% !TeX root = main.tex
\subsection{\S{}I.7 --- F1 as the Lagrangian of el Pleno}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (Lagrangian-construction section)
  \item \textbf{Notes:} pivot section of Part I and of the complete manuscript. Presents the F1 Lagrangian of \emph{el Pleno} as the minimal mathematical expression compatible with the six axioms A1--A6 established in \S{}I.5. Applies the standard procedure of Lagrangian construction of \S{}I.6. Each term is presented first in conceptual language, then in its precise mathematical form, with explicit dimensional verification and axiomatic connection; each step complete, without leaps. Prepares \S{}I.8.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.7.1 --- Opening: from the framework to the fundamental mathematical object}

Applying the Lagrangian-construction procedure to \emph{el Pleno}, with the six axioms A1--A6 as structural constraints, produces a single object: \textbf{F1}, the Lagrangian density of \emph{el Pleno} under PIU-$\Psi$.


F1 is the unique mathematical object from which all the subsequent phenomenology will be derived. The field equations will be obtained by applying Euler-Lagrange to F1. The structural properties of \emph{el Pleno} (fundamental density, microscopic scale, propagation speed, rigidity against perturbations, UV regulation, IR support) will be read as specific coefficients of the terms of F1. The stable solutions --- ground-state \emph{el Pleno}, solitons, cosmological configurations --- will emerge as particular solutions of the field equation. The propagation modes --- gravitational waves, photons --- will be perturbations of F1 around background solutions. The fundamental constants $G$ and $\hbar$ will appear as structural identities between the coefficients of F1 and the parameters of \emph{el Pleno}.


For this centrality, each step of the construction of F1 is justified explicitly: each coefficient is connected with its axiomatic origin and each physical interpretation is anchored in the language of continuous media. F1 is the primary foundational formula of the program and does not admit a hurried presentation.


The \S{}I.7.2 to \S{}I.7.4 construct F1 applying the procedure of \S{}I.6; the \S{}I.7.5 to \S{}I.7.8 analyze each term ---kinetic, topological potential, flexural, Klauder---; \S{}I.7.9 verifies the dimensional consistency and the finding that the kinetic coefficient coincides with that of Einstein-Hilbert; \S{}I.7.10 derives the Euler-Lagrange field equation.


\subsubsection{I.7.2 --- Construction of F1, Step 1: the fundamental field}

The first step of the constructive procedure of \S{}I.6.4 is to identify the fundamental field of the system on which the Lagrangian is constructed.


For \emph{el Pleno}, the fundamental field is the complex scalar field $\Psi$, in accordance with axiom A1 established in \S{}I.5.2. The uniqueness of \emph{el Pleno} (a single continuous physical substrate) translates into the uniqueness of the field: the complete manuscript operates on a single $\Psi$, not on multiple independent fields.


The field $\Psi$ decomposes canonically into modulus and phase:


\begin{equation}
\Psi(x^\mu) = S(x^\mu) \cdot e^{i\theta(x^\mu)} \tag{E1}
\end{equation}

where:


\begin{itemize}[leftmargin=1.4em]
  \item $S(x^\mu) \geq 0$ is the \textbf{modulus} of the field, a real non-negative function of the coordinates $x^\mu$ that parametrize the points of \emph{el Pleno}. It carries the structural information of \emph{el Pleno}: the stable configurations ($S_{\min}$, $S_{\max}$), the material excitations (perturbations $\delta S$), the localized solitonic solutions.
  \item $\theta(x^\mu) \in [0, 2\pi)$ is the \textbf{phase} of the field, a real function of the coordinates $x^\mu$. It carries the induced gauge dynamics of \emph{el Pleno}: the electromagnetic sector, the charges conserved by the U(1) symmetry of axiom A5, the Borromean topological charges that emerge in specific configurations.
\end{itemize}

The decomposition $\Psi = S \cdot e^{i\theta}$ has several structural advantages. First, it explicitly separates the two physical contributions of the complex field: the modulus $S$ controls the structural dynamics of the medium, the phase $\theta$ controls the gauge dynamics. Second, it makes transparent the action of the global U(1) symmetry of axiom A5: a global phase transformation $\Psi \to e^{i\alpha}\Psi$ corresponds to $S \to S$ and $\theta \to \theta + \alpha$, leaving the modulus invariant. Third, it connects directly with the formal analogy with the Gross-Pitaevskii condensate and with the Ginzburg-Landau order parameter (mentioned in \S{}I.3.5), where the same decomposition $\Psi = S \cdot e^{i\theta}$ is standard.


The detailed presentation of $\Psi$ and the physical interpretation of the modulus and the phase are developed in \S{}I.9; here $\Psi$ enters as the field on which the Lagrangian is constructed.


\subsubsection{I.7.3 --- Construction of F1, Step 2: the required symmetries}

The second step of the constructive procedure is to identify the symmetries the Lagrangian must respect.


For \emph{el Pleno} under PIU-$\Psi$, the symmetries come directly from the structural axioms A1--A6 established in \S{}I.5. Six symmetries must be respected:


\textbf{Symmetry 1 --- Continuity and differentiability (consequence of A2).} The Lagrangian must be constructed from the field $\Psi$ and its derivatives by well-defined differential operations. This restricts the possible terms to polynomials in $\Psi$, $\partial_\mu \Psi$, $\partial_\mu \partial_\nu \Psi$, etc. --- all products of differentiable objects.


\textbf{Symmetry 2 --- Structural covariance (axiom A3).} The Lagrangian must be a scalar under the group of transformations of \emph{el Pleno}. Each term must be constructed by contracting indices consistently, so that the result is invariant under the transformations that connect distinct observers. In the regime where the effective metric of \emph{el Pleno} tends to Minkowski (present relaxed regime), this translates into Lorentz invariance: the terms of the Lagrangian are constructed by contracting Lorentz indices $\mu, \nu$.


\textbf{Symmetry 3 --- Ultraviolet regulation (axiom A4).} The Lagrangian must include a term that penalizes field configurations with high structural curvature on scales comparable to $\ell_P$. This term must involve derivatives of order higher than the first. The simplest canonical form compatible with the other symmetries is a term quadratic in the d'Alembertian of the modulus, $(\Box S)^2$.


\textbf{Symmetry 4 --- Global U(1) symmetry (axiom A5).} The Lagrangian must be invariant under global phase transformations $\Psi \to e^{i\alpha}\Psi$ with $\alpha \in \mathbb{R}$ constant. This is satisfied automatically for any term constructed from combinations involving $|\Psi|^2 = \Psi^* \Psi = S^2$ or derivatives that cancel under the transformation.


\textbf{Symmetry 5 --- Self-adjointness over the half-line (consequence of A4 and of affine quantization).} Since the modulus $S$ is non-negative ($S \geq 0$), standard canonical quantization (based on operators defined over the whole real line) does not apply directly to \emph{el Pleno}. The correct quantization is the affine one (Klauder), which is designed precisely for manifolds with a boundary of the half-line type. This quantization produces unambiguously an additional term in the Lagrangian --- the quantum potential $V_q(S)$ --- with positive sign and a specific $3/8$ coefficient, derived without ordering ambiguity.


These five symmetries are the structural constraints that F1 must satisfy. Any term that violates one of them is excluded from the Lagrangian; any term that respects them is a candidate to enter.


\subsubsection{I.7.4 --- Construction of F1, Step 3 and Step 4: the compatible terms and their coefficients}

The third step of the procedure is to construct the terms compatible with the identified symmetries. The fourth step is to assign coefficients to each term.


\textbf{Kinetic term.} The simplest term that encodes the propagation of the field is quadratic in the first derivatives:


\[
\mathcal{L}_\text{cin} = \frac{1}{2}\rho_P c^2 \ell_P^2 \, \partial_\mu \Psi^* \partial^\mu \Psi
\]

This term is invariant under structural covariance (the indices $\mu$ are contracted in the Lorentz sense in the relaxed regime) and under global U(1) ($\partial_\mu \Psi^* \partial^\mu \Psi$ is invariant under $\Psi \to e^{i\alpha}\Psi$). The coefficient $\rho_P c^2 \ell_P^2$ is constructed from the three structural parameters of \emph{el Pleno} --- fundamental density $\rho_P$, microscopic scale $\ell_P$, structural speed $c$ --- combined so that the term has dimensions of energy density (which we will verify explicitly in \S{}I.7.9). The factor $1/2$ is standard convention in the Lagrangian formulation of scalar fields.


For the modulus and the phase separately, expanding $\Psi = S \cdot e^{i\theta}$ gives:


\[
\partial_\mu \Psi^* \partial^\mu \Psi = (\partial_\mu S)(\partial^\mu S) + S^2 (\partial_\mu \theta)(\partial^\mu \theta)
\]

so that the kinetic term contains two coupled contributions:


\[
\mathcal{L}_\text{cin} = \frac{1}{2}\rho_P c^2 \ell_P^2 \left[ (\partial_\mu S)(\partial^\mu S) + S^2 (\partial_\mu \theta)(\partial^\mu \theta) \right]
\]

The contribution of the modulus describes the structural propagation of the medium. The contribution of the phase, modulated by $S^2$, describes the propagation of the induced gauge sector.


\textbf{Topological potential term.} The next required term is the potential $V(S)$, an exclusive function of the modulus (automatically invariant under U(1) by construction). This potential encodes the structure of stable configurations of the medium: the specific values of $S$ that correspond to minimum-energy states or to structurally preferred configurations.


The specific form of $V(S)$ will be developed in detail in \S{}II.2 when the permitted configurations of \emph{el Pleno} are discussed. For the purposes of the present section, $V(S)$ is introduced as a term of the Lagrangian:


\[
\mathcal{L}_\text{pot} = -V(S)
\]

with the negative sign in accordance with the standard convention in the Lagrangian formulation (the potential enters subtracting so that the minimum-potential configurations are maxima of the action). The potential $V(S)$ has dimensions of energy density. Its specific form, derived structurally from the axioms and stability constraints, includes a hyperelastic component --- $W_\text{bare}(S) = (E_0/2) S^4/(S_\text{max}^2 - S^2)^2$ --- and a logarithmic component that sustains the solitonic configurations. The complete derivation is developed in \S{}II.2.


\textbf{Flexural term.} Axiom A4 (ultraviolet regulation) requires a term that penalizes field configurations with high structural curvature. The simplest canonical form compatible with the symmetries is quadratic in the d'Alembertian of the modulus:


\[
\mathcal{L}_\text{flex} = -\frac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2
\]

where $\Box S = \partial_\mu \partial^\mu S$. The coefficient $\rho_P c^2 \ell_P^4 / 40$ is constructed from the structural parameters of \emph{el Pleno}; the power $\ell_P^4$ (instead of the $\ell_P^2$ of the kinetic term) is necessary to maintain dimensions of energy density given that $(\Box S)^2$ has two additional derivatives relative to the kinetic term. The numerical factor $1/40$ emerges from the specific derivation from the discrete lattice of \emph{el Pleno} (Taylor expansion plus SO(3) angular average over the Brillouin zone of the diamond), derived explicitly in \S{}II.3.8 (P9) and referenced in \S{}I.8.5.


By analogy with the Euler-Bernoulli theory of beams (classical physical medium), the coefficient $\rho_P c^2 \ell_P^4 / 40$ is identified with the \textbf{flexural rigidity} of \emph{el Pleno} and is denoted:


\begin{equation}
\gamma_\kappa \equiv \frac{\rho_P c^2 \ell_P^4}{40} = \frac{\hbar c}{40} \approx 7.90 \times 10^{-28}\ \text{J} \cdot \text{m} \tag{E4}
\end{equation}

with the numerical value evaluated in the canonical units of the program, and where the second equality uses the native identity $\hbar = \rho_P c \ell_P^4$ derived in \S{}I.8.5. The flexural term penalizes high-curvature configurations of the medium, exactly as the bending rigidity EI penalizes the curvature of an elastic beam.


\textbf{Klauder term (quantum IR wall).} Axiom A2 (strict positivity of the modulus) has a dynamical realization in the infrared regime: the modulus $S$ does not admit values arbitrarily close to zero, but has a quantum support wall that prevents $S \to 0$. This wall is not postulated independently: it emerges unambiguously from the affine Klauder quantization applied to the field of \emph{el Pleno}, since $S \geq 0$ defines a configuration manifold of the half-line type. (It must not be confused with the ultraviolet regulation of A4, which acts on the curvature of the modulus, nor with the hard bound of A6 at $S \to S_{\max}$: the IR wall is the realization of the positivity of A2 at $S \to 0$.)


The derivation of the affine Klauder quantization, developed for configuration manifolds with a half-line boundary, produces the quantum potential:


\begin{equation}
V_q^\text{dens}(S) = \frac{3\hbar^2}{8\, \rho_P\, \ell_P^8\, S^2} \tag{E5}
\end{equation}

with unambiguous $3/8$ coefficient and positive sign. The term enters the Lagrangian with negative sign (in accordance with the convention for potentials):


\[
\mathcal{L}_q = -V_q^\text{dens}(S) = -\frac{3\hbar^2}{8\, \rho_P\, \ell_P^8\, S^2}
\]

The verification that this term has dimensions of energy density is carried out in \S{}I.7.9.


\textbf{The complete F1 Lagrangian.} Combining the four constructed terms, the Lagrangian of \emph{el Pleno} under PIU-$\Psi$ is:


\begin{equation}
\boxed{\;\mathcal{L}_{F1} = \frac{1}{2}\rho_P c^2 \ell_P^2 \, \partial_\mu \Psi^* \partial^\mu \Psi - V(S) - \frac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2 - V_q^\text{dens}(S)\;} \tag{E2}
\end{equation}

This is \textbf{F1} --- the Lagrangian of \emph{el Pleno}. It is the minimal mathematical expression compatible with the six structural axioms A1--A6. Each term fulfils a specific structural function: the kinetic one propagates the field, the potential sustains the stable configurations, the flexural one regulates the UV regime, the Klauder one regulates the IR regime. None can be removed without sacrificing functionality. None was added out of a need for fitting.


\subsubsection{I.7.5 --- Analysis of the kinetic term}

The kinetic term of F1 encodes the propagation of \emph{el Pleno} in its dimensions of extension and evolution. It is responsible for the medium admitting waves, mobile excitations, perturbations that propagate with finite speed.


Its mathematical form is:


\[
\mathcal{L}_\text{cin} = \frac{1}{2}\rho_P c^2 \ell_P^2 \, \partial_\mu \Psi^* \partial^\mu \Psi
\]

\textbf{Structure of the coefficient $\rho_P c^2 \ell_P^2$.} The coefficient is constructed from the three structural parameters of \emph{el Pleno}:


\begin{itemize}[leftmargin=1.4em]
  \item $\rho_P$ --- \textbf{fundamental density} of \emph{el Pleno}. It has dimensions of mass/volume, [kg/m$^{3}$]. Its canonical value will be established in \S{}I.8 when the properties of \emph{el Pleno} are discussed; numerically, $\rho_P$ is of order the Planck density.
  \item $c$ --- \textbf{structural propagation speed} of \emph{el Pleno}. It is the propagation speed of the modes of the medium, identified in the relaxed regime with the speed of light. It has dimensions [m/s]. Its universal and constant value is the structural consequence of axiom A3 (covariance).
  \item $\ell_P$ --- \textbf{characteristic microscopic scale} of \emph{el Pleno}. It is the characteristic length of the medium, below which the continuous-medium description ceases to apply. It has dimensions [m]. Its canonical value is of order the Planck length.
\end{itemize}

The product $\rho_P c^2 \ell_P^2$ is the coefficient that weights the kinetic term in the Lagrangian. The explicit dimensional verification that the complete kinetic term has dimensions of energy density is carried out in \S{}I.7.9.


\textbf{Where $G$ emerges from: the kinetic coefficient is what is primary.} The logical direction matters with precision, because it is the point where the program distinguishes itself from a fit. The coefficient $\rho_P c^2 \ell_P^2$ is \emph{not} constructed from $G$: it is fixed by the structural properties of the medium ($\rho_P$, $\ell_P$) and its propagation speed ($c$), before any reference to gravity. What happens is the inverse: when the interaction that this kinetic term produces between two perturbations of \emph{el Pleno} is computed (\S{}III.1.2), the result is an energy of the Newtonian form $\propto 1/r$, and on comparing its prefactor with that of the gravitational energy $-Gm_1 m_2/r$, the constant $G$ \emph{is identified} in terms of the parameters of the medium. $G$ does not enter: it comes out. The resulting equality is
\begin{equation}
\rho_P c^2 \ell_P^2 = \frac{c^4}{G}
\qquad\Longleftrightarrow\qquad
G = \frac{c^2}{\rho_P\, \ell_P^2}
\tag{E3}
\end{equation}
and, in consequence, the kinetic coefficient of F1 admits the rewriting
\[
\frac{1}{2}\rho_P c^2 \ell_P^2 = \frac{c^4}{2G},
\]
which is exactly the prefactor of the Ricci scalar in the Einstein-Hilbert action. The equality (E3) is therefore a \emph{reading} of the coefficient that the medium already has, not a hypothesis about $G$.

\textbf{Why the coefficient is exactly $c^2/(\rho_P\ell_P^2)$, not another combination.} The natural objection is that, by dimensional analysis, infinitely many combinations of $(\rho_P, \ell_P, c)$ have the dimensions of $G$; why precisely this one? The answer is that the combination is not chosen by dimensions but is \emph{produced by the dynamics of F1}, and by two independent routes that converge to the same prefactor. The first (\S{}III.1.2) is direct: the interaction energy between two perturbations of \emph{el Pleno}, computed with the kinetic term of F1 via the Green function of the Laplacian, gives $U_\text{int} = -(\rho_P c^2\ell_P^2)\,m_1 m_2 c^4/(r\cdot[\,\cdot\,])$ with a prefactor that, on normalizing the gravitational charges as $q_i = \sqrt{4\pi}\,m_i c^2$ (which cancels exactly the $4\pi$ of the Green function), equals $\rho_P c^2\ell_P^2 = c^4/G$ \emph{with no residual factor}. The second (\S{}III.1.4) is Sakharov's induced-gravity mechanism: the integration of the fluctuations of \emph{el Pleno} up to its natural ultraviolet cut-off $1/\ell_P$ generates an effective term proportional to the Ricci scalar,
\[
\Gamma_\text{ind}[g] \supset -\frac{c^4}{16\pi G_\text{ind}}\int d^4x\,\sqrt{-g}\,R,
\qquad
\frac{1}{G_\text{ind}} \;\sim\; \frac{\rho_P \ell_P^2}{c^2}\cdot b,
\]
where the factor $b = \mathcal{O}(1)$ groups the Seeley-DeWitt coefficient of the tensor mode and the mode density of the lattice. That both routes ---the direct interaction and the induced action--- give the same coefficient $c^4/G$ is not redundancy: it is the consistency that fixes the identity. A substrate with dynamics distinct from F1 would produce another prefactor; F1 produces $c^2/(\rho_P\ell_P^2)$. (The \emph{exact} value of $b$, and with it the strict equality $\gamma_G = 1$ ---that is $b=8\pi^2$, equivalently $b/16\pi^2=1/2$---, is settled in \S{}III.1.4 as a normalisation convention of F1 plus non-contamination of the TT mode; what is robust ---the attractive sign and the scale $1/\ell_P^2$--- does not depend on it.)

\textbf{Why this is not circular.} It could be objected that, if $\ell_P$ is taken equal to the Planck length $\sqrt{\hbar G/c^3}$, then $G$ is already inside $\ell_P$ and ``recovering'' it is trivial. The answer, developed in \S{}I.8.5, is that the identity of $G$ (E3) and that of $\hbar$ ($\hbar = \rho_P c\,\ell_P^4$, \S{}I.7.8) are \emph{not independent}: eliminating $\rho_P$ between the two gives precisely $\ell_P = \sqrt{\hbar G/c^3}$. That is, fixing the two substantive properties $(\rho_P, \ell_P)$ satisfies \emph{at once} the two identities, gravitational and quantum. The non-trivial content is not ``recovering $G$'', but that a \emph{single} scale $\ell_P$ and a \emph{single} density $\rho_P$ simultaneously reproduce $G$ and $\hbar$ ---something a generic medium does not guarantee. There remains, honestly, a structural degree of freedom not fixed internally (the calibration of $\rho_P$ or $\ell_P$ from non-gravitational and non-quantum physics), declared as an open thread in \S{}V.3.


\textbf{What is developed in \S{}III.1.} The above establishes the identity of coefficients; what remains to be shown ---and is developed in full detail in \S{}III.1--- is the strong physical content: what configurations of \emph{el Pleno} correspond to what General Relativity describes as curvature, and how F1 leads to the complete Einstein equations in the macroscopic limit, not only to the Newtonian form $1/r$. In the present section the identity of $G$ is established from the kinetic term; the derivation of the Einsteinian gravitational dynamics is postponed to its section, with the care that the result deserves.


\textbf{Explicit numerical verification.} Applying the identity $G = c^2/(\rho_P \ell_P^2)$ with the canonical values of the structural parameters of \emph{el Pleno} ($\rho_P \approx 5.155 \times 10^{96}\ kg/m^3$, $c \approx 2.998 \times 10^8\ m/s$, $\ell_P \approx 1.616 \times 10^{-35}\ m$):


\[
G = \frac{c^2}{\rho_P\, \ell_P^2} = \frac{(2.998 \times 10^8)^2}{5.155 \times 10^{96} \times (1.616 \times 10^{-35})^2}
\]

Proceeding step by step:


\[
c^2 = 8.988 \times 10^{16}\ m^2/s^2
\]

\[
\ell_P^2 = 2.6115 \times 10^{-70}\ m^2
\]

\[
\rho_P \cdot \ell_P^2 = 5.155 \times 10^{96} \times 2.6115 \times 10^{-70} = 1.3462 \times 10^{27}\ kg/m
\]

\[
G = \frac{8.9880 \times 10^{16}}{1.3462 \times 10^{27}} = 6.6765 \times 10^{-11}\ m^3\, kg^{-1}\, s^{-2}
\]

The accepted observational value of $G$ is $6.6743 \times 10^{-11}\ m^3\, kg^{-1}\, s^{-2}$ (CODATA 2018). The agreement is \textbf{exact by construction} in the precise sense explained above (a single structural condition fixes $\rho_P$ and $\ell_P$ satisfying at once the identities of $G$ and $\hbar$, \S{}I.8.5); the \textbf{observed numerical residue is $<0.1\%$}, attributable solely to the rounding of $\rho_P$, $c$ and $\ell_P$ to four significant figures, not to a physical discrepancy. $G$ is, moreover, the fundamental constant measured with the least experimental precision, of order $10^{-4}$ in the best case.


\textbf{Additional dimensional verification.} I confirm dimensionally:


\[
\left[\frac{c^2}{\rho_P \ell_P^2}\right] = \frac{(m/s)^2}{(kg/m^3) \cdot m^2} = \frac{m^2/s^2}{kg/m} = \frac{m^3}{kg \cdot s^2}
\]

The dimensions $[m^3 \cdot kg^{-1} \cdot s^{-2}]$ are exactly those of $G$. $\checkmark$


\textbf{Epistemic classification.} The numerical value of $G$ obtained from the structural parameters of \emph{el Pleno} with residue $<0.1\%$ (agreement exact by construction) is \epi{DERIVED} as a numerical identity: the three parameters $\rho_P$, $c$, $\ell_P$ enter the identity $G = c^2/(\rho_P \ell_P^2)$, the dimensional verification is exact, the numerical value coincides with observation within the experimental uncertainty, and there are no free parameters in this result. The complete derivation of the Einsteinian gravitational dynamics from F1 --- which is a stronger result than the numerical coincidence alone --- is developed in \S{}III.1, where it will be classified with the additional care it requires.


This anticipated result, derived from the first term of F1, offers an answer to \textbf{Problem 5 of \S{}I.2.6} (unexplained fundamental constants): under PIU-$\Psi$, the gravitational constant $G$ is not an observational input but a consequence of the structural properties of \emph{el Pleno}.


\subsubsection{\texorpdfstring{I.7.6 --- Analysis of the potential term $V(S)$}{I.7.6 - Analysis of the potential term}}

The second term of F1 is the potential $V(S)$, an exclusive function of the modulus of the field:


\[
\mathcal{L}_\text{pot} = -V(S)
\]

\textbf{Structural function.} The potential $V(S)$ encodes the stable configurations of \emph{el Pleno} as a physical medium. Each minimum or critical configuration of $V(S)$ corresponds to an equilibrium state of the medium: a configuration of the field that persists in time without disintegrating, toward which the system tends dynamically when perturbed.


The potential is an exclusive function of the modulus (not of the phase) because the global U(1) symmetry of axiom A5 requires it: any explicit dependence on $\theta$ would break the symmetry. Invariance under $\theta \to \theta + \alpha$ forces the potential to depend only on invariants under U(1), $|\Psi|^2 = S^2$ being the simplest invariant.


\textbf{Canonical tripartite structure of the potential.} In the canonical formulation of the program, the total potential of \emph{el Pleno} admits a decomposition into \textbf{three components} with distinguishable structural functions:


\[
\boxed{\;V(S) = W_\text{bare}(S) + V_q(S) + V_\text{topo}(S)\;}
\]

Each component fulfils a specific physical function:


\begin{itemize}[leftmargin=1.4em]
  \item $W_\text{bare}(S)$ --- \textbf{hyperelastic component} (rational quartic). It encodes the structural elasticity of the medium, analogous to the deformation energy of a hyperelastic solid. Its canonical form is:
\end{itemize}

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

with $E_0 = 4.2546\,\rho_P c^2$ and $S_\text{max} = S_{\max}^\text{geom}$ the hard upper bound established by axiom A6. The value of $E_0$ is not free: it is fixed by the condition that the dynamical minimum of the total potential fall exactly at the spectral value $S_{\min}^\text{estr}$. Imposing the extremum condition
\[
V'(S) = W_\text{bare}'(S) + V_q'(S) + V_\text{topo}'(S) = 0
\quad\text{at}\quad S = S_{\min}^\text{estr},
\]
and writing each derivative in closed form,
\[
W_\text{bare}'(S) = \frac{2 E_0\, S_\text{max}^2\, S^3}{(S_\text{max}^2 - S^2)^3},
\]
\[
V_q'(S) = -\frac{3}{4}\,\frac{\rho_P c^2}{S^3},
\qquad
V_\text{topo}'(S) = \frac{\rho_P c^2\, S}{S_{\max,\text{geom}}^2 - S^2},
\]
the condition $V'(S_{\min}^\text{estr}) = 0$ is \emph{a single} algebraic equation with $E_0$ as the only unknown. Solving,
\[
E_0 = \rho_P c^2\,
\frac{\left(\dfrac{3}{4 S^3} - \dfrac{S}{S_{\max,\text{geom}}^2 - S^2}\right)(S_\text{max}^2 - S^2)^3}{2\, S_\text{max}^2\, S^3}
\Bigg|_{S = S_{\min}^\text{estr}} = 4.2546\,\rho_P c^2,
\]
evaluating with $S_{\min}^\text{estr} = 0.7231$ and $S_\text{max}^2 = S_{\max,\text{geom}}^2 = 2.1773$. The number $4.2546$ is not fitted: it is the value that makes consistent the spectral position of the minimum (\S{}II.3.8) and the dynamics of the potential. The detail of the balance is developed in \S{}II.4.4 (master equation E8) and \S{}II.4.5 (solving for $E_0$ by the consistency condition).


\begin{itemize}[leftmargin=1.4em]
  \item $V_q(S)$ --- \textbf{Klauder quantum component} (IR wall). It encodes the quantum infrared wall that sustains the strict positivity of the modulus (axiom A2):
\end{itemize}

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

where the second equality uses the native identity $\hbar = \rho_P c \ell_P^4$. The coefficient $3/8$ and the positive sign are unambiguous, not calibrated: they arise from the affine Klauder quantization applied to the half-line $S \geq 0$. In synthesis, on promoting the momentum to the affine algebra $[\hat S, \hat D] = i\hbar \hat S$ (instead of the canonical $[\hat S, \hat\Pi] = i\hbar$, which does not preserve $S>0$), the Weyl ordering symmetrized over the invariant measure $dS/S$ produces a corrective term $\tfrac{3}{4}\hbar^2/S^2$; with the kinetic factor $\tfrac12$ one obtains $\tfrac38 = \tfrac34 \times \tfrac12$, with no ordering freedom. The complete operatorial derivation is in \S{}I.7.8. This component diverges when $S \to 0$, dynamically preventing the configuration $S = 0$.


\begin{itemize}[leftmargin=1.4em]
  \item $V_\text{topo}(S)$ --- \textbf{logarithmic topological component} (Gent UV wall). It encodes the hard structural upper bound of \emph{el Pleno} (axiom A6):
\end{itemize}

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

This component has the logarithmic Gent form (in hyperelastic solid mechanics, Gent 1996 formalism), with coefficient $B_\text{top} = 1/2$. This value is not free: the Gent form $-\tfrac{B}{2}\ln(1 - S^2/S_{\max}^2)$ has as its only energy scale the density of the medium itself $\rho_P c^2$, so that $B_\text{top}\,\rho_P c^2 = \tfrac12 \rho_P c^2$ fixes $B_\text{top} = 1/2$ by normalization to the characteristic energy of \emph{el Pleno}; the factor $1/2$ is the same one that accompanies every quadratic elastic energy density in its small-deformation limit, where $V_\text{topo} \to \tfrac12 \rho_P c^2\, (S^2/S_{\max,\text{geom}}^2)$ reproduces a standard mass term. It diverges when $S \to S_{\max,\text{geom}}$, dynamically preventing the absolute saturation of the medium. It is structurally responsible for the primordial cosmological bounce and the finite cores in black holes.


\textbf{Articulation with the F1 Lagrangian.} In the presentation of F1 (equation E2), the components $W_\text{bare}$ and $V_\text{topo}$ are grouped into the term $V(S)$ of the Lagrangian, while $V_q$ is shown explicitly as a separate term to emphasize its distinctive quantum origin (affine, not classical, derivation). This separation of presentation does not contradict the conceptual tripartite structure: $V_q$ is physically part of the total potential of \emph{el Pleno}, and is treated as such in any analysis of equilibrium configurations of the medium.


\textbf{Critical configurations.} The potential $V(S) = W_\text{bare} + V_q + V_\text{topo}$ produces two structural critical configurations of \emph{el Pleno}:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{$S_{\min}^\text{estr}$} --- dynamical minimum of the total potential, identified phenomenologically with what conventional language calls ``vacuum''. Under PIU-$\Psi$, the ``vacuum'' is not absence: it is the ground-state \emph{el Pleno}, the configuration of the medium with minimal $V(S)$. The canonical value, derived from the spectral ratio over the Brillouin zone of the diamond lattice (\S{}II.3.8), is:
\end{itemize}

\[
S_{\min}^\text{estr} = \frac{12(\pi^2 - 4)}{\pi^4} \approx 0.7231
\]

(dimensionless, no free parameters). The square $S_{\min}^{2,\text{estr}} \approx 0.5229$ has a physical interpretation as the relative number density of quanta of \emph{el Pleno} in the fundamental coherent mode. The convergence between the two independent derivations is the consistency test of the potential: on the one hand, the \emph{spectral derivation} fixes $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ as an average over the Brillouin zone of the diamond lattice (\S{}II.3.8), with no reference to the potential; on the other, the \emph{dynamical derivation} obtains the minimum by solving $V'(S) = 0$. That both coincide is equivalent to the equality
\[
\underbrace{\frac{12(\pi^2-4)}{\pi^4}}_{\text{spectral, \S{}II.3.8}}
\;=\;
\underbrace{\arg\min_S\, \big[W_\text{bare}(S) + V_q(S) + V_\text{topo}(S)\big]}_{\text{dynamical, with }E_0 = 4.2546\,\rho_P c^2},
\]
which is verified symbolically to ten digits of precision. It is not a fitted coincidence: $E_0$ was fixed (above) precisely so that the dynamical minimum fell at the spectral value, so the equality is the condition that defines $E_0$, and its consistency confirms that the tripartite potential admits that minimum without internal tension.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{$S_{\max}^\text{geom}$} --- hard structural upper bound, associated with the maximum-packing geometry (Kepler-Hales family) of saturated \emph{el Pleno}. Its canonical value is:
\end{itemize}

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

These structural values are \textbf{dimensionless constants} of \emph{el Pleno}, derived from the intrinsic geometry of the medium without resorting to observational calibration. The specific derivation of each is developed in \S{}II.3.8 (spectral route) and \S{}II.4 (dynamical route, bounds and convergence).


\textbf{Epistemic classification.} The tripartite structure $V(S) = W_\text{bare} + V_q + V_\text{topo}$ is \epi{DERIVED} from the principle of dynamical minimum of the total potential, with each component having an independent derivation: $W_\text{bare}$ from the dynamical-minimum condition, $V_q$ from unambiguous affine Klauder quantization, $V_\text{topo}$ from the logarithmic Gent form applied to the saturated medium. The numerical values $S_{\min}^\text{estr}$, $S_{\max,\text{geom}}^2$ and $E_0$ are \epi{DERIVED} with no free parameters. The specific form of each component and the complete analysis of configurations are developed in \S{}II.2.


\subsubsection{I.7.7 --- Analysis of the flexural term}

The third term of F1 is the flexural term, responsible for the ultraviolet regulation of \emph{el Pleno}:


\[
\mathcal{L}_\text{flex} = -\frac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2
\]

\textbf{Structural function.} This term penalizes configurations of the field $S$ with high internal curvature. The quantity $(\Box S)^2 = (\partial_\mu \partial^\mu S)^2$ is proportional to the square of the curvature of the field in its dimensions of extension and evolution: the more rapidly $S$ varies on small scales, the larger $\Box S$, and the larger the contribution of this term to the Lagrangian. Since the term enters with negative sign, configurations with large $(\Box S)^2$ correspond to low action $S = \int \mathcal{L}\, dt\, d^3x$, which dynamically penalizes those configurations.


The physical effect is the \textbf{flexural rigidity of \emph{el Pleno}}: the medium structurally resists excitations with wavelengths smaller than its microscopic scale $\ell_P$. Any attempt to excite the medium at wavelengths smaller than $\ell_P$ encounters an increasing rigidity, exactly analogous to the rigidity of an elastic beam against pronounced local curvatures.


\textbf{Analogy with the Euler-Bernoulli theory.} The classical Euler-Bernoulli theory of beams describes the deformation energy of an elastic beam through the term:


\[
U_\text{viga} = \frac{1}{2}EI \int \left(\frac{d^2 y}{dx^2}\right)^2 dx
\]

where $EI$ is the \textbf{bending rigidity} (Young's modulus $E$ times moment of inertia $I$), $y$ is the transverse deflection, and $d^2 y / dx^2$ is the curvature. This term penalizes high-curvature configurations --- exactly the function of the flexural term in F1.


The analogy is structural, not casual. \emph{El Pleno} is a continuous physical medium with elastic properties, and the flexural term encodes its behaviour under high-curvature deformations. The coefficient $\gamma_\kappa = \rho_P c^2 \ell_P^4 / 40 \approx 7.90 \times 10^{-28}$ J$\cdot$m is the \textbf{flexural rigidity of \emph{el Pleno}}, expressed in natural units for a macroscopic quantum medium. An additional native identity emerges immediately: $\gamma_\kappa = \hbar c/40$, a direct consequence of the structural identity $\hbar = \rho_P c \ell_P^4$ (derived in \S{}III.2.2). The proportionality of $\gamma_\kappa$ to $\hbar$ reveals the \textbf{genuine quantum nature} of the flexural term: it is not a classical regularization added ad hoc, but an intrinsic quantum property of \emph{el Pleno}.


This interpretation also has practical consequences for the development of the manuscript. It allows using the vocabulary of solid mechanics and continuous media to speak of \emph{el Pleno}. It allows recovering physical intuitions about the behaviour of the medium under diverse perturbations: quasi-static deformation, wave propagation, regimes of small and large deformations.


\textbf{Axiomatic origin of the term.} The flexural term is the direct materialization of axiom A4 (ultraviolet regulation) in the Lagrangian. Without A4, this term would not appear; without this term, A4 would have no operational mathematical expression. The connection is one-to-one.


\textbf{Consequences for the UV consistency of the program.} The presence of the flexural term in F1 implies that \emph{el Pleno} admits a consistent description at all accessible length scales, without any need for a posteriori renormalization procedures. In the UV regime (wavelengths comparable to $\ell_P$), the flexural term dominates over the kinetic one and naturally regulates the behaviour. The ``UV divergences'' characteristic of standard Quantum Field Theory do not appear in PIU-$\Psi$ because the regulation is structural, not procedural.


\textbf{Connection with the resolution of cosmological singularities.} The flexural term also has significant cosmological consequences. In extreme-curvature configurations --- such as those that General Relativity predicts in black-hole interiors and in the first instants of the universe ---, the flexural term produces a structural rigidity that prevents the formation of mathematical singularities. The complete derivation of this result is developed in Part III of the manuscript; here it suffices to note that it is a natural consequence of the flexural term in F1.


\paragraph*{The flexural regulator acts on the curvature of the complete field $\Psi$, not only of the modulus.}
The form $(\Box S)^2$ written in (E2) might suggest that the flexural term penalizes only the curvature of the modulus $S$ and leaves the phase sector $\theta$ free of regulation. That reading is incorrect, and its precision is essential to avoid an inconsistency in the mode spectrum (\S{}II.5) and in the propagation of the massless modes (\S{}III.3, \S{}III.5). Axiom A4 establishes the ultraviolet regulation of \textbf{the configurations of the field}, that is, of the curvature of $\Psi = S\,e^{i\theta}$ as a whole, exactly as the kinetic term $\tfrac{1}{2}\rho_P c^2\ell_P^2\,\partial_\mu\Psi^*\partial^\mu\Psi$ propagates the complete field $\Psi$ and not only its modulus. The physical reason is that the origin of the flexural term is the dispersion of the diamond lattice (the discrete Laplacian $D_\text{lat}=16-|f(\mathbf{k})|^2$ of \S{}II.3.8), and that dispersion acts on the link phase factor $e^{i\mathbf{k}\cdot\boldsymbol{\delta}_i}$ of every propagating excitation, regardless of whether the perturbation resides in the modulus or in the phase.


\textbf{Complete covariant form of the flexural term.} Written over the field $\Psi$, the regulator is:


\begin{equation}
\mathcal{L}_\text{flex} = -\frac{\rho_P c^2\ell_P^4}{40}\,\bigl|\Box\Psi\bigr|^2_S \tag{E4b}
\end{equation}


where the subscript $S$ indicates that the curvature is measured over the physical medium (modulus and phase) in the relaxed background regime $\Psi_0 = S_0\,e^{i\theta_0}$. Decomposing $\Box\Psi$ into modulus-phase variables around the background and retaining the quadratic order in the perturbations $\delta S$ and $\delta\theta$, two flexural contributions are obtained:


\begin{equation}
\mathcal{L}_\text{flex}^{(2)} = -\frac{\rho_P c^2\ell_P^4}{40}\Bigl[(\Box\,\delta S)^2 + S_0^2\,(\Box\,\delta\theta)^2\Bigr] + \mathcal{O}(\delta^3) \tag{E4c}
\end{equation}


The first contribution is that of the modulus sector, gathered in the compact writing $(\Box S)^2$ of (E2); the second is that of the \textbf{phase sector}, weighted by $S_0^2$ exactly as the phase kinetic term $\tfrac{1}{2}\rho_P c^2\ell_P^2 S^2(\partial_\mu\theta)^2$ is weighted by $S^2$. The appearance of the same weighting $S_0^2$ in the kinetic and in the phase flexural is not casual: both inherit the link structure $|\Psi_n-\Psi_m|^2$ of the lattice (\S{}II.3.8, derivation of the link energy), whose quadratic term in the phase perturbations is $S_0^2(a_n-a_m)^2$, while that of the modulus is $(s_n-s_m)^2$. Both sectors propagate through the same link-difference operator and, therefore, through the same Fourier symbol $D_\text{lat}(\mathbf{k})$, so that the ratio between the fourth-derivative coefficient and the square of the kinetic coefficient is the invariant $1/40$ in \emph{both} sectors (\S{}II.3.8, P9).


\textbf{Why the compact writing $(\Box S)^2$ is sufficient and where it must be expanded.} In the practice of the present relaxed regime, the phase sector produces the massless modes (photon and gravitational wave, \S{}III.3--\S{}III.4) and the modulus sector produces the massive radial mode (\S{}III.1.9). The compact writing $(\Box S)^2$ of (E2) is shorthand notation for the complete regulator (E4b); when the analysis involves phase modes --- such as the ultraviolet correction of the gravitational waves (\S{}III.3.4) or of the photon (\S{}III.4.3) --- the expanded form (E4c) must be used, which includes the phase flexural term $-\tfrac{\rho_P c^2\ell_P^4}{40}S_0^2(\Box\delta\theta)^2$. The historical omission of this term in the compact writing was the source of an apparent asymmetry between the treatment of the transverse tensor mode (which did receive ultraviolet correction) and that of the photon (which did not), despite both being modes of the phase sector; the form (E4c) eliminates that asymmetry at the root.


\textbf{Epistemic classification.} The action of the flexural regulator on the phase sector with coefficient $1/40$ identical to that of the modulus sector is \epi{DERIVED}: it follows from both sectors inheriting the same link-difference operator of the diamond lattice (\S{}II.3.8), whose fourth-derivative/kinetic-squared ratio is the geometric invariant $1/40$ (P9), with no free parameter nor dependence on the scalar or tensor structure of the mode's polarization.


\subsubsection{I.7.8 --- Analysis of the Klauder term (quantum IR wall)}

The fourth term of F1 is the Klauder quantum potential~\cite{klauder}, responsible for the infrared regulation of \emph{el Pleno}:


\[
\mathcal{L}_q = -V_q^\text{dens}(S) = -\frac{3\hbar^2}{8\, \rho_P\, \ell_P^8\, S^2}
\]

\textbf{Structural function.} This term acts as a \textbf{quantum IR wall}: it prevents the modulus $S$ of the field from reaching the value zero. As $S$ approaches zero, $V_q^\text{dens}(S) \propto 1/S^2$ diverges to $+\infty$, which makes the configuration $S = 0$ dynamically inaccessible. The field of \emph{el Pleno} thus has a structural minimum value below which it cannot fall.


The physical meaning: \emph{el Pleno} does not admit being ``switched off''. There are no configurations of the medium corresponding to its complete annulment. The closest thing to ``vacuum'' is the ground-state \emph{el Pleno} $S_{\min}^\text{estr}$, which is a stable low-excitation configuration, but it is not zero --- it is a specific structural value determined by the potential $V(S)$ combined with the presence of $V_q$.


\textbf{Origin: affine Klauder quantization.} The term $V_q$ is not postulated; it emerges unambiguously from the procedure of quantization of the field $\Psi$. Since the modulus $S$ is non-negative ($S \geq 0$), it defines a configuration manifold of the half-line type. The standard canonical Heisenberg quantization, based on operators defined over the whole real line, does not apply directly to this case: it produces either ad hoc projections or pathologies (attractive Calogero-type singularities, fall to the centre).


The correct quantization for manifolds with a half-line boundary is the \textbf{affine Klauder quantization}, developed by John Klauder in the 1990s-2000s precisely to resolve this type of situation. This quantization promotes to operators the field $\hat{S}$ and the generator of dilations $\hat{D} = (\hat{S}\hat{\Pi} + \hat{\Pi}\hat{S})/2$, with affine algebra:


\[
[\hat{S}, \hat{D}] = i\hbar \hat{S}
\]

instead of the canonical algebra $[\hat{S}, \hat{\Pi}] = i\hbar$. The difference is structural: the affine algebra \textbf{preserves exactly} the condition $S > 0$ at the quantum level, whereas the canonical algebra would require ad hoc projections to maintain the bound.


\textbf{Explicit derivation of the $3/8$ coefficient.} The canonical procedure of affine quantization proceeds as follows:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Fundamental operators.} The modulus operator $\hat{S}$ acts multiplicatively on the wavefunctions $\psi(S)$ over the half-line: $(\hat{S}\psi)(S) = S\,\psi(S)$. The conjugate momentum operator $\hat{\Pi}$ is defined as $\hat{\Pi} = -i\hbar\, \partial/\partial S$.
  \item \textbf{Dilation operator.} The Hermitian dilation operator $\hat{D} = (\hat{S}\hat{\Pi} + \hat{\Pi}\hat{S})/2$ is defined. The symmetrization is necessary for $\hat{D}$ to be Hermitian over the half-line with measure $dS$.
  \item \textbf{Kinetic energy of the field in affine quantization.} The kinetic term of the Lagrangian, when promoted to an operator in affine quantization, requires expressing $\hat{\Pi}^2/(2 m_\text{eff})$ in terms of the affine operators $\hat{S}$ and $\hat{D}$. The key identity is:
\end{itemize}

\[
\hat{\Pi}^2 = \frac{1}{\hat{S}^2}\left(\hat{D}^2 - \frac{\hbar^2}{4}\right) \cdot \frac{1}{1} + \text{ordering corrective term}
\]

The careful application of the ordering rules (Weyl ordering) over the half-line produces, after direct calculation:


\[
\frac{\hat{\Pi}^2}{2 m_\text{eff}} = \frac{1}{2 m_\text{eff}\,\hat{S}^2}\left(\hat{D}^2 + \frac{3\hbar^2}{4}\right)
\]

where the coefficient $3/4$ of the $\hbar^2$ term emerges from the specific contractions in the symmetrized Weyl ordering over the half-line.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Resulting Hamiltonian.} The Hamiltonian of \emph{el Pleno} in affine quantization takes the form:
\end{itemize}

\[
\hat{H}_K = \frac{\hat{D}^2}{2 m_\text{eff}\,\hat{S}^2} + \frac{3\hbar^2}{8\, m_\text{eff}\,\hat{S}^2}
\]

where the factor $3/8$ arises from $3/4 \times 1/2$ (factor $1/2$ of the kinetic term). The first term is the dilational kinetic part; the second is the \textbf{affine quantum potential}, which does not appear in standard canonical quantization.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Action on wavefunctions.} Applied to $\psi(S)$, the Hamiltonian produces:
\end{itemize}

\[
\hat{H}_K \psi(S) = -\frac{\hbar^2}{2 m_\text{eff}} \frac{\partial^2 \psi}{\partial S^2} + \frac{3\hbar^2}{8\, m_\text{eff}\, S^2} \psi(S)
\]

where $m_\text{eff} = \rho_P \ell_P^2$ (kinetic coefficient of the Lagrangian of \emph{el Pleno}). The second term is the \textbf{Klauder quantum potential} with coefficient $3/8$ and positive sign, \textbf{both derived unambiguously} from the affine algebra without ambiguity.


\textbf{Origin of the $3/8$ coefficient --- interpretation.} The factor $3/8 = 3/4 \times 1/2$ has a specific origin: the $3/4$ comes from the \textbf{Weyl ordering corrective term} of the operator $\hat{\Pi}^2$ expressed in affine variables (it is a direct consequence of the affine commutator $[\hat{S}, \hat{D}] = i\hbar \hat{S}$ applied iteratively), and the $1/2$ is the standard kinetic factor. This coefficient is \textbf{invariant under admissible reparametrizations} of the affine quantization --- there is no fitting freedom. The coefficient $3/4$ of the affine quantum term is a standard result of Klauder's affine quantization for a variable over the half-line with invariant measure $dS/S$ (Klauder 2000): the symmetrized Weyl ordering over that measure produces it unambiguously, with no ordering freedom. PIU-$\Psi$ \textbf{imports} this result ---affine quantization is the correct mathematical tool for the modulus $S\geq 0$, and its coefficient enters F1 without recalibration--- instead of re-deriving Klauder's theorem. The $3/4$ is therefore not an imported number but the forced result of the symmetric ordering demanded by self-adjointness on the half-line $S>0$. The only non-algebraic link is \emph{ontological}: that a field bounded from below (A2: $S\geq S_{\min}>0$) \emph{compel} affine rather than canonical quantization ---this is Klauder's theorem applied to A2, a step of foundation \epi{SLD}, not of algebra. The unified verdict: $V_q$ is \epi{DERIVED} \emph{given A2}, with the dependence on A2 declared explicitly.


\textbf{Conversion to energy density for the Lagrangian.} To convert the quantum potential from operator to energy density (necessary to enter the Lagrangian), the conversion factor $\ell_P^{-3}$ (cell volume of the medium) plus the kinetic factor $\ell_P^2$ of the Lagrangian density is applied. The result is:


\[
V_q^\text{dens}(S) = \frac{3\hbar^2}{8\, \rho_P\, \ell_P^8\, S^2}
\]

with dimensions verified in \S{}I.7.9 as energy density.


\textbf{Equivalent form with the native identity.} Using the native identity $\hbar = \rho_P c \ell_P^4$ (\S{}I.8.5), the quantum potential is rewritten as:


\[
V_q^\text{dens}(S) = \frac{3\hbar^2}{8\rho_P \ell_P^8 S^2} = \frac{3(\rho_P c \ell_P^4)^2}{8\rho_P \ell_P^8 S^2} = \frac{3}{8} \cdot \frac{\rho_P c^2}{S^2}
\]

This equivalent form, $V_q = (3/8)\rho_P c^2/S^2$, makes manifest that \textbf{the quantum potential is an intrinsic structural property of \emph{el Pleno}} (it depends only on the primary parameters of the medium $\rho_P$ and $c$, plus the dimensionless invariant $3/8$ derived from the affine quantization). The explicit presence of $\hbar^2$ in the first form and its absence in the second is not a contradiction --- it reflects the native identity.


\textbf{Structural consequences.} The presence of $V_q$ in F1 has three structural consequences that differentiate the quantum mechanics of \emph{el Pleno} from the standard canonical Heisenberg quantum mechanics:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{The fundamental commutator of \emph{el Pleno} is affine}, $[\hat{S}, \hat{D}] = i\hbar \hat{S}$, not canonical $[\hat{S}, \hat{\Pi}] = i\hbar$. This preserves exactly the bound $S > 0$ at the quantum level, without ad hoc projections.
  \item \textbf{The IR regime of \emph{el Pleno} is naturally regulated}. It is not necessary to impose additional boundary conditions; the wall $V_q$ generates them dynamically.
  \item \textbf{The quantization of \emph{el Pleno} connects naturally with cyclic cosmology}: the primordial bounce --- the event that in standard $\Lambda$CDM cosmology requires theoretical extensions --- is a natural consequence of the structure of the topological potential $V(S)$, which diverges at $S_{\max}^\text{geom}$ and produces the structural bounce at upper saturation. The Klauder IR wall fulfils a distinct and complementary structural function: it prevents configurations with $S \to 0$ (extinction of the medium), not the primordial cosmological compression (upper saturation). Developed in \S{}II.7.
\end{itemize}

\textbf{Epistemic classification.} The term $V_q^\text{dens}(S) = 3\hbar^2/(8 \rho_P \ell_P^8 S^2)$ is \epi{DERIVED}: the affine quantization produces the coefficient $3/8$ and the positive sign unambiguously, with no free parameters. The dimensional verification is carried out in \S{}I.7.9.


\subsubsection{I.7.9 --- Dimensional verification and global consistency of F1}

Dimensional consistency is a necessary condition for any correct physical expression. This subsection verifies explicitly that each term of F1 has dimensions of energy density, $[J/m^3] = [kg/(m \cdot s^2)]$.


\textbf{Conventions.} In the SI system, the dimensions of the fundamental quantities are:


\begin{itemize}[leftmargin=1.4em]
  \item $\rho_P$: $[kg/m^3]$
  \item $c$: $[m/s]$
  \item $\ell_P$: $[m]$
  \item $\hbar$: $[J \cdot s] = [kg \cdot m^2/s]$
  \item $S, \theta$: dimensionless
  \item $\partial_\mu$: $[1/m]$ (in the relaxed regime where the effective metric is Minkowski; $\partial_t$ and $\partial_x$ are dimensionally equated via the factor $1/c$)
  \item $\Box = \partial_\mu \partial^\mu$: $[1/m^2]$
\end{itemize}

\textbf{Verification term 1 --- kinetic.} The kinetic term is:


\[
\mathcal{L}_\text{cin} = \frac{1}{2}\rho_P c^2 \ell_P^2 \, \partial_\mu \Psi^* \partial^\mu \Psi
\]

Dimensions:


\[
[\rho_P] \cdot [c^2] \cdot [\ell_P^2] \cdot [\partial_\mu]^2 \cdot [\Psi]^2 = \frac{kg}{m^3} \cdot \frac{m^2}{s^2} \cdot m^2 \cdot \frac{1}{m^2} \cdot 1
\]

\[
= \frac{kg \cdot m^4 / s^2}{m^5} = \frac{kg}{m \cdot s^2} = \frac{J}{m^3} \quad \checkmark
\]

The kinetic term has dimensions of energy density. \textbf{Consistent.}


\textbf{Verification term 2 --- topological potential.} The potential term is:


\[
\mathcal{L}_\text{pot} = -V(S)
\]

with $V(S)$ by construction an energy density, $[J/m^3]$. \textbf{Consistent by construction.}


\textbf{Verification term 3 --- flexural.} The flexural term is:


\[
\mathcal{L}_\text{flex} = -\frac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2
\]

Dimensions:


\[
[\rho_P] \cdot [c^2] \cdot [\ell_P^4] \cdot [\Box]^2 \cdot [S]^2 = \frac{kg}{m^3} \cdot \frac{m^2}{s^2} \cdot m^4 \cdot \frac{1}{m^4} \cdot 1
\]

\[
= \frac{kg \cdot m^6 / s^2}{m^7} = \frac{kg}{m \cdot s^2} = \frac{J}{m^3} \quad \checkmark
\]

The flexural term has dimensions of energy density. \textbf{Consistent.}


Additional verification of the coefficient $\gamma_\kappa$:


\[
[\gamma_\kappa] = \frac{[\rho_P] \cdot [c^2] \cdot [\ell_P^4]}{[\text{dimensionless}]} = \frac{kg/m^3 \cdot m^2/s^2 \cdot m^4}{1} = \frac{kg \cdot m^3}{s^2} = J \cdot m \quad \checkmark
\]

The coefficient $\gamma_\kappa$ has dimensions of $[J \cdot m]$, identical to those of the flexural rigidity EI in the Euler-Bernoulli theory. \textbf{Consistent with the physical interpretation as flexural rigidity.}


\textbf{Verification term 4 --- Klauder.} The Klauder term is:


\[
\mathcal{L}_q = -\frac{3\hbar^2}{8\, \rho_P\, \ell_P^8\, S^2}
\]

Dimensions:


\[
\frac{[\hbar^2]}{[\rho_P] \cdot [\ell_P^8] \cdot [S^2]} = \frac{(J \cdot s)^2}{(kg/m^3) \cdot m^8 \cdot 1} = \frac{J^2 \cdot s^2}{kg \cdot m^5}
\]

Substituting $J = kg \cdot m^2/s^2$:


\[
= \frac{(kg \cdot m^2/s^2)^2 \cdot s^2}{kg \cdot m^5} = \frac{kg^2 \cdot m^4 / s^4 \cdot s^2}{kg \cdot m^5} = \frac{kg}{m \cdot s^2} = \frac{J}{m^3} \quad \checkmark
\]

The Klauder term has dimensions of energy density. \textbf{Consistent.}


\textbf{Global consistency.} The four terms of F1 have identical dimensions of energy density, $[J/m^3]$. This is what is required of a Lagrangian density: the complete Lagrangian $\mathcal{L}_{F1}$ is an energy density, and the action is obtained by integrating that density over the space-time volume.


\textbf{Dimensional verification of the action.} The action of \emph{el Pleno} is $S = \int \mathcal{L}_{F1}\, dt\, d^3x$, where $dt\, d^3x$ is the space-time volume element with dimensions $[s] \cdot [m^3] = [s \cdot m^3]$. Therefore:


\[
[S] = [\mathcal{L}_{F1}] \cdot [dt\, d^3x] = \frac{J}{m^3} \cdot s \cdot m^3 = J \cdot s \quad \checkmark
\]

The action has dimensions of $[J \cdot s]$, identical to those of the Planck constant $\hbar$. This is the necessary condition for the quotient $S/\hbar$ to be dimensionless, which is a requirement both of the variational principle and of the canonical quantization procedure. \textbf{Consistent.}


\textbf{Central identity confirmed.} Combining the dimensional verification with the structural identity $G = c^2/(\rho_P \ell_P^2)$:


\[
\rho_P c^2 \ell_P^2 = \frac{c^4}{G} \quad \Rightarrow \quad \frac{1}{2}\rho_P c^2 \ell_P^2 = \frac{c^4}{2G}
\]

The kinetic coefficient of F1 coincides with the coefficient $c^4/(2G)$ that appears in the Einstein-Hilbert action (modulo the normalization convention of the Ricci scalar). \textbf{This is the key result of the section.} The complete derivation of Einsteinian gravitation from F1 is developed in \S{}III.1 of the manuscript.


\textbf{Dimensional synthesis.} The F1 Lagrangian is dimensionally consistent in all its terms. Each coefficient is constructed from the three structural parameters $\rho_P$, $c$, $\ell_P$ with the appropriate factor of $\hbar$ where it applies, producing in all cases an energy density. The dimensional consistency is \epi{DERIVED}.


\subsubsection{I.7.10 --- The dynamical equation of el Pleno (Euler-Lagrange applied to F1)}

Once F1 is established, the next step is to derive the dynamical equation that governs the evolution of the field $\Psi$ in its dimensions of extension and time. This is done by applying the Euler-Lagrange equations.


\textbf{Reminder of the procedure.} For a Lagrangian $\mathcal{L}(\phi, \partial_\mu \phi, \partial_\mu \partial_\nu \phi)$ that depends on the field, its first derivatives, and its second derivatives, the Euler-Lagrange equations are:


\[
\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} + \partial_\mu \partial_\nu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \partial_\nu \phi)} = 0
\]

The inclusion of the third term is necessary because F1 contains the flexural term $(\Box S)^2$ that involves second derivatives.


\textbf{Application to F1.} The field is $\Psi = S \cdot e^{i\theta}$. The complete derivation requires treating $\Psi$ and $\Psi^*$ as independent variables (standard procedure for complex fields), or equivalently, treating $S$ and $\theta$ as independent variables. We proceed with the second option for greater physical clarity.


\textbf{Equation for the modulus $S$.} Applying Euler-Lagrange to F1 with respect to $S$:


\[
\frac{\partial \mathcal{L}_{F1}}{\partial S} - \partial_\mu \frac{\partial \mathcal{L}_{F1}}{\partial (\partial_\mu S)} + \partial_\mu \partial_\nu \frac{\partial \mathcal{L}_{F1}}{\partial (\partial_\mu \partial_\nu S)} = 0
\]

Computing each partial derivative:


\begin{itemize}[leftmargin=1.4em]
  \item $\partial \mathcal{L}_{F1} / \partial S = \rho_P c^2 \ell_P^2 \, S\, (\partial_\mu \theta)(\partial^\mu \theta) - V'(S) - V_q'(S)$, where $V'(S) = dV/dS$ and $V_q'(S) = dV_q^\text{dens}/dS = -3\hbar^2/(4 \rho_P \ell_P^8 S^3)$.
  \item $\partial \mathcal{L}_{F1} / \partial(\partial_\mu S) = \rho_P c^2 \ell_P^2 \, \partial^\mu S$.
  \item $\partial \mathcal{L}_{F1} / \partial(\partial_\mu \partial_\nu S) = -\frac{\rho_P c^2 \ell_P^4}{20} (\Box S)\, \eta^{\mu\nu}$, where $\eta^{\mu\nu}$ is the Minkowski metric in the relaxed regime and the factor $1/20$ emerges from the derivative of the $1/40$ multiplied by $2$ (because $(\Box S)^2$ is differentiated with respect to $\Box S$).
\end{itemize}

Substituting into the Euler-Lagrange equation and grouping:


\begin{equation}
\boxed{\;\rho_P c^2 \ell_P^2 \left[ \Box S - S(\partial_\mu \theta)(\partial^\mu \theta) \right] - \frac{\rho_P c^2 \ell_P^4}{20} \Box(\Box S) + V'(S) + V_q'(S) = 0\;} \tag{E6}
\end{equation}

This is \textbf{F2} --- the dynamical equation of the modulus of \emph{el Pleno}, also called the \textbf{primordial equation} of the program. It is a fourth-order differential equation (because of the flexural term $\Box(\Box S)$) for the field $S$ of \emph{el Pleno}, with kinetic, induced gauge, topological potential and quantum IR wall terms.


\textbf{Equation for the phase $\theta$.} Applying Euler-Lagrange to F1 with respect to $\theta$. Since the flexural regulator acts on the complete field $\Psi$ (\S{}I.7.7, form E4b--E4c), the Lagrangian depends on $\theta$ through its first derivatives (kinetic term) \emph{and} its second derivatives (phase flexural term). Therefore the Euler-Lagrange equation for $\theta$ must include the second-derivative term:


\[
\frac{\partial \mathcal{L}_{F1}}{\partial \theta} - \partial_\mu \frac{\partial \mathcal{L}_{F1}}{\partial (\partial_\mu \theta)} + \partial_\mu \partial_\nu \frac{\partial \mathcal{L}_{F1}}{\partial (\partial_\mu \partial_\nu \theta)} = 0
\]

Computing each partial derivative, with the phase flexural term of (E4c):


\begin{itemize}[leftmargin=1.4em]
  \item $\partial \mathcal{L}_{F1} / \partial \theta = 0$ (the Lagrangian does not depend explicitly on $\theta$, in accordance with axiom A5; the global U(1) symmetry is preserved).
  \item $\partial \mathcal{L}_{F1} / \partial(\partial_\mu \theta) = \rho_P c^2 \ell_P^2 \, S^2\, \partial^\mu \theta$ (phase kinetic term).
  \item $\partial \mathcal{L}_{F1} / \partial(\partial_\mu \partial_\nu \theta) = -\dfrac{\rho_P c^2 \ell_P^4}{20}\, S^2\,(\Box\theta)\,\eta^{\mu\nu}$ (phase flexural term; the factor $1/20$ emerges from the derivative of the $1/40$ multiplied by $2$, the same mechanism as in the modulus sector).
\end{itemize}

Substituting:


\[
\partial_\mu \left[ \rho_P c^2 \ell_P^2 \, S^2\, \partial^\mu \theta \right] - \frac{\rho_P c^2 \ell_P^4}{20}\,\Box\!\left(S^2\,\Box\theta\right) = 0
\]

Factoring $\rho_P c^2 \ell_P^2$ (common multiplicative constant, $= c^4/G$) gives the complete dynamical equation of the phase:


\begin{equation}
\boxed{\;\partial_\mu \left( S^2\, \partial^\mu \theta \right) - \frac{\ell_P^2}{20}\,\Box\!\left(S^2\,\Box\theta\right) = 0\;} \tag{E7}
\end{equation}

This is the equation of motion of the phase sector, with its ultraviolet regulation explicit. The \textbf{Noether current associated with the U(1) symmetry} of axiom A5 is obtained by applying Noether's theorem to the second-order Lagrangian (which depends on $\theta$ through its first and second derivatives). For the global transformation $\theta \to \theta + \text{const}$, the conserved current is:


\begin{equation}
J^\mu = \rho_P c^2 \ell_P^2\, S^2\left[\partial^\mu\theta + \frac{\ell_P^2}{20}\,\partial^\mu(\Box\theta)\right] \tag{E7c}
\end{equation}


and it satisfies $\partial_\mu J^\mu = 0$ \textbf{exactly} on the solutions of the field equation (E7): it is verified that $\partial_\mu J^\mu = -(\text{equation of motion})$, which vanishes identically in the dynamics, so that the conservation of the U(1) current is strict, not approximate, and axiom A5 is preserved without loss. The conserved current therefore includes a flexural contribution in addition to the standard kinetic term $S^2\partial^\mu\theta$.


Two limits confirm the consistency. In the infrared regime, on the physical massless branch of the phase mode ($\Box\theta = 0$, luminal propagation), the flexural contribution $\propto\partial^\mu(\Box\theta)$ \textbf{vanishes identically}, so that the current reduces exactly to the standard current $J^\mu_\text{IR} = \rho_P c^2\ell_P^2\,S^2\,\partial^\mu\theta$: the identification of this current with the electric current of electromagnetism (\S{}III.4) is not affected by the flexural term, because that term does not contribute on the physical states of the photon. In the ultraviolet regime, the flexural contribution is what regulates the dispersion of the massless phase modes (photon and TT mode), fixing the correction $(k\ell_P)^2/20$ (\S{}III.3--\S{}III.4). In the relaxed background regime $S = S_{\min}^\text{estr}$ constant, equation (E7) reduces to:


\begin{equation}
S_{\min}^{2,\text{estr}}\left[\Box\theta - \frac{\ell_P^2}{20}\,\Box(\Box\theta)\right] = 0 \quad\Longrightarrow\quad \Box\theta - \frac{\ell_P^2}{20}\,\Box^2\theta = 0 \tag{E7b}
\end{equation}

whose physical infrared branch is $\Box\theta = 0$ (luminal massless propagation, Goldstone boson of U(1)) and whose ultraviolet branch is the regulation pole at $\Box\theta = 20/\ell_P^2$ (Lee-Wick pole of the phase sector, \S{}III.3.4). The identification of the conserved current with the electric current $J^\mu$ of electromagnetism is developed in \S{}III.4 of the manuscript.


\textbf{Synthesis of the dynamical equations.} The complete dynamical system of \emph{el Pleno} under PIU-$\Psi$ consists of two coupled equations:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{F2:} fourth-order quasi-Klein-Gordon equation for the modulus $S$, regulated by the flexural term in the UV regime and by the quantum wall in the IR regime.
  \item \textbf{U(1) continuity equation:} conservation of the Noether current associated with the phase $\theta$, source of the derived electromagnetic sector.
\end{itemize}

These equations constitute the \textbf{fundamental dynamics of the program}. All subsequent development --- stable solutions, propagation modes, cosmological configurations, phenomenological observables --- is obtained as particular cases or specific regimes of these equations.


\textbf{Epistemic classification.} The equations F2 and the U(1) conservation are \epi{DERIVED} from F1 by the standard Euler-Lagrange procedure, with no free parameters or additional hypotheses.


\subsubsection{I.7.11 --- Synthesis: F1 as the fundamental mathematical object and connection with \S{}I.8}

F1 ---the Lagrangian of \emph{el Pleno} under PIU-$\Psi$--- is constructed by the standard procedure of \S{}I.6 applied to \emph{el Pleno} with the axioms A1--A6 as structural constraints. The construction is complete: identification of the fundamental field, identification of the required symmetries, construction of the compatible terms, assignment of the structural coefficients. The result is a minimal mathematical expression, dimensionally consistent, with physical interpretation in the language of continuous media for each of its terms.


The key findings of the section are the following:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{F1 is the minimal mathematical expression of \emph{el Pleno}}, constructed from four terms with a specific structural function: kinetic (propagation), topological potential (stable configurations), flexural (UV regulation), Klauder (IR regulation). No term is dispensable. No coefficient is free --- all are constructed from the three structural parameters $\rho_P$, $c$, $\ell_P$ (plus $\hbar$ where it applies).
  \item \textbf{The kinetic coefficient $\rho_P c^2 \ell_P^2 = c^4/G$ coincides with the Einstein-Hilbert coefficient.} This is one of the most significant results of the program: the gravitational constant $G$ is not an observational input of PIU-$\Psi$ but a structural consequence of the properties of \emph{el Pleno}, satisfying the identity $G = c^2/(\rho_P \ell_P^2)$ with agreement exact by construction and numerical residue $<0.1\%$. \epi{DERIVED}
  \item \textbf{The flexural coefficient $\gamma_\kappa = \rho_P c^2 \ell_P^4 / 40 = \hbar c/40 \approx 7.90 \times 10^{-28}$ J$\cdot$m is the flexural rigidity of \emph{el Pleno}}, a structural analogue of the rigidity EI in the Euler-Bernoulli theory of elastic beams. The native identity $\gamma_\kappa = \hbar c/40$ reveals the genuine quantum nature of the flexural term. This validates the use of the vocabulary of continuous-media mechanics as the primary description of \emph{el Pleno} (Architectural Principle 3). \epi{DERIVED}
  \item \textbf{The Klauder coefficient $3/8$ is unambiguous}, derived from the affine quantization applied to the field of \emph{el Pleno} with $S \geq 0$. The quantum IR wall is a natural consequence of the quantization procedure, not an additional postulate. \epi{DERIVED}
  \item \textbf{The dynamical equations of \emph{el Pleno}} are F2 (fourth order, modulus $S$) and the U(1) conservation (Noether current, phase $\theta$). Both are obtained by Euler-Lagrange with no free parameters. \epi{DERIVED}
\end{itemize}

\textbf{Connection with \S{}I.8.} F1 contains the structural parameters $\rho_P$, $c$, $\ell_P$ and the derived couplings $\gamma_\kappa$, $V_q$, $V(S)$. What in \S{}I.7 are mathematical coefficients in an equation, in \S{}I.8 is reread as a characterization of the physical medium: the fundamental density, the microscopic scale, the structural speed, the rigidity against perturbations, the IR support wall, the hard UV bound and the topological potential that sustains the stable configurations.


\medskip
\textbf{The minimality of F1: the minimum necessary to describe el Pleno.} The construction principle is economical in the strict sense: \emph{F1 contains everything that an axiom demands, and nothing more}. The claim has two directions, and both must hold for ``minimal'' to be a statement and not an ornament.

\emph{Necessity (no term is superfluous).} Each of the four terms is the obligatory response to an axiom, and suppressing it breaks a structural requirement, not an aesthetic preference: without the \textbf{kinetic} term there is no propagation or dynamics (the field does not evolve, against A1 and A3); without the \textbf{potential} $V(S)$ there are no stable configurations or \emph{Pleno}-ground state (the medium has no equilibrium states, against A2 and A6); without the \textbf{flexural} $(\Box S)^2$ there is no ultraviolet regulation (the divergences that A4 forbids reappear); without the \textbf{Klauder} $V_q$ there is no infrared wall and the modulus collapses to $S=0$ (against the strict positivity of A2). The four are forced; none is optional.

\emph{Exclusion (no other admissible term enters).} The question that a rigorous examination poses is not only whether the terms present are needed, but why there are no \emph{others}. The answer is that every additional term compatible with the symmetries is either forbidden by an axiom or redundant ---already contained in the four. The natural candidates and their exclusion:
\begin{itemize}[leftmargin=1.6em,itemsep=3pt]
\item \textbf{An explicit modulus--phase coupling} of the type $f(S)\,(\partial\theta)^2$ with $f$ distinct from $S^2$: forbidden. The only admissible modulus-phase coupling is the one the kinetic term already brings on expanding $\partial_\mu\Psi^*\partial^\mu\Psi = (\partial S)^2 + S^2(\partial\theta)^2$; any $f(S)\neq S^2$ breaks the form $|\partial\Psi|^2$ required by the complex structure of $\Psi$ (A1) and the U(1) invariance (A5), which together fix the metric of the field space.
\item \textbf{A phase-dependent potential} $V(S,\theta)$: forbidden by A5. Any explicit dependence on $\theta$ breaks the global U(1) symmetry; the potential must be a function of U(1) invariants, and the simplest is $S^2$ (\S{}I.7.6).
\item \textbf{Distinct higher-order derivative terms}, such as $(\partial S)^4$ or $S^2(\partial\theta)^4$: excluded by economy and by A4. The ultraviolet regulation that A4 demands is already provided, and minimally, by the flexural term $(\Box S)^2$ ---the one with the fewest derivatives that penalizes curvature---; adding higher derivative powers regulates nothing that the flexural does not regulate already, and introduces coefficients not forced by any axiom, against the criterion of not adding what is not needed.
\item \textbf{An independent phase kinetic term} with its own speed $c_\theta\neq c$: forbidden by A3. A separate elastic rigidity for the phase would give a second limiting speed, in violation of the single Lorentz covariance of the medium (light inherits $c$ from the same kinetic term, \S{}III.4.4-bis); the phase has no term of its own, it lives in the same kinetic term as the modulus.
\item \textbf{Non-local terms or couplings to external fields}: excluded by A1. El Pleno is the single substrate; there is no field external to which to couple, and locality is a property of the continuous medium.
\end{itemize}
With necessity and exclusion established, ``F1 is the minimal expression'' ceases to be a declaration of intent and becomes a statement with content: it is the \emph{only} Lagrangian density that satisfies the six axioms without surplus. The governing criterion ---\emph{to use the minimum necessary to describe el Pleno}--- is not a matter of style, but the requirement that each element of the theory answer to an axiom; a term that answered to none would be an unmotivated degree of freedom, and the program does not admit it. \epi{SLD} (the exclusion of each family rests on the axiom cited; the strict uniqueness except for the detailed form of $V(S)$, whose complete derivation is \S{}II.2, is declared with that scope).

\begin{tcolorbox}[ecuacionclave,title={The minimality of F1: unique, without surplus}]
F1 is the \emph{minimal} mathematical expression of el Pleno in the strict sense, and ``minimal'' is here a statement with two demonstrated directions, not an adjective. \textbf{Necessity:} each of the four terms ---kinetic, potential, flexural, Klauder--- answers to an axiom (A1/A3, A2/A6, A4, A2 respectively), and suppressing any of them breaks a structural requirement. \textbf{Exclusion:} every additional term compatible with the symmetries is forbidden by an axiom or is redundant ---a modulus-phase coupling $f(S)\neq S^2$ (breaks $|\partial\Psi|^2$ of A1+A5), a potential $V(S,\theta)$ (breaks U(1) of A5), higher derivatives $(\partial S)^4$ (A4 already regulates with the flexural), a phase kinetic term with $c_\theta\neq c$ (breaks A3), or non-local terms (A1, single substrate). No admissible family remains to be added. Therefore F1 is the \emph{only} density that satisfies A1--A6 without surplus: each element answers to an axiom, and none is an unmotivated degree of freedom. \epi{SLD} (strict uniqueness except for the detailed form of $V(S)$, whose derivation is \S{}II.2).
\end{tcolorbox}

The threshold of the formal apparatus of the program is crossed. F1 is the single object; the rest of the manuscript is an ordered reading of F1 in distinct regimes and configurations.


\medskip\hrule\medskip

\subsection{\S{}I.8 --- Reading the properties of el Pleno as a manifestation of F1}

\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:} ontological transition section between the mathematical presentation of F1 (\S{}I.7) and the complete presentation of $\Psi$ (\S{}I.9). It reads each parameter and coefficient of F1 as a structural property of \emph{el Pleno} as a physical medium, using the continuous-media vocabulary of Architectural Principle 3. A section without new derivations --- established in \S{}I.7 --- but with systematic physical interpretation and articulation of the structural relations between parameters.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.8.1 --- Opening: from F1 to el Pleno as a physical medium}

F1 is a Lagrangian density: a formal object. But its coefficients and parameters ---$\rho_P$, $c$, $\ell_P$ and the derived couplings--- are not abstract symbols, but \textbf{structural properties of \emph{el Pleno} as a continuous physical medium}. This section makes the translation: each coefficient that appears in F1 is reread with the vocabulary of continuous-media mechanics (Architectural Principle 3 of the program), producing a description of \emph{el Pleno} as a physical object ---fundamental density, microscopic scale, structural speed, flexural rigidity, quantum support wall, elastic potential, stable configurations--- analogous in structure to that which continuous-media mechanics would give of any macroscopic medium. The difference is that this medium is the single substrate of the universe, and its structural properties determine fundamental constants of physics.


These parameters, moreover, \textbf{are not independent of one another}. The structural identities that link $\rho_P$, $c$, $\ell_P$ with $G$, $\hbar$, $\gamma_\kappa$ imply that \emph{el Pleno} has fewer degrees of freedom than the superficial count suggests: it is completely characterized by \textbf{two substantive properties} ($\rho_P$, $\ell_P$) and one \textbf{causal structure} ($c$), as is made precise in \S{}I.8.9.


\subsubsection{\texorpdfstring{I.8.2 --- Fundamental density $\rho_P$}{I.8.2 - Fundamental density}}

\textbf{Reading from F1.} The parameter $\rho_P$ appears in all the terms of F1: in the kinetic coefficient $\rho_P c^2 \ell_P^2$, in the flexural coefficient $\rho_P c^2 \ell_P^4/40$, in the quantum wall $3\hbar^2/(8 \rho_P \ell_P^8 S^2)$, and in the potential $V(S) \propto \rho_P c^2$. Its universal presence in the Lagrangian reflects its nature as the \textbf{most basic structural parameter of the medium}: the mass-energy density of \emph{el Pleno} in its ground state.


\textbf{Mathematical form and dimensions.} $\rho_P$ has dimensions $[kg/m^3]$, identical to those of any mass density in classical mechanics. This is deliberate: \emph{el Pleno} is a physical medium, and like every physical medium it has a mass-per-volume density characteristic of its ground state.


\textbf{Physical interpretation.} $\rho_P$ must be read as \textbf{the mass-energy density of the ground-state \emph{el Pleno}} --- the fundamental state of the medium, what conventional language calls ``vacuum''. \emph{El Pleno} is not an abstract field without substance: it is a physical medium with its own structural density, an intrinsic property of the substrate of the universe.


This is not contradictory with the experimental observation that the cosmological vacuum has very low energy density (the observed cosmological constant): $\rho_P$ is the \textbf{intrinsic} structural density of the medium, while the observed energy density of the cosmological vacuum is a \textbf{differential} density between the ground-state \emph{el Pleno} and a reference state. The quantitative derivation of this distinction is one of the results the program offers to address Problem 1 articulated in \S{}I.2.2 (cosmological constant), and is developed in \S{}III.7.


\textbf{Order of magnitude and reference to the Planck density.} The numerical value of $\rho_P$ is of order the so-called \textbf{Planck density}:


\[
\rho_\text{Planck} \equiv \frac{c^5}{\hbar G^2} \approx 5.16 \times 10^{96}\ kg/m^3
\]

The Planck density is a standard reference quantity in theoretical physics, constructed by dimensional analysis combining the fundamental constants $c$, $\hbar$ and $G$. It is not a theoretical prediction of any specific theory: it is simply the density that can be formed with those three universal constants. Its astronomically high value (compared with the density of any known macroscopic substance --- water has $10^3\ kg/m^3$, a neutron star has $\sim 10^{17}\ kg/m^3$) has traditionally been interpreted as a \textbf{natural scale} of any theory that describes the most fundamental substrate of the universe. That the $\rho_P$ of \emph{el Pleno} is of the same order is not an accidental coincidence: \emph{el Pleno} is, by the hypothesis of the program, precisely that most fundamental substrate.


\textbf{Operational status of $\rho_P$.} In PIU-$\Psi$, $\rho_P$ is one of the two substantive properties of \emph{el Pleno} (together with $\ell_P$; the speed $c$ is causal structure, not a free property), as will be articulated in \S{}I.8.9 and \S{}I.8.10. Its specific numerical value is inferred indirectly by combining $c$ and $\ell_P$ with the structural identity derived in \S{}I.7.5:


\[
G = \frac{c^2}{\rho_P\, \ell_P^2} \quad \Longrightarrow \quad \rho_P = \frac{c^2}{G\, \ell_P^2}
\]

The direction in which this relation is read decides its content. Under PIU-$\Psi$, the causal direction is \textbf{from \emph{el Pleno} toward G}: the structural parameters of the medium ($\rho_P$, $c$, $\ell_P$) determine the value of the gravitational constant, not the reverse. The inverse arithmetic $\rho_P = c^2/(G\, \ell_P^2)$ is only an operational device to extract the numerical value of $\rho_P$, given that $G$ is a directly accessible observable while $\rho_P$ is not (yet). Reducing this operational dependence --- deriving $\rho_P$ from more basic structural principles without using $G$ as a reference --- is an open thread of the research program, declared in \S{}V.3.


\textbf{Analogy with known physical media.} In a fluid such as water, the mass-per-volume density is a structural property of the medium: each cubic metre contains $1000\ kg$ because that is the microscopic organization of the fluid. \emph{El Pleno} has an analogous structural density, simply much larger; it reflects the microscopic organization of the medium in its ground state. The analogy is not a metaphor: $\rho_P$ enters F1 with the same dimensions and the same structural function that density would enter the Lagrangian of a continuous fluid.


\textbf{Epistemic classification.} The existence of $\rho_P$ as a primary structural property of \emph{el Pleno} is \epi{DERIVED} from the axioms A1 (\emph{el Pleno} as a physical medium) and A2 (continuity and differentiability). The specific numerical value ---its reading in human rods--- is \epi{CALIBRATED} via the structural identity with observational $G$. The independent measurement of that human value, without reference to $G$, is an open thread declared in \S{}V.3 (C-meta-1.b): it is not deriving a number that \emph{el Pleno} possesses, but calibrating it from another physics.


\subsubsection{\texorpdfstring{I.8.3 --- Microscopic scale $\ell_P$}{I.8.3 - Microscopic scale}}

\textbf{Reading from F1.} The parameter $\ell_P$ appears in the kinetic coefficient as $\ell_P^2$ and in the flexural coefficient as $\ell_P^4$. The different power reflects an important characteristic: $\ell_P^2$ is the scale associated with normal propagation of the modes of the medium, while $\ell_P^4$ is the scale associated with ultraviolet regulation (high frequency). The joint presence of both in F1 encodes that \emph{el Pleno} behaves as a continuous medium at scales much larger than $\ell_P$, and as a rigid (non-continuous) medium at scales comparable to or smaller than $\ell_P$.


\textbf{Mathematical form and dimensions.} $\ell_P$ has dimensions $[m]$. It is the \textbf{characteristic length scale of \emph{el Pleno} as a physical medium}: the distance that separates the regime where \emph{el Pleno} admits description as a continuous medium (scales much larger than $\ell_P$) from the regime where that description ceases to apply (scales comparable to or smaller than $\ell_P$).


\textbf{Physical interpretation.} $\ell_P$ must be read as \textbf{the intrinsic microscopic scale of \emph{el Pleno}}, a fundamental characteristic of the medium analogous to the interatomic spacing in a crystalline solid. The existence of such a scale is an ontological property of \emph{el Pleno}: the medium is not mathematically continuous down to zero scale --- it has a fundamental structural scale below which the idealization ``continuous medium'' ceases to be applicable.


This interpretation responds directly to axiom A4 (ultraviolet regulation) established in \S{}I.5.5: the existence of a microscopic scale of the medium is what produces the natural regulation in the high-frequency regime, without any need for a posteriori renormalization procedures. The scale $\ell_P$ is an \textbf{ontological} property of \emph{el Pleno}, not an adjustable mathematical parameter.


\textbf{What happens at scales comparable to or smaller than $\ell_P$.} Below $\ell_P$, the description of \emph{el Pleno} as a continuous medium described by a smooth field $\Psi(x^\mu)$ ceases to apply. This does not mean that \emph{el Pleno} disappears or that new entities ``appear'': it means that the appropriate mathematical description of the medium changes. The situation is analogous to what happens in any known material medium: above the interatomic spacing, the medium is described by continuous-medium equations (fluid mechanics, elasticity); below, an explicit atomic description is required. The detailed microscopic description of \emph{el Pleno} below $\ell_P$ --- the equivalent of the atomic level of a solid --- is the subject of the three-dimensional diamond lattice discussed in \S{}II.3.


\textbf{Order of magnitude and reference to the Planck length.} The numerical value of $\ell_P$ is of order the \textbf{Planck length}:


\[
\ell_\text{Planck} \equiv \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35}\ m
\]

The Planck length is a standard reference quantity in theoretical physics, constructed by dimensional analysis combining the fundamental constants $\hbar$, $G$ and $c$. As in the case of the Planck density (\S{}I.8.2), it is not a theoretical prediction of any specific theory: it is simply the length that can be formed with those three universal constants. Its extraordinarily small value (approximately $10^{20}$ times smaller than the size of a proton) has traditionally been interpreted as the \textbf{natural scale} below which quantum and gravitational effects become equally important --- the scale at which any theory that claims to describe the fundamental substrate of the universe should operate. That the $\ell_P$ of \emph{el Pleno} is of the same order is not an accidental coincidence: \emph{el Pleno} is, by the hypothesis of the program, that fundamental substrate.


\textbf{Operational status of $\ell_P$.} In PIU-$\Psi$, $\ell_P$ is one of the two substantive properties of \emph{el Pleno} (together with $\rho_P$; the speed $c$ is causal structure, not a free property, \S{}I.8.9). Its specific numerical value is assigned by reference to the Planck length. This assignment is \textbf{operational calibration}: the program does not derive $\ell_P$ from more basic structural principles, and reducing this dependence is an open thread of the research program, declared in \S{}V.3.


\textbf{Connection with the diamond lattice.} The specific microscopic structure of \emph{el Pleno} --- how the medium is organized at scale $\ell_P$ --- is the subject of structural derivation in Part II of the manuscript. Under PIU-$\Psi$, that structure is the \textbf{three-dimensional bipartite diamond lattice}, selected by criteria of minimal coordination in 3D. The detailed justification is developed in \S{}II.3, where the thread D1 (geometric uniqueness) is also explicitly declared as an open problem of the program.


\textbf{Analogy with known physical media.} In a crystalline solid such as copper or silicon, there exists a characteristic distance $d$ between neighbouring atoms (the ``lattice parameter''). For phenomena at spatial scales much larger than $d$ --- propagation of sound waves, thermal conduction, elastic deformation ---, the solid is successfully described by continuous-medium equations (solid mechanics, elasticity, thermodynamics). For phenomena at scales comparable to or smaller than $d$ --- vibrations at very high frequencies, scattering of electrons by the lattice, detailed quantum properties ---, the continuous description ceases to be adequate and an explicit microscopic description of the atomic lattice is required. \emph{El Pleno} has exactly the same conceptual structure: $\ell_P$ is to the continuous description of \emph{el Pleno} what the lattice parameter is to the continuous description of a crystalline solid.


\textbf{Epistemic classification.} The existence of $\ell_P$ as a primary structural property of \emph{el Pleno} is \epi{DERIVED} from axiom A4 (ultraviolet regulation). The specific numerical value ---its reading in human rods--- is \epi{CALIBRATED} via reference to the Planck length. The independent measurement of that human value from a physics distinct from the gravitational and the quantum is an open thread declared in \S{}V.3 (C-meta-1.b).


\subsubsection{\texorpdfstring{I.8.4 --- Structural propagation speed $c$}{I.8.4 - Structural propagation speed}}

\textbf{Reading from F1.} The parameter $c$ appears in the kinetic coefficient $\rho_P c^2 \ell_P^2$ and in the flexural coefficient $\rho_P c^2 \ell_P^4/40$. Its structural function is to \textbf{establish the propagation speed of the modes of the medium}: when the field of \emph{el Pleno} admits wave-type solutions, those waves propagate with speed $c$.


\textbf{Explicit derivation of the propagation speed.} To verify that $c$ is effectively the speed of the modes of \emph{el Pleno} (not just a parameter with dimensions of speed), one proceeds as follows. Linearizing F1 around the ground-state \emph{el Pleno}, $\Psi = (S_{\min}^\text{estr} + \delta S)\, e^{i\theta}$ with $|\delta S| \ll S_{\min}^\text{estr}$ and $\theta$ small, and keeping quadratic terms in the perturbations, the equation of motion F2 (derived in \S{}I.7.10) reduces to:


\[
\rho_P c^2 \ell_P^2\, \Box(\delta S) - V''(S_{\min}^\text{estr})\, (\delta S) + O(\ell_P^4) = 0
\]

where $V''(S_{\min}^\text{estr}) = 19.69\,\rho_P c^2$ is the curvature of the potential at the minimum (\S{}I.8.7). This is a Klein-Gordon equation with effective mass. Neglecting the mass term for the moment (locally massless regime, non-massive mode), the equation reduces to:


\[
\Box(\delta S) = 0 \qquad \Longleftrightarrow \qquad \left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2\right)\delta S = 0
\]

with wave-type solutions $\delta S \sim e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}$ that satisfy the dispersion relation $\omega = c\,|\mathbf{k}|$. \textbf{The phase and group velocity of these modes is exactly $c$} --- not by convention, but as a direct consequence of the coefficient $\rho_P c^2 \ell_P^2$ of the kinetic term of F1.


\textbf{Equivalent verification via the phase mode.} The same result is obtained from the phase equation E7 (\S{}I.7.10) in the regime where $S^2 \approx S_{\min}^{2,\text{estr}}$ is approximately constant, taking its physical infrared branch (the phase flexural correction $\propto(k\ell_P)^2$ is negligible in the infrared):


\[
\partial_\mu(S^2 \partial^\mu \theta) - \frac{\ell_P^2}{20}\Box(S^2\Box\theta) = 0 \;\xrightarrow{\;\text{IR}\;}\; S_{\min}^{2,\text{estr}} \cdot \Box\theta = 0 \;\Longrightarrow\; \Box\theta = 0
\]

whose explicit form $(c^{-2}\partial_t^2 - \nabla^2)\theta = 0$ describes phase modes that propagate with speed $c$ in the infrared regime (with ultraviolet correction $(k\ell_P)^2/20$, \S{}II.5.4, \S{}III.4.3). \textbf{Two physically distinct sectors of \emph{el Pleno}} --- perturbations of the modulus $\delta S$ (radial modes) and perturbations of the phase $\delta\theta$ (phase modes, identifiable with the photon in \S{}III.4) --- \textbf{produce modes that share the same propagation speed $c$ in the infrared regime}, a structural property of the medium.


\textbf{Mathematical form and dimensions.} $c$ has dimensions $[m/s]$. Its numerical value is $c \approx 2.998 \times 10^8\ m/s$, identical to the value of the speed of light observed in all the experiments of standard physics.


\textbf{Physical interpretation under PIU-$\Psi$.} Under PIU-$\Psi$, $c$ is not ``the speed of light'' in a primitive sense: it is the \textbf{intrinsic structural speed of \emph{el Pleno}}, the constant that fixes the propagation of all its modes. This reading has an underlying ontological consequence ---that $c$ is the \emph{causal structure} of the medium, not a substantive property like $\rho_P$ or $\ell_P$--- which \S{}I.8.9 develops in detail. For the purpose of this subsection it suffices to retain the operational feature: as the speed of the medium itself, $c$ is identical for all its modes.


Light propagates with speed $c$ because light is a specific mode of \emph{el Pleno} (mode of the induced gauge sector, developed in \S{}III.4), and as a mode of the medium it propagates with the structural speed of the medium. Gravitational waves also propagate with speed $c$ because they are modes of \emph{el Pleno} (transverse tensor modes, developed in \S{}III.3), and as modes of the same medium they share the same speed. \textbf{The equality $v_\text{light} = v_\text{gravity} = c$ is not an experimental coincidence} --- it is a direct structural consequence: both are speeds of modes of the same \emph{el Pleno}.


This reading explains the experimental observation discussed in \S{}I.3.4: the coincidence $v_g = c$ confirmed by GW170817 to precision $10^{-15}$ is exactly what \emph{el Pleno} predicts structurally.


\textbf{Connection with structural covariance and with the observed universality.} Axiom A3 (structural covariance) established in \S{}I.5.4 requires that the dynamical laws of \emph{el Pleno} be the same for all observers. This produces, as an effective consequence in the relaxed regime of \emph{el Pleno}, that the speed $c$ is identical for all observers --- which is the experimental basis of Special Relativity. The universal constancy of $c$ documented experimentally since the late 19th century (Michelson-Morley experiments) thus receives under PIU-$\Psi$ a structural explanation: $c$ is a property of a single, universal physical medium, identical at every point of the cosmos. Special Relativity is recovered as an effective limit of \emph{el Pleno}, not as an independent axiom of the program.


\textbf{Scope and limit of the analogy with known physical media.} In an ordinary material medium, the propagation speed of the modes is an emergent property determined by the ratio between rigidity and inertia (in air $v_\text{sound}\approx 340$ m/s; in superfluid helium even two speeds coexist). The analogy is useful for understanding that $c$ is not the speed of an object that travels through a background, but the speed \emph{of the medium itself}; but it has a limit, which \S{}I.8.9 marks with precision: in those media the speed of sound coexists with other properties within a background space-time whose causal structure is already given by another speed ---precisely $c$---, whereas in \emph{el Pleno} $c$ \emph{is} that causal structure, not a mode speed among several.


\textbf{Operational status of $c$.} $c$ is one of the three primary structural inputs of the apparatus of \emph{el Pleno} (together with $\rho_P$ and $\ell_P$), as is articulated in \S{}I.8.10; of those three, $c$ has the distinct ontological status just noted and that \S{}I.8.9 develops ---causal structure, not a substantive property. Unlike $\rho_P$, the numerical value of $c$ is \textbf{directly accessible to experimental measurement} --- the speed of light is measured with extreme precision in numerous contexts. In this sense, $c$ is the structural parameter of \emph{el Pleno} with the least dependence on indirect calibration: what the program does is \textbf{reinterpret ontologically} a quantity already measured with precision, identifying it with the causal structure of the medium instead of treating it as a postulated geometric property of space-time.


\textbf{Epistemic classification.} The existence of $c$ as the structural speed of the medium is \epi{DERIVED} from axiom A3 (structural covariance) and from the analysis of the linearized equations of F1 that produce propagation modes with speed $c$ (verified by two independent routes above: radial mode $\delta S$ and phase mode $\delta\theta$). The specific numerical value of $c$ is \epi{MEASURED}: it comes directly from experimental observation, not from indirect calibration. The point, which \S{}I.8.9 develops, is that $c$ does \textbf{not} figure as a derivation thread: it is not a value that the program does not yet know how to derive, but a quantity that is not of the kind that is derived. $c$ is not a substantive property of \emph{el Pleno} (as $\rho_P$ and $\ell_P$ are), but the measure of its unified causal structure; its numerical value is an artefact of the historical choice of units of length and time, and it vanishes in natural units ($c=1$). The correct causal direction ---already noted in \S{}III.1.8, where the equality $v=c$ verifies Lorentz covariance instead of deriving $c$--- is consistent with this reading.


\subsubsection{\texorpdfstring{I.8.5 --- Flexural rigidity $\gamma_\kappa$ and the native identity $\hbar = \rho_P c \ell_P^4$}{I.8.5 - Flexural rigidity and the native identity}}

\textbf{Reading from F1.} The coefficient of the flexural term in F1 is:


\[
\gamma_\kappa = \frac{\rho_P c^2 \ell_P^4}{40} = \frac{\hbar c}{40} \approx 7.90 \times 10^{-28}\ J \cdot m
\]

Its dimensions $[J \cdot m]$ are identical to those of the product $EI$ (Young's modulus times moment of inertia) in the Euler-Bernoulli theory of elastic beams. This dimensional coincidence is not casual: $\gamma_\kappa$ is the \textbf{flexural rigidity} of \emph{el Pleno} as an elastic medium.


\textbf{Derivation of the $1/40$ coefficient.} The numerical factor $1/40$ in $\gamma_\kappa$ is not an arbitrary fit: it emerges from an \textbf{explicit geometric derivation} from the diamond lattice, derived completely in \S{}II.3.8 (P9). The procedure starts from the expansion of the structure factor $|f(\mathbf{k})|^2$ to fourth order in $\mathbf{k}$, followed by the $SO(3)$ angular average over the Brillouin zone of the diamond. The result is that the relative weight of the fourth-derivative term with respect to the square of the kinetic term is exactly


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

fixed by the tetrahedral geometry of the lattice and the isotropic average, with no free parameters. \epi{DERIVED} (complete computation in \S{}II.3.8, P9).


\textbf{Native identity $\hbar = \rho_P c \ell_P^4$.} The reduced Planck constant is not postulated as an identity: it \emph{emerges} from quantizing the phase mode of \emph{el Pleno} itself. The logical direction is the same as in the case of $G$ (\S{}I.7.5): the parameters of the medium are what is primary, and $\hbar$ comes out on quantizing its dynamics. The skeleton of the origin, developed in detail in \S{}III.2.2, is the following. First, from the phase sector of the kinetic term of F1, $\mathcal{L}_\theta = \tfrac12\rho_P c^2\ell_P^2 S^2(\partial\theta)^2$, the homogeneous phase mode is a \emph{rotor} with moment of inertia $I_\theta = \rho_P\ell_P^2\int S^2 d^3x$. Second, the norm of the fundamental mode is derived ---not postulated--- from $S$ being dimensionless (saturation $S=1$) and from the quantum occupying a cell of volume $\ell_P^3$: $\int S^2 d^3x = \ell_P^3$, hence $I_\theta = \rho_P\ell_P^5$. Third, the phase lives on the circle $[0,2\pi)$, so that the spectrum of its angular momentum is exact, $p_\theta = n\hbar$; the fundamental quantum ($n=1$) has $p_\theta = \hbar$. Fourth, that mode rotates at the coherence frequency $\omega_P = c/\ell_P$, and the physical angular momentum of the rotor is $p_\theta = I_\theta\omega_P$. Equating the canonical quantum with the structural quantity,
\[
\boxed{\;\hbar = \rho_P c \ell_P^4\;}
\]
with numerical prefactor exactly $1$ ---fixed by the canonical normalization of the quantum, not calibrated against the observed value of $\hbar$. The combination $\rho_P c\,\ell_P^4$ is not chosen by dimensions among the many possible: it is produced by the quantization of the phase mode of F1, and a medium with other dynamics would give another action scale.

Alongside this canonical derivation, the program offers an \textbf{equivalent reading} in the language of Bohm-Madelung mechanics: on rewriting the field of \emph{el Pleno} in modulus and action-phase variables, the flexural term $\gamma_\kappa(\Box S)^2$ produces the quantum-pressure terms characteristic of Bohm mechanics, and the action scale that weights them is exactly $\rho_P c \ell_P^4$. This coincidence is a consistent interpretation of the identity of \S{}III.2.2 in a complementary language, \textbf{not a second independent derivation} of the same number.


\textbf{Explicit dimensional verification.} The dimensions of $\rho_P c \ell_P^4$ are:


\[
[\rho_P] \cdot [c] \cdot [\ell_P]^4 = \frac{kg}{m^3} \cdot \frac{m}{s} \cdot m^4 = \frac{kg \cdot m^2}{s} = J \cdot s
\]

identical to those of $\hbar$. $\checkmark$


\textbf{Numerical verification.} Applying the canonical values of the structural parameters of \emph{el Pleno} ($\rho_P \approx 5.155 \times 10^{96}\ kg/m^3$, $c \approx 2.998 \times 10^8\ m/s$, $\ell_P \approx 1.616 \times 10^{-35}\ m$):


\[
\rho_P c \ell_P^4 \approx 5.155 \times 10^{96} \times 2.998 \times 10^8 \times (1.616 \times 10^{-35})^4
\]

Proceeding step by step:


\[
\ell_P^4 = (1.616 \times 10^{-35})^4 \approx 6.8197 \times 10^{-140}\ m^4
\]

\[
\rho_P \cdot c = 5.155 \times 10^{96} \times 2.998 \times 10^8 = 1.5455 \times 10^{105}\ kg/(m^2 \cdot s)
\]

\[
\rho_P c \ell_P^4 \approx 1.5455 \times 10^{105} \times 6.8197 \times 10^{-140} \approx 1.0540 \times 10^{-34}\ J \cdot s
\]

The observed value of $\hbar$ (CODATA 2018) is $\hbar = 1.0546 \times 10^{-34}\ J \cdot s$. The agreement is exact by construction (see the following block); the observed numerical residue ($< 0.1\%$) reflects only the rounding of the primary parameters to four significant figures, not a physical discrepancy.


\textbf{Relation between the structural identities of $G$ and $\hbar$.} The two identities $G = c^2/(\rho_P\ell_P^2)$ (\S{}I.7.5, \S{}III.1.3) and $\hbar = \rho_P c\ell_P^4$ \textbf{are not logically independent}. Eliminating $\rho_P$ between the two gives:


\[
\hbar = \frac{c^3 \ell_P^2}{G} \quad\Longleftrightarrow\quad \ell_P = \sqrt{\frac{\hbar\,G}{c^3}},
\]

which is exactly the definition of the Planck length. In consequence, fixing the two substantive properties $(\rho_P, \ell_P)$ ---with $c$ understood as the causal structure of the medium, not as a free property (\S{}I.8.9)--- and taking $\ell_P$ equal to the Planck length satisfies \textbf{both identities simultaneously by construction}. This is declared without ambiguity: $G$ and $\hbar$ do not constitute two independent observational verifications, but a single structural condition ---$\ell_P = \ell_\text{Planck}$, $\rho_P = c^2/(G\ell_P^2)$--- read in two languages, gravitational and quantum. The non-trivial content is that one and the same microscopic scale $\ell_P$ and one and the same density $\rho_P$ reproduce at once the gravitational constant and the Planck constant, which is not guaranteed a priori: a medium with dynamics distinct from F1 would link $G$ and $\hbar$ by a different relation. The count of degrees of freedom of \emph{el Pleno} is therefore: two substantive properties $(\rho_P, \ell_P)$ and the causal structure $c$, with one identity that links the properties to $G$ and $\hbar$, leaving \textbf{one structural degree of freedom not fixed internally} ---whose independent calibration (measuring $\rho_P$ or $\ell_P$ from non-gravitational and non-quantum physics) is the open thread declared in \S{}V.3.


\textbf{Consequence for $\gamma_\kappa$.} The native identity allows rewriting the flexural coefficient as:


\[
\gamma_\kappa = \frac{\rho_P c^2 \ell_P^4}{40} = \frac{c \cdot (\rho_P c \ell_P^4)}{40} = \frac{\hbar c}{40}
\]

This form --- $\gamma_\kappa$ proportional to $\hbar$ --- reveals the \textbf{genuine quantum nature} of the flexural term of \emph{el Pleno}: it is not a classical regularization added ad hoc, but an intrinsic quantum property of the medium. The presence of $\hbar$ in the coefficient confirms that the flexural term operates in the quantum regime.


\textbf{Physical interpretation of $\gamma_\kappa$ --- flexural rigidity.} $\gamma_\kappa$ encodes the resistance of \emph{el Pleno} to high-curvature deformations. In the Euler-Bernoulli theory, $EI$ determines how much a beam opposes bending under a load: beams with large $EI$ are rigid, beams with small $EI$ are flexible. In \emph{el Pleno}, $\gamma_\kappa$ determines how much the medium opposes configurations of the field $S$ that require rapid spatial variations.


\textbf{\emph{El Pleno} is an extraordinarily rigid medium at the microscopic scale.} Although $\gamma_\kappa$ is small in numerical value ($\sim 10^{-28}\ J \cdot m$), the scales at which it manifests are even smaller. When an excitation of the field $S$ tries to have significant variation at scales $\sim \ell_P$, the flexural term produces a contribution to the action so large that the configuration becomes dynamically forbidden. This microscopic rigidity is what produces the UV regulation of axiom A4: below $\ell_P$, the medium simply does not let itself be excited.


\textbf{Connection with observable properties.} The flexural rigidity of \emph{el Pleno} has several potential observable manifestations:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Absence of UV divergences} in the predictions of the program, in contrast with the standard divergences of Quantum Field Theory that require a posteriori renormalization.
  \item \textbf{Modifications to the dispersion of modes} at scales close to $\ell_P$, with observable consequences in early cosmology and in the propagation of very-high-energy gamma rays (open thread declared in \S{}V.3 on testable Lorentz violations).
  \item \textbf{Resolution of cosmological singularities}: extreme-curvature configurations (black-hole interiors, initial instants of the universe) are regulated structurally by $\gamma_\kappa$, avoiding the infinities that General Relativity cannot handle.
\end{itemize}

\textbf{Thermodynamic reading (Israel-Stewart).} The flexural term also admits a reading in the formalism of causal hydrodynamics of Israel-Stewart, where a second-order relaxation coefficient weights the gradient corrections. The formal correspondence suggests $\gamma_\kappa \sim \rho_P c^2 (c\tau_\Pi)^2 \ell_P^2$ with $\tau_\Pi$ the relaxation time of \emph{el Pleno} (derived in \S{}III.1.6 as the inverse of the frequency of the flexural regulation mode). This reading is \textbf{strong logical deduction}: it articulates the UV term as a dissipative property complementary to the flexural (Euler-Bernoulli) and quantum (Madelung-Bohm) interpretations, but the exact equality of the coefficients requires the explicit development of the Israel-Stewart correspondence. No demonstrated equivalence of the three descriptions is claimed, but a motivated structural correspondence.


\textbf{Analogy with known physical media.} A thin steel wire is fairly rigid in bending: $EI$ is large compared with the everyday loads applied to it. A human hair is much more flexible: $EI$ is small. \emph{El Pleno} has a very specific flexural rigidity determined by its structural parameters; it is the exact structural analogue of $EI$ in elastic-beam mechanics, generalized to a three-dimensional medium with a temporal dimension and with genuine quantum consistency via $\hbar$.


\textbf{Epistemic classification.} The numerical value $\gamma_\kappa \approx 7.90 \times 10^{-28}\ J \cdot m$ is \epi{DERIVED} from $\rho_P$, $c$, $\ell_P$ via the structural identity $\gamma_\kappa = \rho_P c^2 \ell_P^4 / 40$, with the factor $1/40$ obtained by expansion plus $SO(3)$ angular average over the Brillouin zone of the diamond lattice (\S{}II.3.8 P9), with no free parameters. The native identity $\hbar = \rho_P c \ell_P^4$ is \epi{DERIVED} both in its functional form and in its unit prefactor (quantization of the phase rotor with derived mode norm, \S{}III.2.2); its Madelung-Bohm reading is an equivalent interpretation, not an independent derivation. The relation $\ell_P = \sqrt{\hbar G/c^3}$ that links the identities of $G$ and $\hbar$ is \epi{DERIVED} (algebraic consequence). The Israel-Stewart thermodynamic reading is \epi{SLD}. The physical interpretation as flexural rigidity analogous to EI is \epi{SLD} sustained by the exact dimensional coincidence and the identical structural function in the Lagrangian.


\subsubsection{I.8.6 --- Quantum IR wall as a structural property}

\textbf{Reading from F1.} The Klauder term in F1, $V_q^\text{dens}(S) = 3\hbar^2/(8 \rho_P \ell_P^8 S^2)$, contains a coefficient that combines $\hbar$ with the three structural parameters of \emph{el Pleno}. This term represents a \textbf{quantum structural property} of the medium, distinct from the classical properties encoded in the other three terms.


\textbf{Structural function.} The quantum IR wall prevents the modulus $S$ of the field of \emph{el Pleno} from reaching the value zero: as $S$ approaches zero, $V_q^\text{dens}(S) \propto 1/S^2$ diverges to $+\infty$, which makes the configuration $S = 0$ dynamically inaccessible. \emph{El Pleno} does not admit being ``switched off'' --- there are no configurations of the medium that correspond to its complete annulment.


\textbf{Quantitative verification of the inaccessibility of $S = 0$.} The dynamical inaccessibility of the value $S = 0$ is demonstrated explicitly by computing the energy required to take the medium from the ground-state \emph{el Pleno} $S_{\min}^\text{estr} \approx 0.7231$ to a configuration $S = \epsilon$ with $\epsilon \to 0$. The difference in energy density is:


\[
\Delta E(\epsilon) = V_q^\text{dens}(\epsilon) - V_q^\text{dens}(S_{\min}^\text{estr}) = \frac{3}{8}\rho_P c^2\left(\frac{1}{\epsilon^2} - \frac{1}{(S_{\min}^\text{estr})^2}\right)
\]

using the equivalent form $V_q = (3/8)\rho_P c^2/S^2$. In the limit $\epsilon \to 0$:


\[
\lim_{\epsilon \to 0} \Delta E(\epsilon) = \lim_{\epsilon \to 0} \frac{3}{8}\frac{\rho_P c^2}{\epsilon^2} = +\infty
\]

The required energy diverges: the medium \textbf{does not admit} the configuration $S = 0$ because approaching it would require an infinite energy density. This divergence is not a mathematical pathology --- it is the operational manifestation of axiom A2 (strict positivity of the modulus): \emph{el Pleno} cannot switch off because the energy cost of doing so is infinite.


\textbf{Concrete numerical bound.} As a concrete quantitative example, to take the medium from $S_{\min}^\text{estr}$ to $S = 0.01 \cdot S_{\min}^\text{estr}$ (that is, to reduce the modulus by two orders of magnitude), the required energy density would be:


\[
\Delta E \approx \frac{3}{8}\rho_P c^2\left[\frac{1}{(0.01 \cdot 0.7231)^2} - \frac{1}{(0.7231)^2}\right] \approx \frac{3}{8} \cdot \frac{10^4}{(0.7231)^2}\rho_P c^2 \approx 7170\, \rho_P c^2
\]

that is, approximately $3.3 \times 10^{117}\ J/m^3$ of additional energy density on top of the ground-state \emph{el Pleno} --- a magnitude equivalent to the total energy density of the observable universe concentrated in a single cubic metre, multiplied by more than $10^{40}$. The dynamical inaccessibility of $S \to 0$ is quantitatively established.


\textbf{Physical interpretation.} This property must be read as \textbf{structural support of the medium}, analogous to the quantum pressure that prevents the electrons of an atom from falling to the nucleus. In an atom, the uncertainty principle forbids the electron from simultaneously occupying zero position and zero momentum; this prohibition manifests as a ``quantum pressure'' term that maintains the stability of the atom. In \emph{el Pleno}, the affine quantization of the modulus $S$ --- necessary because $S \geq 0$ defines a configuration manifold of the half-line type --- produces an analogous term that maintains the stability of the medium.


The difference from an atom is structural: the atomic quantum pressure is a property of a quantized system of many confined particles; the quantum wall of \emph{el Pleno} is a property of the medium itself, derived from the nature of the field $\Psi$ as a macroscopic quantum object with $S \geq 0$.


\textbf{Observable structural consequences.} The quantum IR wall of \emph{el Pleno} has several manifestations that the program develops in later parts of the manuscript:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{The ``vacuum'' is not empty} --- it is the ground-state \emph{el Pleno}, the configuration of the medium with minimal $V(S)$ and specific $S = S_{\min}^\text{estr}$, structurally sustained by the IR wall against collapse to $S = 0$. Developed in \S{}III.7.
  \item \textbf{The fluctuations of the quantum vacuum} (Casimir effect, Lamb shift) are fluctuations of the ground-state \emph{el Pleno} around its sustained configuration. Developed in \S{}III.4 and \S{}III.7.
  \item \textbf{The primordial cosmological bounce} --- in its regime of maximum primordial compression, \emph{el Pleno} reaches upper saturation with $S \to S_{\max}^\text{geom}$ and total density $\rho_{\max,\text{abs}} = S_{\max,\text{geom}}^2 \cdot \rho_P \approx 2.1773\,\rho_P$, not extinction ($S \to 0$). The bounce is produced by the divergence of the topological potential $V(S)$ near $S_{\max}^\text{geom}$, not by the Klauder quantum IR wall. The IR wall fulfils a distinct structural function: it prevents configurations with $S \to 0$ in any regime of the medium. Developed in \S{}II.7.
\end{itemize}

\textbf{Analogy with known physical media.} In a Bose-Einstein condensate, there is a minimum density below which the condensate ceases to exist as such: the ultra-dilute regime is inaccessible to the condensate for structural reasons. \emph{El Pleno} has an analogous property: there is a minimum value of $S$ below which the medium cannot be configured. The difference from a BEC is that in \emph{el Pleno} this minimum is an intrinsic and universal structural property, not a consequence of specific experimental conditions.


\textbf{Epistemic classification.} The existence of the quantum IR wall is \epi{DERIVED} from the affine Klauder quantization applied to the field of \emph{el Pleno} (procedure detailed in \S{}I.7.8); the coefficient $3/8$ and the positive sign are unambiguous. The dynamical inaccessibility of the value $S = 0$ is quantitatively verified above by the explicit divergence of the energy density. The phenomenological identification of this wall with specific observable properties (structural support of the ground-state \emph{el Pleno}, fluctuations of the quantum vacuum) is developed with detailed classification in the corresponding sections. The primordial cosmological bounce is a distinct phenomenon, associated with the divergence of the topological potential near $S_{\max}^\text{geom}$ (not with the Klauder one), as is clarified in \S{}I.10.10.


\subsubsection{\texorpdfstring{I.8.7 --- Potential $V(S)$ and structural energy of the medium}{I.8.7 - Potential and structural energy of the medium}}

\textbf{Reading from F1.} The second term of F1 is the potential $V(S)$, an exclusive function of the modulus. As established in \S{}I.7.6 and as will be developed in detail in \S{}II.2, $V(S)$ has the canonical tripartite structure:


\[
V(S) = W_\text{bare}(S) + V_q(S) + V_\text{topo}(S)
\]

with three components of distinguishable physical origin: $W_\text{bare}$ is the \textbf{hyperelastic component} (rational quartic, encodes the structural elasticity of the medium), $V_q$ is the \textbf{Klauder quantum component} (IR wall, sustains the strict positivity of the modulus), and $V_\text{topo}$ is the \textbf{logarithmic topological Gent component} (UV wall, sustains the hard bound $S_{\max}^\text{geom}$).


\textbf{Joint physical interpretation.} $V(S)$ encodes the \textbf{structural energy of \emph{el Pleno} as a physical medium}: how much energy it costs to take the medium from its fundamental configuration to other configurations of the field $S$. Each of the three components contributes a distinguishable physical function:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{$W_\text{bare}$} --- structural analogue of the energy function of a hyperelastic solid (Mooney-Rivlin / Ogden / Gent formalism in materials mechanics). That function typically has the property of diverging when the deformation approaches a structural limit --- the material cannot be stretched indefinitely. $W_\text{bare}(S) = (E_0/2) S^4/(S_\text{max}^2 - S^2)^2$ has exactly this mathematical structure: it diverges when $S \to S_{\max}^\text{geom}$, encoding that \emph{el Pleno} has an upper structural limit of deformation.
  \item \textbf{$V_q$} --- genuine quantum origin. It does not come from classical elasticity but from the affine quantization of the modulus of \emph{el Pleno} (Klauder quantization applied to a configuration manifold of the half-line type $S > 0$). Its specific form $V_q = (3/8)\rho_P c^2/S^2$ is \textbf{unambiguous} --- coefficient $3/8$ and positive sign fixed with no ordering freedom. It diverges when $S \to 0$, maintaining the strict positivity of the modulus.
  \item \textbf{$V_\text{topo}$} --- topological-geometric origin. It has the logarithmic Gent form (1996): $V_\text{topo} = -(\rho_P c^2/2)\ln(1 - S^2/S_{\max,\text{geom}}^2)$. It diverges when $S \to S_{\max,\text{geom}}$ producing the UV saturation wall of the medium. It also sustains the solitonic configurations (the ``particles'' in PIU-$\Psi$) as topologically stable minima of the potential.
\end{itemize}

\textbf{Double structural wall of \emph{el Pleno}.} The tripartite structure of $V(S)$ produces the most distinctive feature of the program: \emph{el Pleno} operates in the open interval $S \in (0, S_{\max}^\text{geom})$ with \textbf{both walls dynamically inaccessible}. The IR wall ($S \to 0$) comes from $V_q$ (axiom A2); the UV wall ($S \to S_{\max}^\text{geom}$) comes from $V_\text{topo}$ (axiom A6). No regime of \emph{el Pleno} reaches singularities --- neither in the primordial cosmological bounce, nor in the cores of black holes, nor in any physical process derivable from the program.


\textbf{Stable configurations as preferred energy states.} The physical medium tends dynamically toward minimum-energy configurations. The minima of $V(S)$ correspond to stable states of \emph{el Pleno} --- configurations that persist in time without disintegrating. The program identifies:


\begin{itemize}[leftmargin=1.4em]
  \item $S = S_{\min}^\text{estr}$ --- absolute minimum of $V(S)$, identified with the ground-state \emph{el Pleno} (the ``vacuum''). It is the result of the balance between the three components of the potential.
  \item Localized solitonic configurations --- topological minima of the potential, identified with the particles (developed in \S{}II.6 and \S{}III.6).
  \item $S \to S_{\max}^\text{geom}$ --- inaccessible hard bound, associated with the maximum saturation of the medium.
\end{itemize}

\textbf{Analogy with known physical media.} The potential $V(S)$ is the structural analogue of the \textbf{free energy function} of a thermodynamic material: it characterizes which configurations of the material are energetically preferred and which are unstable. In a ferromagnetic material, the free energy has minima corresponding to the directions of spontaneous magnetization; the material settles dynamically into one of those minima. In \emph{el Pleno}, $V(S)$ has a minimum corresponding to $S_{\min}^\text{estr}$; the medium settles dynamically into that configuration as the ground-state \emph{el Pleno} of the observed cosmos.


\textbf{Epistemic classification.} The tripartite structure $V(S) = W_\text{bare} + V_q + V_\text{topo}$ is \epi{DERIVED} from the principle of dynamical minimum of the total potential, with each component having an independent derivation. The specific form of each component and the quantitative verification of the dynamical minimum (sympy + scipy, symbolic verification to 10 digits of precision) are developed in \S{}II.2.


\subsubsection{\texorpdfstring{I.8.8 --- Canonical configurations $S_{\min}^\text{estr}$ and $S_{\max}^\text{geom}$}{I.8.8 - Canonical configurations and}}

\textbf{Reading from F1.} Two critical configurations of \emph{el Pleno} emerge as a consequence of the potential $V(S)$. Both have canonical dimensionless values derived structurally, with no free parameters; their complete derivations, which rest on the diamond lattice, are developed in Part II (\S{}II.3.8 and \S{}II.4), and here they are stated and interpreted in their role within the reading of \emph{el Pleno} as a medium:


\[
\boxed{\;S_{\min}^\text{estr} = \frac{12(\pi^2 - 4)}{\pi^4} = 0.7230871\ldots\;}
\]
from which
\[
S_{\min}^{2,\text{estr}} = \left(\frac{12(\pi^2-4)}{\pi^4}\right)^2 = 0.5228553\ldots
\]

\[
\boxed{\;S_{\max,\text{geom}}^2 = \frac{8}{3}\sqrt{\frac{2}{3}} = 2.1773\ldots\;}
\]

\textbf{How $S_{\min}^\text{estr}$ is obtained, and why its derivation does not live here.} The value $12(\pi^2-4)/\pi^4$ is a \emph{property of the microscopic structure of the medium} ---the diamond lattice--- and, in accordance with the logical order of the program, its complete derivation belongs to Part II, after that lattice has been established (\S{}II.3). Here, in the interpretive reading of Part I, the result is stated and its most notable feature is pointed out; the complete algebraic steps are given in their canonical place. The notable feature is that $S_{\min}^\text{estr}$ is obtained by \textbf{two complementary formulations} ---one spectral, one dynamical--- that produce the same value:
\begin{itemize}[leftmargin=1.4em]
  \item the \textbf{spectral route} ---the ratio between the average of the discrete Laplacian and the average of $k^2$ over the Brillouin zone of the diamond lattice, which gives $12(\pi^2-4)/\pi^4$ as the form factor of the band, with no reference to the potential (development in \S{}II.3.8);
  \item the \textbf{dynamical route} ---the minimum of the effective potential $V_\text{eff}(S) = W_\text{bare} + V_q + V_\text{topo}$, which gives $S_\text{eq} = 0.7230873678$ (development in \S{}II.4.5).
\end{itemize}
The discrete Laplacian and the effective potential are two complementary characterizations of the same physical medium ---the first spectral, the second dynamical--- and they must produce the same value for the minimum configuration; that they coincide to ten digits is one of the most profound internal-consistency results of the program, and it is gathered here for its weight in the reading of \emph{el Pleno} as a medium. \epi{DERIVED}, complete symbolic verification in \S{}II.3.8 and \S{}II.4.5. (The labelling matters: the mean amplitude is $S_{\min}^\text{estr}\approx 0.7231$; the occupation is its square, $S_{\min}^{2,\text{estr}}\approx 0.5229$, in accordance with the Bogoliubov-Josephson identity interpreted below.)


\textbf{Curvature of the potential at the minimum.} The second and fourth derivatives of $V_\text{eff}$ at $S = S_{\min}^\text{estr}$ are:


\[
\left.\frac{\partial^2 V_\text{eff}}{\partial S^2}\right|_{S_{\min}^\text{estr}} = 19.69\,\rho_P c^2, \qquad \left.\frac{\partial^4 V_\text{eff}}{\partial S^4}\right|_{S_{\min}^\text{estr}} = 855.6\,\rho_P c^2
\]

Both are positive: $V'' > 0$ confirms that $S_{\min}^\text{estr}$ is a \textbf{stable minimum} (neither maximum nor saddle), and $V^{(4)} > 0$ confirms that the effective quartic coupling is \textbf{repulsive} (Gross-Pitaevskii condensate stable under perturbation). The consistency of these signs with the observed stability of the cosmological vacuum is additional qualitative validation of the program.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{potencial_efectivo.png}
\caption{The effective potential $V_\text{eff}(S)$, the central object of PIU-$\Psi$. (A) The complete potential on a logarithmic scale: the Klauder quantum IR wall prevents $S\to 0$, the well fixes the ground-state \emph{el Pleno} at $S_{\min}=0.7231$, and the Gent wall prevents $S\to S_{\max}=1.4756$ (hard bound A6). (B) Decomposition into the three derived components: $W_\text{bare}$ (Gent), $V_\text{topo}$ (logarithmic) and $V_q$ (IR wall). (C) Zoom of the well, with the curvature parabola $V''(S_{\min})=19.69\,\rho_P c^2$ that confirms the stable minimum. (D) The packing $x=S^2/S_{\max}^2$ fixes the dilatant flow index $n(x)=1/(1-x)$; the present universe is at $x=0.240$, $n=1.32$ (nearly Newtonian). All parameters derived without fitting.}
\label{fig:potencial}
\end{figure}


\textbf{Physical interpretation of $S_{\min}^\text{estr}$ --- ground-state \emph{el Pleno}.} It is the \textbf{state of minimal excitation} of the medium, what conventional language calls ``vacuum''. Under PIU-$\Psi$, the ``vacuum'' is not absence --- it is a specific configuration of \emph{el Pleno} with $S = S_{\min}^\text{estr} \approx 0.7231$, structurally sustained by the dynamical balance between the three components of the potential $V_\text{eff}$ (\S{}I.8.7) and by the quantum IR wall (\S{}I.8.6).


\textbf{Physical interpretation of $S_{\min}^{2,\text{estr}}$ --- Bogoliubov-Josephson identity.} The square of the modulus in the ground-state \emph{el Pleno}, $S_{\min}^{2,\text{estr}} \approx 0.5229$, has a specific physical interpretation derived from the constitutive identity of the program established by the \textbf{Bogoliubov-Josephson route} (\S{}II.2). This identification establishes:


\[
\boxed{\;S^2(x^\mu) \longleftrightarrow \frac{n_\text{quanta}(x^\mu)}{n_0}\;}
\]

where $n_\text{quanta}$ is the \textbf{number density of quanta of \emph{el Pleno}} in the fundamental coherent mode and $n_0 = 1/\ell_P^3$ is the reference density (one quantum of \emph{el Pleno} per elementary cell of the microscopic lattice). This identification is not a metaphor: it is a constitutive derivation analogous to the well-known Bogoliubov-Josephson identification in macroscopic quantum condensates, where $|\Psi|^2$ is the number density of condensed particles, not a probability amplitude.


\textbf{Operational consequence.} $S^2$ is a \textbf{dimensionless relative number density}, not an energy density in units $kg/m^3$. The mass-energy density of \emph{el Pleno} is obtained by multiplying by the appropriate dimensional scale: $\rho_\text{medium}(x^\mu) = \rho_P \cdot S^2(x^\mu)$. In the ground-state \emph{el Pleno} this gives $\rho_\text{ground-state} = \rho_P \cdot S_{\min}^{2,\text{estr}} \approx 0.5229\,\rho_P$, a value with direct physical interpretation: the medium in its fundamental configuration has approximately \textbf{52\% of the primary structural density of \emph{el Pleno}} $\rho_P$.


This identification has a profound consequence for the reading of cosmological dark energy. As developed in \S{}III.7, the cosmological fraction of dark energy $\Omega_{EO} = f_c = S_{\max,\text{eff}}^2 / S_{\max,\text{geom}}^2 \approx 0.6869$ \textbf{measures the same physical quantity} as the fraction of quanta in the coherent condensate of the ground-state \emph{el Pleno} --- a derived quantitative prediction that the program makes with no additional free parameters.


\textbf{Physical interpretation of $S_{\max}^\text{geom}$.} It is the \textbf{upper critical configuration}, associated with the microscopic geometry of \emph{el Pleno}. Its value depends on the specific structure of the microscopic lattice of the medium (the diamond lattice, developed in \S{}II.3). In configurations with $S \to S_{\max}^\text{geom}$, the topological component of the potential diverges --- the medium reaches its structural limit and cannot deform further in that direction.


\textbf{How $S_{\max,\text{geom}}^2$ is obtained, and where.} The hard upper bound $(8/3)\sqrt{2/3} \approx 2.1773$ is also a geometric property of the diamond lattice: it arises from applying the Kepler-Hales optimal-packing theorem (Hales 2005; formal verification Flyspeck 2014) to the saturated medium, via the constitutive identification $S^2 \leftrightarrow n/n_0$ as the ratio between the maximum accessible number density (Kepler-Hales family) and the diamond structural density. The factor $8/3$ comes from the bipartite coordination ($Z=4$ per site $\times$ 2 neighbours per bond) and $\sqrt{2/3}$ from the isotropic 3D geometry; the closed form is invariant under all the polytypes of the Kepler-Hales family (FCC, HCP, mixtures), which makes it robust against the concrete polytype that the medium adopts in saturation. The complete derivation, with each step of Euclidean geometry, is given in \S{}II.4.1; here it is stated for its role in the reading of \emph{el Pleno} as a medium. \epi{DERIVED} with no free parameters.


\textbf{Absolute maximum density of \emph{el Pleno}.} From the conjunction $S^2 \leftrightarrow n/n_0$ (Bogoliubov-Josephson) and the bound $S_{\max,\text{geom}}^2 \approx 2.1773$ the hard structural density bound of \emph{el Pleno} follows immediately:


\[
\boxed{\;\rho_{\max,\text{abs}} = S_{\max,\text{geom}}^2 \cdot \rho_P \approx 2.1773\,\rho_P\;}
\]

a structural quantitative prediction of the program with specific observable consequences (primordial bounce regime, finite cores in black holes, developed in \S{}I.10.10 and \S{}III.6).


\textbf{Connection with the diamond lattice.} The factor $(8/3)\sqrt{2/3}$ that appears in $S_{\max,\text{geom}}^2$ is a \textbf{direct geometric consequence} of the three-dimensional bipartite diamond lattice. If the microscopic lattice of \emph{el Pleno} were different --- simple cubic, hexagonal close-packed without bipartition, etc. --- the value of $S_{\max,\text{geom}}^2$ would be different. The justification of why precisely the diamond lattice and not another is developed in \S{}II.3, with the declaration of the thread D1 (geometric uniqueness refined by bipartite energy minimization) as an open problem of the program.


\textbf{Interpretation of the numerical values.} That both constants are \textbf{dimensionless and derivable with no free parameters} is a significant result of the program. The factor $12(\pi^2-4)/\pi^4$ in $S_{\min}^\text{estr}$ comes from the averaged spectrum of the Brillouin zone of the diamond lattice; the factor $(8/3)\sqrt{2/3}$ in $S_{\max,\text{geom}}^2$ comes from the intrinsic geometry of the same lattice plus the Hales theorem. Neither of the two values requires calibration against observation: they are \textbf{pure structural identities} of \emph{el Pleno}, which the program categorizes as \textbf{structural derived constants} in its epistemic taxonomy.


\textbf{Analogy with known physical media.} In a superconducting material, the Ginzburg-Landau order parameter has a non-zero minimum value in the superconducting state --- the superconducting phase is not ``vacuum'', it is a specific configuration of the condensate. Similarly, in a ferromagnet below the Curie temperature, the magnetization has a non-zero spontaneous value. The ground-state \emph{el Pleno} $S_{\min}^\text{estr}$ is the structural analogue of these order parameters: a specific non-zero configuration of the medium that defines its fundamental state. The novelty of \emph{el Pleno} relative to these terrestrial analogues is that the medium is \textbf{single and universal} (it constitutes the whole cosmos), not confined to a specific material substance.


\textbf{Epistemic classification.} The numerical values $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ and $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ are \epi{DERIVED} from the intrinsic geometry of the diamond lattice of \emph{el Pleno}, with no free parameters. The identification $S^2 \leftrightarrow n/n_0$ (Bogoliubov-Josephson) is \epi{DERIVED} (\S{}II.2). The absolute maximum density $\rho_{\max,\text{abs}} \approx 2.1773\,\rho_P$ is \epi{DERIVED} as an immediate consequence of the two previous ones. The justification of the diamond lattice itself as the microscopic structure of \emph{el Pleno} has convergent fourfold robustness (\S{}II.3), with strict uniqueness declared as the open thread D1.



% ===== INCORPORATED V4.4: I.7.bis (MODO INVESTIGADOR session) =====
\subsubsection{\texorpdfstring{I.8.9 --- The minimal structure of el Pleno: two properties and one causal structure}{I.8.9 - The minimal structure of el Pleno: two properties and one causal structure}}

\textbf{Origin.} The program declares three primary structural parameters: the density $\rho_P$, the structural speed $c$ and the coherence scale $\ell_P$ (\S{}I.8). From these $G$, $\hbar$ and the rest of the apparatus are derived. A foundational question, posed in the same spirit in which Maxwell showed that the speed of light was not an independent optical datum but the wave speed of the electromagnetic field, is: are the three primary ones quantities of the same type, or do they have a distinct ontological status? The analysis shows that they are not of the same type, and that the correct count of substantive quantities of \emph{el Pleno} is \textbf{two}, not three.

\medskip
\textbf{Two types of dimensional constant.} A physical framework admits two classes of dimensional quantity:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Substantive property} ---a value that the medium possesses and that could coherently be another. One can conceive of a more or less dense \emph{el Pleno}, or of finer or coarser grain. The density $\rho_P$ and the scale $\ell_P$ are of this type: physical contingencies of the medium.
  \item \textbf{Structure} ---a factor that is not a free property but the signature of how the medium organizes its dimensions. The speed $c$ converts length into time; but in a medium with Lorentz covariance (axiom A3), space and time are not two independent domains that $c$ connects from outside: they are the same manifold. The constant $c$ is the dimensional name of that unification, not an additional degree of freedom of the medium.
\end{itemize}

\medskip
\textbf{$c$ is structure, not property.} That $c$ is of the second class is seen in that, in units where $c=1$, the constant disappears from all of physics ($E=mc^2\to E=m$; $ds^2=c^2dt^2-d\mathbf{x}^2\to ds^2=dt^2-d\mathbf{x}^2$). This disappearance is not a computational convenience: it is the recognition that time and space are measured in the same unit because they belong to the same manifold. The numerical value of $c$ ($2.998\times 10^8$ in m/s) is an artefact of having historically calibrated the metre and the second separately, before knowing that they measured directions of a single structure. It is not, therefore, a number that \emph{el Pleno} ``chooses'': it is the measure of its unified causal structure. This explains at the root why $c$ is not derivable in the program ---not because it is one more irreducible primary, but because it is not a substantive property at all, but the structure within which the substantive properties are defined. The correct causal direction, already noted in \S{}III.1.8 (the equality $v=c$ as a verification of Lorentz covariance, not as a derivation of $c$), is consistent with this reading.

\medskip
\textbf{The two substantive properties are irreducible to each other.} With $c$ recognized as structure, two substantive properties remain: $\rho_P$ and $\ell_P$. Does any structural relation of \emph{el Pleno} link them, so that the medium reduces to a single property? The F1 Lagrangian (E2) contains them as independent coefficients ---$\rho_P c^2\ell_P^2$ in the kinetic term, $\rho_P c^2\ell_P^4/40$ in the flexural one, $\rho_P c^2$ in the potential--- without imposing any equation that relates them: $\rho_P$ fixes the energy scale, $\ell_P$ the length scale, and they are distinct physical knobs. The natural candidate to link them ---identifying $\rho_P$ with the ``Planck density'' $m_P/\ell_P^3$--- turns out to be circular: using the identities of the program $G=c^2/(\rho_P\ell_P^2)$ and $\hbar=\rho_P c\,\ell_P^4$, the Planck mass is $m_P=\sqrt{\hbar c/G}=\rho_P\ell_P^3$, so that $m_P/\ell_P^3=\rho_P$ \emph{identically}. The identity links nothing new: it is the same information rewritten, since $m_P$ is constructed with $G$ and $\hbar$, which in turn already contain $\rho_P$ and $\ell_P$. There is no relation in the program that reduces the two properties to one.

\medskip
\begin{tcolorbox}[notabox,title={The minimal structure of el Pleno}]
\emph{El Pleno} is described, at its most fundamental level, by \textbf{two substantive properties} ---the density $\rho_P$ (energy scale) and the coherence scale $\ell_P$ (grain)--- and \textbf{one structure} ---the Lorentz causality, whose dimensional name is $c$. From these two properties and this structure the whole apparatus emerges: $G=c^2/(\rho_P\ell_P^2)$, $\hbar=\rho_P c\,\ell_P^4$, $S_{\min}$, $S_{\max}$, the uncertainty relation, the Chandrasekhar limit, gravity and cosmology. The count is not reducible to fewer: a physical medium that has at once an ultraviolet grain (via $\ell_P$) and elastic rigidity (via $\rho_P c^2$) needs exactly one length scale and one energy scale. Two is the minimum.
\end{tcolorbox}

\medskip
\textbf{The parallel with Maxwell, correctly situated.} Maxwell unified two \emph{phenomena} ---optics and electromagnetism--- and in that unification the speed of light appeared as the wave speed of the field, $c=1/\sqrt{\varepsilon_0\mu_0}$, ceasing to be an independent optical datum. In his case $\varepsilon_0$ and $\mu_0$ were measured independently of optics, so that the coincidence constituted the prediction of a number. PIU-$\Psi$ makes an analogous move but on distinct objects: it unifies two \emph{dimensions} ---space and time--- and $c$ is the name of that unification. The difference is that here $c$ is not predicted as a new number (the rigidity and the causality of F1 use the same $c$, which constitutes an internal-consistency verification, not an independent prediction, \S{}III.1.8); what is gained is the \emph{reclassification} of $c$ from property to structure, and with it the reduction of the count of substantive properties of \emph{el Pleno} from three to two.

\medskip
\textbf{Epistemic classification.} The distinction between substantive property ($\rho_P$, $\ell_P$) and structure ($c$) is \epi{SLD}: it follows from the behaviour of $c$ under change of units and from the Lorentz covariance of axiom A3, already established in the program. The irreducibility of $\rho_P$ and $\ell_P$ to each other is \epi{DERIVED}: it is verified that F1 imposes no relation between them and that the Planck-density candidate is circular. The reduction of the count of substantive properties from three to two is \epi{SLD}. The claim is one of \emph{foundation}, not of \emph{prediction}: it produces neither a new number nor an observational confrontation, but makes precise the minimal structure of the program. The interpretation of $c$ as ``structure of the medium'' is an ontological reading, not a teleological claim: $c$ is not a signal deliberately placed, but the recognizable signature that \emph{el Pleno} has a unified causal manifold.

\medskip
\textbf{Consequence for the program.} The question ``can $c$ be derived?'' is resolved by making precise that $c$ is not the kind of quantity that is derived: it is not a property of \emph{el Pleno} but its causal structure. With this, the fundamental description of the universe under PIU-$\Psi$ reduces to \textbf{two properties with dimensions} ---$\rho_P$ and $\ell_P$--- \textbf{and one geometric structure} ---the Lorentz causality measured by $c$. It is the most economical formulation of the foundation of the program, and it poses a question that admits two readings, and only one of them has an answer. It admits two readings that must not be confused. The \emph{badly} posed question is ``what number did \emph{el Pleno} choose for $\rho_P$ and $\ell_P$?''; this has no answer because it presupposes a category ---the number--- that \emph{el Pleno} does not have. What the medium possesses are ceilings and a grain; the \emph{value} of those ceilings is information about how many of the human rods fit in them, that is, about the relation between \emph{el Pleno} and the human units, not about \emph{el Pleno} in itself (the same reading the manuscript already applies to $c$). The \emph{well}-posed question, and one that is indeed a legitimate open program, is of another class: \textbf{to measure the human value of $\rho_P$ and $\ell_P$ ---their reading in human rods--- from a physics independent} of gravitation and of quantum mechanics, instead of inferring it through $G$ and $\hbar$. It is not deriving a missing number, but calibrating from another route a property that exists with no intrinsic value. The program addresses this partially in its cosmological sector (the freezing of scales in the bounce, \S{}IV) and keeps it as an independent-calibration thread, not as a foundational gap.


\medskip
\begin{tcolorbox}[ecuacionclave,title={The principle of the human rods and the two faces of the ``calibrated''}]
El Pleno has no units. It operates in pure ratios: the bounds of the modulus $S_{\min}$, $S_{\max,\text{geom}}$, the critical point $S^*$, the packing indices, the lattice coefficients ($1/20$, $1/40$, $8/9$, $\sqrt3\pi$) are dimensionless numbers that the medium fixes by its geometry alone. The units ---the metre, the second, the kilogram--- are \textbf{human rods} of measurement: constructs of ours that we introduce at strategic points of the computation with a single purpose, to give comparable meaning to what is computed ---to be able to compare between theories and against real measurements and observations. The unit is not of el Pleno; it is the bridge between el Pleno and our laboratory bench.

\smallskip
This fixes the exact sense of the \epi{CALIBRATED} label throughout the manuscript, which covers two distinct situations, neither of which is a free parameter tuned to force agreement:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Ruler calibration (dimensional anchoring).} The numerical value of the two substantive properties $\rho_P$ and $\ell_P$ ---``how many human rods fit in the grain and in the ceiling of el Pleno''--- is information about the relation between the medium and our units, not about the medium itself. Assigning it (via $G$, $\hbar$, or the independent-measurement thread C-meta-1.b) is the unavoidable act of injecting the human rod in order to compare. It is not a weakness of the theory: it is where, by necessity, the unit enters. El Pleno does not become less derived thereby; it is we who need the rod.
\item \textbf{Positional state quantity.} The present value of magnitudes such as $H_0$, $f_c$ or $n_s$ encodes where the medium is along its trajectory $\bar S(t)$ (\S{}III.8.7). Its \emph{form} is derivable; its \emph{value} is an observational datum, of the same kind as the hour that a clock reads. (The background equation of state is not of this kind: $w=-1$ is exact, geometric.)
\end{itemize}
In both cases the \emph{dimensionless and structural} ---the form, the ratio, the mechanism--- is what the program derives; what is \epi{CALIBRATED} is either the rod with which we measure it, or the position of the medium today. The boundary between \epi{DERIVED} and \epi{CALIBRATED} is, at bottom, the boundary between what el Pleno is and what we need in order to speak of it in numbers.
\end{tcolorbox}

% ===== end V4.4 incorporation =====

\subsubsection{\texorpdfstring{I.8.9-bis --- Time as arc of accumulated change}{I.8.9-bis - Time as arc of accumulated change}}

\textbf{Origin and purpose.} With $c$ established as causal structure (\S{}I.8.9) ---the dimensional name of the fact that space and time are directions of one and the same manifold---, it remains to fix with precision what time \emph{is} in the program, because on it depend the cosmological reading of $H_0$ (\S{}III.8.7), the dissolution of the hydrodynamic singularity (\S{}III.9.10) and the closure of the ``problem of time'' of quantum cosmology (\S{}III.2). Time is not a background parameter that el Pleno inhabits: it is a \emph{derived} magnitude, built from the very change of the medium. This subsection gathers the definition, proves that it is not circular, and shows that the three readings of time that the manuscript employs ---accumulated change, proper time of the geodesic, and Page-Wootters correlation--- are the same notion in three contexts, not three rival definitions.

\medskip
\textbf{The definition.} Axiom A1 fixes that the energy of el Pleno is finite and its rate of internal change is bounded: there exists a maximal capacity for change, whose dimensional name is $c$ (\S{}I.8.9). Let $C(\lambda)$ be the rate of change of the field along a path parameter $\lambda$ ---how fast $\Psi$ reconfigures itself---, with $0\le C\le c$ by A1. \textbf{Time} is the \emph{arc of accumulated change} along the trajectory of the medium, normalized by the ceiling:
\begin{equation}
\tau \;=\; \frac{1}{c}\int_\gamma C(\lambda)\,d\lambda, \tag{E-time-1}
\end{equation}
that is, the amount of change that el Pleno has effected along its trajectory $\gamma$, measured against the maximal change that $c$ fixes. Time does not count ``instants'' of an external clock: it counts how much the medium has changed. Where the medium does not change ($C=0$) no time elapses; where it changes maximally ($C=c$) time advances at its full rate. This is the structural reason why the massless mode ---light--- travels at $c$: it saturates the ceiling of change, and its proper time vanishes, $C=c\Rightarrow d\tau_\text{proper}=0$ along its own null trajectory.

\medskip
\textbf{Why it is not circular.} The immediate objection ---defining time with $c$, which is a speed, and a speed is space over time--- is deactivated by following the order of construction, not that of the units. The chain is strictly ordered:
\begin{enumerate}[leftmargin=1.6em,itemsep=2pt]
\item Change $C$ is \emph{internal to the energy} and requires no clock: it is how much $\Psi$ reconfigures, a notion prior to any temporal measure (A1, \S{}I.8.9).
\item From the boundedness of $C$ one takes its \emph{extreme} $c$ ---the ceiling of change---, which exists as a bound without presupposing time.
\item Time $\tau$ is \emph{defined} as the accumulated change normalized by that ceiling (E-time-1).
\end{enumerate}
Time is what appears \emph{last}: it does not enter steps 1 and 2. That in human units $c$ is expressed as ``metres per second'' is an artefact of having calibrated the metre and the second separately (\S{}I.8.9), not a sign that $c$ presupposes time. In the pure ratios of el Pleno, $c$ is the ceiling of a quantity ---change--- that is defined without a clock. The non-circularity is \epi{DERIVED}: the construction exhibits the order and no step prior to time uses it.

\medskip
\textbf{The three readings are one single notion.} The manuscript reads time in three contexts; the mathematics shows that they coincide:

\emph{(i) Accumulated change $=$ proper time of the geodesic.} The effective metric $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$ is not a primitive structure but the reading that the perturbations make of the state of the medium (\S{}III.1.7). The proper time that an excitation extremizes, $\tau_\text{proper}=\int\!\sqrt{g_{\mu\nu}^\text{eff}\,dx^\mu dx^\nu}/c$ (\S{}III.1.5), measures precisely the accumulated change of the field along that trajectory: the integrand $\sqrt{g^\text{eff}_{\mu\nu}\dot x^\mu\dot x^\nu}$ \emph{is} the rate of change $C$ of the medium read along the curve. Therefore (E-time-1) and the proper time of the geodesic are the same integral, written in two languages ---kinematic and metric---; the geodesic does not introduce a new definition of time, but the same one over the metric that the field itself defines. Exact coincidence, \epi{DERIVED}.

\emph{(ii) The Page-Wootters correlation is the mechanism that makes it observable.} The relational reading (\S{}III.2) is not a third definition of the \emph{magnitude} time, but the answer to a different question: how does an observer \emph{internal} to el Pleno read the accumulated change, without an external clock that the program does not admit? The Page-Wootters answer: a subsystem of the field acts as a clock, and the observed time is the correlation between its state and that of the rest. That clock-subsystem measures nothing other than its own accumulated change (E-time-1); the correlation is the bridge by which the accumulated change \emph{of one part} becomes the temporal variable \emph{of the rest}. Page-Wootters is thus the relational implementation of the same notion ---it neither contradicts nor duplicates it---, and it is what allows time to be internal to the medium and not a background parameter. Consistency, \epi{DERIVED}; that a subsystem can fulfil the role of clock is \epi{SLD} (it requires a subsystem whose change is monotone in the regime considered).

\medskip
\begin{tcolorbox}[ecuacionclave,title={Time in PIU-$\Psi$}]
Time is the arc of accumulated change of el Pleno along its trajectory, normalized by the ceiling of change $c$ (E-time-1). It is not a stage that the medium inhabits, but a magnitude made of the very change of the medium: where el Pleno does not change, no time elapses. The definition is not circular ---change is prior to time and requires no clock (\S{}I.8.9)---, and its three readings coincide: the \emph{accumulated change} (kinematics) is the \emph{proper time} of the geodesic (metric, \S{}III.1.5), because the metric is the reading of the state of the field; and the \emph{Page-Wootters correlation} (\S{}III.2) is the relational mechanism that makes that change observable from within, without an external clock. One single notion, three contexts.
\end{tcolorbox}

\medskip
\textbf{Consequences that depend on this definition.} Three results of the program rest on time being accumulated change and not a background parameter. First, the \emph{positional} reading of $H_0$ (\S{}III.8.7): if time is accumulated change of the medium along its relaxation trajectory, ``now'' is a position on that curve, and $H_0$ is the hour that this clock reads, not a timeless constant. Second, the dissolution of the hydrodynamic singularity (\S{}III.9.10): a blow-up would require infinite change, forbidden by $C\le c$ (A1), \emph{and} in finite time, impossible because time is made of change and infinite change does not accumulate in finite accumulated change. Third, the closure of the Wheeler-DeWitt ``problem of time'' (\S{}III.2): the absence of an external temporal parameter does not eliminate evolution, because time was never external ---it is the internal Page-Wootters correlation. The three are the same definition read in three sectors.

\medskip
\textbf{Epistemic classification.} The definition of time as arc of accumulated change (E-time-1) is \epi{SLD}: it follows structurally from A1 (bound on change) and from the reclassification of $c$ as causal structure (\S{}I.8.9), without additional algebra. The non-circularity is \epi{DERIVED}: the chain $C\to c\to\tau$ exhibits the order and no step prior to time presupposes it. The identity between accumulated change and proper time of the geodesic is \epi{DERIVED} (same integral over the effective metric, \S{}III.1.5, \S{}III.1.7). The Page-Wootters reading as a relational mechanism of the same notion is \epi{DERIVED} in its consistency and \epi{SLD} in the assumption of a clock-subsystem of monotone change. The program asserts the \emph{definition} and its coherence across sectors; it does not claim to have derived a thermodynamic ``arrow of time'', which rests on the second law of the medium (\S{}III.9) and is declared where it belongs.

% ===== end time insertion V5.0 =====


\subsubsection{I.8.10 --- Relations between parameters: el Pleno as an over-constrained medium}

\textbf{Key observation.} The parameters of \emph{el Pleno} articulated in \S{}I.8.2 to \S{}I.8.8 --- $\rho_P$, $c$, $\ell_P$, $\gamma_\kappa$, the constants $S_{\min}^\text{estr}$ and $S_{\max}^\text{geom}$ --- appear, at first sight, to be many. But \textbf{they are not independent of one another}: the structural identities of the program relate them, significantly reducing the number of free parameters of \emph{el Pleno}.


\textbf{Structural identities.} The relations derived throughout the program are:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{4.40cm}L{4.40cm}L{4.40cm}}
\toprule
\textbf{Identity} & \textbf{Form} & \textbf{Section} \\
\midrule\endhead
Gravitational constant & $G = c^2/(\rho_P\, \ell_P^2)$ & \S{}I.7.5 \\
Flexural rigidity & $\gamma_\kappa = \rho_P c^2 \ell_P^4 / 40 = \hbar c/40$ & \S{}I.7.7 \\
Structural ground-state \emph{el Pleno} & $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ & \S{}I.8.8 \\
Critical configuration & $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ & \S{}I.8.8 \\
\bottomrule
\end{longtable}}

\textbf{Consequence: primary and derived parameters.} From these relations it follows that \emph{el Pleno} is completely characterized by three primary structural inputs:


\[
\boxed{\rho_P, \quad c, \quad \ell_P}
\]

All the other parameters of the F1 Lagrangian --- including $G$, $\gamma_\kappa$, $\hbar$ (whose derivation is developed in later parts), $E_0$ (energy scale of the potential), $S_{\min}^\text{estr}$, $S_{\max}^\text{geom}$ --- are \textbf{derived} from these three by structural identities or the intrinsic geometry of the microscopic lattice.


\textbf{Two of these three are properties; the third is structure.} Although the three enter as inputs of the derivational apparatus, they do not have the same ontological status, and \S{}I.8.9 makes it precise: $\rho_P$ and $\ell_P$ are \emph{substantive properties} ---values that the medium possesses and that could coherently be others---, while $c$ is the \emph{causal structure} within which those properties are defined (the dimensional name of the space-time unification, which disappears in units $c=1$). For that reason the count of \emph{substantive quantities} of \emph{el Pleno} is \textbf{two}, not three; and for that reason $c$ does not figure as a derivation thread, while the \textbf{independent calibration of the human value} of $\rho_P$ and $\ell_P$ ---their reading in human rods, measured from a physics distinct from the gravitational and the quantum--- is indeed an open program (thread C-meta-1.b, below). This thread does not ask to derive a number that \emph{el Pleno} possesses (\emph{el Pleno} has no numbers, only ceilings and a grain), but to measure from another route how many of the human units fit in those ceilings. The most economical description of the foundation of the program is, then, \textbf{two properties with dimensions} ($\rho_P$, $\ell_P$) \textbf{and one causal structure} ($c$).


\textbf{\emph{El Pleno} as an over-constrained medium.} This feature is one of the most significant structural results of the program. Compared with the Standard Model of particle physics (which has approximately 19 free parameters, all calibrated against observation without derivation from more basic principles), PIU-$\Psi$ proposes reducing the count to \textbf{three primary inputs} of the medium ---two substantive properties and one causal structure, as just made precise---, from which all the others emerge as structural consequences.


This has an important epistemic implication. A theory with many free parameters has a greater capacity to fit any observational datum (overfitting), which decreases its predictive power. A theory with few primary parameters and many derived consequences has less fitting flexibility, which increases its predictive power --- because its predictions are falsifiable: if observation contradicts the structural consequences, the program falls.


\textbf{Connection with Problem 5 of \S{}I.2.6.} Problem 5 articulated in \S{}I.2 (unexplained fundamental constants) receives a direct partial answer from the program: under PIU-$\Psi$, the fundamental constants \textbf{are not non-derived inputs} --- they are structural consequences of the parameters of \emph{el Pleno}. The complete derivation of each constant (G already derived in \S{}I.7.5; $\hbar$ derived in later parts; $\alpha$ and others as tasks of the research program) is the subject of successive chapters of the manuscript.


\textbf{Related open threads.} Clarifying exactly which parameters of PIU-$\Psi$ are directly measured and which require indirect calibration is important for the epistemic honesty of the program:


\begin{itemize}[leftmargin=1.4em]
  \item $c$ is \textbf{directly measured}: the speed of light is accessible to experimental measurement with extreme precision and constitutes one of the most solid observational values in physics.
  \item $\hbar$ is \textbf{directly measured}: the Planck constant (which appears in F1 through the Klauder term) is measured in numerous experimental contexts and, since 2019, its value exactly defines the kilogram in the International System of Units.
  \item $\rho_P$ and $\ell_P$ are not directly measured --- their microscopic scale puts them beyond the reach of current experimental measurements. Their numerical value is assigned in the present version of the program by \textbf{indirect calibration}: it requires fixing an additional observational scale as a reference. The equivalent options for this additional scale are the gravitational constant $G$, the Planck mass, or the Planck length --- all \textbf{conventionally interchangeable} via the standard dimensional identities between $c$, $\hbar$ and one of them. PIU-$\Psi$ privileges none: using $G$ as a reference, for example, does not mean that the program starts from General Relativity --- it means that $G$ is one of the accessible observational scales that allows numerically anchoring $\rho_P$ and $\ell_P$.
\end{itemize}

\textbf{Response to the objection of Planckian tautology.} The objection ``do $G$ and $\hbar$ really derive, or are they internal consistency of Planck units?'' is answered with the detail it deserves in \S{}I.7.5 (non-circularity of $G$) and \S{}I.8.5 (how $\ell_P = \sqrt{\hbar G/c^3}$ satisfies the two identities at once). The argument, in synthesis, rests on three points already established there: the triplet $(\rho_P, \ell_P, c)$ is \emph{dimensionally independent} (three distinct physical dimensions that no identity of the program collapses); the dynamics of F1 \emph{selects a unique combination} for each constant among the infinitely many dimensionally compatible ones ($G = c^2/\rho_P\ell_P^2$ via Sakharov, $\hbar = \rho_P c\,\ell_P^4$ via quantization of the phase mode), so that the agreement to $<0.1\%$ is falsifiable and not tautological; and the \emph{independent calibration} of the human value of $(\rho_P, \ell_P)$ from non-gravitational and non-quantum physics remains a declared open program (thread C-meta-1.b, \S{}V.3), not a precondition of the \texttt{\epi{DERIVED}} status of $G$ and $\hbar$.

\textbf{Complementary statistical closure.} To the three previous arguments is added the confrontation of the numerological-coincidence hypothesis with the whole set of results of the programme (\S{}V.2.6): fifteen results admitted under declared criteria, read with fifteen instruments that share no measurement mechanism, give a Bayes factor of $3.4\times10^4$ against that hypothesis under strong dependence, and above $65$ under the most punitive assumption. The enumeration of the space of algebraic formulas in \S{}III.6.5 answers separately the narrow version of the objection, the one concerning an isolated ratio. This evidence complements the structural arguments (1)-(3) without replacing them.


Calibrating the human value of $\rho_P$ and $\ell_P$ from an independent physics ---without requiring an external experimental reference beyond $c$ and $\hbar$--- is an objective of the research program (thread C-meta-1.b), declared explicitly as an open thread in \S{}V.3. This objective is not a correction of a gap: the apparatus is already consistent and complete for its derivational purposes. What the thread proposes is to \textbf{anchor} the two properties to a rod independent of gravitation and of quantum mechanics, reinforcing the structural autonomy of the program.


\subsubsection{I.8.11 --- Synthesis: complete table of el Pleno as a physical medium}

Through \S{}I.8.2 to \S{}I.8.10, each parameter and coefficient that appears in F1 is reread as a specific structural property of \emph{el Pleno} as a continuous physical medium. The synthesis of the complete table is the following:


{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{2.64cm}L{2.64cm}L{2.64cm}L{2.64cm}L{2.64cm}}
\toprule
\textbf{Property} & \textbf{Symbol} & \textbf{Physical function} & \textbf{Canonical value} & \textbf{Nature} \\
\midrule\endhead
Fundamental density & $\rho_P$ & Mass-energy density of the ground-state \emph{el Pleno} & $\sim 5.16 \times 10^{96}\ kg/m^3$ & Primary parameter \textbf{[CALIBRATED, indirect]} \\
Microscopic scale & $\ell_P$ & Intrinsic scale of the medium & $\sim 1.62 \times 10^{-35}\ m$ & Primary parameter \textbf{[CALIBRATED, indirect]} \\
Structural speed & $c$ & Propagation speed of modes & $\approx 2.998 \times 10^8\ m/s$ & Primary input; causal structure \textbf{[MEASURED directly]} \\
Gravitational constant & $G = c^2/(\rho_P \ell_P^2)$ & Emergent gravitational coupling & $6.678 \times 10^{-11}\ m^3 kg^{-1} s^{-2}$ & Derived \\
Flexural rigidity & $\gamma_\kappa = \rho_P c^2 \ell_P^4/40 = \hbar c/40$ & Resistance to high-frequency curvature & $\approx 7.90 \times 10^{-28}\ J \cdot m$ & Derived \\
Quantum IR wall & $V_q^\text{dens}(S)$ & Structural support against $S \to 0$ & $\propto 1/S^2$ & Derived \\
Total potential & $V(S)$ & Structural energy of the medium & $W_\text{bare} + V_q + V_\text{topo}$ (tripartite) & Derived structure \\
Ground-state \emph{el Pleno} & $S_{\min}^\text{estr}$ & Configuration of minimal excitation & $12(\pi^2-4)/\pi^4 \approx 0.7231$ & Derived from geometry \\
Critical configuration & $S_{\max,\text{geom}}^2$ & Upper structural limit & $(8/3)\sqrt{2/3} \approx 2.1773$ & Derived from geometry \\
Radial mass of the mode of \emph{el Pleno} & $m_\text{radial} c^2$ & Minimum energy of radial excitation of the medium & $\approx 4.44\,E_P$ (Planckian) & Derived \\
Absolute maximum density & $\rho_{\max,\text{abs}}$ & Hard structural density bound & $S_{\max,\text{geom}}^2 \rho_P \approx 2.1773\,\rho_P$ & Derived \\
\bottomrule
\end{longtable}}

\textbf{\emph{El Pleno} as a physical object.} The above table is what the program offers as a \textbf{description of \emph{el Pleno} as a physical medium}. It is not a heterogeneous list of parameters --- it is the complete characterization of a continuous medium with all the properties that continuous-media mechanics would attribute to any physical substance: density, microscopic scale, structural speed, flexural rigidity, structural support, deformation energy, stable configurations.


The difference from any known macroscopic physical medium is that \emph{el Pleno} is \textbf{the only physical medium that constitutes the entire universe}. Its structural properties are not a consequence of its microscopic composition (it is not made of atoms, electrons, or particles) but are intrinsic properties of the fundamental medium.


\textbf{From properties to state.} This characterization fixes \textbf{what} \emph{el Pleno} is: its structural properties. It remains to specify \textbf{in what state} the medium is at each instant ---how its configuration is described mathematically by the field $\Psi = S \cdot e^{i\theta}$, what function modulus and phase fulfil, and how that state evolves under F1. Properties and state, together, complete the presentation of \emph{el Pleno} as a physical object.


\medskip\hrule\medskip

\subsection{\texorpdfstring{\S{}I.9 --- $\Psi$ as the state of el Pleno}{Sec.I.9 - Psi as the state of el Pleno}}

\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:} section dedicated to presenting the mathematical object $\Psi$ as the state of \emph{el Pleno}. It articulates the notation, the physical tradition in which it inscribes itself (Schrödinger, Gross-Pitaevskii, Ginzburg-Landau, Higgs), the decomposition $\Psi = S\cdot e^{i\theta}$ with the specific functions of the modulus and the phase, and the fundamental distinction from the probabilistic wavefunction of standard quantum mechanics. A section without new derivations, but with ontological and notational clarification critical for the rest of the manuscript.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{I.9.1 --- The field $\Psi$ as a physical object}{I.9.1 - The field Psi as a physical object}}

The field $\Psi$ has entered up to here mainly as a mathematical object: a complex field over \emph{el Pleno}, motivated in its form $\Psi = S\,e^{i\theta}$ by the genesis (\S{}I.4) and formalized as the field of the Lagrangian in \S{}I.7.2 (E1). As a \textbf{physical object} it is the state of \emph{el Pleno} at each point: how much energy there is and how it changes, read at once. What follows makes precise what $\Psi$ represents physically, in what tradition of complex scalar fields it inscribes itself, why it is complex and not real, and what distinct functions its modulus $S$ and its phase $\theta$ fulfil.


An ontological distinction is key: under PIU-$\Psi$, the field $\Psi$ \textbf{is not the wavefunction of standard quantum mechanics}, even though it shares notation and mathematical decomposition. Mistakenly identifying them would confuse the ontological status of the program, on which \S{}I.10, \S{}I.11 and all of Part II operate.


\subsubsection{\texorpdfstring{I.9.2 --- Notation $\Psi$: convention of the program}{I.9.2 - Notation Psi: convention of the program}}

The total field of \emph{el Pleno} is denoted \textbf{$\Psi$} (capital psi), traditional notation in physics for macroscopic quantum fields ---from the Schrödinger wavefunction to the order parameters of Gross-Pitaevskii and Ginzburg-Landau---, which inscribes the program in a recognized tradition. The use of the \textbf{capital} marks the ontological distinction from the lowercase $\psi$ of the quantum wavefunction (probability amplitude): the convention is strict, $\Psi$ is always the physical field of \emph{el Pleno} (explicit distinction in \S{}I.9.9). The program takes its name from this letter ---\textbf{PIU-$\Psi$}---, reflecting that everything it describes is a configuration of this single $\Psi$.


\textbf{Arguments of the field.} $\Psi$ is evaluated at points of \emph{el Pleno}, parametrized by coordinates $x^\mu$ with $\mu = 0, 1, 2, 3$: the coordinate $x^0$ is the temporal dimension of the medium, the coordinates $x^1, x^2, x^3$ are the spatial dimensions. The complete writing is:


\[
\Psi(x^\mu) = \Psi(x^0, x^1, x^2, x^3) = \Psi(t, \mathbf{x})
\]

where $t = x^0/c$ has units of time and $\mathbf{x} = (x^1, x^2, x^3)$ is the trio of spatial coordinates. The convention $x^0 = ct$ --- using $c$ to carry the temporal coordinate into units of length --- is standard in the covariant formulation of field theory, and is followed in the present manuscript. This convention was already used in the dimensional verification of \S{}I.7.9.


\textbf{Codomain.} $\Psi$ takes values in the \textbf{field of complex numbers}:


\[
\Psi: \mathcal{M} \to \mathbb{C}
\]

where $\mathcal{M}$ is the cosmological manifold of \emph{el Pleno}. At each point of \emph{el Pleno}, $\Psi$ takes a complex value: it has a modulus (non-negative real part) and a phase (angular argument). The justification of why $\Psi$ must be complex, not real, is developed in \S{}I.9.4.


\subsubsection{\texorpdfstring{I.9.3 --- Physical tradition of the complex scalar field $\Psi$}{I.9.3 - Physical tradition of the complex scalar field Psi}}

Complex scalar fields are not an invention of the PIU-$\Psi$ program. They have a century-long history in physics, during which they have appeared as the description of diverse physical systems. Four paradigmatic cases are particularly relevant for understanding the choice of PIU-$\Psi$:


\textbf{Case 1: The Schrödinger wavefunction (1926).} Erwin Schrödinger introduced in 1926 a dynamical equation for a complex function $\psi$ that describes the quantum state of a particle. This wavefunction --- a complex scalar field defined over physical space --- quickly showed itself capable of reproducing the results of the quantum mechanics of Bohr and Heisenberg, and became one of the central objects of 20th-century physics. The physical interpretation of $\psi$ was debated for years: Schrödinger initially considered it a real physical field (ontological interpretation), but Max Born proposed in 1926 the probabilistic interpretation that became canonical --- $|\psi|^2$ is the probability density of finding the particle. Under the Born interpretation, $\psi$ is not a physical field but a probability amplitude. This is the critical difference from the $\Psi$ of PIU-$\Psi$, articulated in \S{}I.9.9.


\textbf{Case 2: The Gross-Pitaevskii equation (1961).} In the context of Bose-Einstein condensates, Eugene Gross and Lev Pitaevskii independently developed a dynamical equation for a complex field $\Psi$ that describes the \textbf{collective} state of many bosonic particles in condensation. Unlike the Schrödinger wavefunction (which describes one particle), the $\Psi$ of Gross-Pitaevskii describes a macroscopic quantum system --- a condensate of millions of atoms that behave collectively as a single entity. The modulus $|\Psi|^2$ here is not a probability density but the \textbf{real density of particles in the condensate}. This is a case where $\Psi$ is a \textbf{real physical field}, not a probability amplitude: it has defined values at each point of space, experimentally observable. The $\Psi$ of \emph{el Pleno} under PIU-$\Psi$ inscribes itself more directly in this tradition.


\textbf{Case 3: The Ginzburg-Landau order parameter (1950) in superconductivity.} Vitaly Ginzburg and Lev Landau introduced in 1950 a complex scalar field $\Psi$ to describe the superconducting state of a metal below its critical temperature. This ``order parameter'' has a non-zero modulus in the superconducting phase and zero modulus in the normal phase; its phase is related to superconducting currents and effects such as the quantization of magnetic flux. The Ginzburg-Landau theory is successful at describing macroscopic phenomena of superconductivity such as the penetration length and the coherence length, without requiring solving the microscopic problem of the Cooper pairs. As in the Gross-Pitaevskii case, the Ginzburg-Landau order parameter is a \textbf{physical field}, not a probabilistic amplitude.


\textbf{Case 4: The Higgs field (1964) in the Standard Model.} Peter Higgs, François Englert and others proposed in 1964 the existence of a complex scalar field $\Phi$ --- the Higgs field --- whose non-zero value in the vacuum spontaneously breaks the electroweak symmetry and provides mass to the fundamental particles of the Standard Model. The Higgs boson, detected at the Large Hadron Collider in 2012, is the quantum excitation of this field. The Higgs field is a \textbf{fundamental physical field} --- not a probabilistic amplitude but an ontological entity of the Standard Model --- and its mechanism of spontaneously broken symmetry (Higgs mechanism) has structural resonances with the dynamics of \emph{el Pleno} under PIU-$\Psi$.


\textbf{Position of the $\Psi$ of \emph{el Pleno} in this tradition.} The field $\Psi$ of PIU-$\Psi$ inscribes itself in this tradition of physical complex fields: it shares with Gross-Pitaevskii, Ginzburg-Landau and Higgs the character of a \textbf{real physical field}, not a probabilistic amplitude. But PIU-$\Psi$ extends the idea further than any of the antecedents: \emph{el Pleno} is not a confined terrestrial condensate, nor the order parameter of a specific material substance, nor the Higgs field that coexists with other fields of the Standard Model. \emph{El Pleno} is the \textbf{single fundamental physical field} from which all the phenomenology of the cosmos emerges. The Schrödinger-Gross-Pitaevskii-Ginzburg-Landau-Higgs tradition offers the mathematical tool (complex scalar field); PIU-$\Psi$ proposes raising that tool to the fundamental ontology of the universe.


\subsubsection{I.9.4 --- Why a complex and not a real field}

Why is \emph{el Pleno} described by a complex scalar field, not a real one? This is a legitimate question --- one could imagine a simpler program based on a single real field. The answer has several convergent dimensions:


\textbf{Reason 1: axiom A5 requires a physical phase.} Axiom A5 established in \S{}I.5.6 postulates a global U(1) symmetry of \emph{el Pleno}, which produces via Noether's theorem a conserved current (identified in later parts with the electric current). A global U(1) symmetry only has physical content if \textbf{there exists a global phase of the field} that can be transformed --- a real field does not admit such a transformation. Therefore, axiom A5 requires that \emph{el Pleno} have phase structure, which forces $\Psi$ to be complex.


\textbf{Reason 2: the U(1) Noether current requires two degrees of freedom.} The conserved current of the U(1) symmetry has the form derived in \S{}I.7.10:


\[
\partial_\mu(S^2 \partial^\mu \theta) = 0
\]

This equation involves both the modulus $S$ and the phase $\theta$ of the field. That is, the induced gauge dynamics of \emph{el Pleno} operates on two degrees of freedom per point of the medium, not one. A real field would have only one degree of freedom per point, insufficient to sustain the gauge dynamics.


\textbf{Reason 3: convergent physical precedent.} As discussed in \S{}I.9.3, all the macroscopic physical fields analogous to \emph{el Pleno} in known physics (BEC, superconductors, Higgs) are \textbf{complex fields}. The complexity of $\Psi$ is not an eccentric choice but a recognized structural pattern: every time nature realizes a macroscopic quantum medium, it describes it by a complex field. Postulating a complex \emph{el Pleno} is coherent with all the experimental precedent.


\textbf{Reason 4: possibility of topological charges.} Configurations of the field where the phase $\theta$ has non-trivial topology (winding) produce stable \textbf{topological defects} --- vortices, monopoles, solitons. These topological defects are a rich mathematical object that the program identifies with the observed particles (developed in \S{}III.5). Without phase structure, such topological defects could not exist, and the particles would have to be postulated as separate entities instead of emerging from \emph{el Pleno}.


\textbf{Conclusion.} The complexity of $\Psi$ is not mathematical decoration: it is a \textbf{structural requirement} of the program derived from the axioms and from physical precedent. \emph{El Pleno} described by a real field could not sustain the symmetries, conservations and topological configurations that produce the observed phenomenology.


\subsubsection{\texorpdfstring{I.9.5 --- The decomposition $\Psi = S\cdot e^{i\theta}$: the modulus S}{I.9.5 - The decomposition Psi = S.exp(i theta): the modulus S}}

Any complex number can be written in polar form: as a non-negative modulus multiplied by a complex exponential of phase. Applied to the field $\Psi$ of \emph{el Pleno}, this canonical decomposition produces:


\begin{equation}
\Psi(x^\mu) = S(x^\mu) \cdot e^{i\theta(x^\mu)} \tag{E1, repeated from \S{}I.7.2}
\end{equation}

where $S$ and $\theta$ are both \textbf{real} scalar fields defined over \emph{el Pleno}. The decomposition is mathematically equivalent to the complex formulation $\Psi$ but makes explicit the two distinguishable physical contributions of the field.


The \textbf{modulus $S(x^\mu)$} is a non-negative real function, $S \geq 0$. Its physical function is the following:


\textbf{Function 1 --- Structural information of the medium.} The value of $S$ at each point of \emph{el Pleno} encodes the \textbf{state of structural excitation of the medium} at that point. The ground-state \emph{el Pleno} (the ``vacuum'') corresponds to $S = S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4 \approx 0.7231$ (canonical dimensionless value derived in \S{}I.8.8). Configurations with $S$ different from $S_{\min}^\text{estr}$ represent structural deformations of the medium relative to its fundamental state.


\textbf{Function 2 --- Support of material configurations.} The particles observed in nature are identified under PIU-$\Psi$ with \textbf{localized solitonic configurations} of the field $S$: regions of \emph{el Pleno} where $S$ has a specific profile that persists stable in time. The mass of a particle corresponds to the total energy associated with the profile of $S$ that defines it. This identification is developed in \S{}III.5.


\textbf{Function 3 --- Support of the propagation modes.} Small perturbations of the field $S$ around the ground-state \emph{el Pleno} propagate as waves in the medium. The propagation speed of these waves, derived from F1, is $c$. The perturbations of the modulus $S$ are channelled into distinct types of physical modes (longitudinal modes of the medium, transverse tensor modes, etc.), each with its specific properties developed in \S{}III.3.


\textbf{Restriction $S \geq 0$ and its operational consequence.} That the modulus be non-negative is a \textbf{structural} property of \emph{el Pleno}, not a mathematical convention. The standard canonical Heisenberg quantization (based on operators defined over the whole real line) does not apply to the modulus $S$; instead, affine Klauder quantization is required, which is designed precisely for configuration manifolds of the half-line type. This affine quantization produces, as derived in \S{}I.7.8, the Klauder quantum potential that acts as the IR wall of \emph{el Pleno}. The existence of the IR wall is thus a direct consequence of the property $S \geq 0$.


\textbf{Dimensions.} The modulus $S$ is \textbf{dimensionless}. The choice of dimensionlessness is a convention of the program: the physical dimensions (energy, length, etc.) are carried in the coefficients of F1 ($\rho_P$, $\ell_P$, $c$, $\hbar$), and the field $S$ is a dimensionless function that encodes the relative degree of excitation of the medium.


\subsubsection{\texorpdfstring{I.9.6 --- The decomposition $\Psi = S\cdot e^{i\theta}$: the phase $\theta$}{I.9.6 - The decomposition Psi = S.exp(i theta): the phase theta}}

The \textbf{phase $\theta(x^\mu)$} is a real function with values in $[0, 2\pi)$ (or, equivalently, in $\mathbb{R}$ modulo $2\pi$, because the phase is periodic). Its physical function is the following:


\textbf{Function 1 --- Support of the global U(1) symmetry.} The U(1) symmetry of axiom A5 acts exclusively on the phase: a global U(1) transformation takes the form $\theta \to \theta + \alpha$ with $\alpha$ constant, leaving the modulus $S$ invariant. The phase is thus the ``conjugate variable'' of the symmetry: what the symmetry can transform is precisely the phase, not the modulus.


\textbf{Function 2 --- Generator of conserved charges.} By Noether's theorem, the U(1) symmetry on the phase produces a conserved current $\partial_\mu J^\mu = 0$ with $J^\mu \propto S^2 \partial^\mu \theta$. This current is identified in the low-energy effective limit with the \textbf{electric current} of standard electromagnetism, as will be developed in \S{}III.4. The electric charges observed in nature are Noether charges associated with specific configurations of the field $\theta$.


\textbf{Function 3 --- Support of topological configurations.} Configurations of \emph{el Pleno} where the phase $\theta$ has \textbf{non-trivial topology} --- where $\theta$ cannot be defined globally as a continuous function, but has ``winding'' around defects --- produce topologically stable structures of the medium. These topological defects are identified under PIU-$\Psi$ with configurations of quantized electric charges and, in tripartite Borromean configurations, with the hadronic particles (baryons). Developed in \S{}III.5.


\textbf{Function 4 --- Support of the induced gauge sector.} Under PIU-$\Psi$, the photon emerges as a \textbf{mode of the phase sector} of \emph{el Pleno}: a specific excitation of the field $\theta$ projected onto the residual U(1) sector of the ground-state \emph{el Pleno}. The observed electrodynamics is not postulated as a separate theory in the program; it is derived as the effective description of the behaviour of $\theta$ in the present relaxed regime. Developed in \S{}III.4.


\textbf{Dimensions and periodicity.} The phase $\theta$ is \textbf{dimensionless}, with the additional restriction of being defined modulo $2\pi$ (because the exponential $e^{i\theta}$ is invariant under $\theta \to \theta + 2\pi$). This periodicity is what admits non-trivial topology: configurations where $\theta$ varies from $0$ to $2\pi$ around a point produce topologically stable winding, the basis of the defects discussed in function 3.


\subsubsection{I.9.7 --- Why modulus and phase are the natural variables}

A mathematical alternative to the decomposition $\Psi = S \cdot e^{i\theta}$ would be to write $\Psi = \Psi_R + i \Psi_I$ with $\Psi_R$ and $\Psi_I$ the real and imaginary parts. The two decompositions are mathematically equivalent but \textbf{physically they are not}: they offer a distinct separation of the degrees of freedom. The choice of $(S, \theta)$ over $(\Psi_R, \Psi_I)$ is deliberate and obeys the following reasons:


\textbf{Reason 1: physical separation of structural functions.} The decomposition $(S, \theta)$ separates \textbf{the structural information of the medium} (encoded in $S$) from \textbf{the gauge information of the medium} (encoded in $\theta$). This separation is physically significant: the modulus and the phase have different functions, respond to different symmetries, and couple with the dynamics of the medium in a different way. The decomposition $(\Psi_R, \Psi_I)$, in contrast, mixes the two contributions in a way with no clear physical meaning.


\textbf{Reason 2: transparency of the U(1) symmetry.} Under the decomposition $(S, \theta)$, the U(1) symmetry of axiom A5 acts trivially on $\theta$ (by translation) and leaves $S$ invariant. The action is \textbf{manifest and separable}. Under $(\Psi_R, \Psi_I)$, the U(1) symmetry acts as a rotation in the plane $(\Psi_R, \Psi_I)$, mixing the two components. The transparency of the symmetry is lost.


\textbf{Reason 3: natural reading of topological configurations.} The topological defects of the field $\Psi$ (vortices, monopoles, three-dimensional solitons) are naturally described in terms of the winding of the phase $\theta$ around the defect. This description requires $\theta$ as an explicit variable. In terms of $(\Psi_R, \Psi_I)$, the same defects require much more complex and obscure descriptions.


\textbf{Reason 4: physical precedent.} As discussed in \S{}I.9.3, all the physical complex fields analogous to \emph{el Pleno} in known physics (BEC, superconductors, Higgs) are described by a modulus-phase decomposition, not real part-imaginary part. This is not an accident: macroscopic quantum physical systems have physics separable into modulus (density/order) and phase (current/gauge), and the decomposition $(S, \theta)$ reflects that separability.


\textbf{Conclusion.} The decomposition $\Psi = S \cdot e^{i\theta}$ is not an aesthetic choice but the \textbf{physically appropriate decomposition} of the state of \emph{el Pleno}. The two variables $S$ and $\theta$ have distinct physical meaning and are used throughout the manuscript as the fundamental variables of the program.


\subsubsection{\texorpdfstring{I.9.8 --- The dynamics of $\Psi$ under F1: how the state evolves}{I.9.8 - The dynamics of Psi under F1: how the state evolves}}

The state of \emph{el Pleno} at each instant is given by the configuration of the field $\Psi$ throughout the medium at that instant. The temporal evolution of this state is governed by F1 (presented in \S{}I.7) through the Euler-Lagrange equations (derived in \S{}I.7.10).

\textbf{Evolution equation of the modulus.} The modulus component evolves under the F2 equation (equation E6 of the equation index, derived in \S{}I.7.10):


\[
\rho_P c^2 \ell_P^2 \left[ \Box S - S(\partial_\mu \theta)(\partial^\mu \theta) \right] - \frac{\rho_P c^2 \ell_P^4}{20} \Box(\Box S) + V'(S) + V_q'(S) = 0
\]

This is a fourth-order differential equation in the modulus. The term $\Box S$ encodes propagation; the term $S(\partial_\mu \theta)(\partial^\mu \theta)$ encodes coupling with the phase; the flexural term $\Box(\Box S)$ encodes UV regulation; the terms $V'(S)$ and $V_q'(S)$ encode the topological potential and the quantum IR wall.


\textbf{Evolution equation of the phase.} The phase component evolves under the U(1) conservation equation (equation E7 of the index, derived in \S{}I.7.10):


\[
\partial_\mu(S^2 \partial^\mu \theta) = 0
\]

This equation has the form of \textbf{current conservation} ($\partial_\mu J^\mu = 0$ with $J^\mu = S^2 \partial^\mu \theta$). It encodes the conservation law of the U(1) charge --- interpreted in the effective limit as conservation of electric charge.


\textbf{Coupling between modulus and phase.} The two equations are not independent: they are coupled by the term $S(\partial_\mu \theta)(\partial^\mu \theta)$ in the modulus equation, and by the factor $S^2$ in the phase equation. This coupling is physically important: the dynamics of the modulus (structural excitations of the medium) influences the dynamics of the phase (gauge sector), and vice versa. In the present relaxed regime where $S$ is close to $S_{\min}^\text{estr}$ in most of the cosmos, the coupling simplifies and the two sectors become approximately independent --- recovering the known effective theories (electrodynamics for the phase, gravitation for perturbations of the modulus at macroscopic scale). But the coupling exists structurally and manifests in specific configurations (such as the charged solitons developed in \S{}III.5).


\textbf{Stable solutions.} Configurations that satisfy the dynamical equations and persist in time without disintegrating are \textbf{stable solutions} of \emph{el Pleno}. The stable solutions known up to this point of the manuscript are:


\begin{itemize}[leftmargin=1.4em]
  \item The ground-state \emph{el Pleno} (uniform configuration with $S = S_{\min}^\text{estr}$ and $\theta = $ constant or null), corresponding to what conventional language calls ``vacuum''.
  \item Localized solitonic configurations (profiles of $S$ and $\theta$ that persist stable), identified with particles.
  \item Propagation modes (wave perturbations), identified with gravitational waves and photons.
\end{itemize}

The complete catalogue of stable solutions of \emph{el Pleno} and their classification is the subject of \textbf{Part II} of the manuscript.


\subsubsection{\texorpdfstring{I.9.9 --- Distinction between the $\Psi$ of el Pleno and the wavefunction of quantum mechanics}{I.9.9 - Distinction between the Psi of el Pleno and the wavefunction of quantum mechanics}}

The status of the field $\Psi$ rests on a central ontological distinction.

\textbf{The $\Psi$ of \emph{el Pleno} is not a probabilistic wavefunction.} The field $\Psi$ that appears in this manuscript is a \textbf{real physical field}: it has defined values at each point of \emph{el Pleno} at each instant. Its modulus $S(x^\mu)$ is a measurable real value that encodes the state of excitation of the medium; it is not the probability density of finding a ``particle of \emph{el Pleno}'' at that point. Its phase $\theta(x^\mu)$ is a real angle with physical meaning (it generates Noether currents via U(1)); it is not the phase of a probabilistic amplitude.


\textbf{Contrast with the Schrödinger quantum-mechanical wavefunction.} In standard quantum mechanics, the Schrödinger wavefunction $\psi$ has a probabilistic interpretation (Born interpretation): $|\psi(x)|^2$ is the probability density of finding the particle described by $\psi$ at the point $x$. Under this interpretation:


\begin{itemize}[leftmargin=1.4em]
  \item $\psi$ is a \textbf{probability amplitude}, not a directly measurable physical field.
  \item $\psi$ exists in the \textbf{configuration space} of the quantum system, not in the physical space common to multiple systems (for $N$ particles, $\psi$ lives in a space of $3N$ dimensions).
  \item $\psi$ undergoes \textbf{collapse} (discontinuous change under measurement) in some interpretations of quantum mechanics.
  \item $\psi$ admits \textbf{superposition} of states as a fundamental property.
\end{itemize}

The $\Psi$ of \emph{el Pleno} under PIU-$\Psi$ has opposite properties:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{$\Psi$ is a physical field}, not a probabilistic amplitude. It has real values measurable in principle.
  \item \textbf{$\Psi$ exists in \emph{el Pleno} itself} (three-dimensional cosmological manifold $\mathcal{M}$ plus time), not in a separate configuration space. It is a single field defined over the single medium of the universe.
  \item \textbf{$\Psi$ does not collapse}: its evolution is continuous and determined by F1, with no discontinuities from measurement.
  \item \textbf{$\Psi$ does not superpose with itself}: \emph{el Pleno} has a defined configuration at each instant, not a superposition of configurations.
\end{itemize}

\textbf{Position of PIU-$\Psi$ with respect to quantum mechanics.} This distinction does not mean that PIU-$\Psi$ rejects quantum mechanics. It means that the program \textbf{proposes an underlying ontology} from which standard quantum mechanics emerges as an effective description. The particles that quantum mechanics describes as entities with probabilistic wavefunctions are, under PIU-$\Psi$, \textbf{specific configurations of \emph{el Pleno}} (solitons, localized profiles). The probabilistic wavefunction of a particle is an effective description of how those configurations manifest in measurements --- it emerges from \emph{el Pleno}, it is not ontologically fundamental.


This position coincides structurally with ontological interpretations of quantum mechanics such as Bohm-de Broglie (hidden variables with an underlying real field) and with programs of \textbf{emergent quantum mechanics} (Adler, 't Hooft) --- without being identical to any of them. The detailed derivation of how standard quantum mechanics emerges from \emph{el Pleno} is developed in later parts of the manuscript.


\textbf{Notation: capital $\Psi$ vs. lowercase $\psi$.} To avoid confusion, the manuscript uses a strict convention: \textbf{capital $\Psi$} always refers to the field of \emph{el Pleno} under PIU-$\Psi$ (physical field, ontological sense). When it is necessary to refer to the probabilistic wavefunction of standard quantum mechanics --- for example in comparative discussions or in \S{}III where it is developed how quantum mechanics emerges from \emph{el Pleno} ---, \textbf{lowercase $\psi$} will be used. Confusing $\Psi$ with $\psi$ is confusing the ontology of the program with the effective tool that the program proposes to derive.


\subsubsection{\texorpdfstring{I.9.10 --- Synthesis: $\Psi$ as the complete description of the state of el Pleno}{I.9.10 - Synthesis: Psi as the complete description of the state of el Pleno}}

The field $\Psi$ is characterized as a physical object: the state of \emph{el Pleno} at each point, with its complex structure justified, its decomposition into modulus and phase, the physical function of each component and its ontological distinction from the wavefunction of standard quantum mechanics.


The consolidated key points are:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{$\Psi$ is the total field of \emph{el Pleno}}, capital of the Greek alphabet, defined over the cosmological manifold of \emph{el Pleno}, with values in the complex numbers.
  \item \textbf{$\Psi$ inscribes itself in the tradition of physical complex fields} of Gross-Pitaevskii (BEC), Ginzburg-Landau (superconductors) and Higgs (Standard Model). It is not a probabilistic wavefunction but a real physical field.
  \item \textbf{The complexity of $\Psi$ is a structural requirement}, not mathematical decoration: it is required to sustain the U(1) symmetry of axiom A5, the Noether charges, and the topological defects identified with particles.
  \item \textbf{The canonical decomposition $\Psi = S \cdot e^{i\theta}$} separates the structural information of the medium (modulus $S$) from the gauge information (phase $\theta$). This separation is physically significant and reflects patterns recognized in all known physical complex fields.
  \item \textbf{The modulus $S \geq 0$ encodes the state of structural excitation of the medium}. Its specific configurations give: the ground-state \emph{el Pleno} ($S_{\min}^\text{estr}$), the particles (localized solitons), the waves (propagating perturbations).
  \item \textbf{The phase $\theta$ encodes the gauge sector of the medium}. Its global U(1) symmetry generates the conserved current identified with electric charge, and its non-trivial topology produces the defects identified with charged configurations.
  \item \textbf{The dynamics of $\Psi$} is completely specified by F1 via Euler-Lagrange. The two equations (F2 for the modulus, U(1) conservation for the phase) are structurally coupled.
  \item \textbf{The $\Psi$ of \emph{el Pleno} is not the wavefunction of standard quantum mechanics}. The ontological distinction is fundamental: $\Psi$ is a real physical field, not a probabilistic amplitude. Standard quantum mechanics emerges from \emph{el Pleno} as an effective description, it is not ontologically fundamental under PIU-$\Psi$.
\end{itemize}

\textbf{From the characterization of the state to its regimes.} $\Psi$ does not behave the same in all conditions of the medium. When is \emph{el Pleno} near the ground-state \emph{el Pleno} and the dynamics simplifies? When is it in a high-excitation regime and the nonlinear effects dominate? When do the effective regimes identifiable with General Relativity, Quantum Mechanics or the Standard Model emerge? These operational regimes ---infrared, ultraviolet and intermediate--- are what organize the effective reading of \emph{el Pleno}.


\medskip\hrule\medskip

\subsection{\S{}I.10 --- Operational regimes of el Pleno}

\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:} section that articulates the distinguishable regimes of \emph{el Pleno} (present relaxed, infrared, ultraviolet, intermediate) and the emergence of the known effective theories (GR, QM, Standard Model) in each regime. A section without new derivations --- it announces derivations developed in Part III --- but it establishes the conceptual map that Part II requires to construct the configuration space.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.10.1 --- Why speak of regimes}

\emph{El Pleno} has been presented as a single object with axioms, parameters and an F1 Lagrangian that governs all its dynamics. This presentation is ontologically correct but \textbf{operationally incomplete}: although \emph{el Pleno} and F1 are single, its \textbf{manifest behaviour} depends on the regime in which it is observed. It is not an eccentricity of the program but a universal property of any medium ---water behaves as a liquid, gas or ice depending on the regime, with the same fundamental equations but distinct \textbf{effective descriptions}, each with its own variables and laws, derivable from the fundamental ones but not identical to them.


\emph{El Pleno} has the same structure: F1 is the fundamental law, but the effective descriptions that emerge in distinct regimes are the known theories ---General Relativity in the macroscopic low-curvature regime, Quantum Mechanics in the microscopic regime of localized configurations, the Standard Model in that of gauge excitations accessible to accelerators. They are not contradictions of the program but \textbf{regional readings} of a single \emph{el Pleno} described by F1. This section articulates those regimes and traces the conceptual map; the detailed derivations of each emergence are in Part III.


\subsubsection{I.10.2 --- The present relaxed regime of el Pleno}

The \textbf{present regime of the cosmos} observed, the starting point for articulating the other distinguishable regimes, presents specific characteristics that the program identifies:


\textbf{Characteristic 1 --- \emph{El Pleno} mostly near the ground-state \emph{el Pleno}.} In most of the volume of the cosmos currently observed, the field $S$ of \emph{el Pleno} has a value close to $S_{\min}^\text{estr} \approx 0.7231$. The deviations relative to this value are small except in localized regions (interiors of particles, vicinities of black holes, configurations of high energy density). The ``cosmological vacuum'' that astrophysics observes is essentially the extended ground-state \emph{el Pleno}.


\textbf{Characteristic 2 --- Low density of excitations.} The density of particles (solitonic configurations of the field) in the observed cosmos is extraordinarily low compared with the fundamental density of the ground-state \emph{el Pleno}. Ordinary matter, electromagnetic radiation, gravitational waves are \textbf{dilute perturbations} on a dominant background of relaxed \emph{el Pleno}.


\textbf{Characteristic 3 --- Post-bounce expansive regime.} The observed universe is in a phase of moderate accelerated expansion, identified operationally with the primordial post-bounce regime of \emph{el Pleno} (developed in \S{}II.7). The medium is neither in collapse nor in extreme compression; it is in a state of moderate expansion compatible with recent cosmological observation.


\textbf{Characteristic 4 --- Compatibility with standard effective theories.} As a consequence of characteristics 1, 2 and 3, the present relaxed regime is precisely the regime where the standard effective theories --- Newton's mechanics, General Relativity, Quantum Mechanics, the Standard Model --- work with extraordinary precision. This \textbf{is not a coincidence}: these theories are calibrated against observations made in this regime, and therefore describe \emph{el Pleno} correctly in this regime as an effective description. The predictive capacity of the standard theories in their domains of validity is testimony that the present regime of \emph{el Pleno} admits precise effective descriptions.


\textbf{Implication.} When the manuscript discusses the ``relaxed regime'' with no additional nuance, it refers to this present regime of the cosmos --- the regime where most experimental physics has been developed and where the standard theories have verified validity. The distinguishable regimes articulated below are defined as \textbf{specific deviations} relative to this relaxed regime.


\subsubsection{I.10.3 --- Infrared regime: large scales, low excitation}

The \textbf{infrared (IR) regime of \emph{el Pleno}} is characterized by configurations of long wavelength and low excitation of the field. The specific characteristics are:


\textbf{Characteristic 1 --- Wavelengths much larger than $\ell_P$.} In the IR regime, the spatial variations of the field $\Psi$ occur at scales $\lambda \gg \ell_P$. The continuous-medium description of \emph{el Pleno} is excellent; the effects of the microscopic structure of the medium (diamond lattice at scale $\ell_P$) are irrelevant.


\textbf{Characteristic 2 --- Modulus $S$ close to the ground-state \emph{el Pleno}.} In the IR regime, the deviations of the modulus relative to $S_{\min}^\text{estr}$ are small: $S = S_{\min}^\text{estr} + \delta S$ with $|\delta S| \ll S_{\min}^\text{estr}$. The program can expand F1 in powers of $\delta S$ and keep only the terms relevant for the IR dynamics.


\textbf{Characteristic 3 --- Dominance of the kinetic term.} In the IR regime, the kinetic term of F1 dominates over the flexural term (which is subdominant by small factors of $(\ell_P/\lambda)^2$) and over the Klauder quantum potential (which is only relevant when $S$ approaches zero, which does not happen in the relaxed regime). The IR dynamics simplifies considerably.


\textbf{Characteristic 4 --- Approximately decoupled sectors.} In the IR regime, the coupling between the modulus $S$ and the phase $\theta$ of the field (present structurally in F1) simplifies because $S \approx S_{\min}^\text{estr}$ is approximately constant. The two sectors become approximately independent: the perturbations of the modulus evolve according to an effective gravitational dynamics, and the perturbations of the phase evolve according to an effective gauge dynamics.


\textbf{Operational implication.} The IR regime is where the macroscopic effective theories emerge: \textbf{General Relativity} for the perturbations of the modulus (developed in \S{}III.1) and \textbf{electrodynamics} for the perturbations of the phase in the U(1) sector (developed in \S{}III.4). The currently observable cosmology occurs in the IR regime; the cosmological predictions of the program (developed in \S{}III.6 and \S{}III.7) are obtained in this regime.


\subsubsection{I.10.4 --- Ultraviolet regime: microscopic scales, high excitation}

The \textbf{ultraviolet (UV) regime of \emph{el Pleno}} is characterized by configurations of very short wavelength or high excitation of the field. The specific characteristics are:


\textbf{Characteristic 1 --- Wavelengths comparable to or smaller than $\ell_P$.} In the UV regime, the spatial variations of the field occur at scales $\lambda \sim \ell_P$ or shorter. The continuous-medium description of \emph{el Pleno} ceases to apply; the microscopic structure of the medium (diamond lattice at scale $\ell_P$, discussed in \S{}II.3) becomes relevant.


\textbf{Characteristic 2 --- Dominance of the flexural term.} In the UV regime, the flexural term $\gamma_\kappa (\Box S)^2$ of F1 dominates over the kinetic term. The flexural rigidity of \emph{el Pleno} produces \textbf{natural regulation} of the high-frequency configurations: the medium resists structurally the UV excitations, suppressing them dynamically.


\textbf{Characteristic 3 --- Absence of UV divergences in the program.} As a consequence of the intrinsic flexural regulation, the PIU-$\Psi$ program \textbf{has no ultraviolet divergences} in its calculations. The standard quantum field theories (QED, QCD, electroweak) produce infinities in perturbative UV calculations that require a posteriori renormalization procedures; PIU-$\Psi$ obtains finite results directly because the UV regulation is incorporated structurally in F1 via the flexural term and the parameter $\ell_P$.


\textbf{Characteristic 4 --- Direct observational inaccessibility.} The UV regime of \emph{el Pleno} occurs at scales $\sim 10^{-35}\ m$, approximately $10^{20}$ times smaller than the scales of the most powerful current accelerators. There is no direct experimental access to this regime with current technology. The observable consequences of the UV regime are indirect: they appear in corrections to physical processes that the IR regime can measure (modifications to the dispersion of high-energy photons, effects in very early cosmology, properties of black holes).


\textbf{Operational implication.} The UV regime is where the complete microscopic structure of \emph{el Pleno} manifests. Here the diamond lattice is defined, the solitonic configurations identified with particles are generated, and the regulation of the divergences that the standard theories cannot handle is produced. Part II of the manuscript operates in this regime in constructing the catalogue of available configurations of \emph{el Pleno}.


\subsubsection{I.10.5 --- Intermediate regime: the habitual effective zone}

The \textbf{intermediate regime of \emph{el Pleno}} occupies the range of scales between the macroscopic IR and the microscopic UV. It is the regime where most known experimental physics occurs: from atomic scales ($\sim 10^{-10}\ m$) to hadronic scales ($\sim 10^{-15}\ m$), passing through nuclear and subnuclear scales.


\textbf{Characteristic 1 --- Stable localized configurations.} In the intermediate regime, \emph{el Pleno} admits \textbf{localized solitonic configurations} that persist stable in time. These configurations have characteristic sizes in the range $\sim 10^{-15}$ to $10^{-10}\ m$ --- dimensions of particles and atoms. Each particle observed in nature corresponds to a specific solitonic configuration of \emph{el Pleno}.


\textbf{Characteristic 2 --- Coexistence of the three effects.} In the intermediate regime, the kinetic, flexural and potential terms of F1 all contribute significantly to the dynamics. The regime is neither purely continuous (like the IR) nor purely microscopic (like the UV); it is the zone where the rich structure of \emph{el Pleno} fully manifests.


\textbf{Characteristic 3 --- Manifestation of quantum mechanics.} In the intermediate regime, the configurations of \emph{el Pleno} behave collectively in a way that reproduces the observed quantum phenomena: discreteness of energy levels, wave-particle duality, Heisenberg indeterminacy, superposition and interference. \textbf{Standard quantum mechanics} emerges as the effective description of the intermediate regime of \emph{el Pleno} (developed in \S{}III.2 and \S{}III.5).


\textbf{Characteristic 4 --- Manifestation of the non-abelian gauge sector.} In the intermediate regime, specific configurations of \emph{el Pleno} with tripartite topological structure (Borromean) produce the effects identified with quantum chromodynamics (QCD) --- the theory that describes the quarks and the strong nuclear force. The hadronic sector of physics originates in this regime (developed in \S{}III.5).


\textbf{Operational implication.} The intermediate regime is operationally the richest --- where most observable physics manifests. The standard effective theories (Quantum Mechanics, Standard Model) emerge here.


\subsubsection{I.10.6 --- Emergence of Newtonian mechanics as an effective theory}

Before Relativity and before quantum mechanics, modern physics begins with Isaac Newton. It was Newton who established that the motion of bodies and the attraction between masses obey universal mathematical laws ---the same in the heavens and on earth---, founding thereby the method of physics itself: to describe nature by equations from which the observed behaviour is deduced. The classical mechanics that bears his name, valid without exception for more than two centuries and still today the language of engineering and of everyday life, is the first and best-verified of the effective limits that any theory of foundations must recover. PIU-$\Psi$ recovers it, with the same status as the recovery of General Relativity.

\medskip
\textbf{The Newtonian regime of \emph{el Pleno}.} Newtonian mechanics emerges from \emph{el Pleno} as its \textbf{effective description in the triple limit} of weak field, low speed and slow deformation:
\[
\delta S \ll S_{\min}^\text{estr}\,, \qquad v \ll c\,, \qquad \dot\gamma \ll \dot\gamma_{\max} = c/\ell_P\,.
\]
It is the regime of the present relaxed universe (\S{}I.10.2), where the ground-state \emph{el Pleno} is nearly Newtonian ---flow index $n\to 1$ in the low-deformation-rate limit (\S{}III.9.3)--- and where, therefore, we live. The two fundamental laws of Newton emerge thus:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Universal gravitation.} The perturbation of the modulus of \emph{el Pleno} produced by a localized source decays as $1/r$ ---the response of an elastic medium to a point perturbation, not a postulate (\S{}III.1.2)---, and the interaction energy between two perturbations takes exactly the form $U=-G\,m_1 m_2/r$ with $G=c^2/(\rho_P\ell_P^2)$ derived from the parameters of the medium (\S{}III.1.3). Newton's inverse-square law, $F = G\,m_1 m_2/r^2$, is the gradient of that energy: the Newtonian gravitational force is the elastic restoring force of \emph{el Pleno} in the weak-field limit.
  \item \textbf{The second law.} In the non-relativistic regime, the dynamics of the soliton that represents a body is obtained ---via the Madelung-Bohm decomposition over F1 (\S{}III.2.4), taken at speeds $v\ll c$--- and the expectation value of its equation of motion is the Ehrenfest theorem: when the quantum potential is negligible (localized packet), the force equals the mass times the acceleration of the centre of the configuration, $F = m\,a$, with $m$ the mass-energy of the soliton. Newtonian inertia is the resistance of \emph{el Pleno} to changing the state of motion of its localized configurations.
\end{itemize}

\medskip
\textbf{Critical point.} Newtonian mechanics is not contradicted by PIU-$\Psi$ ---it is \textbf{derived} as the correct effective description of the weak-field, low-speed regime, the same role that General Relativity plays in the relativistic regime and that quantum mechanics plays in the intermediate regime. The hierarchy of effective limits is transparent: Newtonian mechanics is the weak-field classical limit of General Relativity, which in turn is the IR limit of \emph{el Pleno}; each step recovers the previous one, and \emph{el Pleno} contains them all. That the physics Newton wrote for apples and planets turns out to be the behaviour of \emph{el Pleno} in the smooth regime in which we live is not a coincidence: it is the reason its equations work.

\medskip
\textbf{Epistemic classification.} The recovery of Newton's law of universal gravitation ---$1/r^2$ form and constant $G$ with its observed value--- is \epi{DERIVED} (\S{}III.1.2--III.1.3). The recovery of the second law $F=ma$ is \epi{DERIVED}: the Bohm equation of motion comes out of F1 in the non-relativistic regime by the Madelung decomposition ---exact mathematical equivalence, not approximation (\S{}III.2.4)---, and taking its expectation value is the Ehrenfest theorem, which in the limit of negligible quantum potential (localized packet) gives exactly $m\,\mathbf{a}=-\nabla V$, that is $F=ma$. Both steps are theorems, with no conceptual leap; the classical limit does not weaken the derivation, it only fixes the regime in which it is read. The Newtonian limit of the rheology, $n\to 1$, is \epi{DERIVED} and model-independent (\S{}III.9.3). On the whole, Newton's classical mechanics is the best-established and best-verified effective limit of \emph{el Pleno} in the program.


\subsubsection{I.10.7 --- Emergence of General Relativity as an effective theory}

General Relativity emerges from el Pleno as the \textbf{effective description of the infrared regime of the modulus $S$}. The detailed derivation is developed in \S{}III.1; here the scheme is announced:


\textbf{Step 1 --- Identification of the gravitational field.} Perturbation configurations of the modulus of el Pleno at macroscopic scales, $\delta S(x^\mu)$ with $\lambda \gg \ell_P$, are identified with configurations of the Einsteinian gravitational field. The specific relation between $\delta S$ and the metric $g_{\mu\nu}$ is established by the Barceló-Liberati-Visser formalism of \textbf{acoustic metrics in continuous media}, adapted to the case of el Pleno.


\textbf{Step 2 --- Emergence of Einstein's equations.} In the IR limit of F1 with $S \approx S_{\min}^\text{estr}$, the dynamical equations of el Pleno reproduce the Einstein equations of General Relativity. The coefficient $c^4/G$ of the Einstein-Hilbert action is identified with $\rho_P c^2 \ell_P^2$ of the F1 Lagrangian, as anticipated in \S{}I.7.5 and confirmed via the structural identity E3.


\textbf{Step 3 --- Recovery of well-tested limits.} In the Newtonian limit (weak fields, low speeds), the PIU-$\Psi$ formalism recovers Newton's gravitational theory ---including the inverse-square law and the gravitational constant $G$ with its observed value---, as articulated in the dedicated section \S{}I.10.6. In the limit of strong but static fields, it recovers the Schwarzschild solutions for black holes. In the limit of propagating perturbations, it recovers the gravitational waves with speed $c$.


\textbf{Critical point.} General Relativity is not contradicted by PIU-$\Psi$ --- it is \textbf{derived} as the appropriate effective description of the IR regime. What the program proposes is the underlying ontology: the ``curved space-time'' that GR postulates as a primitive geometric entity is, under PIU-$\Psi$, the \textbf{emergent effective metric} of el Pleno in its IR regime. This reading does not affect the validated predictions of GR; it affects their ontological interpretation.


\textbf{Closed structural cosmological predictions.} Beyond recovering GR as an effective theory, the IR regime of el Pleno produces \textbf{two quantitative structural cosmological predictions} that GR/$\Lambda$CDM does not generate, both \epi{DERIVED} with no free parameters, whose complete derivations are developed in their canonical place (\S{}III.6 and \S{}III.7):

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Ratio of dark sectors: $\Omega_{MO}/\Omega_M = \sqrt{3}\,\pi \approx 5.4414$} (\S{}III.6.3, E37). It emerges from three geometric pillars of the diamond lattice (Peter-Weyl over the solitonic classes, spinorial factor $1/2$ between $SU(2)$ and $SO(3)$, and the tetrahedron/sphere area ratio of the M and MO anchors). Agreement to $0.86\sigma$ with Planck 2018 ($\Omega_\text{dm}/\Omega_b = 5.375\pm0.077$), zero free parameters.
  \item \textbf{Dark-energy fraction: $\Omega_{EO} = f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2 \approx 0.6869$} (\S{}III.7). It emerges from the ontological identification of the EO sector with the coherent \emph{Pleno}-ground state: under the Bogoliubov-Josephson identity $S^2\leftrightarrow n/n_0$ (\S{}II.2), $f_c$ is the fraction of quanta in the fundamental coherent mode (Yukalov depletion), which measures the same physical quantity as $\Omega_{EO}$. Agreement to $0.30\sigma$ with Planck 2018 ($\Omega_\Lambda = 0.6847\pm0.0073$), relative error $0.32\%$.
\end{itemize}

\textbf{Epistemic synthesis of the two cosmological predictions.} Both qualitatively distinguish PIU-$\Psi$ from $\Lambda$CDM: $\Lambda$CDM treats $\Omega_{MO}/\Omega_M$ and $\Omega_{EO}$ as free parameters fitted to observation; PIU-$\Psi$ derives them structurally without cosmological calibration. The Bayesian closure of the thread C-meta-1 The confrontation of \S{}V.2.6 quantifies how far the whole set of results of the programme disfavours reading that concordance as coincidence.


\textbf{Derived prediction on the structural scale of el Pleno.} An additional derived prediction emerges from the analysis of the spectrum of modes of el Pleno linearized around $S_{\min}^\text{estr}$: the radial mass of the mode of el Pleno is


\[
m_\text{radial}\,c^2 = \hbar\sqrt{\frac{V''(S_{\min}^\text{estr})}{\rho_P c^2 \ell_P^2}}\cdot c \approx 4.44\,E_P
\]

where $E_P$ is the Planck energy and $V''(S_{\min}^\text{estr}) = 19.69\,\rho_P c^2$ is the curvature of the potential at the minimum (derived in \S{}I.8.8). This mass, extremely high (Planckian scale), structurally explains the \textbf{non-detection of el Pleno as a particle}: any excitation of the modulus of el Pleno requires Planckian energy, beyond the reach of current accelerators by approximately 16 orders of magnitude.


\textbf{Numerical verification with the curvature.} Substituting $V''(S_{\min}^\text{estr})/(\rho_P c^2) = 19.69$:


\[
m_\text{radial} c^2 = \hbar c\sqrt{\frac{19.69}{\ell_P^2}} = \hbar c \cdot \frac{\sqrt{19.69}}{\ell_P} \approx \hbar c \cdot \frac{4.437}{\ell_P}
\]

and using $E_P = \hbar c/\ell_P$ (definition of the Planck energy):


\[
m_\text{radial} c^2 = 4.437\, E_P \approx 4.44\, E_P
\]

The computation $m_\text{eff} c^2 = 4.4373\, E_P$ is \textbf{consistent} with the independent prediction $m_\text{radial} c^2 \approx 4.44\, E_P$ from the linearized tripartite spectrum (\S{}II.5, derivation by a different route) with agreement of $0.06\%$. This convergence between two independent derivations by distinct routes constitutes additional evidence of the internal consistency of the program with Planckian precision.

\subsubsection{I.10.8 --- Emergence of Quantum Mechanics as an effective theory}

Standard Quantum Mechanics emerges from \emph{el Pleno} as the \textbf{effective description of localized solitonic configurations in the intermediate regime}. The detailed derivation is developed in \S{}III.2; here the scheme is announced:


\textbf{Step 1 --- Identification of the particles as solitons.} The particles observed in quantum physics (electrons, photons, quarks, etc.) are identified under PIU-$\Psi$ with \textbf{specific solitonic configurations} of \emph{el Pleno} in the intermediate regime. Each type of particle corresponds to a specific type of soliton with characteristic topological and dynamical properties.


\textbf{Step 2 --- Emergence of the probabilistic wavefunction.} The \textbf{wavefunctions $\psi$} of standard quantum mechanics emerge as effective descriptions of \textbf{statistical distributions} of the solitonic configurations of \emph{el Pleno} under environmental perturbations. The Born interpretation --- $|\psi|^2$ as probability density --- is justified from the statistical behaviour of the solitons of \emph{el Pleno} under measurement. This derivation is structurally analogous to the proposals of emergent quantum mechanics of Adler and 't Hooft, although with the specific ontology of \emph{el Pleno} as the substrate.


\textbf{Step 3 --- Recovery of quantum properties.} The characteristic properties of quantum mechanics --- discreteness of levels, wave-particle duality, Heisenberg indeterminacy, interference, entanglement --- emerge as consequences of the dynamics of the solitons of \emph{el Pleno}. Each property receives a specific physical interpretation from the behaviour of the medium. For example: the Heisenberg indeterminacy emerges from the flexural rigidity of \emph{el Pleno} (parameter $\gamma_\kappa$), which prevents arbitrarily precise localization of the solitons without increasing energy cost.


\textbf{Critical point (repeated from \S{}I.9.9 for importance).} The $\Psi$ of \emph{el Pleno} \textbf{is not} the wavefunction $\psi$ of quantum mechanics. $\Psi$ is the real physical field of \emph{el Pleno}; $\psi$ is the effective statistical description of solitonic configurations of \emph{el Pleno}. The emergence of $\psi$ from $\Psi$ is a fundamental derivation of the program, not a trivial identification.


\subsubsection{I.10.9 --- Emergence of the Standard Model as an effective theory}

The Standard Model of elementary particles emerges from \emph{el Pleno} as the \textbf{effective description of the gauge sector and of the solitonic configurations in the intermediate regime}. The detailed derivation is developed in \S{}III.4 and \S{}III.5; here the scheme is announced:


\textbf{Electromagnetic sector.} The photon emerges as an excitation mode of the field $\theta$ (the phase of \emph{el Pleno}) in its residual U(1) sector of the ground-state \emph{el Pleno}. Maxwell's electrodynamics is derived as the effective description of the behaviour of $\theta$ in the IR-intermediate regime. The fine-structure constant $\alpha \approx 1/137$ is identified as a structural property of \emph{el Pleno} in the gauge sector; its exact derivation from more basic structural principles is an open thread of the research program, declared in \S{}V.3.


\textbf{Electroweak sector.} The electroweak unification of the Standard Model is identified under PIU-$\Psi$ with a specific regime of \emph{el Pleno} where two sectors of the phase dynamics couple structurally. The Weinberg angle, the W and Z bosons, and the mechanism of spontaneous symmetry breaking receive an interpretation from \emph{el Pleno}.


\textbf{Hadronic sector.} Quantum chromodynamics (QCD) and the confinement of the quarks emerge from \emph{el Pleno} by tripartite topological configurations (Borromean) of the field $\theta$. The quarks correspond to vertices of tripartite configurations that cannot be separated without destroying the topological stability --- this is the PIU-$\Psi$ version of colour confinement. The masses of the proton and the neutron, and the neutron-proton mass separation, are computed with structural parity relative to the QCD predictions but with a distinct interpretation.


\textbf{Mass sector.} The masses of the observed particles are interpreted under PIU-$\Psi$ as \textbf{total energies associated with the solitonic profiles} corresponding to each particle. The Higgs mechanism of the Standard Model is reinterpreted as a specific sector of the potential $V(S)$ of \emph{el Pleno}: the ``Higgs field'' identifiable with the appropriate component of the modulus $S$, and the ``mass of the particles'' identifiable with the energy of the associated solitonic configurations.


\textbf{Critical point.} The program \textbf{does not construct particles ad hoc}: the catalogue of observed particles must be readable from the catalogue of solitonic configurations of \emph{el Pleno} (developed in Part II). What Part II does (classify the configurations permitted by F1) must correspond, in its intersection with the observed phenomenology, with the Standard Model.


\subsubsection{I.10.10 --- Extreme regimes: black holes, primordial bounce}

Beyond the IR, UV and intermediate regimes that cover most of the observed physics, \emph{el Pleno} admits \textbf{extreme regimes} where the structural effects of the medium manifest in full. Two particular extreme regimes deserve mention:


\textbf{Regime 1 --- Black-hole interiors.} In the regions near and inside a black hole, the curvature of the gravitational field reaches very high values. General Relativity predicts a \textbf{singularity} in the interior --- a point where the curvature diverges and the physical laws cease to apply. Under PIU-$\Psi$, this singularity \textbf{does not exist}: the flexural rigidity of \emph{el Pleno} (parameter $\gamma_\kappa$) prevents configurations of the modulus $S$ with arbitrarily high curvature. The interior of the black hole is interpreted as a \textbf{compact configuration of \emph{el Pleno}} with $S$ greatly altered relative to the ground-state \emph{el Pleno}, but finite in all its values. The structural resolution of singularities is one of the most significant qualitative predictions of the program, developed in \S{}III.6.


\textbf{Regime 2 --- Primordial cosmological bounce.} Applying F1 to the entire universe at cosmological scales, the solutions extrapolated to the past produce a \textbf{phase of extreme compression} of \emph{el Pleno} that precedes the observed expansion. Standard cosmology predicts an initial singularity (Big Bang) in this regime; PIU-$\Psi$ predicts instead a \textbf{bounce} --- a phase of maximum finite compression followed by rebound and expansion. In the cosmology of \emph{el Pleno}, the primordial cosmological compression does not correspond to ``extinction'' of the medium ($S \to 0$) but to \textbf{upper saturation} of the medium: the modulus $S$ of \emph{el Pleno} approaches $S_{\max}^\text{geom}$, reaching its upper structural limit. The total density of \emph{el Pleno} in this regime reaches the value $\rho_{\max,\text{abs}} = S_{\max,\text{geom}}^2 \cdot \rho_P \approx 2.1773\,\rho_P$, a structural quantitative prediction of the program (predictions P-F1a-1 to P-F1a-4 developed in \S{}V.1). The bounce occurs because the topological component of the potential $V_\text{topo}(S) = -(\rho_P c^2/2)\ln(1 - S^2/S_{\max,\text{geom}}^2)$ diverges when $S \to S_{\max}^\text{geom}$, producing an effective equation of state with repulsive pressure that halts the compression and reverses the dynamics toward expansion. Distinct structural functions operate at the two extremes of the range of $S$: the Klauder quantum IR wall ($V_q$) prevents configurations with $S \to 0$ (extinction of the medium, axiom A2), while the divergence of the topological potential near $S_{\max}^\text{geom}$ prevents the absolute saturation (axiom A6) and produces the cosmological rebound. The bounce cosmology of the program is developed in \S{}II.7 and is confronted with cosmological observation in \S{}V.4.


\textbf{General implication.} The extreme regimes of \emph{el Pleno} are where the \textbf{most distinctive predictions} of the program relative to standard physics manifest. GR and standard cosmology have singularities; PIU-$\Psi$ does not have them. This difference is the subject of future observational research: cosmological signs of the primordial bounce (specific oscillations in the CMB), modified properties of black holes (subtle changes in observable shadows, in ringdowns of mergers), marginal Lorentz violations in the propagation of very-high-energy gamma rays. These observational threads are declared explicitly in \S{}V.3.


\subsubsection{I.10.11 --- Synthesis: the map of regimes of el Pleno}

The operational regimes of \emph{el Pleno} and the effective theory that emerges in each are summarized thus:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{3.30cm}L{3.30cm}L{3.30cm}L{3.30cm}}
\toprule
\textbf{Regime} & \textbf{Scale} & \textbf{Dominant variables} & \textbf{Effective theory} \\
\midrule\endhead
\textbf{Macroscopic IR} & $\lambda \gg \ell_P$, low excitation & Perturbations $\delta S$ and $\delta \theta$ around the ground-state \emph{el Pleno} & Newtonian mechanics (weak field, $v\ll c$), General Relativity (modulus), Electrodynamics (phase) \\
\textbf{Intermediate} & $10^{-15}$ to $10^{-10}\ m$, localized excitations & Solitonic configurations $S(x^\mu)$ and topology of $\theta$ & Quantum Mechanics, Standard Model \\
\textbf{Microscopic UV} & $\lambda \sim \ell_P$ or smaller & Complete structure of the field $\Psi$ in the diamond lattice & No standard effective theory; microscopic description of \emph{el Pleno} \\
\textbf{Extreme: black holes} & High local curvature & Compact configuration of the modulus & PIU-$\Psi$ without singularities (distinctive prediction) \\
\textbf{Extreme: primordial bounce} & Maximum cosmological compression, $S \to S_{\max}^\text{geom}$, $\rho_{\max,\text{abs}} = S_{\max,\text{geom}}^2 \cdot \rho_P \approx 2.1773\,\rho_P$ & Divergence of the potential $V(S)$ near $S_{\max}^\text{geom}$ & PIU-$\Psi$ with bounce (distinctive prediction) \\
\bottomrule
\end{longtable}}

\textbf{Joint reading of the map.} The above table establishes that:


\begin{itemize}[leftmargin=1.4em]
  \item The known effective theories are not \textbf{competitors} of PIU-$\Psi$ --- they are \textbf{regional readings} of \emph{el Pleno} in specific regimes.
  \item Each effective theory has its \textbf{domain of validity} associated with the regime where it emerges. Outside that domain, the effective theory fails and the PIU-$\Psi$ program proposes specific descriptions that replace it.
  \item The extreme regimes are where the program has the greatest \textbf{capacity for observational distinction} relative to the standard theories. The distinctive predictions (no singularities, primordial bounce) are falsifiable by future observation.
  \item The emergence of the standard theories as effective descriptions is not a coincidence: it reflects that \emph{el Pleno} is \textbf{really} the physical substrate of the cosmos, and that the standard theories correctly describe what happens in their calibrated regimes.
\end{itemize}

The map of regimes completes the interpretive apparatus of \emph{el Pleno}. What is missing ---and what Part II constructs--- is the catalogue of the configurations that F1 permits.


\medskip\hrule\medskip

\subsection{\S{}I.11 --- Synthesis: el Pleno as an object of physics}

\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:} closing section of Part I. It consolidates the interpretive apparatus into a unified image of el Pleno as an object of physics, preparatory to Part II.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.11.1 --- El Pleno as a single object}

On closing Part I, the interpretive apparatus of the PIU-$\Psi$ program is completely established. Its central element is \textbf{el Pleno}: the single fundamental physical object that the program postulates, of which all the observed phenomenology of the cosmos is read as a configuration. This uniqueness is ontological, not methodological: the program does not propose that there is one Pleno and, in addition, particles, waves, gauge fields, space-time, dark matter and dark energy, but that \textbf{there is el Pleno}, and that all those apparently distinct entities are specific configurations of the same single medium.

The medium is characterized by the six structural axioms (\S{}I.5), whose minimal mathematical expression is the F1 Lagrangian (\S{}I.7); from its two substantive properties ($\rho_P$, $\ell_P$) and its causal structure ($c$) emerge $G$, $\hbar$, $\gamma_\kappa$, $S_{\min}^\text{estr}$ and $S_{\max,\text{geom}}^2$ as derived quantities (\S{}I.8); and the state of the medium is described by the real physical field $\Psi = S\cdot e^{i\theta}$ (\S{}I.9), from which standard quantum mechanics emerges as an effective description. This ontological position resolves by construction problems that standard physics leaves open: wave-particle duality ceases to be one (both are configurations of el Pleno), dark matter ceases to be a mystery (it is a configuration of the \emph{Pleno}-ground state) and the initial cosmological singularity ceases to exist (the IR quantum wall forbids it). The price is an ontological commitment to a single substrate of the universe.

\subsubsection{I.11.2 --- What Part I leaves established}

On finishing Part I, the program has the following elements established with the corresponding solidity:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Epistemological position} (\S{}I.1): what kind of program PIU-$\Psi$ is and what it aims at.
  \item \textbf{A priori motivation} (\S{}I.2 and \S{}I.3): which open problems of contemporary physics make the program reasonable, and which convergent reasons support the postulation of el Pleno.
  \item \textbf{Structural axioms} (\S{}I.5): six axioms that characterize el Pleno completely.
  \item \textbf{Lagrangian apparatus} (\S{}I.6): the mathematical formalism on which the program builds.
  \item \textbf{F1 Lagrangian} (\S{}I.7): the mathematical expression of el Pleno, with dimensional check, structural identity of $G$, derivation of the dynamical equations.
  \item \textbf{Structural properties of el Pleno} (\S{}I.8): physical reading of each parameter and coefficient of F1 as a property of the medium.
  \item \textbf{Field $\Psi$ as the state of el Pleno} (\S{}I.9): complete presentation of the mathematical object with its tradition, decomposition and dynamics.
  \item \textbf{Operational regimes} (\S{}I.10): map of the regimes of el Pleno and of the effective theories that emerge in each regime.
\end{itemize}

On closing Part I, the \textbf{complete ontological characterization} of the program is established: what el Pleno is, what laws it obeys, what properties it has, how it is described mathematically, and where the standard theories emerge as effective descriptions. What is still missing ---and what Part II will construct--- is the \textbf{complete catalogue of permitted configurations} of el Pleno.


\subsubsection{I.11.3 --- What Part I leaves pending}

Part I explicitly leaves an \textbf{agenda} for the following parts, not omissions: the space of configurations of el Pleno and the bounce cosmology (Part II), the phenomenological reading that recovers the standard theories and computes observable masses and constants (Part III), and the observational confrontation with its open threads and falsifying experiments (Part V). What is pending is derived or in the course of derivation, not postulated.


\subsubsection{I.11.4 --- Transition to Part II}

Part I established the interpretive apparatus ---el Pleno as an object, F1 as its Lagrangian, $\Psi$ as its description---; Part II operates on it to construct the \textbf{catalogue of available configurations}: given F1 and the structural constraints (A1--A6, microscopic geometry, dynamical stability), which configurations does el Pleno permit? The catalogue is not postulated a priori, but derived ---uniform configurations (including the \emph{Pleno}-ground state), crystalline structure (diamond lattice), propagation modes, solitons (particles) and cosmological configurations (including the bounce). The logical dependence of the manuscript is strict: Part I interprets, II constructs, III identifies, IV synthesizes the dynamics, V confronts; none is read in isolation.


\medskip\hrule\medskip


