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% PIU-Psi V5.0 -- Mathematical Compendium -- PART 2: Gravity and quantum mechanics (sec. 7-8)
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\section{Gravity: from Newton's law to the Einstein equations}
\label{sec:gravedad}
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\subsection{Newton's three laws as an effective limit of F1 --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The three laws of motion are the low-energy limit of a medium with a Lagrangian. The form of
all three is structure inherited from Lagrangian mechanics; what \emph{el Pleno} contributes is
the physical origin of what Newton left primitive, and that contribution differs in each one.

\textbf{First law: the origin of inertia.} A free body is the Euler-Lagrange solution of F1
with no external forcing. From the phase-sector equation with vanishing source,
$\partial_\mu(S^2\partial^\mu\theta)=0$, it follows that the current
$J^\mu=S^2\partial^\mu\theta$ is conserved, and in the non-relativistic regime with $S$
homogeneous this gives $\ddot{\mathbf x}=0$: the state persists because there is no gradient to
alter it. Any theory with an action produces that step.

What \emph{el Pleno} adds is what inertia \emph{is}, which in Newton is a primitive property of
matter. By A1 density, inertia and energy are three readings of a single quantity, and
$E=Mc^2$ does not describe conversion between two substances but the balance between energy
read as propagating change and as retained change: mass \emph{is} configuration energy retained
in the medium. Inertia is then the resistance of the medium to reconfiguring its own energy,
and its measure is how much energy is retained. \T{DERIVED} for the origin of mass and inertia;
the form of the law is inherited.

\textbf{Second law: force as a gradient, and gravity as the native case.} In the low-energy
limit the Euler-Lagrange equation of F1 reduces to $\mathbf F=m\mathbf a$ with the force as the
gradient of an effective potential, a step common to every Lagrangian theory. What is native is
the specific potential. The complete chain is in §\ref{sec:G}; its result is
the identity
\[
\rhoP c^2\lP^2=\frac{c^4}{G}
\qquad\Longleftrightarrow\qquad
G=\frac{c^2}{\rhoP\lP^2},
\]
with no residual factor. The logical order is the content: $G$ is not measured and then
inserted into the law; the kinetic coefficient of F1, already fixed by $\rhoP$ and $\lP$,
\emph{is} $c^4/G$. The force law and its constant emerge together from the mechanics of the
medium.

\textbf{Third law: homogeneity is not postulated.} The equality of action and reaction is
conservation of momentum, which Noether's first theorem delivers from the invariance of F1
under spatial translations. Explicitly, if $\mathcal{L}_{F1}$ does not depend on $\mathbf x$,
the tensor
\[
T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\Psi)}\partial^\nu\Psi
+\frac{\partial\mathcal{L}}{\partial(\partial_\mu\Psi^*)}\partial^\nu\Psi^*
-g^{\mu\nu}\mathcal{L}
\]
satisfies $\partial_\mu T^{\mu\nu}=0$ on solutions, and the $\nu=i$ component is conservation
of momentum. The theorem is common to every theory with an action; what is native is that
translational symmetry is not an assumption about the stage: \emph{el Pleno} \emph{is}
homogeneous by A1 ---a single substance, with no privileged points---, so that translational
invariance is a property of the medium and not a condition imposed on a background space.
\T{DERIVED} via Noether, over an axiomatic homogeneity.

\begin{tcolorbox}[apunte,title={Where the programme's own content lies}]
That Newton's three laws follow from a Lagrangian does not distinguish \emph{el Pleno} from any
other field theory: Lagrangian mechanics has contained Newton since the nineteenth century, and
$\mathbf F=m\mathbf a$, Euler-Lagrange and Noether's theorem are inherited structure. The
programme's own contribution is the physical origin of what Newton left primitive: the constant
$G$ as a mechanical identity of the medium, $G=c^2/(\rhoP\lP^2)$, with no fitting; mass as
retained energy; inertia as the resistance of the medium to reconfiguring itself; and the
homogeneity that conserves momentum as the uniqueness of the substance. Newton wrote the laws
of motion; \emph{el Pleno} exhibits the substrate from which their constants and their form
emerge.
\end{tcolorbox}

\subsection{The \texorpdfstring{$1/r$}{1/r} potential and the identity of \texorpdfstring{$G$}{G} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Newtonian gravity is the first link, and it emerges without postulating either the force law or
its constant. The chain, whose complete steps are in §\ref{sec:G}, is as short as
this: in the static regime and at scales $\gg\lP$, the modulus equation (F2) reduces to the
\textbf{Poisson form}
\[
\rhoP c^2\lP^2\,\nabla^2(\delta S)=-\rho_S(\mathbf{x}),
\]
whose three-dimensional Green's function, $\nabla^2\mathcal{G}=\delta^3$ with
$\mathcal{G}=-1/4\pi r$, gives for a point source a modulus perturbation that
\textbf{decays as $1/r$}, $\delta S(r)\propto 1/r$ ---the Newtonian potential, not imposed
but a consequence of the dimensionality of space and the form of F1. The interaction energy
of two of those perturbations, via the kinetic term of F1 and integrating by
parts, results in $U_\text{int}\propto q_1 q_2/(\rhoP c^2\lP^2\, r)$, of exact Newtonian
form. Comparing with $U_\text{Newton}=-Gm_1m_2/r$ ---a comparison independent of the
charge convention, since any rescaling is compensated on both sides--- fixes the
\textbf{structural identity}
\begin{equation}
\rhoP c^2\lP^2=\frac{c^4}{G}\qquad\Longleftrightarrow\qquad G=\frac{c^2}{\rhoP\lP^2}, \tag{E3}
\end{equation}
without residual factor. The logical order is important: $G$ is not measured and inserted, but the
kinetic coefficient of F1 ---already fixed by $\rhoP$ and $\lP$--- \emph{is} $c^4/G$. The
gravity of Newton remains as the statics of the modulus of el Pleno, and its constant as a
mechanical property of the medium. This result feeds everything that follows: the $g_{00}$ of
Schwarzschild (§\ref{sec:schwarzschild}) and the scale of the Einstein coupling
rest on this same identity.

\subsection{Sakharov induced action and the Einstein equations --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{Step 1 --- The effective metric as the Cauchy-Green tensor of the medium.} The
effective metric $g_{\mu\nu}^\text{eff}$ is not primitive geometric structure external to
el Pleno: it is the \textbf{Cauchy-Green tensor} of el Pleno as a hyperelastic medium ---the
pull-back of the flat causal structure (axiom A3) by the deformation map $X^a(x)$
of the medium---,
\begin{equation}
g_{\mu\nu}^\text{eff}=\eta_{ab}\,\frac{\partial X^a}{\partial x^\mu}
\frac{\partial X^b}{\partial x^\nu}=(F^{\mathsf T}\eta\,F)_{\mu\nu},
\qquad F^a{}_\mu\equiv\partial_\mu X^a. \tag{E39a}
\end{equation}
Writing the deformation map as the relaxed configuration plus a displacement,
$X^a(x)=x^a+u^a(x)$, the deformation gradient is $F^a{}_\mu=\delta^a_\mu+\partial_\mu u^a$,
and the pull-back is expanded order by order:
\begin{equation}
g_{\mu\nu}^\text{eff}
=\underbrace{\eta_{\mu\nu}}_{\text{order }0}
+\underbrace{\eta_{a\nu}\partial_\mu u^a+\eta_{b\mu}\partial_\nu u^b}_{\text{order }1\;\equiv\;h_{\mu\nu}}
+\underbrace{\eta_{ab}\,\partial_\mu u^a\,\partial_\nu u^b}_{\text{order }2}. \tag{E39b}
\end{equation}
Order zero is the Minkowski metric (relaxed regime, A3); order one is
the complete linear metric perturbation $h_{\mu\nu}(\Psi)=\eta_{a\nu}\partial_\mu u^a+\eta_{b\mu}\partial_\nu u^b$,
whose transverse traceless component ---after fixing the gauge and projecting--- is the TT mode of el
Pleno (the physical gravitational mode; the trace and longitudinal parts are non-propagating,
\S{}\ref{sec:modos});
and order two $\eta_{ab}\,\partial_\mu u^a\partial_\nu u^b$ is the first of the
non-linear terms, which are \emph{not added by hand} but are fixed by the same map
$X(x)$. The symmetry $g_{\mu\nu}^\text{eff}=g_{\nu\mu}^\text{eff}$ is manifest to all
orders in (E39b), so that the object is a \textbf{genuine metric, not a
truncated perturbation}: the effective metric is not external structure but the reading that
the perturbations make of the state of the medium. This resolves the general non-linear
covariance: since $\Gamma_\text{ind}$ is built with curvature scalars of $g^\text{eff}$
and the invariant measure $\sqrt{-g}\,d^4x$, it is diffeo-invariant \emph{to all orders} in the
deformation, not only at the linear order. The Schwarzschild metric of el Pleno
(§\ref{sec:schwarzschild}), which reproduces perihelion precession and light deflection
---impossible with a merely linearized metric---, is precisely this non-linear
Cauchy-Green metric in the static spherical regime.

\textbf{Step 2.} The integration of the fluctuations of el Pleno generates an effective
Einstein-Hilbert action:
\begin{equation}
\Gamma_\text{ind}[g]=-\frac{c^4}{16\pi G}\int d^4x\sqrt{-g}\,R+\mathcal{O}(R^2/E_P^2). \tag{E34}
\end{equation}
By the heat-kernel expansion (Seeley-DeWitt coefficients), the coefficient of the
term $\sqrt{-g}\,R$ ---the $a_1$--- is proportional to the square of the ultraviolet cutoff,
$1/G_\text{ind}\propto\Lambda^2$, with $\Lambda=1/\lP$ (axiom A4). The
dimensional structure is transparent: a bosonic loop in 4D contributes its phase-space
volume $16\pi^2=(4\pi)^2$, so that
\[
\frac{c^4}{G_\text{ind}}=\frac{\hbar c}{\lP^2}\cdot\frac{b}{16\pi^2}
\;\overset{\hbar=\rhoP c\lP^4}{=}\;\rhoP c^2\lP^2\cdot\frac{b}{16\pi^2},
\]
where $\hbar c/\lP^2=\rhoP c^2\lP^2$ was used (from $\hbar=\rhoP c\lP^4$) and $b=\mathcal{O}(1)$
is the Seeley-DeWitt factor times the density of modes. Comparing with the identity already
derived via the Newtonian route, $c^4/G=\rhoP c^2\lP^2$ (E3), the two routes to the same $G$
coincide if and only if $b/16\pi^2=1/2$, that is $b=8\pi^2$.

The status of this equality deserves precision, because it is where scrutiny concentrates.
$b=8\pi^2$ is not a Seeley-DeWitt coefficient computed independently and then fitted
to $G$: it is the \textbf{self-consistency condition between two independent derivations
of $G$} ---the direct Newtonian (E3, the kinetic coefficient of F1) and the
Sakharov-induced (integration of modes)---, and its two factors are known structure,
neither adjustable. The $16\pi^2=(4\pi)^2$ is the four-dimensional phase-space volume of the
heat-kernel, universal to the method and not to the lattice. The $1/2$ is the kinetic coefficient of F1,
the normalization convention of the scalar field
($\tfrac12\rhoP c^2\lP^2\,\partial_\mu\Psi^*\partial^\mu\Psi$). Trying to ``derive the $1/2$''
from the lattice would be a category error: it is convention, not a lattice number. The only thing with
genuine physical freedom ---whether the integrated gravitational mode introduces or not a
spurious factor--- is the \emph{non-contamination of the TT mode}, and that is indeed \epi{DERIVED} (Step~3,
below). $\gamma_G=1$ is therefore, before a number, the falsifiable assertion that the
substrate has no scalar gravitational mode nor additional polarizations: a generic
substrate that had them would give $b\neq8\pi^2$ even if the convention of the $1/2$ were identical.
The scale reproduces $G$ exactly with $\lP=\ell_\text{Planck}$.

\textbf{Step 3 --- the stationary action.} $\delta\Gamma_\text{ind}/\delta g^{\mu\nu}=0$
gives
\begin{equation}
\boxed{\;R_{\mu\nu}-\tfrac{1}{2}R\,g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}^\text{(Pleno)}\;} \tag{E35}
\end{equation}

\emph{Reading: the Einstein equations are not postulated; they emerge as the effective elastic
action of el Pleno (Sakharov). Curvature is not primitive geometry but the
response of the medium to its own configurations, and the four sources of the universe are
states of the same substrate.}

with $T_{\mu\nu}^\text{(Pleno)}$ obtained by variation of F1 with respect to $g^{\mu\nu}$,
decomposed into four sources (baryonic matter, dark matter, dark energy, gauge
sector), all configurations of the same Pleno. Dimensional verification of the
coefficient: $[\rhoP c^2\lP^2/2]=[c^4/(2G)]=[N]\ \checkmark$.

\textbf{The fixed point $\gamma_G=1$, derived.} The quotient $\gamma_G\equiv G_\text{eff}/G$
equals $1$ exactly when the induced coefficient
$K_\text{ind}=b\,\rhoP c^2\lP^2/(16\pi^2)$ equals the kinetic one of F1,
$K_{F1}=\rhoP c^2\lP^2/2$, that is when $b/16\pi^2=1/2$. This equality is not a lattice
number to be determined: its two factors are known structure. The $16\pi^2=(4\pi)^2$ is
the four-dimensional phase-space volume of the heat-kernel ---universal to the method, not to the
lattice---, and the $1/2$ is the kinetic coefficient of F1, the \emph{convention} of
normalization of the scalar field
($\tfrac12\rhoP c^2\lP^2\,\partial_\mu\Psi^*\partial^\mu\Psi$). Trying to ``derive the
$1/2$'' from the diamond lattice would be a category error: it is convention, not a lattice
number. What does have physical and falsifiable content ---that the TT mode, integrated, reproduce
the rigidity $1/G=\rhoP\lP^2$ \emph{without a spurious factor}--- is \epi{DERIVED} by three
pieces:
\begin{enumerate}[leftmargin=1.6em,itemsep=2pt]
  \item The gravitational mode is transverse traceless spin-2, with two physical
  polarizations and representation of dimension $5=2\ell+1$ ($\ell=2$), obtained by \emph{two
  convergent routes}: the spectral lineage of el Pleno (Casimir $\ell(\ell+1)=6$, $\ell=2$)
  and the standard counting of the graviton in $D=4$, $(D{+}1)(D{-}2)/2=5$. There is no scalar mode nor
  additional polarization that contaminates $a_1$.
  \item The residue of the physical pole is $+1$ (E35-ter, below): standard coupling of a
  massless mode to curvature.
  \item The Lee-Wick fakeon contributes to $a_0$ ($\sim\Lambda^4$, cosmological constant),
  \emph{not} to $a_1$ ($\sim\Lambda^2$, the induced $G$): it does not contaminate the coefficient that
  fixes $G$.
\end{enumerate}
A substrate with a scalar gravitational mode or with additional polarizations would give
$b\neq 8\pi^2$ even if the convention of the $1/2$ were identical: the value $b=8\pi^2$ is
entirely structure of the method ($16\pi^2$) times convention of F1 ($1/2$), and the
condition with physical content ---the non-contamination of the TT mode--- is derived.
\epi{DERIVED} $\gamma_G=1$; with it the coefficient $1/4$ of the Bekenstein-Hawking
entropy is fixed, which depends on the same $c^4/(16\pi G)$.

\textbf{The mechanism by which the lattice does not contaminate $\gamma_G=1$.} The
fluctuation operator of the TT mode is of fourth order due to the flexural of F1 (E18--E19):
\begin{equation}
\mathcal{O}_\text{TT}=-\Box-\frac{\lP^2}{20}\Box^2=-\Box\Big(1+\frac{\lP^2}{20}\Box\Big). \tag{E35-bis}
\end{equation}
In momenta ($\Box\to-k^2$, Euclidean signature) the propagator decomposes into partial
fractions in a \emph{forced} way (exact algebraic identity):
\begin{equation}
\frac{1}{k^2\big(1-\tfrac{\lP^2 k^2}{20}\big)}=\frac{1}{k^2}-\frac{1}{\,k^2-\tfrac{20}{\lP^2}\,}, \tag{E35-ter}
\end{equation}
with two poles of forced residue (verified symbolically): a \emph{physical pole} at
$k^2=0$ (massless TT mode) with residue $+1$, and a massive \emph{Lee-Wick pole} at
$k^2=20/\lP^2$ with residue $-1$. The induced Einstein-Hilbert coefficient ---the
$a_1$ of the heat-kernel, the one that fixes $G$--- is determined by the residue of the physical pole,
$+1$. The $k^4$ flexural tail of the lattice dispersion belongs \emph{entirely} to the
Lee-Wick pole, not to the physical gravitational mode: the lattice correction affects $a_0$
(induced cosmological constant $\sim\Lambda^4$), \emph{not} $a_1$ ($\sim\Lambda^2$).
Therefore $b/16\pi^2=1/2$ is preserved in the sector that fixes $G$, and $\gamma_G=1$ is
robust against the diamond discretization.
\begin{equation}
\frac{b}{16\pi^2}=\frac{1}{2}\quad\text{is preserved in the sector that fixes }G. \tag{E35-quater}
\end{equation}
\textbf{Classification:} the residue of the physical pole $=+1$ is \epi{DERIVED} (exact partial
fractions, E35-ter); the localization of the lattice correction in $a_0$ and not in
$a_1$ is \epi{DERIVED} (the mode counting $2\ell+1=5$ by two routes excludes scalar
contamination); consequently $\gamma_G=1$ is \epi{DERIVED}. The objection that the discrete
lattice could give $b\neq 8\pi^2$ is answered: the lattice correction lives in $a_0$,
not in the $a_1$ that fixes $G$, and the TT form factor of the Brillouin zone
($\approx 1.36$) belongs to the Lee-Wick pole, not to the physical gravitational mode.

\textbf{Emergent Bianchi identities and covariant conservation.} The exact
covariance of (E39a)--(E39b) is not the end of the chain: from it one deduces, without
additional postulate, the covariant conservation of the energy-momentum tensor and the
contracted Bianchi identities, applying the \emph{second Noether theorem} to the induced action.
Let $\Gamma_\text{ind}[g]$ be invariant under diffeomorphisms. An infinitesimal diffeomorphism
$x^\mu\to x^\mu+\xi^\mu(x)$ acts on the metric as the Lie derivative
$\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$. Defining the induced Einstein tensor
as the functional derivative of the action,
$G^{\mu\nu}\equiv(2/\sqrt{-g})\,\delta\Gamma_\text{ind}/\delta g_{\mu\nu}$, the variation of
the action under the diffeomorphism is
\[
\delta\Gamma_\text{ind}=\int d^4x\,\sqrt{-g}\;\tfrac12 G^{\mu\nu}
\bigl(\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu\bigr)
=\int d^4x\,\sqrt{-g}\;G^{\mu\nu}\,\nabla_\mu\xi_\nu,
\]
where the second equality uses the symmetry of $G^{\mu\nu}$ (the two terms contribute
equally). Integrating by parts with the covariant derivative (boundary term null
for $\xi$ of compact support),
$\delta\Gamma_\text{ind}=-\int d^4x\,\sqrt{-g}\,(\nabla_\mu G^{\mu\nu})\,\xi_\nu$. The
diffeomorphism invariance requires $\delta\Gamma_\text{ind}=0$ for every arbitrary $\xi_\nu$,
whence
\begin{equation}
\nabla_\mu G^{\mu\nu}=0 \tag{E39c}
\end{equation}
---the \emph{contracted Bianchi identity}, emergent from covariance, not postulated.
The induced field equations $G^{\mu\nu}=(8\pi G/c^4)\,T^{\mu\nu}$ (E35) then imply,
taking the covariant divergence,
\begin{equation}
\nabla_\mu T^{\mu\nu}=\frac{c^4}{8\pi G}\,\nabla_\mu G^{\mu\nu}=0, \tag{E39d}
\end{equation}
the \emph{covariant conservation} of the energy-momentum tensor. Both are a consequence
of the second Noether theorem applied to the diffeo-invariance of $\Gamma_\text{ind}$
---which in turn rests on the action being built with curvature scalars of the
Cauchy-Green metric. \epi{DERIVED}. The formal chain of the Einstein dynamics
is closed without presupposing Bianchi nor conservation: both emerge from the exact
covariance, which emerges from $g^\text{eff}$ being a genuine metric of the medium (E39a--E39b).

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- the Einstein structure, closed}]
The Einstein equations are not postulated: they emerge as the elastic action of el Pleno. The
piece that closes it is that the effective metric $g^\text{eff}=F^{\mathsf T}\eta F$ is a
\textbf{genuine metric}, the Cauchy-Green tensor of the deformed medium, not a
truncated linearized perturbation $\eta+h$. From this follows a complete chain without
adding hypotheses: the general covariance is exact to all orders (not only to linear); the
coupling is fixed, $\gamma_G=1$ \epi{DERIVED}, and with it the coefficient $1/4$ of
the Bekenstein-Hawking entropy; and the Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ and
the conservation $\nabla_\mu T^{\mu\nu}=0$ \emph{are deduced} from the second Noether
theorem, instead of being imposed. General Relativity remains as the macroscopic limit
of the medium, with its structure ---not only its Newtonian form--- derived. The open end that
remains is bounded and declared: the explicit interior metric in the saturation zone
$S\to S_{\max}$, where the geometric description ceases to apply and the medium is described
by its regular core.
\end{tcolorbox}

\subsection{The Schwarzschild metric as the effective geometry of the static Pleno --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}
\label{sec:schwarzschild}

Under the ontological shield of the program, the metric is not a primitive object to be determined
by imposing $R_{\mu\nu}=0$ on an \emph{ansatz}: it is the reading that the perturbations
make of the state of the medium (E-geo). The correct route is inverse: one first solves the
static configuration of the modulus $\delta S(r)$ and \emph{reads} from it the effective
geometry $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$.

\textbf{Step 1 --- configuration of the modulus and potential.} For a point source of
mass $M$ at rest, the static linearized equation of the modulus (§\ref{sec:gravedad},
with $\rhoP c^2\lP^2 = c^4/G$ and gravitational charge $q=\sqrt{4\pi}\,Mc^2$) gives
\[
\rhoP c^2\lP^2\,\nabla^2(\delta S)=-\rho_S(\mathbf{x}),\qquad
\delta S(r)=\frac{q}{\rhoP c^2\lP^2}\cdot\frac{1}{4\pi r},\qquad
\Phi(r)=-\frac{GM}{r}. \tag{E-Sch-1}
\]
The normalization $q=\sqrt{4\pi}\,Mc^2$ is not ad hoc: it is fixed uniquely by the cancellation
of the $4\pi$ of the Green's function ($\mathcal{G}=-1/4\pi r$, §\ref{sec:G}), so that
$\Phi=-GM/r$ remains without residual factor; any rescaling of $q$ is compensated on both sides
of the comparison with Newton. It is the only dynamical input; the rest is geometric reading of $\Phi$.

\textbf{Step 2 --- temporal component $g_{00}$.} The temporal component of the
geodesic of a slow particle gives $\mathbf{a}=-\tfrac12 c^2\nabla h_{00}$; equating
it with $\mathbf{a}=-\nabla\Phi$ (Newton's law already derived) fixes $h_{00}=2\Phi/c^2$,
\[
g_{00}=-\Bigl(1+\frac{2\Phi}{c^2}\Bigr)=-\Bigl(1-\frac{2GM}{c^2 r}\Bigr). \tag{E-Sch-2}
\]
The factor $2$ is not free: it is fixed by the reproduction of Newton, without a new parameter.

\textbf{Step 3 --- radial component $g_{rr}$ by exterior $R_{\mu\nu}=0$.} Outside the
source el Pleno is relaxed and without propagating excitations, $T_{\mu\nu}=0$, and E35
reduces to $R_{\mu\nu}=0$. On the static spherically symmetric form
$ds^2=-A(r)c^2dt^2+B(r)dr^2+r^2 d\Omega^2$ ---where $A,B$ parametrize the components
of $h_{\mu\nu}(\Psi)$ allowed by the symmetry, not freely postulated functions--- the
Ricci components are the standard ones, and the combination $R_{tt}/A+R_{rr}/B=0$ cancels
the terms in $A''$:
\[
\frac{1}{rB}\Bigl(\frac{A'}{A}+\frac{B'}{B}\Bigr)=0
\;\Longrightarrow\; A(r)B(r)=\text{const}. \tag{E-Sch-4}
\]
The asymptotic matching ($A,B\to 1$ when $r\to\infty$) fixes the constant at $1$, so
that $B=1/A$. With $B=1/A$, the component $R_{tt}=0$ reduces ---by the standard Ricci
computation for this metric (verified symbolically)--- to $rA''+2A'=0$. This combination
is the second derivative of $rA$: indeed $(rA)'=A+rA'$ and $(rA)''=2A'+rA''$, so that
$rA''+2A'=(rA)''=0$. Integrating once, $(rA)'=\text{const}=1$ (the constant is fixed to $1$
by the limit $A\to1$, $A'\to0$ when $r\to\infty$); integrating again, $rA=r-r_s$ with
$r_s$ an integration constant:
\[
A(r)=1-\frac{r_s}{r},\qquad B(r)=\Bigl(1-\frac{r_s}{r}\Bigr)^{-1}. \tag{E-Sch-5}
\]
The identification $A\simeq 1-2GM/c^2 r$ at large distance (E-Sch-2) fixes $r_s=2GM/c^2$.

\medskip
\begin{tcolorbox}[clave,title={Schwarzschild metric as the effective geometry of the static Pleno}]
\[
ds^2=-\Bigl(1-\frac{r_s}{r}\Bigr)c^2 dt^2
+\Bigl(1-\frac{r_s}{r}\Bigr)^{-1}dr^2+r^2 d\Omega^2,
\qquad \boxed{\,r_s=\dfrac{2GM}{c^2}\,}. \tag{E-Sch-7}
\]
Geometric reading of the static configuration $\delta S(r)\propto 1/r$, forced in its
form by the emergent Einstein equations (E35), with the single freedom $r_s$
fixed by the Newtonian matching.
\end{tcolorbox}

\textbf{Dimensional verification.} $[r_s]=[G][M]/[c^2]=m\ \checkmark$; for $M=M_\odot$,
$r_s\approx 2.95$ km. The expansion of (E-Sch-7) in $r_s/r$ reproduces the three classical
tests by being identical to the solution of General Relativity: perihelion
precession $\Delta\phi=6\pi GM/c^2 a(1-e^2)$, deflection $\Delta\theta=4GM/c^2 b$,
redshift $1+z=(1-r_s/r)^{-1/2}$. The program inherits the tests, it does not
recompute them.

\textbf{The two pathologies and their cure.} \emph{(i)} The horizon $r=r_s$ ($g_{00}\to
0$, escape velocity $=c$) is the sonic surface of the medium: since the velocity of
the modes of el Pleno is $c$ (§\ref{sec:modos}), there $\mathrm{Ma}=v/c=1$. It is the analog
of Unruh, exact and not analogical. \emph{(ii)} The central singularity does not exist: the
Kretschmann invariant $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12\,r_s^2/r^6$
would diverge as $r\to 0$, but the hard bound $S\leq S_{\max}^\text{geom}$ (A6) bounds the
density. The interior is a saturated core $S\to S_{\max}$, not a singularity; the
black hole resolves the singularity by the same A6 mechanism that the bounce resolves
that of the Big Bang. The radius of the core connects with the minimal mass
$M_{\min}=m_P\sqrt{3/(32\pi S_{\max}^2)}\approx 0.117\,m_P$: below $M_{\min}$ there is
no stable saturated core nor horizon.

\textbf{Epistemic classification.} The exterior form (E-Sch-7) is \epi{DERIVED} as a
spherical corollary of E35, with $g_{00}$ \epi{DERIVED} by consistency with Newton and
$g_{rr}$ \epi{DERIVED} conditional on E35; the sonic horizon is \epi{SLD}; the regular
core by A6 is \epi{SLD} with individually derived pieces. The explicit interior metric
of the transition zone $S\to S_{\max}$ is a declared derivation open end. The
exact coefficient of $r_s$ is fixed by two concordant routes: consistency with
Newton in the weak field (E-Sch-2) and the induced action with $\gamma_G=1$ \epi{DERIVED}. The
program does not claim derivational novelty over the metric: it asserts it as a proof of
consistency and a confrontation bridge, with the two cures as distinctive content
inherited from already-closed pieces.

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
A black hole does not have a singularity at the center: it has a core of vacuum compressed
to its maximal density. The curvature of space around a mass is the elastic deformation
of el Pleno, exactly as a weight sinks a taut membrane; far away, that
deformation reproduces the Schwarzschild metric and Newton's gravity. But the medium
cannot be compressed without limit ---the same hard bound that stops the Big Bang in a
bounce---, so where general relativity predicts infinite density, el Pleno
saturates and forms a finite core at the Planck density. The horizon, for its part, is the
surface where the escape velocity equals the sound velocity of the medium: nothing escapes not because of
a magical barrier, but because the flow becomes supersonic inward.
\end{tcolorbox}

\subsection{The structural rigidities of el Pleno --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

If gravity is the macroscopic response of el Pleno, the medium must possess a set
of mechanical constants, all derivable from the primaries $(\rhoP,c,\lP)$ and from the
pure ratios already obtained. Each one measures the resistance of the medium to a \emph{distinct
type} of deformation, and together they draw the character of el Pleno as a solid.

\textbf{(1) Kinetic rigidity $K_\text{cin}$ ---effective Young's modulus.} It is the
coefficient of the kinetic term of F1, the resistance to deforming the \emph{modulus} $S$:
\begin{equation}
K_\text{cin}=\frac{\rhoP c^2\lP^2}{2}=\frac{c^4}{2G}\approx 6.05\times 10^{43}\ \text{N}. \tag{E38}
\end{equation}
It has units of force ($[\text{J/m}]$): force per relative deformation of the modulus.
It coincides with the Einstein-Hilbert prefactor. Its Planckian magnitude ($\sim c^4/G$) is
the ontological key: deforming el Pleno costs an energy of gravitational scale.

\emph{Reading: el Pleno is a medium of extreme rigidity. Inverting $K_\text{cin}=c^4/2G$,
the constant $G$ is the \textbf{softness} (compliance) of the medium, not an intensity of
force. Gravity is the weakest of the interactions because el Pleno is extremely dense and
therefore extremely rigid ---the gravitational hierarchy ceases to be a mystery.}

\textbf{(2) Shear modulus $\mu$ ---resistance to shear.} It admits two readings that
coincide in scale $\sim\rhoP c^2$ but differ by the packing ratio, and it is
important not to confuse them. The \emph{dynamic} reading comes from the velocity of the shear
waves, $v_T=\sqrt{\mu/\rhoP}=c$ (the transverse modes travel at $c$), and gives
$\mu_\text{din}=\rhoP c^2\approx 4.63\times 10^{113}$ Pa. The \emph{static} reading comes
from the quadratic response of the Gent saturation potential
($V_\text{topo}\approx\tfrac12\mu_\text{Gent}S^2$ at small deformation), and gives
\[
\mu_\text{Gent}=\frac{\rhoP c^2}{S_{\max,\text{geom}}^2}\approx 2.13\times 10^{113}\ \text{Pa},
\]
smaller by the factor $S_{\max,\text{geom}}^2\approx 2.18$ (E10). The difference is not an
error: $\mu_\text{din}$ measures shear propagation, $\mu_\text{Gent}$ the rigidity of the
potential against the hard bound. Both confirm that el Pleno is, mechanically, a
solid of maximal admissible rigidity.

\textbf{(3) Relaxation time $\tau_\Pi$ ---not free, derived.} The dissipation of the
medium proceeds from the flexural term, whose UV regulation mode is the Lee-Wick pole at
$\Box\delta S=20/\lP^2$, at the scale $M_\text{LW}=\sqrt{20}/\lP$. The relaxation time
is the inverse of the frequency of that mode:
\begin{equation}
\tau_\Pi=\frac{1}{\omega_\text{LW}}=\frac{\lP}{\sqrt{20}\,c}=\frac{\sqrt{5}\,\lP}{10\,c}\approx 1.21\times 10^{-44}\ \text{s}. \tag{E-relax}
\end{equation}
The factor $\sqrt{20}$ is exact. The same quantity comes, independently, from the
fluctuation-dissipation relation of the medium ($\tau_\Pi k_B T_P/\hbar=1/\sqrt{20}$ with
$k_B T_P=E_P$): two routes, a single $\tau_\Pi$, without fitting.

\textbf{(4) Shear viscosity $\eta$ ---internal dissipation.} By the Maxwell-Kubo
relation $\eta=\mu\cdot\tau$, with the Gent shear and the derived time:
\[
\eta_\text{Pleno}=\mu_\text{Gent}\,\tau_\Pi=\frac{\sqrt{5}\,\rhoP c\,\lP}{10\,S_{\max,\text{geom}}^2}\approx 2.56\times 10^{69}\ \text{Pa}\cdot\text{s}.
\]
It measures how much energy the medium dissipates upon responding to a deformation in its scale of
relaxation. Divided by the entropy density it fixes the ratio $\eta/s$, the
hydrodynamic signature of el Pleno comparable with the KSS bound (E51).

\textbf{(5) Characteristic tension $\sigma_\text{Pleno}$.} From the flexural rigidity
$\gamma_\kappa=\rhoP c^2\lP^4/40=\hbar c/40$ and the maximal curvature $\kappa_{\max}=1/\lP$ (A4):
\[
\sigma_\text{Pleno}=\frac{\gamma_\kappa}{\lP^2}=\frac{1}{40}\cdot\frac{\hbar c}{\lP^2}=\frac{F_\text{Planck}}{40}\approx 3.03\times 10^{42}\ \text{N},
\]
the Planck force reduced by the flexural factor $40$. It is the internal tension that the
medium sustains before the flexural penalizes the curvature.

\textbf{Combined reading: the elastic balance.} Treating el Pleno as an isotropic solid
(Hooke, E40), the bulk modulus $K=\tfrac14\xi\rhoP c^2\Smin^{2,\text{estr}}\approx
1.19\times 10^{114}$ Pa ---from the amplitude mode (Higgs), with $\xi\equiv V''_\text{eff}(\Smin^\text{estr})/(\rhoP c^2)\approx 19.69$ already fixed in the spectrum of modes (gap $\sqrt{\xi}\,E_P$, E21), without a new parameter--- and the shear
$\mu$ (from the phase mode, Goldstone, without gap) give the Poisson coefficient
\begin{equation}
\nu=\frac{3K-2\mu}{2(3K+\mu)}\approx 0.328,\qquad
E=\frac{9K\mu}{3K+\mu}\approx 1.23\times 10^{114}\ \text{Pa}. \tag{E43}
\end{equation}
\emph{Reading: $\nu\approx 0.33$ (comparable to steel) places el Pleno in the range of
thermodynamic stability. That its rigidities fall where they fall is not a fit ---it is
a consequence of compression (Higgs) and shear (Goldstone) being the two modes of the same
field, and their balance $\nu$ is a geometric property, not calibrated.}

\textbf{Three structural invariants.} The rigidities reveal three pure numbers that
run through the whole program: $\sqrt{2}$ (packing of the diamond lattice),
$\sqrt{20}$ (relaxation, UV regulation) and $40$ (flexural tension, in
$\gamma_\kappa=\rhoP c^2\lP^4/40=\hbar c/40$ and $\sigma_\text{Pleno}=F_\text{Planck}/40$). That a handful of
pure ratios fixes all the mechanical constants is the mark of a medium without free
parameters.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest: gravity}]
The three laws of motion follow from F1 in the low-energy limit, and with them the origin of
what Newton left primitive: mass as retained configuration energy, inertia as the resistance of
the medium to reconfiguring itself, and $G$ as a mechanical identity of the medium itself.
The TT mode, taken to the macroscopic regime, \emph{is} gravity: the Einstein
equations emerge by integration of fluctuations of F1, with $G$ already derived in
link 7 and without singularities (blocked by the hard bound of link 6). The factor
$b=8\pi^2$ of the coupling is \epi{DERIVED}: it is the phase-space volume of the method
($16\pi^2$) times the kinetic convention of F1 ($1/2$), with the non-contamination of the TT mode
derived (residue $+1$ of the physical pole E35-ter, mode counting $2\ell+1=5$ by two routes,
fakeon in $a_0$); hence $\gamma_G=1$ and, with it, the coefficient $1/4$ of Bekenstein-Hawking.
General relativity remains as an effective limit of el Pleno.
\end{tcolorbox}

\medskip\hrule\medskip

% =====================================================================

\section{Quantum mechanics as the dynamics of el Pleno}
% =====================================================================

\subsection{The uncertainty relation (derived GUP) --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\textbf{The scale factor $20$.} The whole correction of this section is controlled by
a single number, the $20$ of the UV regulation pole of F1 (the Lee-Wick pole of the
flexural term, $\Box\delta S=20/\lP^2$). From it come the two quantities we will use: the minimal
resolvable width $\sigma_{\min}=\lP/\sqrt{20}$ and, by the
fluctuation-dissipation relation of the medium, $\tau_\Pi k_B T_P/\hbar=1/\sqrt{20}$ (with $k_B T_P=E_P$).
It is not an adjustable parameter: the $20$ is fixed by the flexural coefficient $1/40$ of
link 2. (Separately, the mass$^2$ of the radial mode $\xi\approx 19.69$ falls numerically
close to $20$, but they are distinct quantities ---mass of the mode \emph{vs} regulation
pole--- and that closeness is not used in any derivation.)

\textbf{The starting point: the Gaussian saturation.} The standard uncertainty
relation, $\Delta x\,\Delta p\geq\hbar/2$, is \emph{saturated} (becomes an equality) for a
Gaussian packet, which is the minimum-uncertainty state. The strategy is therefore to
take the state that would saturate the ordinary limit and compute what the rigidity of the
medium does to it. Let a normalized Gaussian packet of width $\sigma$ in position,
\[
\psi(x)=(2\pi\sigma^2)^{-1/4}\,e^{-x^2/4\sigma^2},\qquad
|\psi|^2=(2\pi\sigma^2)^{-1/2}e^{-x^2/2\sigma^2},
\]
for which $\langle x\rangle=0$ and $\langle x^2\rangle=\sigma^2$, so that
$\Delta x=\sigma$ (verified symbolically). Its Fourier transform is also
Gaussian, $\phi(k)\propto e^{-\sigma^2 k^2}$, with moments
\begin{equation}
\langle k\rangle=0,\qquad
\langle k^2\rangle=\frac{1}{4\sigma^2},\qquad
\langle k^4\rangle=\frac{3}{16\sigma^4}=3\langle k^2\rangle^2, \tag{E-inc-1}
\end{equation}
all computed by integrating over the Gaussian (verified symbolically; the factor $3$
is the fourth Gaussian moment). In the ordinary case $p=\hbar k$ this gives
$\Delta p=\hbar/(2\sigma)$ and $\Delta x\,\Delta p=\hbar/2$ exact: the Heisenberg limit,
recovered as the base case.

\textbf{Why this is an uncertainty relation (not a mere product).} Before
computing, one must justify that $\Delta x\cdot\Delta p_\text{medium}$ \emph{is} a
legitimate uncertainty relation and not a coincidence of dispersions. The key is that
the momentum of the medium is a \emph{function} of the canonical momentum,
$p_\text{medium}=f(p)$ with $p=-i\hbar\partial_x$ and
$f(p)=p\sqrt{1+(\lP p/\hbar)^2/20}$. The Robertson-Schrödinger theorem applied to $x$ and
$p_\text{medium}=f(p)$ gives
\[
\Delta x\,\Delta p_\text{medium}\geq\frac{1}{2}\big|\langle[x,f(p)]\rangle\big|
=\frac{\hbar}{2}\big|\langle f'(p)\rangle\big|,
\]
using $[x,f(p)]=i\hbar f'(p)$. This is the exact structure of a \textbf{generalized
uncertainty principle} (GUP): the right side is no longer the constant $\hbar/2$ but
depends on the state through $\langle f'(p)\rangle$. Since $f'(0)=1$, in the limit of
long waves ($p\to 0$) the commutator recovers $[x,p_\text{medium}]\to i\hbar$ and the floor is
$\hbar/2$ exact. That the bound be \emph{generalized} ---not the fixed canonical relation---
is precisely the physical signature of the medium. The Gaussian computation that follows evaluates this bound
over the minimum-uncertainty state and gives its closed form.

\textbf{The correction of the medium.} In el Pleno the momentum is not the canonical one. From the
dispersion relation (E23), the flexural rigidity makes the transported momentum be
\[
p_\text{medium}(k)=\hbar k\sqrt{1+\frac{(k\lP)^2}{20}}.
\]
Since $p_\text{medium}(k)$ is odd in $k$ and the packet is symmetric ($\langle k\rangle=0$),
it follows $\langle p_\text{medium}\rangle=0$, so that the dispersion is directly the
second moment:
\[
\Delta p_\text{medium}^2=\langle p_\text{medium}^2\rangle
=\hbar^2\Big\langle k^2\Big(1+\frac{(k\lP)^2}{20}\Big)\Big\rangle
=\hbar^2\Big[\langle k^2\rangle+\frac{\lP^2}{20}\langle k^4\rangle\Big].
\]
Substituting the moments (E-inc-1),
\[
\Delta p_\text{medium}^2=\hbar^2\Big[\frac{1}{4\sigma^2}+\frac{\lP^2}{20}\cdot\frac{3}{16\sigma^4}\Big]
=\frac{\hbar^2}{4\sigma^2}\Big[1+\frac{3}{80}\frac{\lP^2}{\sigma^2}\Big].
\]
Multiplying by $\Delta x^2=\sigma^2$ and taking the root, the product is obtained in closed
form:
\begin{equation}
\boxed{\;\Delta x\cdot\Delta p=\frac{\hbar}{2}\sqrt{1+\frac{3}{80}\Big(\frac{\lP}{\sigma}\Big)^2}\;} \tag{E-inc-3}
\end{equation}
The coefficient $3/80=\tfrac{1}{20}\cdot 3\cdot\tfrac{1}{4}$ combines the flexural rigidity
($1/20$), the fourth Gaussian moment ($3$) and the normalization of the second moment
($1/4$). \textbf{Symbolic verification}: the difference between $\sigma^2\Delta p_\text{medium}^2$
and the closed form $(\hbar/2)^2[1+\tfrac{3}{80}(\lP/\sigma)^2]$ is exactly zero.

\textbf{Recovery of Heisenberg.} The result has the two correct limits.
\emph{(i) Ordinary regime}: for $\sigma\gg\lP$ the correction term vanishes and
\[
\Delta x\,\Delta p\to\frac{\hbar}{2}\quad\text{exact},
\]
independent of $\xi$ and of any parameter of the medium: the Heisenberg limit is not
an input of the model but the floor that the Gaussian saturation produces when the flexural
rigidity is negligible. \emph{(ii) Extreme regime}: at the minimal width
$\sigma=\sigma_{\min}=\lP/\sqrt{20}$ (the Lee-Wick pole), substituting
$(\lP/\sigma_{\min})^2=20$,
\[
\Delta x\,\Delta p=\frac{\hbar}{2}\sqrt{1+\frac{3}{80}\cdot 20}=\frac{\hbar}{2}\sqrt{\frac{7}{4}}=\frac{\sqrt7}{4}\,\hbar\approx 0.6614\,\hbar,
\]
an excess of $32.3\%$ over the floor: the maximum possible deviation, at the limit of
resolution of the medium.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{incertidumbre_pleno.png}
\caption{The uncertainty relation as a property of the medium. Panel B: the product
$\Delta x\,\Delta p$ falls to the exact floor $\hbar/2$ for $\sigma\gg\lP$ and rises to
$+32\%$ at $\sigma_{\min}=\lP/\sqrt{20}$, with the UV excess of derived coefficient
$3/80$. Panel A: the factor $\sqrt{20}$ converges from three independent sectors of the
Lagrangian (elastic, flexural, dissipative).}
\end{figure}

\textbf{The sign $\geq$.} Since $1+\tfrac{3}{80}(\lP/\sigma)^2\geq 1$ by the strictly
positive flexural rigidity (A4), the Heisenberg inequality \emph{is} the
positivity of the curvature rigidity of el Pleno. The closed form and the recovery
of $\hbar/2$ are \epi{DERIVED}; the reading of the sign is \epi{SLD}; the GUP
as a prediction is suppressed $\sim 10^{-50}$ at atomic scales (not tested).

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest: the quantum}]
If gravity is the medium in the large, the quantum is the medium in the small: the
uncertainty relation is the phenomenology of a medium with positive curvature rigidity
(the flexural of F1). The floor $\hbar/2$ is exact (Gaussian saturation of
long waves) and independent of $\xi$; the ultraviolet excess $(3/80)(\lP/\sigma)^2$
is a prediction with a derived coefficient, not parametrized. The same rigidity that regulates
F1 in the UV produces the quantum limit ---gravity and the quantum, a single substrate.
\end{tcolorbox}

\subsection{The Chandrasekhar limit as an internal balance --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\textbf{The chain, internal to el Pleno.} (1) The fermionic statistics is not imported:
it is Finkelstein-Rubinstein over the solitons ($(-1)^B$, §\ref{sec:solitones}). (2)
The degeneracy pressure is the uncertainty (flexural rigidity). (3) Gravity
is the macroscopic response of el Pleno (§\ref{sec:gravedad}). (4) The equilibrium of
a self-gravitating degenerate relativistic gas is the polytrope of index $n=3$
(Lane-Emden), which fixes the coefficient.

\textbf{The Lane-Emden equation} $\theta''+\tfrac{2}{\xi}\theta'+\theta^3=0$, with
$\theta(0)=1$, $\theta'(0)=0$, integrated numerically up to the first root, gives
(verified): first root $\xi_1=6.8968$ and
$(-\xi^2\theta')_{n=3}=2.0182$. The balance produces a limiting mass independent of the
radius:
\begin{equation}
\boxed{\;M_\text{Ch}=\frac{\sqrt{3\pi}}{2}\big(-\xi^2\theta'\big)_{n=3}\frac{(\hbar c/G)^{3/2}}{(\mu_e m_u)^2}\approx 1.456\,M_\odot\;} \tag{E-Ch-1}
\end{equation}
with $\mu_e=2$ (C-O matter). Factored with $m_P=\sqrt{\hbar c/G}$:
\begin{equation}
M_\text{Ch}=m_P\cdot\Big(\frac{m_P}{\mu_e m_u}\Big)^2\cdot\underbrace{\frac{\sqrt{3\pi}}{2}\big(-\xi^2\theta'\big)_{n=3}}_{=3.098},
\end{equation}
where $m_P$ is the scale of gravity-uncertainty equalization, $(m_P/\mu_e m_u)^2$ the
Planck-hadronic hierarchy (the same as the mass of the fundamental soliton), and $3.098$
the pure coefficient of the $n=3$ polytrope. Three mass limits in one framework:
$M_{\min}\approx 0.117\,m_P$ (black hole), $M_\text{Ch}\approx 1.456\,M_\odot$,
$M_Q=M_P/S_{\max,\text{eff}}$.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{limites_masa_pleno.png}
\caption{The Chandrasekhar limit as an internal balance. Panel A: anatomy of the
coefficient in three factors ---the Planck scale $m_P$, the hierarchy
$(m_P/\mu_e m_u)^2$ and the pure coefficient of the $n=3$ polytrope, $\tfrac{\sqrt{3\pi}}{2}(-\xi^2\theta')=3.098$.
Panel B: the three mass limits ($M_Q$, $M_{\min}$, $M_\text{Ch}$) as balances of the
same medium, spanning 57 orders of magnitude.}
\end{figure}

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The Chandrasekhar limit is not borrowed from astrophysics: it is the maximum mass that the
flexural rigidity of el Pleno sustains against its own gravitational response ---gravity
and the quantum, the two sectors of the same medium, together fixing a critical stellar mass. The
polytropic framework ($n=3$, Lane-Emden) is imported, and is declared as such; the coefficient
$2.0182$ is a pure ratio of the polytropic profile, and the scales that enter ($m_P$, the
mass hierarchy) are those of el Pleno itself. The result, $1.456\,M_\odot$, comes out without
a fitting parameter.
\end{tcolorbox}

\subsection{Born, Bell and the Tsirelson bound --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

Three features that make quantum mechanics ``strange'' ---the Born rule, the
Bell non-locality and the quantum Tsirelson~\cite{tsirelson} limit--- emerge here from the same medium,
without being postulated. The wave function $\psi$ is not an abstract probability object:
it \emph{is} the perturbation $\delta S\,e^{i\theta}$ of el Pleno.

\textbf{The Born rule as accounting of conserved charge.} The localization
probability $|\psi|^2$ is not a separate law: it is the fraction of the U(1) Noether charge
(axiom A5) of the quantum that is in each region. For a quantum
$\psi=\delta S\,e^{i\theta}$, the Noether current of F1 (E7c) and its charge density
are
\[
j^\mu=\rhoP c^2\lP^2\,S^2\partial^\mu\theta,\qquad
\rho_Q=\rhoP\lP^2\,\omega\,S^2\propto|\psi|^2,
\]
so that the probability of finding the quantum in a region $R$ is the fraction of
charge localized there:
\begin{equation}
P(R)=\frac{Q_R}{Q_\text{total}}=\frac{\int_R S^2\,d^3x}{\int S^2\,d^3x}
=\int_R|\psi_\text{norm}(\mathbf{x})|^2\,d^3x. \tag{E-Born}
\end{equation}
The normalization $\int|\psi|^2=1$ is not imposed: it is closed by the topological quantization of
the total charge (§\ref{sec:gauge}). The Born rule is thus \emph{accounting of something
that is conserved}, not an independent probabilistic axiom. \epi{DERIVED}.

\emph{Reading: the quantum probability ceases to be a mysterious postulate. $|\psi|^2$
measures what fraction of a physically conserved quantity ---the phase charge of the medium---
is in each region; the ``randomness'' of the measurement is the macroscopic reading of the
internal noise of el Pleno, not an external collapse.}

\textbf{Bell correlations, step by step.} A pair of subsystems of el Pleno in
antisymmetric configuration realizes the analog of the singlet state; from it the
correlation function is \emph{derived}, not postulated.
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Step 1 --- singlet state.} Two complementary phase modes form
$|\Psi^-\rangle=\tfrac{1}{\sqrt2}(|{\uparrow\downarrow}\rangle-|{\downarrow\uparrow}\rangle)$,
the total-zero-angular-momentum configuration of the pair.
\item \textbf{Step 2 --- measurement operators.} To measure in the direction $\mathbf{a}$ is
the operator $\boldsymbol{\sigma}\cdot\mathbf{a}$; the correlation is the expectation value of the
product:
\[
E(\mathbf{a},\mathbf{b})=\langle\Psi^-|(\boldsymbol{\sigma}_1\cdot\mathbf{a})(\boldsymbol{\sigma}_2\cdot\mathbf{b})|\Psi^-\rangle .
\]
\item \textbf{Step 3 --- evaluation.} The Pauli-algebra identity over the singlet
gives $E=-\mathbf{a}\cdot\mathbf{b}=-\cos\theta_{ab}$. Parametrizing the directions by
measurement angles, $\theta_{ab}=2(a-b)$ ---the factor $2$ is the signature that the phase
mode is a spinor (a physical rotation $\alpha$ rotates the spinor by $\alpha/2$):
\begin{equation}
E(a,b)=-\cos\big(2(a-b)\big). \tag{E-Bell}
\end{equation}
\item \textbf{Step 4 --- Tsirelson bound.} With the CHSH combination
$\mathcal{S}=E(a,b)-E(a,b')+E(a',b)+E(a',b')$, optimizing over the angles
(optimum at separations of $22.5^\circ$) gives
\begin{equation}
\boxed{\;|\mathcal{S}|_\text{max}=2\sqrt{2}\;}\qquad(\text{Tsirelson bound}),
\end{equation}
verified numerically. It violates the classical Bell limit ($|\mathcal{S}|\leq 2$) but
respects the quantum one, exactly as observed.
\item \textbf{Step 5 --- no-signaling.} The reduced state of a subsystem is
$\rho_B=\operatorname{Tr}_A|\Psi^-\rangle\langle\Psi^-|=\tfrac12\mathbb{I}$, independent
of the direction measured at A: no measurement at A alters the local statistics of B.
There is no superluminal communication; no-signaling is verified exactly.
\end{itemize}

\textbf{Why $2\sqrt2$: the tetrahedral connection.} The bound is not an arbitrary number:
it is anchored in the geometry of the lattice. The tetrahedral angle
$\theta_\text{tet}=\arccos(-\tfrac13)=109.47^\circ$ satisfies exactly
\[
\sin\theta_\text{tet}=\frac{2\sqrt2}{3},\qquad \tan\frac{\theta_\text{tet}}{2}=\sqrt2,
\]
and the binary tetrahedral group $2T$ (order 24) is the symmetry of the diamond lattice. The
$\sqrt2$ of Tsirelson is the same $\sqrt2$ of the packing (structural invariant
of el Pleno): the quantum limit of the correlations is the imprint of the geometry of the
medium. \epi{SLD}.

\emph{Reading: the quantum ``non-locality'' is not action at a distance. It is a single object
---el Pleno--- whose configuration is global by construction, prior to the metric that
defines the distances. EPR is resolved by the pre-metric uniqueness of the medium: there are not two
systems that communicate, there is one whose state manifests in two regions.}

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Quantum mechanics, at its core, is the phenomenology of a physical medium. Its three
puzzling features have here a mechanical origin and a clear meaning: the
\textbf{uncertainty} is the positive curvature rigidity of el Pleno (localizing costs
energy, floor $\hbar/2$ exact); the \textbf{Born rule} is accounting of conserved
charge ($|\psi|^2$ = fraction of localized phase charge); the \textbf{Bell
correlations} up to $2\sqrt2$ are the tetrahedral geometry of the lattice read in the
measurement space. There is no collapse, no fundamental randomness, no superluminal signal: there is a
single substrate whose deterministic dynamics is perceived as probabilistic at the level of its
perturbations. Three features, three distinct origins within F1, a single medium.
\end{tcolorbox}

\subsection{The non-relativistic limit: F1 \texorpdfstring{$\rightarrow$}{->} Gross-Pitaevskii --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:gp}

A structural correspondence closes the quantum link: in the
\textbf{non-relativistic} limit of F1 ---modes of frequency much lower than the Planckian
$\omega_P=c/\lP$ and slow dynamics against the microscopic scale--- F1 reduces
formally to the \textbf{Gross-Pitaevskii equation} (Gross 1961, Pitaevskii 1961),
the best-validated macroscopic quantum field equation of physics (Bose-Einstein
condensates, superfluids, superconductors). The same decomposition
$\Psi=Se^{i\theta}$ of el Pleno is that of the order parameter of those systems, so
that the reduction is not an analogy but a limit.

\textbf{Step 1 --- Regime.} $S=\Smin^\text{estr}+\delta S$ with
$\delta S\ll \Smin^\text{estr}$ and slow dynamics against $\omega_P$.

\textbf{Step 2 --- Field substitution.} The fast phase of the
rest mass of the radial mode is factored out and the modulus is normalized to its reading as
\emph{number density} of the medium. By the constitutive identification
$S^2\leftrightarrow n_\text{quanta}/n_0$ with $n_0=1/\lP^3$ (E-dens, §\ref{sec:potencial}), the perturbation
$\delta S$ has the number-density reading $\delta S^2/\lP^3$, which fixes the
prefactor of the effective wave function:
\[
\psi(\mathbf{x},t)=\frac{\delta S(\mathbf{x},t)}{\lP^{3/2}}\,e^{-i\omega_0 t},
\qquad \omega_0=\frac{m_\text{radial}c^2}{\hbar}=\sqrt{\xi}\,\frac{c}{\lP},
\]
so that $|\psi|^2=\delta S^2/\lP^3$ is \textbf{number density}
($[\,|\psi|^2\,]=\text{m}^{-3}$), the canonical Gross-Pitaevskii convention in which
$|\psi|^2$ counts quanta per unit volume ---coherent with the reading of the
modulus of el Pleno established in E-dens (§\ref{sec:potencial}), not with a normalization to energy density.
The phase $e^{-i\omega_0 t}$ separates the fast dynamics ($\omega_0=\sqrt{\xi}\,\omega_P\approx 4.44\,\omega_P$,
ultra-Planckian) from the non-relativistic one ($\omega_{\delta S}\ll\omega_0$).

\textbf{Step 3 --- Expansion of the potential to fourth order} around
$\Smin^\text{estr}$, retaining the cubic term:
\[
V_\text{eff}(\Smin+\delta S)=V_\text{eff}(\Smin)+\tfrac12 V''_\text{eff}\,\delta S^2
+\tfrac16 V'''_\text{eff}\,\delta S^3+\tfrac{1}{24}V^{(4)}_\text{eff}\,\delta S^4
+\mathcal{O}(\delta S^5).
\]
The cubic term has a nonzero coefficient at the minimum. The three derivatives of the
potential at $\Smin^\text{estr}$ are obtained from the same
$V=W_\text{bare}+V_q+V_\text{topo}$ (§\ref{sec:potencial}, first subsection, where they are computed and verified
symbolically from the closed form of the potential):
\begin{equation}
V''_\text{eff}=19.69\,\rhoP c^2,\qquad
V'''_\text{eff}=27.92\,\rhoP c^2,\qquad
V^{(4)}_\text{eff}=855.6\,\rhoP c^2. \tag{E24-bis}
\end{equation}
With $\delta S=\tfrac12(A\,e^{-i\omega_0 t}+A^*e^{+i\omega_0 t})$, the square
$\delta S^2=\tfrac14 A^2 e^{-2i\omega_0 t}+\tfrac14 A^{*2}e^{+2i\omega_0 t}+\tfrac12|A|^2$
has two components at $2\omega_0$ and one DC component. The reason the cubic
does not contribute \emph{at first order} is not that it averages to zero, but that none of
those components is in resonance with $\omega_0$, the carrier frequency of $\psi$.
\emph{At second order}, in contrast, those components fed back by the cubic generate
a resonant term at $\omega_0$ that \textbf{renormalizes} the effective quartic: the
coupling $g_\text{eff}$ receives a cubic correction that is computed explicitly
in Step 6.

\textbf{Step 4 --- the reduction to Schrödinger (rotating average).} Here is the heart
of the limit, and the sense of ``non-relativistic'' here is precise. The
perturbation $\delta S$ obeys a Klein-Gordon-type equation with mass, that of the radial
mode (E18). Substituting $\delta S\propto\operatorname{Re}[\psi\,e^{-i\omega_0 t}]$ with
$\psi$ slow and $\omega_0=m_\text{eff}c^2/\hbar$, the second time derivative gives
\[
e^{i\omega_0 t}\,\partial_t^2(\psi\,e^{-i\omega_0 t})
=\ddot\psi-2i\omega_0\dot\psi-\omega_0^2\psi .
\]
The term $-\omega_0^2\psi$ cancels \emph{exactly} with the mass one
$(m_\text{eff}c/\hbar)^2\psi=\omega_0^2\psi/c^2$ (that is the factored-out rest mass).
The \emph{non-relativistic} condition is $|\ddot\psi|\ll\omega_0|\dot\psi|$ ---the residual
dynamics of $\psi$ is slow against $\omega_0$, not that $m_\text{eff}$ be small; on the
contrary, it is precisely the enormity of $\omega_0$ (ultra-Planckian) that guarantees
the scale separation. Neglecting $\ddot\psi$ remains
\[
-\frac{2i\omega_0}{c^2}\dot\psi=-\nabla^2\psi+(\text{non-linear term})
\;\Longrightarrow\;
i\hbar\,\dot\psi=-\frac{\hbar^2}{2m_\text{eff}}\nabla^2\psi+\dots,
\]
using $\omega_0=m_\text{eff}c^2/\hbar$. The corresponding effective Lagrangian, in
which the effective quartic $g_\text{eff}$ already incorporates the renormalization of the cubic at
second order (Step 6), is
\[
\mathcal{L}_\text{eff}=\frac{i\hbar}{2}(\psi^*\partial_t\psi-\psi\partial_t\psi^*)
-\frac{\hbar^2}{2m_\text{eff}}|\nabla\psi|^2-\frac{g_\text{eff}}{2}|\psi|^4.
\]

\textbf{Step 5 --- Effective mass.} Equating the rest-mass term
$\hbar\omega_0=m_\text{eff}c^2$, with
$\omega_0=\sqrt{V''_\text{eff}(\Smin)/(\rhoP\lP^2)}$ from (E21):
\begin{equation}
\boxed{\;m_\text{eff}\,c^2=\hbar\,\sqrt{\frac{V''_\text{eff}(\Smin^\text{estr})}{\rhoP\,\lP^2}}
=\sqrt{\xi}\,\frac{\hbar c}{\lP}=\sqrt{\xi}\,E_P\;}\tag{E24}
\end{equation}
With $\xi\approx 19.69$: $\sqrt{\xi}=4.4373$, so that
$m_\text{eff}c^2=\sqrt{\xi}\,E_P\approx 4.44\,E_P\approx 5.42\times10^{19}$ GeV
(with $E_P\approx 1.22\times10^{19}$ GeV), an ultra-Planckian excitation that justifies that
the observable regime be that of the massless phase mode, not the radial one. Dimensional
verification: $[\sqrt{V''_\text{eff}/(\rhoP\lP^2)}]=[\sqrt{(J/m^3)/((kg/m^3)\,m^2)}]
=[1/\text{s}]$, whence $[(\hbar/c^2)\cdot(1/\text{s})]=[\text{kg}]$ $\checkmark$.

\textbf{Step 6 --- Coupling constant $g_\text{eff}$ at second order.} The
direct quartic contributes $V^{(4)}_\text{eff}/12$, but the second-order cubic
contribution renormalizes it (Step 3). The derivation proceeds by adiabatic elimination
(multiscale) of the equation of motion of the radial mode around the minimum.
Writing it in units $\rhoP\lP^2=1$ with $a\equiv V''_\text{eff}$,
$b\equiv V'''_\text{eff}$, $c\equiv V^{(4)}_\text{eff}$:
\begin{equation}
\ddot{\delta}+a\,\delta=-\tfrac12 b\,\delta^2-\tfrac16 c\,\delta^3, \tag{E25a}
\end{equation}
with the multiscale expansion $\delta=\varepsilon\delta_1+\varepsilon^2\delta_2
+\varepsilon^3\delta_3$ and slow time $T_2=\varepsilon^2 t$ where the amplitude
$A(T_2)$ lives.

\emph{Order $\varepsilon$:} $\delta_1=A\,e^{i\omega_0 t}+\text{c.c.}$, with
$\omega_0^2=a$.

\emph{Order $\varepsilon^2$ (modes generated by the cubic).} The source
$-\tfrac12 b\,\delta_1^2$ has a DC component and one at $2\omega_0$, which excite
respectively
\begin{equation}
\text{DC mode:}\quad d=-\frac{b\,|A|^2}{a},\qquad
\text{mode }2\omega_0:\quad p=\frac{b\,A^2}{6a}, \tag{E25b}
\end{equation}
where $p$ comes from $(a-4\omega_0^2)\,p=-\tfrac12 b A^2$ with $\omega_0^2=a$, so that
$a-4a=-3a$.

\emph{Order $\varepsilon^3$ (non-secularity condition).} The resonant terms at
$\omega_0$ that determine the amplitude equation are: from the quartic,
$-\tfrac16 c\cdot 3|A|^2 A=-\tfrac12 c\,|A|^2 A$; and from the crossed cubic
$-b\,\delta_1\delta_2$, whose component at $\omega_0$ is $-b(A\,d+A^*p)$, with two channels,
\begin{equation}
\text{DC channel:}\;\;-b\,A\,d=+\frac{b^2|A|^2 A}{a}\;\;(+6/6),\qquad
\text{channel }2\omega_0:\;\;-b\,A^*p=-\frac{b^2|A|^2 A}{6a}\;\;(-1/6), \tag{E25c}
\end{equation}
whose sum is $+\tfrac{5}{6}\,b^2|A|^2 A/a$ (the $2\omega_0$ channel \emph{subtracts}, being
out of phase). The net coefficient of $|A|^2 A$ combines quartic and cubic:
\begin{equation}
\text{coef}=-\tfrac12 c+\tfrac{5}{6}\frac{b^2}{a}
=-\tfrac12 c\left[1-\frac{5}{3}\frac{b^2}{a\,c}\right]. \tag{E25d}
\end{equation}
Restoring the normalization to number density ($n_0=1/\lP^3$, factor $1/(12\,n_0^2)$
of the quartic combinatorics), the complete coupling constant is:
\begin{equation}
\boxed{\;g_\text{eff}=\frac{V^{(4)}_\text{eff}\,\lP^6}{12}
\left[1-\frac{5}{3}\frac{(V'''_\text{eff})^2}{V''_\text{eff}\,V^{(4)}_\text{eff}}\right]\;}
\tag{E25}
\end{equation}
The coefficient $5/3$ is exact and invariant under the normalization of the amplitude (a
rescaling $A\to\lambda A$ multiplies numerator and denominator of the quotient
$g_\text{corr}/g_\text{naive}$ by $|\lambda|^2\lambda$ and cancels), and coincides with the
canonical result of the anharmonic oscillator (Landau-Lifshitz, \emph{Mechanics} §28). The
static minimization of the potential would give instead a coefficient $3$, distinct, because
it does not capture the dynamic channel $2\omega_0$; that GP requires $5/3$ confirms that that channel is
real.

\textbf{Dimensional verification.} $[V^{(4)}_\text{eff}\,\lP^6]=[J/m^3]\cdot[m^6]
=[J\,m^3]$, the contact dimension in 3D; the factor in brackets is dimensionless
$\checkmark$. The factor $1/(12\,n_0^2)=\lP^6/12$ combines the combinatorics of the quartic
term ($\delta S^4/24$ differentiated twice) with the conversion of $\delta S^2$ to
number density $|\psi|^2$ of Step 2.

\textbf{Numerical verification.} With the three derivatives (E24-bis),
$V''_\text{eff}=19.69\,\rhoP c^2$, $V'''_\text{eff}=27.92\,\rhoP c^2$,
$V^{(4)}_\text{eff}=855.6\,\rhoP c^2$, the cubic factor is
\[
1-\frac{5}{3}\frac{(27.92)^2}{19.69\times 855.6}
=1-\frac{5}{3}(0.04627)=0.9229,
\]
whence
\[
g_\text{eff}=\frac{855.6\,\rhoP c^2\,\lP^6}{12}\times 0.9229
=65.8\,\rhoP c^2\,\lP^6\approx 5.43\times10^{-94}\ \text{J}\cdot\text{m}^3,
\]
a correction of $-7.7\%$ with respect to the direct quartic
($71.3\,\rhoP c^2\lP^6\approx 5.88\times10^{-94}$ J$\cdot$m$^3$). The positive sign is
preserved ($g_\text{eff}>0$): the self-interaction $|\psi|^4$ remains repulsive and the
ground-Pleno condensate remains stable. \epi{DERIVED-ALGEBRA}: $g_\text{eff}$ is
a pure ratio of the potential derivatives, zero free parameters.

\textbf{Step 7 --- Euler-Lagrange.} Varying $\mathcal{L}_\text{eff}$:
\begin{equation}
\boxed{\;i\hbar\,\partial_t\psi=-\frac{\hbar^2}{2m_\text{eff}}\nabla^2\psi+g_\text{eff}|\psi|^2\psi\;}\tag{E26}
\end{equation}
\textbf{Closure dimensional verification (the three terms).} With
$[\psi]=\text{m}^{-3/2}$, $[m_\text{eff}]=\text{kg}$ and $[g_\text{eff}]=\text{J}\,\text{m}^3$,
the three terms of (E26) share units
$[\hbar\,\partial_t\psi]=[\hbar^2\nabla^2\psi/m_\text{eff}]=[g_\text{eff}|\psi|^2\psi]
=\text{kg}\,\text{m}^{1/2}\text{s}^{-2}$ (verified symbolically): the equation is
dimensionally consistent term by term, which fixes the prefactor of $\psi$ of
Step 2 uniquely to $\lP^{-3/2}$. The effective mass
$m_\text{eff}=\sqrt{19.69}\,m_P=4.4373\,m_P$ coincides with the radial mode of (E21),
obtained by an independent route (linearized spectrum): the frequency
$\omega_0$ that is factored out in Step 4 is the same as that of the massive mode, which confirms the
internal coherence of the reduction.

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- what this limit means}]
The implication is ontological, not only formal. That F1 contain Gross-Pitaevskii as an
\emph{exact} non-relativistic limit ---not as an analogy--- means that the same
equation that governs Bose-Einstein condensates, superfluids and
superconductors in the laboratory governs el Pleno: the universe \emph{is}, in its
slow regime, a condensate. The decomposition $\Psi=Se^{i\theta}$ of el Pleno is
literally the order parameter of those systems. Three consequences: \emph{(i)} the
normalization of the effective field is not free ---the reading of the modulus as number
density (E-dens) fixes the prefactor $\psi=\delta S/\lP^{3/2}$ and closes the three terms of
(E26) without fitting; \emph{(ii)} the effective mass of the radial mode coincides with the
linearized spectrum by two independent routes, a proof of internal consistency; \emph{(iii)}
all laboratory-validated GP phenomenology ---Bogoliubov-de Gennes spectra,
bright solitons, collapse-rebound in \emph{Bosenovas}--- is then \textbf{indirect
experimental validation} of el Pleno: tabletop experiments that probe the structure of the
cosmic vacuum. Status \epi{DERIVED} (dimensionally consistent computation, E24--E26),
with the complete coupling $g_\text{eff}$ at second order ---the direct quartic plus the
cubic renormalization (factor $5/3$, E25a--E25d) leave it as a pure ratio of the
potential derivatives, without free parameters, \epi{DERIVED-ALGEBRA}; as a parametric
mapping, conditional on the GP phenomenology, experimentally established.
\end{tcolorbox}

\subsection{The Dirac structure as an effective theory of the phase modes --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}
\label{sec:dirac}

The \emph{structure} of the Dirac equation ---first-order operator, Clifford
algebra, relativistic energy-momentum relation, spin-$\tfrac12$--- is derived as an effective
low-energy theory of the phase modes $\theta_-$ over the bipartite diamond lattice.
The structure is \epi{DERIVED}: two routes converge on it. The \emph{band
route} (developed here) derives each link ---the doublet A/B, the bands and the nodal
line, the $SU(2)$ phase, the covariance by $v_F=c$--- from the geometry of the medium. The
\emph{covariant route} (corpus §V.7.2) establishes it independently of the value of
$\lambda_\text{SO}$: the two binary structures of the medium (bipartition A/B and modulus-phase
duality) fix a unique representation of the algebra by the Pauli theorem, so
that the Dirac form does not depend on any fine parameter. What remains as an open end is
the \emph{value} of two constants ($\lambda_\text{SO}$ and the masses $m$), tied to the
open ends of fermionic masses and chirality (§\ref{sec:cabos}).

\textbf{Step 1 --- doublet and bipartite Bloch Hamiltonian.} The bipartition A/B
(§\ref{sec:gauge}) provides the doublet $\Phi=(\varphi_A,\varphi_B)^T$. Since an excitation
on an A site only hops to its four B neighbors (tetrahedral geometry), the Bloch
Hamiltonian is strictly off-diagonal:
\[
H(\mathbf{k})=t\begin{pmatrix} 0 & f(\mathbf{k}) \\ f^*(\mathbf{k}) & 0 \end{pmatrix},
\qquad f(\mathbf{k})=\sum_{j=1}^{4} e^{\,i\,\mathbf{k}\cdot\mathbf{d}_j},
\qquad E(\mathbf{k})=\pm\,t\,|f(\mathbf{k})|. \tag{E-Dirac-1}
\]
They are the two bands that the bipartition already produces (§\ref{sec:1sobre40}, form factor
of $\Smin^\text{estr}$).

\textbf{Step 2 --- nodal lines.} The set where the bands touch is
$f(\mathbf{k})=0$, with
\[
|f(\mathbf{k})|^2=4+4\cos\!\tfrac{a k_x}{2}\cos\!\tfrac{a k_y}{2}
+4\cos\!\tfrac{a k_x}{2}\cos\!\tfrac{a k_z}{2}
+4\cos\!\tfrac{a k_y}{2}\cos\!\tfrac{a k_z}{2}, \tag{E-Dirac-2}
\]
whose zero set is \emph{one-dimensional}: nodal lines, not points. The pure lattice gives
a nodal-line semimetal, not the 3D Dirac cone; the ingredient that opens the
line into isolated points is missing.

\textbf{Step 3 --- the spin-orbit is forced $\neq 0$.} The intrinsic
second-neighbor spin-orbit coupling, the only one allowed by $Fd\bar3m$ (Fu-Kane),
\[
H_\text{SO}=i\,\lambda_\text{SO}\!\!\sum_{\langle\langle ij\rangle\rangle}\!\!
(\mathbf{d}_{il}\times\mathbf{d}_{lj})\cdot\boldsymbol{\sigma}, \tag{E-Dirac-3}
\]
is forced to be nonzero by the $SU(2)$ character of the mode $\theta_-$
(§\ref{sec:1sobre40}). The transport over a plaquette of two non-collinear bonds,
with $A_1=\theta\sigma_x$ and $A_2=\theta\sigma_y$, gives the holonomy
\[
W=e^{iA_1}e^{iA_2}e^{-iA_1}e^{-iA_2}
=\mathbb{1}-\theta^2[\sigma_x,\sigma_y]+\mathcal{O}(\theta^3)
=\mathbb{1}-2i\theta^2\sigma_z+\mathcal{O}(\theta^3), \tag{E-Dirac-4}
\]
using $[\sigma_x,\sigma_y]=2i\sigma_z$: proportional to $\boldsymbol{\sigma}$ and distinct
from the identity. Hence the equivalence
\[
\lambda_\text{SO}=0 \;\Longleftrightarrow\; \text{holonomy } SU(2)\text{ trivial}
\;\Longleftrightarrow\; \theta_-\text{ lives in } U(1). \tag{E-Dirac-5}
\]
Since $\theta_-$ lives in $SU(2)$ by derivation ---the same property from which the
fermions hang---, $\lambda_\text{SO}\neq 0$ is forced. It is the \emph{same} holonomy
$SU(2)$ that produces the non-abelian structure of Yang-Mills (§\ref{sec:gauge}): they are not
independent facts, but a single property of the medium read in two sectors.

\textbf{Step 4 --- opening of the node and the Weyl operator.} With $\lambda_\text{SO}\neq 0$
the nodal line opens into 3D Dirac points. Near a node,
\[
H(\mathbf{q})=v_F\,(q_x\Gamma_1+q_y\Gamma_2+q_z\Gamma_3),\qquad
\Gamma_1=\sigma_x\otimes\mathbb{1},\;\Gamma_2=\sigma_y\otimes\mathbb{1},\;
\Gamma_3=\sigma_z\otimes\sigma_x, \tag{E-Dirac-6}
\]
with $4\times 4$ matrices (two sublattices $\times$ two spin components) that satisfy
the Euclidean Clifford algebra, verified symbolically,
\[
\{\Gamma_i,\Gamma_j\}=2\delta_{ij}\mathbb{1}_4,\qquad i,j\in\{1,2,3\}, \tag{E-Dirac-7}
\]
whence $H^2=v_F^2\mathbf{q}^2\mathbb{1}_4$ and the isotropic linear cone $E=\pm v_F|\mathbf{q}|$.

\textbf{Step 5 --- mass and complete equation.} A fourth matrix
$\Gamma_0=\sigma_z\otimes\sigma_z$ ($\Gamma_0^2=\mathbb{1}$, anticommutes with the three
$\Gamma_i$), originating in the A--B imbalance (Peierls dimerization), opens a gap. The
complete Hamiltonian and its spectrum are
\[
H=v_F\,\boldsymbol{\Gamma}\cdot\mathbf{q}+mc^2\,\Gamma_0, \tag{E-Dirac-8}
\]
\[
H^2=\bigl(v_F^2\,\mathbf{q}^2+m^2 c^4\bigr)\mathbb{1}_4
\;\Longrightarrow\;
E=\pm\sqrt{v_F^2\,\mathbf{q}^2+m^2 c^4}. \tag{E-Dirac-9}
\]
Since the phase modes propagate at velocity $c$ (§\ref{sec:modos}), with $v_F=c$ the
relation is exactly $E^2=c^2\mathbf{q}^2+m^2c^4$ and the operator is the Dirac
equation, with Clifford-Lorentz algebra $\{\Gamma_\mu,\Gamma_\nu\}=2\eta_{\mu\nu}$. The
covariance is not imposed: it is inherited from $v_F=c$ (axiom A3).

\medskip
\begin{tcolorbox}[clave,title={The Dirac equation as an effective theory of el Pleno}]
\[
\boxed{\;\bigl(i\hbar\,\Gamma^\mu\partial_\mu-mc\bigr)\Psi_D=0\;}
\qquad\Longleftrightarrow\qquad
H=c\,\boldsymbol{\Gamma}\cdot\mathbf{q}+mc^2\,\Gamma_0,
\qquad \{\Gamma_\mu,\Gamma_\nu\}=2\eta_{\mu\nu}. \tag{E-Dirac-10}
\]
The doublet comes from the bipartition A/B; the Clifford matrices, from the generators
$SU(2)$ of $\theta_-$; the first order, from the spin-orbit opening of the nodal line;
the covariance, from $v_F=c$.

\emph{Reading: the spin-$\tfrac12$ and the relativistic structure of the electron are not
postulated ---they emerge from the topology of the lattice. The fermion is a mode of el Pleno, not a
fundamental entity; what remains as an open end is the value of the masses, not the form of the
equation.}
\end{tcolorbox}

\textbf{Dimensional verification.} In (E-Dirac-9), $[v_F q\,\hbar]=[J]$ and $[mc^2]=[J]$:
energy $\checkmark$. The covariant form has $[\hbar\Gamma^\mu\partial_\mu]=[mc]
=kg\cdot m/s$: momentum $\checkmark$.

\textbf{Epistemic classification.} The \emph{structure} of Dirac is \epi{DERIVED}: the
covariant route (corpus §V.7.2) establishes it from the two binary structures of the medium with
a unique representation by the Pauli theorem, without depending on the value of $\lambda_\text{SO}$;
the band route (here) confirms it with each link \epi{DERIVED} (doublet A/B, bands and
nodal line, phase $SU(2)$, $v_F=c$) or \epi{SLD} ($\lambda_\text{SO}\neq 0$ by holonomy,
opening of the node). The emergent spin-$\tfrac12$ is \epi{DERIVED} as a property of the
doublet. The \emph{value} of $\lambda_\text{SO}$ and of the masses $m$ is a declared open end; the
chirality ($SU(2)_L$ vs $SU(2)_R$) and the number of generations remain open
ends. The program affirms the Dirac structure as a derived result and does not claim
to have derived masses, chirality or generations.

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Matter with spin ---the electron, the quarks--- is a torsion vibration of the medium.
The diamond lattice has two interlaced sublattices (A and B), and a phase mode that oscillates
out of phase between them cannot be described with a single number: it needs two coupled
components, exactly the Dirac spinor. The first-order structure of the equation,
the Clifford algebra and the spin-$\tfrac12$ are not imposed: they are the obligatory form of
propagation of an excitation that lives over two sublattices at once. Spin is thus a
real geometric property ---the way in which the vibration is distributed between the two faces
of the crystal of the vacuum---, not an abstract quantum label. What the medium does not yet fix
by itself alone (the value of the masses, the chirality, how many generations) remains
explicitly as the pending part; what it does deliver is the physical reason that matter
be spinorial.
\end{tcolorbox}

\medskip\hrule\medskip

% =====================================================================
