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% APPENDIX AG: MASTER CLAIMS CONFRONTATION MATRIX
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\section{Master Claims Confrontation Matrix}\label{app:confrontation-matrix}

\providecommand{\TheoremGrade}{\textsc{T}}
\providecommand{\DerivedGrade}{\textsc{D}}
\providecommand{\ConjecturedGrade}{\textsc{C}}
\providecommand{\EmpiricalGrade}{\textsc{E}}
\providecommand{\ValidatedGrade}{\textit{validated}}
\providecommand{\PendingGrade}{\textit{pending}}
\providecommand{\OpenGrade}{\textit{open}}

This appendix consolidates every quantitative empirical claim in this monograph into a single navigational catalog. The intent is to let a referee or independent reader assess at a glance which claims are \emph{theorem-grade}, which are \emph{derived under stated auxiliary assumptions}, and which are \emph{conjectured or empirical-fit-grade}; what each claim predicts; what is observed; the current agreement; and the explicit threshold that would falsify the claim.

This appendix introduces no new physics. Every entry cross-references a theorem, derivation, table, or section already present in the monograph.

\subsection{Tier Definitions}\label{app:matrix-tiers}

\begin{description}[itemsep=2pt, leftmargin=*, labelsep=4pt]
\item[\textbf{\TheoremGrade{} (Theorem):}] Result follows from a formal proof using only the core DFD axioms ($n=e^\psi$, $a=(c^2/2)\nabla\psi$, $\mu=x/(1+x)$) plus stated mathematical lemmas.

\item[\textbf{\DerivedGrade{} (Derived):}] Result follows from the core axioms plus an explicit auxiliary closure framework whose assumptions are displayed (spectral action, Chern-Simons truncation $k_{\max}=60$, manifold $X=\mathbb{C}P^2\times S^3$, generation count $N_{\rm gen}=3$, PDE-validated numerical results).

\item[\textbf{\ConjecturedGrade{} (Conjectured):}] Result is plausible from DFD principles but not yet derived from the core axioms or a fully-displayed closure framework. Open program items.

\item[\textbf{\EmpiricalGrade{} (Empirical):}] Result is a phenomenological match using physically-motivated correction factors at literature-bounded values, where the closure depends on those corrections.

\item[\textbf{\ValidatedGrade{} (Validated):}] An observational test that has \emph{already} been performed and is consistent with DFD at stated $\sigma$.

\item[\textbf{\PendingGrade{} (Pending):}] An observational test where the experimental capability exists but the measurement is not yet published.

\item[\textbf{\OpenGrade{} (Open):}] A predicted observable for which no comparable measurement currently exists.
\end{description}

The cluster entry deliberately uses the \EmpiricalGrade{} label because the 14/16 closure depends on the five-factor correction stack (Appendix~\ref{app:cluster-full}), even though the individual factors are independently published or PDE-derived. This reflects v4.0's tier-grading discipline: a claim's tier is set by its \emph{weakest} link, not its strongest.

\subsection{Block 1: Fundamental Constants and Microsector}\label{app:matrix-block1}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{Fine-structure constant $\alpha^{-1}$ from microsector closure} (Sec.~\ref{sec:alpha-locked}). DFD: $\alpha^{-1}=137.03599985$ ($0.0056$~ppm). SM: free parameter. Observation: $137.035999084(21)$. Agreement: $-0.006$ ppm. Falsifier: $|\Delta\alpha^{-1}|>0.1$ ppm not from QED loops. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Higgs VEV from $\alpha^8\sqrt{2\pi}$ scaling} (App.~\ref{app:complete-params}). DFD: $v=246.09$~GeV. SM: free parameter. Observation: $246.22$~GeV. Agreement: $0.05\%$. Falsifier: $|\Delta v/v|>1\%$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Strong CP: $\bar\theta=0$ to all loops} (App.~\ref{app:strong-cp}). DFD: $\bar\theta=0$ identically. SM: needs axion or fine-tuning. Observation: $|\bar\theta|<10^{-10}$ (nEDM). Agreement: exact. Falsifier: detection of $|\bar\theta|>10^{-19}$ from QCD alone. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Fermion mass hierarchy (9 masses)} (Sec.~\ref{sec:G-derivation}). DFD: $m_f=A_f\alpha^{n_f}v/\sqrt{2}$ with integer $n_f$, $1.42\%$ leading-order mean error. SM: 9 free parameters. Observation: PDG values. Agreement: $1.42\%$ mean (leading order). Falsifier: individual mass off by $>10\%$ for any $f$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Weinberg angle from gauge partition} (App.~\ref{app:weinberg-angle}). DFD: $\sin^2\theta_W=3/13=0.2308$. SM: free parameter. Observation: $0.23121(4)$ at $M_Z$. Agreement: $0.16\%$. Falsifier: $|\Delta\sin^2\theta_W/\sin^2\theta_W|>1\%$ at tree level. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{CKM Cabibbo angle from $\mathbb{C}P^2$ cohomology} (App.~\ref{app:ckm-update}). DFD: $\lambda=31\alpha=0.2262$. SM: free parameter. Observation: $0.22501(68)$ (PDG~2024). Agreement: $0.54\%$ ($1.8\sigma$). Falsifier: $|\Delta\lambda|>0.005$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Gatto--Sartori--Tonin closure $|V_{us}|=\sqrt{m_d/m_s}$} (App.~\ref{app:crosssector}, Thm~\ref{thm:ay-gatto}). DFD: $|V_{us}|/\sqrt{m_d/m_s}=\sqrt{31^2\alpha/7}=1.0009$ (i.e.\ $m_s/m_d=1/(7\alpha)=19.58$). SM: unexplained. Observation: relation holds to $0.66\%$; $m_s/m_d=19.9(\pm1)$ (FLAG). Agreement: $0.09\%$ closure, $<0.5\sigma$ on $m_s/m_d$. Falsifier: $m_s/m_d$ outside $[17,23]$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{TM1 solar mixing angle} (App.~\ref{app:crosssector}, Thm~\ref{thm:ay-tm1}). DFD: $\cos^2\theta_{12}\cos^2\theta_{13}=\tfrac23$ exact, $\sin^2\theta_{12}=(1-9\alpha)/(3(1-3\alpha))=0.3184$. SM: free parameters. Observation: $\sin^2\theta_{12}=0.307(^{+12}_{-11})$ (NuFIT~6.0). Agreement: $+0.95\sigma$. Falsifier: $\sin^2\theta_{12}\notin[0.30,0.34]$ or non-maximal $\theta_{23}$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{PMNS matrix structure} (Sec.~\ref{sec:G-derivation}). DFD: Tribimaximal $+$ corrections. SM: free parameters. Observation: global fit values. Agreement: $\sim 5\%$. Falsifier: off-diagonal element $>20\%$ off. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Hubble--Planck closure: $G\hbar H_0^2/c^5=\alpha^{57}$} (App.~\ref{app:O_alpha57_clean}). DFD: $H_0/M_P=\alpha^{28.5}$ ($H_0=72.09$ km/s/Mpc). $\Lambda$CDM: free quantity, no theoretical prediction. Observation: SH0ES $73.0\pm 1.0$, JWST $\sim 72.6\pm 2.0$. Agreement: $0.3\sigma$ (local), $\sim 3\sigma$ from CMB. Falsifier: $H_0$ converging on $\Lambda$CDM-CMB value $67.4$ at $>5\sigma$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Cosmological constant scale: $(H_0/M_P)^2=\alpha^{57}$}. DFD: $\rho_\Lambda/M_P^4\sim 10^{-122.7}$ as $\alpha^{57}$ topological identity. $\Lambda$CDM: ``$10^{122}$ problem''. Observation: matches observed $\Omega_\Lambda H_0^2$. Agreement: exponent-level identity; the $O(1)$ prefactor enters via the clock dictionary (Rem.~\ref{rem:clock-credit}) and the identity is microsector-scoped (Rem.~\ref{rem:au-scope}). Falsifier: topology-derived exponent fails to match $H_0^2/M_P^2$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Clock coupling: $k_\alpha=\alpha^2/(2\pi)$} (App.~\ref{app:P_kalpha_MR}). DFD: $k_\alpha=8.5\times 10^{-6}$ (unscreened); $\lambda_\alpha\approx 2.1\times 10^{-8}$ (screened). SM: $0$ (LPI exact). Observation: Yb$^+$ E3/E2 bound $|k_\alpha|<3.2\times 10^{-8}$. Agreement: consistent at $66\%$ of bound. Falsifier: detection of LPI null at $<10^{-8}$ in pure-$\alpha$ channel. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{Majorana scale: $M_R=M_P\alpha^3$} (App.~\ref{app:P_kalpha_MR}). DFD: $M_R\approx 4.7\times 10^{12}$~GeV. SM: free seesaw parameter. Observation: inferred from $m_\nu$. Agreement: matches mass scale. Falsifier: $m_\nu$ pattern inconsistent with $M_R\alpha^3$ structure. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\item \textbf{Proton stability: $\tau_p=\infty$ topologically} (App.~\ref{app:proton-stability}). DFD: exact (no decay channel), \emph{conditional on the baryon-number-equals-winding-integer identification (axiom V7)} --- theorem-grade for the winding-integer tautology itself; the existence of the required Derrick-stable soliton is posited, not proven. SM: $\tau_p>10^{34}$~yr observed. Observation: Super-K bounds. Agreement: exact. Falsifier: $p\to e^+\pi^0$ detection at any rate (a clean kill-test vs GUTs, conditional on V7). Tier: \DerivedGrade{} (conditional). Status: \PendingGrade{}.

\end{itemize}

\subsection{Block 2: Gravitational Tests (PPN, Pulsars, GW, Strong-Field)}\label{app:matrix-block2}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{PPN $\gamma$} (Sec.~\ref{sec:ppn}). DFD: $\gamma=1$ exactly. GR: $\gamma=1$. Observation: Cassini $1.000021(23)$. Agreement: exact. Falsifier: $|\gamma-1|>10^{-5}$. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{PPN $\beta$} (Sec.~\ref{sec:ppn}, Pad\'e Thm~\ref{thm:pade-identity}). DFD: $\beta=1$ exactly. GR: $\beta=1$. Observation: LLR $1.0000(11)$. Agreement: exact. Falsifier: $|\beta-1|>10^{-3}$. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{All 10 PPN parameters}. DFD: match GR identically. GR: baseline. Observation: all bounds satisfied. Agreement: exact. Falsifier: any PPN parameter differs from GR at solar-system precision. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{GR as Pad\'e approximant of DFD} (App.~\ref{app:pade-identity}, Thm~\ref{thm:pade-identity}). DFD: $\mathcal{L}_{\rm GR}=[P_{1,1}(u)]^2$ in isotropic coords, with $P_{1,1}(u)$ the $[1,1]$ Pad\'e approximant of $e^u$. GR: structural identity. Observation: ---. Agreement: exact identity. Falsifier: failure of identity at next Pad\'e order. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Mercury perihelion advance}. DFD: $42.98''$/century. GR: $42.98''$/century. Observation: $42.98(4)''$/century. Agreement: exact. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Shapiro time delay}. DFD: matches GR. GR: matches GR. Observation: Cassini consistent. Agreement: exact. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Light deflection}. DFD: reproduces the leading (1PN) GR deflection in the declared physical branch. GR: matches GR. Observation: VLBI consistent. Agreement: exact at 1PN (the 2PN coefficient is branch-dependent, App.~\ref{app:alpha57-closure}, Def.~\ref{ap:def:lensing-branches}, Thm~\ref{ap:thm:lensing-statement}). Tier: \TheoremGrade{} (1PN). Status: \ValidatedGrade{}.

\item \textbf{Frame dragging (Lense-Thirring)}. DFD: matches GR. GR: matches GR. Observation: Gravity Probe B consistent. Agreement: exact. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{GW propagation speed: $c_T=c$ exactly} (Sec.~\ref{sec:gw}). DFD: $c_T=c$. GR: $c_T=c$. Observation: GW170817: $|c_T/c-1|<10^{-15}$. Agreement: exact. Falsifier: $|c_T/c-1|>0$ at any precision. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{GW170817 tidal deformability and EM counterpart timing}. DFD: matches GR. GR: matches GR. Observation: consistent with GR + DFD. Agreement: exact. Falsifier: inconsistency with GR-equivalent at LIGO sensitivity. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{ppE GW phase deviations}. DFD: all deviations zero at leading order. GR: all zero. Observation: all bounds satisfied. Agreement: exact. Falsifier: nonzero $\delta\hat\varphi_b$ in any GW band. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Binary pulsar orbital decay} (App.~\ref{app:pulsar_data}). DFD: matches GR (B1913+16, J0737-3039). GR: matches GR. Observation: all consistent. Agreement: exact. Falsifier: decay rate off by $>0.1\%$ for any pulsar. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Black hole shadow} (Sec.~\ref{sec:strong}). DFD: $4.6\%$ larger than Schwarzschild. GR: Schwarzschild $b=3\sqrt{3}\,GM/c^2$. Observation: M87* $42\pm 3\,\mu$as. Agreement: $0.6\sigma$ from GR; consistent with DFD. Falsifier: next-gen VLBI $<1\,\mu$as precision rules out $4.6\%$ deviation. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\item \textbf{No-wormhole theorem} (App.~\ref{app:em-psi-coupling}, Thm~\ref{thm:no-wormhole}). DFD: no traversable wormholes. GR: allows with exotic matter. Observation: ---. Agreement: exact theorem. Falsifier: exhibition of DFD wormhole solution. Tier: \TheoremGrade{}. Status: \OpenGrade{}.

\end{itemize}

\subsection{Block 3: Galactic Dynamics}\label{app:matrix-block3}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{$\mu(x)=x/(1+x)$ from $S^3$ composition} (App.~\ref{app:mu-derivation}, Thm~\ref{thm:mu-theorem}). DFD: unique $\mu$ at all scales. $\Lambda$CDM: dark matter halos. Observation: SPARC RAR. Agreement: $<5\%$ residuals. Falsifier: SPARC residual scatter exceeds $0.15$ dex floor. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{$a_*=2\sqrt{\alpha}\,cH_0$ from scaling stationarity} (App.~\ref{app:mu-derivation}, Thm~\ref{thm:a-star-final}). DFD: $1.20\times 10^{-10}$~m/s$^2$. MOND: free parameter. Observation: McGaugh+ 2016 fit $\sim 1.2\times 10^{-10}$. Agreement: exact. Falsifier: SPARC global $a_0$ inconsistent with $2\sqrt{\alpha}\,cH_0$ at $>3\%$. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{SPARC RAR head-to-head} (Sec.~\ref{sec:galactic}). DFD: beats Newton in $100\%$ of 175 galaxies; $n_{\rm opt}=1.15\pm 0.12$ centered on $n=1$. $\Lambda$CDM: DM halo per galaxy. Observation: McGaugh+ 2016. Agreement: $n=1$ confirmed; standard MOND $n=2$ disfavored. Falsifier: best-fit $n$ exits $[1.0,1.5]$ at $95\%$ CL on full sample. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Baryonic Tully-Fisher: $M_{\rm bar}\propto v_f^4$} (Sec.~\ref{sec:galactic}). DFD: exact slope and normalization from $a_*$. $\Lambda$CDM: DM halo prediction. Observation: McGaugh+ slope $3.94\pm 0.08$. Agreement: matches. Falsifier: BTFR slope outside $[3.7, 4.3]$ at high significance. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Wide binary $42\%$ velocity boost at $10^4$~AU} (Sec.~\ref{sec:galactic}). DFD: $\Delta v/v=42\%$ at $\sim 10^4$~AU. Newton: no boost. Observation: Chae 2023 \emph{Gaia}. Agreement: matches at $\sim 5\sigma$. Falsifier: precise WB studies show null at $<5\%$ at $10^4$~AU. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Classical dwarf spheroidals}. DFD: two-regime (isolated/EFE) Jeans model fits all 9 classical dSphs with $M/L\in[5,30]$. $\Lambda$CDM: DM halo per dSph. Observation: observed $M/L$ ratios. Agreement: consistent within systematics. Falsifier: inconsistent dynamical $M/L$ in any single isolated classical dSph at $>5\sigma$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Ultra-faint dwarf $M/L$ extreme values}. DFD: systematic effects (binary contamination, tidal heating) inflate inferred $M/L$ by factors of $10$--$100$. $\Lambda$CDM: dark matter. Observation: UF dSph $M/L\sim 100$--$1000$. Agreement: consistent with systematics correction. Falsifier: UF dSph kinematics confirmed pristine yet retain anomalous $M/L$. Tier: \EmpiricalGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Galactic-scale extension $a_*(z)=2\sqrt{\alpha}\,cH(z)$} (companion paper). DFD: redshift-evolving $a_*$. MOND: static $a_0$. Observation: not yet measured. Falsifier: high-$z$ rotation curves show $a_*$ inconsistent with $2\sqrt{\alpha}\,cH(z)$ at $>3\sigma$. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\end{itemize}

\subsection{Block 4: Cluster Scale}\label{app:matrix-block4}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{Universal $\mu$ at all scales} (App.~\ref{app:cluster-full}). DFD: $n=1$ at cluster scale, no scale-dependent $\mu$. $\Lambda$CDM: DM halo. Observation: cluster sample under five-factor budget. Agreement: consistent (14/16 within $\pm 10\%$; Bullet and El Gordo remain $13$--$19\%$ low, at $\le1.05\sigma$ of their published lensing-mass errors). Falsifier: cluster RAR with strong-lensing-resolved subhalos shows $n\neq 1$ at $>3\sigma$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{PDE-validated Jensen factor $J_{\rm PDE}=1+0.39 f_{\rm sub}$ from 3D AQUAL solver}. DFD: $J\simeq 1.07$--$1.12$ for $f_{\rm sub}=0.15$--$0.30$, robust across $M_{200}\in[3,12]\times 10^{14}\,M_\odot$. $\Lambda$CDM: not applicable. Observation: validated against algebraic-MOND for spherical NFW within $1\%$. Agreement: theorem-grade (PDE). Falsifier: high-resolution multigrid AQUAL solver gives $J$ outside $[1.05, 1.15]$ at $f_{\rm sub}=0.20$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Cluster five-factor decomposition: $C_i=B\,J_{\rm PDE}\,T\,M\,P$} (App.~\ref{app:cluster-full}). DFD: 14/16 within $\pm 10\%$ under uniformly applied literature-bounded factors. $\Lambda$CDM: dark matter halos. Observation: 16-cluster sample. Agreement: relaxed ($n{=}10$) $1.01\pm 0.05$; merging ($n{=}6$) $0.95\pm 0.08$ --- Bullet and El Gordo remain $13$--$19\%$ low, at $\le1.05\sigma$ of their published lensing-mass errors. Falsifier: cluster sample with all 5 factors at literature-mean values fails $>50\%$ within $\pm 15\%$. Tier: \EmpiricalGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Bullet Cluster lensing offset} (Sec.~\ref{sec:galactic}). DFD: $129$~kpc main cluster offset. $\Lambda$CDM: DM halo offset. Observation: observed $155$~kpc. Agreement: $83\%$ match. Falsifier: lensing peak coincides with gas at any reasonable resolution. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Galaxy groups: EFE-suppressed enhancement}. DFD: Obs/DFD $<1$ for groups embedded in dense environments. $\Lambda$CDM: DM halos predict $>1$ uniformly. Observation: Virgo, Fornax, NGC5044 all $<1$. Agreement: matches sign and magnitude. Falsifier: isolated groups show $<1$ ratio without environmental embedding. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{CMB peak ratio: $R=2.34$ from baryon loading alone} (App.~\ref{app:psi-cmb}). DFD: $R=2.34$. $\Lambda$CDM: $R\sim 2.4$ with $\Omega_{\rm CDM}\sim 0.27$. Observation: $R=2.34\pm 0.02$ (Planck 2018). Agreement: exact. Falsifier: precise peak ratio measurement requires DM at $>3\sigma$ to fit. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{CMB third-peak height from derived $\chi$-matter} (App.~\ref{app:chi-matter}). DFD: $H_3/H_2\approx1.0$ from the cold $\chi$ field, $\Omega_\chi h^2\simeq0.12$ (derived decay constant $f_\chi=\bar M_{\rm P}\alpha^3$ and mass $m_\chi\simeq5$~eV; abundance \emph{also derived} --- post-inflation (App.~\ref{app:chi-matter} Step~5b) the misalignment angle is removed and the amplitude is the finite $SU(2)_{60}$ CS-vacuum Casimir expectation, $\Omega_\chi h^2=0.118$ ($-1.5\sigma$), theorem-grade (lone non-DFD input: the standard cosmological relic-redshift); the former $\sim$9--44$\times$ overshoot was a classical-continuum-measure artifact). $\Lambda$CDM: same height from a \emph{postulated} $\Omega_{\rm CDM}\sim0.27$ particle. Observation: $H_3/H_2\approx0.97$ (Planck 2018). Agreement: matches. Falsifier (decisive): the $\Delta\psi\times\kappa$ cross-spectrum carries the third-peak height (which would make it line-of-sight optical, not $\chi$-clustering), or a $244$~nm dielectric-haloscope null persists once next-generation sensitivity reaches $g_{\chi\gamma}\sim10^{-15}$, or a halo-recoil DM detection appears in any current-reach channel. Tier: \DerivedGrade{}. Status: \PendingGrade{} (derived prediction; decisive $244$~nm / cross-spectrum test outstanding).

\item \textbf{RAR intrinsic scatter $\approx0.037$~dex} (App.~\ref{app:chi-matter}, Remark~\ref{rem:rar-scatter-prediction}). DFD: $\sigma_{\rm RAR}\approx0.037$~dex (range $0.03$--$0.05$), because the derived constant $a_0=2\sqrt\alpha\,cH_0$ forces the $a_0$-drift channel to zero. $\Lambda$CDM: $\sim0.06$--$0.08$~dex (the $a_0$-universality is a fine-tuning puzzle requiring per-galaxy feedback tuning). Observation: \emph{contested} --- Lelli--Li $\sim0.057$~dex (tight) vs Stone+2019 $\sim0.11$~dex. Agreement: DFD predicts the Lelli tight end is the true intrinsic floor; beats untuned $\Lambda$CDM structurally, ties best-tuned $\Lambda$CDM on raw value. Falsifier: true intrinsic RAR scatter $>\sim0.06$~dex (Stone's $0.11$ confirmed at the intrinsic level) falsifies DFD. Tier: \DerivedGrade{}. Status: \PendingGrade{} (medium confidence; harden via full SPARC-matched $\mu$-AQUAL synthesis).

\end{itemize}

\subsection{Block 5: Cosmology}\label{app:matrix-block5}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{$H_0$ from local SNe + $\psi$-screen}. DFD: $H_0=72.09$~km/s/Mpc. $\Lambda$CDM: ``Hubble tension'' ($67.4$ Planck vs $73.0$ SH0ES). Observation: SH0ES $73.0\pm 1.0$, JWST $72.6\pm 2.0$. Agreement: $0.3\sigma$ from local. Falsifier: $H_0$ converging on $67.4$ at $>5\sigma$ from any consistent local probe. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{CMB acoustic scale: $\ell_1=220$ from $\psi$-lensing} (App.~\ref{app:psi-cmb}). DFD: $\ell_1=220$ with $\Delta\psi(z=1)\approx 0.30$. $\Lambda$CDM: $\ell_1=220$ with DM. Observation: Planck $\ell_1=220.0\pm 0.5$. Agreement: exact. Falsifier: $\ell_1$ shifted from $220$ at higher precision. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{$\sigma_8$ closure under optical-metric reweighting} (App.~\ref{app:sigma8-optical}). DFD: $\sigma_8=0.811$ at $z=0$ (apparent amplitude, optical closure). $\Lambda$CDM: $0.81$ (Planck) vs $0.78$ (KiDS). Observation: matches Planck within errors. Agreement: consistent. Falsifier: RSD measurement gives $\sigma_8<0.78$ at $>3\sigma$ from any independent probe. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Modified distance duality: $D_L/(1+z)^2 D_A=1$ (DDR violation $=0$)}. DFD: exact identity. $\Lambda$CDM: exact identity (in standard FLRW). Observation: not yet directly tested. Falsifier: detection of DDR violation $>0.01$ at any $z$. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{$\Delta\psi$-screen reconstruction from SNe Ia}. DFD: $\Delta\psi(z=1)=0.27\pm 0.02$. $\Lambda$CDM: dark energy $\Lambda$. Observation: SNe distance excess $\sim 32\%$ at $z=1$. Agreement: matches. Falsifier: SNe distance excess inconsistent with $\Delta\psi$ at $>3\sigma$ across redshift bins. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{DESI BAO: evolving dark energy hint}. DFD: qualitative match to $\Delta\psi$ evolution. $\Lambda$CDM: static $\Lambda$ inconsistent with DESI DR2. Observation: DESI DR2 evolving $w$. Agreement: qualitative match. Falsifier: DESI DR3 confirms static $w=-1$ at $>5\sigma$. Tier: \ConjecturedGrade{}. Status: \PendingGrade{}.

\item \textbf{CMB-foreground correlation: $\Delta\psi(\hat n)$ correlates with structure}. DFD: nonzero correlation at the model's $\sigma_\psi$ level. $\Lambda$CDM: no correlation predicted. Observation: not yet measured. Falsifier: no correlation found at predicted $\sigma_\psi$ across multiple frequencies. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\item \textbf{UVCS double-transit $\Gamma$ ratio} (Sec.~\ref{sec:solar-corona}, App.~\ref{app:double-transit}). DFD: $\Gamma=4$. Standard: $\Gamma=1$ (single-transit). Observation: SOHO/UVCS $\Gamma=4.4\pm 0.9$. Agreement: $0.4\sigma$ from DFD; $>3\sigma$ from standard. Falsifier: follow-up observation gives $\Gamma\le 1.5$ at high significance. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\item \textbf{ESPRESSO $\Delta\alpha/\alpha$ at $z\sim 1$}. DFD: $+2.3\times 10^{-6}$. SM: $0$ (no evolution). Observation: $+1.3\pm 1.3\times 10^{-6}$. Agreement: $0.8\sigma$. Falsifier: $\Delta\alpha/\alpha$ inconsistent with $+2.3\times 10^{-6}$ at $>3\sigma$. Tier: \DerivedGrade{}. Status: \ValidatedGrade{}.

\end{itemize}

\subsection{Block 6: Laboratory Tests (Clocks, Cavities, Matter Waves, Antimatter)}\label{app:matrix-block6}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{Cross-species clock LPI: $K_A=k_\alpha S_A^\alpha$ species-dependent}. DFD: nonzero, species-dependent at $\sim 10^{-15}$. SM: exactly zero (Einstein equivalence principle). Observation: current null at $10^{-17}$ in pure-$\alpha$ channel. Agreement: consistent (screened). Falsifier: cross-species LPI null at $<10^{-15}$ on cross-species channel involving strong sector. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{Pure-$\alpha$ residual $\lambda_\alpha\approx 2.1\times 10^{-8}$ in same-ion E3/E2} (Sec.~\ref{sec:clocks}). DFD: $\lambda_\alpha\approx 2.1\times 10^{-8}$. SM: $0$. Observation: Yb$^+$ E3/E2 bound $|k_\alpha|<3.2\times 10^{-8}$. Agreement: at $66\%$ of bound. Falsifier: E3/E2 same-ion bound $<10^{-8}$ at $>3\sigma$. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{Composition/strong-sector residual $\lambda_{N,e,s}\approx 4.2\times 10^{-7}$}. DFD: $4.2\times 10^{-7}$ in cross-species channels. SM: $0$. Observation: BACON-II Al$^+$/Sr/Yb at $\sim 8\times 10^{-18}$. Agreement: consistent (screened). Falsifier: cross-species LPI null at $<5\times 10^{-7}$ at multi-$\sigma$. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\item \textbf{Cavity-atom residual $\xi_{\rm LPI}^{\rm res}$}. DFD: screened residual after geometric cancellation. SM: $\xi=0$. Observation: not yet at sensitivity. Falsifier: next-gen cavity-atom test reaches predicted sensitivity and finds null. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\item \textbf{Th-229 nuclear clock anomaly}. DFD: $\lambda_{\rm strong}\sim 10^{-7}$ enhancement vs electronic. SM: $0$. Observation: in development (Ooi+, JILA). Falsifier: Th-229 clock at $\sim 10^{-19}$ stability finds null in strong-sector channel. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\item \textbf{Matter-wave $T^3$ phase: $\Delta\phi_{\rm DFD}\sim 2\times 10^{-11}$ rad} (Sec.~\ref{sec:matter_wave}). DFD: $T^3$ scaling with $a_0$-dependent prefactor. SM: $0$ ($T^2$ only). Observation: not yet at sensitivity. Falsifier: $T^3$-search reaches $\sigma_\phi<7\times 10^{-14}$~rad and finds null. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\item \textbf{Antimatter gravity: $a_{\bar H}=a_H$ at metric level}. DFD: equal. SM: equal (Einstein). Observation: ALPHA-g consistent. Agreement: exact. Falsifier: antimatter shows $a\neq g$ at metric. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Antimatter species-dependent residual $\sigma_A$} (Sec.~\ref{sec:antimatter-sigma}). DFD: non-metric residual at $10^{-7}$--$10^{-8}$. SM: $0$ exactly. Observation: not yet at sensitivity. Falsifier: ALPHA-g + species-resolved measurements show null at predicted level. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\item \textbf{LPI test $\xi=0$ vs DFD $\xi\neq 0$ (cavity-atom)}. DFD: nonzero residual $\xi^{\rm res}$. SM: $\xi=0$ exactly. Observation: current bound $|\xi|<10^{-5}$. Agreement: consistent (screened). Falsifier: cavity-atom $\xi$ null at predicted screened residual level. Tier: \DerivedGrade{}. Status: \PendingGrade{}.

\item \textbf{Fiber-loop Sagnac residual}. DFD: $\sim\mu$rad-level frame-dragging analog. SM: $0$ for static lab. Observation: not at sensitivity. Falsifier: fiber-loop bound at $<\mu$rad shows null. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\end{itemize}

\subsection{Block 7: Strong-Field, Hyperbolicity, and Open Items}\label{app:matrix-block7}

\begin{itemize}[itemsep=4pt, leftmargin=*]

\item \textbf{No-BKL theorem near $\psi$-singularities} (Sec.~\ref{sec:strong}). DFD: no chaos, no modular invariance, no automorphic L-functions. GR: predicts BKL chaos. Observation: untested regime. Falsifier: numerical evolution of strong-field DFD shows BKL-type chaos. Tier: \TheoremGrade{}. Status: \OpenGrade{}.

\item \textbf{Yilmaz-type exterior solution} (App.~\ref{app:em-psi-coupling}, Thm~\ref{thm:no-exotic}). DFD: no finite-radius horizon; $r=2GM/c^2$ is photon sphere. GR: event horizon at $r_s=2GM/c^2$. Observation: EHT consistent with both. Agreement: exact identity. Falsifier: high-precision dark-spot imaging at $r<r_s$ shows event horizon. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{Hyperbolicity for small perturbations on smooth backgrounds}. DFD: well-posed IVP. GR: well-posed IVP. Observation: ---. Agreement: exact theorem. Falsifier: numerical instability for any DFD perturbation evolution. Tier: \TheoremGrade{}. Status: \PendingGrade{}.

\item \textbf{Full hyperbolicity for nonlinear strong-field dynamics}. DFD: open problem. GR: well-posed (with constraint propagation). Observation: ---. Falsifier: DFD evolution code finds genuine ill-posedness for any physical configuration. Tier: \ConjecturedGrade{}. Status: \OpenGrade{}.

\item \textbf{Uniqueness of $\mathbb{C}P^2\times S^3$} (App.~\ref{app:uniqueness-manifold}, Thm~\ref{thm:uniqueness}). DFD: unique compact Riemannian manifold satisfying V1--V9. Observation: internal consistency. Agreement: exact theorem. Falsifier: alternative compact manifold satisfies V1--V9 with different topological invariants. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Higgs matching scale $\lambda_H=1/8$} (current best read: dim spinor rep on 7D). DFD: $\lambda_H=1/8$. SM: input parameter. Observation: $\lambda_H(M_h)\approx 0.13$. Agreement: match within $4\%$. Falsifier: full NCG matching shows $\lambda_H\neq 1/8$ at high precision. Tier: \ConjecturedGrade{}. Status: \OpenGrade{}.

\item \textbf{Mach's principle: rest-mass inertia from cosmic $\psi$ background} (Sec.~\ref{sec:openproblems}). DFD: functional of cosmic $\psi$ background. GR: inertia is fundamental. Observation: not currently testable. Falsifier: Mach's-principle predictions for inertial-frame anomalies tested at high precision. Tier: \ConjecturedGrade{}. Status: \OpenGrade{}.

\item \textbf{PDE uniqueness theorem for $\psi$ rest frame} (Sec.~\ref{sec:openproblems}). DFD: unique up to gauge. GR: boost invariance. Observation: not yet proven. Falsifier: demonstration of multiple inequivalent rest frames in any DFD configuration. Tier: \ConjecturedGrade{}. Status: \OpenGrade{}.

\item \textbf{Standard quantum unitarity preserved (Mott problem)} (Sec.~\ref{sec:quantum}). DFD: no modification to QM. SM: standard QM. Observation: all QM tests consistent. Agreement: exact. Falsifier: QM modification detected at any precision. Tier: \TheoremGrade{}. Status: \ValidatedGrade{}.

\item \textbf{Diosi-Penrose collapse rate vs DFD} (Sec.~\ref{sec:openproblems}). DFD: no anomalous collapse. DP: mass-scaling decoherence. Observation: MAQRO will discriminate. Falsifier: MAQRO confirms DP scaling at predicted level. Tier: \DerivedGrade{}. Status: \OpenGrade{}.

\end{itemize}

\subsection{Summary statistics}\label{app:matrix-summary}

The Master Claims Confrontation Matrix contains 73 distinct claims spanning fundamental constants, gravitational tests, galactic and cluster dynamics, cosmology, and laboratory probes.

\paragraph{Aggregating by tier.}
\begin{itemize}[itemsep=2pt, leftmargin=*]
\item \textbf{Theorem-grade (\TheoremGrade{}):} 26 claims, representing the load-bearing structural identities of the theory (PPN parameters, $\mu$-derivation, $a_*$-derivation, Strong CP, Pad\'e identity, no-wormhole and no-exotic theorems, GW propagation, BTFR, $\sin^2\theta_W$, $\bar\theta=0$, antimatter gravity at metric level, manifold uniqueness). (Proton stability is graded conditional --- see its entry.)
\item \textbf{Derived (\DerivedGrade{}):} 40 claims, including all microsector-derived quantities ($\alpha^{-1}$, Higgs VEV, fermion masses, CKM, PMNS, $H_0$ closure, $\sigma_8$ closure, cluster $J_{\rm PDE}$).
\item \textbf{Conjectured (\ConjecturedGrade{}):} 5 claims, all explicitly open program items ($\lambda_H=1/8$, full hyperbolicity, Mach's principle, $\psi$ rest-frame uniqueness, DESI evolving-$w$ qualitative match).
\item \textbf{Empirical (\EmpiricalGrade{}):} 2 claims, the cluster five-factor decomposition and the UF-dwarf $M/L$ bound; the cluster closure depends on multiple literature-bounded systematic factors.
\end{itemize}

\paragraph{Aggregating by status.}
\begin{itemize}[itemsep=2pt, leftmargin=*]
\item \textbf{Validated (\ValidatedGrade{}):} 45 claims, where current observational data supports DFD at the stated agreement level. This includes essentially all established gravitational tests, the parameter-free predictions for $\alpha^{-1}$ and the Higgs VEV, the SPARC galactic dynamics tests, the cluster five-factor closure, the CMB peak structure, and the UVCS double-transit observation.
\item \textbf{Pending (\PendingGrade{}):} 16 claims, where the experimental capability exists or is imminent but the relevant measurement is not yet published. These include cross-species clock LPI tests, Th-229 nuclear clock anomalies, ESPRESSO $\alpha$-evolution refinements, DDR violation tests, and matter-wave $T^3$ phase searches.
\item \textbf{Open (\OpenGrade{}):} 12 claims with no comparable observational test currently available. These include strong-field BKL absence, full hyperbolicity, and Mach's-principle inertia derivations.
\end{itemize}

\subsection{Reading guide for referees}\label{app:matrix-reading}

A referee assessing v4.0 should focus principal scrutiny on the following high-leverage entries:

\begin{enumerate}[itemsep=2pt, leftmargin=*]
\item \textbf{Theorem-grade structural claims:} Verify that the proofs cited (e.g., Theorem~\ref{thm:strong_cp_all_loops} for Strong CP, Theorem~\ref{thm:mu-theorem} for $\mu$-derivation, Theorem~\ref{thm:pade-identity} for the Pad\'e identity, Theorem~\ref{thm:uniqueness} for $\mathbb{C}P^2\times S^3$ uniqueness) are mathematically sound and use only the stated assumptions. The theorems are independent of the rest of the theory; failure of any single one would force a substantial revision.

\item \textbf{Derived microsector results:} Verify that the closure framework (spectral action, Chern-Simons on $S^3$, $k_{\max}=60$ truncation) is internally consistent and reproduces the listed quantities. The microsector framework is the closure that promotes ``conjectures'' to ``derived,'' so its assumptions matter.

\item \textbf{Cluster five-factor decomposition (\EmpiricalGrade{}):} The 14/16 within $\pm 10\%$ closure (relaxed $1.01\pm0.05$; mergers $0.95\pm0.08$; Bullet and El Gordo remain $13$--$19\%$ low, at $\le1.05\sigma$ of their published lensing-mass errors) is the most-correction-dependent claim in the matrix. A referee may legitimately demand: (a) that each factor's range is defended cluster-by-cluster in App.~\ref{app:cluster-full}, (b) that no single factor dominates the closure, and (c) that the five-factor stack is not overfit to the 16-cluster sample. The PDE-validated $J_{\rm PDE}$ is theorem-grade; the surrounding factors are bounded by published cluster-mass-systematics literature.

\item \textbf{Pending experimental tests:} Cross-species clock LPI, Th-229 nuclear clock, BACON-II next iteration, KiDS-Legacy lensing/$E_G$ result, Yb$^+$ same-ion E3/E2 next-generation. These will determine whether DFD survives or is falsified within the next 1--3 years. They are the decisive tests, not the historical data.
\end{enumerate}

\subsection{Use of this matrix}\label{app:matrix-use}

This matrix is intended for three audiences:

\begin{itemize}[itemsep=2pt, leftmargin=*]
\item \textbf{Referees}: To provide a single navigational catalog summarizing the falsifiability structure of the theory. Failure of any \TheoremGrade{}-class entry forces structural revision; failure of any \PendingGrade{} entry would falsify a specific prediction. The matrix is designed to be a referee's first stop, not a substitute for the detailed sections.

\item \textbf{Experimenters}: To identify which open or pending tests offer the highest discriminating power between DFD and competing frameworks. The Yb$^+$ E3/E2 same-ion null at $|k_\alpha|<3.2\times 10^{-8}$ is currently the sharpest in the laboratory regime; the DESI DR3 evolving-$w$ result is currently the sharpest in cosmology.

\item \textbf{Independent theorists}: To distinguish the theorem-grade structural identities from the auxiliary closure-framework derivations. The theorem-grade items (PPN, $\mu$-derivation, $a_*$-derivation, Pad\'e identity, Strong CP, manifold uniqueness, no-wormhole) carry forward independent of the closure framework; alternative closure frameworks for the microsector would change the \DerivedGrade{} entries but not the \TheoremGrade{} entries.
\end{itemize}

The discipline of maintaining this matrix --- forcing every empirical claim to wear a tier label and a falsifier --- is itself a contribution: it prevents structural claims from being mistaken for hidden first-principles proofs, and it prevents auxiliary closure derivations from being mistaken for empirical fits. The matrix should be updated whenever new claims are added or old claims are revised; a per-row history can be tracked in the patch notes.
