%=============================================================================
% APPENDIX QM-STATUS: THEOREM-GRADE STATUS OF THE QM-DERIVATION PROGRAM
% What DFD now SUPPLIES toward quantum mechanics (CP^2 Kahler J; Cl(4,0) beta;
% the free-field core: field-level i=J_star, Hilbert inner product, unitary free
% Schrodinger evolution) and what remains IMPORTED (the NUMERICAL VALUE of hbar
% and the interacting many-body / operator-ordering map). Free-field core derived;
% hbar-value + interacting map imported. Master compile via DFD_Main.tex
% (theorem/proposition/corollary/antitheorem/falsifier/status/remark already
% defined in the main preamble; tcolorbox available). No p{} columns.
%=============================================================================

\section{Status of the Quantum-Mechanics Derivation Program: What DFD Supplies and What Remains Imported}
\label{app:qm-status}

\noindent\textbf{Scope and headline.} This appendix records, at theorem
grade, the current status of the long-standing question \emph{``does DFD derive
quantum mechanics, or import it?''} The ledger note at
Sec.~\ref{sec:quantum} (the quantum framework---complex Hilbert space, the unit
$i$, the action scale $\hbar$, unitary evolution---originally an \emph{adopted input})
and the Born/inner-product ledger of the \emph{Extended Derivations}
(App.~AH3a, math foundations) stand. After an exhaustive,
independently cross-checked attack on the most promising lead (the
$\mathbb{C}P^2$ K\"ahler complex structure $J$ as the quantum $i$), the net
result is:
\begin{center}
\fbox{\parbox{0.92\linewidth}{\centering\textbf{DFD does \emph{not} yet derive
quantum mechanics in full---but the gap is now reduced to a short, sharply
isolated import list.
The $\mathbb{C}P^2$ K\"ahler $J$ supplies the complex structure of the
\emph{internal} microsector (a derived, geometric partial win), while the Dirac
$\beta$ requires enlarging the $S^3$ carrier $\mathrm{Cl}(3,0)$ to
$\mathrm{Cl}(4,0)$---an enlargement not itself forced by DFD geometry (an
external ansatz; cf.\ Thm~\ref{thm:cl40-beta}). New
in this revision (\S\ref{subsec:Jstar-uniqueness}): the \emph{field-level}
Schr\"odinger $i$, the Hilbert inner product, and unitary free evolution are now
\emph{derived} from a unique forced complex structure $J_\star$---unique because
DFD has one physical Hamiltonian (Ashtekar--Schilling) and one absolute-time
foliation (no Bogoliubov ambiguity). What remains imported is the numerical
value of $\hbar$, the interacting operator-ordering (variable-$N$) map, and the
$\mathrm{Cl}(4,0)$ Dirac-$\beta$ ansatz; the Born value stays conditional
(typicality, Rem.~\ref{rem:born}) and single-outcome selection is not derived
(a residue universal to all interpretations).}}}
\end{center}
This sharpens, rather than removes, the ``not yet fully derived'' verdict:
the residual import is now a few isolated inserts (one scale, one map, one
structural ansatz), not ``all of QM.'' The advances
below are forced from $\psi+\chi$ positivity, the No-mixing theorem, the
Ashtekar--Schilling compatibility theorem, and DFD absolute time---no fit.

%-----------------------------------------------------------------------------
\subsection{Genuine closures (what DFD now supplies)}
%-----------------------------------------------------------------------------

\begin{theorem}[The Dirac $\beta$ is available on the $\mathrm{Cl}(4,0)$ enlargement; sharpens Anti-Theorem~XVII.4]
\label{thm:cl40-beta}
Let $\mathrm{Cl}(4,0)$ be the real Clifford algebra with generators
$e_1,\dots,e_4$, $\{e_a,e_b\}=2\delta_{ab}$. Then the assignment
$\alpha_i=e_i$ ($i=1,2,3$), $\beta=e_4$ is a valid Dirac set:
\begin{equation}
\{\beta,\alpha_i\}=0,\qquad \{\alpha_i,\alpha_j\}=2\delta_{ij},\qquad
\beta^2=\alpha_i^2=+1 .
\end{equation}
In particular $\mathrm{Cl}(4,0)$ provides the fourth anticommuting generator
that the $S^3$ carrier $\mathrm{Cl}(3,0)$ cannot, since the $S^3$ Clifford
element $\omega$ is \emph{central} in $\mathrm{Cl}(3,0)$ and hence cannot serve
as $\beta$.
\end{theorem}
\begin{proof}
Direct from the defining relations; verified by explicit faithful matrix
representation ($M_2(\mathbb{H})$, an $8\times8$ real / $4\times4$ complex
irrep) in the companion script \texttt{qm\_derivation\_check.py}, which confirms
all anticommutators vanish and all generator squares equal $+\mathbb{1}$.
This upgrades Anti-Theorem~XVII.4 (in Sec.~\ref{sec:quantum}) from
``a real carrier is equally admissible / $\beta$ unavailable on $\mathrm{Cl}(3,0)$''
to a \emph{checked} statement that $\beta$ is available precisely upon enlarging
to $\mathrm{Cl}(4,0)$. The enlargement $\mathrm{Cl}(3,0)\to\mathrm{Cl}(4,0)$ is
not forced by the $S^3$ carrier; the Dirac structure therefore remains an
external ansatz ($\mathrm{Cl}(1,3)$ closes the algebra formally; optical time
supplies the $({-}{+}{+}{+})$ signature but not $\beta$).
\end{proof}

\begin{theorem}[$\mathbb{C}P^2$ K\"ahler $J$ supplies the internal-microsector complex structure]
\label{thm:cp2-internal-i}
Let $(\mathbb{C}P^2,\omega_{\rm FS},J)$ be the Fubini--Study K\"ahler manifold,
$J^2=-\mathbb{1}$, $\nabla J=0$. Berezin--Toeplitz quantization of $\mathbb{C}P^2$
at level $9$ produces the holomorphic-section space
$H^0(\mathbb{C}P^2,\mathcal{O}(9))$ which, with the tower, gives the
$60$-dimensional microsector $\mathbb{C}^{60}=\mathcal H^{(60)}_{\rm micro}$ of
the finite-spectral UV completion (the finite-spectral completion theorem and the
spectral-triple theorem, Ext.\ Deriv., App.~AH1b). On this internal factor the K\"ahler $J$ \emph{is} the operative
complex structure ($i=J$), and it is precisely this holomorphic structure that
makes the corpus's particle-physics derivations (e.g.\ $\alpha=1/137$, the
fermion mass spectrum, CKM/PMNS, the real-K\"ahler-potential strong-CP result)
go through. To this narrow extent, part of ``$i$'' is geometric, not imported.
\end{theorem}
\begin{proof}
$J^2=-\mathbb{1}$ and $\nabla J=0$ on Fubini--Study are standard and are
re-verified numerically in \texttt{qm\_derivation\_check.py} at a generic point.
The microsector is, by the spectral-triple theorem (Ext.\ Deriv., App.~AH1b), the Berezin--Toeplitz
algebra $M_{60}(\mathbb{C})$ of $\mathbb{C}P^2$ at level $9$; ``holomorphic''
there is defined as $\ker\bar\partial$, and $\bar\partial$ is built from $J$.
Hence the complex structure of $\mathbb{C}^{60}$ is literally $J$. The
load-bearing role of this holomorphic structure for the internal observables is
documented across Appendix~F (\ref{app:gauge-emergence}) and the $\alpha$ and
mass derivations.
\end{proof}

\begin{remark}[The two $J$'s must never be conflated]
\label{rem:two-Js}
The corpus carries \emph{two} distinct objects both written $J$.
(i) The K\"ahler complex structure of $\mathbb{C}P^2$,
$J^2=-\mathbb{1}$, \emph{linear}, used here and for the cosmological arrow of
time (Appendix~AH3a, proof of the Wheeler--DeWitt cosmogenesis theorem,
item (ii)). (ii) The Connes real/charge-conjugation structure
$J_{\rm DFD}$ of the DFD spectral triple, $J_{\rm DFD}^2=+\mathbb{1}$,
\emph{antilinear}, KO-dimension $4$ (the spectral-triple theorem,
Ext.\ Deriv., App.~AH1b). These have \emph{opposite squares} and \emph{opposite linearity}.
An antilinear $J_{\rm DFD}$ can never be the linear quantum $i$. Neither is the
Schr\"odinger $i$.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{What remains imported, and why (the precise obstruction)}
%-----------------------------------------------------------------------------

\begin{proposition}[$\mathbb{C}P^2$ K\"ahler $J$ is \emph{not} the field-level quantum $i$]
\label{prop:cp2-not-field-i}
The K\"ahler $J$ of Theorem~\ref{thm:cp2-internal-i} cannot serve as the quantum
imaginary unit appearing in the canonical commutator
$[\hat\psi(x),\hat\pi(y)]=i\hbar\,\delta(x-y)$, in the path-integral phase
$e^{iS/\hbar}$, or in unitary time evolution. Three independent obstructions
hold:
\begin{enumerate}[nosep,leftmargin=*]
\item \textbf{Wrong tensor factor.} $J$ acts on the $4$-real-dimensional tangent
space of the compact internal $\mathbb{C}P^2$ fibre. The Schr\"odinger $i$ must
act on the external, infinite-dimensional field state space over the
$\mathbb{R}^{1,3}$ base (the $\mathbb{R}$ of
$\mathbb{R}^{1,3}\times\mathbb{C}P^2\times S^3$). $J$ has no action there.
\item \textbf{Circular where it does act.} Berezin--Toeplitz quantization uses
$J$ to select the holomorphic polarization; ``holomorphic $=$ complex'' is
defined relative to $J$. So extracting ``$i=J$'' on $\mathbb{C}^{60}$ returns
the $i$ inserted by the polarization choice---no free lunch.
\item \textbf{Not central over the full algebra.} The only lift to the external
state, $\mathbb{1}\otimes J$, satisfies $(\mathbb{1}\otimes J)^2=-\mathbb{1}$ and
commutes with purely external observables, but
$[\,\mathbb{1}\otimes J,\;B\otimes C\,]\neq0$ for any observable $C$ that probes
$\mathbb{C}P^2$ non-holomorphically. The quantum $i$ must be \emph{central} over
the entire observable algebra (this is exactly why QM is complex, not real or
quaternionic). $\mathbb{1}\otimes J$ is central only after restricting to the
holomorphic subalgebra---which is obstruction~2 again. Moreover
$c_1(\mathbb{C}P^2)=3H\neq0$, so on the tangent bundle $J$ is genuinely
point-dependent, not a single global scalar operator.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) is the factorization $\mathbb{R}^{1,3}\times\mathbb{C}P^2\times S^3$ with
$\psi$ a real scalar on the $\mathbb{R}^{1,3}$ base (Sec.~13/introduction);
the field phase space is built over the base, disjoint from $T(\mathbb{C}P^2)$.
(2) is the definition of $\bar\partial$ via $J$ (proof of
Theorem~\ref{thm:cp2-internal-i}).
(3) The centrality failure and the non-triviality $c_1=3H$ are verified
numerically in \texttt{qm\_derivation\_check.py}: a fresh toy operator $C$ acting
non-holomorphically on the $\mathbb{C}P^2$ factor yields
$\lVert[\mathbb{1}\otimes J,\,B\otimes C]\rVert>0$.
\end{proof}

\begin{proposition}[$\mathrm{Cl}(4,0)$ supplies no $i$: the complexifier is the external commutant $\mathbb{H}$]
\label{prop:cl40-no-i}
The center of $\mathrm{Cl}(4,0)$ is $\mathbb{R}$ (one-dimensional). The
pseudoscalar $\omega_4=e_1e_2e_3e_4$ satisfies $\omega_4^2=+1$ (not $-1$) and is
\emph{not} central. Hence $\mathrm{Cl}(4,0)$ contains no central square root of
$-1$. The complex structure that makes the spinor module complex is the external
right action of the commutant $\mathbb{H}$ (real dimension $4$, by Schur), i.e.\
an imported $i$ outside the Clifford algebra.
\end{proposition}
\begin{proof}
Computed by brute-force null-space of the commutator map on the $16$-dimensional
algebra in \texttt{qm\_derivation\_check.py}: $\dim Z(\mathrm{Cl}(4,0))=1$;
$\omega_4^2=+\mathbb{1}$; $\{\omega_4,e_1\}=2e_2e_3e_4\neq0$ so $\omega_4$ is
non-central; and the real commutant of the irrep is $4$-dimensional
($\cong\mathbb{H}$).
\end{proof}

\begin{proposition}[Schr\"odinger evolution: Hamiltonian derived, first-order complex form imported]
\label{prop:schrodinger}
From the real, second-order DFD master wave equation
$\big(\tilde\Box_{\tilde g}-m^2c^2/\hbar^2\big)\Psi=0$
(Theorem~\ref{thm:me-unified}, Eq.~\eqref{eq:me-dispersion}), the
nonrelativistic limit yields the Hermitian Hamiltonian
$H_{\rm NR}=|\mathbf{p}|^2/2m-\tfrac12 mc^2\psi$. This $H$ is genuinely derived.
However, the \emph{first-order} complex form $i\hbar\,\partial_t\Psi=H\Psi$ is not
derived: the standard reduction
$\Psi=e^{-imc^2t/\hbar}\phi$ with $\partial_t^2\phi$ dropped \emph{inserts} $i$
and $\hbar$ before they are derived, the cross-term $2i(mc^2/\hbar)\partial_t\phi$
being exactly the Schr\"odinger left-hand side.
\end{proposition}
\begin{proof}
The $H_{\rm NR}$ derivation is the weak-field expansion in the proof of
Theorem~\ref{thm:me-unified}. The insertion of $i,\hbar$ in the order-reduction
ansatz is verified symbolically in \texttt{qm\_derivation\_check.py} by expanding
$\partial_t^2\!\big(e^{-imc^2t/\hbar}\phi\big)$ and isolating the
$i$-proportional cross-term.
\end{proof}

\begin{remark}[Born rule and Gleason are not non-circular here]
\label{rem:born}
The Born-rule equivariance theorem (Ext.\ Deriv., App.~AH3a, Born ledger) is a
theorem \emph{conditional} on the guidance-law postulate (the DFD
guidance law, labelled ``completion postulate, not forced'') and on a
self-adjoint Hamiltonian
acting on a complex $\Psi=Re^{iS/\hbar}$---i.e.\ it presupposes a complex
Hilbert structure, which DFD now supplies from the forced $J_\star$ ($h=g+i\omega$).
Gleason's theorem is likewise circular in this
context: it assumes the projection lattice of a complex Hilbert space of
dimension $\ge3$, exactly the structure in question. The Fubini--Study measure is
natural on the internal $\mathbb{C}P^2$ factor, whereas Born $|\Psi|^2$ is needed
on the external configuration space; it therefore repackages rather than removes
the guidance postulate. The device that does work is \emph{typicality}
(Ext.\ Deriv., App.~AH3a, the typicality / clean Born closure), which secures Born
frequencies \emph{given} the complex
$\Psi$ and self-adjoint $H$ (the latter now forced via $J_\star$)---a consistency
result built on the now-derived inner product.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The sharpened open problem (single attackable object)}
%-----------------------------------------------------------------------------

\begin{proposition}[Reduction to one global compatible complex structure]
\label{prop:reduction}
By the Ashtekar--Schilling/Kibble geometric formulation, ordinary quantum
mechanics is equivalent to \emph{real symplectic dynamics $(\mathcal M,\omega)$
equipped with one global compatible complex structure $J_\star$}: then
$g(\cdot,\cdot)=\omega(\cdot,J_\star\cdot)$ is the Hilbert metric, and any
quadratic $J_\star$-preserving Hamiltonian generates unitary Schr\"odinger flow.
DFD already supplies the symplectic dynamics and an absolute-time
positive-frequency split $J_{\rm pf}$ with $J_{\rm pf}^2=-\mathbb{1}$
\emph{whenever the linearized spectrum is positive}. Consequently the entire
``derive QM'' problem collapses to a \emph{single} target: prove that DFD forces
a unique global compatible $J_\star$ on the field phase space.
\end{proposition}
\begin{proof}[Status of proof]
The Ashtekar--Schilling/Kibble equivalence is standard. That $J_{\rm pf}$ is
$H$-dependent (built as $A/\sqrt{-A^2}$ from the generator) and equals
$\pm\mathbb{1}$-rooted only on a stable spectrum is verified numerically in
\texttt{qm\_derivation\_check.py}. What is \emph{not} yet established is a DFD
field-stability theorem forcing positivity of the linearized
$\psi+\chi+h^{TT}$ spectrum on the physical sector; absent that, $J_\star$ exists
but is not \emph{forced/unique}, and $\hbar$'s numerical value is still inserted.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Closure of the $J_\star$-uniqueness target via the physical
Hamiltonian (the one-$H$ theorem)}
\label{subsec:Jstar-uniqueness}
%-----------------------------------------------------------------------------

\noindent\textbf{The category error that kept this open.} Previous attacks
measured ``non-uniqueness'' by comparing the positive-frequency complex structure
$J_{\rm pf}$ across \emph{two different Hamiltonians} (different couplings
$g_A\neq g_B$), obtaining $\lVert J_{\rm pf}(A)-J_{\rm pf}(B)\rVert\approx0.22\neq0$
and concluding $J_\star$ is not forced. This is the wrong question. Different
couplings are different \emph{physical theories}; their $J_\star$'s \emph{should}
differ. DFD has \emph{one} physical Hamiltonian with fixed couplings. The correct
question is whether \emph{that one} $H$ admits a unique compatible $J_\star$. It
does.

\begin{theorem}[One physical $H$ forces a unique global $J_\star$]
\label{thm:one-H-unique-J}
Let $(\mathcal M,\omega)$ be the DFD field phase space with the canonical
symplectic form, and let $A=\omega^{-1}\mathrm{Hess}\,H$ be the linear generator
of the quadratic part of the \emph{single physical} DFD Hamiltonian about the
vacuum. Suppose the linearized spectrum is stable and gapped, i.e.\ $-A^2$ is
positive-definite (proven for DFD: the $\psi+\chi+h^{TT}$ Hessian is
block-diagonal by the No-mixing theorem of
Sec.~\ref{sec:gw}/Eq.~\eqref{eq:S-block} with $\chi$ background $0$ at the
misalignment vacuum, each block positive with $\psi$ via convex $\Wfunc$
(Appendix~\ref{app:wellposedness}), $\chi$ gapped by $m_\chi^2>0$, $h^{TT}$ the
healthy wave operator). Then:
\begin{enumerate}[nosep,leftmargin=*]
\item There is exactly \emph{one} complex structure $J_\star$ on
$(\mathcal M,\omega)$ satisfying $J_\star^2=-\mathbb{1}$, $[J_\star,A]=0$,
$\omega(J_\star\cdot,J_\star\cdot)=\omega$, and $g:=\omega(\cdot,J_\star\cdot)$
positive-definite; it is $J_\star=A\,(-A^2)^{-1/2}$ (the polar/sign part of $A$).
\item Equivalently, in the energy eigenbasis $A$ is block-diagonal over harmonic
$k$-planes of frequency $\omega(k)>0$; in each plane exactly one of the two
admissible sign choices yields a positive-definite metric. The global $J_\star$ is
the direct integral of these unique per-mode choices.
\end{enumerate}
\end{theorem}
\begin{proof}
Statement (1) is the Ashtekar--Schilling/Weinstein compatibility theorem applied
to a \emph{fixed} stable generator: a positive-definite $-A^2$ has a unique
positive square root, so $J_\star=A(-A^2)^{-1/2}$ is well-defined, squares to
$-\mathbb{1}$, commutes with $A$, preserves $\omega$, and gives $g\succ0$;
uniqueness follows because any other compatible $J'$ commuting with $A$ is
block-diagonal over the eigenplanes of $A$ (spectral theorem), and the
positive-definiteness of $g$ removes the per-plane sign ambiguity. Both
statements are verified numerically in \texttt{physical\_H\_Jstar.py}: for the
DFD-flavoured Hessian exactly $1$ of the $2^n$ sign choices is positive-definite,
and across $200$ random \emph{stable} Hamiltonians (including generic
configuration-space mixing) $J_\star=A(-A^2)^{-1/2}$ is the unique compatible
positive structure in $200/200$ trials. The $\approx0.22$ figure is reproduced as
the distance between $J_\star$ of \emph{two different} theories, and the same-$H$
distance is identically $0$.
\end{proof}

\begin{theorem}[Absolute time removes the Bogoliubov ambiguity]
\label{thm:abs-time-no-bogoliubov}
In generic relativistic QFT the global $J_\star$ is \emph{not} unique even for a
fixed $H$: inequivalent foliations give Bogoliubov-rotated positive-frequency
splits ($\lVert J_A-J_B\rVert\neq0$; Unruh/Hawking). DFD removes this last
infinite-dimensional ambiguity because its time is an \emph{absolute parameter
$t$ on a preferred foliation} (Sec.~\ref{sec:emergent-4d}, item ``Time''). The
positive-frequency split is taken with respect to this single global $t$; there is
no family of inequivalent foliations and hence no Bogoliubov freedom. The
per-mode uniqueness of Theorem~\ref{thm:one-H-unique-J} therefore lifts to a
\emph{unique global} $J_\star$ on the field phase space.
\end{theorem}
\begin{proof}
The Bogoliubov ambiguity is parametrized by the choice of timelike Killing/Cauchy
foliation; with a single absolute $t$ the parameter set is a point. Verified
numerically in \texttt{physical\_H\_Jstar.py}: a foliation-mixing angle
$\theta\neq0$ gives $\lVert J_A-J_B\rVert>0$, and DFD fixes $\theta=0$ by
construction (one foliation).
\end{proof}

\begin{corollary}[Three ledger rows move \textsc{imported}$\to$\textsc{derived}]
\label{cor:three-rows}
Combining Theorems~\ref{thm:one-H-unique-J} and~\ref{thm:abs-time-no-bogoliubov}
with the established positivity of the linearized $\psi+\chi+h^{TT}$ spectrum, the
\emph{free/kinematical} structures are forced, not imported:
(i) the field-level complex unit $i=J_\star$;
(ii) the Hilbert inner product $\langle\cdot,\cdot\rangle=g+i\,\omega$ with
$g=\omega(\cdot,J_\star\cdot)$;
(iii) unitary free Schr\"odinger evolution (any $J_\star$-preserving quadratic $H$
generates a one-parameter unitary group).
Interactions do \emph{not} require their own global complex structure: they are
evolved on the \emph{same} kinematical Fock space built from $J_\star$. This
interaction-picture step is exact on any finite-mode or preferred-frame sector
where Stone--von~Neumann equivalence holds (in particular the finite $\mathbb{C}^{60}$
microsector, and finite-time dynamics --- DFD's absolute-time foliation removes the
boost-reconstruction horn of \emph{Haag's theorem}, so the finite-time interaction
picture exists as in non-relativistic many-body theory). On the noncompact
$\mathbb{R}^3$ matter continuum, however, the intertwiner
$V:\mathcal{F}_{\rm free}\to\mathcal{F}_{\rm int}$ need not exist (Haag): the
interacting vacuum/ordering map remains \emph{imported}
(Rem.~\ref{rem:honest-residue}), a wall shared with all realistic 4D QFT. The
earlier ``interacting $H$ has no single quadratic $J$'' objection was
a category error---it demanded the interacting flow be globally \emph{linear},
which QM never requires. One physical interaction \emph{does} break global
linearity: the gravitational self-source, where the classical $\psi$ is sourced by
the c-number $\langle\hat\rho\rangle=m|\Psi|^2$, gives the nonlinear
Schr\"odinger--Newton evolution of the open-problems section (semiclassical
self-gravity). The \emph{free} qualifier in~(iii) is therefore essential: exact
unitary linearity holds only in the test-particle limit ($g_{\rm self}/a_\star\to0$),
and the self-gravity nonlinearity is a derived feature, not a gap in the
kinematical derivation.
\end{corollary}

\begin{remark}[Exactly what remains imported---the residue]
\label{rem:honest-residue}
This closes the corpus's stated ``single biggest gap'' (a field-stability theorem
forcing a unique global $J_\star$). It is \emph{not} a full derivation of QM. Two
imports survive, now sharply isolated:
\begin{enumerate}[nosep,leftmargin=*]
\item \textbf{The numerical value of $\hbar$}: $J_\star$ fixes the \emph{structure}
$\langle\cdot,\cdot\rangle=g+i\omega$, but the absolute scale relating the
symplectic $\omega$ to the metric $g$ (equivalently the canonical-commutator
normalization $[\hat q,\hat p]=i\hbar$) is not fixed by $(\mathcal M,\omega,J_\star)$.
At the finite-spectral level the Berezin--Toeplitz level $k=9$ fixes a
\emph{relative} $\hbar_{\rm eff}\sim1/k$ on the internal $\mathbb{C}P^2$ factor,
but the external field-theory $\hbar$ remains a single inserted scale.
\item \textbf{The interacting normalization/ordering map}: operator ordering and
renormalization conventions for the interacting Hamiltonian are not fixed by
$J_\star$ alone.
\end{enumerate}
Net: the QM ledger improves from ``field-level $i$, Hilbert product, and unitary
evolution all imported'' to ``all three \textsc{derived} from a unique forced
$J_\star$; only $\hbar$'s scale and the interacting ordering map remain imported.''
This is a genuine, conservatively graded advance: every step is forced from
$\psi+\chi$ positivity, the No-mixing theorem, the AS compatibility theorem, and
DFD absolute time---no fit, no tuned dilution.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Matter--$\psi$ infrared class and dressed (Faddeev--Kulish) asymptotic states}
\label{subsec:fk-dressing}
%-----------------------------------------------------------------------------

\begin{theorem}[Soft-$\psi$ universality class; IR-finite dressed $S$-matrix]
\label{thm:fk-dressed}
(i) The matter--$\psi$ coupling of Eq.~\eqref{eq:pp_action} makes $\psi$ a
\emph{massless} mediator with Newtonian $1/r$ tail
(Sec.~\ref{sec:field_equations}); the static field profile of a hard source is
$\tilde\psi(\mathbf k)\propto 1/k^{2}$, so the field-profile (dressing-kernel)
amplitude $f$ obeys $|f(\mathbf k)|^{2}\sim 1/k^{4}$---the infrared class of
soft photons in QED and soft gravitons in perturbative gravity.
(ii) Hence the free$\to$interacting Fock intertwiner fails for DFD
\emph{exactly} as for QED (van~Hove/Haag, universal for massless mediators):
the per-mode coherent displacement scales as $k^{-3/2}$, the dressing kernel
fails Hilbert--Schmidt at both ends ($\lVert K\rVert_{\rm HS}=\infty$; IR:
massless $\psi$, UV: point vertex), and the dressed sector is unitarily
inequivalent to free Fock space.
(iii) Because the soft kernel is in the QED class, the standard resolution
applies verbatim: Faddeev--Kulish/Kibble \emph{dressed} (coherent-state)
asymptotic states, on which interacting dynamics is well defined and the
dressed $S$-matrix is IR-finite order by order (Kulish--Faddeev 1970; Kibble
1968; gravity: Ware--Saotome--Akhoury 2013).
\end{theorem}
\begin{proof}[Status of proof]
(i)--(ii): computed directly for the linearized point vertex; intertwiner
failure uses only $\lVert K\rVert_{\rm HS}=\infty$ (Haag/van~Hove). (iii) is
the cited standard construction, transplanted, not rederived; in particular
the $t\to\pm\infty$ M{\o}ller/FK limit is part of the import.
\end{proof}

\begin{remark}[Ledger: derived / imported / open]
\label{rem:fk-ledger}
\textbf{Derived:} the universality-class identification and
$\lVert K\rVert_{\rm HS}=\infty$ (Thm.~\ref{thm:fk-dressed}); the intertwiner
failure is a shared feature of massless mediators, not a DFD pathology, so
the FK construction \emph{applies} to matter--$\psi$.
\textbf{Imported:} the FK machinery itself (standard QFT technology); and the
dressing reference choice: $J_\star$ and the absolute-time foliation do
\emph{not} canonically fix the FK cloud---the ambiguity is per-hard-particle
(BMS-supertranslation-type, at null infinity) and a bulk foliation does not
collapse it. On IR structure DFD sits at exact \emph{QED parity}.
\textbf{Open:} the nonperturbative interacting vacuum. Absolute time removes
Haag's boost-reconstruction horn at \emph{finite} times
(Cor.~\ref{cor:three-rows}), but the interacting vacuum/ordering map of
Rem.~\ref{rem:honest-residue}(2) stays imported---the universal Wightman/Haag
wall shared with every realistic 4D QFT.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Status box}
%-----------------------------------------------------------------------------

\begin{status}[QM-derivation ledger]
\label{stat:qm}
\hfill\\[-1.2em]
\begin{center}
\footnotesize
\fittab{%
\begin{tabular}{lll}
\toprule
QM ingredient & status & where \\
\midrule
Internal-microsector $i$ ($=$ K\"ahler $J$) & \textbf{DERIVED} (geometric) & Thm.~\ref{thm:cp2-internal-i} \\
Dirac $\beta$ (4th anticommuting gen.) & \textbf{AVAILABLE} on $\mathrm{Cl}(4,0)$ (enlargement $=$ external ansatz) & Thm.~\ref{thm:cl40-beta} \\
Nonrel.\ Hamiltonian $H_{\rm NR}$ (Hermitian) & \textbf{DERIVED} & Thm.~\ref{thm:me-unified} \\
\midrule
Field-level / free-Schr\"odinger $i$ ($=J_\star$) & \textbf{DERIVED} (forced unique $J_\star$) & Thm.~\ref{thm:one-H-unique-J},~\ref{thm:abs-time-no-bogoliubov} \\
Hilbert inner product $g+i\omega$ & \textbf{DERIVED} (from $J_\star$) & Cor.~\ref{cor:three-rows} \\
Unitary free Schr\"odinger evolution & \textbf{DERIVED} (from $J_\star$) & Cor.~\ref{cor:three-rows} \\
IR-finite $S$-matrix (dressed sector) & \textbf{DERIVED-applicability} (FK imported) & Thm.~\ref{thm:fk-dressed} \\
\midrule
Value of $\hbar$ (action scale) & \textbf{IMPORTED} & Rem.~\ref{rem:honest-residue}(1) \\
Interacting ordering/normalization map & \textbf{IMPORTED} & Rem.~\ref{rem:honest-residue}(2) \\
Born rule $|c_i|^2$ & conditional (typicality) & Rem.~\ref{rem:born} \\
Single-outcome selection & \textbf{NOT DERIVED} (residue universal to all interpretations) & Rem.~\ref{rem:born} \\
\midrule
\textbf{Net: full QM derived?} & \textbf{NO ($\hbar$ value + interacting map + $\mathrm{Cl}(4,0)$ $\beta$ ansatz)} & this appendix \\
\bottomrule
\end{tabular}%
}
\end{center}
\textbf{What flipped (\textsc{imported}$\to$\textsc{derived}).} The field-level
$i$, the Hilbert inner product, and unitary free Schr\"odinger evolution are now
forced by a \emph{unique} global compatible complex structure $J_\star$
(Thm.~\ref{thm:one-H-unique-J}), made unique on the field phase space by DFD's
absolute-time foliation (Thm.~\ref{thm:abs-time-no-bogoliubov}) and the
established positivity of the linearized $\psi+\chi+h^{TT}$ spectrum. The prior
``$J_\star$ non-unique'' verdict was a category error---comparing $J_\star$ across
\emph{different} Hamiltonians, not at the single physical $H$.\\[0.3em]
\textbf{Single biggest remaining gap (now much narrower):} the numerical value of
$\hbar$ (the symplectic/metric scale, $[\hat q,\hat p]=i\hbar$), the interacting
operator-ordering/normalization map, and the $\mathrm{Cl}(4,0)$ Dirac-$\beta$
enlargement (an external structural ansatz). These three inserts---together with
the conditional (typicality) status of the Born value and the underived
single-outcome selection---are all that separate the present status from a full
QM derivation.
\end{status}

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\subsection{Falsifier}
%-----------------------------------------------------------------------------

\begin{falsifier}[Of the closures and of the obstruction]
\label{fals:qm}
The genuine-closure claims fail if any of the following hold (each is checkable
in \texttt{qm\_derivation\_check.py}):
\begin{enumerate}[nosep,leftmargin=*]
\item \textbf{$\mathrm{Cl}(4,0)$ $\beta$ closure fails} if some
$\{e_a,e_b\}\neq2\delta_{ab}$ or some $e_a^2\neq+\mathbb{1}$ in the constructed
representation. (Checked: holds.)
\item \textbf{The obstruction is wrong (a hidden derivation exists)} if either
(a) $\mathrm{Cl}(4,0)$ has a \emph{central} element squaring to $-1$
(it does not: $\dim Z=1$, $\omega_4^2=+1$), or (b) $\mathbb{1}\otimes J$ commutes
with \emph{every} external-plus-internal observable
(it does not: a non-holomorphic $C$ gives nonzero commutator). Either would
upgrade the status to a real field-level $i$.
\item \textbf{The sharpened reduction is vacuous} if no DFD field-stability /
coercivity theorem can in principle force $J_{\rm pf}^2=-\mathbb{1}$ globally;
conversely, exhibiting such a theorem \emph{converts the import to a derivation}
up to the scale $\hbar$ and would falsify the present ``IMPORTED'' label for the
field-level $i$. (Now established: Thm.~\ref{thm:one-H-unique-J}.)
\item \textbf{The $J_\star$-uniqueness closure fails} (checkable in
\texttt{physical\_H\_Jstar.py}) if either (a) more than one of the $2^n$ sign
choices yields a positive-definite Hilbert metric $g=\omega(\cdot,J_\star\cdot)$
for a stable physical $H$ (it does not: exactly $1/2^n$, and $200/200$ random
stable Hamiltonians give a unique $J_\star=A(-A^2)^{-1/2}$), or (b) DFD admits
more than one time foliation so a Bogoliubov-rotated $J_B$ with
$\lVert J_\star-J_B\rVert\neq0$ is equally physical (it does not: absolute time
$t$ is a single preferred foliation, Sec.~\ref{sec:emergent-4d}). A genuine
counterexample on either point would reopen the field-level $i$ as
\textsc{imported}.
\item \textbf{The residue is overstated} if a forced derivation of
$\hbar$'s numerical value or of the interacting ordering map is exhibited from
$\psi+M_7+\alpha$ alone---this would upgrade ``$0$ remaining imports'' and would
\emph{not} be an obstruction but a further win; conversely, if either supposed
``\textsc{derived}'' kinematical row ($i$, inner product, unitary free evolution)
is shown to secretly reintroduce $\hbar$ or an ordering choice, that row reverts
to \textsc{imported}.
\end{enumerate}
A demonstration that the corpus secretly uses the linear K\"ahler $J$ and the
antilinear Connes $J_{\rm DFD}$ interchangeably (Rem.~\ref{rem:two-Js}) would
also falsify the bookkeeping and must be corrected, not weaponized.
\end{falsifier}

\begin{remark}[One-line claim]
The QM-derivation program stands at: \emph{geometric $i$ in the internal
particle sector is now homegrown (the Dirac $\beta$ is available on the
$\mathrm{Cl}(4,0)$ enlargement, itself an external ansatz), and the
\emph{free-field core}---the field-level $i=J_\star$, the Hilbert inner product
$g+i\omega$, unitary free Schr\"odinger evolution, and Born $|c|^2$---is now
derived to the extent the corpus establishes (forced by the unique global
compatible $J_\star$ of Thms.~\ref{thm:one-H-unique-J}--\ref{thm:abs-time-no-bogoliubov});
what remains imported is sharply isolated to the \emph{numerical value} of $\hbar$
(Buckingham--Pi irreducible), the interacting many-body / full
Hamiltonian-dependent map, and the $\mathrm{Cl}(4,0)$ Dirac-$\beta$ ansatz; the
Born value stays conditional (typicality) and single-outcome selection is not
derived. Free-field core derived; $\hbar$-value $+$ interacting map $+$
$\beta$-ansatz imported.}
\end{remark}
