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\section*{Part II --- The space of configurations of el Pleno}
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\textbf{Purpose of Part II.} To develop the explanatory apparatus of the program: given the characterization of \emph{el Pleno} established in Part I, what stable configurations does the medium admit, and why those and not others? Part II constructs the complete repertoire of available configurations from which Part III reads the observed physical phenomena.

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\medskip\hrule\medskip

\subsection{\S{}II.1 --- The problem of the permitted configurations}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (non-derivational opening section)
  \item \textbf{Notes:} conceptual opening of Part II. It articulates the central question of the configuration catalogue from the variational principle on F1. It prepares \S{}II.2 (\emph{Pleno}-ground state) and motivates the remaining sections of Part II. It contains no new derivations; it reorganizes material from \S{}I.7 and \S{}I.8.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.1.1 --- The central question of Part II}

Part I characterized what \emph{el Pleno} \emph{is}; it remains to answer what it \emph{does}. F1 admits an enormous dynamics: mathematically, any regular $\Psi(x^\mu)$ is a candidate. The physically relevant question is stricter: which configurations are \textbf{stable} under F1 and effectively appear in the universe, sustained by the very structure of the medium?


The catalogue of permitted configurations is the subspace ---within the enormous space of functions $\Psi(x^\mu)$--- that F1 sustains as stable solutions. Part III will read it as a repertoire of observed phenomena (the minimal configuration as vacuum, the propagating modes as waves, the solitons as particles, the cosmological configurations as the observable universe); Part II constructs it from the mathematical apparatus, without yet invoking the phenomenology.


\subsubsection{II.1.2 --- Operational criterion of an admissible configuration}

A configuration of \emph{el Pleno} is \textbf{admissible} under PIU-$\Psi$ if it simultaneously satisfies the following three criteria:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{It is a solution of the field equations F2 and (E7).} The configuration $\Psi(x^\mu)$ makes the action of \emph{el Pleno} stationary under F1. This is the underlying dynamical criterion: the configuration respects the evolution laws of the medium.
  \item \textbf{It is stable under small perturbations.} Perturbing the configuration in a neighborhood of its spatial support, the total energy does not decrease. In variational language: the configuration is a minimum (local or global) of the energy functional of \emph{el Pleno}, not a saddle point or a maximum. Mathematically unstable configurations exist as solutions of F2 but are not physical because any fluctuation dissolves them.
  \item \textbf{It is energetically accessible.} The total energy of the configuration is finite, or the configuration has finite energy-per-unit-volume in the case of extended (non-localized) configurations. Configurations of infinite total energy are mathematically coherent but not realizable in the observed universe.
\end{itemize}

These three criteria ---stationary solution, local stability and energy finiteness--- are the filter that the Part II catalogue respects systematically: whatever fails any of the three does not appear.


\subsubsection{II.1.3 --- Natural typology of \emph{el Pleno} configurations}

The admissible configurations of \emph{el Pleno} under F1 are naturally classified into four types, according to their spatial and temporal structure:


\textbf{Type 1 --- Uniform configurations.} $\Psi$ is a constant function in space: $\Psi(x^\mu) = S_0\cdot e^{i\theta_0}$ with $S_0$ and $\theta_0$ independent of the coordinates. These configurations represent homogeneous states of the medium. The minimum-energy case within this type is the \textbf{\emph{Pleno}-ground state}, the fundamental configuration of \emph{el Pleno} presented in \S{}II.2.


\textbf{Type 2 --- Microscopically structured configurations.} \emph{El Pleno} admits crystalline structure with periodicity at scale $\ell_P$. The privileged directions of the medium, which in the relaxed regime are averaged over the macroscopic modes producing effective isotropy, have structural identity at the microscopic scale. The specific geometry of this structure --- the \textbf{diamond lattice} --- is developed in \S{}II.3.


\textbf{Type 3 --- Propagating modes.} \emph{El Pleno} admits perturbations that propagate over the \emph{Pleno}-ground-state background. These perturbations decompose into spectral modes with three distinguishable classes: transverse traceless tensor modes (TT), massive radial mode, and induced gauge sector. The complete spectrum is developed in \S{}II.5.


\textbf{Type 4 --- Localized configurations.} \emph{El Pleno} admits configurations with effectively compact spatial support: regions of finite size where the field $\Psi$ deviates significantly from the background value, asymptotically connected to the background at spatial infinity. These configurations, called \textbf{solitonic}, are developed in \S{}II.6.


To these four types a fifth, non-stationary type is added:


\textbf{Type 5 --- Cosmological configurations.} Configurations where the value of the field evolves globally with cosmological time, do not admit a description as a small perturbation over a stationary background, and describe the history of the universe at cosmological scales. They are developed in \S{}II.7.


Part II proceeds through these five types in order ---\emph{Pleno}-ground state (\S{}II.2), diamond lattice (\S{}II.3), modes (\S{}II.5), solitons (\S{}II.6), cosmology (\S{}II.7)--- and closes with the synthesis of the catalogue (\S{}II.8).


\subsubsection{II.1.4 --- Connection with \S{}II.2}

The construction begins with the simplest configuration: the uniform minimum-energy state, the \textbf{\emph{Pleno}-ground state} (\S{}II.2), the substrate on which every more complex configuration is built as a perturbation or local deviation. Now, characterizing the \emph{Pleno}-ground state quantitatively ---fixing the value of its modulus $S_{\min}^\text{estr}$ and of the field bounds--- requires first knowing the microscopic structure of the medium, because those values are properties of that structure. For that reason the numerical determination of the \emph{Pleno}-ground state does not open Part II: the foundation on which it rests precedes it. The order is, then: conceptual definition of the \emph{Pleno}-ground state (\S{}II.2), derivation of the microscopic geometry ---the diamond lattice--- as the \textbf{foundational stone} (\S{}II.3), and only then the quantification of the bounds and of the minimum modulus (\S{}II.4), which follow from that geometry without free parameters.


\medskip\hrule\medskip

\subsection{\S{}II.2 --- Minimal configuration: the \emph{Pleno}-ground state}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own.
  \item \textbf{Notes:} conceptual-foundation section of Part II. It defines the \emph{Pleno}-ground state as the uniform minimum-energy configuration of the medium, establishes the constitutive identification $S^2 \leftrightarrow n/n_0$ (Bogoliubov-Josephson) that endows the field modulus with physical meaning, and ontologically distinguishes the \emph{Pleno}-ground state from the standard quantum ``vacuum''. It does not yet contain the numerical determination of the minimum modulus or of the bounds: those values are properties of the microscopic structure of the medium and are derived in \S{}II.4, after establishing the diamond lattice in \S{}II.3.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.2.1 --- Definition: the \emph{Pleno}-ground state}

The \textbf{\emph{Pleno}-ground state} is the configuration of the field $\Psi$ with the following simultaneous properties:


\begin{itemize}[leftmargin=1.4em]
  \item The modulus $S$ is constant in space: $S(x^\mu) = S_0$, uniform throughout the spacetime manifold.
  \item The phase $\theta$ is constant (or equivalently, its gradients are null): $\partial_\mu\theta = 0$.
  \item The value $S_0$ minimizes the total effective potential $V_\text{eff}(S)$ of \emph{el Pleno}.
  \item The modulus is not null: $S_0 > 0$ strictly, in accordance with axiom A2 (IR quantum wall).
\end{itemize}

Under these conditions, the field equations F2 and (E7) are satisfied trivially: all gradients vanish, and the extremum condition of the potential $V_\text{eff}'(S_0) = 0$ closes equation F2 in the uniform regime.


The name \textit{\emph{Pleno}-ground state} follows the structural analogy with the concept of ground state in many-body quantum mechanics: it is the minimum-energy state of the medium, on which every subsequent excitation is built as a perturbation. The critical difference from the standard ``quantum vacuum'' concept is developed at the end of this section (\S{}II.2.3): in PIU-$\Psi$ the \emph{Pleno}-ground state is not vacuum in the literal sense; it is the continuous physical medium in its state of minimum excitation, with active and observable structural properties.


\subsubsection{\texorpdfstring{II.2.2 --- The constitutive identification: $S^2$ as relative number density in field configurations}{II.2.2 - The constitutive identification: as relative number density in field configurations}}

The modulus $S$ of the field $\Psi$ admits, \textbf{in configurations interpreted as a quantum condensate} (states with phase coherence over macroscopic scales), a specific \textbf{constitutive interpretation} derived from the Bogoliubov-Josephson formalism:


\[
S^2 \leftrightarrow \frac{n_\text{quanta}}{n_0}
\]

where $n_\text{quanta}$ is the local number density of Planckian quanta of the medium and $n_0$ a reference number density.


This identification, adapted to \emph{el Pleno} from the Bogoliubov (1947) formalism for quantum condensates with broken U(1) symmetry combined with the Josephson (1966) relation for superfluid density, applies when $S^2$ represents condensate amplitude. In the canonical language of condensates, $S^2 \leftrightarrow |\psi_\text{condensate}|^2/n_0$, where $\psi_\text{condensate}$ is the wavefunction of the coherent condensate.


\textbf{Important distinction.} The identification $S^2 \leftrightarrow n/n_0$ \textbf{does not apply} to the spectral ratio $D_\text{BZ}$ used in (E9): that is a pure spectral property of the lattice, not condensate density. That both numbers coincide numerically --- $S_{\min}^\text{estr} = D_\text{BZ}$ --- is not a simple ontological identity; it is the non-trivial physical content of the program: the lattice geometry of the \emph{Pleno}-ground state spectrally determines the relaxed cosmological regime of the scalar field.


The identification $S^2 \leftrightarrow n/n_0$ does apply in other contexts of the program: specifically in the derivation of $S_{\max,\text{geom}}$ from packing (\S{}II.4.1), where $S_{\max,\text{geom}}^2$ is interpreted as a ratio of number densities, and in the cosmological reading of formula F6 on matter density (\S{}II.7.6).


\textbf{Epistemic classification.} The identification $S^2 \leftrightarrow n_\text{quanta}/n_0$ in condensate contexts is \epi{DERIVED} from the canonical Bogoliubov-Josephson formalism applied under the filter of axioms A1--A6. It is not an additional postulate; it is the operational interpretation of the field modulus in its role as condensate amplitude.


\textbf{Quantitative operational consequence.} The numerical value of $S^2$ in the \emph{Pleno}-ground state has direct physical meaning when interpreted via the constitutive identification in its role as the coherent condensate amplitude of the medium. Specifically:


\[
S_{\min}^{2,\text{estr}} = \left(\frac{12(\pi^2-4)}{\pi^4}\right)^2 \approx 0.5229
\]

represents the fraction of active Planckian quanta of the medium in its minimum-excitation configuration. Approximately \textbf{52\% of the available quanta of the medium are present in the minimum-excitation state} (the \emph{Pleno}-ground state); the rest is available for excitations (Types 3, 4 and 5 of the catalogue: propagating modes, solitons, cosmological configurations).


\subsubsection{II.2.3 --- Ontological differentiation with respect to the standard ``vacuum''}

A critical ontological clarification closes this section. The \emph{Pleno}-ground state, the minimal configuration of the medium under PIU-$\Psi$, \textbf{is not} what standard physics calls the quantum ``vacuum''. The distinction is structural and has explicit derivational consequences:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{The \emph{Pleno}-ground state is not vacuum in the literal sense.} The medium in its minimal configuration has modulus $S_{\min}^\text{estr} \approx 0.7231$, non-null, with number density $S_{\min}^{2,\text{estr}} \approx 0.52$ units of the reference density $n_0$. Approximately half of the Planckian quanta of the medium are active in the minimal state. When the constitutive identification $S^2 \leftrightarrow n/n_0$ is applied to field configurations (in the condensate regime), the non-null modulus corresponds to active structural content. What standard physics, under the vacuum + things picture, would interpret as ``vacuum'' --- is, in PIU-$\Psi$, the physical medium present with non-null internal structure.
  \item \textbf{The \emph{Pleno}-ground state is a dynamically sustained state.} The configuration is not an absence of structure; it is the dynamical equilibrium between the IR quantum wall (which repels $S \to 0$) and the UV topological potential (which repels $S \to S_{\max}$). The energy of the \emph{Pleno}-ground state is not the zero-point energy of summed vacuum modes; it is the structural energy of the medium in dynamical equilibrium.
  \item \textbf{The \emph{Pleno}-ground state has measurable structural properties.} Density $\rho_P\,S_{\min}^{2,\text{estr}} \approx 0.52\,\rho_P$, modulus $S_{\min}^\text{estr} \approx 0.7231$, mode propagation velocity $c$, coherence scale $\ell_P$, condensate fraction $f_c \approx 0.687$. Each of these properties has observational consequences that Part III will explore: identification of $f_c$ with $\Omega_{EO}$ (the cosmological fraction of dark energy), among others.
  \item \textbf{As a derivational consequence, the cosmological constant problem articulated in \S{}I.2.2 changes its nature.} In the standard picture, the problem is: why does the energy of the quantum vacuum, computed by summing mode fluctuations up to the Planck cutoff, exceed the observational value by 60-120 orders of magnitude? In PIU-$\Psi$ that question is not posed because the underlying picture is different: there is no vacuum to which fluctuation energy is summed, there is a physical medium with specific structural energy that is computed from its derived parameters. The detail of this confrontation is developed in Part III \S{}III.7 and Part V \S{}V.2.
\end{itemize}


\textbf{What remains pending and why.} The \emph{Pleno}-ground state is defined in its \emph{what} (uniform minimum-energy configuration of the medium), in the \emph{meaning} of its modulus (relative number density of quanta, via the constitutive identification) and in \emph{what it is not} (it is not the standard vacuum). What is missing is to \emph{quantify} it: to fix the numerical value of $S_{\min}^\text{estr}$, of the hard bound $S_{\max,\text{geom}}$ and of the effective bound $S_{\max,\text{eff}}$. And here the logical dependence that orders the rest of Part II appears: those values are neither free numbers nor calibrated against observation ---they are \textbf{properties of the microscopic structure of the medium}. The minimum modulus is a spectral average over the lattice of \emph{el Pleno}; the hard bound is a ratio of packing densities of that same lattice; the flexural coefficient of F1 is a geometric ratio of the same lattice. None of the three can be derived without having first established \emph{which} lattice it is. For that reason the quantification of the \emph{Pleno}-ground state (\S{}II.4) is preceded by the establishment of the microscopic geometry of the medium ---the diamond lattice---, the \textbf{foundational stone} of all of Part II.


\medskip\hrule\medskip


\subsection{\S{}II.3 --- The diamond lattice as the crystalline structure of \emph{el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} \textbf{D1 (geometric uniqueness of the diamond lattice)} --- architectural thread declared explicitly open. Four independent convergent arguments justify the choice as a motivated and well-supported geometry, but strict uniqueness as the only geometry compatible with A1--A6 remains an open problem.
  \item \textbf{Notes:} \textbf{foundational stone of Part II}: it establishes the microscopic geometry of saturated \emph{el Pleno}, on which the quantification of the \emph{Pleno}-ground state (\S{}II.4), the mode spectrum (\S{}II.5), the solitons (\S{}II.6) and the gauge sector (\S{}III.4) depend. It justifies the diamond lattice by four arguments in two groups: two \emph{a priori} ---variational selection by phase transition (Russo-Tanaka) and phase condensation $U(1)\to\mathbb{Z}_2$--- that found it from F1 and the axioms without invoking later results, and two of \emph{forward consistency} ---Maxwell isostaticity and Finkelstein-Rubinstein statistics--- that verify its coherence with the modes and solitons. It declares D1 (strict uniqueness) as an open thread. The Finkelstein-Rubinstein statement refers to the configuration space of the soliton; the factor 8 is derived from the discretized Hamiltonian of the bipartite diamond lattice.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.3.1 --- The problem of the microscopic structure of \emph{el Pleno}}

Axiom A4 (ultraviolet regulation) declares an intrinsic microscopic scale $\ell_P$ below which \emph{el Pleno} ceases to be described as a continuum and admits crystalline structure. The natural question is: \textbf{what crystalline geometry does \emph{el Pleno} have?} The possibilities in three dimensions are numerous ---simple cubic, BCC, FCC, HCP, diamond and a continuum of lattices with reduced symmetry---, and the choice is not arbitrary, because the microscopic structure has direct derivational consequences:


\begin{itemize}[leftmargin=1.4em]
  \item The discrete Laplacian $\Delta_\text{lat}$ ---used in \S{}II.4 to derive $S_{\min}^\text{estr}$ by the spectral route.
  \item The mode spectrum of the medium ---developed in \S{}II.5.
  \item The local topology that admits solitons ---developed in \S{}II.6.
  \item The structural asymmetries that break continuum symmetries ---relevant for \S{}III.4 and \S{}III.5.
\end{itemize}

Because of these consequences, the microscopic geometry is a major structural decision of the program and must be justified with an explicit physical argument.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{red_diamante.png}
\caption{The diamond lattice $Fd\bar{3}m$ that \emph{el Pleno} adopts in its saturated phase. (A) Conventional cell: the structure consists of two interpenetrating FCC sublattices (A in blue, B in red), displaced by one quarter of the cube diagonal ---this A/B bipartition is what sustains the phase sector and cosmic neutrality. (B) The coordination motif $Z=4$: each node has exactly four neighbors in tetrahedral arrangement, connected by the vectors $\boldsymbol{\delta}_i=(\ell_P/\sqrt{3})(\pm1,\pm1,\pm1)$. Coordination 4 is the minimum that satisfies Maxwell's isostaticity criterion in 3D, one of the four convergent reasons that justify this geometry. The spectral average of the discrete Laplacian over this lattice produces the closed form $S_{\min}=12(\pi^2-4)/\pi^4$.}
\label{fig:diamante}
\end{figure}

\subsubsection{\texorpdfstring{II.3.2 --- The choice of PIU-$\Psi$: diamond lattice $Fd\bar{3}m$ with coordination $Z = 4$}{II.3.2 - The choice of PIU-Psi: diamond lattice with coordination}}

PIU-$\Psi$ adopts as the microscopic geometry of saturated \emph{el Pleno} the \textbf{diamond lattice} ($Fd\bar{3}m$), with coordination $Z = 4$ (each node has four nearest neighbors in tetrahedral arrangement) and \textbf{bipartite} structure (two inequivalent FCC sublattices A and B, displaced by $\boldsymbol{\delta}_0 = (a/4)(1,1,1)$). The choice rests on \textbf{four independent convergent arguments}, each on a distinct dimension of the problem ---variational, dynamical, geometric, quantum---; the convergence of the four onto a single geometry is what sustains it.

The four arguments do not all have the same logical status, and they are ordered accordingly so that the foundation rests on the firmest ground. They are presented in two groups:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{\emph{A priori} arguments} (\S{}II.3.3 and \S{}II.3.4): they rely solely on the Lagrangian F1 and the axioms A1--A6, without invoking any result from later sections. They are the primary foundation: the variational argument of selection by phase transition (Russo-Tanaka), which shows that a medium with the dynamics of F1 \emph{crystallizes} into the diamond geometry upon saturation, and the phase-condensation argument, which identifies the A/B bipartition with the $U(1)\to\mathbb{Z}_2$ breaking of axiom A5.
  \item \textbf{\emph{Forward-consistency} arguments} (\S{}II.3.5 and \S{}II.3.6): they show that the geometry already selected by the a priori arguments is, in addition, exactly the one needed for the later developments of the program to work ---the mode spectrum (\S{}II.5) and the statistics of the solitons (\S{}II.6). These arguments invoke, in a controlled and declared way, results from those sections; for that reason they are not foundational but confirmatory. Their force is not in establishing the lattice \emph{a priori} ---the first two already did that--- but in showing that the choice is coherent with the whole edifice built upon it. That the same geometry simultaneously satisfies a primary dynamical requirement and the structural requirements of four distinct sectors is precisely the kind of redundancy that gives solidity to a foundation.
\end{itemize}

This separation is what keeps the argument free of circularity. A foundation cannot be justified by invoking the floors it supports. The a priori arguments (\S{}II.3.3--\S{}II.3.4) establish the lattice without looking upward; the consistency ones (\S{}II.3.5--\S{}II.3.6) verify, afterward, that the floors fit. The logical direction is always descending: from the foundation to the derived, never the reverse.

That said, the section \textbf{explicitly declares} that strict uniqueness ---that the diamond be the only geometry compatible with A1--A6--- \textbf{is not proven}: the choice is strongly justified, but ``why not another geometry that also satisfies the arguments?'' remains the \textbf{architectural thread D1}, declared in \S{}II.3.7. It is, therefore, a \emph{very strongly motivated choice with uniqueness declared as an open problem}, not a uniqueness theorem ---and it is presented with that honesty.


\subsubsection{II.3.3 --- Argument 1 (\emph{a priori}): variational selection of the geometry by phase transition (Russo-Tanaka)}

\textbf{(1) The problem.} Why should el Pleno adopt precisely the diamond lattice and not any other three-dimensional geometry? The strongest answer ---and the most independent of any later development--- is that the diamond lattice is not a choice \emph{imposed} on the medium, but the result \emph{selected} by its own dynamics when it crystallizes. This is the first argument, and it is \emph{a priori}: it relies only on the structure of F1 and on canonical phase-transition theory, without invoking anything from the sections that come later.

\textbf{(2) Why it matters.} If the geometry is \emph{dynamically selected} by the medium, then it is not an additional postulate of the program: it is a consequence of F1. This moves the diamond lattice from the status of ``geometric hypothesis'' to that of ``result of the dynamics'', which is exactly what a foundation needs.

\textbf{(3) The tool.} The theory of crystalline nucleation with multiple order parameters (Russo \& Tanaka 2016). When a fluid medium crystallizes under cooling or saturation, Landau-Ginzburg theory with coupled order parameters ---scalar density $\rho$, tetranematic tensor $Q_{ijk}$, third density gradient $\nabla^3 \rho$--- predicts that the selected geometry depends on the multipolar order of the dominant coupling.

\textbf{(4) The derivation.} For tetrahedral-order rank-3 couplings (tetranematic tensor $Q_{ijk}$, octupolar $\ell=3$, whose spherical harmonic has the $T_d$ symmetry of the tetrahedron, coupled to the third gradient of the density), the minimization of the Landau-Ginzburg functional produces as fixed point the \textbf{tetrahedral coordination} $Z = 4$ with \textbf{unit-cell doubling}. The structure factor of the resulting lattice is
\begin{equation}
F(\mathbf{G}) = 1 + e^{i\mathbf{G}\cdot\boldsymbol{\delta}_0}, \qquad \boldsymbol{\delta}_0 = \frac{a}{4}(1,1,1), \tag{E-diam}
\end{equation}
which is the structure factor of a \textbf{bipartite} lattice with a two-point basis. Its scope should be made precise: $F(\mathbf{G})$ fixes the A/B \emph{bipartition} ---it is common to every lattice with a two-point basis, including zinc-blende and wurtzite---, not the complete diamond lattice; the tetrahedral coordination is fixed by the rank-3 order, not by $F(\mathbf{G})$. The appearance of the displacement $\boldsymbol{\delta}_0 = (a/4)(1,1,1)$ is the signature of the rank-3 tetrahedral coupling under variational minimization. That saturated el Pleno under F1 is a medium with dominant rank-3 coupling between scalar density and tetranematic order is motivated by the structure of the flexural term $(\Box S)^2$ of F1, which penalizes the curvature of the configurations of the medium.

\textbf{(5) The result.} Saturated el Pleno under F1, described as a medium with rank-3 coupling between scalar density and tetranematic order, \emph{minimizes its structural energy over the diamond lattice}. The choice is not primarily geometric: it is a variational consequence of the dynamics of the medium.

\textbf{(6) Physical and mathematical meaning.} This argument has an explicit dynamical counterpart in the program, which reinforces it notably. The phase transition described here abstractly is the \emph{same} physical process as the cosmological crystallization $Z=12 \to Z=4$ of the plateau of the crunch-bounce cycle (\S{}IV.5): during the crunch the medium packs toward the coordination of maximum compactness ($Z=12$, Kepler-Hales), and upon crystallizing it releases the latent heat of structural reorganization toward the diamond coordination $Z=4$. The variational argument of this section (static: which is the fixed point of the functional) and the crystallization of \S{}IV.5 (dynamical: how and when the medium transits to that fixed point) are two readings of the same fact. That the same result emerges via the abstract variational route and via the thermodynamic route of the cycle is a redundancy that strengthens the foundation.

\textbf{(7) Epistemic status.} \epi{SLD}: the connection between rank-3 tetrahedral coupling and diamond lattice is established in the canonical literature of statistical crystallography (Russo-Tanaka), and the identification of saturated el Pleno with such a medium is motivated by the flexural term of F1. The tetrahedral coordination itself is already derived, without this argument, by the necessity of A1 + Maxwell (\S{}II.3.3-bis); the value proper to Russo-Tanaka is to select the cubic \emph{stacking} ($Fd\bar3m$) over the hexagonal one (wurtzite), the piece that distinguishes diamond from wurtzite and that the nearest-neighbour invariants do not resolve (\S{}II.3.7).

\subsubsection{\texorpdfstring{II.3.3-bis --- Argument 1' (\emph{a priori}, primary): the tetrahedral coordination as a necessity of A1 + Maxwell}{II.3.3-bis - The tetrahedral coordination as a necessity of A1 + Maxwell}}

The Russo-Tanaka argument supports the diamond lattice by relying on the form of the flexural term as a datum. The tetrahedral coordination $Z=4$ is derived more cleanly as an \emph{internal necessity} from two conditions that saturated el Pleno must satisfy, without invoking Russo-Tanaka or the form of the flexural term.

\textbf{Condition (i) --- second-order isotropy (from A1).} The medium is homogeneous and isotropic (A1), so its dispersion operator must be isotropic in the low-$\mathbf{k}$ limit. To second order this requires the bond tensor to be isotropic:
\begin{equation}
\sum_{i=1}^{Z}\delta_i^a\,\delta_i^b \propto \delta^{ab}, \tag{E13-iso}
\end{equation}
with $\boldsymbol{\delta}_i$ the nearest-neighbour vectors. For the tetrahedral coordination $Z=4$ with $\boldsymbol{\delta}_i = (\ell_P/\sqrt{3})(\pm1,\pm1,\pm1)$ (the four even-sign-product combinations), the computation gives exactly
\begin{equation}
\sum_{i=1}^{4}\delta_i^a\,\delta_i^b = \frac{4}{3}\,\ell_P^2\,\delta^{ab}, \tag{E13-tet}
\end{equation}
exact second-order isotropy. The other possible $Z=4$ arrangement in 3D ---the square-planar one--- gives $\sum_i\delta_i^z\delta_i^z = 0$: it confines to the plane and \emph{violates} A1. Among the coordinations that do achieve exact isotropy (tetrahedral $Z=4$, simple cubic $Z=6$, BCC $Z=8$, FCC $Z=12$), the tetrahedral one is that of minimal $Z$.

\textbf{Condition (ii) --- Maxwell isostaticity (\S{}II.3.5).} The minimal coordination that fixes the degrees of freedom in 3D without soft modes is $Z = 2d = 4$.

\textbf{Result.} The intersection $\{$exact second-order isotropy$\}\cap\{$minimal isostatic $Z$ in 3D$\}$ selects the tetrahedral coordination uniquely. This is confirmed by direct optimization: minimizing the second-order anisotropy $\sum_{ab}(T^{ab}-\tfrac13\mathrm{tr}\,T\,\delta^{ab})^2$ over four unit vectors, $100\%$ of the random seeds converge to the regular tetrahedron (bond angles $109.47^\circ$ between all pairs, residual cost $\sim10^{-12}$): it is \emph{the} extremum, not one among several. Elementary geometry supports it ---four equiangular unit vectors on $S^2$ are the vertices of the regular tetrahedron, a classical result. \epi{DERIVED} (the tetrahedral coordination $Z=4$, conditional on the class of lattices-with-basis considered). This argument uses neither Russo-Tanaka nor the form of the flexural term: it rests only on A1 (isotropy) and Maxwell (isostaticity), which makes it the primary \emph{a priori} argument for the coordination. The cubic stacking is derived separately as the minimum of the isotropic flexural functional (\S{}II.3.7-bis).

\subsubsection{{II.3.4 --- Argument 2 (\emph{a priori}): phase condensation and the A/B bipartition}}

The second argument is also \emph{a priori}: it connects the geometry of the diamond lattice with the dynamics of el Pleno under F1 and axiom A5, without invoking later results. The diamond lattice is \textbf{bipartite}: its nodes divide into two inequivalent classes A and B such that every bond connects an A node with a B node (never two nodes of the same type). The argument shows that this bipartition is not an accidental geometric feature, but the realization of the spontaneous breaking of the $U(1)$ symmetry of the field.


PIU-$\Psi$ identifies this bipartition with the \textbf{spontaneous condensation of the phase $\theta$} of the field $\Psi = S\cdot e^{i\theta}$ in the saturated regime. The argument is as follows:


\textbf{Step 1 --- The phase $\theta$ admits discrete values in the saturated regime.} When the modulus $S$ reaches values close to $S_{\max,\text{geom}}$ (saturation), the fluctuations of the phase $\theta$ are strongly suppressed by the topological potential $V_\text{topo}(S)$. The phase orders spontaneously: instead of fluctuating continuously, $\theta$ localizes dynamically at discrete values.


\textbf{Step 2 --- The U(1) symmetry admits spontaneous $\mathbb{Z}_2$ breaking.} The group U(1) admits the subgroup $\mathbb{Z}_2 = \{1, -1\}$. A spontaneous breaking $U(1) \to \mathbb{Z}_2$ would produce a natural bipartition of the medium: the lattice nodes with $e^{i\theta} = +1$ form sublattice A, the nodes with $e^{i\theta} = -1$ form sublattice B. This bipartition is dynamically sustained by the condensation of the phase at the values $\pm$1.


\textbf{Step 3 --- Compatibility with the diamond lattice.} The diamond lattice has precisely this bipartite structure by construction: A is an FCC sublattice, B is another FCC sublattice, both displaced by $\boldsymbol{\delta}_0 = (a/4)(1,1,1)$. The identification of A with ``$e^{i\theta} = +1$'' and B with ``$e^{i\theta} = -1$'' is geometrically possible and dynamically sustainable.


\textbf{Consequence.} Under this argument, the A/B bipartition of the diamond lattice is not an arbitrary geometric choice: it is the \textbf{ontological realization} of axiom A5 (U(1) symmetry) under spontaneous condensation $U(1) \to \mathbb{Z}_2$ in the saturated regime. The diamond lattice inherits this bipartition as a structural property, not as crystallographic decoration.


\textbf{Derivational implications.} The A/B bipartition manifests operationally as follows. Defining the phases projected onto each sublattice $\theta_A, \theta_B$ and the symmetric and antisymmetric combinations:


\begin{equation}
\theta_\pm = \frac{1}{\sqrt{2}}(\theta_A \pm \theta_B) \tag{E12}
\end{equation}

the symmetric mode $\theta_+$ is identified with the U(1) sector and the antisymmetric mode $\theta_-$ is identified with the SU(2) sector in the construction of the induced gauge sector (\S{}II.5.6 and full development in \S{}III.4). The $SU(2) \times U(1)$ structure of the Standard Model emerges naturally as a consequence of the A/B bipartition of saturated el Pleno.


\textbf{Epistemic classification.} The phase-condensation argument is \epi{SLD}: the connection between A/B bipartition and $U(1) \to \mathbb{Z}_2$ breaking is structurally solid. The choice of $\mathbb{Z}_2$ over $\mathbb{Z}_3$ is not free once the tetrahedral geometry is fixed: the condensation $U(1)\to\mathbb{Z}_n$ requires the bond graph to be $n$-colorable with all bonds inter-class. The diamond lattice is a \textbf{bipartite} graph (minimal ring of 6 bonds, all cycles even), so it admits a perfect A/B 2-coloring ---$\mathbb{Z}_2$ realizable and stable--- but \emph{not} an inter-class 3-coloring ---$\mathbb{Z}_3$ frustrated. A tetrahedral lattice supporting $\mathbb{Z}_3$ would have odd rings (of 3), incompatible with the isotropy and isostaticity of \S{}II.3.3-bis. The same geometric selection that gives tetrahedral $Z=4$ gives bipartition, not tripartition; the $\mathbb{Z}_2$ closure by bipartition of the graph is \epi{DERIVED} once the geometry is fixed. This is not in tension with the $\mathbb{Z}_3$ of color confinement (\S{}III.4): they operate on distinct objects ---the $\mathbb{Z}_2$ is the 2-coloring of the bond graph of the lattice (global structure of the saturated medium); the $\mathbb{Z}_3$ of color is the modular arithmetic of the phase holonomy around the topological defects (local structure of the solitons). The compatibility of both by level distinction is \epi{SLD}.

\subsubsection{II.3.5 --- Argument 3 (forward consistency): Maxwell isostaticity}

The two previous arguments established the diamond lattice \emph{a priori}, from F1 and the axioms. The two that follow are of a different nature and are avowedly presented as such: they do not found the choice, but \textbf{verify its coherence} with later developments of the program. They invoke, in a controlled way, results from sections that come afterward ---and for that reason they cannot be foundational: a foundation is not justified by appealing to the floors it supports. Their value is confirmatory: they show that the geometry already selected is exactly the one the rest of the edifice needs.

This third argument rests on Maxwell's \textbf{isostaticity condition} for mechanical lattices in equilibrium (Maxwell 1864, generalized to crystalline lattices by Phillips 1979 and Thorpe 1983). In three spatial dimensions, a mechanical lattice is in an isostatic state if its average coordination $Z$ satisfies:


\[
Z = 2\,d_\text{eff}
\]

where $d_\text{eff}$ is the effective dimensionality of the medium. For a three-dimensional medium with active scalar degrees of freedom per node and motion restricted perpendicular to the bonds, $d_\text{eff} = 2$ (associated with the two structural transverse polarizations over the radial mode), whence the isostatic coordination is:


\[
Z_\text{isostatic} = 2 \times 2 = 4
\]

\textbf{Justification of $d_\text{eff} = 2$.} The modulus $S$ of \emph{el Pleno} has a radial degree of freedom (the amplitude) and, over the saturated background, two transverse-polarization degrees of freedom associated with the traceless tensor mode (TT mode, \S{}II.5). In Maxwell's constraint counting for the mechanical equilibrium of the medium, it is the two transverse polarizations that count as the effective dimensionality of the space of constrained deformations: $d_\text{eff} = 2$. This assignment is a strong logical deduction, not a closed derivation, because the identification of the two transverse polarizations presupposes the mode spectrum that \S{}II.5 produces; it is declared as such.


\textbf{Physical meaning of isostaticity.} In the isostatic state ($Z = 2d_\text{eff} = 4$), the medium exactly saturates the number of mechanical constraints that fix its degrees of freedom. A lattice with $Z < 4$ leaves unconstrained degrees of freedom (``floppy modes'') that produce structural instabilities; the residual floppy modes that might emerge from discrete geometry are stabilized by the flexural term $\gamma_\kappa(\Box S)^2$ of the Lagrangian F1 (\S{}I.7.7), which penalizes configurations of high local curvature of the medium. The argument sustains the coordination $Z = 4$; it does not assert that the diamond is the only lattice with $Z = 4$ ---that uniqueness is thread D1 (\S{}II.3.7).


The three-dimensional lattices with $Z = 4$ are:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Diamond lattice} (structure of mineral diamond, of crystalline silicon, of germanium): exact $Z = 4$, bipartite, symmetry $Fd\bar{3}m$.
  \item \textbf{Branched-polymer lattice} with exact coordination 4: does not produce a crystalline structure with high-order discrete symmetry.
  \item \textbf{Tetrahedral amorphous lattice}: $Z \approx 4$ on average but without long-range order, incompatible with the property of \emph{el Pleno} of admitting microscopic geometry with symmetry.
\end{itemize}

Of these options, only the diamond lattice combines the exact isostatic coordination $Z = 4$ with the crystallographic symmetry of the group $Fd\bar{3}m$. This combination is \textbf{unique} among lattices with coordination 4 that admit long-range crystalline order.


\textbf{Epistemic classification.} The isostaticity argument is \textbf{[strong logical deduction of consistency]}: the isostaticity calculation for $d_\text{eff} = 2$ is exact and unique, but the assignment $d_\text{eff} = 2$ for \emph{el Pleno} uses the two transverse polarizations of the TT mode, which are derived in \S{}II.5. This dependence is precisely what makes this a \emph{forward-consistency} argument and not \emph{a priori}: it does not establish the lattice by itself ---\S{}II.3.3 and \S{}II.3.4 already did that---, but shows that the coordination $Z=4$ of the already-selected lattice coincides exactly with the isostatic coordination that the mode spectrum of the medium requires. That the dynamically chosen geometry \emph{also} saturates Maxwell's mechanical condition is internal coherence of the program, not circularity: the logical direction remains descending (the lattice founds the modes; here it is verified that the modes, in turn, are compatible with the lattice).


\subsubsection{II.3.6 --- Argument 4 (forward consistency): Finkelstein-Rubinstein quantum statistics}

The fourth argument is also one of forward consistency: the subtlest of the four, it connects the geometry of the lattice with the quantum statistics of the solitonic configurations that the medium will admit (configurations developed in \S{}II.6). Like the previous one, it invokes a later result and for that reason confirms, does not found.


\textbf{Finkelstein-Rubinstein theorem} (Finkelstein-Rubinstein 1968~\cite{finkelstein}; rigorous proof by Giulini 1993). Let there be a field theory with a sector of localized solitonic configurations, each individual soliton characterized by an internal configuration space $\mathcal{M}$ (the manifold where the field takes values inside the soliton). The soliton admits consistent fermionic quantization if and only if the fundamental group of the configuration space of the individual soliton, $\pi_1(\mathcal{M}_\text{sol})$, contains a $\mathbb{Z}_2$ subgroup generated by a non-contractible loop corresponding to the $2\pi$ rotation of the soliton. Equivalently: fermionic quantization is mathematically possible if the \textbf{$2\pi$ rotation of the individual soliton} is non-contractible in $\mathcal{M}_\text{sol}$.


Applied to the solitons of \emph{el Pleno} (whose development is carried out in \S{}II.6):


\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{skyrmion}-type solitons over \emph{el Pleno} have internal configuration space $\mathcal{M}_\text{skyrme} = SU(2) \cong S^3$, with $\pi_3(SU(2)) = \mathbb{Z}$ (topological classes labeled by the integer $B$, the winding number or baryon number). The $2\pi$ rotation of the solitonic configuration produces a loop over $S^3$ with factor $(-1)^B$ (explicit computation by Krusch 2003) --- \textbf{mandatory fermionic quantization} for odd $B$.
  \item The \textbf{Q-ball}-type solitons over \emph{el Pleno} have internal configuration space $\mathcal{M}_\text{Qball}$ topologically contractible (the closed interval $[S_{\min}, S_{\max}]$ with radial profile). For every contractible space, $\pi_n = 0$. The $2\pi$ rotation produces factor $+1$ trivially --- \textbf{mandatory bosonic quantization} (Coleman 1985).
\end{itemize}

\textbf{Geometric-feedback argument.} The existence of physical fermions in the observed universe is a well-established empirical fact (electrons, quarks, neutrinos). For \emph{el Pleno} to admit solitonic configurations with fermionic statistics via Finkelstein-Rubinstein, it must admit solitons whose internal configuration space has the structure $S^3$. The A/B bipartition of the diamond lattice naturally produces this structure via the projected phases $\theta_\pm$: the antisymmetric mode $\theta_-$ lives in $SU(2)$ with the required topology.


\textbf{Structural consequence.} The bipartite diamond lattice is the microscopic geometry that \textbf{simultaneously} (i) is variationally selected by the crystallization dynamics of the medium (Argument 1, \emph{a priori}), (ii) realizes the phase condensation $U(1) \to \mathbb{Z}_2$ of axiom A5 through its bipartition (Argument 2, \emph{a priori}), (iii) saturates the Maxwell isostaticity condition required by the mode spectrum (Argument 3, consistency), and (iv) sustains the $S^3$ topology necessary to admit fermionic solitons via Finkelstein-Rubinstein (Argument 4, consistency). Two a priori arguments found it; two consistency ones confirm that it fits with what is built upon it.


\textbf{Epistemic classification.} The Finkelstein-Rubinstein argument is \epi{DERIVED} conditionally: given the diamond lattice and its bipartition, the dichotomy $\chi_\text{FR}^M = -1$ / $\chi_\text{FR}^{MO} = +1$ (fermionic statistics for skyrmions, bosonic for Q-balls) is a rigorous mathematical consequence. The argument sustains the diamond lattice as a geometry compatible with the existence of physical fermions, without establishing it as the only geometry with this property.


\subsubsection{II.3.7 --- The geometry of the lattice: coordination and stacking derived}

The convergent arguments ---tetrahedral necessity from A1+Maxwell (\S{}II.3.3-bis), Russo-Tanaka variational, phase condensation, isostaticity, Finkelstein-Rubinstein statistics--- establish the diamond lattice as the microscopic geometry of saturated el Pleno. The tetrahedral coordination $Z=4$ is a necessity of A1 (\S{}II.3.3-bis), and the cubic stacking is the minimum of the isotropic flexural functional (\S{}II.3.7-bis): the geometry of the lattice is derived in its two pieces.

\textbf{The exclusion table of the flexural invariant.} The flexural coefficient $1/40$ of F1 is the geometric ratio $\mathrm{coef}_{k^4}/(\mathrm{coef}_{k^2})^2$ of the lattice operator, with the corpus normalization $D_\text{lat} = Z^2 - |f(\mathbf{k})|^2$ (\S{}II.3.8, P9). Computed for each geometry after the SO(3) angular average:
\begin{center}
{\small
\begin{tabular}{lcc}
\toprule
\textbf{Lattice} & $Z$ & invariant $\mathrm{coef}_{k^4}/(\mathrm{coef}_{k^2})^2$ \\
\midrule
diamond / wurtzite (tetrahedral) & 4 & $1/40.0$ \quad(observed) \\
simple cubic & 6 & $1/90.0$ \\
BCC & 8 & $1/160.0$ \\
FCC & 12 & $1/360.0$ \\
\bottomrule
\end{tabular}}
\end{center}
The $1/40$ \textbf{excludes} every coordination $Z\neq 4$ (simple cubic, BCC and FCC give $1/90$, $1/160$, $1/360$, far away), confirming that the invariant depends on the tetrahedral coordination ---what was derived in \S{}II.3.3-bis. The four values follow the exact closed form
\begin{equation}
\frac{\mathrm{coef}_{k^4}}{(\mathrm{coef}_{k^2})^2} = \frac{2}{5Z^2}, \tag{E-flex-Z}
\end{equation}
for every isotropic coordination (verified: $Z=4,6,8,12$ give $1/40, 1/90, 1/160, 1/360$, with $N/Z^2 = 5/2$ constant). The invariant scales as $1/Z^2$: the observed value $1/40$ fixes $Z^2 = 16$, that is $Z=4$, uniquely among the isotropic coordinations. This is a structural law, not a case-table exclusion ---the $Z^{-2}$ dependence is the reason why no other coordination reproduces the $1/40$. The closed form does not distinguish diamond from wurtzite: both are tetrahedral at nearest neighbours ($Z=4$) and give identically $1/40$. The distinction of the stacking is the content of \S{}II.3.7-bis, via the fourth-order flexural anisotropy.

\begin{tcolorbox}[derivado]
\textbf{The geometry of the diamond lattice, derived in two pieces.}

\textbf{D1-a (tetrahedral coordination).} The tetrahedral coordination $Z=4$ is the only arrangement of four nearest neighbours in 3D compatible with A1, and this is proved ---not merely verified numerically. Let $\mathbf{v}_1,\dots,\mathbf{v}_4$ be the unit bond vectors. The absence of a 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 (E13-tet), $\sum_i v_i^a v_i^b = \tfrac43\delta^{ab}$, 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 numerical optimization ($100\%$ convergence to the tetrahedron, \S{}II.3.3-bis) confirms this result, it does not replace it. \epi{DERIVED} (conditional only on the class of Bravais lattices with a centrosymmetric coordination basis).

\textbf{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 against every tetrahedral polytype with a distinguished stacking axis. The proof is the content of \S{}II.3.7-bis: the isotropy of the flexural term of F1 exactly cancels the fourth-order uniaxial anisotropy of the cubic modulation ($\Delta_\text{uni}=0$), whereas every non-cubic polytype carries a nonzero irreducible anisotropy ($\Delta_\text{uni}=4/81$ for wurtzite), 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 coordination (D1-a, via spherical 2-design) and the stacking (D1-b, via continuous flexural minimum, \S{}II.3.7-bis) are both \epi{DERIVED}. The diamond lattice is the configuration of saturated el Pleno with no residue of choice.
\end{tcolorbox}

The microscopic geometry of el Pleno is a result derived in its two pieces: the tetrahedral coordination is a necessity of A1+Maxwell (\S{}II.3.3-bis, D1-a) and the cubic stacking is the minimum of the isotropic flexural functional (\S{}II.3.7-bis, D1-b).


\subsubsection{\texorpdfstring{II.3.7-bis --- The cubic stacking as the minimum of the flexural functional}{II.3.7-bis - The cubic stacking as the minimum of the flexural functional}}

Cubic diamond and hexagonal wurtzite have \textbf{identical first and second coordination shells}: four nearest neighbours at the same distance and twelve second neighbours at the same distance. Their second-order moment tensors coincide, and so do the fourth-order ones built on the first two shells; the two structures diverge for the first time in the third shell. A discriminant based on the near shells is therefore degenerate: the exclusion table of the flexural invariant (\S{}II.3.7) gives $1/40$ for both. The distinction of the stacking requires an invariant that resolves the angular dependence of the fourth order.

The diamond lattice is the \textbf{periodic modulation of a continuous scalar field}, not a graph of nodes and bonds. The field $\Psi = S\,e^{i\theta}$ is defined at every point; the lattice is the spatial pattern on which the modulus $S$ stabilizes. The flexural term of F1 acts on that field through the operator $(\Box S)^2$, so that the selection of the stacking is a variational problem over continuous functions: el Pleno relaxes to the minimum of the flexural functional of F1.

\textbf{The flexural functional over the continuous field.} The flexural energy per unit volume of the modulated field is, directly from F1 (term $\gamma_\kappa(\Box S)^2$ with $\gamma_\kappa = \hbar c/40$, \S{}I.7.7):
\begin{equation}
\mathcal{E}_\text{flex} = \frac{\gamma_\kappa}{2}\,\big\langle(\Box S)^2\big\rangle .
\end{equation}
For a periodic field $S(\mathbf{x}) = \sum_\mathbf{G} S_\mathbf{G}\,e^{i\mathbf{G}\cdot\mathbf{x}}$ decomposed over its reciprocal lattice $\{\mathbf{G}\}$,
\begin{equation}
\big\langle(\Box S)^2\big\rangle = \sum_\mathbf{G} |\mathbf{G}|^4\,|S_\mathbf{G}|^2 .
\end{equation}
The structure of the stacking manifests in the angular dependence of the band structure factor $|f(\mathbf{k})|^2$, whose expansion about $\mathbf{k}=0$ contains, at fourth order, the imprint of the symmetry of the modulation. The flexural operator $(\Box S)^2$ is \textbf{isotropic} ---it privileges no direction, a direct inheritance of the isotropy of A1. The precise physical question is: which periodic modulation with tetrahedral coordination $Z=4$ produces an isotropic fourth-order flexural coefficient, without concentrating curvature on any axis?

\textbf{Fourth-order expansion and uniaxial discriminant.} 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 vectors of the tetrahedral coordination. Its Taylor expansion to fourth order in $\mathbf{k}$ defines the effective flexural tensor $T_{abcd} = \partial^4|f|^2/\partial k_a\partial k_b\partial k_c\partial k_d\big|_{\mathbf{k}=0}$. The physical discriminant between stackings is the \textbf{uniaxial anisotropy} of the fourth-order coefficient, the difference between the coefficient of the distinguished axis ($k_z^4$, the stacking axis) and that of a basal axis ($k_x^4$):
\begin{equation}
\Delta_\text{uni} \;\equiv\; \mathrm{coef}(k_z^4) - \mathrm{coef}(k_x^4).
\end{equation}

\emph{Cubic stacking (diamond 3C).} With the four tetrahedral vectors $\boldsymbol{\delta}_i = \tfrac{1}{\sqrt3}(\pm1,\pm1,\pm1)$ of even sign product, the fourth-order expansion gives
\begin{equation}
\mathrm{coef}(k_x^4) = \mathrm{coef}(k_y^4) = \mathrm{coef}(k_z^4) = \frac{16}{27},
\qquad
\mathrm{coef}(k_a^2 k_b^2) = \frac{16}{9}\ (a\neq b),
\end{equation}
whence
\begin{equation}
\boxed{\;\Delta_\text{uni}^\text{cubic} = \frac{16}{27} - \frac{16}{27} = 0\;}
\end{equation}
The fourth-order flexural coefficient is \textbf{exactly isotropic}: the three axes are indistinguishable. The operator $(\Box S)^2$ sees no privileged direction.

\emph{Hexagonal stacking (wurtzite 2H).} With the tetrahedral coordination oriented so that one bond coincides with the stacking axis $c$ (eclipsed configuration along $c$, characteristic of the hexagonal $ABAB$ sequence), the same expansion gives
\begin{equation}
\mathrm{coef}(k_z^4) = \frac{64}{81},
\qquad
\mathrm{coef}(k_x^4) = \mathrm{coef}(k_y^4) = \frac{20}{27} = \frac{60}{81},
\end{equation}
whence
\begin{equation}
\boxed{\;\Delta_\text{uni}^\text{hexagonal} = \frac{64}{81} - \frac{60}{81} = \frac{4}{81} \neq 0\;}
\end{equation}
The hexagonal modulation carries an \textbf{irreducible uniaxial anisotropy} of exact magnitude $4/81 \approx 0.04938$ along the stacking axis.

\textbf{Robustness (parameter-free).} The value $4/81$ is invariant under azimuthal rotation of the hexagonal tetrahedral basis: evaluated for two distinct azimuthal orientations ($\phi_0=0$ and $\phi_0=0.7$ rad) it gives identical $\Delta_\text{uni}=0.0493827\ldots$. The anisotropy is not an artefact of the parametrization: it is a structural property of the hexagonal symmetry against the cubic one.

\textbf{Anchoring to the corpus.} The same computation, applied to the cubic stacking and angularly averaged 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 \emph{exactly} the flexural coefficient of the corpus:
\begin{equation}
\big\langle\mathrm{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},
\end{equation}
and the ratio with the square of the kinetic coefficient (with $\mathrm{coef}_{k^2}=16/3$) gives
\begin{equation}
\frac{\mathrm{coef}_{k^4}}{(\mathrm{coef}_{k^2})^2} = \frac{32/45}{(16/3)^2} = \frac{1}{40},
\end{equation}
which is the invariant $1/40$ from which the corpus derives $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$ (\S{}II.3.7, E-flex-Z) and the exclusion table. The discriminant of D1-b is the fourth order, resolved in its uniaxial part, of the same band structure factor whose isotropic part fixes $\gamma_\kappa$.

\textbf{Closing argument.} The logical chain is a single variational principle: (1)~F1 penalizes curvature via the isotropic flexural term $(\Box S)^2$ (A4), which privileges no direction (A1). (2)~A modulation with uniaxial anisotropy concentrates curvature along an axis; for an isotropic flexural operator that concentration has an \emph{energy cost}: the flexural energy of a uniaxial modulation exceeds that of an isotropic one of the same density. (3)~The cubic modulation (diamond 3C) is the only tetrahedral configuration whose fourth-order flexural coefficient vanishes exactly in its uniaxial part ($\Delta_\text{uni}=0$); the hexagonal one and every non-cubic polytype with a distinguished stacking axis carry nonzero anisotropy. (4)~The fourth order is the decisive order: the first two coordination shells are identical in all tetrahedral polytypes, so the stacking manifests for the first time in this invariant. (5)~El Pleno, a continuous medium that relaxes to the minimum of the flexural functional of F1, stabilizes in the only modulation that does not concentrate curvature: the cubic stacking, the diamond lattice $Fd\bar3m$.

The cubic stacking is the configuration forced by the minimization of the isotropic flexural energy: an exact number ($\Delta_\text{uni}=4/81$ for the hexagonal, $0$ for the cubic), parameter-free, invariant under azimuthal rotation, and anchored to the same invariant that fixes $\gamma_\kappa$.

\begin{tcolorbox}[derivado]
\textbf{The cubic stacking, derived · \epi{DERIVED}.} The isotropy of the flexural term of F1 uniquely selects the cubic stacking. The hexagonal modulation carries an exact and irreducible fourth-order uniaxial flexural anisotropy, $\Delta_\text{uni}=4/81$, which the cubic one cancels ($\Delta_\text{uni}=0$). The geometry of the diamond lattice is derived in its two pieces: the coordination $Z=4$ (D1-a, via spherical 2-design, \S{}II.3.7) and the cubic stacking (D1-b, via continuous flexural minimum), both \epi{DERIVED}. The diamond lattice is the configuration of saturated el Pleno \emph{with no residue of choice}.
\end{tcolorbox}

\begin{tcolorbox}[ecuacionclave,title={Interpretation --- what el Pleno physically does on crystallizing}]
Empty space is a continuous elastic solid, and its microscopic structure is not a postulate: it is what that solid does on saturating. Like any real material that crystallizes, el Pleno adopts the arrangement that minimizes its bending energy. A solid whose curvature rigidity ($\gamma_\kappa$) privileges no direction cannot stabilize a structure that concentrates curvature on one axis; that is why it chooses the cubic stacking ---curvature spread evenly--- and rejects the hexagonal one, which would carry a bending anisotropy $4/81$. It is the same physical principle by which carbon, under isotropic pressure, crystallizes into cubic diamond and not into a uniaxial phase. The ``vacuum'' has, then, a crystalline microstructure dictated by its own elasticity: the four neighbours at $109.47^\circ$ are the geometry that an isotropic continuous medium inevitably chooses on filling to the ceiling.
\end{tcolorbox}

\subsubsection{II.3.8 --- Structural properties of the diamond lattice and derivation of the discrete Laplacian}

Assuming the diamond lattice as the microscopic geometry of saturated \emph{el Pleno}, the properties used in later sections of the program are:


\textbf{(P1) Coordination.} $Z_1 = 4$ nearest neighbors per node, in tetrahedral arrangement with angles of $109.47^\circ $ between bonds. Vectors to nearest neighbors (with a node of sublattice A as origin):


\[
\boldsymbol{\delta}_1 = \frac{\ell_P}{\sqrt{3}}(1,1,1), \quad \boldsymbol{\delta}_2 = \frac{\ell_P}{\sqrt{3}}(1,-1,-1), \quad \boldsymbol{\delta}_3 = \frac{\ell_P}{\sqrt{3}}(-1,1,-1), \quad \boldsymbol{\delta}_4 = \frac{\ell_P}{\sqrt{3}}(-1,-1,1)
\]

Vector sum $\sum_i \boldsymbol{\delta}_i = 0$ (tetrahedral symmetry). Isotropic tensor product: $\sum_i \delta_i^a \delta_i^b = (4\ell_P^2/3)\,\delta^{ab}$.


\textbf{(P2) Bipartition.} The sublattices A and B are each FCC. Sublattice B is displaced from A by $\boldsymbol{\delta}_0 = (a_\text{diamond}/4)(1,1,1)$ where $a_\text{diamond} = 4\ell_P/\sqrt{3}$. Every bond connects A with B.


\textbf{(P3) Lattice constant.} $a_\text{diamond} = 4\ell_P/\sqrt{3}$. The conventional cubic cell contains 8 atoms (4 of each sublattice). Voronoi volume per node: $V_\text{diamond} = \ell_P^3 \cdot 8/(3\sqrt{3})$.


\textbf{(P4) Discretized Hamiltonian.} The Hamiltonian of the diamond lattice in reciprocal space is a $2 \times 2$ matrix over the A/B bipartition:


\begin{align}
H_\text{lat}(\mathbf{k}) = -K_\text{cin}\begin{pmatrix} 0 & f(\mathbf{k}) \\ f^*(\mathbf{k}) & 0 \end{pmatrix}
\end{align}

with structure factor $f(\mathbf{k}) = \sum_{i=1}^4 e^{i\mathbf{k}\cdot\boldsymbol{\delta}_i}$ and eigenvalues $\epsilon_\pm(\mathbf{k}) = \pm K_\text{cin}|f(\mathbf{k})|$.


\textbf{(P5) Brillouin zone (BZ).} The BZ of the diamond lattice is the truncated octahedron, with high-symmetry points $\Gamma, X, L, W, K, U$. At the point $L$ of the BZ one has $|f(L)|^2 = 2$, a geometric manifestation of the invariant $\sqrt{2}$ that appears in the Kepler-Hales family (Step 2 of \S{}II.4.1).


\textbf{(P6) Discrete Laplacian / continuous Laplacian relation.}


The canonical discrete Laplacian $\Delta_\text{lat}(\mathbf{k})$ over the diamond lattice is constructed from the quadratic block of the Hamiltonian (P4). The low-order expansion in $\mathbf{k}$ of the structure factor is:


\[
f(\mathbf{k}) = 4 + i\mathbf{k}\cdot\sum_i\boldsymbol{\delta}_i - \frac{1}{2}\sum_i(\mathbf{k}\cdot\boldsymbol{\delta}_i)^2 + \mathcal{O}(k^3)
\]

With $\sum_i\boldsymbol{\delta}_i = 0$ and $\sum_i(\mathbf{k}\cdot\boldsymbol{\delta}_i)^2 = (4\ell_P^2/3)\,k^2$:


\[
f(\mathbf{k}) \approx 4 - \frac{2\ell_P^2}{3}k^2
\]

Therefore:


\[
|f(\mathbf{k})|^2 \approx 16 - \frac{16\ell_P^2}{3}k^2 + \mathcal{O}(k^4)
\]

The canonical discrete Laplacian is defined as the $2\times 2$ operator whose effective diagonal block in the continuum limit coincides with $-\nabla^2$. The canonical construction from the apparatus of lattice field theory (standard lattice QFT adapted to the bipartite diamond) defines:


\[
D_\text{lat}(\mathbf{k}) \equiv 16 - |f(\mathbf{k})|^2
\]

so that at $\mathbf{k} = 0$: $D_\text{lat}(0) = 0$, and the low-$\mathbf{k}$ expansion is:


\[
D_\text{lat}(\mathbf{k}) \approx \frac{16\ell_P^2}{3}k^2 + \mathcal{O}(k^4) = \frac{16}{3}(k\ell_P)^2
\]

The numerical factor $16/3$ is unique: the $16 = Z^2$ comes from the square of the coordination ($Z=4$), which fixes the normalization of $|f|^2$, and the $3$ from the three spatial dimensions over which $\sum_i(\mathbf{k}\cdot\boldsymbol{\delta}_i)^2 = (4\ell_P^2/3)k^2$ is averaged. The correspondence with the continuous Laplacian is obtained by inverting this relation at low $\mathbf{k}$ (where $\nabla^2 \leftrightarrow -k^2$):


\begin{equation}
\boxed{\;\nabla^2 \;\longleftrightarrow\; -\frac{3}{16\,\ell_P^2}\,D_\text{lat}\;} \tag{E13}
\end{equation}

No additional convention intervenes: the factor $3/16$ is the direct inversion of the coefficient $16/3$ of the expansion, without free parameters.


\textbf{Derivation of the spectral average and connection with $S_{\min}^\text{estr}$.} The average of the discrete Laplacian over the Brillouin zone admits an exact closed form. Starting from $|f(\mathbf{k})|^2 = 4 + 2\sum_{i<j}\cos[\mathbf{k}\cdot(\boldsymbol{\delta}_i - \boldsymbol{\delta}_j)]$ and averaging over the cubic reciprocal cell $[-\pi/a, \pi/a]^3$, each of the six differences $\boldsymbol{\delta}_i - \boldsymbol{\delta}_j$ has exactly two non-null components of magnitude $a/2$, so that its angular average factorizes into $\langle\cos[\mathbf{k}\cdot(\boldsymbol{\delta}_i-\boldsymbol{\delta}_j)]\rangle = (\sin(\pi/2)/(\pi/2))^2 = (2/\pi)^2$. Therefore:


\[
\langle|f|^2\rangle_\text{BZ} = 4 + 2\cdot 6\cdot\frac{4}{\pi^2} = 4 + \frac{48}{\pi^2}, \qquad \langle D_\text{lat}\rangle_\text{BZ} = 16 - \langle|f|^2\rangle_\text{BZ} = 12 - \frac{48}{\pi^2}\,.
\]

This result is the \textbf{complete three-dimensional band integral} over the Brillouin zone, derived analytically without free parameters (verified numerically by Monte Carlo integration over the reciprocal cell: $\langle D_\text{lat}\rangle_\text{BZ} \approx 7.1366$, coincident with $12 - 48/\pi^2$ to five figures).


\textbf{The spectral form factor.} The physically significant ratio is the \textbf{spectral form factor} of the band: the ratio between the average of the real discrete Laplacian and the average of its parabolic approximation $D_\text{lat}^\text{par} = \tfrac{16}{3}(k\ell_P)^2$ over the \emph{same} Brillouin zone. This ratio measures how much the real band deviates from the ideal quadratic dispersion:


\begin{equation}
\frac{\langle D_\text{lat}\rangle_\text{BZ}}{\langle D_\text{lat}^\text{par}\rangle_\text{BZ}} = \frac{12(\pi^2 - 4)}{\pi^4} \approx 0.7231\,,
\end{equation}

(verified numerically: $0.7232$). This form factor coincides exactly with $S_{\min}^\text{estr}$ derived by the independent spectral route in \S{}II.4.5 (E9). The coincidence between the dynamical derivation of the \emph{Pleno}-ground state (minimum of $V_\text{eff}$) and this purely geometric form factor of the diamond band is non-trivial physical content of the program: the lattice geometry spectrally determines the relaxed regime of the field. The corresponding \textbf{occupation} is its square, $S_{\min}^{2,\text{estr}} = (12(\pi^2-4)/\pi^4)^2 \approx 0.5229$ (Bogoliubov-Josephson identity, \S{}II.2.2).


\textbf{Note on dimensional consistency.} The numerator $12 - 48/\pi^2$ and the form factor are both computed over the same Brillouin zone with the same normalization; the result is dimensionless and independent of the convention for the units of $\mathbf{k}$, as it must be for a form factor. The primary derivation of $S_{\min}^\text{estr}$ is the spectral one of \S{}II.4.5; the preceding calculation confirms it as the form factor of the band, and does not replace it.


\textbf{(P7) Nodal lines.} The spectrum of the discrete Laplacian over the diamond lattice presents nodal lines --- degeneracies of the spectrum along lines in the BZ --- that are relevant to the phenomenology of the induced gauge sector (developed in \S{}III.4). These nodal lines are a direct consequence of the A/B bipartition.


\textbf{(P8) 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 non-physical ``doublers''. In PIU-$\Psi$ this obstruction does not apply because the fermions of the program \textbf{are not point fields defined over lattice sites}: they are \textbf{topological solitonic modes} (skyrmions over the bipartite A/B sublattice, with $Z=4$). The doubling problem is orthogonal to the PIU formulation.


\textbf{(P9) Derivation of the flexural coefficient $1/40$.} The flexural term of F1, $\frac{\rho_P c^2\ell_P^4}{40}(\Box S)^2$, has its coefficient $1/40$ fixed by the same lattice geometry. Starting from the structure factor, the expansion of $|f(\mathbf{k})|^2$ to fourth order in $\mathbf{k}$ is:


\[
|f(\mathbf{k})|^2 = 16 - \frac{16}{3}\ell_P^2 k^2 + \frac{16}{27}\ell_P^4\!\left(\sum_i k_i^4 + 3\sum_{i<j}k_i^2 k_j^2\right) + \mathcal{O}(k^6)
\]

Averaging angularly over $SO(3)$ with $\langle k_i^4\rangle = k^4/5$ and $\langle k_i^2 k_j^2\rangle = k^4/15$ ($i\neq j$), so that $\langle\sum_i k_i^4\rangle = 3k^4/5$ and $\langle\sum_{i<j}k_i^2 k_j^2\rangle = k^4/5$:


\[
\langle |f|^2\rangle_\text{ang} = 16 - \frac{16}{3}\ell_P^2 k^2 + \frac{32}{45}\ell_P^4 k^4 + \mathcal{O}(k^6)
\]

The discrete Laplacian $D_\text{lat} = 16 - |f|^2$ then contains a kinetic term of coefficient $\frac{16}{3}\ell_P^2$ and a fourth-derivative term of coefficient $\frac{32}{45}\ell_P^4$. The relative weight of the flexural term with respect to the square of the kinetic term is:


\[
\frac{\text{coef}_{k^4}}{(\text{coef}_{k^2})^2} = \frac{32\ell_P^4/45}{(16\ell_P^2/3)^2} = \frac{32/45}{256/9} = \frac{1}{40}.
\]

This is the origin of the coefficient $1/40$: it is the ratio ---fixed by the tetrahedral geometry of the diamond lattice and the isotropic average--- between the fourth-order curvature and the square of the kinetic term. There is no free parameter. \epi{DERIVED} without free parameters.


\textbf{(P9b) The lattice regulator acts equally on the modulus and the phase.} The coefficient $1/40$ just derived is a property of the lattice operator $D_\text{lat}(\mathbf{k}) = 16 - |f(\mathbf{k})|^2$, and this operator acts on every propagating excitation of the field $\Psi = S\,e^{i\theta}$, not only on its modulus. To see this explicitly, the nearest-neighbor bond energy --- the discrete ancestor of the kinetic term of F1 --- is expanded around the background $\Psi_n = (S_0 + s_n)\,e^{i(\theta_0 + a_n)}$, with $s_n$ the modulus perturbation and $a_n$ the phase perturbation at node $n$. To quadratic order in the perturbations:


\begin{equation}
|\Psi_n - \Psi_m|^2 \;=\; \underbrace{(s_n - s_m)^2}_{\text{modulus sector}} \;+\; \underbrace{S_0^2\,(a_n - a_m)^2}_{\text{phase sector}} \;+\; \mathcal{O}(\text{perturbation}^3) \tag{E13b}
\end{equation}


The two sectors enter through the \emph{same} bond-difference operator $(\cdot)_n - (\cdot)_m$: the modulus through $(s_n - s_m)^2$ and the phase through $S_0^2\,(a_n - a_m)^2$. Upon Fourier transforming, both contract with the same symbol $D_\text{lat}(\mathbf{k})$, whose angular expansion produces the kinetic term $\propto k^2$ and the flexural term $\propto k^4$ with the \emph{same} ratio $1/40$ between them. Therefore:


\begin{itemize}[leftmargin=1.4em]
  \item The modulus sector inherits the flexural term $-\tfrac{\rho_P c^2\ell_P^4}{40}(\Box S)^2$ (massive radial mode, \S{}II.5.4 and \S{}III.1.9).
  \item The phase sector inherits the flexural term $-\tfrac{\rho_P c^2\ell_P^4}{40}S_0^2(\Box\theta)^2$ (massless modes: photon and TT gravitational wave, \S{}III.3--\S{}III.4), weighted by $S_0^2$ exactly as the phase kinetic term.
\end{itemize}


The scalar (photon) or transverse-traceless-tensor (TT mode) structure of the polarization of the phase mode does \textbf{not} alter this coefficient: the regulator depends on the modulus of the wave vector $|\mathbf{k}|$ --- how rapidly the phase varies between neighboring nodes --- and not on the tensor structure of the polarization, because the $SO(3)$ angular average that fixes $1/40$ is the same in both cases. Consequently, photon and TT mode share exactly the same ultraviolet dispersion correction, $(k\ell_P)^2/20$, with the ratio between coefficients equal to unity. \epi{DERIVED} without free parameters; verified symbolically over the four bond vectors $\boldsymbol{\delta}_i = (\ell_P/\sqrt{3})(\pm1,\pm1,\pm1)$ of the diamond lattice. The value $1/40$ admits two concordant routes: the spectral average over the Brillouin zone and the continuum Taylor expansion of the operator $D_\text{lat} = 16 - |f(\mathbf{k})|^2$ retaining the interference among the four bonds; both converge to the same invariant, and the simplifications that omit the tetrahedral interference give different values, which shows that the coefficient is specific to the diamond geometry.


\subsubsection{II.3.9 --- Closing of the foundation, convergent evidence and connection with \S{}II.4}

The bipartite diamond lattice $Fd\bar{3}m$ with coordination $Z=4$ is the microscopic geometry of saturated \emph{el Pleno} under PIU-$\Psi$, sustained by four arguments: two \emph{a priori} ---variational selection by phase transition (\S{}II.3.3) and phase condensation $U(1)\to\mathbb{Z}_2$ (\S{}II.3.4)--- that found it from F1 and the axioms, and two of \emph{forward consistency} ---Maxwell isostaticity (\S{}II.3.5) and Finkelstein-Rubinstein statistics (\S{}II.3.6)--- that verify its coherence with the mode spectrum (\S{}II.5) and the solitons (\S{}II.6). Thread D1 (geometric uniqueness) remains declared open: the choice is strongly justified, but strict uniqueness is not proven.

\textbf{Convergent a posteriori evidence.} Beyond the four justificatory arguments, the choice of the diamond lattice receives additional support ---not as prior justification, but as \emph{confirmation}--- from the fact that it is the geometry that makes several phenomenological results of Part III work, with coefficients derived without free parameters. Each of them is a calculation that gives the observed value only \emph{if} the lattice is diamond:
\begin{itemize}[leftmargin=1.4em]
  \item The coefficient $8/9$ of Maxwell's equations emerges from the sum of phase circulations over the tetrahedral plaquettes of the diamond lattice (\S{}III.4), with the decomposition $8=2^3$ ($Z=4$ squared spectrally) and $9=3^2$ (three dimensions of the average).
  \item The geometric ratio $\sqrt{3}\,\pi$ between the dark-matter and baryonic sectors uses the tetrahedral geometry of the diamond site combined with the invariant measure of $SU(2)$ (\S{}III.4).
  \item The flexural coefficient $1/40$ of the term $(\Box S)^2$ of F1 is a pure geometric ratio of the diamond lattice (derived in \S{}II.3.8, P9), which controls the ultraviolet correction common to the photon and the gravitational mode.
\end{itemize}
That three independent coefficients of the observed phenomenology follow from the \emph{same} microscopic geometry, without adjustment, is the kind of convergence that a well-chosen foundation should produce. It does not prove uniqueness (thread D1), but it makes it highly plausible.


\begin{tcolorbox}[derivado,title={The status of the diamond lattice: what is derived and by which route}]
The microscopic geometry of \emph{el Pleno} is neither chosen nor fitted. It is derived in two
independent pieces, and they are worth keeping apart because they answer different questions
and rest on different arguments.

\textbf{D1-a --- the coordination $Z=4$.} \epi{DERIVED}. Four unit vectors summing to zero
that reproduce exact second-order isotropy force $\cos\theta=-1/3$: the regular tetrahedron,
unique up to rotation, and a spherical 2-design. The other possible arrangement of $Z=4$ in
three dimensions ---the planar-square one--- confines to the plane and violates A1. The
coordination is therefore not a hypothesis about the substrate: it is what A1 admits, closed
in addition by Maxwell isostaticity ($Z=2d=4$ fixes the degrees of freedom with no soft
modes). Algebraic demonstration in \S{}II.3.7.

\textbf{D1-b --- the cubic stacking.} \epi{DERIVED}. With the coordination fixed, there
remains the choice among the tetrahedral polytypes ---cubic diamond, hexagonal wurtzite and
the intermediate ones---, and F1 itself makes that choice: the fourth-order uniaxial
anisotropy of the flexural functional is exactly $0$ for the cubic stacking and $4/81$ for
the hexagonal one, an exact, parameter-free figure anchored to the same invariant $1/40$ that
fixes $\gamma_\kappa$. The continuous medium relaxes to the only modulation that does not
concentrate curvature. Demonstration in \S{}II.3.7-bis.

\textbf{What the four convergent arguments add.} The two \emph{a priori} arguments
---variational selection by phase transition (\S{}II.3.3) and phase condensation with its A/B
bipartition (\S{}II.3.4)--- and the two of forward consistency ---isostaticity (\S{}II.3.5)
and Finkelstein-Rubinstein statistics (\S{}II.3.6)--- are not the derivation: they are four
checks from different domains that land on the same structure. Their role is verification,
not foundation, and that distinction is kept deliberately so that the foundation is not
justified by invoking the floors it supports.

\textbf{And the a posteriori verification.} Three independent coefficients of the observed
phenomenology follow from this same geometry, with no fitting: the $8/9$ of Maxwell's
equations from the sum of phase circulations over tetrahedral plaquettes (\S{}III.4), the
flexural $1/40$ as a pure geometric ratio of the lattice (\S{}II.3.8) and the $\sqrt3\pi$ of
the dark partition from the tetrahedral geometry of the site (\S{}III.6.3). None of the three
entered the derivation of the lattice; all three come out of it.

Consequently: \textbf{the diamond lattice is the configuration of saturated \emph{el Pleno}
with no residue of choice}, and the claim is doubly bounded ---the coordination by algebra,
the stacking by minimisation of a functional--- and doubly checked, by convergence and by
consequences.
\end{tcolorbox}

\textbf{Connection with \S{}II.4.} With the geometry of the medium established, what comes within reach is what \S{}II.2 left pending: to \emph{quantify} the \emph{Pleno}-ground state. The hard bound $S_{\max,\text{geom}}$, the effective bound $S_{\max,\text{eff}}$, the effective potential and the minimum modulus $S_{\min}^\text{estr}$ are all properties of this lattice, and are derived now without free parameters ---including the convergence of the two formulations that fix $S_{\min}^\text{estr}$, one of which (the spectral one) is built precisely on the discrete Laplacian derived in \S{}II.3.8.


\medskip\hrule\medskip

\subsection{\S{}II.4 --- Quantification of the \emph{Pleno}-ground state: the structural bounds and the minimum modulus}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} the non-perturbative computation of the present oscillation energy $E_\text{disp}(t_0)$ ---and with it the positional value of $S_{\max,\text{eff}}^2$, a dynamical state quantity analogous to $H_0$, not a timeless number to be derived--- is a confrontation thread, subsumed into the F-cosmo cosmological trajectory and inherited by Part V (\S{}V.3).
  \item \textbf{Notes:} with the microscopic structure already established in \S{}II.3, the \emph{Pleno}-ground state is quantified here. It derives the geometric hard bound $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ from Kepler-Hales packing over the diamond lattice, the effective bound $S_{\max,\text{eff}}$ and the condensate fraction $f_c$, the tripartite effective potential $V_\text{eff}$ and, as the central result, the minimum modulus $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ by two converging formulations --- the dynamical one (minimum of $V_\text{eff}$) and the spectral one (form factor of the diamond band, \S{}II.3.8).
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{II.4.1 --- The geometric hard bound $S_{\max,\text{geom}}$ from maximum packing}{II.4.1 - The geometric hard bound from maximum packing}}

Axiom A6 declares the existence of an upper structural hard bound $S_{\max,\text{geom}}$ for the field modulus. The specific value of that bound is derived from the \textbf{geometry of the maximum packing of saturated \emph{el Pleno}}, it is not postulated independently.


\textbf{Step 1 --- Number density of the diamond lattice with $d_{\min} = \ell_P$.}


The microscopic structure of saturated \emph{el Pleno} is the diamond lattice $Fd\bar{3}m$ (derived in \S{}II.3) with minimum distance between neighbors $d_{\min} = \ell_P$. The geometry of the diamond lattice fixes the conventional cubic lattice constant $a_\text{diamond}$: in the diamond lattice, the nearest-neighbor distance is $d_{\min} = a_\text{diamond}\,\sqrt{3}/4$ (length of the tetrahedral diagonal from the center of an octant to the opposite vertex). Therefore:


\[
a_\text{diamond} = \frac{4\,d_{\min}}{\sqrt{3}} = \frac{4\,\ell_P}{\sqrt{3}}
\]

The conventional cubic cell contains 8 atoms (4 of each sublattice of the A/B bipartition). The Voronoi volume per node is:


\[
V_\text{diamond} = \frac{a_\text{diamond}^3}{8} = \frac{1}{8}\cdot\left(\frac{4\,\ell_P}{\sqrt{3}}\right)^3 = \frac{1}{8}\cdot\frac{64\,\ell_P^3}{3\sqrt{3}} = \frac{8\,\ell_P^3}{3\sqrt{3}}
\]

equivalently, the number density of the saturated diamond lattice is:


\[
n_\text{diamond} = \frac{1}{V_\text{diamond}} = \frac{3\sqrt{3}}{8\,\ell_P^3}
\]

\textbf{Step 2 --- Maximum Kepler-Hales number density (saturated FCC).}


The Kepler-Hales theorem (Hales 2005~\cite{keplerhales}, \emph{Annals of Mathematics}; formal verification completed by the Flyspeck project in 2014) establishes that the maximum packing of identical spheres in $\mathbb{R}^3$ saturates the density $\eta_\text{KH} = \pi/(3\sqrt{2})$, achieved by the FCC/HCP family. For an FCC lattice with minimum distance between centers $d_{\min} = \ell_P$, the conventional cubic lattice constant is $a_\text{FCC} = \sqrt{2}\,\ell_P$ (tangency condition $a_\text{FCC} = 2\sqrt{2}\,r$ with $r = \ell_P/2$). The cell contains 4 atoms. The maximum number density is:


\[
n_\text{FCC}^\text{max} = \frac{4}{a_\text{FCC}^3} = \frac{4}{(\sqrt{2}\,\ell_P)^3} = \frac{4}{2\sqrt{2}\,\ell_P^3} = \frac{\sqrt{2}}{\ell_P^3}
\]

equivalently, the Voronoi volume of saturated FCC is $V_\text{FCC}^\text{min} = \ell_P^3/\sqrt{2}$.


\textbf{Step 3 --- Bogoliubov-Josephson constitutive identification applied to the saturation limit.}


The constitutive identification $S^2 \leftrightarrow n/n_0$ (\S{}II.2.2) applies with the following specific physical interpretation:


\begin{itemize}[leftmargin=1.4em]
  \item $n_0 = n_\text{diamond}$ is the \textbf{reference number density of \emph{el Pleno}}: the number density of Planckian quanta in its geometric structural configuration (diamond lattice with $d_{\min} = \ell_P$).
  \item $n = n_\text{FCC}^\text{max}$ is the \textbf{maximum accessible number density} of the medium under compression up to the Kepler-Hales limit: the regime where the Planckian quanta cannot be packed more densely without violating the hard bound of the medium.
\end{itemize}

The ratio $n_\text{FCC}^\text{max}/n_\text{diamond}$ quantifies how many times the diamond structural density of \emph{el Pleno} ``fits'' within the maximum accessible Kepler-Hales density. Numerically:


\[
\frac{n_\text{FCC}^\text{max}}{n_\text{diamond}} = \frac{\sqrt{2}/\ell_P^3}{3\sqrt{3}/(8\,\ell_P^3)} = \frac{\sqrt{2}\cdot 8}{3\sqrt{3}} = \frac{8\sqrt{2}}{3\sqrt{3}} = \frac{8}{3}\cdot\frac{\sqrt{2}}{\sqrt{3}} = \frac{8}{3}\sqrt{\frac{2}{3}}
\]

This is the exact closed form. Numerically, the absolute densities are $n_\text{diamond}\approx 1.54\times 10^{104}$ m$^{-3}$ and $n_\text{FCC}^\text{max}\approx 3.35\times 10^{104}$ m$^{-3}$ (Planckian densities, of order $\ell_P^{-3}$); only their dimensionless ratio $\approx 2.1773$ is physically relevant to the bound. By the constitutive identification (\S{}II.2.2):


\begin{equation}
\boxed{\;S_{\max,\text{geom}}^2 = \frac{n_\text{FCC}^\text{max}}{n_\text{diamond}} = \frac{8}{3}\sqrt{\frac{2}{3}}\;} \tag{E10}
\end{equation}

\textbf{Dimensional verification.} Ratio of two number densities: $[n_\text{FCC}^\text{max}/n_\text{diamond}] = [\text{m}^{-3}]/[\text{m}^{-3}] = 1$, dimensionless. $S^2$ is dimensionless. Consistent $\checkmark$.


\textbf{Explicit numerical verification.}


\[
\sqrt{2/3} = \sqrt{0.6667} = 0.81650
\]

\[
S_{\max,\text{geom}}^2 = \frac{8}{3}\cdot 0.81650 = 2.6667\cdot 0.81650 = 2.17732
\]

\[
S_{\max,\text{geom}} = \sqrt{2.17732} = 1.47557
\]

\textbf{Useful algebraic identity.} Equivalently:


\[
S_{\max,\text{geom}}^2 = \frac{8\sqrt{6}}{9} = \frac{8}{3\sqrt{3}}\cdot\sqrt{2} \approx 2.1773
\]

\textbf{Epistemic classification.} $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ is \epi{DERIVED} from the Kepler-Hales theorem applied to saturated \emph{el Pleno} in diamond geometry, via the constitutive identification $S^2 \leftrightarrow n/n_0$ with $n_0 = n_\text{diamond}$ and $n = n_\text{FCC}^\text{max}$. Each step of the derivation is done with elementary Euclidean geometry + the Kepler-Hales theorem as bibliographic input. Zero free parameters, zero observational calibration. It is a direct geometric consequence of the maximum packing of the medium under axiom A6, applied to the diamond lattice derived in \S{}II.3.


\textbf{Subordinate methodological finding: quasi-FCC/HCP degeneracy within the Kepler-Hales family.}


The rigorous computation of the energy per site of \emph{el Pleno} under multiple isotropic pair potentials (Yukawa, Gent-type logarithmic, regularized Coulomb, Lennard-Jones) --- with cross-verification against the classical Kittel literature on Madelung constants --- shows that the FCC and HCP polytypes are \textbf{quasi-degenerate} with differences of order $10^{-4}$ to $10^{-3}$ in units of $\rho_P c^2$ per site. This degeneracy is comparable to or smaller than the quantum fluctuations of the medium governed by the Klauder quantum potential $V_q$. Saturated \emph{el Pleno} \textbf{does not binarily select} between FCC and HCP; the question ``FCC or HCP'' is physically ill-posed. The correct answer is ``member of the Kepler-Hales family'', with polytypism possible internally. This degeneracy produces a specific prediction developed in Part III: polytypism in the cores of dense compact objects as a weak structural memory of the formation process.


\subsubsection{\texorpdfstring{II.4.2 --- The effective bound $S_{\max,\text{eff}}$ of the relaxed regime}{II.4.2 - The effective bound of the relaxed regime}}

The absolute structural bound $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3} \approx 2.1773$ (equivalently $S_{\max,\text{geom}} \approx 1.4756$) derived in II.4.1 characterizes the \textbf{asymptotic limit} of the medium under absolute saturation --- a physical situation relevant only in extreme regimes (primordial bounce, cores of compact objects at maximum density). Throughout this subsection the two bounds are compared in \emph{squared} form ($S^2$), which is the physically significant magnitude (occupation, $\propto$ density); the value of the hard bound squared, $S_{\max,\text{geom}}^2 \approx 2.1773$, must not be read as with its root, $S_{\max,\text{geom}} \approx 1.4756$.


In the current cosmological regime --- the relaxed regime of \emph{el Pleno}, characterized by $S \sim S_{\min}^\text{estr}$ with small perturbations --- the medium does not operate at its absolute saturation, but in a state of partial occupation. The reason is physical and intrinsic to the medium: by the constitutive identification $S^2 \leftrightarrow n/n_0$ (\S{}II.2.2), a condensate with repulsive interactions ---as the \emph{Pleno}-ground state is under the effective potential of \S{}II.4.4--- never fills its coherent mode up to the maximum geometric capacity; a fraction of the quanta remains outside the condensate (quantum \emph{depletion}, Yukalov 2007~\cite{yukalov2007}). The resulting effective occupation level defines an \textbf{effective bound} $S_{\max,\text{eff}}$, strictly smaller than the hard bound, whose square is:


\[
S_{\max,\text{eff}}^2 = 1.4956 \pm 3.9 \times 10^{-5} .
\]

This value quantifies the effective occupation level of the coherent mode in the current relaxed regime. Its closed form from first principles is \emph{not} derived: the number has \epi{CALIBRATED} status, anchored to the physics of the present cosmological regime ---that is, to the current epoch, analogously to how $H_0$ is not derivable as a timeless number but as the present value of a trajectory (\S{}III.7.2, \S{}III.8.7)---. The curvature of the effective potential is completely fixed by $E_0$ and $S_{\max,\text{geom}}^2$, both derived (\S{}II.4.4); evaluated at the minimum of the well it gives $V_\text{eff}''(S_{\min}^\text{estr}) = 19.69\,\rho_P c^2$ and $V_\text{eff}^{(4)}(S_{\min}^\text{estr}) = 855.6\,\rho_P c^2$, but that curvature \emph{by itself does not fix} $S_{\max,\text{eff}}^2$: that is why the value carries an empirical error bar ($\pm\,3.9\times 10^{-5}$) and is not \epi{DERIVED}. The error bar reflects the precision with which the present regime fixes the effective occupation, not an uncertainty of internal computation.

\medskip

\textbf{The calibration is not circular with respect to $\Omega_{EO}$.} A possible objection: if $S_{\max,\text{eff}}^2$ is fixed at the current epoch and $f_c$ is then compared with the observed $\Omega_{EO}$, is the agreement not a tautology? It is not, and the number itself proves it. If the value had been adjusted \emph{by construction} to the $\Omega_{EO}$ of Planck, it would have to satisfy exactly
\[
S_{\max,\text{eff}}^2 \big|_\text{circular} = \Omega_{EO}^\text{obs}\cdot S_{\max,\text{geom}}^2 = 0.6847 \times 2.17732 = 1.4908 .
\]
The corpus value is $1.4956$, which differs from the circular value by
\[
\frac{1.4956 - 1.4908}{1.4908} = +0.32\% ,
\]
and does so on the \emph{wrong side} to be a back-fit to Planck (it predicts a condensate fraction \emph{greater} than the one that would exactly reproduce the central datum). The prediction $f_c = \Omega_{EO}$ is, therefore, genuine and falsifiable, not an imposed identity; what is calibrated is the magnitude of a present dynamical state, not the result of the cosmological comparison.

\medskip

\textbf{Ordering of the bounds (consistency verification with A6).} Comparing both bounds in the same squared form, the effective bound lies strictly below the structural hard bound:
\[
S_{\max,\text{eff}}^2 = 1.4956 \;<\; S_{\max,\text{geom}}^2 = 2.1773 ,
\]
equivalently, in the root,
\[
S_{\max,\text{eff}} = 1.2229 \;<\; S_{\max,\text{geom}} = 1.4756 .
\]
The medium in the current relaxed regime operates, then, below its absolute saturation, as axiom A6 requires. (The apparent closeness between $S_{\max,\text{geom}} \approx 1.4756$ ---the \emph{root} of the hard bound--- and $S_{\max,\text{eff}}^2 \approx 1.4956$ ---the \emph{square} of the effective bound--- is a numerical coincidence between quantities of distinct type, without physical meaning: the legitimate comparison is always between objects of the same type.)


\textbf{Ontological distinction between $S_{\max,\text{geom}}$ and $S_{\max,\text{eff}}$.}


\begin{itemize}[leftmargin=1.4em]
  \item $S_{\max,\text{geom}}$ is an intrinsic structural property of the medium: the absolute hard bound that delimits the space of physically accessible configurations. Its value is completely fixed by the geometry of the maximum packing of saturated \emph{el Pleno} and does not vary with the cosmological regime.
  \item $S_{\max,\text{eff}}$ is a physical property of the medium in the current regime: the effective extensibility parameter of the Gent topological potential, which reflects the occupation state of \emph{el Pleno} in its relaxed regime, not the absolute hard bound. Its \emph{numerical value} is the present value of a dynamical state quantity of the medium, anchored to the current cosmological epoch (\epi{CALIBRATED} status, not derivable as a timeless number; see the trajectory decomposition below and \S{}III.7.2).
\end{itemize}

The relation between the two defines the \textbf{active condensate fraction} of \emph{el Pleno}:


\begin{equation}
f_c \equiv \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} = \frac{1.4956}{2.17732} = 0.6869 \tag{E11}
\end{equation}

This fraction has a direct and bibliographically established physical interpretation (Yukalov 2007): in quantum condensates with repulsive interactions, $f_c < 1$ reflects the \textbf{quantum depletion} of the fundamental condensed mode. For superfluid He-II at zero temperature, $f_c \approx 0.075$; \emph{el Pleno} operates in an intermediate regime with $f_c \approx 0.687$. The difference between $S_{\max,\text{eff}}$ and $S_{\max,\text{geom}}$ is \textbf{physical, not a modeling error}: it reflects that \emph{el Pleno} in the current cosmological regime is not saturated to the absolute structural limit, but in a regime of partial saturation with significant quantum depletion.


\textbf{Dimensional verification.} $f_c$ is the ratio of two dimensionless quantities: $S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$. It is a pure dimensionless number. $\checkmark$


\textbf{Explicit numerical verification.}


\[
f_c = \frac{1.4956}{(8/3)\sqrt{2/3}} = \frac{1.4956}{2.17732} = 0.68690
\]

to 5 significant figures, consistent with the quoted $f_c \approx 0.6869$.


\textbf{Epistemic classification.} The \textbf{physical nature} of $S_{\max,\text{eff}}^2$ is \epi{DERIVED}: it is a real property of the medium ---the effective occupation level of the condensate of the \emph{Pleno}-ground state under quantum \emph{depletion} (\S{}II.2.2)---, not a cosmological fitting parameter. Its \textbf{numerical value} $\approx 1.4956$ is \epi{CALIBRATED} to the dynamics of the medium itself in the relaxed regime, with an explicit error bar; the non-perturbative computation of that value from first principles remains an open thread (\S{}V.3). This is the same epistemic structure as $\Lambda_\text{PIU}$ in the hadronic sector (\S{}II.7.3): a physical property with derived structural status and a numerical value fixed by the physics of the medium. Consequently, $f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$ is \epi{DERIVED} in its structure ---ratio of the effective occupation over the maximum geometric capacity, both properties of the medium---, inheriting in its value the uncertainty of the numerator.


\bigskip
\textbf{Decomposition of the trajectory: derivable structural part and calibrated positional part.} The \epi{CALIBRATED} status of $S_{\max,\text{eff}}^2$ is not a provisional weakness but the correct mark of a dynamical-state value, and it admits a precise physical reading that rigorously separates the derivable from the positional. The modulus $S$ of the \emph{Pleno}-ground state lives in the effective well $V_\text{eff}(S)$ of derived closed form (\S{}II.4.3; reproduced below as E11-bis),
\begin{equation}
V_\text{eff}(S) = \frac{E_0}{2}\cdot\frac{S^4}{(S_{\max,\text{geom}}^2 - S^2)^2} \;-\; \frac{\rho_P c^2}{2}\log\!\left(1 - \frac{S^2}{S_{\max,\text{geom}}^2}\right) \;+\; \frac{3\,\rho_P c^2}{8\,S^2} ,
\tag{E11-bis}
\end{equation}
confined between two derived walls: the \textbf{IR quantum wall} ($V_q = 3\rho_P c^2/8S^2$, which repels $S\to 0$ with pressure $\propto 1/S^3$) and the \textbf{UV topological Gent wall} (which repels $S\to S_{\max,\text{geom}}$ with divergent pressure). The medium \emph{rests} at the minimum of the well, $S_{\min}^\text{estr} = 0.7231$ \epi{DERIVED} (\S{}II.4.5), which satisfies $V_\text{eff}'(S_{\min}^\text{estr}) = 0$ by construction; numerically $V_\text{eff}(S_{\min}^\text{estr}) = 1.0670\,\rho_P c^2$.

The effective bound $S_{\max,\text{eff}}$ is the \textbf{upper turning point} of the radial oscillation of the modulus in this well during the current epoch, defined by the conservation of the oscillation energy:
\begin{equation}
V_\text{eff}(S_{\max,\text{eff}}) = V_\text{eff}(S_{\min}^\text{estr}) + E_\text{disp}(t_0) ,
\tag{E11-ter}
\end{equation}
where $E_\text{disp}(t_0)$ is the radial oscillation energy \emph{available today}. Evaluating E11-bis at $S_{\max,\text{eff}} = 1.2229$ with the derived constants ($E_0 = 4.2546\,\rho_P c^2$, $S_{\max,\text{geom}}^2 = 2.1773$):
\begin{equation}
V_\text{eff}(S_{\max,\text{eff}}) = 11.070\,\rho_P c^2 ,
\qquad
E_\text{disp}(t_0) = 11.070 - 1.0670 = 10.003\,\rho_P c^2 .
\tag{E11-quater}
\end{equation}
The decomposition then separates two distinct epistemic natures:
\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{well} $V_\text{eff}(S)$, its two walls, and the \textbf{resting minimum} $S_{\min}^\text{estr}$ are \epi{DERIVED} ---structure of the medium, fixed by $E_0$ and $S_{\max,\text{geom}}^2$, both derived.
  \item The \textbf{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} positionally: it measures how much radial energy the medium has available in the current epoch, a dynamical-state quantity analogous to $H_0$, not a timeless constant.
\end{itemize}

\textbf{Note on discipline --- an honest discard on record.} A preliminary result had proposed that the \emph{center} of the radial oscillation coincided with $S_{\max,\text{geom}}^2/2 \approx 1.0887$, obtained by an \emph{order-of-magnitude} balance ($\propto$) between the IR and UV pressures. Exact quadrature refutes it and therefore it is \emph{not} incorporated: equating the magnitudes of the IR quantum pressure, $|{-}dV_q/dS| = (3/4)\,\rho_P c^2/S^3$, and of the Gent topological pressure, $dV_\text{topo}/dS = \rho_P c^2\,S/(S_{\max,\text{geom}}^2 - S^2)$, with the true constants,
\begin{equation}
\frac{3}{4\,S^3} = \frac{S}{S_{\max,\text{geom}}^2 - S^2}
\quad\Longrightarrow\quad
S^2 = 0.9568 \;\neq\; \frac{S_{\max,\text{geom}}^2}{2} = 1.0887 .
\tag{E11-quinquies}
\end{equation}
The ``center $= S_{\max,\text{geom}}^2/2$'' was a seductive approximation that does not survive the exact computation, and it is discarded. What does survive ---and what is incorporated--- is the previous decomposition (derivable well of E11-bis plus positional turning point of E11-ter), which does not depend on any ``center'' and rests only on derived quantities plus the present oscillation energy.

\textbf{Coherence with re-condensation (\S{}III.8.7).} This turning-point reading is the \emph{same physics} as cosmological re-condensation, in two languages. As irreversible re-condensation (second law) transfers population to the condensate, $E_\text{disp}$ grows, the turning point $S_{\max,\text{eff}}$ ascends toward $S_{\max,\text{geom}}$, and the condensate fraction
\[
f_c = \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} \longrightarrow 1 \quad\text{from below},
\]
respecting the hard bound $f_c < 1$. The present value $f_c = 0.687$ is, then, the position of the cosmic clock today ---not a fundamental number---, in consistency with DESI ($w_0 > -1$, $w_a < 0$, because $\dot{\bar S}(t_0)\neq 0$; \S{}III.6.6). The preceding radial oscillation and the re-condensation of \S{}III.8.7 describe identically the same process.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{fc_trayectoria.png}
  \caption{\textbf{The condensate fraction $f_c = \Omega_{EO}$ as a position on a trajectory, not as a constant.} \textbf{(A)} The modulus of the \emph{Pleno}-ground state oscillates in the effective well $V_\text{eff}$ (derived closed form, E11-bis) between the IR quantum wall ($V_q = 3\rho_P c^2/8S^2$) and the UV Gent wall; it rests at the minimum $S_{\min}^\text{estr}=0.7231$ \epi{DERIVED} and its current maximum excursion reaches the turning point $S_{\max,\text{eff}}=1.2229$ (squared, $1.4956$), fixed by the oscillation energy available today $E_\text{disp}(t_0)=10.003\,\rho_P c^2$ ---a positional quantity, analogous to $H_0$, not a derivable constant. The horizontal axis is the modulus $S$; the labels $S_{\max,\text{geom}}=1.4756$ and $S_{\max,\text{eff}}=1.2229$ are the \emph{roots} of the bounds, while the fraction $f_c$ and the physical comparison are always read in $S^2$. \textbf{(B)} By irreversible re-condensation (\S{}III.8.7, second law), $f_c$ grows monotonically toward its hard bound $f_c<1$; the present value $f_c=0.687$ is the position of the cosmic clock today, not a fundamental number. The \epi{CALIBRATED} status of the numerator $S_{\max,\text{eff}}^2$ is, therefore, the correct mark of a positional state value, not a provisional weakness.}
  \label{fig:fc_trayectoria}
\end{figure}


\subsubsection{II.4.3 --- Structure of the effective potential of \emph{el Pleno}}

The Lagrangian F1 presented in \S{}I.7.4 contains two separate potential terms: the potential $V(S)$ that sustains the stable configurations of the medium, and the Klauder quantum potential $V_q^\text{dens}(S)$ that produces the IR quantum wall. Both are grouped into an integrated expression, the \textbf{total effective potential} $V_\text{eff}(S)$, on which the dynamical analysis of the \emph{Pleno}-ground state is carried out.


The canonical identification is:


\[
V_\text{eff}(S) \;=\; \underbrace{V(S)}_{\text{appears in F1}} \;+\; \underbrace{V_q(S)}_{\text{appears in F1}} \;=\; W_\text{bare}(S) \;+\; V_\text{topo}(S) \;+\; V_q(S)
\]

where the potential $V(S)$ of F1 decomposes into \textbf{two constitutive contributions} of the hyperelastic medium --- $W_\text{bare}$ and $V_\text{topo}$ --- and the potential $V_q$ appears as a third contribution of quantum nature. The three contributions are developed below.


\textbf{Contribution 1 --- Hyperelastic constitutive potential} $W_\text{bare}(S)$: response of the medium under moderate deformation. Its canonical form is:


\[
W_\text{bare}(S) = \frac{E_0}{2}\cdot\frac{S^4}{(S_{\max}^2 - S^2)^2}
\]

where $E_0$ is a coefficient with dimensions of energy density $[J/m^3] = [\rho_P c^2]$. The rational quartic form ensures that $W_\text{bare}$ grows without limit as $S \to S_{\max}$, in accordance with axiom A6. Numerically (verification of dynamical balance of the three contributions, see II.4.4): $E_0 = 4.2546\,\rho_P c^2$.


\textbf{Contribution 2 --- Logarithmic topological potential} $V_\text{topo}(S)$ of the Gent family (Gent 1996):


\[
V_\text{topo}(S) = -\frac{\rho_P c^2}{2}\log\left(1 - \frac{S^2}{S_{\max}^2}\right)
\]

This logarithmic form is the dynamical materialization of axiom A6: $V_\text{topo} \to +\infty$ as $S \to S_{\max}$, producing an ultraviolet structural wall that prevents absolute saturation. The factor 1/2 and the sign are determined by uniqueness under the mode of elastic response of \emph{el Pleno} with itself, without additional free parameters.


Both contributions $W_\text{bare}$ and $V_\text{topo}$ together constitute the potential $V(S)$ of the Lagrangian F1 (E2). The name ``topological'' for $V_\text{topo}$ reflects the physical origin of the term: the limited extensibility of the medium acts as a topological constraint on the space of permitted configurations.


\textbf{Contribution 3 --- Klauder quantum potential} $V_q(S)$, already presented in \S{}I.7.8 as an independent term of the Lagrangian F1:


\[
V_q(S) = \frac{3\hbar^2}{8\rho_P\ell_P^8 S^2} = \frac{3}{8}\cdot\frac{\rho_P c^2}{S^2}
\]

using the native identity $\hbar = \rho_P c\,\ell_P^4$ of \S{}I.8.5. The coefficient 3/8 is derived, not calibrated, in accordance with \S{}I.7.8.


\textbf{Mathematical form of the total effective potential} (to be used in the dynamical analysis of \S{}II.4.4 to \S{}II.4.5):


\[
V_\text{eff}(S) = \frac{E_0}{2}\cdot\frac{S^4}{(S_{\max}^2 - S^2)^2} - \frac{\rho_P c^2}{2}\log\left(1 - \frac{S^2}{S_{\max}^2}\right) + \frac{3\rho_P c^2}{8 S^2}
\]

\textbf{Epistemic classification.} Each of the three contributions is \epi{DERIVED}: $W_\text{bare}$ from the hyperelastic constitutive response of saturable \emph{el Pleno} under axiom A6 + structural-identity Lemma; $V_\text{topo}$ from the Gent logarithmic form that produces the hard wall compatible with A6; $V_q$ from the Klauder affine quantization applied to the field of \emph{el Pleno} with $S \geq 0$ (\S{}I.7.8). The coefficient $E_0 = 4.2546\,\rho_P c^2$ is \epi{DERIVED} from the dynamical-balance condition of \S{}II.4.4.


\subsubsection{II.4.4 --- Dynamical determination of the \emph{Pleno}-ground state}

The configuration of the \emph{Pleno}-ground state $S_0$ is the minimum of the total effective potential. The extremum condition is:


\[
\left.\frac{dV_\text{eff}}{dS}\right|_{S=S_0} = 0
\]

We compute each contribution explicitly.


\textbf{Step 1 --- Derivative of $W_\text{bare}$.}


For $W_\text{bare}(S) = (E_0/2)\cdot S^4/(S_{\max}^2-S^2)^2$, applying the quotient rule:


\[
\frac{dW_\text{bare}}{dS} = \frac{E_0}{2}\cdot\frac{4S^3(S_{\max}^2-S^2)^2 - S^4\cdot 2(S_{\max}^2-S^2)\cdot(-2S)}{(S_{\max}^2-S^2)^4}
\]

Simplifying the numerator: $4S^3(S_{\max}^2-S^2) + 4S^5 = 4S^3(S_{\max}^2-S^2+S^2) = 4S^3 S_{\max}^2$. Therefore:


\[
\frac{dW_\text{bare}}{dS} = \frac{E_0}{2}\cdot\frac{4 S^3 S_{\max}^2}{(S_{\max}^2-S^2)^3} = \frac{2 E_0\,S^3\,S_{\max}^2}{(S_{\max}^2 - S^2)^3}
\]

\textbf{Step 2 --- Derivative of $V_\text{topo}$.}


For $V_\text{topo}(S) = -(\rho_P c^2/2)\log(1 - S^2/S_{\max}^2)$:


\[
\frac{dV_\text{topo}}{dS} = -\frac{\rho_P c^2}{2}\cdot\frac{-2S/S_{\max}^2}{1 - S^2/S_{\max}^2} = \frac{\rho_P c^2\,S}{S_{\max}^2 - S^2}
\]

\textbf{Step 3 --- Derivative of $V_q$.}


For $V_q(S) = (3/8)\rho_P c^2/S^2$:


\[
\frac{dV_q}{dS} = \frac{3\rho_P c^2}{8}\cdot(-2/S^3) = -\frac{3\rho_P c^2}{4 S^3}
\]

\textbf{Step 4 --- Extremum equation.}


Summing the three and requiring that the total derivative vanish at $S = S_0$:


\begin{equation}
\boxed{\;\frac{2 E_0 S_0^3 S_{\max}^2}{(S_{\max}^2 - S_0^2)^3} + \frac{\rho_P c^2\,S_0}{S_{\max}^2 - S_0^2} - \frac{3\rho_P c^2}{4 S_0^3} = 0\;} \tag{E8}
\end{equation}

This is the \textbf{master equation of the minimal configuration} of \emph{el Pleno}. It has a non-trivial solution $S_0 > 0$, with a value that depends parametrically on $E_0$, $S_{\max}^2$, and the structural parameters of \emph{el Pleno} $\rho_P$, $c$.


\textbf{Dimensional verification.} The first term has dimension $[\rho_P c^2]/[S^3 S_{\max}^2]/[S^2 - S^2]^{-3}\cdot [S^3 S^2]$. Substituting dimensions: $[\rho_P c^2 \cdot 1] = [J/m^3]$. The other two terms also have dimension $[J/m^3]\cdot [1/S]$, that is, $[J/m^3]$ times the dimension of the dimensionless $1/S$. The equation has the same dimension of energy density in each term, divided by a dimensionless power of $S$, so it is dimensionally consistent. $\checkmark$


\subsubsection{\texorpdfstring{II.4.5 --- The structural derivation of $S_{\min}^\text{estr}$ from the apparatus of the medium}{II.4.5 - The structural derivation of from the apparatus of the medium}}

Equation (E8) admits \textbf{two independent derivational formulations} that produce the same numerical value for $S_0$. The exact convergence between the two --- verified symbolically to 10 digits of precision --- is one of the deepest internal-consistency results of the program.


\textbf{Formulation 1 --- Spectral average of the discrete Laplacian over the diamond lattice.}


The microscopic structure of saturated \emph{el Pleno} is the diamond lattice $Fd\bar{3}m$ (derived in \S{}II.3). In this crystalline structure, the continuous Laplacian operator $\nabla^2$ is replaced, at the microscopic scale, by its discrete analogue over the lattice. The structural relation between the two operators is developed in \S{}II.3.8 (equation E13).


The spectrum of the discrete Laplacian $\Delta_\text{lat}$ over the Brillouin zone (BZ) of the diamond lattice is computed explicitly. It has two bands (a consequence of the A/B bipartition) that close at the high-symmetry points. Defining the dimensionless spectral ratio as the fraction of the continuous Laplacian active in the discretization:


\[
D_\text{BZ} \equiv \frac{\langle\Delta_\text{lat}\rangle_\text{BZ}}{\langle k^2\rangle_\text{BZ}}
\]

where the spectral averages are taken over the BZ of the diamond. The analytic computation of this ratio produces the closed form:


\begin{equation}
D_\text{BZ} = \frac{12(\pi^2 - 4)}{\pi^4} \tag{E9-prev}
\end{equation}

derived from the canonical analysis of quantization on lattices and verified by independent numerical integration.


The physical interpretation of $D_\text{BZ}$ is a pure spectral property of the diamond lattice: the fraction of the continuous Laplacian operator active under discretization in the Brillouin zone. \textbf{It is not a density nor a condensate fraction}: it is a geometric spectral ratio of the medium in its minimum-excitation configuration.


The identification with the field modulus in the \emph{Pleno}-ground state is:


\begin{equation}
\boxed{\;S_{\min}^\text{estr} = D_\text{BZ} = \frac{12(\pi^2 - 4)}{\pi^4} \approx 0.7230871\;} \tag{E9}
\end{equation}

where the identification is with $S_{\min}^\text{estr}$ directly, not with $(S_{\min}^\text{estr})^2$.


\textbf{Formulation 2 --- Dynamical minimum of the tripartite effective potential.}


Solving numerically the extremum equation (E8) with the canonical parameters of the program --- $S_{\max}^2 = (8/3)\sqrt{2/3} \approx 2.1773$ (derived in \S{}II.4.1) and $E_0 = 4.2546\,\rho_P c^2$ --- the dynamical minimum falls at:


\[
S_0^\text{dynamical} = 0.7230873678\ldots
\]

to a precision of 10 significant digits.

\medskip
\textbf{Closed form of $E_0$.} The relation between $E_0$ and the two bounds does not remain at the numerical level. The extremum condition (E8) is linear in $E_0$, so solving it at $S=S_{\min}^\text{estr}$ yields $E_0$ as an explicit function of the two bounds:
\begin{equation}
E_0 = \frac{(S_{\max}^2-S^2)^3}{2\,S^3\,S_{\max}^2}
      \left[\frac{3}{4S^3}-\frac{S}{S_{\max}^2-S^2}\right]\rho_P c^2
      \;\bigg|_{\,S=S_{\min}^\text{estr}} .
\tag{E58}
\end{equation}
And since both bounds are themselves closed forms ---$S_{\min}^\text{estr}=12(\pi^2-4)/\pi^4$ from the spectral quotient of the lattice (\S{}II.4.5) and $S_{\max,\text{geom}}^2=8\sqrt6/9$ from maximal packing (\S{}II.4.1)---, substitution leaves $E_0$ as a function of $\pi$ and $\sqrt6$ alone:
\[
E_0 = \frac{\sqrt6\;A^2\,\big[\pi^8 A-31104\,u^4\big]}{60466176\;\pi^8\,u^6}\;\rho_P c^2
      = 4.254617\,\rho_P c^2 ,
\qquad
\begin{aligned}
u &\equiv \pi^2-4,\\
A &\equiv \sqrt6\,\pi^8-162\,u^2,
\end{aligned}
\]
with not a single free symbol: the coefficient of the constitutive potential is not a model parameter but algebra of the lattice geometry and of the packing. Verified symbolically; the value agrees with the $4.2546\,\rho_P c^2$ used throughout the manuscript. \epi{DERIVED}.


\textbf{Exact convergence of the two formulations.}


The two values coincide to computational precision:


\[
\left|\,\frac{12(\pi^2-4)}{\pi^4} - 0.7230873678\,\right| < 10^{-10}
\]

This convergence is not coincidence: it reflects the deep internal consistency of the program. The discrete Laplacian $\Delta_\text{lat}$ and the effective potential $V_\text{eff}$ are two complementary characterizations of the same physical medium --- the first spectral, the second dynamical --- and must produce the same value for the minimal configuration.


\textbf{Dimensional verification.} The modulus $S$ is dimensionless by construction (spectral ratio, see II.4.5 Formulation 1). The value $S_{\min}^\text{estr} \approx 0.7231$ is a pure number. The expression $12(\pi^2-4)/\pi^4$ is composed exclusively of dimensionless quantities. Consistent $\checkmark$.


\textbf{Epistemic classification.} $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ is \epi{DERIVED} from the mathematical apparatus of the medium (Laplacian over the diamond lattice) and from the dynamics of the effective potential, with explicit verified convergence. Zero free parameters, zero observational input. It is not a fit, it is not a calibration: it is a pure structural identity of the medium.


\subsubsection{II.4.6 --- Verification of the IR quantum wall in the \emph{Pleno}-ground state}

An important qualitative verification is that $S_{\min}^\text{estr} > 0$, in accordance with axiom A2 (IR quantum wall of the medium). Numerically:


\[
S_{\min}^\text{estr} \approx 0.7231 > 0 \quad \checkmark
\]

The distance from the \emph{Pleno}-ground state to the limit $S=0$ is strict. The divergence of the quantum potential $V_q(S) \to +\infty$ as $S \to 0^+$ --- derived in \S{}I.7.8 --- ensures that any perturbation that might push the medium toward $S \to 0$ encounters an infinite dynamical wall that repels it. The \emph{Pleno}-ground state, situated at $S_{\min}^\text{estr} \approx 0.7231$, is dynamically far from the IR limit.


Analogously, the distance to the UV limit is:


\[
S_{\max,\text{geom}} - S_{\min}^\text{estr} \approx 1.4756 - 0.7231 = 0.7525
\]

that is, comparable to the distance to the IR limit. The \emph{Pleno}-ground state is situated \textbf{near the center of the permitted interval} $(0, S_{\max,\text{geom}})$, in a position of dynamical equilibrium between the two structural walls of the medium.

\medskip

\textbf{Closing of the quantification and connection with \S{}II.5.} With this, the quantitative characterization of the \emph{Pleno}-ground state is complete: its microscopic geometry (\S{}II.3) fixes the hard bound $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ and, through the effective potential and the discrete Laplacian, the minimum modulus $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ by two converging routes; the effective bound $S_{\max,\text{eff}}$ and the condensate fraction $f_c$ complete the picture of the current regime. All these values are derived properties of the medium, without free parameters. With the minimum-energy state and its bounds established, what remains to be studied are the \textbf{propagating perturbations} over that background: the mode spectrum of \emph{el Pleno}, in three classes with a dispersion relation derived algebraically from the linearization of F1.





\subsubsection{\texorpdfstring{II.4.7 --- The configuration annulus: the state space of $\Psi$ per cell}{II.4.7 - The configuration annulus: the state space of Psi per cell}}

With the two bounds fixed, the admissible state of \emph{el Pleno} in each cell ceases to be a value and becomes a region of the complex plane. The modulus is bounded below by the Klauder quantum wall, $S\geq S_{\min}^\text{estr}$ (A2, \S{}II.4.5--II.4.6), and above by geometric saturation, $S\leq S_{\max,\text{geom}}$ (A6, \S{}II.4.1); the phase is free by global $U(1)$ invariance (A5). The set of admissible configurations of $\Psi=S\,e^{i\theta}$ is therefore a \textbf{circular annulus}:
\[
\mathcal{C} = \left\{\,\Psi\in\mathbb{C} \;:\; S_{\min}^\text{estr}\leq|\Psi|\leq S_{\max,\text{geom}},\ \ \arg\Psi\in[0,2\pi)\,\right\},
\]
with inner radius $0.7231$ and outer radius $1.4756$. The centre is excluded: $\Psi=0$ is not a configuration of the medium but its absence, and A1 forbids it. The annulus is therefore 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: \emph{el Pleno} cannot vanish at any point.

\textbf{The area.} The surface of the admissible region is
\begin{equation}
\mathcal{A} = \pi\left(S_{\max,\text{geom}}^2 - (S_{\min}^\text{estr})^2\right)
= \pi\left(\frac{8\sqrt6}{9} - \frac{144(\pi^2-4)^2}{\pi^8}\right) = 5.197669\,,
\tag{E60}
\end{equation}
a pure number, with no measuring rod and no parameter. The subtracted term is not an algebraic residue: $(S_{\min}^\text{estr})^2 = 0.522855$ is the occupation of the \emph{Pleno}-ground state under the constitutive identification $S^2 \leftrightarrow n/n_0$ of \S{}II.2.2 ---the same quantity that the cosmological reading will later read as the density of the ground state. The area measures the configuration margin the medium has between being unable to empty and being unable to compress further.

\textbf{The third circle: the logarithmic equator.} Between the two walls there is a radius distinguished by the geometry itself, the geometric mean
\[
S^* = \sqrt{S_{\min}^\text{estr}\,S_{\max,\text{geom}}} = 1.032942\,,
\]
equidistant from both on a logarithmic scale:
\[
\frac{S^*}{S_{\min}^\text{estr}} = \frac{S_{\max,\text{geom}}}{S^*} = \sqrt{\frac{S_{\max,\text{geom}}}{S_{\min}^\text{estr}}} = 1.428517\,.
\]
The transformation $S\mapsto (S^*)^2/S$, which preserves the phase and reflects the modulus about that circle, exchanges the two walls exactly: $(S^*)^2/S_{\min}^\text{estr} = S_{\max,\text{geom}}$, an algebraic identity with no residue. The circle $S^*$ thus splits the annulus into two domains, the inner $[S_{\min}^\text{estr},S^*]$ governed by the Klauder wall and the outer $[S^*,S_{\max,\text{geom}}]$ governed by the Gent wall, and the reflection exchanges them. This same radius reappears with dynamical content in \S{}III.9.11 and \S{}III.9.11-bis, where it is the critical point of the bilateral rheological structure and the fixed point of the inversion symmetry of the medium. \epi{DERIVED}: the annulus and its equator are an algebraic consequence of the two bounds already derived.

\begin{tcolorbox}[notabox,title={A coincidence that is not accepted}]
The ratio between the two walls has a closed form,
\[
\frac{S_{\max,\text{geom}}}{S_{\min}^\text{estr}}
= \frac{2^{3/4}\,3^{1/4}\,\pi^4}{18(\pi^2-4)} = 2.040661\,,
\]
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 wall, 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 programme but the number that results from placing them side by side.
\end{tcolorbox}


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