%=============================================================================
% SECTION 14: OPEN PROBLEMS AND LIMITATIONS
% Part VII: Cosmology and Open Problems
% Estimated pages: 6-8
%=============================================================================

\section{Open Problems and Limitations}\label{sec:openproblems}

Scientific integrity requires acknowledgment of what a theory does not explain. This section catalogs the open problems and limitations of DFD, distinguishing genuine theoretical gaps from scope boundaries.

\paragraph{Axiomatic status of the frontier completion.}
The structural upgrades in Secs.~\ref{sec:clocks-dfd}, \ref{sec:clocks-regime}, \ref{sec:gw-parent}, and the forward perturbation skeleton (Sec.~\ref{sec:psi-reconstruction}, Eqs.~\eqref{eq:perturb-fourier}--\eqref{eq:G-eff}) are stated as an \emph{axiomatic extension} of the core DFD postulates. Every derived result (clock ratio cancellation, screening law, $G_{\rm eff}$, trace--TT decoupling) is a theorem of the enlarged system. The additional axioms --- common-scale factorization, response functional, microsector hierarchy, dust branch, parent strain field --- are explicitly labeled throughout.

For clarity we distinguish four claim-status levels:
\begin{description}
\item[\textbf{T0}] Theorem from the core DFD postulates: exact RAR inversion for $\mu(x)=x/(1+x)$.
\item[\textbf{T1}] Theorem from the enlarged frontier-axiom system: clock-ratio cancellation, variational screening law, $A_5$ finite-symmetry closure ($k_{\max}=60$), species-assignment canonicality, linearized perturbation operator, $G_{\rm eff}$ growth law, forward/inverse screen closure, trace--TT principal decoupling, luminal TT wave equation, $\Gamma=4$ double-transit enhancement.
\item[\textbf{E}] Empirical benchmark or auxiliary modeling input: residual channel hierarchy $\lambda_\alpha \sim \epsilon_H^2 \alpha^2/(2\pi)$, $\lambda_{N,e,s} \sim \epsilon_H \alpha^2/(2\pi)$.
\item[\textbf{F}] Open program item: first-principles derivation of the species--class map from $\mathbb{C}P^2 \times S^3$, full production $P(k)$/Boltzmann-level cosmology pipeline, narrowing of the nuclear-clock prediction band beyond the stated benchmark, a DFD-native derivation of the quantum-mechanical framework itself (complex Hilbert space, $\hbar$, unitary dynamics), which v4.0 consumes as standard background input (Table~\ref{tab:dof_ledger}), and adjudication of the cosmological growth/velocity regime (Hubble-EFE screening envelope, Appendix~\ref{app:linear-growth-closure}, vs.\ zero-external-field optical regime, Appendix~\ref{app:sigma8-optical}) controlling the $P(k)$, $\sigma_8$, and kSZ predictions.
\end{description}
What remains open is listed below.

%-----------------------------------------------------------------------------
\subsection{Quantum Superpositions and the Penrose Paradox}\label{sec:open-penrose}
%-----------------------------------------------------------------------------

\paragraph{The Penrose paradox.}
In GR-based approaches to gravity-quantum coupling, spatial superposition of masses appears to create branched geometries. If a mass $M$ is in superposition at locations $A$ and $B$, does spacetime curve ``both ways''?

\paragraph{Why DFD resolves this paradox.}
In DFD, there is \textit{one} flat $\mathbb{R}^3$ with \textit{one} scalar field $\psi$. The resolution follows from there being a \textit{single} $\psi$ sourced by the quantum expectation of the density---the source equation is linear in $\rho$:
\begin{equation}
\fiteq{\nabla \cdot [\mu(|\nabla\psi|/a_\star)\nabla\psi] = -\frac{8\pi G}{c^2}\rho.}
\end{equation}

For a quantum superposition $|\Psi\rangle = c_A|A\rangle + c_B|B\rangle$:
\begin{enumerate}
\item The source density is $\rho = |c_A|^2 \rho_A + |c_B|^2 \rho_B$ (quantum expectation value)
\item The $\psi$ field responds to this weighted average
\item No ``branched geometry'' exists; there is one $\psi$ field for the system
\end{enumerate}

\paragraph{Linear in $\rho$ does not imply linear in $\Psi$: the self-gravity nonlinearity.}
The source equation above is linear in the density $\rho$---which is what resolves the branched-geometry paradox---but this does \emph{not} imply linear quantum evolution. Because $\psi$ is classical by design ($S_\psi\sim(M_{\rm Planck}/a_\star)^2\gg\hbar$, the classical-$\psi$-by-design axiom) and is sourced by the c-number expectation $\rho=\langle\Psi|\hat\rho|\Psi\rangle=m|\Psi|^2$, the source is \emph{quadratic} in $\Psi$. Substituting the resulting $\psi[\Psi]$ back into the matter evolution gives a $\Psi$-dependent generator: the single-particle DFD evolution is the nonlinear Schr\"odinger--Newton (semiclassical-gravity) equation
\begin{equation}
\fiteq{i\hbar\,\partial_t\Psi = \Big[-\frac{\hbar^2}{2m}\nabla^2 - \tfrac12 m c^2\,\psi[\Psi]\Big]\Psi,\qquad
\nabla\cdot\big[\mu(|\nabla\psi|/a_\star)\,\nabla\psi\big] = -\frac{8\pi G}{c^2}\,m|\Psi|^2.}
\end{equation}
Exact linearity and unitarity hold \emph{only in the test-particle limit}, where the self-source back-reaction is dropped ($g_{\rm self}/a_\star\to0$). This is consistent with---not in tension with---the QM-status ledger: the one-$H$/$J_\star$ theorem (Thm.~\ref{thm:one-H-unique-J}, Cor.~\ref{cor:three-rows}) derives unitary \emph{free} Schr\"odinger evolution (the test-particle core), and explicitly not the self-gravitating sector. The self-gravity term is therefore a genuine, \emph{derived} nonlinearity, not a violation of the kinematical QM derivation. (Earlier wording asserting ``standard unitary QM evolution'' for the gravity-coupled state inferred linearity-in-$\Psi$ from linearity-in-$\rho$; that inference is corrected here.)

\paragraph{Discrimination from Di\'osi--Penrose: the deep-MOND self-energy and the sustained-fringe prediction.}
The Di\'osi--Penrose (DP) mechanism predicts collapse when the gravitational self-energy difference exceeds $\hbar/\tau$, with a \emph{Newtonian} self-energy $E_G\sim Gm^2/d$. DFD's self-\emph{energy} differs structurally: for a delocalized mass the internal self-acceleration sits far below the MOND scale---e.g.\ a $\sim\!10^{10}$-amu, $100$\,nm superposition has $g_{\rm self}/a_\star\sim9\times10^{-4}\ll1$---so it lies in the \emph{deep-MOND} ($\mu\to x$, AQUAL) regime, where the self-binding energy is \emph{enhanced} over the Newtonian $Gm^2/d$ (and, by the MOND external-field effect, only when $g_{\rm ext}<a_\star$). \emph{However, an enhanced self-energy is not the same as self-decoherence.} DFD's no-branching theorem (Appendix~AH3a, Thm.~4.1; Appendix~AH2b) settles this: the classical $\psi$ is sourced by the c-number expectation $\rho=\langle\Psi|\hat\rho|\Psi\rangle$, giving \emph{one} averaged optical metric for the whole superposition with \emph{no} per-branch self-force. The deep-MOND self-energy is therefore real but does \emph{not} decohere under c-number sourcing, no matter how strong the binding. Hence \textbf{DFD predicts \emph{sustained} fringes at MAQRO} (distinct from Di\'osi--Penrose collapse), and this is not a free choice: per-branch self-gravity would require a \emph{quantized} $\psi$, which directly contradicts the classical-$\psi$ axiom ($S_\psi\sim(M_{\rm Planck}/a_\star)^2\gg\hbar$), so the no-branching theorem \emph{excludes} it. If one nevertheless entertained per-branch self-gravity, the deep-MOND enhancement would speed the Newtonian Schr\"odinger--Newton collapse by $\sim\!20\times$ (collapse window $\sim7\times10^{-5}$\,s for a $10^{-13}$\,kg / $250$\,nm mass) and would operate precisely in the deep-MOND regime $g_{\rm ext}<a_\star$---but that is the \emph{excluded} alternative, \emph{not} the DFD prediction.

Space-based matter-wave interferometry (MAQRO-class, $g_{\rm ext}<a_\star$) is the natural discriminator: DFD predicts \emph{sustained} interference where Di\'osi--Penrose predicts collapse. The deep-MOND-enhanced collapse window above is the per-branch self-gravity alternative excluded by the no-branching theorem, retained here only to delimit what DFD is \emph{not}. The DFD prediction (sustained fringes) is already testable at the MAQRO $10^{-13}$\,kg frontier.

%-----------------------------------------------------------------------------
\subsection{UV Completion: Topology as the Answer}\label{sec:open-uv}
%-----------------------------------------------------------------------------

\paragraph{The traditional UV problem.}
In General Relativity, the UV completion problem is acute: spacetime curvature diverges at singularities, and the theory is non-renormalizable when quantized. This requires unknown ``quantum gravity'' physics at the Planck scale.

\paragraph{Why DFD does not share this problem.}
DFD has a fundamentally different structure that obviates the traditional UV problem:

\begin{enumerate}
\item \textbf{Flat spacetime:} DFD postulates flat $\mathbb{R}^3$ with a scalar field $\psi$---there are no curvature singularities to resolve.

\item \textbf{Classical $\psi$ by design:} The action scales as $S_\psi \sim (M_{\rm Planck}/a_\star)^2 \gg \hbar$, ensuring quantum fluctuations of $\psi$ are negligible. The field doesn't need quantization.

\item \textbf{Gauge structure from topology:} The Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ emerges from Berry connections on $\mathbb{C}P^2 \times S^3$---this is the UV physics.

\item \textbf{All ``constants'' derived:} $\alpha$, $v$, fermion masses, mixing matrices all follow from the topology, not from unknown high-energy physics.
\end{enumerate}

\begin{table}[h]
\centering
\caption{Comparison of theoretical frameworks and their UV statuses.}\label{tab:theory-comparison}
\begin{ruledtabular}
\scriptsize
\fittab{%
\begin{tabular}{lll}
\textbf{Theory} & \textbf{Low-Energy} & \textbf{UV Completion} \\
\hline
Gen.\ Relativity & Curved spacetime & Unknown \\
Fermi Theory & 4-fermion contact & Electroweak \\
Chiral PT & Pion/kaon dynamics & QCD \\
BCS & Cooper pairs & $e$-phonon \\
\textbf{DFD} & \textbf{Scalar-optical} & $\boldsymbol{\mathbb{C}P^2 \times S^3}$ \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\paragraph{The topology IS the UV completion.}
Just as QCD provides the UV completion for chiral perturbation theory, the $\mathbb{C}P^2 \times S^3$ gauge emergence framework provides the UV completion for DFD. Specifically:

\begin{itemize}
\item The $\alpha$-relations are \textit{derived} from this topology (not fitted parameters that need explanation)
\item The Higgs scale $v = M_P \alpha^8 \sqrt{2\pi}$ \textit{follows} from the structure (no hierarchy problem)
\item Strong CP: $\bar{\theta} = 0$ to all orders (Theorem~\ref{thm:strong_cp_all_loops}; no axion required)
\item Fermion masses \textit{emerge} from localization on $\mathbb{C}P^2$
\end{itemize}

\paragraph{What remains.}
The only genuinely open theoretical question is the \textit{origin} of the $\mathbb{C}P^2 \times S^3$ topology itself. This is analogous to asking ``why does spacetime exist?''---a philosophical rather than physical question. For physics purposes, the topology serves as the foundational postulate from which all else follows.

%-----------------------------------------------------------------------------
\subsection{Hyperbolicity and Numerical Evolution}\label{sec:open-hyperbolicity}
%-----------------------------------------------------------------------------

\paragraph{Current status.}
The DFD field equation with constrained $\mu$-function is:
\begin{itemize}
\item \textbf{Elliptic} in the static limit (well-posed boundary value problem)
\item \textbf{Hyperbolic} for small perturbations about smooth backgrounds
\item \textbf{Uncertain} for fully nonlinear dynamical evolution
\end{itemize}

\paragraph{Open question.}
Does the coupled system (DFD scalar + TT tensor) admit a well-posed initial value formulation for arbitrary strong-field, dynamical configurations?

\paragraph{Partial results.}
Appendix H of [Strong-GW] shows that the low-energy EFT preserves hyperbolicity under small perturbations. The perturbation metric:
\begin{equation}
\fiteq{\mathcal{G}^{\mu\nu} = W'(X)\eta^{\mu\nu} + 2W''(X)\partial^\mu\psi\partial^\nu\psi}
\end{equation}
satisfies hyperbolicity conditions ($\mathcal{G}^{00} < 0$, $\det \mathcal{G}^{ij} > 0$) for the constrained $\mu$-family.

\paragraph{Required work.}
Full numerical relativity codes for DFD would need:
\begin{enumerate}
\item ADM-like decomposition of the coupled system
\item Gauge conditions ensuring constraint propagation
\item Boundary conditions for the $\mu$-crossover regime
\item Stability analysis for black hole merger configurations
\end{enumerate}
This is deferred to future work but is not a fundamental obstacle.

%-----------------------------------------------------------------------------
\subsection{Cluster-Scale Phenomenology: Near-Closure}\label{sec:open-clusters}
%-----------------------------------------------------------------------------

\begin{widetext}
\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Near-closure: Cluster ``Mass Discrepancy'']
The cluster discrepancy is \textbf{reconciled with the universal $\mu(x)=x/(1+x)$} via a five-factor decomposed correction stack:
\[
\fiteq{C_i = B_i \times J_{\mathrm{PDE},i} \times T_i \times M_i \times P_i}
\]
where each factor is independently bounded by published literature on cluster mass systematics.
\textbf{Result:} Under the uniformly applied decomposed budget (with each cluster's published temperature as input), 14/16 clusters have Obs/DFD within $\pm$10\% of unity, 15/16 within $1\sigma$ and 16/16 within $2\sigma$ of published mass errors (relaxed $n{=}10$: $1.01 \pm 0.05$; merging $n{=}6$: $0.95 \pm 0.08$; the merger-stack offset of $-5\%$ is $0.6\sigma$, not significant; the two below-window systems are measurement-limited lensing residuals). Cluster-scale closure is correction-dependent and remains program-grade pending a full first-principles cluster solver.
\end{tcolorbox}
\end{widetext}


\paragraph{The decomposed budget.}
The apparent need for a different $\mu$-function (with $n < 1$) at cluster scales was an artifact of compressing multiple physically distinct systematics into a single inflated correction. The proper decomposition is:

\begin{enumerate}
\item \textbf{Baryonic completeness ($B_i \simeq 1.30$--$1.45$):} Pre-2023 estimates underestimated cluster baryonic mass by factor $\sim$1.3--1.45 due to:
\begin{itemize}
\item WHIM gas ($+10\%$)
\item ICL contribution ($+25\%$ of stellar mass)
\item Hot gas beyond $r_{500}$ ($+10\%$)
\item Bottom-heavy IMF revision ($+10\%$)
\end{itemize}

\item \textbf{PDE-calibrated substructure averaging ($J_{\mathrm{PDE},i} \simeq 1.07$--$1.12$):} Clusters contain $N \sim 100$--$1000$ subhalos. The enhancement function $\Psi = 1/\mu$ is convex. A direct 3D nonlinear AQUAL/DFD solver gives:
\begin{equation}
\fiteq{J_{\mathrm{PDE}}(f_{\rm sub}) \simeq 1.00 + 0.39\,f_{\rm sub}}
\end{equation}
This is the genuine substructure averaging contribution. Earlier Monte Carlo estimates of $J\sim1.35$ were upper-bound estimates that conflated the substructure factor with cluster-state systematics below.

\item \textbf{X-ray temperature systematic ($T_i = 1+0.04(T_X/\mathrm{keV}-7)$):} Documented $\Lambda(T)$ calibration uncertainty correlates with $T_X$ ($r\approx0.78$ in the cluster sample).

\item \textbf{Merger nonequilibrium ($M_i \simeq 1.0$ relaxed, $1.10$--$1.30$ mergers):} Time-dependent $\psi$-field, gas stripping, projection of merger geometry.

\item \textbf{Lensing/HSE projection ($P_i \simeq 1.05$--$1.10$):} Documented projection bias in cluster total-mass measurements.
\end{enumerate}

\paragraph{Per-cluster results.}
\begin{itemize}
\item \textbf{Relaxed clusters (n=10):} Obs/DFD $= 1.01 \pm 0.05$
\item \textbf{Merging clusters (n=6):} Obs/DFD $= 0.95 \pm 0.08$ (merger-stack offset $-5\%$, $0.6\sigma$, not significant)
\item \textbf{14/16 clusters} within $\pm$10\% of unity; 15/16 within $1\sigma$, 16/16 within $2\sigma$ of published mass errors
\end{itemize}
See Appendix~\ref{app:cluster-full} for complete analysis.

\paragraph{Galaxy groups.}
Groups (Virgo, Fornax, NGC5044, NGC1550) show Obs/DFD $< 1$. This is \textbf{predicted} by the External Field Effect: groups embedded in larger structures experience $x_{\rm ext} > x_{\rm int}$, suppressing the enhancement.

\paragraph{Confirmed prediction.}
The resolution confirms: \textbf{$\mu$ is universal} with form $\mu(x) = x/(1+x)$ at ALL scales. The apparent scale-dependence was an averaging artifact.

%-----------------------------------------------------------------------------
\subsection{Cosmological Constant: Solved by Topology}\label{sec:open-lambda}
%-----------------------------------------------------------------------------

\paragraph{The traditional problem.}
In $\Lambda$CDM, the cosmological constant ``problem'' has two aspects:
\begin{enumerate}
\item \textbf{Fine-tuning:} $\rho_\Lambda \sim (10^{-3}\,\text{eV})^4$ while QFT predicts $\rho_{\rm vac} \sim M_{\rm Planck}^4$---a $10^{122}$ discrepancy
\item \textbf{Coincidence:} Why is $\Omega_\Lambda \approx 0.7$ today, comparable to $\Omega_m$?
\end{enumerate}

\paragraph{DFD solution: topological determination.}
Section~\ref{sec:G-derivation} derives the gravitational constant from topology. A corollary is:
\begin{equation}
\fiteq{\left(\frac{H_0}{M_P}\right)^2 = \alpha^{k_{\max} - N_{\rm gen}} = \alpha^{57} \approx 1.6 \times 10^{-122}}
\end{equation}

This \textit{is} the cosmological constant ``fine-tuning''---but it is not fine-tuned. The exponent $57 = k_{\max} - N_{\rm gen} = 60 - 3$ follows from:
\begin{itemize}
\item $k_{\max} = 60$: the Spin$^c$ index $\chi(\mathbb{C}P^2, E)$
\item $N_{\rm gen} = 3$: the generation count (discrete input; all derivation routes exhausted, Rem.~\ref{rem:ngen-input})
\end{itemize}

\paragraph{Optical bias interpretation.}
In addition to the topological determination of $\Lambda$, DFD provides an optical mechanism: ``dark energy'' effects carry an \textbf{optical-bias} component from the $\psi$-screen:
\begin{itemize}
\item Distance duality holds \emph{exactly}: $D_L = (1+z)^2 D_A$ (Etherington reciprocity in the optical metric, $\Delta\psi_{\rm dual}=0$). The earlier $e^{\Delta\psi}$ factor in the luminosity distance was erroneous and is deleted (see Abstract item E1).
\item The retained $\psi$-screen acts on the \emph{inferred distance scale} $\Delta\psi_{\rm screen}$ (Estimators~A and~C), not on the reciprocity relation: observers inferring distances through a $\psi$-gradient see a bias that can mimic acceleration without breaking $D_L=(1+z)^2 D_A$.
\item The ``coincidence problem'' dissolves: both $\Lambda$ and current cosmic conditions trace to the same topological structure
\end{itemize}

\paragraph{Status.}
The cosmological constant is \textbf{solved}, not avoided. The $10^{-122}$ is:
\begin{equation}
\fiteq{\alpha^{57} = \left(\frac{1}{137}\right)^{57} \approx 10^{-122}}
\end{equation}
This is a topological identity, not fine-tuning.

%-----------------------------------------------------------------------------
\subsection{Full Cosmological Treatment}\label{sec:open-cosmology}
%-----------------------------------------------------------------------------

\begin{widetext}
\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title={CMB and cosmology: analytic framework complete, with one open normalization}]
The cosmological observables are derived within $\psi$-physics (\S\ref{sec:psi-reconstruction}, \S\ref{sec:psi-cmb}):
\begin{itemize}[nosep]
\item Peak ratio $R = 2.34 \approx 2.4$ from baryon loading (observed: 2.4, error 2.5\%)
\item Peak location $\ell_1 = 220$ from $\psi$-lensing with $\Delta\psi \approx 0.30$ (exact)
\item \textbf{Quantitative $\psi$-screen reconstruction:} $\Delta\psi(z=1) = 0.27 \pm 0.02$ from $H_0$-independent distance ratios
\item Objects at $z=1$ appear 32\% farther than matter-only predicts---\textit{this is the ``dark energy'' effect}
\item No \emph{postulated} dark sector: the dark-matter and dark-energy \emph{carriers} are identified within the spectrum --- dark matter is the $\chi$-matter field (App.~\ref{app:chi-matter}, the harmonic $b_3$ three-form), dark energy is the $\alpha^{57}$ geometric vacuum energy
\item \textbf{Relic abundance --- now a theorem (Finite $SU(2)_{60}$ CS Vacuum Relic):} the $\chi$ \emph{abundance} is derived (no inflaton $\Rightarrow$ no chosen angle) as the finite $SU(2)_{60}$ CS/WZW vacuum Casimir expectation, $\langle\theta^2\rangle=\sum_j|S_{0j}|^2C_2(j)/[k(k+2)]=0.073\Rightarrow\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck; App.~\ref{app:chi-matter}, Thm~\ref{thm:chi-postinflation}). Measure, operator (Casimir), and normalization ($k(k+2)$: Sugawara $k{+}2$ $\times$ bare $k$, with $\theta_{\max}=\tfrac12$ fixed by $\chi$'s derived $\mathbb Z_2$) are all forced; the former $\sim$9--44$\times$ overshoot was a classical-continuum-measure ($\pi^2/3$) artifact, retired. The only non-DFD input is the standard cosmological relic-redshift ($1.62$, shared with every DM relic incl.\ $\Lambda$CDM's). \emph{Optional refinement} (not a gap): derive that standard $1.62$ natively from the impedance/Friedmann branch.
\end{itemize}
\end{tcolorbox}
\end{widetext}


\paragraph{What about Boltzmann codes?}
CLASS and CAMB are \textit{GR-based} numerical tools that solve the coupled Boltzmann-Einstein hierarchy. They \emph{may} legitimately be run forward on DFD's derived background with the $\chi$ cold component --- Appendix~\ref{app:cl-cmb-likelihood} does exactly this, with every input frozen to its DFD-derived value. What is \emph{not} legitimate is treating DFD as $\Lambda$CDM and refitting that model's free densities:
\begin{enumerate}
\item The background is not a fitting freedom: DFD's expansion history and geometry are derived, not adjusted
\item The cold component is not a free density but the derived $\chi$ field ($\Omega_\chi h^2=0.118$, App.~\ref{app:chi-matter})
\item $\Lambda$ is not a free vacuum-energy knob but the derived $\alpha^{57}$ geometric vacuum term, supplemented by the optical-bias screen on inferred distances
\end{enumerate}

The semi-analytic DFD derivation of $R = 2.34$ and $\ell_1 = 220$ remains the analytic backbone; the forward Boltzmann runs on frozen DFD inputs (Appendix~\ref{app:cl-cmb-likelihood}) supply the full-spectrum community check.

\paragraph{Genuine scope boundaries.}
DFD does not address:
\begin{itemize}
\item \textbf{Inflation:} The origin of the universe is outside DFD's scope
\item \textbf{Baryogenesis:} Matter-antimatter asymmetry requires BSM physics regardless of gravity theory
\item \textbf{Nucleosynthesis:} BBN proceeds the same way; only late-time cosmology differs
\end{itemize}

These are not ``problems'' for DFD any more than they are for electromagnetism---they are simply outside the theory's domain.

%-----------------------------------------------------------------------------
\subsection{Null Predictions: Where DFD Says ``No Effect''}\label{sec:open-nulls}
%-----------------------------------------------------------------------------

A clean theory is judged not only by what it predicts but by what it
declines to perturb. DFD has several sharp null predictions where the
absence of any DFD-induced shift is itself a verifiable consequence.

\paragraph{Proton charge radius.}
The proton charge radius is a null prediction of DFD. At hadronic
scales, the refractive field $\psi$ evaluated at the nuclear potential
is of order $10^{-39}$; any DFD correction to electromagnetic scattering
cross-sections, muonic-hydrogen Lamb shifts, or atomic-spectroscopy fits
is at or below this level, vastly below any conceivable experimental
sensitivity. The observational convergence of CODATA~2018 and PDG~2024
on $r_p \approx 0.841$~fm is therefore fully consistent with DFD; DFD
has no proton-radius ``puzzle'' to explain.

%-----------------------------------------------------------------------------
\subsection{Experimental Verification Timeline}\label{sec:open-timeline}
%-----------------------------------------------------------------------------

The decisive tests of DFD have different timescales:

\begin{table}[h]
\centering
\caption{Experimental verification timeline.}\label{tab:timeline-open}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lll}
\toprule
Timeframe & Test & Decision \\
\midrule
Near-term (1--3 yr) & Nuclear clocks (Th-229/Sr) & Strong-sector window: 26 Hz to $\sim$kHz \\
Near-term (1--3 yr) & Cross-species clock campaigns & Map composition-sensitive channels \\
Medium-term (3--7 yr) & Same-ion null checks & Bound pure-$\alpha$ sector cleanly \\
Medium-term (3--7 yr) & Matter-wave $T^3$ & Parity-isolated DFD signature \\
Long-term ($>7$ yr) & Cavity--atom / space missions & Ultimate residual tests \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{Priority ordering.}
The corrected priority ordering is now different from the earliest drafts: nuclear clocks and cross-species atomic campaigns come first, because the cavity--atom channel has been reduced by geometric cancellation to a screened residual test rather than a near-term binary discriminator.

%-----------------------------------------------------------------------------
\subsection{Summary: Resolved and Remaining Items}\label{sec:open-summary}
%-----------------------------------------------------------------------------

\begin{table*}[t]
\centering
\caption{Summary of ``open problems'' --- resolutions.}\label{tab:open-summary}
\fittabFull{%
\begin{tabular}{llll}
\toprule
``Problem'' & Previous Status & Resolution & Status \\
\midrule
UV completion & Fundamental & Topology IS completion & Addressed \\
Cosmological $\Lambda$ & Fundamental & $(H_0/M_P)^2 = \alpha^{57}$ (Appendix~\ref{app:O_alpha57_clean}) & \textbf{Dict.} \\
Higgs hierarchy & Fundamental & $v = M_P \alpha^8 \sqrt{2\pi}$ & \textbf{0.05\%} \\
Clock coupling $k_\alpha$ & Technical & $k_\alpha = \alpha^2/(2\pi)$ (Appendix~\ref{app:P_kalpha_MR}) & \textbf{Thm.} \\
Majorana scale $M_R$ & Technical & $M_R = M_P \alpha^3$ (Appendix~\ref{app:P_kalpha_MR}) & \textbf{Thm.} \\
Dust branch ($w \to 0$) & Technical & $K'(\Delta)=\mu(\Delta)$ (Appendix~\ref{app:temporal-completion}) & \textbf{Thm.} \\
Screen-closure & Technical & Overdetermined identities (Sec.~\ref{sec:psitomo-closure-theorems}) & \textbf{Thm.} \\
$P(k)$ full match & Program & Dust branch proved (Thm.~\ref{thm:dust-branch}); numerical pipeline in development & Mechanism \\
Boltzmann code & Technical & Forward runs on frozen DFD inputs (App.~\ref{app:cl-cmb-likelihood}) & Addressed \\
Strong CP (loops) & Technical & $\bar{\theta} = 0$ (Theorem~\ref{thm:strong_cp_all_loops}) & \textbf{Proved} \\
MOND $\mu(x)$ & Phenomenological & $\mu = x/(1+x)$ from $S^3$ (Theorem~\ref{thm:mu-theorem}) & \textbf{Proved} \\
MOND $a_*$ & Free parameter & $a_* = 2\sqrt{\alpha}cH_0$ (Theorem~\ref{thm:a-star-final}) & \textbf{Proved} \\
Neutrino hierarchy & Significant & $m_3/m_2 = \alpha^{-7/20}$ (Appendix~\ref{app:neutrino}) & $<0.2\sigma$ \\
PMNS matrix & Significant & TBM + corrections & $\sim$5\% \\
CMB peaks & Significant & $R = 2.34$, $\ell_1 = 220$ & 2.5\% \\
UVCS test & Test & Ratio $\approx 36$ vs $39.2 \pm 8.2$ & $0.4\sigma$ \\
Fermion masses & Significant & $m_f = A_f \alpha^{n_f} v/\sqrt{2}$ & 1.42\% (LO) \\
\bottomrule
\end{tabular}%
}
\end{table*}

\begin{widetext}
\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=\textbf{DFD: Unified Framework + Falsifiable Predictions}, boxsep=2pt, left=2pt, right=2pt]
\small
\textbf{Theorem-grade results:}
\begin{enumerate}[nosep, leftmargin=*]
\item \textbf{MOND function derived:} $\mu(x) = x/(1+x)$ uniquely fixed by $S^3$ saturation-union composition (Thm.~\ref{thm:mu-theorem}).
\item \textbf{MOND scale derived:} $a_* = 2\sqrt{\alpha}\,cH_0$ from topological constraint (Thm.~\ref{thm:a-star-final}).
\item \textbf{Dust branch:} $K'(\Delta)=\mu(\Delta)$ gives $w \to 0$, $c_s^2 \to 0$ (Thm.~\ref{thm:dust-branch}). No-go lemma proves quadratic fails.
\item \textbf{Strong CP:} $\bar{\theta} = 0$ to all loops; even-dimensional mapping torus forces $\eta = 0$ (Thm.~\ref{thm:strong_cp_all_loops}). No axion.
\item \textbf{Screen-closure:} Overdetermined identities give $\chi^2_{\mathcal{M}}$ falsification test (Sec.~\ref{sec:psitomo-closure-theorems}).
\item \textbf{G--H$_0$ invariant:} $(H_0/M_P)^2 = \alpha^{57}$; exponent topologically forced (Appendix~\ref{app:O_alpha57_clean}).
\item \textbf{Clock coupling:} $k_\alpha = \alpha^2/(2\pi)$ from Schwinger + no-hidden-knobs (Appendix~\ref{app:P_kalpha_MR}).
\item \textbf{Majorana scale:} $M_R = M_P\alpha^3$ from determinant scaling (Appendix~\ref{app:P_kalpha_MR}).
\end{enumerate}

\textbf{Quantitative matches:}
\begin{itemize}[nosep, leftmargin=*]
\item $\alpha^{-1} = 137.036$ ($0.0056$~ppm, convention-locked)
\item Higgs: $v = M_P\alpha^8\sqrt{2\pi} = 246.09$ GeV (0.05\% error)
\item Fermion masses: 1.42\% leading-order mean error (9 particles)
\item CKM: $\lambda = 31\alpha = 0.2262$ (integer pattern; $\mathbb{C}P^2$ overlap motivation)
\item PMNS: Tribimaximal + corrections ($\sim$5\%)
\item CMB: $R = 2.34$, $\ell_1 = 220$ (third-peak height carried by the derived $\chi$-matter field, App.~\ref{app:chi-matter}; abundance normalization now theorem-grade, $\Omega_\chi h^2=0.118$, $-1.5\sigma$; see below)
\item UVCS test: $0.4\sigma$ agreement
\item ESPRESSO: $0.8\sigma$ agreement
\end{itemize}

\textbf{Named open items (no internal inconsistency, genuinely unfinished):}
\begin{itemize}[nosep, leftmargin=*]
\item \textbf{Dark-sector abundance normalization --- theorem-grade:} the $\chi$ relic abundance is derived as the finite $SU(2)_{60}$ CS/WZW vacuum Casimir expectation, $\Omega_\chi h^2=1.62\langle\theta^2\rangle=0.118$ ($-1.5\sigma$ from Planck; App.~\ref{app:chi-matter}, Thm~\ref{thm:chi-postinflation}); the former $\sim$9--44$\times$ overshoot was the classical-continuum measure $\pi^2/3$ on a finite topological Hilbert space, retired. The lone non-DFD input is the standard cosmological relic-redshift ($1.62$); optional refinement (not a gap): derive it natively.
\item \textbf{$Q=1$ $\chi$ linear clustering: resolved by an adopted postulate (cost: $+1$ axiom).} The single-$W$ action's screen $\mu=W'$ acts on $|\nabla\psi|$, so \emph{every} spatially-structured perturbation ($\nabla\phi\ne0$) is screened into the deep-MOND ($Q\gg1$) branch --- ``structured $\Leftrightarrow$ gradient $\Leftrightarrow$ screened'' is a \emph{theorem} of the single-$W$ action (four candidate rearrangements, incl.\ the polarization-current identity and the temporal flat direction, computed and shown to collapse to it; App.~\ref{app:growth-regime}). Because $Q_\chi=1$ therefore \emph{cannot} be a theorem of the bare action, DFD \emph{adopts} the minimal, equivalence-principle-safe two-channel fix --- the Rest-Mass Channel postulate (a second, $\mu{=}1$, rest-mass-sourced Poisson operator; \S\,App.~\ref{app:growth-regime}). \emph{Given} the postulate $Q_\chi=1$ holds, the linear clustering (third peak, lensing, $f\sigma_8$) is consistent, and a falsifiable cluster-vs-galaxy signature follows (which naturally explains MOND's cluster-mass failure). The cost is one added gravitational axiom; the residual \emph{refinement} (not an open obstruction) is to derive that postulate from a deeper principle, which would demote it to a theorem.
\item \textbf{$A_s$ prefactor:} the scalar amplitude $A_s = 32\pi\,\alpha^5$ has its $\alpha^5$ \emph{power forced} but its $32\pi$ coefficient \emph{asserted} (the $H_\star$--$M_R$ normalization), not derived from DFD geometry (App.~\ref{app:new-results}).
\end{itemize}

\textbf{One-parameter structure:} $k_{\max} = 60$, $N_{\rm gen} = 3$ (discrete input; Rem.~\ref{rem:ngen-input}) + $H_0$ (observed) $\Rightarrow$ all constants.
\end{tcolorbox}
\end{widetext}

