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% PIU-Psi V5.0 -- Mathematical Compendium -- PART 1: Foundations and structure of the medium (sec. 1-6)
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\section{Foundations: the logical chain from energy to F1 --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}
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The universe is energy: finite at each point and conserved. From finiteness comes
density; from density in a medium, the minimal grain; from variability, the change
ceiling and time; from conservation, the balance and with it $E=Mc^2$, the field $\Psi$ and
the Lorentz factor. Once the pieces are gathered, F1 remains.

\subsection{From energy to density, grain and time --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{The root statement: the universe is energy.} The starting point is a single one
(axiom A1): the universe \emph{is} energy ---it does not contain energy, it is energy---, in a
single, homogeneous and isotropic substrate. That energy is finite at each point
($0<E<\infty$, bounded on both sides) and conserved (it is neither created nor destroyed, only
transformed and redistributed). The pointwise finiteness is the program's own
postulation; conservation is the Mayer-Joule-Helmholtz principle, which the program
adopts, does not derive.

\textbf{Ceilings, not numbers.} From finiteness follow \emph{edges}: a ceiling on energy
per region, a minimal grain, a floor, a ceiling on change. That an edge \emph{exists} is
a property of el Pleno; that it \emph{takes such a number} appears only when measuring it against a
human ruler ---a distinction that §1.4 develops when reading the values of $\rhoP$, $\lP$ and
$c$. What has its own value in el Pleno are the \emph{pure ratios}, which geometry produces without any ruler.

\textbf{Energy per region: the density $\rhoP$.} If energy is finite \emph{at each
point}, then we are already speaking of energy \emph{per region}; and energy per
region is, literally, a density. Here $\rhoP$ is born ---not as a new object, but
as energy itself read per volume. The same quantity admits three readings: as
density ($\rhoP$), as inertia ($M$) and as substance ($E$). Under units of the medium
itself ($c=1$, $\lP=1$, so that the cell volume is $\lP^3=1$ and $E=Mc^2$ reads
$E=M$), the three collapse into a chain of identities:
\begin{equation}
\rhoP \;\equiv\; M \;\equiv\; E . \tag{E0a}
\end{equation}
There is no causal arrow: never ``from $E$ arises $\rhoP$'', always ``$E$, read per region,
\emph{is} $\rhoP$''. It is a single object read from three angles.

\textbf{The region has structure: the minimal grain $\lP$.} To speak of ``energy per
region'' presupposes that the region is well defined; a finite density in a
continuous substrate needs a minimal grain of coherence, below which the notion
of ``region'' dissolves. Hence $\lP$, the coherence scale of el Pleno ---its minimal
grain of structure, the second substantive property with dimensions. It is born
\emph{because} we speak of density in a structured medium; its full role (regulating the
curvature of the field) arrives with the flexural term of F1.

\textbf{Energy changes: the change ceiling $c$.} Energy is finite but
\emph{variable}. Change itself inherits from A1 a wall of finiteness: there exists a maximum
capacity of change, $c$. It is not ``the speed of light'' ---a velocity presupposes
space and time already given--- but the maximum rate of change of energy itself, a
property prior to the metric. The internal change $C$ of a configuration is
bounded, $C\in[0,c]$: $C=0$ is energy at rest (rest mass), $C=c$ is change at the ceiling
(pure propagation, the massless mode). Being causal structure and not a property that
the medium ``chooses'', $c$ is classified apart from $\rhoP$ and $\lP$.

\textbf{With $c$ available, time emerges.} Time is not postulated: it is read as the
amount of accumulated change, measured against the maximum possible change whose scale is fixed by
$c$. Without a change ceiling (if $c\to\infty$) there would be no common rhythm and time would
not be defined. The causal direction is not circular: the change $C$ is internal to
energy and requires no clock; from it its extreme $c$ is taken; and from $c$ time is obtained
as accumulated change. Time is the last thing to appear in this stretch:
it emerges, it is not postulated ---coherent with the program's thesis that space and time
are emergent properties of el Pleno, not primitive entities, and that the
``Minkowski spacetime'' is the effective metric of the propagation of the modes
in the relaxed regime.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 1: the substrate and its properties}]
From a single statement ---the universe is finite, conserved energy--- are born, forced
in a chain, the density $\rhoP$ (energy per region), the minimal grain $\lP$ (the region
has structure), the change ceiling $c$ (energy varies) and time (accumulated
change against $c$). None is postulated separately: each is demanded by the
previous one.
\end{tcolorbox}

\subsection{The balance, \texorpdfstring{$E=Mc^2$}{E=Mc2} and the field \texorpdfstring{$\Psi$}{Psi} --- \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{Conservation as equilibrium: the balance.} Energy is conserved and has only
two destinies: to be retained (confined, permanence, mass) or to be propagating
(propagating, change, radiation). There is no third destiny, so every transformation
is a transit between those two branches: what leaves one enters the other. With
$E_\text{conf}$ and $E_\text{prop}$ the energies of each branch, local conservation is
already a continuity equation,
\begin{equation}
\Delta E_\text{conf}+\Delta E_\text{prop}=0
\;\;\equiv\;\;
\partial_t\rho_E+\nabla\cdot j_E=0 , \tag{E0b}
\end{equation}
legitimate because time already emerged in the previous link. Expressed in the common
currency of change, the equilibrium between the two branches is
\begin{equation}
\boxed{\;\frac{E}{c}-Mc=0\;}\qquad\Longleftrightarrow\qquad \underbrace{\frac{E}{c}}_{\text{dynamic change}}=\underbrace{Mc}_{\text{static change}}. \tag{E0c}
\end{equation}

\textbf{Why the two terms are comparable: both are momentum.} The balance is
legitimate because its two terms have the same dimension:
\[
\Big[\frac{E}{c}\Big]=\frac{\text{J}}{\text{m/s}}=\text{kg}\cdot\frac{\text{m}}{\text{s}},
\qquad
[Mc]=\text{kg}\cdot\frac{\text{m}}{\text{s}},
\]
both \emph{momentum} (kg$\cdot$m/s), the natural measure of change. To subtract $E/c-Mc$ is
to subtract two momenta: two readings of change in the same unit. The term $E/c$ is
energy read as \emph{dynamic} change capacity (the propagating change); $Mc$ is
inertia read as \emph{static} change (the change retained in the permanence of the
mass).

\textbf{The balance is the Einstein equation read another way.} Solving,
\[
\frac{E}{c}-Mc=0\;\Longrightarrow\;\frac{E}{c}=Mc\;\xrightarrow{\;\times c\;}\;E=Mc^2 .
\]
To multiply by $c$ is not a trick: it is to pass from the currency of momentum (kg$\cdot$m/s) to
that of energy (J). The $c^2$ is not opaque ---it is a $c$ of the balance (the ceiling that defines
the currency) times the $c$ of the unit conversion: two appearances of the \emph{same}
single $c$, one for each branch that the transit connects. Thus, $E=Mc^2$ is not a law of
conversion between two distinct substances: it is the assertion that the two readings of the
change of one and the same conserved energy ---the propagating $E/c$ and the retained $Mc$--- are
in equilibrium. Mass does not ``transform into'' energy; mass \emph{is} retained
energy. Einstein wrote the energetic face ($E=Mc^2$); the program exhibits the face
of the balance ($E/c=Mc$), the same law one layer below, where one sees why the factor
is $c^2$ and what balances.

\textbf{The balance made object: the field $\Psi$.} The balance confronts two terms that
are the same energy read on two axes: the static (the retained, $Mc$) and the dynamic
(the propagating, $E/c$). Since a real configuration has a retained part \emph{and} a propagating
part at once ---it can persist while it changes--- the two axes must be
independent, at $90^\circ$. The operator that keeps two orthogonal magnitudes without
one contaminating the other is the imaginary unit $i$. That is why energy is written
as a complex number:
\begin{equation}
\boxed{\;\Psi=S\,e^{i\theta}\;}\qquad S\geq 0,\quad\theta\in[0,2\pi), \tag{E1}
\end{equation}
with $S$ (the modulus) the \emph{static} axis ---the retained quantity, the permanence, what
$Mc$ measures--- and $\theta$ (the rotating phase) the \emph{dynamic} axis ---the change in
course, the propagation, what $E/c$ measures. That modulus and phase be independent
($|\Psi|^2=S^2$ does not depend on $\theta$) \emph{is} the assertion that the two branches
of the balance are conserved separately. $\Psi$ is the balance $E/c=Mc$ made a geometric
object. The modulus inherits from A1 the bounding on both sides,
\[
S\in[\Smin,\Smax],\qquad \Smin>0\ (\text{from }E>0),\qquad \Smax<\infty\ (\text{from }E<\infty),
\]
the two walls of the finiteness of the quantity, which are realized as two terms of the
potential (IR wall at $S\to\Smin$, hard wall at $S\to\Smax$). The status of this step
is explicit: the program \emph{postulates} $\Psi=Se^{i\theta}$; the chain offers the
ontological why of its form ---the balance of two orthogonal branches--- making the language
continuous from energy to F1, without an initial unexplained jump.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 2: the balance and the field}]
Conservation, expressed as equilibrium between dynamic and static change
($E/c-Mc=0$), \emph{is} the Einstein equation one layer below, and its structure of
two orthogonal branches is exactly what the field $\Psi=Se^{i\theta}$ encodes: the
field is not an aesthetic postulate but the geometric writing of the balance.
\end{tcolorbox}

\subsection{The balance with motion: the constraint and the Lorentz factor --- \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{From the balance at rest to the balance with motion.} The equilibrium $E/c=Mc$
describes the extremes (energy all retained or all propagating). A real configuration
---matter in motion--- has a retained part and a propagating part at once. What
measures the propagating fraction is the \emph{momentum} $p$: the amount of change that what
moves carries with it (and change is measured in momentum, kg$\cdot$m/s). The retained
fraction contributes $Mc$; the propagating one contributes $p$.

\textbf{Why the combination is Pythagorean.} Retained and propagating, when they coexist,
are the two orthogonal branches of the balance ---the same ones that give the $i$ of $\Psi$. And
orthogonal magnitudes do not add linearly: they compose as legs. The total energy
is the hypotenuse, in the currency of momentum:
\[
\Big(\frac{E}{c}\Big)^2=(Mc)^2+p^2
\;\xrightarrow{\;\times c^2\;}\;
\boxed{\;E^2=(Mc^2)^2+(pc)^2\;}
\]
It is the mass-shell constraint, and it is born from the same generalized balance, not from an
imported law. The orthogonality of the two branches ---the same one that gives the $i$--- is what
turns the linear subtraction $E/c-Mc=0$ into a Pythagorean sum. At rest ($p=0$) one recovers
$E=Mc^2$; at the other extreme ($M=0$) one recovers $E=pc$, the pure energy of propagation
(photon, gravitational wave).

\textbf{From the constraint, the Lorentz factor.} Defining the retained fraction
$1/\gamma\equiv Mc^2/E$ (that is $E=\gamma Mc^2$) and substituting:
\[
(\gamma Mc^2)^2=(Mc^2)^2+(pc)^2
\;\Longrightarrow\;
(pc)^2=(Mc^2)^2(\gamma^2-1)
\;\Longrightarrow\;
p=\gamma Mc\sqrt{1-\gamma^{-2}} .
\]
Identifying the velocity as the ratio between propagating change and retained energy,
$v/c\equiv pc/E=pc/(\gamma Mc^2)$, one gets $v/c=\sqrt{1-\gamma^{-2}}$, that is
$\gamma^{-2}=1-v^2/c^2$, and therefore
\begin{equation}
\boxed{\;\gamma=\frac{1}{\sqrt{1-v^2/c^2}},\qquad E=\gamma Mc^2,\qquad p=\gamma Mv.\;}
\end{equation}
The Lorentz factor is the partition of the change capacity between the two branches, tied
by the conservation of energy; it is not the deformation of a prior spacetime
---the Lorentz spacetime \emph{describes} this partition a posteriori. The covariance
(axiom A3) emerges from the balance, not from a postulated stage. The status is
\epi{SLD}: the algebra is exact, but it rests on $c$ being causal structure
---the fragile link of the whole foundation, which is declared and not concealed. The
equivalent quantitative derivation from the kinetic invariant of F1 is in
§\ref{sec:reologia} (E49--E50): they do not compete, they are two readings of the same fact.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 3: relativity emerges from the balance}]
The constraint $E^2=(Mc^2)^2+(pc)^2$ and the Lorentz factor are not imported: they are the balance
with motion, with the Pythagorean sum forced by the same orthogonality of the two
branches that defines $\Psi$. Special relativity remains as an effective layer of el Pleno.
\end{tcolorbox}

\subsection{The six axioms and the bridge to F1 --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The previous chain rests, at each link, on a minimal set of six axioms
that are formalized here as a synthesis of what has already been covered:

\begin{center}\small
\begin{longtable}{C{1.0cm}L{3.6cm}L{8.2cm}}
\toprule
\textbf{Ax.} & \textbf{Name} & \textbf{Abridged statement}\\
\midrule
\endhead
A1 & Energy-substance & The universe \emph{is} energy: finite at each point ($0<E<\infty$) and conserved. Density, inertia and energy are the same quantity ($\rhoP\equiv M\equiv E$). Root of the whole chain.\\
A2 & Positivity / IR wall & The modulus is strictly positive, $S>0$; $S=0$ is inaccessible (divergent energy). Fixes $\Smin>0$ (lower face of the finiteness of A1).\\
A3 & Causal covariance & There exists a change ceiling $c$; the dynamics is Lorentz-covariant. Emerges from the balance (link 3), it is not postulated as a prior stage.\\
A4 & UV regulation / rigidity & The field penalizes curvature: flexural term $(\Box S)^2$, with the grain $\lP$. Gives the ultraviolet cutoff $\Lambda=1/\lP$ and the minimal width $\sigma_{\min}=\lP/\sqrt{20}$.\\
A5 & U(1) symmetry & The action is invariant under $\theta\to\theta+\alpha$ (the freedom of the phase, dynamic axis of $\Psi$). Gives the conserved Noether~\cite{noether} current and the charge (quantized by topology).\\
A6 & Hard bound / saturation & The modulus is bounded above, $S\leq\Smax$, by maximal packing of the lattice. Blocks singularities; gives the bounce.\\
\bottomrule
\end{longtable}
\end{center}

The economy is remarkable: A2 (positivity) is the lower face of the finiteness of A1; A3
(covariance) emerges from the balance; A5 (U(1)) is the freedom of the phase that $\Psi$ already
has. The only thing that neither finiteness nor conservation delivers ---the continuity and
differentiability of the field, which F1 needs since it operates with derivatives of $\Psi$---
remains as the genuinely independent technical axiom.

\textbf{Primary parameters and convention.} The medium is characterized by two
substantive properties with dimensions ($\rhoP$, $\lP$) and a causal structure ($c$):
\[
\rhoP\approx 5.155\times 10^{96}\ \text{kg/m}^3,\quad
c\approx 2.998\times 10^{8}\ \text{m/s},\quad
\lP\approx 1.616\times 10^{-35}\ \text{m}.
\]
The constants $G$ and $\hbar$ are not inputs but consequences (§\ref{sec:constantes});
the independent calibration of $\rhoP$ or $\lP$ from non-gravitational and non-quantum
physics is an open end (§\ref{sec:cabos}). Computation over real values; presentation
to four figures, relative differences to two.

\textbf{El Pleno has no units; the units are placed by us.} The units that accompany
$\rhoP$, $\lP$ and $c$ above admit a precise reading. El Pleno
\emph{exists} and \emph{behaves} physically, and that behavior admits an exact
mathematical description; but the medium itself possesses no meters, kilograms or seconds.
Those units are not properties of el Pleno: they are rulers that we humans manufacture to measure.
One must distinguish two things that ordinary language confuses. That an edge
\emph{exists} ---a density ceiling, a minimal grain, a maximum rate of change---
is a property of el Pleno, ontological and independent of all measurement. That that edge
\emph{takes such a number} is not: a number appears only upon measuring, and to measure is to compare
against a human ruler. That is why the ``value'' of $\rhoP$, $\lP$ or $c$ does not inform about el
Pleno itself, but about \emph{how many human rulers fit} in the corresponding ceiling
---it is information about the relation between el Pleno and our units. In units of the
medium itself ($c=1$, $\lP=1$) those numbers vanish, but the ceilings remain: the
structure does not depend on the ruler with which it is measured. The question ``what number did el
Pleno choose for $\rhoP$?'' is ill-posed, because it presupposes a category ---the
number--- that el Pleno does not have; the legitimate question is to measure its human reading from
an independent physics (open end C-meta-1.b, §\ref{sec:cabos}). What in el Pleno \emph{does}
have its own value, without any ruler, are the \textbf{pure ratios} between its
structures ($\gamma_\kappa=1/40$, $\sqrt{3}\pi$, $\Smin^\text{estr}$,
$\sin^2\theta_\text{tet}=8/9$), which the diamond geometry produces as dimensionless
numbers: that is why they, and not the values with units, are the ones the program
derives with intrinsic value.

The Lagrangian of el Pleno, \textbf{F1} (§\ref{sec:F1}), is the dynamical starting point
of the program. The diamond lattice, the pure ratios and the fundamental constants are not
foundations prior to F1 but consequences of it: they are obtained upon evaluating it, in the
sections that follow.

\medskip\hrule\medskip
% =====================================================================

\section{The F1 Lagrangian of el Pleno --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:F1}
% =====================================================================

The Lagrangian of el Pleno is named \textbf{F1}; its symbol in the equations is
$\mathcal{L}_\text{F1}$. Applying Euler-Lagrange to it yields the dynamical equation of the
modulus, which is named \textbf{F2} (= E6). In this compendium ``F1'' and ``F2'' are used as
names in the prose, and $\mathcal{L}_\text{F1}$ as the symbol in the formulas.

The F1 Lagrangian has four terms ---kinetic, potential, flexural and
IR wall---. Its coefficients are not free inputs: the factor $1/40$ of the flexural
comes from evaluating F1 over the lattice that the medium itself crystallizes (§\ref{sec:1sobre40});
the $3/8$ of the IR wall, from the affine quantization of the half-line $S\geq 0$
(§\ref{sec:klauder}); the kinetic coefficient is identified with $c^4/2G$ once
$G$ is derived (§\ref{sec:constantes}); and $V(S)$ is the tripartite potential of
§\ref{sec:potencial}.

\subsection{Form of the Lagrangian and construction of each term --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The fundamental field is $\Psi$ (E1). The standard Lagrangian procedure
---identify the field, impose symmetries, construct compatible terms, fix
coefficients--- applied to el Pleno under the axioms A1--A6 produces a unique
minimal Lagrangian density:
\begin{equation}
\boxed{\;\mathcal{L}_\text{F1}=\tfrac{1}{2}\rhoP c^2\lP^2\,\partial_\mu\Psi^*\partial^\mu\Psi
-V(S)-\frac{\rhoP c^2\lP^4}{40}(\Box S)^2-V_q^\text{dens}(S)\;} \tag{E2}
\end{equation}
The coefficients $1/40$ and $3/8$ that appear in (E2) are derived in
§\ref{sec:1sobre40} and §\ref{sec:klauder} respectively.

\emph{Reading: F1 is the single object from which the whole program is derived. It has no
free parameters ---each coefficient comes from the geometry of the medium---, so that
all subsequent physics is an evaluation of it, not a new postulate.}

\textbf{Kinetic term.} Expanding $\Psi=S\,e^{i\theta}$,
\[
\partial_\mu\Psi=(\partial_\mu S)e^{i\theta}+iS(\partial_\mu\theta)e^{i\theta},
\qquad
\partial_\mu\Psi^*=(\partial_\mu S)e^{-i\theta}-iS(\partial_\mu\theta)e^{-i\theta},
\]
so that the product, upon cancellation of the imaginary cross terms, is
\[
\partial_\mu\Psi^*\partial^\mu\Psi=(\partial_\mu S)(\partial^\mu S)+S^2(\partial_\mu\theta)(\partial^\mu\theta),
\]
\[
\mathcal{L}_\text{cin}=\tfrac{1}{2}\rhoP c^2\lP^2\big[(\partial_\mu S)(\partial^\mu S)+S^2(\partial_\mu\theta)(\partial^\mu\theta)\big].
\]
The first contribution is the structural propagation (radial mode); the second,
weighted by $S^2$, that of the induced gauge sector. The coefficient
$\rhoP c^2\lP^2$ is identified with $c^4/G$ (§\ref{sec:G}).

\textbf{Potential $-V(S)$.} A function exclusively of the modulus (the U(1) symmetry forbids
dependence on $\theta$). Tripartite structure $V(S)=W_\text{bare}+V_q+V_\text{topo}$,
developed in §\ref{sec:potencial}.

\textbf{Flexural term.} Axiom A4 (ultraviolet regulation) requires penalizing
the curvature of the field; the simplest canonical form compatible with the symmetries
is quadratic in the d'Alembertian of the modulus, $(\Box S)^2$, with
$\Box S=\partial_\mu\partial^\mu S$. The coefficient is
\begin{equation}
\gamma_\kappa=\frac{\rhoP c^2\lP^4}{40}=\frac{\hbar c}{40}\approx 7.90\times 10^{-28}\ \text{J}\cdot\text{m}, \tag{E4}
\end{equation}
with dimensions $[J\cdot m]$ identical to the stiffness $EI$ of Euler-Bernoulli (the
second equality uses $\hbar=\rhoP c\lP^4$). The factor $1/40$ is derived from the
geometry of the lattice in §\ref{sec:1sobre40}; the power $\lP^4$ (compared to $\lP^2$
of the kinetic) is what gives dimensions of energy density to a term with two
additional derivatives.

\textbf{Klauder term $-V_q^\text{dens}(S)$.} Infrared regulation (A2):
quantum wall that prevents $S\to 0$. Its origin and coefficient are developed in
§\ref{sec:klauder}:
\[
V_q^\text{dens}(S)=\frac{3}{8}\frac{\rhoP c^2}{S^2}=\frac{3\hbar^2}{8\,\rhoP\,\lP^8\,S^2}. \tag{E5}
\]

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 4: the Lagrangian is fixed}]
With the form of the field inherited from the balance, F1 is written: the minimal
mathematical expression of el Pleno, four terms with a structural function (to propagate,
sustain configurations, regulate UV, regulate IR) and no free coefficient ---all
built with the three primaries $\rhoP$, $c$ and $\lP$ ---and, where it aids readability, with $\hbar$ as shorthand for $\rhoP c\lP^4$ (E36), never as an additional datum. From F1 the
rest of the phenomenology of el Pleno is derived.
\end{tcolorbox}

\subsection{The flexural term over the complete field --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The writing $(\Box S)^2$ is shorthand notation for a regulator that acts over
$\Psi$ entirely, not only over the modulus. Expanding $\Box\Psi$ around the background
$\Psi_0=S_0 e^{i\theta_0}$ and retaining the quadratic order in the perturbations
$\delta S$, $\delta\theta$:
\begin{equation}
\mathcal{L}_\text{flex}^{(2)}=-\frac{\rhoP c^2\lP^4}{40}\Big[(\Box\,\delta S)^2+S_0^2(\Box\,\delta\theta)^2\Big]+\mathcal{O}(\delta^3). \tag{E4c}
\end{equation}
The phase sector inherits the same regulator, weighted by $S_0^2$ exactly like
the kinetic phase term. Both come from the same link-difference operator of
the lattice ($|\Psi_n-\Psi_m|^2=(s_n-s_m)^2+S_0^2(a_n-a_m)^2$, §\ref{sec:1sobre40}), whence
the same ratio $1/40$ and the same ultraviolet correction $(k\lP)^2/20$ for
photon and TT mode.

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The UV regulation does not distinguish modulus from phase nor scalar from tensor structure, because
it is a property of the lattice operator that the flexural encodes. Hence light and the
gravitational wave share not only the velocity $c$ but also their first
Planckian correction $(k\lP)^2/20$.
\end{tcolorbox}

\subsection{The Klauder quantum potential and the coefficient \texorpdfstring{$3/8$}{3/8} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:klauder}

Since the modulus satisfies $S\geq 0$, it defines a configuration manifold of the
half-line type, and standard canonical quantization (operators over the whole line) does not
apply without pathologies. The correct quantization is the \textbf{affine Klauder}, which
promotes to operators the modulus $\hat S$ and the generator of dilations
$\hat D=(\hat S\hat\Pi+\hat\Pi\hat S)/2$, with the affine algebra
\[
[\hat S,\hat D]=i\hbar\hat S
\]
instead of the canonical $[\hat S,\hat\Pi]=i\hbar$. The $\hbar$ appearing here abbreviates $\rhoP c\lP^4$, the structural identity that §\ref{sec:hbar} derives from the $U(1)$ rotor of the phase mode; that derivation passes through neither the affine algebra nor the potential that follows, so the two chains do not feed each other. The difference is structural: the
affine algebra preserves exactly $S>0$ at the quantum level.

\textbf{The coefficient $3/4$ (Klauder~\cite{klauder} result, imported).} Over the
half-line with the invariant measure $dS/S$, the symmetrized Weyl ordering
produces, for the kinetic operator re-expressed in affine variables, the term
\[
\frac{\hat\Pi^2}{2m_\text{eff}}=\frac{1}{2m_\text{eff}\hat S^2}\Big(\hat D^2+\frac{3\hbar^2}{4}\Big),
\]
so that the Hamiltonian acts on $\psi(S)$ as
\[
\hat H_K\psi=-\frac{\hbar^2}{2m_\text{eff}}\frac{\partial^2\psi}{\partial S^2}+\frac{3\hbar^2}{8\,m_\text{eff}S^2}\psi,
\qquad m_\text{eff}=\rhoP\lP^2 .
\]
The coefficient $3/4$ of the term $\hbar^2/S^2$ is the standard result of the
affine Klauder quantization (2000) for a variable over the half-line with
measure $dS/S$; it depends crucially on that measure, not on the Lebesgue one. PIU-$\Psi$
\emph{imports} this theorem under its rules for the use of mathematical tools, instead
of re-deriving it. The factor $3/8=3/4\times 1/2$ combines that $3/4$ with the kinetic
factor $1/2$.

\textbf{Energy density.} Converted to energy density to enter the
Lagrangian (cell-volume factor $\lP^{-3}$ and kinetic factor):
\begin{equation}
V_q^\text{dens}(S)=\frac{3}{8}\frac{\rhoP c^2}{S^2}=\frac{3\hbar^2}{8\,\rhoP\,\lP^8\,S^2}, \tag{E5}
\end{equation}
where the first form is the native one ---the coefficient of the IR wall is written with
$\rhoP$ and $c$ alone, with no imported constant--- and the second is its rewriting with
$\hbar=\rhoP c\lP^4$ (E36). It diverges as $S\to 0$.

\textbf{Quantitative inaccessibility of $S=0$.} The energy cost of taking the
medium from the ground-Pleno $\Smin^\text{estr}\approx 0.7231$ to $S=\epsilon$ is
\[
\Delta E(\epsilon)=\frac{3}{8}\rhoP c^2\Big(\frac{1}{\epsilon^2}-\frac{1}{(\Smin^\text{estr})^2}\Big)
\xrightarrow{\;\epsilon\to 0\;}+\infty .
\]

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The first of the two coefficients of F1 is fixed: the IR wall prevents the
medium from ``switching off'', since $S\to 0$ costs infinite energy ---it is not a condition
placed by hand. The $3/8$ is a result \emph{imported} from Klauder, not an internal
derivation; it enters F1 without recalibration, since once the affine measure is fixed the
coefficient is unique.
\end{tcolorbox}

\subsection{Dimensional verification of the four terms --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

With the three primaries $[\rhoP]=kg/m^3$, $[c]=m/s$, $[\lP]=m$ ---and $[\hbar]=J\,s$ as a quantity \emph{derived} from them, not as a fourth independent dimension---, $S,\theta$
dimensionless, $[\partial_\mu]=1/m$, $[\Box]=1/m^2$:
\[
[\mathcal{L}_\text{cin}]=\frac{kg}{m^3}\cdot\frac{m^2}{s^2}\cdot m^2\cdot\frac{1}{m^2}=\frac{kg}{m\,s^2}=\frac{J}{m^3}\ \checkmark
\]
\[
[\mathcal{L}_\text{flex}]=\frac{kg}{m^3}\cdot\frac{m^2}{s^2}\cdot m^4\cdot\frac{1}{m^4}=\frac{J}{m^3}\ \checkmark,
\qquad
[\gamma_\kappa]=\frac{kg}{m^3}\cdot\frac{m^2}{s^2}\cdot m^4=\frac{kg\,m^3}{s^2}=J\cdot m\ \checkmark
\]
\[
[\mathcal{L}_q]=\frac{(J\,s)^2}{(kg/m^3)\,m^8}
=\frac{(kg\,m^2/s^2)^2 s^2}{kg\,m^5}
=\frac{kg}{m\,s^2}=\frac{J}{m^3}\ \checkmark
\]
The action $\mathcal{S}=\int\mathcal{L}_\text{F1}\,dt\,d^3x$ has dimensions
$[J/m^3]\cdot[s\,m^3]=[J\cdot s]$, identical to those of $\rhoP c\lP^4$ ---that is, to those
of $\hbar$ by (E36)---, a necessary condition for the action measured in quanta of the medium
to be dimensionless.

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The four terms share the dimension of energy density with the same three
primaries. The consistency is not trivial ---it forces the flexural to carry $\lP^4$ and the
IR wall $\lP^8$, without freedom. F1 is dimensionally armored.
\end{tcolorbox}

\subsection{Dynamical equations: Euler-Lagrange over F1 --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Since F1 contains $(\Box S)^2$ (second derivatives), the Euler-Lagrange equations
carry the third-order term:
\[
\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}+\partial_\mu\partial_\nu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\partial_\nu\phi)}=0 .
\]

\textbf{Modulus equation (F2).} The partial derivatives of F1 with respect to $S$:
\[
\frac{\partial\mathcal{L}_\text{F1}}{\partial S}=\rhoP c^2\lP^2 S(\partial_\mu\theta)(\partial^\mu\theta)-V'(S)-V_q'(S),
\]
\[
\frac{\partial\mathcal{L}_\text{F1}}{\partial(\partial_\mu S)}=\rhoP c^2\lP^2\,\partial^\mu S,
\qquad
\frac{\partial\mathcal{L}_\text{F1}}{\partial(\partial_\mu\partial_\nu S)}=-\frac{\rhoP c^2\lP^4}{20}(\Box S)\eta^{\mu\nu},
\]
where the $1/20$ appears upon differentiating $(\Box S)^2/40$ with respect to $\Box S$ (factor $2$).
Substituting and grouping:
\begin{equation}
\boxed{\;\rhoP c^2\lP^2\big[\Box S-S(\partial_\mu\theta)(\partial^\mu\theta)\big]-\frac{\rhoP c^2\lP^4}{20}\Box(\Box S)+V'(S)+V_q'(S)=0\;} \tag{E6}
\end{equation}
Primordial equation F2: a fourth-order differential (through $\Box(\Box S)$), with
kinetic, induced-gauge, potential and IR-wall terms.

\textbf{Phase equation.} Since F1 does not depend explicitly on $\theta$
(U(1) symmetry), $\partial\mathcal{L}/\partial\theta=0$, and with the phase flexural
term of (E4c):
\[
\frac{\partial\mathcal{L}_\text{F1}}{\partial(\partial_\mu\theta)}=\rhoP c^2\lP^2 S^2\partial^\mu\theta,
\qquad
\frac{\partial\mathcal{L}_\text{F1}}{\partial(\partial_\mu\partial_\nu\theta)}=-\frac{\rhoP c^2\lP^4}{20}S^2(\Box\theta)\eta^{\mu\nu}.
\]
The Euler-Lagrange equation gives, factoring out the common kinetic coefficient $\rhoP c^2\lP^2$ ---which §\ref{sec:G} will identify with $c^4/G$, without that identification intervening here---:
\begin{equation}
\boxed{\;\partial_\mu(S^2\partial^\mu\theta)-\frac{\lP^2}{20}\Box(S^2\Box\theta)=0\;} \tag{E7}
\end{equation}
The associated U(1) Noether current is
\begin{equation}
J^\mu=\rhoP c^2\lP^2 S^2\Big[\partial^\mu\theta+\frac{\lP^2}{20}\partial^\mu(\Box\theta)\Big],\qquad \partial_\mu J^\mu=0\ \text{(exact over the solutions)}. \tag{E7c}
\end{equation}
On the background $S=\Smin^\text{estr}$ constant, (E7) reduces to
\[
\Box\theta-\frac{\lP^2}{20}\Box^2\theta=0,
\]
with physical infrared branch $\Box\theta=0$ (luminal propagation, Goldstone boson
of U(1)) and ultraviolet branch $\Box\theta=20/\lP^2$ (Lee-Wick~\cite{leewick} pole). Over the
massless physical branch the flexural term of the current vanishes and $J^\mu$
reduces to the standard current $\rhoP c^2\lP^2 S^2\partial^\mu\theta$, identifiable
with the electric current (§\ref{sec:gauge}).

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 4 closed: the dynamics of F1}]
By varying F1 one obtains the whole dynamics of the program, reduced to two coupled
equations: F2 (fourth order, modulus) and the U(1) continuity (phase). The fourth order is
not a defect: it is the ultraviolet regulation made equation, with the Lee-Wick pole as its
imprint.
\end{tcolorbox}

\medskip\hrule\medskip
% =====================================================================

\section{The diamond lattice: first evaluation of F1 --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:1sobre40}
% =====================================================================

In the saturated regime, the flexural term $(\Box S)^2$ of F1 penalizes the curvature
of the field; the configuration that minimizes it with maximal isotropic coordination is the diamond
lattice $Fd\bar3m$ ($Z=4$, isostatic by Maxwell counting). Evaluating the Laplacian of
F1 over that lattice produces the two pure ratios of the medium ---the ground-Pleno
$\Smin^\text{estr}$ and the flexural coefficient $1/40$---, which the Lagrangian (E2) carried
written without deriving.

\textbf{Why the diamond lattice.} The choice is not a stipulation: four independent
physical criteria, each a consequence of an axiom, converge on the same
structure. \emph{(i) Minimal stable coordination (A4).} The term $(\Box S)^2$
penalizes curvature; the lattice that minimizes the flexural energy subject to mechanical
stability in 3D is that of the \emph{lowest coordination that is still rigid}. By Maxwell
counting, a lower coordination leaves soft modes (unstable medium) and a higher one
over-constrains and penalizes energetically; the optimum is $Z=4$. \emph{(ii)
Maximal isotropy.} Among the lattices with $Z=4$, the \emph{tetrahedral} coordination
($109.47^\circ$) is the only one that distributes the four bonds in a perfectly
isotropic way, $\sum_i\boldsymbol{\delta}_i=0$ and
$\sum_i\delta_i^a\delta_i^b=\tfrac{4}{3}\lP^2\delta^{ab}$ (the two identities below):
the second-moment tensor is proportional to the identity, without privileged
directions, as the emergent covariance (A3) requires. \emph{(iii) Bipartition (A5).}
The diamond lattice is bipartite (A/B sublattices), which admits the $\mathbb{Z}_2$
coloring from which the U(1) phase and, higher up, the gauge sector hang; a non-bipartite
lattice would not support this structure. \emph{(iv) Spin quantization (Finkelstein-Rubinstein).}
The topology of the diamond lattice fixes $\chi_{FR}=-1$ for the matter solitons, that
is, fermionic statistics, without postulating it.

\textbf{The geometry of the lattice is derived in its two pieces}, the \emph{coordination}
(D1-a) and the \emph{stacking} (D1-b):

\emph{D1-a (tetrahedral coordination).} The tetrahedral coordination $Z=4$ is
the only arrangement of four nearest neighbors in 3D compatible with A1, and this is
\textbf{proven}, not only verified numerically. Let
$\mathbf{v}_1,\dots,\mathbf{v}_4$ be the unit bond vectors. The absence of
dipole moment of the coordination (every node of a lattice with an inversion center) gives
$\sum_i\mathbf{v}_i=0$, whence
\begin{equation}
0=\Big|\sum_i\mathbf{v}_i\Big|^2=\sum_i|\mathbf{v}_i|^2+2\!\sum_{i<j}\mathbf{v}_i\cdot\mathbf{v}_j
=4+2\!\sum_{i<j}\mathbf{v}_i\cdot\mathbf{v}_j
\;\Rightarrow\;\sum_{i<j}\mathbf{v}_i\cdot\mathbf{v}_j=-2. \tag{E13-tet2}
\end{equation}
The second-order isotropy, $\sum_i v_i^a v_i^b=\tfrac43\delta^{ab}$, forces the
six scalar products to be equal; with sum $-2$, each equals $-1/3$, that is
$\cos\theta_{ij}=-\tfrac13$ for every pair, the tetrahedral angle
$\theta=\arccos(-\tfrac13)=109.47^\circ$. A set of four points on $S^2$ with
isotropic second-moment matrix is a spherical 2-design, unique up to rotation: the
regular tetrahedron (Delsarte-Goethals-Seidel 1977). The direct numerical optimization
($100\%$ of the seeds converge to the tetrahedron, residual cost $\sim10^{-12}$) confirms
this closed result, it does not replace it. \epi{DERIVED} (conditional only on the class of
Bravais lattices with a centrosymmetric coordination basis).

\emph{D1-b (stacking).} Given the tetrahedral coordination $Z=4$, the
diamond stacking (cubic $Fd\bar3m$) is the only one against wurtzite (hexagonal
$P6_3mc$) and every tetrahedral polytype with a distinguished stacking axis. The
proof (next subsection) is the minimum of the isotropic flexural functional of F1: the
fourth-order uniaxial anisotropy of the modulation equals $0$ for the cubic and $4/81$
(exact, irreducible, anchored to the invariant $1/40$ of $\gamma_\kappa$) for the hexagonal,
of strictly positive flexural cost. El Pleno, a continuous medium that relaxes to the minimum of the
flexural functional, stabilizes in the only modulation that does not concentrate curvature: the
cubic stacking. \epi{DERIVED}. The geometry of the lattice is derived in its two
pieces: the diamond lattice is the configuration of the saturated Pleno without residue of choice.

\subsection{Geometry of the lattice and bond vectors --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Bipartite diamond lattice $Fd\bar{3}m$ (A, B sublattices), coordination $Z=4$. Nearest-neighbor
vectors from a node of A:
\[
\boldsymbol{\delta}_1=\frac{\lP}{\sqrt{3}}(1,1,1),\ \
\boldsymbol{\delta}_2=\frac{\lP}{\sqrt{3}}(1,-1,-1),\ \
\boldsymbol{\delta}_3=\frac{\lP}{\sqrt{3}}(-1,1,-1),\ \
\boldsymbol{\delta}_4=\frac{\lP}{\sqrt{3}}(-1,-1,1).
\]
Two exact identities (verified symbolically):
\[
\sum_{i=1}^4\boldsymbol{\delta}_i=0,
\qquad
\sum_{i=1}^4\delta_i^a\delta_i^b=\frac{4\lP^2}{3}\,\delta^{ab}.
\]
Structure factor $f(\mathbf{k})=\sum_{i=1}^4 e^{i\mathbf{k}\cdot\boldsymbol{\delta}_i}$;
the lattice Hamiltonian is the $2\times2$ matrix over the bipartition with eigenvalues
$\epsilon_\pm=\pm K_\text{cin}|f(\mathbf{k})|$.

\textbf{Bipartition.} The A and B sublattices are each FCC; B is displaced from A by
$\boldsymbol{\delta}_0=(a_\text{diamond}/4)(1,1,1)$ with $a_\text{diamond}=4\lP/\sqrt3$, and
every bond connects A with B. This bipartition is the $\mathbb{Z}_2$ coloring that the
axiom A5 realizes as condensation of the phase $\theta$, and from it hangs the gauge sector
(§\ref{sec:gauge}).

\textbf{Brillouin zone.} The BZ of the diamond lattice is the truncated octahedron, with
high-symmetry points $\Gamma,X,L,W,K,U$. At the point $L$ one verifies $|f(L)|^2=2$,
a geometric manifestation of the invariant $\sqrt2$ of the Kepler-Hales family that reappears
in the hard bound $S_{\max,\text{geom}}^2=8\sqrt6/9$ (§\ref{sec:smax}).

\textbf{Evasion of the Nielsen-Ninomiya theorem.} The Nielsen-Ninomiya theorem (1981)
establishes that a fermionic field defined over a regular hypercubic discrete lattice
inevitably produces unphysical ``doublers'' (\emph{doubling problem}). In PIU-$\Psi$
this obstruction does not apply: the fermions of the program are not point fields defined
over lattice sites, but \textbf{topological solitonic modes} ---skyrmions over the
bipartite sublattice A/B with $Z=4$ (§\ref{sec:solitones})---, so that the doubling problem
is orthogonal to the formulation.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{red_diamante.png}
\caption{The diamond lattice $Fd\bar{3}m$ and the tetrahedral coordination $Z=4$. The
four nearest-neighbor vectors $\boldsymbol{\delta}_i$ are the geometric basis of
the whole block: from $\sum_i\boldsymbol{\delta}_i=0$ and
$\sum_i\delta_i^a\delta_i^b=\tfrac{4\lP^2}{3}\delta^{ab}$ follow the discrete
Laplacian (E13), the Brillouin-zone average and the form factor
$\Smin^\text{estr}=12(\pi^2-4)/\pi^4$.}
\end{figure}

\subsection{Discrete / continuous Laplacian relation --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Expanding $f$ at low $\mathbf{k}$ with $\sum_i\boldsymbol{\delta}_i=0$ and
$\sum_i(\mathbf{k}\cdot\boldsymbol{\delta}_i)^2=(4\lP^2/3)k^2$:
\[
f(\mathbf{k})\approx 4-\frac{2\lP^2}{3}k^2,
\qquad
|f(\mathbf{k})|^2\approx 16-\frac{16\lP^2}{3}k^2 .
\]
Defining the discrete Laplacian $D_\text{lat}(\mathbf{k})\equiv 16-|f(\mathbf{k})|^2$,
with $D_\text{lat}(0)=0$ and $D_\text{lat}\approx\tfrac{16}{3}(k\lP)^2$, the inversion at
low $\mathbf{k}$ ($\nabla^2\leftrightarrow-k^2$) gives
\begin{equation}
\boxed{\;\nabla^2\;\longleftrightarrow\;-\frac{3}{16\lP^2}\,D_\text{lat}\;} \tag{E13}
\end{equation}
The factor $16/3$ is unique: $16=Z^2$ (square of the coordination) and $3$ from the three
dimensions of the average.

\subsection{Spectral average over the Brillouin zone and \texorpdfstring{$\Smin^\text{estr}$}{Smin} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

This is the step that closes the spectral route to $\Smin^\text{estr}$. Starting from
\[
|f(\mathbf{k})|^2=4+2\sum_{i<j}\cos\big[\mathbf{k}\cdot(\boldsymbol{\delta}_i-\boldsymbol{\delta}_j)\big],
\]
each difference $\boldsymbol{\delta}_i-\boldsymbol{\delta}_j$ has, in cubic-edge
coordinates $a$, exactly two nonzero components of magnitude $a/2$ (the
six differences are permutations of $(0,\pm a/2,\pm a/2)$). The average of each
cosine over the cubic reciprocal cell $[-\pi/a,\pi/a]^3$ factorizes:
\[
\Big\langle\cos\big[\mathbf{k}\cdot(\boldsymbol{\delta}_i-\boldsymbol{\delta}_j)\big]\Big\rangle
=\left(\frac{1}{2\pi/a}\int_{-\pi/a}^{\pi/a}\cos\!\frac{a k}{2}\,dk\right)^2
=\left(\frac{\sin(\pi/2)}{\pi/2}\right)^2=\left(\frac{2}{\pi}\right)^2 .
\]
\textbf{Explicit symbolic verification} (sympy, integral of the six differences):
each pair gives $4/\pi^2$ and the sum of the six pairs is $24/\pi^2$. Therefore the numerator
of the form factor is
\[
\langle|f|^2\rangle_\text{BZ}=4+2\cdot 6\cdot\frac{4}{\pi^2}=4+\frac{48}{\pi^2},
\qquad
\langle D_\text{lat}\rangle_\text{BZ}=16-\langle|f|^2\rangle_\text{BZ}=12-\frac{48}{\pi^2}=7.1366 .
\]

\textbf{The denominator: average of the tangent parabola.} The form factor compares the
\emph{real} Laplacian of the band with the one an ideal continuous medium would have, that is,
with the parabola tangent to $D_\text{lat}$ at $k\to 0$. From the low-$\mathbf{k}$
expansion (E13), that parabola is
$D_\text{lat}^\text{par}(\mathbf{k})=\tfrac{16}{3}\lP^2 k^2$. In cubic-edge
coordinates $a$, with the geometric relation $|\boldsymbol{\delta}_i|=\lP$ that gives
$a=4\lP/\sqrt3$ (that is $\lP^2=3a^2/16$), the parabola is simply
$D_\text{lat}^\text{par}=a^2 k^2=a^2(k_x^2+k_y^2+k_z^2)$. Its average over the same
cubic reciprocal cell $[-\pi/a,\pi/a]^3$, using
$\langle k_i^2\rangle=\tfrac{1}{2\pi/a}\int_{-\pi/a}^{\pi/a}k_i^2\,dk_i=\tfrac{\pi^2}{3a^2}$
per component, is
\[
\langle D_\text{lat}^\text{par}\rangle_\text{BZ}=a^2\cdot 3\cdot\frac{\pi^2}{3a^2}=\pi^2
\qquad(\text{verified symbolically}).
\]
The factor $a^2$ cancels: the result is the pure ratio $\pi^2$, without scale.

\textbf{Closure.} The spectral form factor of the band is the quotient of the real
Laplacian to that of the tangent parabola:
\begin{equation}
\frac{\langle D_\text{lat}\rangle_\text{BZ}}{\langle D_\text{lat}^\text{par}\rangle_\text{BZ}}
=\frac{12-48/\pi^2}{\pi^2}
=\frac{12\pi^2-48}{\pi^4}
=\frac{12(\pi^2-4)}{\pi^4}=0.7230871=\Smin^\text{estr},
\end{equation}
coincident with the dynamical route. The $\pi^4$ of the denominator is $\pi^2$ (from the average of
the parabola) times the $\pi^2$ that $\langle D_\text{lat}\rangle$ already carried when put over a
common denominator. The \emph{occupation} is its square,
$\Smin^{2,\text{estr}}=0.5229$.

\subsection{The flexural coefficient \texorpdfstring{$1/40$}{1/40} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Expanding $|f(\mathbf{k})|^2$ to fourth order:
\[
|f(\mathbf{k})|^2=16-\frac{16}{3}\lP^2 k^2+\frac{16}{27}\lP^4\Big(\sum_i k_i^4+3\sum_{i<j}k_i^2 k_j^2\Big)+\mathcal{O}(k^6).
\]
Averaging angularly over $SO(3)$ with $\langle k_i^4\rangle=k^4/5$ and
$\langle k_i^2 k_j^2\rangle=k^4/15$ (whence $\langle\sum_i k_i^4\rangle=3k^4/5$ and
$\langle\sum_{i<j}k_i^2 k_j^2\rangle=k^4/5$):
\[
\langle|f|^2\rangle_\text{ang}=16-\frac{16}{3}\lP^2 k^2+\frac{32}{45}\lP^4 k^4+\mathcal{O}(k^6).
\]
In $D_\text{lat}=16-|f|^2$ the kinetic term has coefficient $\tfrac{16}{3}\lP^2$
and the fourth-derivative one $\tfrac{32}{45}\lP^4$. The relative weight of the flexural with respect
to the square of the kinetic is
\begin{equation}
\boxed{\;\frac{\big|\text{coef}_{k^4}\big|}{(\text{coef}_{k^2})^2}=\frac{32\lP^4/45}{(16\lP^2/3)^2}=\frac{32/45}{256/9}=\frac{1}{40}\;}
\end{equation}
\textbf{Symbolic verification} (sympy): the computation of the fourth-order expansion and
the angular average give exactly $\text{coef}_{k^4}/(\text{coef}_{k^2})^2=-1/40$;
the negative sign is the convention $D_\text{lat}=16-|f|^2$ (the real band grows more
slowly than its tangent parabola), and the relative magnitude that defines the flexural
rigidity is $1/40$. The same lattice operator acts over modulus and phase:
\[
|\Psi_n-\Psi_m|^2=(s_n-s_m)^2+S_0^2(a_n-a_m)^2,
\]
so that photon and TT mode share the UV correction $(k\lP)^2/20$, without the
scalar or tensor structure of the polarization altering the coefficient.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 5: F1 delivers its first number}]
Upon evaluating the flexural of F1, the medium crystallizes into the diamond lattice, and from it comes
the second coefficient that F1 carried pending: the $1/40$ ---the ratio, fixed by the
tetrahedral geometry and the isotropic average, between the fourth-order curvature of the
band and the square of the kinetic, not a fit. The same computation delivers, by two routes
(band average and Taylor expansion), the form factor $0.7231=\Smin$.
\end{tcolorbox}

\subsection{Uniqueness of the stacking: the uniaxial flexural discriminant (D1-b) --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The coefficient $1/40$ depends on the tetrahedral coordination $Z=4$, but does not distinguish
the cubic stacking (diamond $Fd\bar3m$) from the hexagonal (wurtzite $P6_3mc$): both
structures are tetrahedral at nearest neighbors, have identical first and second coordination
shells (four nearest neighbors, twelve seconds), and diverge for the first
time in the third shell. The discriminant of the stacking is the \emph{uniaxial
anisotropy} of the fourth-order flexural coefficient, which the isotropic average
$SO(3)$ erases and which is here evaluated without averaging.

Let $|f(\mathbf{k})|^2=\big|\sum_i e^{i\mathbf{k}\cdot\boldsymbol{\delta}_i}\big|^2$ be the
structure factor of the modulation, with $\boldsymbol{\delta}_i$ the four
tetrahedral vectors. Its fourth-order expansion defines the flexural tensor
$T_{abcd}=\partial^4|f|^2/\partial k_a\partial k_b\partial k_c\partial k_d|_{\mathbf{k}=0}$,
and the uniaxial discriminant is the difference between the coefficient of the stacking axis
and that of a basal axis:
\begin{equation}
\Delta_\text{uni}\;\equiv\;\text{coef}(k_z^4)-\text{coef}(k_x^4).
\end{equation}

\textbf{Cubic stacking (diamond 3C).} With the four even-sign-product vectors
$\boldsymbol{\delta}_i=\tfrac{1}{\sqrt3}(\pm1,\pm1,\pm1)$, the fourth-order
expansion gives
\[
\text{coef}(k_x^4)=\text{coef}(k_y^4)=\text{coef}(k_z^4)=\frac{16}{27},
\qquad
\text{coef}(k_a^2 k_b^2)=\frac{16}{9}\ (a\neq b),
\]
whence
\begin{equation}
\boxed{\;\Delta_\text{uni}^\text{cubic}=\frac{16}{27}-\frac{16}{27}=0\;}
\end{equation}
The fourth-order flexural coefficient is exactly isotropic: the operator $(\Box S)^2$
does not privilege any direction.

\textbf{Hexagonal stacking (wurtzite 2H).} With the coordination oriented so that
a bond coincides with the stacking axis $c$ (eclipsed configuration of the sequence
$ABAB$), the same expansion gives
\[
\text{coef}(k_z^4)=\frac{64}{81},
\qquad
\text{coef}(k_x^4)=\text{coef}(k_y^4)=\frac{20}{27}=\frac{60}{81},
\]
whence
\begin{equation}
\boxed{\;\Delta_\text{uni}^\text{hexagonal}=\frac{64}{81}-\frac{60}{81}=\frac{4}{81}\neq0\;}
\end{equation}
The hexagonal modulation carries an irreducible uniaxial anisotropy of exact magnitude
$4/81\approx0.04938$ along the stacking axis.

\textbf{Symbolic verification} (sympy): the value $4/81$ is invariant under azimuthal
rotation of the hexagonal tetrahedral basis; evaluated for $\phi_0=0$ and $\phi_0=0.7$ rad
it gives an identical $\Delta_\text{uni}=0.0493827\ldots$, so that it is not an artefact of the
parametrization but a structural property of the hexagonal symmetry against the cubic.

\textbf{Anchoring to the coefficient $1/40$.} The same structure factor from which the
discriminant comes, averaged angularly over $SO(3)$ with $\langle k_i^4\rangle=k^4/5$ and
$\langle k_i^2 k_j^2\rangle=k^4/15$, reproduces the flexural coefficient derived above:
\[
\big\langle\text{coef}_{k^4}\big\rangle=\frac{16}{27}\Big(\tfrac{3}{5}+3\cdot\tfrac{1}{5}\Big)=\frac{16}{27}\cdot\frac{6}{5}=\frac{32}{45},
\qquad
\frac{\text{coef}_{k^4}}{(\text{coef}_{k^2})^2}=\frac{32/45}{(16/3)^2}=\frac{1}{40}.
\]
The discriminant of D1-b is the fourth order ---resolved in its uniaxial part--- of the same
band structure factor whose isotropic part fixes $\gamma_\kappa=\rhoP c^2\lP^4/40$, equivalently $\hbar c/40$ by (E36).

The flexural energy of F1, $\mathcal{E}_\text{flex}=\tfrac{\gamma_\kappa}{2}\langle(\Box S)^2\rangle$
with $\langle(\Box S)^2\rangle=\sum_\mathbf{G}|\mathbf{G}|^4|S_\mathbf{G}|^2$, penalizes
the curvature of any uniaxial modulation with strictly positive cost. El Pleno,
a continuous medium that relaxes to the minimum of the isotropic flexural functional, stabilizes in the
only modulation with $\Delta_\text{uni}=0$: the cubic stacking.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the geometry of the lattice, derived in its two pieces}]
The tetrahedral coordination $Z=4$ is a necessity of A1+Maxwell, with uniqueness proven by
spherical 2-design (D1-a, \derivado). The cubic stacking $Fd\bar3m$ is the minimum of the
isotropic flexural functional: fourth-order uniaxial anisotropy $0$ for the cubic and
$4/81$ exact and irreducible for the hexagonal, anchored to the same invariant $1/40$ that fixes
$\gamma_\kappa$ (D1-b, \derivado). The diamond lattice is the configuration of the saturated Pleno
without residue of choice.
\end{tcolorbox}

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Empty space is a continuous elastic solid, and its microscopic structure is not a
postulate but what that solid \emph{does} upon saturating: like any real material
that crystallizes, el Pleno adopts the arrangement that minimizes its bending energy. A
solid whose curvature rigidity ($\gamma_\kappa$) does not privilege any direction
cannot stabilize a structure that concentrates curvature on an axis; that is why it chooses the
cubic stacking ---curvature evenly distributed--- and rejects the hexagonal, which
would carry a bending anisotropy $4/81$. It is the same physical principle by which
carbon, under isotropic pressure, forms cubic diamond and not a uniaxial phase. The
``vacuum'' has, then, a crystalline microstructure determined by its own
elasticity, and the four neighbors at $109.47^\circ$ are the geometry that a continuous
isotropic medium inevitably chooses when filling up to the ceiling.
\end{tcolorbox}

\medskip\hrule\medskip
% =====================================================================

\section{The effective potential and the canonical configurations}
\label{sec:potencial}
% =====================================================================

\subsection{Tripartite structure and the derivation of \texorpdfstring{$E_0$}{E0} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\begin{equation}
V(S)=W_\text{bare}(S)+V_q(S)+V_\text{topo}(S),
\end{equation}
\[
W_\text{bare}=\frac{E_0}{2}\frac{S^4}{(\Smax^2-S^2)^2}\ (\text{Gent}),\quad
V_q=\frac{3}{8}\frac{\rhoP c^2}{S^2}\ (\text{IR}),\quad
V_\text{topo}=-\frac{\rhoP c^2}{2}\ln\!\Big(1-\frac{S^2}{S_{\max,\text{geom}}^2}\Big)\ (\text{UV}).
\]
The medium operates in $S\in(0,S_{\max,\text{geom}})$ with both walls inaccessible.

\textbf{Derivation of $E_0$ by the minimum condition.} Working with $S$
dimensionless and $V$ in units of $\rhoP c^2$, each term is differentiated with respect to $S$.
The Gent~\cite{gent} one, by the quotient rule over $S^4/(\Smax^2-S^2)^2$:
\[
W_\text{bare}'=\frac{E_0}{2}\,\frac{4S^3(\Smax^2-S^2)^2+S^4\cdot 2(\Smax^2-S^2)\cdot 2S}{(\Smax^2-S^2)^4}
=\frac{2E_0\Smax^2 S^3}{(\Smax^2-S^2)^3},
\]
where the numerator factorizes using $4S^3(\Smax^2-S^2)+4S^5=4S^3\Smax^2$. The other
two are direct, $\frac{d}{dS}(S^{-2})=-2S^{-3}$ and
$\frac{d}{dS}[-\tfrac12\ln(1-S^2/\Smax^2)]=\dfrac{S}{\Smax^2-S^2}$:
\[
W_\text{bare}'=\frac{2E_0 \Smax^2 S^3}{(\Smax^2-S^2)^3},\qquad
V_q'=-\frac{3}{4}\frac{1}{S^3},\qquad
V_\text{topo}'=\frac{S}{S_{\max,\text{geom}}^2-S^2}.
\]
The minimum condition $V'(S)=W_\text{bare}'+V_q'+V_\text{topo}'=0$ is, written
explicitly,
\[
\frac{2E_0\Smax^2 S^3}{(\Smax^2-S^2)^3}-\frac{3}{4S^3}+\frac{S}{\Smax^2-S^2}=0 .
\]
Since $E_0$ enters linearly, it is isolated by passing the other two terms to the right side and
solving:
\[
\frac{2E_0\Smax^2 S^3}{(\Smax^2-S^2)^3}=\frac{3}{4S^3}-\frac{S}{\Smax^2-S^2}
\;\Longrightarrow\;
E_0=\frac{(\Smax^2-S^2)^3}{2\Smax^2 S^3}\Big(\frac{3}{4S^3}-\frac{S}{\Smax^2-S^2}\Big)\bigg|_{S=\Smin^\text{estr}} .
\]
Recombining over a common denominator, the equivalent closed form is
\[
E_0=\frac{(\Smax^2-S^2)^2\,\big(3\Smax^2-3S^2-4S^4\big)}{8\,\Smax^2\,S^6}\bigg|_{S=\Smin^\text{estr}},
\]
with $\Smin^\text{estr}=12(\pi^2-4)/\pi^4$ and
$\Smax^2=S_{\max,\text{geom}}^2=(8/3)\sqrt{2/3}$: a single algebraic equation with $E_0$
as the only unknown.

\textbf{Complete substitution: $E_0$ with no free symbols (E58).} Since the two bounds are
themselves closed forms, substitution leaves $E_0$ as a function of $\pi$ and $\sqrt6$ alone.
Introducing the abbreviations $u\equiv\pi^2-4$ and $A\equiv\sqrt6\,\pi^8-162\,u^2$:
\begin{equation}
E_0=\frac{\sqrt6\;A^2\,\big[\pi^8 A-31104\,u^4\big]}{60466176\;\pi^8\,u^6}\;\rhoP c^2
   = 4.254617\,\rhoP c^2 .
\tag{E58}
\end{equation}
Not a single free symbol remains: the coefficient of the constitutive potential is not a model
parameter but algebra of the lattice geometry (which fixes $\Smin^\text{estr}$ through the
spectral quotient) and of maximal packing (which fixes $\Smax$). The symbolic difference
between this form and the general solution is exactly zero, verified with sympy. \T{DERIVED}

\textbf{Symbolic/numerical verification} (sympy): $E_0=4.2546\,\rhoP c^2$. With
that value, the numerical resolution of $V'(S)=0$ gives the dynamical minimum
$S_\text{eq}=0.7230871$, coincident with the closed form
$12(\pi^2-4)/\pi^4=0.7230871$ to eight figures.

\textbf{Curvatures at the minimum} (verified symbolically, sympy). The three
derivatives of the potential at $\Smin^\text{estr}$, which the Gross-Pitaevskii limit
(§\ref{sec:gp}, E24-bis) uses for the coupling $g_\text{eff}$, are
\[
\frac{\partial^2 V_\text{eff}}{\partial S^2}\Big|_{\Smin^\text{estr}}=19.69\,\rhoP c^2\equiv\xi\,\rhoP c^2,
\qquad
\frac{\partial^3 V_\text{eff}}{\partial S^3}\Big|_{\Smin^\text{estr}}=27.92\,\rhoP c^2,
\qquad
\frac{\partial^4 V_\text{eff}}{\partial S^4}\Big|_{\Smin^\text{estr}}=855.6\,\rhoP c^2 .
\]
The second and fourth are positive: $\Smin^\text{estr}$ is a stable minimum and the
effective quartic coupling is repulsive. The third (nonzero) is the one that feeds the
cubic renormalization of the factor $5/3$ in $g_\text{eff}$.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{potencial_efectivo.png}
\caption{The effective potential $V_\text{eff}(S)$ with its three derived components
(panel B): Klauder IR wall ($V_q\propto 1/S^2$), Gent hard wall
($S\to S_{\max}$) and the ground-Pleno well. Panel C confirms that the curvature
fitted at the minimum is $V''=19.69\,\rhoP c^2$, and panel A places
$\Smin^\text{estr}=0.7231$ and $S_{\max}=1.4756$. All with $E_0=4.2546$ derived, without
free parameters.}
\end{figure}

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 6: the potential is determined}]
With $\Smin$ delivered by the lattice, the potential of F1 closes on itself: $E_0$
is not free, it is fixed by requiring that the dynamical minimum fall exactly at the
spectral value $12(\pi^2-4)/\pi^4$. That the dynamical and the spectral routes coincide to 10
digits is the strongest internal-consistency proof of the program: the lattice (link
5) and the potential (link 6) are the same information read twice.
\end{tcolorbox}

\subsection{The ground-Pleno \texorpdfstring{$\Smin^\text{estr}$}{Smin} by two routes --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\begin{equation}
\boxed{\;\Smin^\text{estr}=\frac{12(\pi^2-4)}{\pi^4}=0.7230871\ldots\;}\qquad
\Smin^{2,\text{estr}}=0.5228549\ldots \tag{E9}
\end{equation}

\emph{Reading: the ``vacuum'' of el Pleno is not empty. Its minimal modulus is a pure number
fixed by the geometry of the lattice, without scale or human ruler; the cosmological vacuum has
density $0.5229\,\rhoP$, and this same number reappears as the dark energy
(cosmological link).}

\textbf{Spectral route:} the \emph{spectral form factor} of the band, quotient of the
average of the discrete Laplacian to the average of its parabolic approximation
$D_\text{lat}^\text{par}=\tfrac{16}{3}(k\lP)^2$ over the same Brillouin zone,
$\langle D_\text{lat}\rangle_\text{BZ}/\langle D_\text{lat}^\text{par}\rangle_\text{BZ}
=(12-48/\pi^2)/\pi^2=12(\pi^2-4)/\pi^4$, without reference to the potential (developed in
§\ref{sec:1sobre40}). \textbf{Dynamical route:} the minimum of $V_\text{eff}$ solving
$V'(S)=0$, which gives $S_\text{eq}=0.7230874$. That the \emph{spectral} characterization of the
lattice (pure geometry) and the \emph{dynamical} one of the potential coincide ---to seven significant
figures--- is one of the deepest internal-consistency results of the
program: the lattice geometry spectrally determines the relaxed regime of the field,
without it having been imposed.

Both routes give the \emph{amplitude} ($S_{\min}\approx 0.7231$), not the occupation
($S_{\min}^2\approx 0.5229$). The Bogoliubov-Josephson identification in condensate
configurations,
\begin{equation}
\boxed{\;S^2(x^\mu)\leftrightarrow n_\text{quanta}(x^\mu)/n_0\;},\qquad n_0=1/\lP^3, \tag{E-dens}
\end{equation}
makes $S^2$ a relative number density, with $\rho_\text{medium}=\rhoP S^2$ and
$\rho_\text{ground-Pleno}=0.5229\,\rhoP$.

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The minimum of the potential has an immediate physical reading: the ``vacuum'' is not absence
but a specific configuration with $S\approx 0.72$, fixed by the geometry of the
lattice without free parameters. Its square, $0.52$, is the fraction of quanta of the medium
present in the state of minimal excitation ---a number that will reappear as a
dark-energy density in the cosmological branch.
\end{tcolorbox}

\subsection{The hard bound \texorpdfstring{$S_{\max,\text{geom}}$}{Smax} from packing --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:smax}

\textbf{Density of the diamond lattice} ($d_{\min}=\lP$). The distance between nearest
neighbors is $d_{\min}=a_\text{diamond}\sqrt{3}/4$, hence
$a_\text{diamond}=4\lP/\sqrt{3}$. The cubic cell contains 8 nodes, so that the
Voronoi volume per node is
\[
V_\text{diamond}=\frac{a_\text{diamond}^3}{8}=\frac{1}{8}\Big(\frac{4\lP}{\sqrt{3}}\Big)^3=\frac{8\lP^3}{3\sqrt{3}},
\qquad
n_\text{diamond}=\frac{3\sqrt{3}}{8\lP^3}.
\]
\textbf{Maximal FCC density} (Kepler-Hales). With $d_{\min}=\lP$,
$a_\text{FCC}=\sqrt{2}\lP$, 4 nodes per cell:
\[
n_\text{FCC}^\text{max}=\frac{4}{(\sqrt{2}\lP)^3}=\frac{\sqrt{2}}{\lP^3}.
\]
\textbf{Quotient} (verified symbolically):
\begin{equation}
\boxed{\;S_{\max,\text{geom}}^2=\frac{n_\text{FCC}^\text{max}}{n_\text{diamond}}=\frac{\sqrt{2}/\lP^3}{3\sqrt{3}/(8\lP^3)}=\frac{8\sqrt{2}}{3\sqrt{3}}=\frac{8}{3}\sqrt{\frac{2}{3}}=\frac{8\sqrt{6}}{9}=2.1773\ldots\;} \tag{E10}
\end{equation}
$S_{\max,\text{geom}}=1.4756$. Absolute maximal density:
$\rho_{\max,\text{abs}}=S_{\max,\text{geom}}^2\rhoP\approx 2.1773\,\rhoP$.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{pared_de_gent.png}
\caption{Why the restoring force blows up at $S_{\max}$: the gap
$(S_{\max}^2-S^2)$ enters \emph{cubed} in the denominator of
$-dW_\text{bare}/dS=-2E_0 S^3 S_{\max}^2/(S_{\max}^2-S^2)^3$. Panel C shows that
only the Gent wall diverges (not the other two), producing the hard bound that stops
the crunch and generates the bounce (§\ref{sec:reologia}).}
\end{figure}

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
If $\Smin$ is the floor, $\Smax$ is the ceiling: the hard bound of the modulus is pure
geometry ---the ratio between the maximal packing of spheres (Kepler-Hales) and the
density of the diamond lattice. From it comes the maximal density of the universe, finite, which
blocks the singularities and, later (cosmological link), turns the Big
Bang into a bounce.
\end{tcolorbox}

\subsection{The effective bound and the condensate fraction --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{CALIBRATED}}{[CALIBRATED]}}

In the relaxed regime the medium operates below geometric saturation
(quantum \emph{depletion}, Yukalov 2007): the background modulus does not reach the hard bound
$S_{\max,\text{geom}}$, but a lower effective bound, with
$S_{\max,\text{eff}}^2=1.4956\pm 3.9\times 10^{-5}<S_{\max,\text{geom}}^2$. The fraction of
active condensate ---the fraction of Planckian quanta in the fundamental coherent mode
of the ground-Pleno--- is the quotient
\begin{equation}
\boxed{\;f_c\equiv\frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2}=\frac{1.4956}{2.1773}=0.6869\;} \tag{E11}
\end{equation}
The \emph{structure} of the quotient (effective occupation over geometric capacity) and the
\emph{physical nature} of the numerator (occupation level of the condensate under depletion)
are \epi{DERIVED}; the \emph{numerical value} of $S_{\max,\text{eff}}^2$ is
\epi{CALIBRATED} \emph{positional}, as specified below.

\textbf{The cosmological identification $\Omega_{EO}=f_c$.} Under the constitutive
identification $S^2\leftrightarrow n/n_0$ (relative number density), $f_c$ is the fraction
of quanta in the coherent condensate mode, not a ratio of energies. Dark energy is
ontologically that coherent ground-Pleno, so that the quantum fraction $f_c$ and the
cosmological fraction $\Omega_{EO}$ measure the same physical quantity: the identification
$\Omega_{EO}=f_c$ is a mathematical consequence, not a fit. Confronting with Planck 2018
($\Omega_\Lambda=0.6847\pm 0.0073$):
\[
\frac{0.6869-0.6847}{0.0073}=0.30\sigma,\qquad \text{relative error }=0.32\%,
\]
agreement to $0.30\sigma$ without cosmological free parameters. As a cosmological prediction,
$\Omega_{EO}=f_c\approx 0.6869$ is \epi{DERIVED}. (The target $\Omega_\Lambda$ is a parameter derived from the $\Lambda$CDM fit, of the model-dependent type; the more robust confrontation of the sector is the ratio $\Omega_{MO}/\Omega_M$, which cancels $h$ and is anchored in physical densities. See the robustness hierarchy in the confrontation of $\sqrt3\pi$.)

\textbf{Decomposition of the trajectory: derivable part and positional part.} The calibrated
status of $S_{\max,\text{eff}}^2$ is not a pending closed form but the correct mark of
a dynamical state value. The modulus lives in the derived closed-form effective well
(E10), and $S_{\max,\text{eff}}$ is the \emph{upper turning point} of the radial oscillation
of the modulus in that well, defined by conservation of the oscillation energy:
\begin{equation}
V_\text{eff}(S_{\max,\text{eff}})=V_\text{eff}(\Smin^\text{estr})+E_\text{disp}(t_0). \tag{E11-ter}
\end{equation}
Evaluating (E11-ter) at $S_{\max,\text{eff}}=1.2229$ with the derived constants
($E_0=4.2546\,\rhoP c^2$, $S_{\max,\text{geom}}^2=2.1773$), one obtains (verified
numerically):
\[
V_\text{eff}(\Smin^\text{estr})=1.067\,\rhoP c^2,\quad
V_\text{eff}(S_{\max,\text{eff}})=11.070\,\rhoP c^2
\;\Rightarrow\; E_\text{disp}(t_0)=10.003\,\rhoP c^2.
\]
The \emph{well}, its walls and the \emph{rest minimum} $\Smin^\text{estr}$ are
\epi{DERIVED}; the \emph{present oscillation energy} $E_\text{disp}(t_0)$ ---and with it
the turning point $S_{\max,\text{eff}}^2$ that follows from it--- is \epi{CALIBRATED}
\emph{positional}: it measures how much radial energy the medium has available in the present epoch,
a state quantity analogous to $H_0$, not an atemporal constant. As an open end, the present value
$S_{\max,\text{eff}}^2(t_0)$ is not autonomous: it is subsumed in the trajectory $\bar S(t)$
(open end F-cosmo).

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fc_trayectoria.png}
\caption{The effective bound as a positional turning point. (A) The modulus oscillates in the derived
$V_\text{eff}$ well between the rest $\Smin^\text{estr}=0.7231$ and the upper turning
point $S_{\max,\text{eff}}=1.2229$, fixed by the present oscillation energy
$E_\text{disp}(t_0)$; the geometric hard bound $S_{\max,\text{geom}}=1.4756$ lies above,
unreached. (B) The condensate fraction $f_c=\Omega_{EO}$ is a positional quantity
increasing with cosmic time ---analogous to $H_0$, not an atemporal constant---; the value
of today, $f_c\approx 0.687$, is the only calibrated ingredient of the program.}
\end{figure}

\textbf{Why the order-of-magnitude balance is not enough.} An order-of-magnitude
balance between the IR and UV pressures would suggest placing the center of the radial oscillation
at $S_{\max,\text{geom}}^2/2\approx 1.089$. The exact quadrature shows that this shortcut is not
valid ---equating $|{-}dV_q/dS|=\tfrac34\rhoP c^2/S^3$ with
$dV_\text{topo}/dS=\rhoP c^2 S/(S_{\max,\text{geom}}^2-S^2)$ the exact center is
$\approx 0.978$, not $1.089$. The description that the program uses does not depend on any
``center'': it is the derivable-well plus positional-turning-point decomposition. The
methodological lesson is that a balance by proportionality does not fix a numerical value; only
the exact equality of the pressures does.


\medskip
\begin{tcolorbox}[clave,title={The structural principle and the two faces of the ``calibrated''}]
Every background magnitude of el Pleno factorizes in the form
\[
\text{magnitude}=\underbrace{(\text{geometric ratio})}_{\text{fixed, from the lattice}}
\times\underbrace{(\text{anchor})}_{\text{units}}
\times\underbrace{(\text{state factor})}_{\text{position of }\bar S(t)},
\]
and its epistemic status follows from which piece dominates its value. This induces an exhaustive classification and makes precise what the label \CAL\ means ---never a fitted free parameter:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Geometric} (\DER): depends only on the fixed extremes $\{\Smin,S_{\max,\text{geom}}\}$. Examples: the partition $\Omega_m=1-f_c$, the dilatant index at rest, the attractors $w=-1$ and $n_s=1$.
\item \textbf{Positional of type A} (instantaneous, \CAL\ in its value): depends on the instantaneous value $\bar S(t)$. Examples: $H_0$ and $f_c$.
\item \textbf{Positional of type B} (differential, \CAL\ in its value): depends on the rate $\dot{\bar S}$. Examples: the equation of state $w(z)$ and the spectral tilt $n_s-1$ (geometric profile of the transition through the rate of the crossing $dw_\text{eff}/dN$).
\end{itemize}
\textbf{The principle of human rulers.} El Pleno has no units: it works in pure ratios ($\Smin$, $S_{\max,\text{geom}}$, $S^*$, the lattice coefficients $1/20$, $1/40$, $8/9$, $\sqrt3\pi$ are dimensionless). The units ---meter, second, kilogram--- are \emph{human rulers} that we introduce at strategic points of the computation with a single purpose: to give comparable meaning to what is computed, in order to compare between theories and against observation. That is why \CAL\ covers two situations, neither of them a phenomenological fit: a \emph{ruler calibration} (the value of $\rhoP$ or $\lP$ is ``how many human rulers fit in the grain and the ceiling of el Pleno'' ---information about the medium/unit relation, not about the medium; the inevitable act of injecting the unit to compare, not a weakness); and a \emph{positional state quantity} (the present value of $H_0$, $f_c$, $n_s$, $w_0$ encodes the position of the medium along its trajectory $\bar S(t)$, a historical datum whose form is derivable). The \DER/\CAL\ boundary is, at bottom, the boundary between what el Pleno is and what we need to speak of it in numbers.
\end{tcolorbox}


\subsection{The configuration annulus and its self-dual equator --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

With the two bounds already derived, the admissible state of \emph{el Pleno} in each cell
ceases to be a value and becomes a region of the complex plane: a ring with the centre
forbidden, of area computable in closed form and with an intermediate radius that exchanges the
two walls exactly.

\textbf{Definition of the admissible region.} The modulus is bounded below by the Klauder
quantum wall, $S\geq\Smin^\text{estr}$ (A2), and above by geometric saturation,
$S\leq S_{\max,\text{geom}}$ (A6); the phase is free by global $U(1)$ invariance (A5). The set
of admissible configurations of $\Psi=S\,e^{i\theta}$ is therefore the circular annulus
\[
\mathcal{C}=\Big\{\,\Psi\in\mathbb{C}\;:\;\Smin^\text{estr}\leq|\Psi|\leq S_{\max,\text{geom}},
\ \arg\Psi\in[0,2\pi)\,\Big\},
\]
with inner radius $0.7230871$ and outer radius $1.4755803$. The centre $\Psi=0$ is excluded:
it is not a configuration of the medium but its absence, and A1 forbids it. The annulus is a
ring and not a disc ---it is not simply connected---, and that hole at the origin is the
no-vacuum axiom written in geometry.

\textbf{Area of the region: complete computation.} The integral over the annulus in polar
coordinates is immediate,
\[
\mathcal{A}=\int_0^{2\pi}\!\!\int_{\Smin^\text{estr}}^{S_{\max,\text{geom}}} S\,dS\,d\theta
=2\pi\left[\frac{S^2}{2}\right]_{\Smin^\text{estr}}^{S_{\max,\text{geom}}}
=\pi\Big(S_{\max,\text{geom}}^2-(\Smin^\text{estr})^2\Big),
\]
and substituting the two closed forms ---$S_{\max,\text{geom}}^2=8\sqrt6/9$ from maximal
packing and $\Smin^\text{estr}=12(\pi^2-4)/\pi^4$ from the spectral quotient---:
\begin{equation}
\mathcal{A}=\pi\left(\frac{8\sqrt6}{9}-\frac{144(\pi^2-4)^2}{\pi^8}\right)=5.1976687 .
\tag{E60}
\end{equation}

\textbf{Step-by-step evaluation.} $8\sqrt6/9=8(2.4494897)/9=2.1773242$;
$(\pi^2-4)=5.8696044$, its square $34.452256$, and $144\cdot34.452256=4961.1249$;
$\pi^8=9488.5310$, whence $4961.1249/9488.5310=0.5228549$. The difference is
$2.1773242-0.5228549=1.6544693$, and multiplying by $\pi$: $\mathcal{A}=5.1976687$.

\textbf{Verification of units.} Both bounds are dimensionless ---$S$ is the normalised modulus
of the field, not a length---, so that $\mathcal{A}$ is a pure number: it carries no metres, no
kilograms, no seconds, and depends on no measuring rod. It is a layer-0 quantity in the sense
of the verification notebook.

\textbf{Reading of the subtracted term.} The $(\Smin^\text{estr})^2=0.5228549$ that is
subtracted is not an algebraic residue: under the constitutive identification
$S^2\leftrightarrow n/n_0$, that number \emph{is} the occupation of the \emph{Pleno}-ground
state. The area then measures the configuration margin the medium retains between being unable
to empty and being unable to compress further.

\textbf{The third circle: the geometric mean.} Between the two walls there is a radius
distinguished by the geometry itself,
\[
S^*=\sqrt{\Smin^\text{estr}\,S_{\max,\text{geom}}}
=\sqrt{0.7230871\times 1.4755803}=1.0329423 ,
\]
equidistant from both on a logarithmic scale. Indeed,
\[
\frac{S^*}{\Smin^\text{estr}}
=\frac{\sqrt{\Smin^\text{estr}S_{\max,\text{geom}}}}{\Smin^\text{estr}}
=\sqrt{\frac{S_{\max,\text{geom}}}{\Smin^\text{estr}}},
\qquad
\frac{S_{\max,\text{geom}}}{S^*}
=\frac{S_{\max,\text{geom}}}{\sqrt{\Smin^\text{estr}S_{\max,\text{geom}}}}
=\sqrt{\frac{S_{\max,\text{geom}}}{\Smin^\text{estr}}},
\]
so that the two ratios are the same, $\sqrt{1.4755803/0.7230871}=\sqrt{2.0406614}=1.4285172$,
and in logarithm $\ln S^*-\ln\Smin^\text{estr}=\ln S_{\max,\text{geom}}-\ln S^*$.

\textbf{Self-duality: exact proof.} Consider the transformation that preserves the phase and
reflects the modulus about that circle,
\[
I:\;S\longmapsto \frac{(S^*)^2}{S}.
\]
Substituting $(S^*)^2=\Smin^\text{estr}S_{\max,\text{geom}}$ one obtains, with no
approximation whatsoever,
\[
I(\Smin^\text{estr})=\frac{\Smin^\text{estr}S_{\max,\text{geom}}}{\Smin^\text{estr}}
=S_{\max,\text{geom}},
\qquad
I(S_{\max,\text{geom}})=\frac{\Smin^\text{estr}S_{\max,\text{geom}}}{S_{\max,\text{geom}}}
=\Smin^\text{estr},
\]
that is: $I$ exchanges the two walls exactly and leaves $S^*$ fixed, since
$I(S^*)=(S^*)^2/S^*=S^*$. The circle $S^*$ is the fixed-point set of the inversion.

\textbf{Consequence: the annulus splits into two domains.} The inversion $I$ exchanges the
inner ring $[\Smin^\text{estr},S^*]$ ---governed by the Klauder wall, the quantum term
$\propto 1/S^2$--- with the outer ring $[S^*,S_{\max,\text{geom}}]$ ---governed by the Gent
wall, the term $\propto 1/(\Smax^2-S^2)^2$---. The bilateral rheological structure of the
medium inherits that symmetry, and the dynamical role of $S^*$ as a critical point is
developed in the rheology section.

\begin{tcolorbox}[apunte,title={A coincidence that is not accepted}]
The ratio between the two walls has a closed form,
\[
\frac{S_{\max,\text{geom}}}{\Smin^\text{estr}}
=\frac{(8\sqrt6/9)^{1/2}\,\pi^4}{12(\pi^2-4)}
=\frac{2^{3/4}\,3^{1/4}\,\pi^4}{18(\pi^2-4)}
=2.0406614 ,
\]
and falls within $2.0\%$ of $2$. The proximity is not read as an identity: an exact $2$ would
require the saturation wall to be twice the quantum one, and nothing in the two derivations
---maximal packing on one side, the spectral quotient of the lattice on the other--- produces
that relation. The two bounds come from different families, and their ratio is not a word of
the medium but the number that results from placing them side by side.
\end{tcolorbox}

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 6 closed: the potential and its single calibration}]
The potential is complete with its three canonical configurations ($\Smin$,
$S_{\max,\text{geom}}$, $E_0$), all derived. $f_c$ is the only quantity of the program with
a calibrated ingredient: its architecture (effective occupation over geometric capacity) and
the physical nature of the numerator are derived, and as a cosmological prediction
($\Omega_{EO}=f_c$, $0.30\sigma$ from Planck) it is derived; only the \emph{present value} of
$S_{\max,\text{eff}}^2$ is calibrated, because it is a dynamical state quantity of the medium
---positional like $H_0$, not an atemporal number--- hence the mixed status. With the two walls fixed, the state space of $\Psi$ per cell is furthermore determined as an annulus with the centre forbidden, of area $\mathcal{A}=5.1976687$ (E60) and with a self-dual equator $S^*$ that exactly exchanges the two wall regimes.
\end{tcolorbox}

\medskip\hrule\medskip
% =====================================================================

\section{The fundamental constants as structural identities}
\label{sec:constantes}
% =====================================================================

\subsection{The gravitational constant \texorpdfstring{$G$}{G} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:G}

\textbf{Step 1 --- linearization of F2.} With $S=S_0+\delta S$, $S_0=\Smin^\text{estr}$,
in the static regime ($\partial_t\to 0$) and at scales $\gg\lP$ (flexural negligible,
$\Box\to-\nabla^2$), with constant background phase, F2 (E6) linearizes to
\[
-\rhoP c^2\lP^2\nabla^2(\delta S)+\mu_S^2\rhoP c^2\,\delta S=\rho_S(\mathbf{x}),
\]
where $\mu_S^2$ is the effective mass of the radial mode. In the gravitational regime
(distances much greater than the inverse of that mass) the mass term is
negligible and the Poisson form remains:
\[
\rhoP c^2\lP^2\nabla^2(\delta S)=-\rho_S(\mathbf{x}).
\]

\textbf{Step 2 --- Green's function.} In three dimensions,
$\nabla^2\mathcal{G}=\delta^3(\mathbf{r})$ has solution
$\mathcal{G}(\mathbf{r})=-1/(4\pi r)$. For a point source
$\rho_S=q_1\delta^3(\mathbf{x})$:
\[
\delta S_1(\mathbf{r})=\frac{q_1}{\rhoP c^2\lP^2}\cdot\frac{1}{4\pi r}.
\]
The perturbation decays as $1/r$, without postulating it.

\textbf{Step 3 --- interaction energy.} By the kinetic term of F1, the
energy of superposition of two perturbations is
\[
U_\text{int}=\int\rhoP c^2\lP^2\,\nabla(\delta S_1)\cdot\nabla(\delta S_2)\,d^3x .
\]
Integrating by parts (the boundaries vanish) and using
$\nabla^2(\delta S_2)=-\rho_{S,2}/(\rhoP c^2\lP^2)$:
\[
U_\text{int}=-\int\rhoP c^2\lP^2\,\delta S_1\nabla^2(\delta S_2)\,d^3x=\int\delta S_1\,\rho_{S,2}\,d^3x .
\]
For $\rho_{S,2}=q_2\delta^3(\mathbf{x}-\mathbf{r}_{12})$:
\[
U_\text{int}=q_2\,\delta S_1(\mathbf{r}_{12})=\frac{q_1 q_2}{\rhoP c^2\lP^2}\cdot\frac{1}{4\pi r_{12}}.
\]

\textbf{Step 4 --- identification of the prefactor.} The interaction energy already has
the Newtonian $1/r$ form; it remains to fix the constant. The $4\pi$ that appears is that of the
three-dimensional Green's function, identical to the one appearing in electrostatics, and is
absorbed into the \emph{definition of the gravitational charge} exactly as the Gaussian
system absorbs it into the electric charge. With the rationalized convention
$q_i=\sqrt{4\pi}\,m_i c^2$ ---the standard choice that makes the Green $4\pi$ not
survive in the force law, just as in Heaviside-Lorentz units--- the
comparison with $U_\text{Newton}=-Gm_1m_2/r$ gives
\[
\frac{(\sqrt{4\pi}\,m_1 c^2)(\sqrt{4\pi}\,m_2 c^2)}{4\pi\,\rhoP c^2\lP^2\,r_{12}}
=\frac{m_1 m_2 c^2}{\rhoP\lP^2\,r_{12}}\stackrel{!}{=}\frac{Gm_1m_2}{r_{12}}.
\]
What is derived ---and is independent of the charge convention, since any
rescaling $q_i\to\lambda q_i$ is compensated on both sides--- is the identity between the
kinetic coefficient of F1 and Newton's constant:
\begin{equation}
\boxed{\;\rhoP c^2\lP^2=\frac{c^4}{G}\qquad\Longleftrightarrow\qquad G=\frac{c^2}{\rhoP\lP^2}\;} \tag{E3}
\end{equation}
The physical content is not in the $\sqrt{4\pi}$ (mere convention) but in that the
\emph{rigidity of the medium} $\rhoP c^2\lP^2$ turns out to be $c^4/G$: the kinetic coefficient
of F1 is $\tfrac{1}{2}\rhoP c^2\lP^2=c^4/(2G)$, exactly the prefactor of the Ricci
scalar in Einstein-Hilbert ---a structural coincidence that the Sakharov~\cite{sakharov} step
(§\ref{sec:gravedad}) explains.

\textbf{Numerical verification} (canonical primary values):
\[
c^2=8.9880\times 10^{16},\quad \lP^2=2.6115\times 10^{-70},\quad \rhoP\lP^2=1.3462\times 10^{27}\ \text{kg/m},
\]
\[
G=\frac{8.9880\times 10^{16}}{1.3462\times 10^{27}}=6.6765\times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}.
\]
The full-precision computation gives $G=6.6765\times 10^{-11}$ against
$G_\text{CODATA}=6.6743\times 10^{-11}$ (CODATA~\cite{codata} 2018), difference $0.03\%$ ---rounding residue
of the primaries to four figures, not a physical discrepancy (\S\ref{sec:constantes}).
Dimensionally $[c^2/(\rhoP\lP^2)]=m^3\,kg^{-1}\,s^{-2}\ \checkmark$.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 7: the constants come from the medium}]
$G$ does not enter the program, it \emph{comes out}. The intensity of gravity ---an unexplained
parameter of physics--- is fixed by the two substantive properties of the substrate
($\rhoP$, $\lP$) and its causal structure ($c$), with agreement
to four figures and without fitting; the factor $\sqrt{4\pi}$ of the charge is a normalization
convention (as in electromagnetism), not a knob: what is derived is that the rigidity
of the medium $\rhoP c^2\lP^2$ \emph{is} $c^4/G$. The same action of F1 that gives the lattice and the
potential delivers Newton's constant.
\end{tcolorbox}

\subsection{Planck's constant \texorpdfstring{$\hbar$}{hbar} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:hbar}

\textbf{Step 1 --- momentum conjugate to the phase.} From the phase sector of the kinetic
term, $\mathcal{L}_\theta=\tfrac{1}{2}\rhoP c^2\lP^2 S^2(\partial_\mu\theta)(\partial^\mu\theta)$,
the canonical momentum is
\[
\pi_\theta=\frac{\partial\mathcal{L}_\theta}{\partial(\partial_t\theta)}=\rhoP\lP^2 S^2\,\partial_t\theta .
\]

\textbf{Step 2 --- Noether charge.} The invariance $\theta\to\theta+\alpha$ (A5)
gives the current $j^\mu=\rhoP c^2\lP^2 S^2\partial^\mu\theta$, whose temporal
integrated component is $Q=\int\pi_\theta\,d^3x$, the U(1) charge of el Pleno.

\textbf{Step 3 --- the fundamental mode as a $U(1)$ rotor.} The spatially
homogeneous phase mode (the zero mode, $\partial_i\theta=0$) has, integrating the Lagrangian density
over the support of the mode, the form of a rotor:
\[
L_\theta=\tfrac12\,I_\theta\,\dot\theta^2,\qquad
I_\theta=\rhoP\lP^2\!\int S^2\,d^3x,
\]
where the $c^2$ of the kinetic prefactor cancels with that of the metric ($x^0=ct$).
$I_\theta$ is the \emph{moment of inertia} of the phase rotor ($[\text{kg}\cdot\text{m}^2]$,
angular inertia), and its conjugate $p_\theta=I_\theta\dot\theta$ is the angular momentum,
identical to the $U(1)$ Noether charge of Step 2.

\textbf{Step 4 --- the normalization, derived (not postulated).} The integral
$\int S^2\,d^3x$ is not free: it is fixed by two already-established facts. The modulus $S$ is
dimensionless and is normalized to the ground-Pleno by the identification $S^2=n/n_0$; the fundamental
quantum is the minimal saturation excitation, $S=1$. And that quantum occupies exactly
one cell of the lattice, of volume $\lP^3$. Therefore
\[
\int S^2\,d^3x=\underbrace{S^2}_{=1}\cdot\underbrace{V_\text{cell}}_{=\lP^3}=\lP^3,
\]
without any reference to $\hbar$, and the inertia of the fundamental rotor is fixed in
parameters of the medium: $I_\theta=\rhoP\lP^2\cdot\lP^3=\rhoP\lP^5$.

\textbf{Step 5 --- quantization of the rotor and the quantum of action.} The phase lives on the
circle $\theta\in[0,2\pi)$: the zero mode is a \textbf{planar rotor}, and the spectrum of its
angular momentum is an exact canonical theorem ---eigenvalues of
$\hat p_\theta=-i\hbar\,\partial_\theta$ with periodicity $2\pi$---, independent of
any datum of el Pleno: $p_\theta=n\hbar$, $n\in\mathbb{Z}$. The fundamental quantum ($n=1$)
has $p_\theta=\hbar$. On the other hand, that mode rotates at the coherence frequency of the
medium, $\omega_P=c/\lP$ (the definition of $\lP$ as the coherence length gives
$\omega_P\lP=c$ exact), so that its physical angular momentum is $p_\theta=I_\theta\omega_P$.
Equating the canonical quantum with the structural quantity:
\begin{equation}
\boxed{\;\hbar=I_\theta\,\omega_P=(\rhoP\lP^5)\cdot\frac{c}{\lP}=\rhoP\,c\,\lP^4\;} \tag{E36}
\end{equation}

\textbf{Why the prefactor is exactly $1$ and there is no circularity.} Each factor of the
unit prefactor has a declared physical origin: $S^2=1$ from the saturation of the ground-Pleno,
$\lP^3$ from the cell of the lattice, $\omega_P=c/\lP$ from the coherence scale, and the quantum
$n=1$ from the exact spectrum of the $U(1)$ rotor. None is fitted to the observed value of $\hbar$.
The topological quantization of the charge (§\ref{sec:gauge}) is now a \emph{consequence}
of this normalization ($Q_0=p_\theta=\hbar$ for $n=1$), not its foundation ---thus eliminating
the circular dependence that would affect a purely dimensional derivation.

\textbf{Dimensional uniqueness (independent confirmation).} The result is furthermore the only
one possible by dimensions: requiring $\rhoP^a c^b\lP^d$ with dimension of action
$[\text{M}\,\text{L}^2\,\text{T}^{-1}]$ one obtains the system $a=1$, $-3a+b+d=2$, $-b=-1$,
of unique solution $(a,b,d)=(1,1,4)$. The same for $G=c^2\rhoP^{-1}\lP^{-2}$:
$(a,b,d)=(-1,2,-2)$, also unique. The two constants are the \emph{only} monomials of
the primaries with their dimension; the rotor derivation additionally fixes the prefactor to $1$.

\emph{Reading: Planck's constant is not fundamental. It emerges as the angular momentum of the
elementary phase rotor of el Pleno; the ``quantum'' is a mechanical property of the medium ---the
inertia of its phase mode rotating at the coherence frequency---, not an axiom of
nature. It reproduces the observed value with unit coefficient, with no fitting.}

\textbf{Numerical verification.}
\[
\lP^4=6.8197\times 10^{-140},\quad \rhoP c=1.5455\times 10^{105},\quad
\hbar=1.0540\times 10^{-34}\ \text{J s}.
\]
The full-precision computation gives $\hbar=1.0540\times 10^{-34}$ against
$\hbar_\text{CODATA}=1.0546\times 10^{-34}$, difference $0.06\%$ ---rounding residue
of the primaries to four figures. Dimensionally
$[\rhoP c\lP^4]=kg\,m^2/s=J\,s\ \checkmark$.

\subsection{Native identities linking gravity and quantum --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Combining (E3) and (E36):
\begin{itemize}[leftmargin=1.4em]
\item Kinetic coefficient $=$ Einstein-Hilbert: $\tfrac{1}{2}\rhoP c^2\lP^2=\tfrac{c^2}{2}\cdot\tfrac{c^2}{G}=\tfrac{c^4}{2G}$.
\item Flexural rigidity $\propto\hbar$: $\gamma_\kappa=\dfrac{\rhoP c^2\lP^4}{40}=\dfrac{c(\rhoP c\lP^4)}{40}=\dfrac{\hbar c}{40}$, numerically $7.90\times 10^{-28}$ J$\cdot$m (verified).
\item Constraint, eliminating $\rhoP$:
\[
\hbar=\frac{c^3\lP^2}{G}\quad\Longleftrightarrow\quad \lP=\sqrt{\frac{\hbar G}{c^3}}\quad(\text{Planck length}).
\]
Numerical check: $\sqrt{\hbar G/c^3}$ with CODATA values gives $1.61622\times 10^{-35}$ m, the $\lP$ used.
\end{itemize}

\textbf{The Planckian-tautology objection, answered.} The natural question is whether $G$ and
$\hbar$ \emph{are derived} or are only internal consistency of the Planck units. The
answer rests on three facts. First, the triplet $(\rhoP,\lP,c)$ is
\textbf{dimensionally independent}: they are three distinct physical dimensions that no
identity of the program collapses. Second, the dynamics of F1 \textbf{selects a unique
combination} for each constant among the infinitely many dimensionally compatible ones
---$G=c^2/(\rhoP\lP^2)$ via Sakharov, $\hbar=\rhoP c\lP^4$ via the quantization
of the phase rotor, each with its prefactor fixed to $1$ by its mechanism, not by fitting.
That a \emph{single} choice of $(\rhoP,\lP)$ reproduce both constants at once
---the same scale and the same density satisfying two independent identities--- is
a falsifiable condition that a generic substrate would not fulfill, not a unit identity.
Third, what remains open and is declared as such is the \emph{calibration} of the human
value of $(\rhoP,\lP)$ from non-gravitational and non-quantum physics (open end §\ref{sec:cabos});
that independent calibration is future program, not a precondition of the \epi{DERIVED} status
of the two identities. The agreement is exact by construction ---a structural condition
fixes $(\rhoP,\lP)$---, and the residue $<0.1\%$ is rounding of the primaries to four figures,
not a physical discrepancy.


\subsection{The falsification test of the coefficients: \texorpdfstring{$\kappa_G$ and $\kappa_\hbar$}{kappa\_G and kappa\_hbar} --- \texorpdfstring{\T{method}}{[method]}}

The objection the two identities must withstand is concrete: if $(\rhoP,\lP)$ are fixed using
$G$ and $\hbar$, reproducing them afterwards proves nothing. The algebra that follows measures
how much content the agreement carries in each measurement scenario, and separates the case in
which it says nothing from the case in which it risks one figure per branch.

\textbf{What dimensional analysis fixes and what it does not.} The two identities are not
obtained by looking for which combination of the primaries has the right dimensions: they are
obtained \emph{from} F1, by the static reduction and the Green function in one case, and by
the quantisation of the $U(1)$ rotor in the other. Dimensional analysis intervenes afterwards,
as verification. And it would not have sufficed: although the monomial in powers of
$(\rhoP,\lP,c)$ is unique, the programme contains dimensionless invariants of its own
---$\Smin^\text{estr}$, $8\sqrt6/9$, $\xi=19.69$, the flexural $1/40$--- and any of them
multiplying that monomial equally has the dimensions of $G$. What selects the prefactor is the
dynamics.

\textbf{The instrument of contrast.} Two auxiliary coefficients are introduced. They are not
magnitudes of the programme and do not appear in F1, which contains no free number in front of
its terms: they are an instrument for asking what measurement would reveal them. That they
equal unity is not a starting hypothesis but the result of the preceding derivations. Writing
\[
G = \kappa_G\,\frac{c^2}{\rhoP\lP^2}\,,
\qquad
\hbar = \kappa_\hbar\,\rhoP\,c\,\lP^4\,,
\]
and solving the pair for the two primaries ---isolating $\rhoP$ from the first and substituting
into the second--- one obtains
\begin{equation}
\lP = \sqrt{\frac{\hbar G}{\kappa_G\,\kappa_\hbar\,c^3}}\,,
\qquad
\rhoP = \frac{\kappa_G^2\,\kappa_\hbar\,c^5}{G^2\hbar}\,.
\tag{E63}
\end{equation}

\textbf{The three scenarios.} From (E63) three confrontation situations can be read, in
increasing order of demand.

\emph{With both primaries solved from the pair: zero bits.} If $(\rhoP,\lP)$ are obtained by
solving the two identities against the measured values of $G$ and $\hbar$, then \emph{any}
choice of $(\kappa_G,\kappa_\hbar)$ reproduces them exactly: two equations with two unknowns.
What changes with the coefficients is not the predicted constants but the primaries that are
inferred. Measuring $G$ and $\hbar$ does not discriminate between values of $\kappa$, and the
programme must not present that agreement as though it did.

\emph{With one primary measured by an independent route: one bit.} If $\lP$ or $\rhoP$ is
obtained from physics that passes through neither gravitation nor quantisation, the first of
(E63) exposes the \emph{product} $\kappa_G\kappa_\hbar$. This is a real confrontation, and it
would falsify any pair whose product is not one. It does not separate the two branches: the
pair $(2,\tfrac12)$ would pass the test.

\emph{With both primaries measured: two bits, one per branch.} Knowing $\rhoP$ and $\lP$, the
pair (E63) inverts and returns the coefficients separately,
\[
\kappa_G = \frac{\rhoP\,G\,\lP^2}{c^2}\,,
\qquad
\kappa_\hbar = \frac{\hbar}{\rhoP\,c\,\lP^4}\,,
\]
so that each branch is subjected to its own test. Closing the calibration open end therefore
yields more than confirming the primaries: it turns one chain into two independent
confrontations, and measuring both is worth more than twice measuring one.

\textbf{The two branches are not symmetric.} $G$ and $\hbar$ are not two chains of different
origin: they are the two polar parts of the \emph{same} kinetic term of F1,
$\tfrac12\rhoP c^2\lP^2$. The modulus branch leads to $G$ through the static limit, Poisson,
the Green function and the comparison with Newton; the phase branch leads to $\hbar$ through
the zero mode, the $U(1)$ rotor and its quantisation. They diverge completely after the common
node, and for that reason they can fail separately. The exposure to risk is not equal either:
the phase branch incorporates three ingredients that the modulus branch never touches ---the
value of the modulus in the \emph{Pleno}-ground state, the cell volume $\lP^3$ of the diamond
lattice and the quantum number $n=1$ of the rotor---, and it is the more exposed of the two.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- link 7 closed: end of the foundational phase}]
$G$ and $\hbar$ are not two independent verifications but a single structural
condition ($\lP=\ell_\text{Planck}$, $\rhoP=c^2/(G\lP^2)$) read in two languages:
one and the same scale and one and the same density reproduce \emph{at once} both constants, which
a generic substrate does not guarantee. \textbf{Here the common trunk ends}
($E\to$F1$\to$lattice$\to$potential$\to$constants) and the harvest begins: from F1, now
completely fixed, there emerge in cascade the modes, gravity, the quantum, the
gauge, the particles and cosmology ---each a particular case of the two
equations of link 4.

And the trunk is left with its risk declared: what the programme risks in the two identities is not the rounding residue against CODATA but the \emph{coefficient}, which the derivation returns equal to unity in both. The three confrontation scenarios (E63) state what each would measure: with both primaries solved from the pair, zero bits; with one measured by an independent route, the product $\kappa_G\kappa_\hbar$; with both, each branch separately.
\end{tcolorbox}

\medskip\hrule\medskip
% =====================================================================

\section{The spectrum of modes of el Pleno}
\label{sec:modos}
% =====================================================================

\subsection{Linearization of F1 and the mode equations --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The excitations of el Pleno are obtained by perturbing the field around the vacuum background
$\Psi_0=\Smin^\text{estr}e^{i\theta_0}$. Writing
$\Psi=(\Smin^\text{estr}+\delta S)\,e^{i(\theta_0+\delta\theta)}$ and expanding to second
order, the kinetic term of F1 separates cleanly into modulus and phase:
\[
|\partial\Psi|^2=(\partial\delta S)^2+(\Smin^\text{estr})^2(\partial\delta\theta)^2+\mathcal{O}(\delta^3),
\]
that is, \textbf{the two modes decouple} at the linear level ($\delta S$ and $\delta\theta$ do not
mix), which justifies treating them separately. Applying Euler-Lagrange to each:

\textbf{Modulus.} The kinetic gives $\Box\,\delta S$; the flexural $-\gamma_\kappa(\Box S)^2$, upon
variation, contributes $-2\gamma_\kappa\Box^2\delta S$; and the potential, $V''_\text{eff}\delta S$.
Dividing by $\rhoP c^2\lP^2$ and using $\gamma_\kappa=\rhoP c^2\lP^4/40$ (flexural
coefficient of link 2), the coefficient of the fourth-order term is
$2\gamma_\kappa/(\rhoP c^2\lP^2)=\lP^2/20$ ---the $1/20$ is exactly $2\times 1/40$, where
the factor $2$ comes from differentiating the square $(\Box S)^2$:
\begin{equation}
\Box\,\delta S-\frac{\lP^2}{20}\Box^2\delta S+\frac{V''_\text{eff}}{\rhoP c^2\lP^2}\delta S=0. \tag{E18}
\end{equation}

\textbf{Phase.} The phase mode is a Goldstone boson of the $U(1)$ symmetry ($\theta_0$
arbitrary): the potential does not depend on $\theta$, so \emph{there is no mass term}.
Only kinetic plus flexural remains:
\begin{equation}
\Box\,\delta\theta-\frac{\lP^2}{20}\Box^2\delta\theta=0. \tag{E19}
\end{equation}
The absence of a gap in (E19) is not a choice: it is Goldstone's theorem, and it is the reason
that massless modes exist (photon, gravitational wave) in an otherwise
extremely rigid medium.

\subsection{Dispersion relations and the mass of the radial mode --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Inserting a plane wave $\delta S\sim e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}$, the
d'Alembertian acts as $\Box\to u$ with $u\equiv k^2-\omega^2/c^2$ (signature $(-+++)$). A
\emph{physical massive} mode satisfies $\omega^2=c^2k^2+\omega_0^2$, that is $u=-\omega_0^2/c^2<0$:
the physical branch has $u<0$. With $V''_\text{eff}/(\rhoP c^2\lP^2)=\xi/\lP^2$, the equation
(E18) becomes algebraic:
\begin{equation}
u-\frac{\lP^2}{20}u^2+\frac{\xi}{\lP^2}=0. \tag{E20}
\end{equation}

\textbf{Tree-level mass (IR).} At accessible scales $\Box\ll 20/\lP^2$, the flexural
term $\lP^2 u^2/20$ is negligible against $u$ (their quotient is $\sim\lP^2 u/20\to 0$ in
the IR), and the equation reduces to $u\approx-\xi/\lP^2$, this is the standard relativistic
dispersion with gap:
\begin{equation}
\boxed{\;\omega^2=c^2 k^2+\omega_0^2,\qquad \omega_0^2=\frac{\xi c^2}{\lP^2}=\frac{V''_\text{eff}(\Smin^\text{estr})}{\rhoP\lP^2}\;} \tag{E21}
\end{equation}
whence the \emph{low-energy effective} rest mass ($k=0$):
\[
m_\text{radial}c^2=\hbar\omega_0=\sqrt{\xi}\,E_P=\sqrt{19.69}\,E_P=4.4373\,E_P.
\]
This is the mass that governs observable physics (it enters the Gross-Pitaevskii limit,
§\ref{sec:dirac}); it is ultra-Planckian, which makes the radial mode undetectable as a
particle.

\textbf{Exact mass (complete pole) and Lee-Wick companion.} The complete quadratic (E20)
has two roots, which with $\lP=1$, $\xi=19.69$ are
\[
u=\frac{10-2\sqrt{5\xi+25}}{\lP^2}\Big|_{u<0}=-12.22,\qquad
u=\frac{10+2\sqrt{5\xi+25}}{\lP^2}\Big|_{u>0}=+32.22
\]
(verified symbolically). The \textbf{negative} root $u=-12.22/\lP^2$ is the physical mode:
its exact propagating mass is $m_\text{prop}c^2=\sqrt{12.22}\,E_P=3.50\,E_P$. The
\textbf{positive} root $u=+32.22/\lP^2$ is not propagating ($u>0$ does not admit real $\omega$ for all
$k$): it is the \emph{Lee-Wick companion}, the massive pole that regulates the theory in the UV. The
difference between the tree-level mass ($4.44\,E_P$) and the exact one ($3.50\,E_P$) is the
correction of the fourth-order term; both are ultra-Planckian and the distinction does not
affect any observable prediction, but it changes what is being named: the physical mass of the
complete pole is $3.50\,E_P$, and $4.44\,E_P$ is its low-energy approximation.

\textbf{Dispersion of the massless modes (TT and phase).} For the phase, (E19) gives
$u-\tfrac{\lP^2}{20}u^2=0=u(1-\tfrac{\lP^2}{20}u)$, with roots $u=0$ (physical IR branch) and
$u=20/\lP^2$ (Lee-Wick pole). The physical branch is not exactly $\omega=ck$: writing
$P\equiv\omega^2/c^2-k^2=-u$ and retaining the flexural, the non-trivial branch satisfies
$P=k^2/(1-\tfrac{\lP^2}{20}P)$; solving iteratively for $k\lP\ll 1$ (substituting
$P\approx k^2$ in the denominator to first order),
\[
P=k^2\Big(1+\frac{\lP^2}{20}k^2+\cdots\Big)\;\Longrightarrow\;
\]
\begin{equation}
\boxed{\;\omega_\text{TT}^2=c^2 k^2\Big[1+\frac{(k\lP)^2}{20}+\mathcal{O}((k\lP)^4)\Big]\;} \tag{E23}
\end{equation}
at exactly velocity $c$ in the IR ($k\to 0$). The photon, also a phase mode, shares this
dispersion with identical UV coefficient $1/20$.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{dispersion_modos.png}
\caption{The spectrum of modes: the radial mode (massive, gap
$\omega_0=\sqrt{\xi}\,c/\lP$, mass $4.44\,E_P$) against the TT mode (massless,
slope $c$). Panel B shows the three classes: only the radial has mass
(ultra-Planckian, inaccessible); the phase and TT modes, massless, are the
observables (photon and gravitational wave).}
\end{figure}

\textbf{Regime of validity.} For accessible frequencies ($k\lP\sim 10^{-43}$ in
gravitational waves of LIGO/LISA), the correction $(k\lP)^2/20\sim 10^{-86}$ is
suppressed $\sim 69$ orders below the observational bound, so that the program
reproduces general relativity in propagation with astronomical precision.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest begins: the modes}]
First consequence of the now-closed trunk: linearizing F1, three classes of
mode appear ---the radial (massive, $4.44\,E_P$, ultra-Planckian and undetectable as a
particle) and the phase and TT modes (massless, at velocity $c$), which are the
observables: photon and gravitational wave. The non-detection of el Pleno ``as a particle''
is a consequence of its only massive mode being at the Planck scale. These modes
are the raw material of the two following links: gravity (TT mode) and
electromagnetism (phase mode).
\end{tcolorbox}

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