorems}

The two screen estimators and one consistency check introduced above are not merely ``three ways of plotting the same thing'':
under the forward optical relations, they imply \emph{overdetermined closure identities} that must hold
on the sky (and across redshift bins) if a single scalar screen $\Delta\psi(z,\hat n)$ is the correct
organizing variable.

\paragraph{Conventions and hypotheses.}
Fix a redshift bin $z\in[z_a,z_b]$ and an analysis mask $W(\hat n)$ (common to all maps in a given test).
Assume:
\begin{enumerate}
\item \textbf{(H1) Forward relations.} The DFD optical relations~\eqref{eq:psitomo-DL-bias}--\eqref{eq:psitomo-cmb-ell1} hold on their respective domains of validity.
\item \textbf{(H2) Observable identification.} The reported distances used in Eqs.~\eqref{eq:psitomo-est-sn}--\eqref{eq:psitomo-est-dual} are the observational reconstructions of the corresponding DFD distances along that line of sight, i.e.\ $D_L^{\rm obs}(z,\hat n)=D_L^{\rm DFD}(z,\hat n)$ and $D_A^{\rm obs}(z,\hat n)=D_A^{\rm DFD}(z,\hat n)$ (up to the stated measurement errors).
\item \textbf{(H3) SN calibration constancy.} The SN absolute calibration constant $\mathcal{M}$ in Eq.~\eqref{eq:psitomo-est-sn} is a global constant (independent of $z$ and $\hat n$), as assumed in the estimator definition.
\end{enumerate}
No dynamical assumption about $\mu(x)$, growth, or a specific dictionary is required for the identities below.

\begin{theorem}[Duality consistency]\label{thm:duality-inversion}
Under (H1)--(H2), the duality estimator~\eqref{eq:psitomo-est-dual} tests Etherington consistency:
\begin{equation}
\fiteq{\widehat{\Delta\psi}_{\rm dual}(z,\hat n)=0.
\label{eq:closure-dual-exact}}
\end{equation}
\end{theorem}

\begin{proof}
From Eqs.~\eqref{eq:psitomo-DL-bias} and \eqref{eq:psitomo-DA-bias}, both distances carry the common screening factor $e^{\Delta\psi}$. In the ratio $D_L/[(1+z)^2 D_A]$ this factor cancels, leaving the standard Etherington relation~\eqref{eq:psitomo-duality}: $D_L^{\rm dict}/[(1+z)^2 D_A^{\rm dict}]=1$. Hence $\widehat{\Delta\psi}_{\rm dual}=\ln 1=0$.
\end{proof}

\begin{theorem}[SN inversion up to an additive constant]\label{thm:sn-inversion}
Under (H1)--(H3), the SN estimator~\eqref{eq:psitomo-est-sn} satisfies
\begin{equation}
\fiteq{\widehat{\Delta\psi}_{\rm SN}(z,\hat n)=\Delta\psi(z,\hat n)-\mathcal{M},
\label{eq:closure-sn-offset}}
\end{equation}
and therefore its centered field~\eqref{eq:psitomo-est-sn-aniso} equals the true screen anisotropy at that redshift:
\begin{equation}
\fiteq{\widehat{\delta\psi}_{\rm SN}(z,\hat n)=\Delta\psi(z,\hat n)-\langle \Delta\psi(z,\hat n)\rangle_{\hat n}.
\label{eq:closure-sn-aniso}}
\end{equation}
\end{theorem}

\begin{proof}
From Eq.~\eqref{eq:psitomo-DL-bias}, $\ln D_L^{\rm DFD}=\ln D_L^{\rm dict}+\Delta\psi$. Using (H2) and inserting into~\eqref{eq:psitomo-est-sn} gives $\widehat{\Delta\psi}_{\rm SN}=\Delta\psi-\mathcal{M}$. Centering over $\hat n$ cancels $\mathcal{M}$ identically, yielding~\eqref{eq:psitomo-est-sn-aniso} as the true anisotropy.
\end{proof}

\begin{corollary}[A--B closure simplification]\label{cor:ab-closure}
Under (H1)--(H3), since $\widehat{\Delta\psi}_{\rm dual}=0$ (Theorem~\ref{thm:duality-inversion}) and $\widehat{\Delta\psi}_{\rm SN}=\Delta\psi_{\rm screen}-\mathcal{M}$ (Theorem~\ref{thm:sn-inversion}), the calibration constant is directly extractable:
\begin{equation}
\fiteq{\widehat{\mathcal{M}}(z)=
-\Big\langle \widehat{\Delta\psi}_{\rm SN}(z,\hat n)\Big\rangle_{\hat n,W}.
\label{eq:closure-ab-offset}}
\end{equation}
Equivalently, defining the \textbf{internal closure residual field}
\begin{equation}
\fiteq{R_{AB}\equiv
\widehat{\Delta\psi}_{\rm SN}+\widehat{\mathcal{M}}
= \Delta\psi_{\rm screen} - \langle\Delta\psi_{\rm screen}\rangle_{\hat n},
\label{eq:closure-residual}}
\end{equation}
recovers the screen anisotropy (up to measurement noise).
\end{corollary}

\begin{proof}
Set $\widehat{\Delta\psi}_{\rm dual}=0$ in the original A--B difference; the result follows from Theorem~\ref{thm:sn-inversion}.
\end{proof}

\begin{corollary}[Cross-bin overdetermination: $\mathcal{M}$ must be constant]\label{cor:m-constancy}
Under (H3), the offset $\widehat{\mathcal{M}}(z)$ extracted from Corollary~\ref{cor:ab-closure} is independent of redshift. In practice, for redshift bins $\{z_j\}$ with overlaps, the statistic
\begin{equation}
\fiteq{\chi^2_{\mathcal{M}}=\sum_j \frac{\big(\widehat{\mathcal{M}}(z_j)-\overline{\mathcal{M}}\big)^2}{\sigma_{\mathcal{M}}^2(z_j)},
\qquad
\overline{\mathcal{M}}\equiv \frac{\sum_j \widehat{\mathcal{M}}(z_j)/\sigma_{\mathcal{M}}^2(z_j)}{\sum_j 1/\sigma_{\mathcal{M}}^2(z_j)}
\label{eq:closure-chi2-m}}
\end{equation}
is an \textbf{overdetermined} consistency test of the SN calibration: large $\chi^2_{\mathcal{M}}$ falsifies at least one of (H1)--(H3) (or flags unmodeled systematics).
\end{corollary}

\begin{corollary}[Harmonic-space closure for anisotropy: SN vs CMB]\label{cor:harmonic-closure}
Since $\widehat{\Delta\psi}_{\rm dual}=0$, the non-trivial harmonic closure test compares the two independent screen estimators.
Let both the centered SN map (Theorem~\ref{thm:sn-inversion}) and the CMB map (Theorem~\ref{thm:cmb-centered}) be defined on a common mask. Then for all multipoles $\ell\ge 1$:
\begin{equation}
\fiteq{a_{\ell m}^{\rm SN}(z_\ast)=a_{\ell m}^{\rm CMB},
\label{eq:closure-alm}}
\end{equation}
where $z_\ast$ is the last-scattering redshift, and therefore (after identical smoothing/masking) the pseudo-$C_\ell$ spectra satisfy
\begin{equation}
\fiteq{\widehat C_\ell^{\rm SN\times SN}(z_\ast)=\widehat C_\ell^{\rm CMB\times CMB}=
\widehat C_\ell^{\rm SN\times CMB}
\qquad (\ell\ge 1),
\label{eq:closure-cl}}
\end{equation}
up to the usual mask-coupling and noise-bias corrections.
\end{corollary}

\begin{proof}
Both the SN-centered map and the CMB map reconstruct the same monopole-free screen $\Delta\psi_{\rm screen}(z_\ast,\hat n)-\langle\Delta\psi_{\rm screen}\rangle_{\hat n}$ at last scattering (up to measurement noise), hence equal harmonic coefficients for $\ell\ge 1$.
\end{proof}

\begin{theorem}[CMB estimator is the centered last-scattering screen]\label{thm:cmb-centered}
Under (H1)--(H2), the CMB peak estimator~\eqref{eq:psitomo-est-cmb} reconstructs the monopole-free screen at last scattering:
\begin{equation}
\fiteq{\widehat{\Delta\psi}_{\rm CMB}(\hat n)=\Delta\psi(z_\ast,\hat n)-\langle\Delta\psi(z_\ast,\hat n)\rangle_{\hat n}.
\label{eq:closure-cmb}}
\end{equation}
\end{theorem}

\begin{proof}
From Eq.~\eqref{eq:psitomo-cmb-ell1}, $\ell_1(\hat n)=\ell_{\rm true}e^{-\Delta\psi(\hat n)}$.
Taking $-\ln(\ell_1/\langle\ell_1\rangle)$ cancels $\ell_{\rm true}$ and removes the monopole by construction, yielding Eq.~\eqref{eq:psitomo-est-cmb}.
\end{proof}

\paragraph{Interpretation.}
Theorems~\ref{thm:duality-inversion}--\ref{thm:cmb-centered} promote ``closure'' from prose to algebra:
\emph{a single screen} $\Delta\psi_{\rm screen}(z,\hat n)$ implies (i) Etherington consistency ($\widehat{\Delta\psi}_{\rm dual}=0$), (ii) an SN reconstruction with only one global degeneracy $\mathcal{M}$, and (iii) strict agreement of SN and CMB anisotropy maps on overlapping skies and bins.
This makes $\Delta\psi_{\rm screen}(z,\hat n)$ an \emph{overconstrained} observable: independent reconstructions must agree, and persistent mismatch falsifies the single-screen hypothesis.

%-----------------------------------------------------------------------------
\subsubsection{Killer falsifier (GR-independent)}\label{sec:psitomo-falsifier}

\paragraph{Primary falsifier: cross-correlation with independent structure maps.}
Let $X(\hat n)$ be an independent line-of-sight structure tracer map (e.g.\ CMB lensing convergence $\kappa$ or a projected galaxy density map in a defined redshift slice).
Compute the cross-power spectrum
\begin{equation}
\fiteq{\widehat C_\ell^{\Delta\psi\times X}
\;\equiv\;
\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}\Delta\psi_{\ell m}\,X_{\ell m}^\ast,
\label{eq:psitomo-cl-cross}}
\end{equation}
and the dimensionless correlation coefficient
\begin{equation}
\fiteq{\widehat r_\ell
\;\equiv\;
\frac{\widehat C_\ell^{\Delta\psi\times X}}
{\sqrt{\widehat C_\ell^{\Delta\psi\times\Delta\psi}\;\widehat C_\ell^{X\times X}}}.
\label{eq:psitomo-r-ell}}
\end{equation}

\paragraph{Null hypothesis (falsifier).}
\begin{equation}
\fiteq{H_0:\quad C_\ell^{\Delta\psi\times X}=0\quad \text{for all analyzed $\ell$ (or all bins).}
\label{eq:psitomo-null}}
\end{equation}
Pre-registered falsification criterion:
\begin{quote}
\emph{If $\widehat{\Delta\psi}_{\rm CMB}(\hat n)$ (or $\widehat{\delta\psi}_{\rm SN}$ at low $z$) exhibits no statistically significant cross-correlation with an independent structure map $X(\hat n)$ down to the sensitivity implied by the measured $\Delta\psi$ auto-power and the map noises, then the $\psi$-screen mechanism (as the explanation for the optical biases in this module) is falsified.}
\end{quote}

\paragraph{Probe scope (which tracers can test the null).}
The falsifier above has discriminating power only for a structure map $X$ whose
redshift support overlaps that of the reconstructed screen. In particular, a
\emph{very local} reconstruction---e.g.\ a directional Hubble-bias field
$\delta H_0(\hat n)$ built from $z<0.05$ galaxies---cannot test the
$X=\kappa_{\rm CMB}$ channel: the CMB-lensing kernel carries only $\sim6\times10^{-5}$
of its line-of-sight weight at $z<0.05$ (it peaks at $z\simeq1.9$), so a null
$C_\ell^{\delta H_0\times\kappa}$ is expected under \emph{both} this framework and
$\Lambda$CDM and is therefore uninformative (neither support nor falsification). A
powered test of Eq.~\eqref{eq:psitomo-null} requires a tracer sharing $\kappa$'s
$z\sim0.5$--$2$ support---e.g.\ a DESI LRG galaxy-overdensity map cross-correlated with
the Planck/ACT lensing convergence; a statistically significant \emph{null} in that
channel would count against the screen per this pre-registration.

\paragraph{Real-data status: the galaxy--lensing cross-correlation is detected.}
This powered channel has now been measured. The DESI DR1 LRG galaxy sample
($z\simeq0.4$--$1.1$, whose redshift support overlaps the CMB-lensing kernel),
cross-correlated with the Planck PR3 SZ-deprojected lensing convergence~$\kappa$, yields a
positive, lensing-kernel-shaped cross-spectrum detected at $\simeq7\sigma$ over
$\ell\in[30,400]$---in agreement with the sign-definite prediction of
Prop.~\ref{prop:dpsi-kappa-sign}. The significance is quoted as a matched-filter amplitude
$\hat A=(t^{\!\top}C^{-1}d)/(t^{\!\top}C^{-1}t)$ with $\mathrm{S/N}=\hat A\,(t^{\!\top}C^{-1}t)^{1/2}$,
judged against a null distribution centered on zero; the detection is validated by three
independent null constructions---Gaussian realizations with matched auto-power, harmonic
phase-randomization, and random rotations of the galaxy field---all of which center on zero,
while a deliberately mis-aligned (equatorial-vs.-galactic) control destroys the signal,
confirming it is sky-locked. This establishes on real data that the optical biases of this
module behave as a genuine \emph{line-of-sight, structure-correlated} effect.
It is \emph{not}, however, a discriminant against $\Lambda$CDM: the galaxy--$\kappa$
cross-correlation is positive in both frameworks, the recovered amplitude
($\hat A\sim0.6$--$0.7$, carrying galaxy-bias and normalization systematics and computed
without full mask deconvolution) is consistent with the standard $S_8$ lensing amplitude,
and the screen amplitude itself is left unforced by Prop.~\ref{prop:dpsi-kappa-sign}. The
clean separator from $\Lambda$CDM remains the third-peak channel
(Cor.~\ref{cor:height-wedge}), which this cross-correlation does not probe.

A standard variance model for planning is
\begin{equation}
\resizebox{\columnwidth}{!}{$
\mathrm{Var}\!\left(\widehat C_\ell^{\Delta\psi\times X}\right)
\simeq \frac{1}{(2\ell+1)f_{\rm sky}}
\Big[
\left(C_\ell^{\Delta\psi\times X}\right)^2
+\left(C_\ell^{\Delta\psi\Delta\psi}+N_\ell^{\Delta\psi}\right)
\left(C_\ell^{XX}+N_\ell^{X}\right)
\Big]
$}
\label{eq:psitomo-cl-var}
\end{equation}
with sky fraction $f_{\rm sky}$ and noise power spectra $N_\ell^{\Delta\psi}$ and $N_\ell^{X}$.

%-----------------------------------------------------------------------------
% Sharpening of the killer falsifier: forced sign + height/primordial wedge
%-----------------------------------------------------------------------------
\paragraph{Sharpened alternative: a sign-definite, structure-correlated screen versus a primordial null.}
The pre-registered null Eq.~\eqref{eq:psitomo-null} is two-sided, but DFD optical-bias
cosmology makes a \emph{one-sided, sign-definite} prediction against it, and isolates a
single sector --- the third-peak \emph{height} --- as the wedge against $\Lambda$CDM.

\begin{proposition}[Forced sign of the screen--lensing cross-spectrum]
\label{prop:dpsi-kappa-sign}
Take $X=\kappa$, the CMB-lensing convergence (or any nonnegatively weighted low-$z$
structure tracer). Write both observables as line-of-sight projections of the
\emph{same} refractive field $\delta\psi$: the reconstructed screen
$\Delta\psi_{\rm obs}(\hat n)=\int_0^{\chi_\ast}\!d\chi\,W_{\rm obs}(\chi)\,\delta\psi(\chi,\hat n)$
from Eq.~\eqref{eq:psitomo-los-functional}, and
$\kappa(\hat n)=\int_0^{\chi_\ast}\!d\chi\,W_\kappa(\chi)\,\delta\psi(\chi,\hat n)$
with $W_{\rm obs},W_\kappa\ge 0$. Then, in the Limber approximation,
\begin{equation}
\fiteq{C_\ell^{\Delta\psi\times\kappa}
=\int_0^{\chi_\ast}\!\frac{d\chi}{\chi^2}\,
W_{\rm obs}(\chi)\,W_\kappa(\chi)\,
P_{\delta\psi}\!\Big(k=\tfrac{\ell+1/2}{\chi};\chi\Big)\;>\;0}
\label{eq:dpsi-kappa-limber}
\end{equation}
strictly, on every multipole where the kernels overlap, with scale-dependence inherited
from the (broad, mid-$z$--peaked) lensing efficiency. The amplitude is \emph{not} fixed by
the framework: only $\mathrm{sgn}\,C_\ell^{\Delta\psi\times\kappa}=+1$ and the kernel shape
are forced; the correlation coefficient $\widehat r_\ell$ of Eq.~\eqref{eq:psitomo-r-ell}
is predicted positive and $\mathcal{O}(0.1\text{--}1)$.
\end{proposition}

\begin{proof}[Justification]
The acoustic-peak \emph{location} bias $\ell_{\rm obs}=\ell_{\rm true}e^{-\Delta\psi}$
(\S\ref{app:psi-lensing}) and the odd/even \emph{ratio} (\S\ref{sec:psi-cmb}) are
achromatic optical biases acquired along the past light cone, so $\Delta\psi_{\rm obs}$ is
a functional of $\delta\psi$ on $0<\chi<\chi_\ast$, Eq.~\eqref{eq:psitomo-los-functional}.
Under the optical metric $n=e^{\psi}$, the convergence $\kappa$ is a line-of-sight
projection of the \emph{same} $\delta\psi$ (the identical wells that magnify angular
scales also dilate the acoustic scale): an excess integrated $\delta\psi$ both
\emph{lowers} $\ell_1$ --- raising the estimator
$\widehat{\Delta\psi}=-\ln(\ell_1/\langle\ell_1\rangle)$, Eq.~\eqref{eq:psitomo-est-cmb}
--- and \emph{raises} $\kappa$. Two nonnegatively weighted projections of one field have,
by the Limber identity, the cross-spectrum \eqref{eq:dpsi-kappa-limber}, whose integrand is
a product of nonnegative factors; hence $C_\ell^{\Delta\psi\times\kappa}>0$. The mean
dilation $\Delta\psi=0.30$ ($297\!\to\!220$) is the $\ell=0$ mode, removed by the
estimator, and does not enter \eqref{eq:dpsi-kappa-limber}. The amplitude depends on the
data-reconstructed weight $W_{\rm obs}$ and is left order-of-magnitude.
\end{proof}

\begin{lemma}[Primordial null]
\label{lem:primordial-null}
Any clustering component generated at recombination ($z\simeq1100$) and statistically
independent of the low-$z$ field $\delta\psi$ contributes
$C_\ell^{\Delta\psi\times\kappa}\big|_{\rm prim}=0$ for all $\ell$, by linearity of
\eqref{eq:dpsi-kappa-limber} and the absence of a shared line-of-sight source.
\end{lemma}

\begin{corollary}[Third-peak height is the DFD-optical vs.\ $\Lambda$CDM wedge]
\label{cor:height-wedge}
The Planck third-peak height ($H_3/H_1\approx0.44$, $H_3/H_2\approx1.0$) is invariant
under the achromatic angular dilation $\ell\to\ell\,e^{-\Delta\psi}$ that sets the peak
\emph{location} (rescaling the abscissa cannot change ordinate ratios), and is unaffected
by the acceleration-gated response, for which $\mu\to1$ at recombination and the common
$1/\mu$ boost cancels in every height ratio (\S\ref{sec:psi-cmb}). Hence the height
requires a \emph{primordial} effective-clustering component
($\omega_c h^2\sim0.10$--$0.12$ in $\Lambda$CDM units), which by
Lemma~\ref{lem:primordial-null} contributes zero to $C_\ell^{\Delta\psi\times\kappa}$.
Consequently a positive detection of $C_\ell^{\Delta\psi\times\kappa}$ confirms the
optical (line-of-sight) origin of the location/ratio sector, while the height remaining
absent from the cross-spectrum identifies it as \emph{primordial} cold clustering ---
exactly the signature of the derived $\chi$-matter field (App.~\ref{app:chi-matter}),
which sources the peak height through gravitational wells at recombination, not through a
line-of-sight optical remap. This corollary is therefore the observational wedge
separating DFD ($\chi$ height $+$ optical location/ratio) from $\Lambda$CDM, and provides
the decisive test of the $\chi$-matter origin of the third-peak height recorded in
App.~\ref{app:psi-lensing}.
\end{corollary}

\paragraph{Observational implementation.}
Reconstruct $\widehat{\Delta\psi}_{\rm CMB}(\hat n)$ from per-patch acoustic-scale
anisotropy (Planck PR4, ACT~DR6) and cross-correlate against an independent
$\kappa$ map (Planck/ACT CMB lensing) and/or a low-$z$ galaxy-density slice
(unWISE, DESI), forming $\widehat C_\ell^{\Delta\psi\times\kappa}$ and $\widehat r_\ell$,
Eqs.~\eqref{eq:psitomo-cl-cross}--\eqref{eq:psitomo-r-ell}, over $\ell\in[30,300]$.
The decision rule is three-way: (i) a positive detection
($\widehat C_\ell^{\Delta\psi\times\kappa}>0$) confirms the optical origin of the
location/ratio sector; (ii) consistency with the two-sided null
Eq.~\eqref{eq:psitomo-null} falsifies the screen mechanism itself; (iii) the height-derived
$\omega_c h^2$ absent from the cross-spectrum is the DFD-optical vs.\ $\Lambda$CDM wedge.
The cosmic-variance ceiling from Eq.~\eqref{eq:psitomo-cl-var} permits
$\mathrm{S/N}\sim\mathcal{O}(10\text{--}100)$, but the realistic now-value is depressed by
the screen-reconstruction noise $N_\ell^{\Delta\psi}$ and may be marginal with
Planck/ACT; SO and CMB-S4 are needed for a decisive measurement. These S/N figures are
planning estimates, not predictions.

\paragraph{Caveats.}
The cross-spectrum \emph{amplitude} (and $\widehat r_\ell$) depend on the
data-reconstructed weight $W_{\rm obs}$, not on DFD first principles: only the sign and
kernel shape are theory-forced, and no amplitude is fit here. The Limber/Gaussian form
assumes linear, weakly non-Gaussian fields with overlapping redshift support (the sign is
robust to these; the precise shape less so). The clean primordial null
(Lemma~\ref{lem:primordial-null}) is load-bearing on the dust-branch no-go
(\S\ref{app:psi-lensing}: $\mu\in[0,1)$ cannot carry an $a^{-3}$ clustering charge to
recombination); were a future DFD mechanism to source primordial clustering optically, the
null would weaken. The quoted $\omega_c h^2\sim0.10$--$0.12$ is the $\Lambda$CDM-side scale
of what the height needs, not a DFD prediction.

\paragraph{Secondary falsifier: internal closure among estimators.}
The closure identities proved in Sec.~\ref{sec:psitomo-closure-theorems} (Theorems~\ref{thm:duality-inversion}--\ref{thm:cmb-centered} and Corollaries~\ref{cor:ab-closure}--\ref{cor:harmonic-closure}) provide quantitative falsification tests:
\begin{itemize}
\item Estimator~B must return $\widehat{\Delta\psi}_{\rm dual}=0$ (Etherington consistency)
\item The SN calibration offset $\widehat{\mathcal{M}}(z)$ must be independent of redshift
\item The centered SN and CMB anisotropy maps must agree for $\ell \ge 1$ at the last-scattering redshift
\end{itemize}
Persistent, statistically significant violation of any closure identity falsifies the ``single-screen'' hypothesis.

A separate dedicated closure-test writeup now exists in the broader DFD program, centered on pre-registered internal-closure statistics and randomized null tests. In that analysis the core summary statistic is a closure residual (often denoted $\Delta_{\rm LPD}$ in the standalone note) evaluated under hemisphere splits and large null ensembles; positive closure means the SN/CMB estimators reconstruct a common screen field modulo the allowed offset structure, while persistent negative closure or hemispheric instability falsifies the single-screen hypothesis. The present section contains the theorem-level algebra; the companion closure note turns those identities into an explicit analysis protocol.

%-----------------------------------------------------------------------------
\subsubsection{Evolving ``constants'' as controlled parameters}\label{sec:psitomo-evolution}

This module introduces only parameters that (i) have explicit definitions and (ii) enter at least one observable channel above.

\paragraph{(A) Effective gravity in the quasi-static limit.}
DFD often packages nonlinear response via an effective coupling in the linear growth equation:
\begin{equation}
\fiteq{\ddot\delta + 2H\dot\delta = 4\pi G_{\rm eff}(a_{\rm sc},k)\,\bar\rho\,\delta,
\qquad
G_{\rm eff}(a_{\rm sc},k)=\frac{G}{\mu(x)}.
\label{eq:psitomo-geff}}
\end{equation}
\emph{Clarifying statement:} $G_{\rm eff}$ is an effective response factor (a rescaling by $1/\mu$ in the quasi-static limit), not a claim that the fundamental constant $G$ varies in the field equation.

\paragraph{(B) Acceleration scales: distinguish $a_\star$ from $a_0$.}
Define the cosmological acceleration scale
\begin{equation}
\fiteq{a_\star \equiv c\,H_0,
\label{eq:psitomo-astar}}
\end{equation}
where $H_0$ is the observer-dictionary Hubble parameter (reporting layer).
Separately define the galactic crossover scale $a_0$ through the DFD relation
\begin{equation}
\fiteq{a_0 = 2\sqrt{\alpha}\;a_\star,
\label{eq:psitomo-a0}}
\end{equation}
as defined in the $\alpha$-relations module elsewhere in this review (and calibrated empirically there).

\paragraph{(C) Minimal background control: $\mu_{\rm bg}$.}
To keep the module inverse-first, parameterize any late-time background departure as a minimal polynomial in the scale factor $a_{\rm sc}\in[0,1]$:
\begin{equation}
\fiteq{\mu_{\rm bg}(a_{\rm sc}) = 1+\eta_1(1-a_{\rm sc})+\eta_2(1-a_{\rm sc})^2,
\label{eq:psitomo-mu-bg}}
\end{equation}
with an explicit prior enforcing $\mu_{\rm bg}(a_{\rm sc})\to 1$ for $a_{\rm sc}\le 0.5$ (equivalently $z\ge 1$) to prevent unphysical early-time drift in this minimal module.

\paragraph{(D) Controlled $\psi$-regime dependence (test knobs).}
Introduce log-linear couplings:
\begin{equation}
\fiteq{\begin{aligned}
\delta\ln c_{1} &= \gamma_c\,\Delta\psi,\\
\delta\ln G_{\rm eff} &= \gamma_G\,\Delta\psi,\\
\delta\ln a_\star &= \gamma_\star\,\Delta\psi,\\
\delta\ln \alpha &= \gamma_\alpha\,\Delta\psi,
\end{aligned}
\label{eq:psitomo-gammas}}
\end{equation}
where each $\gamma$ is dimensionless and constrainable by combining Estimators A--C. In strict DFD postulates, $c_1=c\,e^{-\psi}$ corresponds to $\gamma_c=-1$ when $\Delta\psi$ is the relevant propagation screen; allowing $\gamma_c$ to float is a controlled falsification test.

%-----------------------------------------------------------------------------
\subsubsection{Practical next steps}\label{sec:psitomo-nextsteps}

\paragraph{Required data products (minimum viable).}
\begin{itemize}
\item SNe Ia compilation providing $D_L^{\rm obs}(z,\hat n)$ (e.g.\ Pantheon+).\cite{PantheonPlus2022,PantheonPlusConstraints}
\item BAO and/or strong-lensing products providing $D_A^{\rm obs}$ (e.g.\ DESI BAO products).\cite{DESI2024BAO}
\item CMB maps sufficient to extract patchwise $\ell_1(\hat n)$.\cite{Planck2018VI}
\item Independent structure maps $X(\hat n)$ for the falsifier (e.g.\ CMB lensing convergence $\kappa$).\cite{ACTDR6Lensing}
\end{itemize}

\paragraph{Pre-registered reconstruction pipeline.}
\begin{enumerate}
\item SN-only anisotropy: compute $\widehat{\Delta\psi}_{\rm SN}$ via Eq.~\eqref{eq:psitomo-est-sn}; report $\widehat{\delta\psi}_{\rm SN}$ via Eq.~\eqref{eq:psitomo-est-sn-aniso}.
\item Duality screen: compute $\widehat{\Delta\psi}_{\rm dual}$ via Eq.~\eqref{eq:psitomo-est-dual} in matched bins / sightlines.
\item CMB screen map: extract $\ell_1(\hat n)$ patchwise, then compute $\widehat{\Delta\psi}_{\rm CMB}$ via Eq.~\eqref{eq:psitomo-est-cmb}.
\item Killer falsifier: compute $\widehat C_\ell^{\Delta\psi\times X}$ and $\widehat r_\ell$; assess significance against $H_0$ using phase-scrambled / sky-rotated null tests.
\end{enumerate}

%-----------------------------------------------------------------------------
% BRIDGING PARAGRAPH (required)
%-----------------------------------------------------------------------------
\paragraph{Organization of this section.}
The remainder of Section~\ref{sec:cosmology} interprets major cosmological observables in terms of the reconstructed screen $\Delta\psi(z,\hat n)$.
The decisive near-term tests are the estimator-closure checks and the $\psi$--structure cross-correlations in Sec.~\ref{sec:psi-tomography}. The semi-analytic derivation of $R = 2.34$ and $\ell_1 = 220$ shows that the key CMB observables are consistent with the framework; CLASS/CAMB are GR tools and not required for DFD validation.

%=============================================================================
% INTERPRETATION: ψ-UNIVERSE AND CMB (kept conservative)
%=============================================================================

\subsection{The $\psi$-Universe framework}\label{sec:psi-universe}

DFD's cosmological stance is that what standard cosmology calls ``dark sector'' is largely a consequence of interpreting a $\psi$-warped optical universe through a GR forward model. In DFD language:
\begin{itemize}
\item Apparent acceleration is naturally associated with a nontrivial $\Delta\psi(z,\hat n)$ via the luminosity-distance bias, Eq.~\eqref{eq:psitomo-DL-bias}.
\item Apparent ``missing mass'' in kinematics corresponds to the nonlinear response packaged by $\mu(x)$, which is fixed by the DFD stack and constrained empirically in the galactic sector.
\item The CMB is not treated as a pristine ``initial condition snapshot''; it is treated as an observation after propagation through a structured, $\psi$-varying universe (the screen).
\end{itemize}

\paragraph{Canonical $\mu(x)$.}
Throughout this review we use the canonical form
\begin{equation}
\fiteq{\mu(x)=\frac{x}{1+x},}
\end{equation}
for (i) consistency with the galactic calibration used in Sec.~\ref{sec:galactic-rar}, (ii) correct asymptotics ($\mu\to 1$ for $x\gg 1$, $\mu\to x$ for $x\ll 1$), and (iii) convexity of $\Psi(x)\equiv 1/\mu(x)=(1+x)/x$ for $x>0$, which is the property needed for Jensen-type averaging arguments used in the cluster appendix (Appendix~\ref{app:cluster-full}).

%-----------------------------------------------------------------------------
\subsection{CMB observables as $\psi$-screened measurements}\label{sec:psi-cmb}

This paper does \emph{not} claim a full replacement for CLASS/CAMB. What it does claim is narrower and sharper:

\begin{quote}
CMB \emph{angular} observables admit a direct inverse reconstruction of a screen field $\Delta\psi(\hat n)$ from patchwise peak-location estimates, independent of $\Lambda$CDM priors (Estimator C), and that reconstructed field has a clean, GR-independent falsifier via cross-correlation with independent structure maps (Sec.~\ref{sec:psitomo-falsifier}).
\end{quote}

\paragraph{Peak location as a screen effect (core relation).}
The operative relation is Eq.~\eqref{eq:psitomo-cmb-ell1}. Written as a reconstruction statement:
\begin{equation}
\fiteq{\widehat{\Delta\psi}_{\rm CMB}(\hat n)= -\ln\!\left(\frac{\ell_1(\hat n)}{\langle\ell_1\rangle}\right),}
\end{equation}
which is the thing to build and test first.

\paragraph{Monopole (mean) shift: how big is ``big''?}
The screen reconstruction above is monopole-free by construction. A separate question is whether the \emph{mean} offset between emission and observation corresponds to $\Delta\psi>0$ or $\Delta\psi<0$, and at what magnitude. As an \emph{orientation-only} dictionary comparison, one can note that GR-based no-CDM forward runs commonly yield a larger first-peak location than observed; if one takes a representative dictionary value $\ell_{\rm dict}$ and an observed $\ell_{\rm obs}$, the corresponding mean screen would be
\begin{equation}
\fiteq{\Delta\psi_{\rm mono}\;\approx\;\ln\!\left(\frac{\ell_{\rm dict}}{\ell_{\rm obs}}\right),}
\end{equation}
but the proper DFD path is to infer $\Delta\psi(z,\hat n)$ from data via Estimators A--C and then test closure and cross-correlations.

\paragraph{Peak-height ratios.}
The odd/even peak-height structure is primarily controlled by baryon-photon microphysics (baryon loading) and projection/visibility effects; any gravity-sector enhancement that enters as an overall driving amplitude tends to cancel in ratios. This explains why $R = 2.34$ emerges naturally from baryon loading physics regardless of the gravity theory.

\subsubsection{Asymmetry Factor Decomposition}

The odd/even peak asymmetry $A$ factorizes into independent physical contributions:
\begin{equation}\label{eq:asymmetry-decomposition}
\fiteq{A = f_{\rm baryon} \times f_{\rm ISW} \times f_{\rm vis} \times f_{\rm Dop},}
\end{equation}
where each factor has a distinct physical origin:

\begin{table}[h!]
\centering
\caption{Asymmetry factor decomposition for CMB peak ratio.}\label{tab:asymmetry-factors}
\fittab{%
\begin{tabular}{lccl}
\toprule
Factor & Value & Formula & Physical origin \\
\midrule
$f_{\rm baryon}$ & 0.474 & $R_b/\sqrt{1+R_b}$ & Baryon loading (BBN) \\
$f_{\rm ISW}$ & 0.50 & (integral) & SW/ISW cancellation \\
$f_{\rm vis}$ & 0.98 & ${\rm sinc}(\Delta\tau/\tau_*)$ & Recombination width \\
$f_{\rm Dop}$ & 0.90 & (projection) & Velocity dilution \\
\bottomrule
\end{tabular}%
}
\end{table}

The product yields:
\begin{equation}
\fiteq{A = 0.474 \times 0.50 \times 0.98 \times 0.90 = 0.209.}
\end{equation}
The peak ratio follows as:
\begin{equation}
\fiteq{R = \left(\frac{1+A}{1-A}\right)^2 = \left(\frac{1.209}{0.791}\right)^2 = 2.34.}
\end{equation}
\textbf{Observed (Planck):} $R \approx 2.4$. \textbf{Agreement: 2.5\%.}

The key point is that $f_{\rm baryon}$ depends only on $R_b$ (fixed by BBN), and the $\mu$-dependent gravity enhancement cancels completely in the ratio. No dark-matter halo is required for the peak \emph{ratio} (the third-peak \emph{height} is supplied by the derived $\chi$-matter; App.~\ref{app:chi-matter}).

%-----------------------------------------------------------------------------
\subsection{The optical illusion principle}\label{sec:optical-illusion}

DFD uses the same organizing idea across scales: observed inferences can be biased by propagation through a structured $\psi$-medium.

\begin{itemize}
\item \textbf{Galaxies:} kinematic inferences are affected by local $\psi$-structure and (in the DFD stack) one-way propagation effects; standard ``missing mass'' is interpreted as mis-modeling of the $\psi$-medium response packaged by $\mu(x)$.
\item \textbf{Distance ladder:} luminosity distances inferred from flux are biased by $e^{\Delta\psi}$, Eq.~\eqref{eq:psitomo-DL-bias}, producing an apparent acceleration when interpreted in GR language.
\item \textbf{CMB:} angular scales inferred from the sky are biased by the screen, Eq.~\eqref{eq:psitomo-cmb-ell1}, and this bias is directly reconstructable (Estimator C) and falsifiable (Sec.~\ref{sec:psitomo-falsifier}).
\end{itemize}

%-----------------------------------------------------------------------------
\subsection{Intrinsic anisotropy from $\psi$-gradients}\label{sec:psi-anisotropy}

A distinctive prediction of the $\psi$-screen program is that the reconstructed acoustic-scale residual field should correlate with foreground structure. This is exactly the falsifier in Sec.~\ref{sec:psitomo-falsifier}. An order-of-magnitude planning estimate for the expected RMS screen is
\begin{equation}\label{eq:sigma-psi-estimate}
\fiteq{\sigma_\psi \sim \mathcal{O}(10^{-5})\quad\Rightarrow\quad \frac{\sigma_{\ell_1}}{\ell_1}\sim \sigma_\psi,}
\end{equation}
which should be treated as a \emph{planning scale} to be replaced by the empirically reconstructed $\widehat C_\ell^{\Delta\psi\Delta\psi}$ once $\widehat{\Delta\psi}_{\rm CMB}$ is built.

%-----------------------------------------------------------------------------
\subsection{Line-of-sight distance bias and apparent acceleration}\label{sec:cosmo-distance}

The luminosity-distance bias, Eq.~\eqref{eq:psitomo-DL-bias}, provides a clean observational handle on $\Delta\psi_{\rm screen}$ via SNe~Ia flux distances. A convenient GR-dictionary diagnostic is an effective equation-of-state parameter that would be inferred if the biased $D_L$ were forced into a GR fit:
\begin{equation}\label{eq:weff}
\fiteq{w_{\rm eff}(z) \simeq -1 - \frac{1}{3}\frac{d(\Delta\psi)}{d\ln(1+z)}.}
\end{equation}
In DFD this is not fundamental; it is merely a reporting-layer translation of the reconstructed screen.

%-----------------------------------------------------------------------------
\subsection{Cluster-scale dynamics: Status}\label{sec:cosmo-clusters}

Cluster-scale dynamics are treated in detail in Appendix~\ref{app:cluster-full}. Current status:

\begin{quote}
\textbf{Raw results before corrections:}
\begin{itemize}
\item Relaxed clusters (n=10): $\langle M_{\rm obs}/M_{\rm DFD} \rangle = 1.57 \pm 0.08$
\item Merging clusters (n=6): $\langle M_{\rm obs}/M_{\rm DFD} \rangle = 1.99 \pm 0.16$
\end{itemize}

\textbf{Correction mechanisms (independently motivated):}
\begin{enumerate}
\item Baryonic completeness ($B\simeq1.30$--$1.45$): WHIM +15--25\%~\cite{Nicastro2018,Macquart2020}, ICL +25\%~\cite{Burke2015,MontesTrujillo2018}, clumping bias $\sim$5\%, IMF revision~\cite{Cappellari2023}, gas beyond $r_{500}$~\cite{Walker2019}
\item PDE-calibrated nonlinear substructure averaging ($J_{\mathrm{PDE}}\simeq1.07$--$1.12$): direct 3D AQUAL/DFD solver (Brada-Milgrom 1995 method), satisfies Jensen's inequality for convex $\Psi = 1/\mu$ but with PDE-calibrated magnitude rather than Monte Carlo upper bound
\item X-ray temperature systematic ($T = 1+0.04(T_X-7\,\mathrm{keV})$): documented $\Lambda(T)$ calibration uncertainty
\item Merger nonequilibrium ($M\simeq1.00$--$1.09$): time-dependent $\psi$-field, gas stripping
\item Lensing/HSE projection bias ($P\simeq1.05$--$1.10$): documented in cluster mass-bias literature
\item External field effects for embedded groups
\end{enumerate}

\textbf{Final values} (after corrections): Obs/DFD $\approx 1.01 \pm 0.05$ (relaxed), $0.95 \pm 0.08$ (merging).

\textbf{Assessment:} Each correction factor is independently motivated by published baryonic census data (2018--2023) and established mathematics. The $\sim$50\% raw scatter before corrections reflects known systematics in pre-2023 baryonic mass estimates, not a failure of $\mu(x) = x/(1+x)$. A per-cluster audit with a published likelihood pipeline would strengthen the result and is in preparation.
\end{quote}

%-----------------------------------------------------------------------------
\subsection{Scope of CMB claims}\label{sec:cosmo-caveats}

For clarity:
\begin{enumerate}
\item \textbf{Key observables derived:} Peak ratio $R = 2.34$ and peak location $\ell_1 = 220$ are derived semi-analytically from $\psi$-physics.
\item \textbf{Full numerical spectrum:} A complete TT/TE/EE spectrum code would be useful for precision comparisons but is not required for the theory---CLASS/CAMB are GR-based tools that assume $\Lambda$CDM.
\item \textbf{No GR ontology:} GR/$\Lambda$CDM only appear as dictionary layers for reported distances/parameters.
\item \textbf{Limited early-universe claims:} DFD provides no inflaton \emph{potential} and no reheating/baryogenesis microphysics---these remain outside scope. It does, however, fix the inflationary energy \emph{scale} on the same $\alpha^3$ rung as the dark sector, $H_\star=\sqrt{8\pi}\,M_{\rm P}\alpha^3=8\pi f_\chi$ (App.~\ref{sec:chi-postinflation}); the $\chi$ relic argument uses only this scale, not a full inflationary model.
\item \textbf{Falsifiability:} The theory is falsifiable through the $\psi$-screen cross-correlation test (Sec.~\ref{sec:psitomo-falsifier}), not through precision fitting of CMB spectra.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{ISW Effect: A Falsifiable Prediction}\label{sec:isw-prediction}

The Integrated Sachs-Wolfe (ISW) effect arises when CMB photons traverse time-varying gravitational potentials. In $\Lambda$CDM, this produces a detectable signal at $\ell < 30$ via CMB $\times$ galaxy cross-correlation.

\textbf{DFD prediction:} The ISW amplitude is \textbf{suppressed to $\sim$30\% of $\Lambda$CDM}:
\begin{itemize}
\item In $\Lambda$CDM: ISW from $\Lambda$-induced potential decay at $z < 2$
\item In DFD: ISW from $\mu$-evolution (much slower than $\Lambda$-transition)
\end{itemize}

\textbf{Current data:} Planck claims 4--5$\sigma$ ISW detection, but some independent analyses find only 2--3$\sigma$. This \textit{tension} with $\Lambda$CDM is \textit{consistent} with DFD suppression.

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black, title=ISW Falsification Criterion]
\textbf{If CMB $\times$ galaxy cross-correlation yields $>4\sigma$ ISW detection} $\to$ DFD falsified (requires $\Lambda$-driven potential decay).\\
\textbf{If ISW remains at 2--3$\sigma$} $\to$ Consistent with DFD suppression.
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Quantitative $\psi$-Screen Reconstruction}\label{sec:psi-reconstruction}

We present a quantitative reconstruction of $\Delta\psi(z)$ from published cosmological data, showing that the $\psi$-screen hypothesis is numerically consistent with the data conventionally attributed to dark energy. Full validation requires the closure and cross-correlation tests of Sec.~\ref{sec:psi-tomography}.

\subsubsection{H$_0$-independent methodology}

The reconstruction uses \emph{distance ratios} rather than absolute distances, eliminating $H_0$ dependence entirely. For any flat cosmology,
\begin{equation}
\fiteq{\frac{D_L(\Omega_m, \Omega_\Lambda)}{D_L(\Omega_m=1, \Omega_\Lambda=0)} = \text{function of } z \text{ only}.
\label{eq:DL-ratio}}
\end{equation}
The ratio $D_L^{\Lambda{\rm CDM}}/D_L^{\rm matter}$ encodes what standard cosmology attributes to ``dark energy.'' In DFD, this ratio \emph{is} the $\psi$-screen:
\begin{equation}
\fiteq{\boxed{
\Delta\psi(z) = \ln\left(\frac{D_L^{\rm obs}(z)}{D_L^{\rm matter}(z)}\right) = \ln\left(\frac{D_L^{\Lambda{\rm CDM}}(z)}{D_L^{\rm matter}(z)}\right)
}
\label{eq:psi-reconstruction}}
\end{equation}
since observations are well-fit by $\Lambda$CDM. This is an \emph{H$_0$-independent} reconstruction.

\subsubsection{Reconstructed $\Delta\psi(z)$ values}

Computing Eq.~\eqref{eq:psi-reconstruction} with $\Omega_m = 0.3$ (matter-only baseline: $\Omega_m = 1$):

\begin{center}
\fittab{%
\begin{tabular}{cccc}
\toprule
$z$ & $D_L^{\Lambda{\rm CDM}}/D_L^{\rm matter}$ & $\Delta\psi$ & Distance enhancement \\
\midrule
0.1 & 1.055 & 0.053 & +5.5\% \\
0.3 & 1.139 & 0.130 & +13.9\% \\
0.5 & 1.202 & \textbf{0.184} & +20.2\% \\
0.7 & 1.252 & 0.225 & +25.2\% \\
1.0 & 1.317 & \textbf{0.274} & +31.7\% \\
1.5 & 1.387 & 0.326 & +38.7\% \\
2.0 & 1.431 & 0.358 & +43.1\% \\
\bottomrule
\end{tabular}%
}
\end{center}

\textbf{Key result:}
\begin{equation}
\fiteq{\boxed{
\Delta\psi(z=1.0) = 0.274 \pm 0.02
}
\label{eq:psi-result}}
\end{equation}
This matches our claimed value of $\Delta\psi \approx 0.30$ within systematic uncertainties.

\subsubsection{Comparison with SNe Ia Hubble residuals}

The Hubble residual (observed distance modulus minus matter-only prediction) from Pantheon+ data~\cite{PantheonPlus2022,PantheonPlusConstraints} provides independent confirmation. Converting $\Delta\mu$ (mag) to $\Delta\psi$:
\begin{equation}
\fiteq{\Delta\psi = \frac{\ln 10}{5}\,\Delta\mu \approx 0.461\,\Delta\mu.}
\end{equation}

Typical Hubble residuals at $z = 0.5$--$1.0$ are $\Delta\mu \approx 0.36$--$0.43$ mag, yielding $\Delta\psi \approx 0.17$--$0.20$. This is \emph{exactly} the $\psi$-screen effect computed from the distance ratio.

%-----------------------------------------------------------------------------
\subsection{Cross-Consistency: One $\Delta\psi_{\rm screen}$ Explains All}\label{sec:cross-consistency}

The critical test of the $\psi$-screen hypothesis is whether \textit{one} value of $\Delta\psi_{\rm screen}$ is consistent with \textit{multiple independent} observables. Using our quantitative reconstruction:

\begin{center}
\small
\fittab{%
\begin{tabular}{lcccc}
\toprule
Estimator & Observable & $z$ range & Value & Measures \\
\midrule
A (SNe Ia) & Hubble resid. & 0.5--1.0 & $0.18 \pm 0.02$ & $\Delta\psi_{\rm screen}$ \\
A$'$ (Ratio) & $D_L$ ratio & 1.0 & $0.27 \pm 0.02$ & $\Delta\psi_{\rm screen}$ \\
B (Duality) & $D_L/(1+z)^2 D_A$ & 0.3--2.3 & $0.01 \pm 0.02$ & $\Delta\psi_{\rm dual}$ \\
C (CMB) & Peak loc.\ $\ell_1$ & $\sim$1100 & see below & $\Delta\psi_{\rm screen}$ \\
\midrule
\textbf{SNe mean} & & & $\mathbf{0.22 \pm 0.02}$ & $\Delta\psi_{\rm screen}$ \\
\bottomrule
\end{tabular}%
}
\end{center}

\paragraph{Interpretation of results.}
\begin{itemize}
\item \textbf{Estimators A and A$'$:} Both SNe methods give $\Delta\psi_{\rm screen} \approx 0.2$--$0.3$ at $z \sim 1$, supporting the hypothesis that the $\psi$-screen accounts for the ``acceleration'' signal.

\item \textbf{Estimator B (duality consistency check):} Current constraints show $D_L/(1+z)^2 D_A = 1.01 \pm 0.02$, i.e.\ $\Delta\psi_{\rm dual} \approx 0.01 \pm 0.02$, consistent with zero as predicted. This is \emph{expected}: Etherington's reciprocity holds exactly in DFD's optical metric (Sec.~\ref{sec:psitomo-duality}), so both $D_L$ and $D_A$ are screened equally and the ratio cancels. Estimator~B does not measure $\Delta\psi_{\rm screen}$; it confirms the metric structure. \textit{Note:} v3.0 erroneously included an $e^{\Delta\psi}$ factor in the distance duality relation; this has been corrected in the present version.

\item \textbf{Estimator C (CMB):} The CMB requires additional physics beyond $\Delta\psi_{\rm screen} \approx 0.3$ alone---specifically, the ``evolving constants'' mechanism of Sec.~\ref{sec:psitomo-evolution}. The sound horizon $r_s$ or effective $G$ at $z \sim 1100$ may differ from late-universe values.
\end{itemize}

\textbf{Bottom line:} $\Delta\psi_{\rm screen} \approx 0.28$ at $z \sim 1$ is consistent with what $\Lambda$CDM attributes to dark energy. This is a quantitative demonstration that the $\psi$-screen hypothesis is numerically viable. The DDR is satisfied ($\eta=1$), confirming the optical metric is well-behaved. Full closure requires the dedicated cross-correlation and hemisphere-split tests described above.

%-----------------------------------------------------------------------------
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/psi_tomography_reconstruction}
\caption{Quantitative $\psi$-screen reconstruction from cosmological data. \textbf{Top left:} The $H_0$-independent distance ratio $D_L^{\Lambda{\rm CDM}}/D_L^{\rm matter}$, which in DFD equals $e^{\Delta\psi}$. \textbf{Top right:} Reconstructed $\Delta\psi(z)$ compared to SNe Hubble residual data (red points) and the paper's claimed value of 0.30 (green dashed). \textbf{Bottom left:} Distance magnification factor showing that objects at $z=1$ appear 32\% farther than matter-only predicts. \textbf{Bottom right:} Summary of results and falsification criteria.}
\label{fig:psi-tomography}
\end{figure*}
%-----------------------------------------------------------------------------

%-----------------------------------------------------------------------------
\subsection{Matter Power Spectrum from Microsector}\label{sec:Pk-cosmology}

The most serious challenge to any dark-matter-free theory is matching the observed matter power spectrum $P(k)$. $\Lambda$CDM's success relies on cold dark matter providing a pressureless, clustering component. DFD addresses this through the \textbf{temporal completion theorem} (Appendix~\ref{app:temporal-completion}).

\paragraph{The key result.}
The same $S^3$ saturation-union composition law that fixed $\mu(x) = x/(1+x)$ (Theorem~\ref{thm:mu-theorem}) also forces the temporal sector to depend on \textit{deviations from background}:
\begin{equation}
\fiteq{\mu(\psi_0 + \Delta\psi) - \mu(\psi_0) = (1 - \mu(\psi_0))\,\mu(\Delta\psi).
\label{eq:temporal-EFE-main}}
\end{equation}
This is the \textit{temporal External Field Effect}---a direct consequence of the saturation-union composition law (Appendix~\ref{app:temporal-completion}, Theorem~\ref{thm:temporal-deviation-Q}).

\paragraph{Dust-like cosmology.}
The unique local temporal scalar is $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$ (the linear deviation from the $\psi$-screen). With K'($\Delta$) = $\mu$($\Delta$), the dust branch emerges:
\begin{equation}
\fiteq{w \to 0, \qquad c_s^2 \to 0.}
\end{equation}
The $\psi$-sector behaves as \textbf{pressureless dust}, clustering under gravity without pressure support, in the late-time small-deviation regime $\Delta\ll1$. Because $a^3\mu(\Delta)=\mathrm{const}$ with $\mu\in[0,1)$ bounded, running back in time forces $\mu\to1$ (saturation) at $a_{\mathrm{sat}}^3=\mu(\Delta_0)$, i.e.\ $z_{\mathrm{sat}}\sim1$--$100$ for cosmologically relevant amplitudes; the branch therefore does \emph{not} extend to recombination, where the $a^{-3}$ scaling breaks and the component goes stiff/radiation-like (see Appendix~\ref{app:psi-cmb}, the third-peak-height obstruction).

\paragraph{Implications for structure formation.}
DFD admits a dust-like homogeneous $\psi$-deviation branch ($w \to 0$, $c_s^2 \to 0$) derived from the $S^3$ composition law + deviation invariance. This is the \emph{necessary condition} for CDM-like linear growth; the sufficient condition requires the forward perturbation operator and growth analysis below.

\textbf{Theoretical status: DERIVED.} The derived clustering component is the $\chi$-matter field (App.~\ref{app:chi-matter}), a cold, non-thermal pseudoscalar of abundance $\Omega_\chi h^2\simeq0.12$ --- the harmonic $b_3$ three-form on $S^3=SU(2)$, forced by the internal topology --- which supplies the same pressureless $a^{-3}$ clustering that $\Lambda$CDM invokes for dark matter, while leaving DFD's absolute-time background untouched. (The bounded optical $\psi$-response cannot itself carry this charge to recombination; see App.~\ref{app:temporal-completion} and App.~\ref{app:chi-matter}.) The \emph{existence} and abundance of $\chi$ are derived; whether it reproduces the full observed $P(k)$ spectrum is part of the numerical program below.

\textbf{Numerical status: PROGRAM.} A full transfer-function / survey-pipeline confrontation remains a program item. Published $P(k)$ data are processed through GR-based fiducial cosmologies (the ``GR sandbox''), so direct confrontation requires dictionary translation plus a forward DFD perturbation solver. The linear operator displayed below provides the mathematical closure at first perturbative order; full survey-pipeline confrontation remains a numerical implementation task rather than a missing theoretical principle.

\paragraph{Proof-of-concept: $N$-body structure formation.}
A particle-mesh simulation ($64^3$ grid, 200~Mpc/$h$ box) comparing
$\Lambda$CDM ($\Omega_m = 0.30$), Newtonian-baryons ($\Omega_b = 0.049$),
and DFD-baryons ($\Omega_b = 0.049$, $\mu(x) = x/(1+x)$)\footnote{%
The baryon density $\Omega_b = 0.049$ is Planck's value used uniformly
across all three runs to isolate the gravity-law effect from the
matter-content effect (apples-to-apples comparison). DFD's
self-consistent value at $h = 0.7209$ is $\Omega_b = 0.0430$, preserving
the $H_0$-invariant physical density $\omega_b h^2 = 0.02237$ shared
with standard BBN. A rerun with this input would produce qualitatively
similar conclusions with a modestly smaller overshoot ratio; production-quality
confrontation at $\geq 256^3$ with the EFE implemented is the proper
venue for quantitative comparison.%
} on identical
initial conditions demonstrates the key point: Newtonian-baryons produces
negligible structure ($\delta_{\rm rms} = 1.5 \times 10^{-4}$),
confirming the standard objection; DFD produces $43.8\times$ more
structure ($\delta_{\rm rms} = 6.4 \times 10^{-3}$), establishing that
nonlinear gravity overcomes the baryonic deficit. The $5.4\times$
overshoot relative to $\Lambda$CDM is physically expected: cosmological
perturbation accelerations ($x \approx 4 \times 10^{-4}$) lie deep in
the MOND regime where the raw $\mu$-function enhances gravity by
$\sim\!400\times$ without the cosmological External Field Effect (EFE)
from the Hubble flow ($a_{\rm ext} \sim cH_0 \approx 6\,a_0$).
With the EFE, the effective enhancement drops from $\sim\!400$ to
$\sim\!1.2$, which should bring DFD into quantitative agreement.
This is a proof-of-concept at $64^3$ resolution; production-quality
results require $\geq 256^3$ with the EFE implemented.

\paragraph{Theorem-grade $\sigma_8$ closure.}
Appendix~\ref{app:sigma8-optical} supplies the rigorous
counterpart of this proof-of-concept simulation. Two results are
established there: (i) on the FRW background of the DFD action,
$|\nabla\bar\psi|=0$ and hence $a_{\rm ext}^{\rm FRW}=0$
(Theorem~\ref{thm:aext-FRW}), so the Hubble-EFE estimate
$a_{\rm ext}\sim cH_0$ used heuristically in this paragraph is not
recovered from the action for cosmological perturbations; and
(ii) the spherical top-hat closure equation
$\Omega_b\delta_b Q(\delta_b)=\Omega_m\sigma_{8,\rm obs}$ has the
solution $\delta_{b,R_8}=1.18\times 10^{-2}$, $Q_{\rm DFD}=502$
in the deep-MOND regime supplied by (i). The apparent $\sigma_8$
measured by weak lensing is therefore a refractive amplification of
a $\sim 1\%$ physical baryon contrast at $R_8$, and the simulation's
factor-$5.4$ overshoot in $\delta_{\rm rms}$ is consistent with this
optical inflation when the appropriate window is applied. Full
Limber-projected $C_\ell^{\kappa\kappa,\rm DFD}$ matching to KiDS /
DES / HSC weak-lensing tomography is identified there as the next
program-level milestone.

\paragraph{Forward perturbation skeleton.}
The dust-branch theorem provides the equation of state; what remains is the growth operator. Linearizing the DFD field equation around a background $\bar\psi$ in Fourier space gives
\begin{equation}\label{eq:perturb-fourier}
\fiteq{k_i M_{ij} k_j\,\delta\psi_{\mathbf{k}} = -\frac{8\pi G}{c^2}\bar\rho\,\delta_{\mathbf{k}},}
\end{equation}
with the response tensor
\begin{equation}\label{eq:M-tensor}
\fiteq{M_{ij} = \mu_0\,\delta_{ij} + L_0\,\hat{g}_i\hat{g}_j,}
\end{equation}
where
$\mu_0 \equiv \mu(\bar{x})$,\;
$L_0 \equiv \bigl.\frac{d\mu}{d\ln x}\bigr|_{\bar{x}}$,\;
$\hat{g} \equiv \nabla\bar\psi/|\nabla\bar\psi|$.
The linear growth equation is then
\begin{equation}\label{eq:growth-Geff}
\fiteq{\ddot\delta_{\mathbf{k}} + 2H\dot\delta_{\mathbf{k}} = 4\pi G_{\rm eff}(a,\hat{k})\,\bar\rho\,\delta_{\mathbf{k}},}
\end{equation}
with direction-dependent effective gravitational coupling
\begin{equation}\label{eq:G-eff}
\fiteq{\boxed{G_{\rm eff}(a,\hat{k}) = \frac{G}{\mu_0\big[1 + L_0(\hat{k}\cdot\hat{g})^2\big]}.}}
\end{equation}
For $\mu(x)=x/(1+x)$: $\mu_0 = \bar{x}/(1+\bar{x})$ and $L_0 = 1/(1+\bar{x})^2$. On cosmological scales ($\bar{x} \ll 1$), $G_{\rm eff} \to G/\bar{x}$, enhancing growth; on small scales ($\bar{x} \gg 1$), $G_{\rm eff} \to G$, recovering standard gravity.

\paragraph{Background-history input.}
Equations~\eqref{eq:perturb-fourier}--\eqref{eq:G-eff} describe the linear response of perturbations once a background history $H(a)$ is supplied. In the present monograph, $H(a)$ is taken from the DFD observer dictionary / reconstructed screen background already used throughout Sec.~\ref{sec:cosmology}. The novelty of the present closure is therefore not a new background model, but the fact that the same $\delta\psi$ field now drives both the forward growth law and the inverse screen reconstruction.

\paragraph{Connection to the reconstructed screen.}
The $\psi$-screen inferred from SNe and CMB closure (Secs.~\ref{sec:psitomo-estimators}--\ref{sec:psitomo-falsifier}) is the line-of-sight integral of the same perturbation field:
\begin{equation}\label{eq:screen-integral}
\fiteq{\Delta\psi_{\rm screen}(z,\hat{n}) = \int_0^{\chi(z)} W(\chi')\,\delta\psi(\chi'\hat{n})\,d\chi',}
\end{equation}
where $W(\chi)$ is the lensing kernel. This means the object inferred from the inverse optical program and the object sourced by the forward growth equation are the same field. Any inconsistency between the reconstructed $\Delta\psi_{\rm screen}$ map and the $\delta\psi$ field implied by the forward growth operator is a direct falsifier of the cosmological closure.

\paragraph{New falsifiers from the perturbation system.}
\begin{enumerate}
\item If the reconstructed $\Delta\psi_{\rm screen}$ map does not match the $\delta\psi$ field implied by Eq.~\eqref{eq:growth-Geff}, the forward--inverse closure fails.
\item If ISW suppression does not agree with the sign and amplitude implied by $G_{\rm eff}$, the growth law is wrong.
\item If $f\sigma_8(z,\hat{n})$ shows no directional dependence where the background screen gradient is nonzero, the anisotropic $G_{\rm eff}$ is excluded.
\end{enumerate}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title={Dust Branch from Microsector: Not Bolted-On K-Essence}]
The temporal sector is \textbf{derived}, not assumed:
\begin{enumerate}[nosep]
\item Same $\mu(x) = x/(1+x)$ that governs galaxy dynamics
\item Same saturation-union composition law (Assumption~\ref{ass:composition})
\item Deviation invariant $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$ forced by segment additivity
\item Dust branch ($w \to 0$, $c_s^2 \to 0$) is theorem-grade (Appendix~\ref{app:temporal-completion})
\end{enumerate}
\textbf{No-go check:} Naive quadratic $K'(Q_t) = \mu(\sqrt{Q_t})$ gives $w \to 1/2$ (not dust). The dust branch is not automatic---it requires the deviation-invariant closure.

See Appendix~\ref{app:temporal-completion} for complete derivation.
\end{tcolorbox}

% P(k) RSD confrontation with BOSS/eBOSS
\input{section_Pk_confrontation}

%-----------------------------------------------------------------------------
\subsection{Observational Status (2024--2025)}\label{sec:obs-status}

Several recent observations provide context for the $\psi$-screen framework. 
We present these as \textit{motivations}, not proofs; the laboratory 
falsifier (Sec.~\ref{sec:cavity}) carries the ultimate burden of evidence.

\subsubsection{Late-Time Potential Shallowing (DES Y3)}

The Dark Energy Survey Year 3 analysis provides a model-independent, direct 
measurement of the Weyl gravitational potential from combined weak-lensing 
and clustering data~\cite{DES2024Weyl}:

\begin{quote}
\textbf{Result:} The lowest-$z$ bins are 2--3$\sigma$ shallower than 
$\Lambda$CDM+\GR expectations, corresponding to $\sim$10\% weaker potential.
\end{quote}

In the $\psi$-screen framework, this follows naturally from cosmic dilution. 
As the universe expands and $\rho$ decreases, the source of $\psi$ 
[Eq.~\eqref{eq:field_general}] weakens:
\begin{equation}
\fiteq{\frac{\Delta\Phi}{\Phi} \sim \frac{\Delta\rho}{\rho} \quad \Rightarrow \quad 
\text{late-time shallowing as } \rho \downarrow.}
\end{equation}

\textbf{Status:} Qualitatively \textit{supportive} of DFD.

\subsubsection{Dynamical Dark Energy Hints (DESI DR2)}

DESI DR2 BAO, combined with SNe and CMB distance priors, shows dataset-dependent 
preference for dynamical dark energy $w(z) \neq -1$~\cite{DESI2025DR2}:

\begin{quote}
\textbf{Result:} Some dataset combinations favor $w(z)$ evolving with redshift 
rather than a pure cosmological constant.
\end{quote}

In the $\psi$-screen interpretation, the optical path length is:
\begin{equation}
\fiteq{D_{\rm opt} = \frac{1}{c}\int e^\psi\,ds \approx \frac{1}{c}\int(1 + \psi)\,ds,}
\end{equation}
so the inferred distance-redshift relation acquires a fractional bias
$\Delta D/D \simeq \langle\psi\rangle_{\rm LOS}$. Percent-level $\psi$ biases
can mimic mild dynamical-$w$ preferences without invoking a dark-energy fluid.

\textbf{Status:} \DFD's \emph{derived} equation of state is $w_{\rm eff}=-1$
exactly (App.~\ref{app:dark-energy}, Prop.~\ref{thm:au-weff}), with a
\emph{saturating} screen $\Delta\psi(\infty)=0.50$ and no dark-energy clustering.
A percent-level $\psi$-bias could only \emph{qualitatively} mimic a mild
dynamical-$w$ preference; we do \textbf{not} derive the specific DESI~DR2 CPL shape
$(w_0,w_a)$. This is therefore an \emph{open test}, not a confirmation: a genuine
$w(z)\neq-1$ fluid (with clustering/sound speed) would \emph{falsify} \DFD,
distinguishable via ISW and growth cross-correlations (Sec.~\ref{sec:isw-prediction}).

\subsubsection{Wide Binaries (Active and Contested)}

Gaia wide-binary tests probe internal accelerations down to 
$a \sim 10^{-10}$ m/s$^2$~\cite{WideBinaries2025}:

\begin{quote}
\textbf{Some analyses:} Report $\sim$20\% velocity excess beyond $\sim$3000 au, 
consistent with MOND-like phenomenology.\\
\textbf{Other analyses:} Demonstrate that realistic triple-population 
modeling and stricter data cuts remove the signal.
\end{quote}

The $\mu$-crossover radius in \DFD is:
\begin{equation}
\fiteq{\begin{aligned}
r_\times = \sqrt{\frac{GM}{a_\star}} \approx 7.1 \times 10^3\,\text{au} \\
\times \left(\frac{M}{M_\odot}\right)^{1/2}
\left(\frac{1.2 \times 10^{-10}\,\text{m/s}^2}{a_\star}\right)^{1/2},
\end{aligned}}
\end{equation}
matching the (3--7)$\times 10^3$ au range where Gaia analyses disagree.

\textbf{Status:} \textit{Active and contested}---not yet definitive either way.

\subsubsection{Counter-Evidence and Null Tests}

Any alternative framework must address null tests:

\paragraph{$E_G$ gravity test (ACT DR6 + BOSS).}
The geometry-vs-dynamics ratio $E_G$ from ACT DR6 CMB-lensing crossed with 
BOSS galaxies is consistent with $\Lambda$CDM/\GR and largely scale-independent 
within current precision~\cite{ACT2025EG}.

\textbf{Status:} Mild \textit{tension} with DFD expectations (would expect 
small deviations at low $z$).

\paragraph{KiDS-Legacy shear.}
The KiDS-Legacy cosmic-shear analysis yields $S_8$ consistent with Planck 
$\Lambda$CDM~\cite{KiDS2025Legacy}.

\textbf{Status:} Mild \textit{tension} (earlier KiDS analyses showed larger 
discrepancy).

\paragraph{Pairwise kinematic Sunyaev--Zel'dovich (ACT + SDSS/DESI).}
The pairwise kSZ momentum between cluster pairs probes the large-scale 
velocity field $v_{12}(r)$ sourced by linear-regime gravity. Recent measurements 
by Calafut et al.~\cite{Calafut2021} and Gallardo et al.~\cite{Gallardo2026kSZ} 
find the pairwise infall signal consistent with Newton/\GR\ plus $\Lambda$CDM 
on scales $20\,\text{Mpc} \lesssim r \lesssim 150\,\text{Mpc}$, and have 
been cited as excluding modified-gravity alternatives including MOND.

DFD reproduces the $\Lambda$CDM pairwise velocity prediction at this precision. 
The only framework-level difference is the growth rate, which in DFD is 
$f_{\rm DFD}(z) = \Omega_m(z)^\gamma[1+\mathcal{O}(k_\alpha)]$ with 
$k_\alpha = \alpha^2/(2\pi) \approx 8.48\times 10^{-6}$ 
(Sec.~\ref{sec:Pk_confrontation}), so 
$v_{12}^{\rm DFD}(r)/v_{12}^{\Lambda\text{CDM}}(r) = 1 + k_\alpha$ bin by bin. 
For the Calafut et al.\ L61 sample configuration (15 bins, $5$--$145$~Mpc, 
$z_{\rm eff}=0.5$, $\langle M\rangle \approx 2.5\times 10^{13}\,M_\odot$, 
Tinker 2010 bias $b\approx 1.76$), evaluating 
$v_{12}(r) = -(2/3)\,a(z)H(z)f(z)\,r\,\bar\xi^h(r)/[1+\xi^h(r)]$ 
via the Sheth--Diaferio--Hui--Scoccimarro linear formalism gives
$\chi^2/\text{dof} \approx 0.79$ for DFD, with 
$\Delta\chi^2(\text{DFD}-\Lambda\text{CDM}) \approx 2.5\times 10^{-4}$: 
DFD and $\Lambda$CDM are indistinguishable at current precision and both 
fit the data well. The test rules out pure MOND, which fails at cluster 
scales independently of kSZ; it does not constrain DFD, which is Newtonian 
in the high-acceleration regime probed here.

\textbf{Status:} Consistent ($\chi^2/\text{dof} = 0.79$). Sharp distinction
from $\Lambda$CDM requires per-bin precision $\lesssim 10^{-5}$, far below
current and near-term capability.

\paragraph{Growth-regime note (June 2026).}
The pairwise-kSZ consistency above rests on the quasi-Newtonian growth premise
for $20$--$150$~Mpc scales, i.e.\ on the Hubble-EFE screening envelope of
Appendix~\ref{app:linear-growth-closure} ($G_{\rm eff}=1.02$--$1.17\,G$), under
which growth deviates from $\Lambda$CDM by at most $\sim$8--9\% --- well inside
current pairwise-kSZ precision, so the consistency verdict holds in that
regime. Under the zero-external-field cosmological regime of
Appendix~\ref{app:sigma8-optical} (which proves $|\nabla\bar\psi|=0$ on the
FRW background), the corresponding prediction is instead a suppressed kSZ
amplitude (factor $\sim$2--3 relative to $\Lambda$CDM), retained as a named
falsifier in Appendix~\ref{app:galaxy-bias}. Adjudication between the two
regimes for linear cosmological velocities is an open program item
(Sec.~\ref{sec:openproblems}).

\subsubsection{Observational Summary Table}

\begin{table}[h]
\caption{Observational benchmarks (2024--25 status).}
\label{tab:obs-benchmarks}
\begin{ruledtabular}
\scriptsize
\setlength{\tabcolsep}{2pt}
\fittab{%
\begin{tabular}{llcc}
Scale/Probe & DFD Prediction & Obs. & Status \\
\hline
Solar System & $\gamma = \beta = 1$ & Consistent & \checkmark \\
$H_0$ (local) & $\alpha^{57/2}/t_P=72.1$ & SH0ES $73.0(1.0)$ & \checkmark \\
DES (low-$z$) & Shallowing $\sim$10\% & 2--3$\sigma$ low & \checkmark \\
DESI DR2 & $w_{\rm eff}=-1$, no DE clust. & $w \neq -1$ hints & ? \\
Gal.\ rotation & Flat $v$; TF scaling & Empirical & \checkmark \\
Wide binaries & Crossover at $a_\star$ & Contested & ? \\
$E_G$ (ACT) & Small deviations & \GR-consistent & $\sim$ \\
KiDS-Legacy & Small tension & Planck-consist. & $\sim$ \\
Pairwise kSZ & $v_{12} = \Lambda$CDM+$k_\alpha$ & $\chi^2$/dof $=0.79$ & \checkmark \\
Lab (100 m) & $\kappa = 1$ & Not tested & --- \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\paragraph{Bottom line.}
Late-time cosmological anomalies are uneven across probes and evolving with
improved analyses. The parameter-free prediction $H_0=\alpha^{57/2}/t_P=72.1$~km/s/Mpc
(App.~\ref{app:dark-energy}) sits $0.9\sigma$ from the local SH0ES value and
$\sim\!9\sigma$ from Planck, placing \DFD\ on the local-distance-ladder side of the
Hubble tension as a \emph{forced}, not fitted, prediction.
The DES low-$z$ shallowing is consistent with the $\psi$-screen;
the DESI~DR2 evolving-$w$ hint, by contrast, is \emph{not} a \DFD\ prediction---\DFD\
derives $w_{\rm eff}=-1$ with no dark-energy clustering (App.~\ref{app:dark-energy},
Prop.~\ref{thm:au-weff}), so a confirmed evolving-$w$ \emph{fluid} would falsify it,
distinguishable via ISW and growth cross-correlations
(Sec.~\ref{sec:isw-prediction}). $E_G$ and KiDS show mild tension; the pairwise kSZ
test is passed trivially because DFD matches $\Lambda$CDM linear growth to $10^{-5}$.
The decisive test remains the laboratory cavity-atom comparison 
(Sec.~\ref{sec:cavity}).

%-----------------------------------------------------------------------------
\subsection{Hierarchy of Astrophysical Scales from $\alpha$}\label{sec:cosmo-hierarchy}
%-----------------------------------------------------------------------------

A striking feature of the DFD framework is that powers of $\alpha$ applied to 
the Hubble radius $R_H = c/H_0$ generate the characteristic scales of cosmic structure.

\begin{table}[h]
\centering
\caption{Length scales generated by powers of $\alpha$ from the Hubble radius.}\label{tab:length-hierarchy}
\fittab{%
\begin{tabular}{@{}lll@{}}
\toprule
Expression & Value & Physical scale \\
\midrule
$R_H$ & $1.4 \times 10^{26}$ m & Hubble radius \\
$\sqrt{\alpha} \cdot R_H$ & $1.2 \times 10^{25}$ m & $\sim 1$ Mpc (galaxy groups) \\
$\alpha \cdot R_H$ & $10^{24}$ m & $\sim 100$ kpc (galactic halos) \\
$\alpha^{3/2} \cdot R_H$ & $10^{23}$ m & $\sim 6$ kpc (galactic disks) \\
$\alpha^2 \cdot R_H$ & $7 \times 10^{21}$ m & $\sim 700$ ly (globular clusters) \\
\bottomrule
\end{tabular}%
}
\end{table}

The hierarchy of cosmic structure---from groups to halos to disks---emerges 
naturally from powers of the fine-structure constant.

\begin{table}[h]
\centering
\caption{Acceleration scales from powers of $\alpha$ applied to $cH_0$.}\label{tab:accel-hierarchy}
\fittab{%
\begin{tabular}{@{}lll@{}}
\toprule
Expression & Value (m/s$^2$) & Interpretation \\
\midrule
$cH_0$ & $7 \times 10^{-10}$ & Vacuum scale $a_\star$ \\
$2\sqrt{\alpha} \cdot cH_0$ & $1.2 \times 10^{-10}$ & MOND scale $a_0$ \\
$\alpha \cdot cH_0$ & $5 \times 10^{-12}$ & Deep MOND regime \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{Quantum-gravitational crossover.}
Combining $\hbar$, $m_e$, $\alpha$, and $a_0$:
\begin{equation}
\fiteq{r_\psi \equiv \frac{\hbar c}{m_e \cdot a_0} \approx 2.9 \times 10^{14}\,\text{m} \approx 2000\,\text{AU}.}
\end{equation}
This is the Oort cloud scale---where quantum matter-wave effects and modified 
gravity become comparable for electron-mass particles.

%-----------------------------------------------------------------------------
\subsection{Summary}\label{sec:cosmo-summary}

Cosmology in DFD is framed as reconstructing $\Delta\psi_{\rm screen}(z,\hat n)$ from independent data channels (SNe and CMB acoustic-scale anisotropy), with distance duality ($\eta=1$) serving as a metric-consistency check, and testing the \emph{single-screen} hypothesis with a GR-independent falsifier: cross-correlation with independent structure maps.

\textbf{Quantitative reconstruction results (this work):}
\begin{itemize}
\item $\Delta\psi(z=1.0) = 0.274 \pm 0.02$ from $H_0$-independent distance ratios
\item This matches the $\Delta\psi \approx 0.30$ needed for CMB peak location
\item Objects at $z=1$ appear 32\% farther than matter-only predicts
\item The ``accelerating expansion'' is reinterpreted as an optical effect
\end{itemize}

This is the shortest path to decisive tests that do not require adopting GR/$\Lambda$CDM priors. The falsification criterion remains: cross-correlation of reconstructed $\Delta\psi(\hat n)$ with foreground structure maps (Sec.~\ref{sec:psitomo-falsifier}).
