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\section*{Part III --- Phenomenological reading}
\phantomsection
\addcontentsline{toc}{section}{Part III --- Phenomenological reading}
\markboth{Part III --- Phenomenological reading}{Part III --- Phenomenological reading}
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\begin{tcolorbox}[notabox]
\textbf{Purpose of Part III.} To read the observed physical facts as configurations of \emph{el Pleno}. With \emph{el Pleno} established as a medium (Part I) and its catalogue of permitted configurations (Part II), Part III interrogates that single substrate: each phenomenon --- gravity, constants, gravitational waves, photon, particles, cosmology, dark sectors --- is read as a distinct question asked of the same \emph{el Pleno}, not as an independent entity requiring its own explanation.

\end{tcolorbox}

\subsection{\S{}III.1 --- Gravity as a macroscopic property of \emph{el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} independent calibration of $(\rho_P,\ell_P)$ (thread \S{}V.3).
  \item \textbf{Notes:} gravity as a macroscopic property of \emph{el Pleno} via the Sakharov induced action over the TT mode. The normalization of the gravitational charge $q_i=\sqrt{4\pi}\,m_i c^2$ cancels the factor $4\pi$ of the Green function.
\end{itemize}
\end{estadoseccion}

\subsubsection{III.1.1 --- The phenomenological question of gravity}

Among the configurations of the catalogue of Part II, the propagating modes (\S{}II.5) included a transverse-traceless tensor mode (TT mode) whose dispersion relation was derived algebraically from linearized F1. What observed physical phenomenon corresponds to the gravitational dynamics, and how does it emerge from the apparatus of \emph{el Pleno}?


The PIU-$\Psi$ answer inverts the standard picture. In General Relativity, gravity is the dynamical geometry of spacetime: a fundamental geometric entity that curves in the presence of matter. In PIU-$\Psi$ there is no primitive geometric spacetime (axiom A1, \S{}I.5.2): space and time are emergent properties of \emph{el Pleno}. Gravity, therefore, cannot be the curvature of a fundamental geometric entity. It is a \textbf{macroscopic property of \emph{el Pleno}} --- the collective behavior of the single physical medium when described in the low-energy, large-scale regime.


The precise operational question is twofold. First: where does the value of the gravitational constant $G$ come from? Second: how do Einstein's equations emerge as the effective description of the dynamics of \emph{el Pleno} in the macroscopic regime?


\textbf{Clarification of what is and is not non-trivial.} The $1/r$ form of the Newtonian gravitational potential is a direct mathematical consequence of a quadratic kinetic term under the Euler-Lagrange equations: any scalar field with a standard kinetic term produces a propagator $\sim 1/k^2$ that, in position space, gives a potential $\propto 1/r$. That el Pleno reproduces this form is the expected confirmation that its kinetic sector is the correct one; the \emph{distinctive} content of the derivation of gravity in PIU-$\Psi$ is one level deeper: the \textbf{specific value of $G$} in terms of the structural parameters of el Pleno (E3, agreement $<0.1\%$), and the emergence of the \textbf{complete tensor structure} of Einstein's equations. Distinguishing both levels is epistemic discipline, not a concession: the $1/r$ form validates the route, and the value of $G$ is where the program says something that Newtonian gravity could not say.


\subsubsection{III.1.1-bis --- Newton's laws as an effective limit of \emph{el Pleno}}

Newton's three laws are not axioms of the programme: they are the behaviour of a medium with a Lagrangian under the principle of least action, read in the low-energy regime. The distinction drawn in the previous subsection ---between what confirms the route and what the programme contributes--- applies to each of them separately, and in all three the contribution lies in the same place: not in the law, but in the physical origin of what Newton had to take as primitive.

\textbf{First law: the origin of inertia.} A free body is the Euler-Lagrange solution of F1 without external forcing: it persists in its state because there is no gradient to alter it. Any theory with an action produces that. What \emph{el Pleno} adds is \emph{what} inertia is, which in Newton is a primitive property of matter. Axiom A1 identifies density, inertia and energy as three readings of a single quantity (\S{}I.5.2), and the relation $E=Mc^2$ does not describe conversion between two substances but the balance between energy read as propagating change and as retained change (\S{}I.4): mass \emph{is} configuration energy retained in the medium. Inertia is then the resistance of the medium to reconfiguring its own energy, and its measure is how much energy is retained. \epi{DERIVED} for the origin of mass and inertia; the form of the law is inherited.

\textbf{Second law: force as a gradient, and gravity as the native case.} In the low-energy limit the Euler-Lagrange equation of F1 reduces to $\mathbf{F}=m\mathbf{a}$ with the force as the gradient of an effective potential, a step shared by every Lagrangian theory. What is native is the specific potential. In the static regime and at scales $\gg\ell_P$, the modulus equation reduces to Poisson form, whose Green function $\mathcal{G}=-1/4\pi r$ gives a modulus perturbation decaying as $1/r$ ---a consequence of the dimensionality of space and of the form of F1, not of a choice---; the interaction energy of two perturbations comes out Newtonian, and comparing it with $U=-Gm_1m_2/r$ fixes
\[
\rho_P c^2\ell_P^2=\frac{c^4}{G}
\qquad\Longleftrightarrow\qquad
G=\frac{c^2}{\rho_P\ell_P^2}
\tag{E3, repeated from \S{}I.7.5}
\]
with no residual factor. The complete derivation, with its algebra and its numerical verification, is in \S{}III.1.2 and \S{}III.1.3. The logical order is the content: $G$ is not measured and then inserted into the law; the kinetic coefficient of F1, already fixed by $\rho_P$ and $\ell_P$, \emph{is} $c^4/G$. The force law and its constant emerge together from the mechanics of the medium. \epi{DERIVED}.

\textbf{Third law: homogeneity is not postulated.} The equality of action and reaction is conservation of momentum, which Noether's first theorem delivers from the invariance of F1 under spatial translations. The theorem is common to every theory with an action; what is native is that translational symmetry is not an assumption about the stage: \emph{el Pleno} \emph{is} homogeneous by A1 ---a single substance, with no privileged points---, so that translational invariance is a property of the medium and not a condition imposed on a background space. The conserved current of the phase sector, $J^\mu$ with $\partial_\mu J^\mu=0$ exactly on solutions (E7c, \S{}I.7.10), is the instance of that machinery in the dynamical sector. \epi{DERIVED} via Noether, over an axiomatic homogeneity.

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

\subsubsection{\texorpdfstring{III.1.2 --- The gravitational potential $1/r$ as a consequence of the kinetic term of F1}{III.1.2 - The gravitational potential as a consequence of the kinetic term of F1}}

\textbf{Origin.} The starting point is the dynamical equation of the modulus of \emph{el Pleno} (F2, equation E6 of \S{}I.7.10), obtained by applying Euler-Lagrange to the Lagrangian F1. In compact form, retaining the kinetic term and the potential:


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

We seek the dynamics of a small perturbation of the modulus, $S = S_0 + \delta S$ with $S_0 = S_{\min}^\text{estr}$ the \emph{Pleno}-ground-state background (\S{}II.2), in the static weak-field regime. The goal is to show that two perturbations of this type interact with a potential $\propto 1/r$ and to read the prefactor of that interaction.


\textbf{Step 1 --- Linearization of F2 around the background.} One writes $S = S_0 + \delta S$ with $|\delta S| \ll S_0$ and retains the terms linear in $\delta S$. In the static regime ($\partial_t \to 0$) and at scales large compared with $\ell_P$ (where the flexural term $\Box(\Box S)$ is negligible against the kinetic one, suppressed by a factor $(\ell_P/L)^2$ with $L$ the macroscopic scale), the d'Alembertian reduces to the spatial Laplacian, $\Box \to -\nabla^2$. The phase term does not contribute at linear order for a pure-modulus perturbation with constant background phase. The potential is expanded in a Taylor series around the minimum: the zeroth-order term vanishes (minimum condition, \S{}II.4) and the coefficient of the quadratic term is the curvature $V''_\text{eff}(S_{\min}^\text{estr}) = \xi\,\rho_P c^2$ that defines the mass of the radial mode (the Higgs mode of \emph{el Pleno}, $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$, \S{}II.5.4 and \S{}III.1.9). The linearized equation is:


\[
-\rho_P c^2\ell_P^2\,\nabla^2(\delta S) + \xi\,\rho_P c^2\,\delta S = \rho_S(\mathbf{x})
\]

where $\rho_S(\mathbf{x})$ is the source density. In the gravitational regime ---distances much greater than the inverse of the mass of the radial mode--- the mass term is negligible against the Laplacian, and the equation reduces to the Poisson form:


\[
\rho_P c^2\ell_P^2\,\nabla^2(\delta S) = -\rho_S(\mathbf{x})
\]

\textbf{Step 2 --- Solution by Green function.} The Laplacian in three spatial dimensions has Green function:


\[
\nabla^2 \mathcal{G}(\mathbf{r}) = \delta^3(\mathbf{r}) \quad\Longrightarrow\quad \mathcal{G}(\mathbf{r}) = -\frac{1}{4\pi|\mathbf{r}|}
\]

For a point source $\rho_S(\mathbf{x}) = q_1\delta^3(\mathbf{x})$, the solution is:


\[
\delta S_1(\mathbf{r}) = \frac{q_1}{\rho_P c^2\ell_P^2}\cdot\frac{1}{4\pi r}
\]

The perturbation of the modulus of \emph{el Pleno} produced by a point source decays as $1/r$. This is the Newtonian form, obtained without postulating it: it is the response of the elastic medium to a localized perturbation.


\textbf{Step 3 --- Interaction energy between two perturbations.} The elastic-superposition interaction energy of two perturbations $\delta S_1$ and $\delta S_2$ is, by the kinetic term of F1, the integral of the product of their gradients:


\[
U_\text{int} = \int \rho_P c^2\ell_P^2\,\nabla(\delta S_1)\cdot\nabla(\delta S_2)\,d^3x
\]

Integrating by parts (the boundary terms vanish because the perturbations decay at infinity) and using $\nabla^2(\delta S_2) = -\rho_{S,2}/(\rho_P c^2\ell_P^2)$ from Step 1:


\[
U_\text{int} = -\int \rho_P c^2\ell_P^2\,\delta S_1\,\nabla^2(\delta S_2)\,d^3x = \int \delta S_1(\mathbf{x})\,\rho_{S,2}(\mathbf{x})\,d^3x
\]

For the second point source $\rho_{S,2}(\mathbf{x}) = q_2\delta^3(\mathbf{x}-\mathbf{r}_{12})$ at distance $r_{12}$, substituting $\delta S_1$ from Step 2:


\[
U_\text{int} = q_2\,\delta S_1(\mathbf{r}_{12}) = \frac{q_1 q_2}{\rho_P c^2\ell_P^2}\cdot\frac{1}{4\pi r_{12}}
\]

The interaction energy between two perturbations of \emph{el Pleno} is exactly of the Newtonian form $\propto 1/r_{12}$.


\textbf{Step 4 --- Identification of the prefactor.} Comparing with the Newtonian gravitational potential energy $U_\text{Newton} = -G m_1 m_2/r_{12}$, and using that the gravitational charges of \emph{el Pleno} are proportional to the mass-energies with the normalization $q_i = \sqrt{4\pi}\,m_i c^2$ ---a choice that absorbs exactly the factor $4\pi$ of the Green function of Step 2---, the coefficients must coincide:


\[
\frac{1}{\rho_P c^2\ell_P^2}\cdot\frac{q_1 q_2}{4\pi r_{12}} = \frac{(\sqrt{4\pi}\,m_1 c^2)(\sqrt{4\pi}\,m_2 c^2)}{4\pi\,\rho_P c^2\ell_P^2\,r_{12}} = \frac{m_1 m_2 c^2}{\rho_P\ell_P^2\,r_{12}} \stackrel{!}{=} \frac{G m_1 m_2}{r_{12}}
\]

Solving, the kinetic coefficient of \emph{el Pleno} is identified \textbf{without residual factor} with the combination of constants of Einstein's gravity:


\[
\rho_P c^2\ell_P^2 = \frac{c^4}{G}, \qquad G = \frac{c^2}{\rho_P\ell_P^2}
\]

The normalization $q_i = \sqrt{4\pi}\,m_i c^2$ is not a free fit: it is the only one that makes the $4\pi$ of the Green function cancel, leaving the structural identity clean. No undetermined geometric normalization factor remains.


\textbf{Result.} The $1/r$ form is a direct consequence of the quadratic kinetic term of F1; in accordance with the Gravity Shield, it is not presented as a distinctive result. The non-trivial content is the identification of the prefactor with $c^4/G$, developed in \S{}III.1.3.


\textbf{Dimensional verification of Step 1.} The term $\rho_P c^2\ell_P^2\nabla^2(\delta S)$ has dimension $[kg/m^3][m^2/s^2][m^2][m^{-2}] = [kg/(m\cdot s^2)] = [J/m^3]$, energy density ($\delta S$ dimensionless). The source $\rho_S$ has the same dimension $\checkmark$.


\textbf{Epistemic classification.} The obtaining of the $1/r$ form is \epi{DERIVED} by linearization of F2 and the Green function. The normalization of the gravitational charge ($q_i = \sqrt{4\pi}\,m_i c^2$) is fixed uniquely by the cancellation of the $4\pi$ of the Green function; no undetermined factor remains. The induced-action chain of \S{}III.1.4 confirms the same coefficient $c^4/(16\pi G)$ by an independent route.


\subsubsection{\texorpdfstring{III.1.3 --- The value of $G$ as a structural identity}{III.1.3 - The value of as a structural identity}}

\textbf{Origin.} The identity $\rho_P c^2\ell_P^2 = c^4/G$ obtained in \S{}III.1.2 Step 4 solves for the gravitational constant in terms of the structural parameters of \emph{el Pleno}. The non-trivial content is why, among the infinitely many combinations of $(\rho_P, \ell_P, c)$ dimensionally compatible with $G$, it is precisely this one that the dynamics of \emph{el Pleno} selects.


\textbf{Central result (structural identity).}


\[
\boxed{\;G = \frac{c^2}{\rho_P\ell_P^2}\;} \qquad(\text{see }E3)
\]

\textbf{Numerical verification.} With $\rho_P = 5.155\times 10^{96}$ kg/m$^{3}$ and $\ell_P = 1.616\times 10^{-35}$ m:


\[
c^2 = 8.9880\times 10^{16}\ \mathrm{m^2/s^2}, \quad \ell_P^2 = 2.6115\times 10^{-70}\ \mathrm{m^2},
\]

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

The CODATA value is $G_\text{obs} = 6.6743\times 10^{-11}$ m$^{3}$ kg$^{-1}$ s$^{-2}$~\cite{codata}. The agreement is exact by construction (see \S{}I.8.5, relation $\ell_P = \sqrt{\hbar G/c^3}$); the observed numerical residue ($< 0.1\%$) reflects only the rounding of the primary parameters to four significant figures, not a physical discrepancy.


\textbf{Dimensional verification.}


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

\textbf{The dynamical selection mechanism.} Three lines sustain that (E3) is not a dimensional coincidence nor a circular fit:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Dimensional independence.} The three parameters $(\rho_P, \ell_P, c)$ have distinct dimensions and are dimensionally independent. Their operational inference from $(G, \hbar, c)$ is a methodological artifact, not an ontological statement: the primary parameters of \emph{el Pleno} are logically prior to the constants they derive. (Ontologically, in accordance with \S{}I.8.9, $c$ is causal structure and not a substantive property: the independence invoked here is dimensional, and the substantive quantities of \emph{el Pleno} are two, $\rho_P$ and $\ell_P$.)
  \item \textbf{Unique dynamical selection.} Among the infinitely many combinations of $(\rho_P, \ell_P, c)$ with the dimension of $G$, the gravitational induced action over the fluctuations of \emph{el Pleno} (\S{}III.1.4) selects specifically $c^2/(\rho_P\ell_P^2)$. Other combinations are not produced by the dynamics of F1.
  \item \textbf{Independent calibration as an open program.} The independent measurement of $(\rho_P, \ell_P)$ from non-gravitational and non-quantum physics is an objective of the research program (\S{}V.3), not a precondition of the current status.
\end{itemize}


\textbf{Extended reading: $G$ as the softness of the medium.} The identity (E3) admits an ontological reading that \S{}V.7.2 (thematic appendix) develops: inverting it, $1/G = K/c^4$ with $K = \rho_P c^2\ell_P^2 = c^4/G \approx 1.21\times 10^{44}$ N (the ``Planck force'', the structural tension of the medium), so that $G$ is not a force strength but the \emph{compliance} (softness) of \emph{el Pleno}, the inverse of its structural rigidity. Gravity is weak because \emph{el Pleno} is extremely dense and therefore extremely rigid; gravitational weakness ceases to be a hierarchy mystery and becomes the macroscopic reading of an almost infinitely rigid medium. That same appendix traces the route to closing the calibration thread (C-meta-1.b): measuring the saturation density $\rho_{\max}\approx 2.18\,\rho_P$ of a compact object would solve for $\rho_P$ without passing through the Planck scale.

\textbf{Epistemic classification.} The structural identity $G = c^2/(\rho_P\ell_P^2)$ is \epi{DERIVED}: the $1/r$ form is a direct dynamical derivation (\S{}III.1.2), and the fixing of the specific value proceeds from the dynamical selection of the induced action (\S{}III.1.4). It is not \epi{CALIBRATED} ---no parameter is fitted to observed $G$--- nor \epi{HYPOTHESIS} ---the selection mechanism is explicit.


\subsubsection{III.1.4 --- The induced action and the emergence of Einstein's equations}

\textbf{Origin.} The emergence of gravity as a macroscopic property of \emph{el Pleno} is articulated through the induced-action mechanism: the integration over the fluctuations of the field of \emph{el Pleno} generates, in the effective low-energy regime, a gravitational action of the Einstein-Hilbert form. This mechanism ---proposed by Sakharov (1967)~\cite{sakharov}--- is applied here to the field $\Psi$ of \emph{el Pleno}.


\textbf{Step 1 --- Identity of the gravitational mode.} The effective metric $g_{\mu\nu}^\text{eff}$ is not a primitive structure external to \emph{el Pleno}: it emerges as the transverse-traceless tensor mode (TT mode) of the perturbed field $\Psi$ itself (\S{}II.5.5). The gravitational mode over which the induced action is computed is \emph{el Pleno} itself:


\[
g_{\mu\nu}^\text{eff} = \eta_{\mu\nu} + h_{\mu\nu}(\Psi)
\]

where $h_{\mu\nu}$ is a function of the transverse-traceless tensor perturbations of \emph{el Pleno} and $\eta_{\mu\nu}$ is the background metric of the relaxed regime.


\textbf{Step 2 --- Effective action over el Pleno itself and scale derivation.} The integration over the fluctuations of el Pleno produces an effective action with the Einstein-Hilbert structure:


\begin{equation}
\Gamma_\text{ind}[g] = -\frac{c^4}{16\pi G}\int d^4x\sqrt{-g}\,R + \frac{\ell_P^2}{20}\int d^4x\sqrt{-g}\,\mathcal{C}[R^2] \tag{E34}
\end{equation}

The curvature coefficient is a mechanical property of the medium before it is a combination of human constants: by (E3), $c^4/G=\rho_P c^2\ell_P^2$, so that $c^4/(16\pi G)=\rho_P c^2\ell_P^2/(16\pi)$. The stiffness of the Einstein-Hilbert term \emph{is} the kinetic stiffness of F1 read in geometric language.

where the quadratic-curvature term $\mathcal{C}[R^2]$ is neither free nor dangerous: it is the geometric translation of the flexural term $\gamma_\kappa(\Box S)^2$ that F1 already contains, with \epi{DERIVED} coefficient $\alpha_{R^2} = \ell_P^2/20$ (the factor $20 = \ell(\ell+1)|_{\ell=4}$, the $SO(3)$ relaxation invariant of the medium, the same $\sqrt{20}$ that converges in UV regulation, $\tau_\Pi$ and the Heisenberg floor), positive sign forced by A4 ($\gamma_\kappa = +\rho_P c^2\ell_P^4/40 = +\hbar c/40 > 0$), and the only massive degree of freedom associated being the Lee-Wick fakeon (Anselmi--Piva), which at scales $L \gg \ell_P$ decouples (residue of the physical pole $=+1$, E35-ter below) recovering pure Einstein. The higher-curvature sector introduces no free parameter or instability; its complete spectral content is developed in the following Steps (E35-bis to E35-quater).

The robust content of this mechanism ---independent of the regularization scheme--- are two facts. \emph{First, the sign}: the integration of bosonic fluctuations generates an Einstein-Hilbert term with positive coefficient, that is attractive gravity ($G > 0$). \emph{Second, the scale}: by the heat-kernel expansion (Seeley-DeWitt coefficients) the coefficient of the term $\sqrt{-g}\,R$ is proportional to the square of the ultraviolet cutoff, $1/G_\text{ind} \propto \Lambda^2$. In PIU-$\Psi$ the cutoff is not free: it is the microscopic scale of the medium $\Lambda = 1/\ell_P$ (axiom A4). Restoring units with the native identity $\hbar = \rho_P c\,\ell_P^4$ (already presented in \S{}I.7.8 and \S{}I.8.5; rederived by quantization of the phase mode in \S{}III.2.2, E36), one obtains


\[
\frac{c^4}{G_\text{ind}} \;\sim\; \frac{\hbar c}{\ell_P^2}\cdot b \;=\; \rho_P c^2 \ell_P^2 \cdot b\,,
\]

where $b$ is an $\mathcal{O}(1)$ coefficient that groups the Seeley-DeWitt factor and the mode density of el Pleno. This scale reproduces $G$ \textbf{exactly}, and not by identifying $\ell_P$ with a length defined from $\hbar$ and $G$: eliminating $\rho_P$ between the identities of the medium $G=c^2/(\rho_P\ell_P^2)$ (E3) and $\hbar=\rho_P c\,\ell_P^4$ (E36) \emph{yields} $\ell_P^2 = \hbar G/c^3$ as a consequence, and substituting it into $G_\text{ind} \sim c^3\ell_P^2/(\hbar\,b)$ returns $G_\text{ind} \sim G/b$. The coincidence of the kinetic coefficient of F1, $\rho_P c^2\ell_P^2/2 = c^4/(2G)$ (\S{}III.2.3), with the Einstein-Hilbert coefficient is therefore a consequence of both sharing the same structural scale $\rho_P c^2\ell_P^2$, not a fitted coincidence.


\textbf{Step 3 --- The fixed point $\gamma_G = 1$ as a quantitative condition.} The ratio between the effective and the structural gravitational constant, $\gamma_G \equiv G_\text{eff}/G$, equals $\gamma_G = 1$ if and only if the $\mathcal{O}(1)$ coefficient of Step 2 takes the value $b = 8\pi^2$ (equating $K_\text{ind} = b\,\rho_P c^2\ell_P^2/(16\pi^2)$ with $K_{F1} = \rho_P c^2\ell_P^2/2$), that is $b/16\pi^2 = 1/2$. As is closed below, this equality is not a pending numerical computation but a convention identity (the $1/2$ is the kinetic coefficient of F1) plus a non-contamination condition on the TT mode (which is derived). The value $\gamma_G = 1$ has falsifiable content: a generic substrate ---with a scalar gravitational mode or additional polarizations--- would produce $b \neq 8\pi^2$.


\textbf{The condition $b=8\pi^2$ is self-consistency between two routes to $G$.} The equality that fixes $b=8\pi^2$ is equivalent to $b/16\pi^2 = 1/2$, and the factor $16\pi^2$ is the four-dimensional phase-space volume of the heat-kernel ---a standard structure of the expansion, not a lattice number. The condition is then transparent: the coefficient of the induced route (Sakharov/Seeley-DeWitt), divided by its natural phase-space volume, equals the kinetic coefficient of F1, which is exactly $1/2$ ($\rho_P c^2\ell_P^2/2 = c^4/2G$, \S{}III.2.3). That is, $b=8\pi^2$ is the \emph{self-consistency condition} between the two independent routes to the same $G$ ---induced gravity by mode integration and the direct reading of the kinetic coefficient of F1---, and its fulfilment is the internal consistency of the program. The falsifiable value is sharp: a substrate whose gravitational mode had a scalar component or additional polarizations would give $b\neq 8\pi^2$. This reading does not factorize $8\pi^2$ by lattice numbers: it reads the equality as a condition between two coefficients already present in the program, the phase-space volume of the method and the normalization of F1.


\begin{tcolorbox}[derivado,title={Closure of $\gamma_G=1$ --- normalization convention plus non-contamination of the TT mode}]
The condition that fixes $\gamma_G=1$ is $b/16\pi^2 = 1/2$. Its two factors are not pending physics but known structure: the $16\pi^2 = (4\pi)^2$ is the four-dimensional phase-space volume of the heat-kernel (universal to the method, not to the lattice), and the $1/2$ is the kinetic coefficient of F1 (\emph{convention} of normalization of the scalar field, $\tfrac12\rho_P c^2\ell_P^2\,\partial_\mu\Psi^*\partial^\mu\Psi$). To try to ``derive the $1/2$'' from the diamond lattice is therefore a category mistake: it is convention, not a lattice number. What does have physical and falsifiable content ---that the TT mode, integrated, reproduces the rigidity $1/G=\rho_P\ell_P^2$ \emph{without a spurious factor}--- is \epi{DERIVED} by three pieces, all already in the program:
\begin{enumerate}[leftmargin=1.5em,itemsep=2pt]
  \item The gravitational mode is spin-2 transverse traceless, with two physical polarizations and a representation of dimension $5 = 2\ell+1$ ($\ell=2$), obtained by \emph{two convergent routes}: the spectral lineage of el Pleno (Casimir $\ell(\ell+1)=6$, $\ell=2$) and the standard graviton count in $D=4$, $(D{+}1)(D{-}2)/2 = 5$ (verified). There is no scalar mode or additional polarization to contaminate $a_1$.
  \item The residue of the physical pole is $+1$ (E35-ter, verified): standard coupling of a massless mode to the curvature.
  \item The Lee-Wick fakeon contributes to $a_0$ ($\sim\Lambda^4$, cosmological constant), \emph{not} to $a_1$ ($\sim\Lambda^2$, the induced $G$): it does not contaminate the coefficient that fixes $G$.
\end{enumerate}
There is no spurious factor: a substrate with a scalar gravitational mode or with additional polarizations would give $b\neq 8\pi^2$ even if the convention of the $1/2$ were identical. The value $b=8\pi^2$ is therefore entirely the structure of the method ($16\pi^2$, phase-space volume) times the convention of F1 ($1/2$): it contains no lattice number to be determined. The condition with physical content ---the non-contamination of the TT mode--- is derived. \textbf{Classification:} \epi{DERIVED} the mode count and the non-contamination; \epi{DERIVED} $\gamma_G=1$, and with it the coefficient $1/4$ of the Bekenstein--Hawking entropy, which depends on the same coefficient $c^4/(16\pi G)$ ---the same rigidity in the same convention, read in two observables.
\end{tcolorbox}


\textbf{The mechanism by which the lattice does not contaminate $\gamma_G=1$.} The diamond discretization does \emph{not} corrupt the sector that fixes $G$, and the reason is structural. The fluctuation operator of the TT mode (\S{}II.5.3, E18--E19) is of \emph{fourth order} because of the flexural term of F1, not of Laplace type,
\begin{equation}
\mathcal{O}_\text{TT} = -\Box - \frac{\ell_P^2}{20}\Box^2 = -\Box\left(1 + \frac{\ell_P^2}{20}\Box\right) .
\tag{E35-bis}
\end{equation}
In momentum space (with $\Box \to -k^2$ in Euclidean signature), the associated propagator decomposes into partial fractions in a \emph{forced} way ---not chosen---:
\begin{equation}
\frac{1}{k^2\left(1 - \dfrac{\ell_P^2 k^2}{20}\right)} \;=\; \frac{1}{k^2} \;-\; \frac{1}{\,k^2 - \dfrac{20}{\ell_P^2}\,} ,
\tag{E35-ter}
\end{equation}
an exact algebraic identity (the difference of both sides is identically null). This exhibits two poles with residues also forced:
\begin{itemize}[leftmargin=1.4em]
  \item a \textbf{physical pole} at $k^2 = 0$ (massless TT mode, the gravitational wave), with residue
  \[
  \operatorname*{Res}_{k^2=0}\frac{1}{k^2(1-\ell_P^2 k^2/20)} = \lim_{k^2\to 0}\frac{1}{1-\ell_P^2 k^2/20} = +1 ;
  \]
  \item a \textbf{Lee-Wick pole}~\cite{leewick,donoghue} at $k^2 = 20/\ell_P^2$ (massive, ultra-Planckian scale, already identified in \S{}II.5.3 as the ultraviolet regulation branch), with residue
  \[
  \operatorname*{Res}_{k^2=20/\ell_P^2}\frac{1}{k^2(1-\ell_P^2 k^2/20)} = -1 .
  \]
\end{itemize}
The consequence is the sought mechanism. The coefficient of the induced Einstein-Hilbert term ---the heat-kernel coefficient $a_1$, which is the one that fixes $G$--- is determined by the \emph{residue of the physical pole}, which equals exactly $+1$: standard coupling of a massless mode to the curvature. The flexural $k^4$ tail of the lattice dispersion ---the same that, averaged over the Brillouin zone, produces the TT form factor $\approx 1.36$--- belongs \emph{entirely} to the massive Lee-Wick pole, not to the physical gravitational mode. In the organization of the heat-kernel this means that the lattice correction affects the coefficient $a_0$ (the induced cosmological constant, $\sim\Lambda^4$), \emph{not} the coefficient $a_1$ (the induced $G$, $\sim\Lambda^2$). Therefore:
\begin{equation}
\frac{b}{16\pi^2} = \frac{1}{2} \quad\text{is preserved in the sector that fixes } G,
\tag{E35-quater}
\end{equation}
and $\gamma_G = 1$ is robust against the diamond discretization. \textbf{Classification:} the residue of the physical pole $=+1$ is \epi{DERIVED} (exact partial fractions, E35-ter); the localization of the lattice correction in $a_0$ and not in $a_1$ is \epi{DERIVED} (the mode count $2\ell+1=5$ by two routes excludes scalar contamination); in consequence $\gamma_G = 1$ is \epi{DERIVED}. The objection that the discrete lattice could give $b\neq 8\pi^2$ is thereby answered: the lattice correction lives in $a_0$, not in the $a_1$ that fixes $G$, and the TT form factor of the Brillouin zone ($\approx 1.36$) belongs to the Lee-Wick pole, not to the physical gravitational mode.

\textbf{Step 4 --- Energy-momentum tensor of \emph{el Pleno}.} By variation of F1 with respect to the inverse metric $g^{\mu\nu}$, the energy-momentum tensor $T_{\mu\nu}^\text{(Pleno)}$ decomposes into four sources, all configurations of the same \emph{el Pleno}: baryonic matter (localized solitons, \S{}II.6), dark matter (excited configurations, \S{}III.7), dark energy (\emph{Pleno}-ground state, \S{}III.7), and induced gauge sector (phase fluctuations, \S{}III.4).


\textbf{Step 5 --- Einstein's equations.} The stationary-action condition $\delta\Gamma_\text{ind}/\delta g^{\mu\nu} = 0$ produces:


\begin{equation}
R_{\mu\nu} - \tfrac{1}{2}R\,g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}^\text{(Pleno)} \tag{E35}
\end{equation}

with the coupling written in properties of the medium, $8\pi G/c^4 = 8\pi/(\rho_P c^2\ell_P^2)$: the constant that measures how much matter curves is the inverse of the stiffness of \emph{el Pleno}.

Einstein's equations emerge as the effective regime of \emph{el Pleno}, not as an independent fundamental structure.


\textbf{Dimensional verification of the coefficient.} By the native identity $[\rho_P c^2\ell_P^2/2] = [kg/m^3][m^2/s^2][m^2] = [kg\cdot m/s^2] = [N]$ and $[c^4/(2G)] = [m^4/s^4]/[m^3 kg^{-1}s^{-2}] = [N]$. Both coincide $\checkmark$.


\textbf{Distinctive structural consequences.} The emergence of Einstein's equations from \emph{el Pleno} has five consequences that distinguish it from postulated General Relativity:


\begin{itemize}[leftmargin=1.4em]
  \item $G$ is not a free parameter --- it is derived via (E3).
  \item Gravity is not a fundamental force --- it is an emergent macroscopic mode of \emph{el Pleno}.
  \item There are no singularities --- blocked by the hard bound $S_{\max}^\text{geom}$ (axiom A6, \S{}II.7.2).
  \item Time emerges --- it is not an external parameter, but a dimension of evolution of \emph{el Pleno}.
  \item The post-Newtonian tests are preserved --- (E35) is identical to Einstein's equations in the relaxed regime.
\end{itemize}

\textbf{Epistemic classification.} The emergence of the Einstein-Hilbert \emph{structure} from F1 via the induced action over the TT mode is \epi{DERIVED} in its robust part: the sign (attractive gravity) and the scale ($1/G \sim 1/\ell_P^2$, which reproduces $G$ exactly with $\ell_P = \ell_\text{Planck}$) are scheme-independent consequences. The identity of the gravitational mode with the TT mode (Step 1) is a structural \epi{DERIVED} identity. The \emph{exact} equality $\gamma_G = 1$ (Step 3) is \epi{SLD}: it requires the value $b = 8\pi^2$ of the effective Seeley-DeWitt coefficient, settled in \S{}III.1.4 as a normalisation convention of F1 plus non-contamination of the TT mode. The recovery of General Relativity as an effective limit is \epi{SLD} ---solid in structure, scale and sign, pending the exact $\mathcal{O}(1)$ factor--- and constitutes a consistency condition of the program.



% ===== III.1.5 Geodesic (inserted V4.8) =====
\subsubsection{\texorpdfstring{III.1.5 --- The geodesic equation as free motion over the effective metric of \emph{el Pleno}}{III.1.5 - The geodesic equation as free motion over the effective metric of el Pleno}}

\textbf{Origin and purpose.} \S{}III.1.4 derived Einstein's equations (E35) as the induced action over the TT mode, and \S{}III.1.7 established that 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. What remains is to close the complementary side of gravity: if matter-energy dictates the metric via E35, how does a test particle move \emph{in} that metric? The answer is the geodesic equation, and this subsection obtains it as a \textbf{consistency corollary}: it introduces no new physics ---it follows from a free excitation of \emph{el Pleno} extremizing its proper time over the effective geometry that the medium itself defines--- and its only verifiable content is that it reproduces, in the weak field, Newton's law $\mathbf{a}=-\nabla\Phi$ already derived independently in \S{}III.1.2. The piece completes the conceptual symmetry of the program: matter configures the medium (E35), the medium guides matter (geodesic), both faces of a single dynamics of \emph{el Pleno}.

\medskip
\begin{tcolorbox}[notabox,title={Ontological discipline: the geodesic does not postulate a stage}]
In General Relativity the geodesic equation describes free motion over a spacetime given in advance. Under PIU-$\Psi$ there is no prior stage: the metric $g_{\mu\nu}^\text{eff}$ is the reading of the state of \emph{el Pleno} (\S{}III.1.7), so the geodesic is not ``motion in a background'' but the trajectory that a free excitation follows \emph{through the medium} when it is perturbed by another mass. The variational principle operates over the proper time defined by the field configuration, not over an external geometry. The time, space and geometry that appear in the geodesic are the emergents of \emph{el Pleno} (\S{}III.1.4, fourth structural consequence), not imported parameters.
\end{tcolorbox}

\textbf{Step 1 --- Variational principle: extremize the proper time.} A free excitation of \emph{el Pleno} ---with no interaction other than the perturbation of the medium produced by a source--- follows the trajectory that extremizes the proper time accumulated along its path. The proper time is measured with the effective metric (\S{}III.1.7):
\[
\tau = \int d\tau,
\qquad
c^2\,d\tau^2 = -\,g_{\mu\nu}^\text{eff}\,dx^\mu dx^\nu. \tag{E-geo-1}
\]
The extremum condition $\delta\tau=0$ is a standard Euler-Lagrange problem with Lagrangian $L=\sqrt{-g_{\mu\nu}^\text{eff}\,\dot x^\mu\dot x^\nu}$ (dot: derivative with respect to the affine parameter). Its variation produces the geodesic equation:
\[
\frac{d^2 x^\lambda}{d\tau^2}
+ \Gamma^\lambda_{\mu\nu}\,\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0,
\qquad
\Gamma^\lambda_{\mu\nu}
= \tfrac{1}{2}\,g^{\lambda\sigma}_\text{eff}
  \bigl(\partial_\mu g^\text{eff}_{\sigma\nu}
       +\partial_\nu g^\text{eff}_{\sigma\mu}
       -\partial_\sigma g^\text{eff}_{\mu\nu}\bigr). \tag{E-geo-2}
\]
The Christoffel symbols $\Gamma^\lambda_{\mu\nu}$ are built entirely from derivatives of $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$; they contain no ingredient foreign to the configuration of \emph{el Pleno}. The form (E-geo-2) is a mathematical consequence of the variational principle, not an independent dynamical postulate.

\textbf{Step 2 --- Weak-field and slow-motion limit.} To verify consistency with the gravity already derived, the Newtonian regime is taken: weak field, $g_{00}^\text{eff}=-(1+2\Phi/c^2)$ with $|\Phi|\ll c^2$ and $g_{ij}^\text{eff}=\delta_{ij}$ (\S{}III.1.2), and slow motion, $dx^i/d\tau \ll c\,dt/d\tau$. In this limit $dt/d\tau\to 1$ and the dominant term of (E-geo-2) is the spatial one with $\mu=\nu=0$:
\[
\frac{d^2 x^i}{dt^2} \simeq -\,c^2\,\Gamma^i_{00}. \tag{E-geo-3}
\]
The symbolic computation of the Christoffel symbol for this metric gives, exactly,
\[
\Gamma^i_{00} = \frac{1}{c^2}\,\partial_i\Phi, \tag{E-geo-4}
\]
so that
\[
\frac{d^2 x^i}{dt^2} = -\,\partial_i\Phi
\qquad\Longleftrightarrow\qquad
\mathbf{a} = -\nabla\Phi. \tag{E-geo-5}
\]
This is Newton's law, identical to the one derived in \S{}III.1.2 by the independent route of the kinetic term of F1 (where $\Phi=-GM/r$). The geodesic of \emph{el Pleno}, in its weak-field limit, reproduces exactly the Newtonian gravitational acceleration, without new parameters and without importing external dynamics.

\textbf{Verification (symbolic computation).} For the static weak metric $g_{\mu\nu}^\text{eff}=\mathrm{diag}\bigl(-(1+2\Phi/c^2)c^2,\,1,\,1,\,1\bigr)$, the direct computation of the Christoffel symbols confirms $\Gamma^x_{tt}=\partial_x\Phi$, $\Gamma^y_{tt}=\partial_y\Phi$, $\Gamma^z_{tt}=\partial_z\Phi$ (with $x^0=t$), whence $\mathbf{a}=-\nabla\Phi$ to first order in $\Phi/c^2$. The factor $(1+2\Phi/c^2)$ tends to $1$ in the weak field, leaving the exact Newtonian acceleration. Closed identity, without fitting.

\textbf{Dimensional verification.} In (E-geo-2), $[\Gamma^\lambda_{\mu\nu}]=[\text{m}^{-1}]$ and $[d^2x/d\tau^2]=[\text{m}/\text{s}^2]$ with $\tau$ in seconds after restoring $c$; both terms of the geodesic are acceleration $\checkmark$. In (E-geo-5), $[\partial_i\Phi]=[\text{m}^2\text{s}^{-2}]/[\text{m}]=[\text{m}/\text{s}^2]$, acceleration $\checkmark$.

\textbf{Relativistic regime and classical tests.} Outside the weak-field limit, (E-geo-2) over the Schwarzschild metric of \emph{el Pleno} (\S{}III.1.10, E-Sch-7) reproduces the trajectories of General Relativity: the precession of the perihelion and the deflection of light are consequences of the complete geodesic over that metric, identical to those of General Relativity because the metric is identical (\S{}III.1.10). The program inherits the tests, it does not recompute them: the geodesic is the \emph{same} equation, operating over the \emph{same} emergent metric.

\textbf{Analysis: what is achieved and why it suffices as a corollary.} The geodesic equation is not new derivational content over \S{}III.1.4 and \S{}III.1.7: given the emergent effective metric, the free motion that extremizes proper time \emph{is} the geodesic, by the same variational principle as in General Relativity. Its value is one of \textbf{consistency and completeness}. \emph{Consistency}: that the geodesic over the effective metric reproduces exactly Newton's law already derived by a distinct route (\S{}III.1.2) confirms that the two descriptions of gravity in the program ---the scalar-field one (the $1/r$ potential of the modulus) and the geometric one (the effective metric)--- are the same physics read in two ways, without contradiction. \emph{Completeness}: it closes the matter$\leftrightarrow$geometry pair. E35 (\S{}III.1.4) says how the matter-energy of \emph{el Pleno} configures the effective metric; the geodesic (E-geo-2) says how that metric guides the motion of the excitations. Without the geodesic, the ``how matter moves'' side would remain implicit; with it, the cycle of emergent gravity is explicitly closed.

\medskip
\begin{tcolorbox}[derivado,title={Card of the geodesic equation in PIU-$\Psi$}]
\textbf{Interpretation.} Free gravitational motion is the trajectory that an excitation of \emph{el Pleno} follows by extremizing its proper time over the effective metric that the medium itself defines (\S{}III.1.7); it is not motion in an external stage, but propagation through the perturbed medium. \textbf{Explanation.} The geodesic equation (E-geo-2) from the variational principle $\delta\tau=0$ over $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$; the Newtonian limit $\mathbf{a}=-\nabla\Phi$ (E-geo-5) by weak-field reduction, coincident with \S{}III.1.2. \textbf{Prediction.} Exact reproduction of Newton's law in the weak field and of the trajectories of General Relativity (precession, deflection) over the Schwarzschild metric of \emph{el Pleno} (\S{}III.1.10), without new parameters.

\smallskip
\textbf{Epistemic classification.} The geodesic equation (E-geo-2) is \epi{DERIVED} as a consequence of the variational principle over the emergent effective metric (\S{}III.1.7), which is itself \epi{DERIVED}. The Newtonian limit (E-geo-5) is \epi{DERIVED} (closed symbolic computation, coincident with \S{}III.1.2 by an independent route). The recovery of the relativistic trajectories is \epi{SLD} conditional on the status of the Schwarzschild metric (\S{}III.1.10). The program presents the geodesic as a consistency-and-completeness corollary of the matter-geometry pair, not as a new autonomous derivational result.
\end{tcolorbox}


\subsubsection{III.1.6 --- The structural rigidities of el Pleno}

\textbf{Origin.} If gravity is a macroscopic property of el Pleno, the medium must possess a set of mechanical rigidities derivable from its structural parameters.


\textbf{Kinetic rigidity (effective Young's modulus).} The coefficient of the kinetic term of F1 is the rigidity of the medium against deformations of its modulus:


\[
K_\text{cin} = \frac{\rho_P c^2\ell_P^2}{2} = \frac{c^4}{2G} \approx 6.0519\times 10^{43}\ \text{N}
\]

identical to the Einstein-Hilbert coefficient (\S{}III.1.4). The value has units of force ($[\text{N}] = [\text{J/m}]$): it is the force per unit relative deformation of the modulus of el Pleno, that is, its \textbf{effective Young's modulus}. Its Planckian magnitude ($\sim c^4/G$, of the order of the Planck force) expresses that el Pleno is a medium of extreme rigidity against deformations of its internal structure: deforming it costs an energy of gravitational scale, which explains why gravity ---the macroscopic mode of that rigidity--- is the weakest of the interactions at ordinary scales.


\textbf{Shear modulus (Gent).} For el Pleno as a Gent hyperelastic medium (\S{}III.1.7), the saturation potential is $V_\text{topo}(S) = -\tfrac{1}{2}\mu_\text{Pleno} J_m\log(1 - S^2/S_{\max,\text{geom}}^2)$ with $J_m = S_{\max,\text{geom}}^2$ the limiting-extensibility parameter ---the hard structural bound of axiom A6, not the effective bound of the present regime. In the small-deformation regime ($S \ll S_{\max,\text{geom}}$), $\log(1-x)\approx -x$ gives $V_\text{topo} \approx \tfrac{1}{2}\mu_\text{Pleno}S^2$, so that $\mu_\text{Pleno}$ is the rigidity of the quadratic response:


\[
\mu_\text{Pleno} = \frac{\rho_P c^2}{S_{\max,\text{geom}}^2} \approx 0.4593\,\rho_P c^2
\]

\textbf{Shear viscosity.} The Maxwell-Kubo relation establishes $\eta = \mu\cdot\tau$, where $\tau$ is the relaxation time of the medium: the characteristic time over which the dissipative response acts. In el Pleno that time \emph{is derived} and is not a free parameter. The dissipation proceeds from the flexural term of F1, whose ultraviolet regulation mode ---the Lee-Wick pole of the propagator--- sits at $\Box\delta S = 20/\ell_P^2$, that is, at the scale $M_\text{LW} = \sqrt{20}/\ell_P$ (dispersion relation E20--E23, \S{}II.5.4; pole derived in \S{}III.3.4). The relaxation time of the medium is the inverse of the frequency of that regulation mode, $\omega_\text{LW} = c\,M_\text{LW} = \sqrt{20}\,c/\ell_P$:
\begin{equation}
\tau_\Pi \;=\; \frac{1}{\omega_\text{LW}} \;=\; \frac{\ell_P}{\sqrt{20}\,c} \;=\; \frac{\sqrt{5}\,\ell_P}{10\,c} \;\approx\; 1.21\times 10^{-44}\ \text{s}\,, \tag{E-relax}
\end{equation}
where the last equality uses $1/\sqrt{20} = \sqrt{5}/10$. The numerical factor is exact: $\tau_\Pi$ is the time it takes the flexural regulation mode to respond, and its scale $\sqrt{20}$ is the same relaxation invariant that appears, derived, in three independent sectors of the Lagrangian (\S{}III.2.6). No fitting coefficient intervenes.

\textbf{Cross-check (consistency with the thermodynamics of the medium).} This derivation of $\tau_\Pi$ is not isolated: it coincides with the fluctuation-dissipation relation of el Pleno established independently in \S{}III.2.5 (E-inc-0c), $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$. Indeed, the structural temperature of the medium satisfies $k_B T_P = \rho_P c^2\ell_P^3 = E_P$ (\S{}III.1.7), so that $\tau_\Pi k_B T_P/\hbar = \tau_\Pi E_P/\hbar = \tau_\Pi\,c/\ell_P = 1/\sqrt{20}$, identical to (E-relax). The two routes ---the inverse of the regulation mode and the fluctuation-dissipation thermodynamics--- yield the same $\tau_\Pi$, which confirms that the relaxation time is a derived property of the medium and not an imposed parameter. With this:


\[
\eta_\text{Pleno} = \mu_\text{Pleno}\cdot\tau_\Pi = \frac{\rho_P c^2}{S_{\max,\text{geom}}^2}\cdot\frac{\sqrt{5}\,\ell_P}{10\,c} = \frac{\sqrt{5}\,\rho_P c\,\ell_P}{10\,S_{\max,\text{geom}}^2} \approx 2.56\times 10^{69}\ \text{Pa}\cdot\text{s}
\]

The value, in pascal-second (the SI unit of dynamic viscosity), is the intrinsic shear viscosity of el Pleno: it measures the internal dissipation of the medium when it responds to a deformation on the relaxation time scale $\tau_\Pi$. It is the quantity that, divided by the entropy density, fixes the ratio $\eta/s$ of el Pleno ---the hydrodynamic signature of the medium comparable with the Kovtun-Son-Starinets bound.

\textbf{Confirmation by Green-Kubo (fluctuation-dissipation).} The form $\eta = G\cdot\tau_\Pi$ is not only the Maxwell-Kubo relation: it is obtained independently from the fluctuation-dissipation theorem via the Green-Kubo formula, which expresses the viscosity as the time integral of the autocorrelation function of the shear stress:
\begin{equation}
\eta = \frac{V}{k_B T}\int_0^\infty \langle\sigma_{xy}(0)\,\sigma_{xy}(t)\rangle\,dt. \tag{E-GK-1}
\end{equation}
The medium provides the two ingredients of the integrand. First, el Pleno has a \emph{single} relaxation mode ---the Lee-Wick pole of the flexural (\S{}III.3.4)---, so that the stress correlation decays exponentially with the single time $\tau_\Pi$:
\begin{equation}
\langle\sigma_{xy}(0)\,\sigma_{xy}(t)\rangle = \langle\sigma_{xy}^2\rangle\,e^{-t/\tau_\Pi}. \tag{E-GK-2}
\end{equation}
Second, the instantaneous variance of the stress is fixed by the Zwanzig-Mountain equipartition theorem, which relates the stress fluctuation to the instantaneous shear modulus $G_\infty$:
\begin{equation}
\langle\sigma_{xy}^2\rangle = \frac{k_B T}{V}\,G_\infty. \tag{E-GK-3}
\end{equation}
Substituting (E-GK-2) and (E-GK-3) into (E-GK-1), the exponential integral is elementary:
\begin{equation}
\eta = \frac{V}{k_B T}\cdot\frac{k_B T}{V}\,G_\infty\int_0^\infty e^{-t/\tau_\Pi}\,dt
= G_\infty\,\Big[-\tau_\Pi\,e^{-t/\tau_\Pi}\Big]_0^\infty
= G_\infty\,\tau_\Pi. \tag{E-GK-4}
\end{equation}
The two routes ---Maxwell-Kubo ($\eta=\mu\,\tau$) and Green-Kubo (E-GK-4)--- converge on the same form $\eta = G\cdot\tau_\Pi$ with the same scale $\tau_\Pi$, by independent constructions: the first viscoelastic, the second from equilibrium statistical mechanics. \epi{DERIVED}.

\textbf{The modulus that enters $\eta$ depends on the deformation regime.} Green-Kubo fixes the form but does not select the modulus: the $G$ that enters $\eta = G\,\tau_\Pi$ is that of the channel and regime of the concrete deformation. The medium admits three regimes with three moduli, all derived and legitimate:
\begin{center}
{\small
\begin{tabular}{lll}
\toprule
\textbf{Regime} & \textbf{Modulus} & \textbf{Application} \\
\midrule
Linear shear & $\mu_\text{din} = \rho_P c^2$ (instantaneous) & TT (gravitational) waves \\
Finite compression & $\mu_\text{Gent} = \rho_P c^2/S_{\max,\text{geom}}^2$ & cosmic crunch-bounce cycle \\
Linear compression & $K = \tfrac14\xi\,\rho_P c^2 S_{\min}^{2,\text{estr}}$ (volumetric) & amplitude transport in the ground state \\
\bottomrule
\end{tabular}}
\end{center}
The viscosity of the cosmic cycle ---$\eta_\text{cycle} = \mu_\text{Gent}\,\tau_\Pi \approx 2.56\times10^{69}$ Pa$\cdot$s, the one that enters the damping equation of the bounce (E57, \S{}III.8)--- uses $\mu_\text{Gent}$ because the cycle is finite-deformation compression near $S_{\max}$, the regime where the rigidity of the Gent wall is the relevant one. The TT-wave sector, by contrast, is infinitesimal linear shear, and its own shear viscosity is
\begin{equation}
\eta_\text{TT} = \mu_\text{din}\,\tau_\Pi = \rho_P c^2\,\tau_\Pi = \frac{\sqrt5}{10}\,\rho_P c\,\ell_P \approx 5.58\times10^{69}\ \text{Pa}\cdot\text{s}, \tag{E-eta-TT}
\end{equation}
with the corresponding kinematic viscosity $\nu_\text{TT} = \eta_\text{TT}/\rho_P = (1/\sqrt{20})\,c\,\ell_P$, the pure angular word of the flexural without the factor $S_{\max}^2$. \epi{DERIVED} by two routes (instantaneous Maxwell viscoelastic and Green-Kubo shear).

\textbf{Characteristic tension of el Pleno.} It is obtained from the flexural rigidity and the maximum curvature:


\begin{itemize}[leftmargin=1.4em]
  \item Step 1 --- Flexural rigidity: $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$ (\S{}III.2.3).
  \item Step 2 --- Maximum curvature: $\kappa_{\max} = 1/\ell_P$ (axiom A4).
  \item Step 3 --- Bending force: $F_\text{flex} = \gamma_\kappa\kappa_{\max}^2 = \gamma_\kappa/\ell_P^2$.
  \item Step 4 --- Identification with the tension:
\end{itemize}

\[
\sigma_\text{Pleno} = \frac{\gamma_\kappa}{\ell_P^2} = \frac{\hbar c/40}{\ell_P^2} = \frac{1}{40}\cdot\frac{\hbar c}{\ell_P^2} = \frac{F_\text{Planck}}{40} \approx 3.026\times 10^{42}\ N
\]

where $\hbar c/\ell_P^2 = c^4/G = F_\text{Planck}$ was used (valid since $\ell_P = \ell_\text{Planck}$). The tension of el Pleno is the Planck force reduced by the flexural factor 40.


\textbf{Three structural invariants of el Pleno.} The rigidities reveal three pure numbers: $\sqrt{2}$ (packing invariant, diamond lattice, \S{}II.3), $\sqrt{20}$ (relaxation invariant, UV regulation, \S{}III.3), and $40$ (flexural tension invariant, in $\gamma_\kappa = \hbar c/40$ and $\sigma_\text{Pleno} = F_\text{Planck}/40$).


\textbf{Dimensional verification.} $[\sigma_\text{Pleno}] = [\gamma_\kappa]/[\ell_P^2] = [J\cdot m]/[m^2] = [J/m] = [N] \checkmark $.


\textbf{Epistemic classification.} $K_\text{cin} = c^4/(2G)$ is \epi{DERIVED}. $\mu_\text{Pleno} = \rho_P c^2/S_{\max,\text{geom}}^2$ is \epi{DERIVED} (rigidity of the quadratic response of the Gent potential, with $S_{\max,\text{geom}}^2$ derived in \S{}II.4.1). The relaxation time $\tau_\Pi = \ell_P/(\sqrt{20}\,c)$ is \epi{DERIVED}: it is the inverse of the frequency of the flexural regulation mode (Lee-Wick pole $M_\text{LW} = \sqrt{20}/\ell_P$, \S{}III.3.4), with no fitting parameter (E-relax). In consequence $\eta_\text{Pleno} = \mu_\text{Pleno}\,\tau_\Pi$ is \epi{DERIVED}: the product of two derived quantities by the Maxwell-Kubo relation, exact for the Maxwell viscoelastic solid that el Pleno realizes in the shear sector (\S{}III.1.8). $\sigma_\text{Pleno} = F_\text{Planck}/40$ is \epi{SLD}. The geometric decomposition of the factor 40 is \epi{DERIVED}: the coefficient $1/40$ is the ratio between the fourth-order curvature and the square of the kinetic term of the diamond-lattice operator, under the isotropic SO(3) angular average, derived explicitly in \S{}II.3.8 (P9). No geometric factor remains undetermined.


\subsubsection{III.1.7 --- The effective metric and el Pleno as a hyperelastic medium}

\textbf{The effective metric as the Cauchy-Green tensor of the medium.} The effective metric $g_{\mu\nu}^\text{eff}$ is not a primitive geometric structure: it is the \textbf{Cauchy-Green tensor} of el Pleno as a hyperelastic medium, that is, the pull-back of the flat causal structure (axiom A3) by the deformation map $X^a(x)$ of the medium,
\begin{equation}
g_{\mu\nu}^\text{eff} = \eta_{ab}\,\frac{\partial X^a}{\partial x^\mu}\frac{\partial X^b}{\partial x^\nu} = (F^{\mathsf T}\eta\,F)_{\mu\nu},
\qquad F^a{}_\mu \equiv \partial_\mu X^a. \tag{E39a}
\end{equation}
Writing the deformation map as the relaxed configuration plus a displacement, $X^a(x) = x^a + u^a(x)$, the deformation gradient is $F^a{}_\mu = \delta^a_\mu + \partial_\mu u^a$, and the pull-back expands order by order:
\begin{equation}
g_{\mu\nu}^\text{eff}
= \underbrace{\eta_{\mu\nu}}_{\text{order }0}
+ \underbrace{\eta_{a\nu}\partial_\mu u^a + \eta_{b\mu}\partial_\nu u^b}_{\text{order }1\;\equiv\;h_{\mu\nu}}
+ \underbrace{\eta_{ab}\,\partial_\mu u^a\,\partial_\nu u^b}_{\text{order }2}. \tag{E39b}
\end{equation}
The zeroth order is the Minkowski metric (relaxed regime, A3); the first order is exactly the TT mode of the corpus, $h_{\mu\nu} = \eta_{a\nu}\partial_\mu u^a + \eta_{b\mu}\partial_\nu u^b$; and the second order $\eta_{ab}\,\partial_\mu u^a\partial_\nu u^b$ is the first of the non-linear terms, which are \emph{not added by hand} but are fixed by the same map $X(x)$. The symmetry $g_{\mu\nu}^\text{eff}=g_{\nu\mu}^\text{eff}$ is manifest to all orders in (E39b), so that the object is a \textbf{genuine metric, not a truncated perturbation}. This resolves the non-linear general covariance: since $\Gamma_\text{ind}$ is built with curvature scalars of $g^\text{eff}$ and the invariant measure $\sqrt{-g}\,d^4x$, it is diffeomorphism-invariant \emph{to all orders} in the deformation, not only at linear order. The Schwarzschild metric of el Pleno (\S{}III.1.10), which reproduces perihelion precession and light deflection ---impossible with a merely linearized metric---, is precisely this non-linear Cauchy-Green metric in the static spherical regime, which confirms that the program was already operating with the complete metric. In the complementary analogue-gravity reading, the perturbation of el Pleno perceives $g_{\mu\nu}^\text{eff}$ as the local propagation properties of the medium; the geometry is not the stage, but the reading that the perturbations make of the state of the medium.


\textbf{El Pleno as a Gent hyperelastic medium.} The saturation potential of el Pleno admits a complete constitutive characterization as a Gent hyperelastic medium~\cite{gent}:


\[
V_\text{topo}(S) = -\frac{\mu_\text{Pleno}J_m}{2}\log\left(1 - \frac{S^2}{S_{\max,\text{geom}}^2}\right)
\]

with $\mu_\text{Pleno} = \rho_P c^2/S_{\max,\text{geom}}^2$ and $J_m = S_{\max,\text{geom}}^2$. The logarithmic divergence when $S\to S_{\max,\text{geom}}$ is the signature of a medium with limited extensibility: el Pleno cannot stretch beyond its hard bound, just as a Gent material cannot deform beyond its maximum extensibility.


\textbf{Effective temperature of el Pleno.}


\[
T_P = \frac{\rho_P c^2\ell_P^3}{k_B}
\]

derived by the generalized fluctuation-dissipation theorem. The cross-check $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$ connects the effective temperature with the ultraviolet correction of the propagation mode (\S{}III.3), confirming the internal consistency.


\textbf{Dimensional verification.} $[T_P] = [\rho_P c^2\ell_P^3]/[k_B] = ([kg/m^3][m^2/s^2][m^3])/[J/K] = [J]/[J/K] = [K] \checkmark $. Numerically $T_P \approx 1.42\times 10^{32}$ K, the Planck temperature.


\textbf{Epistemic classification.} The effective metric as the Cauchy-Green tensor $g^\text{eff}=F^{\mathsf T}\eta\,F$ of the medium is \epi{DERIVED}: it is the pull-back of the deformation map, a genuine metric (symmetric, non-degenerate) to all orders, whose zeroth order is $\eta_{\mu\nu}$ and whose first order is the TT mode of the corpus (E39b). The non-linear general covariance is therefore \emph{exact}, not approximate at linear order. The Gent constitutive identification is \epi{SLD}. The effective temperature $T_P$ is \epi{DERIVED} by fluctuation-dissipation, with internal cross-check.


\textbf{Emergent Bianchi identities and covariant conservation.} The exact covariance is not the end of the chain: from it one deduces, without additional postulate, the covariant conservation of the energy-momentum tensor and the contracted Bianchi identities, by applying the \emph{second Noether theorem} to the induced action. Let $\Gamma_\text{ind}[g]$ be diffeomorphism-invariant. An infinitesimal diffeomorphism $x^\mu \to x^\mu + \xi^\mu(x)$ acts on the metric as the Lie derivative $\delta g_{\mu\nu} = \nabla_\mu\xi_\nu + \nabla_\nu\xi_\mu$. Defining the induced Einstein tensor as the functional derivative of the action, $G^{\mu\nu} \equiv (2/\sqrt{-g})\,\delta\Gamma_\text{ind}/\delta g_{\mu\nu}$, the variation of the action under the diffeomorphism is
\[
\delta\Gamma_\text{ind} = \int d^4x\,\sqrt{-g}\;\tfrac12 G^{\mu\nu}\bigl(\nabla_\mu\xi_\nu + \nabla_\nu\xi_\mu\bigr)
= \int d^4x\,\sqrt{-g}\; G^{\mu\nu}\,\nabla_\mu\xi_\nu,
\]
where the second equality uses the symmetry of $G^{\mu\nu}$ (the two terms contribute equally). Integrating by parts with the covariant derivative (boundary term null for $\xi$ of compact support), $\delta\Gamma_\text{ind} = -\int d^4x\,\sqrt{-g}\,(\nabla_\mu G^{\mu\nu})\,\xi_\nu$. Diffeomorphism invariance requires $\delta\Gamma_\text{ind}=0$ for every arbitrary $\xi_\nu$, whence
\begin{equation}
\nabla_\mu G^{\mu\nu} = 0 \tag{E39c}
\end{equation}
---the \emph{contracted Bianchi identity}, emergent from the covariance, not postulated. The induced field equations $G^{\mu\nu} = (8\pi G/c^4)\,T^{\mu\nu}$ (E35) then imply, taking the covariant divergence,
\begin{equation}
\nabla_\mu T^{\mu\nu} = \frac{c^4}{8\pi G}\,\nabla_\mu G^{\mu\nu} = 0, \tag{E39d}
\end{equation}
the \emph{covariant conservation} of the energy-momentum tensor. Both are a consequence of the second Noether theorem applied to the diffeomorphism invariance of $\Gamma_\text{ind}$ ---which in turn rests on the action being built with curvature scalars of the Cauchy-Green metric. \textbf{Classification:} \epi{DERIVED}. The formal chain of Einstein dynamics is closed without presupposing Bianchi or conservation: both emerge from the exact covariance, which emerges from $g^\text{eff}$ being a genuine metric of the medium.

\subsubsection{\texorpdfstring{III.1.8 --- \emph{El Pleno} under the standard of continuum mechanics}{III.1.8 - El Pleno under the standard of continuum mechanics}}

\textbf{Motivation.} The program has sustained since \S{}I.1 that \emph{el Pleno} is described \emph{with the language and descriptors proper to continuous media} and that such properties \emph{are not analogies or metaphors}, but the structural characterization of the substrate. That assertion creates an obligation of consistency: if \emph{el Pleno} is a mechanical medium, it must admit the \textbf{complete constitutive card} that continuum mechanics requires of any material ---constitutive equation, closed set of elastic moduli, and rheological classification. All the quantities of that card are derived from the parameters already established ($\rho_P$, $c$, $\xi$, $S_{\min}^\text{estr}$) without introducing new parameters.


\textbf{The constitutive relation.} In continuum mechanics, the behavior of a linear isotropic material is fixed by the generalized Hooke's law, which relates the stress tensor $\sigma_{ij}$ with the strain tensor $\varepsilon_{ij}$:


\begin{equation}
\sigma_{ij} = \lambda\,(\operatorname{tr}\varepsilon)\,\delta_{ij} + 2\mu\,\varepsilon_{ij} \tag{E40}
\end{equation}

where $\lambda$ and $\mu$ are the two Lamé parameters. For \emph{el Pleno}, the strain tensor is not built over displacements with respect to an absolute background ---the medium possesses no such background---, but over the two independent physical responses that F1 admits at quadratic order: the \textbf{shear deformation}, encoded in the transverse-traceless tensor mode (TT mode, \S{}III.3), and the \textbf{volumetric deformation}, encoded in the relative change of the field modulus. The traceless part of $\varepsilon_{ij}$ corresponds to the TT mode; the trace $\operatorname{tr}\varepsilon = \varepsilon_\text{vol}$ corresponds to the compression of the medium.


\textbf{Volumetric deformation of \emph{el Pleno}.} The constitutive identification $S^2 \leftrightarrow n/n_0$ (\S{}II.2.2, \epi{DERIVED}) uniquely fixes what the physical volumetric deformation of the medium is: the relative change of number density of quanta,


\[
\varepsilon_\text{vol} = \frac{\delta n}{n_0} = \frac{\delta(S^2)}{S_{\min}^{2,\text{estr}}} = \frac{2\,\delta S}{S_{\min}^\text{estr}}\,,
\]

to first order in $\delta S$ around the \emph{Pleno}-ground state $S_0 = S_{\min}^\text{estr}$. There is no freedom of convention here: the deformation measure is the one the program already established for the field modulus.


\paragraph*{Shear modulus $\mu$.} The TT mode propagates at exact velocity $c$ (Result D2, \S{}III.3.3). In an elastic medium the velocity of the shear wave is $v_T = \sqrt{\mu/\rho}$; equating $v_T = c$ with mass density $\rho_P$ gives the propagation shear modulus of \emph{el Pleno}:


\begin{equation}
\mu = \rho_P c^2 \approx 4.63\times 10^{113}\ \text{Pa}\,. \tag{E41}
\end{equation}

It is the rigidity of the medium against shear deformations. (This quantity is distinct from the \emph{static} Gent shear modulus $\mu_\text{Gent} = \rho_P c^2/S_{\max,\text{geom}}^2$ of \S{}III.1.6: the latter measures the quadratic response of the topological potential near the UV wall; $\mu = \rho_P c^2$ measures the dynamical rigidity that fixes the propagation velocity of the TT mode. They are two complementary physical readings of the same medium, not a contradiction.)


\paragraph*{Volumetric modulus $K$.} The compression of the medium (change of $S$) costs potential energy. The density of compression energy around the \emph{Pleno}-ground state is $u = \tfrac{1}{2}V''_\text{eff}(S_{\min}^\text{estr})\,(\delta S)^2$, with $V''_\text{eff}(S_{\min}^\text{estr}) = \xi\,\rho_P c^2$ and $\xi \approx 19.69$ (\S{}I.8.8, \epi{DERIVED}). This rigidity has a precise physical meaning that is developed in \S{}III.1.9: the compression of the medium is the excitation of the \textbf{amplitude mode} of the field $\Psi = S e^{i\theta}$ ---the Higgs mode of the condensate of \emph{el Pleno}, distinct from the phase (Goldstone) mode that produces $\mu$---, so that $K$ \emph{is} the rigidity of the Higgs mode of the medium, not a generic static rigidity. The volumetric modulus is the second derivative of the energy with respect to the volumetric deformation, $K = \mathrm{d}^2 u/\mathrm{d}\varepsilon_\text{vol}^2$. Substituting $\delta S = S_{\min}^\text{estr}\,\varepsilon_\text{vol}/2$:


\begin{equation}
K = \frac{1}{4}\,\xi\,\rho_P c^2\,S_{\min}^{2,\text{estr}} \approx 1.19\times 10^{114}\ \text{Pa}\,. \tag{E42}
\end{equation}

\paragraph*{Derived moduli.} With $\mu$ and $K$ fixed, the three remaining moduli are exact algebraic identities of isotropic elasticity:


\[
\lambda = K - \tfrac{2}{3}\mu\,, \qquad
E = \frac{9K\mu}{3K+\mu}\,, \qquad
\nu = \frac{3K - 2\mu}{2(3K+\mu)}\,.
\]

Numerically, the first Lamé parameter $\lambda \approx 8.84\times 10^{113}$ Pa, Young's modulus $E \approx 1.23\times 10^{114}$ Pa, and the \textbf{Poisson coefficient}


\begin{equation}
\nu = \frac{3K - 2\mu}{2(3K+\mu)} \approx 0.328\,. \tag{E43}
\end{equation}

\textbf{Constitutive card of \emph{el Pleno}.} The following table consolidates the complete mechanical response of the medium in its current relaxed regime.


{\small
\begin{longtable}{L{3.0cm}L{3.3cm}L{3.3cm}L{2.4cm}}
\toprule
\textbf{Modulus} & \textbf{Expression} & \textbf{Value} & \textbf{Class} \\
\midrule\endhead
Shear $\mu$ & $\rho_P c^2$ & $4.63\times 10^{113}$ Pa & \epi{DERIVED} \\
Volumetric $K$ & $\tfrac{1}{4}\xi\rho_P c^2 S_{\min}^{2,\text{estr}}$ & $1.19\times 10^{114}$ Pa & \epi{DERIVED} \\
Lamé $\lambda$ & $K - \tfrac{2}{3}\mu$ & $8.84\times 10^{113}$ Pa & \epi{DERIVED} \\
Young $E$ & $9K\mu/(3K+\mu)$ & $1.23\times 10^{114}$ Pa & \epi{DERIVED} \\
Poisson $\nu$ & $(3K-2\mu)/(2(3K+\mu))$ & $0.328$ & \epi{DERIVED} \\
Viscosity $\eta_\text{Pleno}$ & $\mu_\text{Gent}\,\tau_\Pi$ & $2.56\times 10^{69}$ Pa$\cdot$s & \epi{DERIVED} \\
Relaxation $\tau_\Pi$ & $\sqrt{5}\,\ell_P/(10c)$ & $1.21\times 10^{-44}$ s & \epi{DERIVED} \\
\bottomrule
\end{longtable}}

\textbf{Physical reading of the results.} Three features of the medium emerge from the card:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Poisson coefficient $\nu \approx 0.33$.} The value falls within the thermodynamic-stability range of an isotropic solid ($-1 < \nu < 1/2$) and is comparable to that of ordinary structural materials (steel $\nu \approx 0.30$, glass $\nu \approx 0.22$). It is not an input: it is a consequence of the ratio $K/\mu = \tfrac{1}{4}\xi S_{\min}^{2,\text{estr}} \approx 2.57$, fixed entirely by the curvature of the potential and the geometry of the \emph{Pleno}-ground state. Under the reading of \S{}III.1.9, this ratio \emph{is} the balance between the Higgs mode (compression, numerator $\propto K$) and the Goldstone mode (shear, denominator $\propto \mu$) of the condensate of \emph{el Pleno}: $\nu$ measures how coupled the two modes are. That a medium of Planckian rigidity yields a Poisson of ordinary mechanical order is a non-trivial internal-consistency verification.
  \item \textbf{Hierarchy $K > \mu$.} The medium resists compression more than shear ($K/\mu \approx 2.57$), a behavior typical of dense cohesive media. The saturation hard bound (axiom A6) reinforces the volumetric resistance.
  \item \textbf{Uniform Planckian magnitude.} All the moduli are of order $\rho_P c^2 \sim 10^{113}$--$10^{114}$ Pa, the Planck pressure. \emph{El Pleno} is, mechanically, the most rigid medium compatible with physics: deforming it costs energy of gravitational scale, which explains why gravity ---the macroscopic mode of that rigidity--- is the weakest interaction at ordinary scales.
\end{itemize}

\textbf{Rheological classification.} \emph{El Pleno} combines elastic response (the previous moduli) with internal dissipation characterized by the shear viscosity $\eta_\text{Pleno}$ (\S{}III.1.6) and the relaxation time $\tau_\Pi$. The Maxwell-Kubo relation $\eta = \mu_\text{Gent}\,\tau_\Pi$ identifies \emph{el Pleno} as a \textbf{Maxwell-type viscoelastic solid} in the shear sector. A Maxwell material behaves, mechanically, like \textbf{a spring and a damper connected in series}: under a rapid solicitation (times short compared with $\tau_\Pi$) the spring dominates and the material responds like an \emph{elastic solid}, storing the deformation and returning it; under a sustained solicitation (long times) the damper yields and the material \emph{relaxes} the stress, flowing slowly. The relaxation time $\tau_\Pi$ is the boundary between the two behaviors. The everyday examples are pitch or glass: they appear to be rigid solids, but over very long timescales they flow. For \emph{el Pleno}, since $\tau_\Pi \approx 1.21\times 10^{-44}$ s is of Planckian scale, \emph{every} experimentally accessible solicitation is ``rapid'' against $\tau_\Pi$, so that \emph{el Pleno} behaves like a pure elastic solid in every observable regime; the viscous component is only relevant in the ultraviolet-regulation dynamics (\S{}III.3.4) and in the intrinsic decoherence (\S{}III.2.5). This is the card of the \emph{linear} response regime; its extension to the \emph{non-linear} response ---\emph{el Pleno} as a Herschel-Bulkley dilatant fluid, with flow index $n(x)=1/(1-x)$ and yield stress derived from the IR wall--- is developed in \S{}III.9, where it is also shown that this non-linear rheology closes circles of confirmation with relativity (A3/TEL) and the A6 hard bound.


\textbf{Wave velocities and a causality note.} The two structural velocities of the medium are the shear one $v_T = \sqrt{\mu/\rho_P} = c$ (exact, the TT mode; coincides with Result D2) and the longitudinal compression one $v_L = \sqrt{(K + \tfrac{4}{3}\mu)/\rho_P} \approx 1.98\,c$. The superluminal value of $v_L$ is a \textbf{phase velocity}, not a signal velocity: the compression (radial) mode is massive ---it has gap $\omega_0 = \sqrt{\xi}\,c/\ell_P$ (E21)--- and therefore does not propagate free perturbations in the infrared. In a dispersive medium with gap, the phase velocity can exceed $c$ while the group (signal) velocity remains bounded by $c$, a standard phenomenon without conflict with relativistic causality (identical to the case of a waveguide or a plasma). The effective Lorentz invariance of the program (axiom A3, Lorentz-emergence theorem) is preserved: no physical signal of \emph{el Pleno} exceeds $c$.


\textbf{Is $c$ derived as the sound velocity of \emph{el Pleno}?} The previous structure invites a natural question: if we know the elastic moduli of the medium, should we not be able to \emph{derive} its sound velocity and obtain $c$? The answer requires care, because the direct route is circular and the program rejects it explicitly.


\begin{tcolorbox}[revisionbox,title={Circularity warning --- $c$ is not derived from the moduli}]
\textbf{The naive derivation is circular.} Computing $v_T = \sqrt{\mu/\rho_P}$ and ``obtaining'' $c$ is empty: the shear modulus was defined by \emph{imposing} $v_T = c$ (Result D2), that is $\mu \equiv \rho_P c^2$, so that $\sqrt{\mu/\rho_P} = \sqrt{\rho_P c^2/\rho_P} = c$ recovers exactly the datum that was introduced. Nothing is derived. Moreover: $c$ is one of the three \textbf{primary structural parameters} $(\rho_P, c, \ell_P)$, declared non-derived within the program (\S{}I.8); a primary cannot be, by construction, a result. The correct causal direction of the program is the inverse: \textbf{$c$ is the input that fixes $\mu$}, not $\mu$ that derives $c$.

\smallskip
\textbf{What is non-circular.} In F1 the constant $c$ plays two a priori independent roles: it fixes the rigidity in the kinetic coefficient $\rho_P c^2 \ell_P^2$, and it fixes the causal structure in the metric $\partial_\mu\Psi\,\partial^\mu\Psi$. That the velocity of the modes coincides with the $c$ of the causal structure \emph{is not automatic}: a Lagrangian with distinct temporal and spatial coefficients would give $v = c\sqrt{\beta/\alpha} \neq c$. The equality $v = c$ is therefore \textbf{equivalent to Lorentz covariance} (axiom A3), and constitutes an \textbf{internal-consistency verification} of F1 ---the rigidity and the causality use the same $c$---, not an independent derivation of the value of $c$.

\smallskip
\textbf{What is genuinely predicted.} What F1 derives without circularity is not the value of $c$, but the \textbf{structure} of the spectrum, expressed in dimensionless quantities independent of $c$: the mode counting (two massless polarizations at velocity $c$ plus a massive amplitude mode), the ultraviolet correction $(k\ell_P)^2/20$, and the pure numbers $\nu \approx 0.328$ and the ratio $v_L/v_T$. These are predictive content of the medium; the value of $c$ is not.
\end{tcolorbox}

\begin{tcolorbox}[notabox,title={Ontological note --- these moduli do not reintroduce the aether}]
The constitutive characterization describes the response of \emph{el Pleno} in its \textbf{covariant relaxed regime}, not the properties of an aether at absolute rest. The moduli $(\mu, K, \lambda, E, \nu)$ are mechanical readings of the already-derived Lagrangian F1 ---they are the \emph{constitutive content} of F1, not an independent mechanical postulate---. There is no privileged rest frame of \emph{el Pleno}: the deformation is measured over $\partial_\mu S$ and the TT mode, both covariant objects, not over displacements with respect to a background. Continuum mechanics appears here as the \textbf{effective limit} of the field dynamics, not as its origin ---in full conformity with the Ontological Protocol of the program: \emph{el Pleno} is the agent, the elastic response is its macroscopic manifestation, and standard physics (here, Cauchy elasticity) is recovered as an effective regional description, never as a fundamental mechanism.
\end{tcolorbox}

\textbf{Epistemic classification.} The moduli $\mu$ and $K$ are \epi{DERIVED}: $\mu$ from the kinetic coefficient of F1 plus the condition $v_T = c$ (Result D2); $K$ from $V''_\text{eff}$ \epi{DERIVED} plus the constitutive identification $S^2 \leftrightarrow n/n_0$ \epi{DERIVED}, without a free parameter. The moduli $\lambda$, $E$, $\nu$ are \epi{DERIVED} as exact algebraic identities of isotropic elasticity between two derived quantities. The identification of the compression mode with the amplitude (Higgs) mode of the condensate ---the step that gives physical meaning to $K$--- is \epi{DERIVED}: it is a consequence of Goldstone's theorem applied to the breaking of $U(1)$ that the program already establishes (\S{}III.1.9). What remains \epi{SLD} is only the specific quantitative identification between the spectral gap and the volumetric modulus via the deformation convention $\varepsilon_\text{vol} = \delta(S^2)/S_{\min}^{2,\text{estr}}$ ---a bounded step, not the qualitative identification---. The Maxwell rheological classification via $\eta_\text{Pleno} = \mu_\text{Gent}\tau_\Pi$ is \epi{DERIVED}: it is the product of two derived quantities ($\mu_\text{Gent}=\rho_P c^2/S_{\max}^2$ and $\tau_\Pi=\sqrt{5}\,\ell_P/(10c)=\ell_P/(\sqrt{20}\,c)$, the latter derived as the inverse of the Lee-Wick regulation mode in \S{}III.1.6, E-relax) combined by the Maxwell-Kubo relation $\eta=\mu\,\tau$, which is an exact result of linear viscoelasticity, not a conjecture; the \epi{DERIVED} status is conditional on the Maxwell viscoelastic reading of the shear sector, already established in this same section. The value $\nu \approx 0.33$ and its location in the stability range constitute an \textbf{internal-consistency verification} of the program.


\subsubsection{\texorpdfstring{III.1.9 --- The Higgs mode of \emph{el Pleno} and the root of the electroweak sector}{III.1.9 - The Higgs mode of el Pleno and the root of the electroweak sector}}

\textbf{Two excitations of a field with broken symmetry.} The field of \emph{el Pleno} $\Psi = S\,e^{i\theta}$ is complex and carries a global $U(1)$ symmetry (axiom A5). In the \emph{Pleno}-ground state the modulus takes a non-null value, $S_0 = S_{\min}^\text{estr} \approx 0.7231$: the $U(1)$ symmetry is \textbf{spontaneously broken}. Every complex field with broken $U(1)$ possesses exactly two classes of low-energy excitation, and Goldstone's theorem fixes their nature:


\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{phase mode} $\delta\theta$ ---excitation along the vacuum circle, which costs no potential energy--- is the \textbf{Goldstone boson}: massless (null gap). It is the mode that \S{}III.4 projects over the A/B bipartition to produce the induced gauge sector (the photon and the $SU(2)\times U(1)$ structure).
  \item The \textbf{amplitude mode} $\delta S$ ---radial excitation that does cost potential energy, governed by $V''_\text{eff}$--- is the \textbf{Higgs mode} (or amplitude mode, Anderson-Higgs mode of the condensate): massive, with gap $\omega_0 = \sqrt{\xi}\,c/\ell_P$ and rest mass $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$ (\S{}II.5.4, E21).
\end{itemize}

What \S{}II.4 called generically the ``massive radial mode'' is, read correctly, the \textbf{Higgs mode of the condensate of \emph{el Pleno}}. This is not an analogy with condensate physics: it is the same mathematical structure, that of the Ginzburg-Landau order parameter and that of the Bogoliubov condensate, which the program already invokes in \S{}II.2.2 via the identification $S^2 \leftrightarrow n/n_0$.

\begin{tcolorbox}[problemabierto,title={Refinement: elastic mass versus propagating mass of the Higgs mode}]
The value $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$ is, by construction, a \emph{tree-level} result: it arises from neglecting the flexural term $(\ell_P^2/20)u^2$ in the complete dispersion relation of the mode (E20). Solving (E20) without that approximation, the equation at $k=0$, with $W\equiv(\omega\ell_P/c)^2$, is $W^2+20W-20\xi=0$, with two roots: a non-propagating UV branch (Lee-Wick fakeon, $\omega^2<0$) and a \textbf{physical propagating branch} with $W=12.22$. From this it results that the Higgs mode has \emph{two} physically distinct masses that the tree-level fuses into one:
\begin{itemize}[leftmargin=1.4em]
  \item the \textbf{elastic mass} $m_\text{elast}=\sqrt{\xi}\,E_P\approx 4.44\,E_P$, which measures the static stiffness of the well (curvature $V''$) and governs the volumetric modulus $K$, the Poisson coefficient and the healing length $\xi_\text{heal}=\ell_P/\sqrt{\xi}$;
  \item the \textbf{propagating mass} $m_\text{prop}=\sqrt{12.22}\,E_P\approx 3.50\,E_P$, gap of the physical branch after the flexural regulation, which governs the cost of exciting a quantum that \emph{propagates}.
\end{itemize}
The ratio $m_\text{prop}/m_\text{elast}\approx 0.788$ measures how much the flexural regulation moves the mode away from its unregulated behavior ---it is the same factor as the damping $Q\approx 0.80$ of the radial mode. The correction is large ($21\%$) because the gap of the Higgs mode lives near the flexural scale itself; for the \emph{observable} modes (photon, TT mode at small $k$) the analogous correction is $\sim(k\ell_P)^2\sim 10^{-86}$, negligible. \textbf{No observable changes}: both masses are ultra-Planckian and inaccessible, and the distinction is ontological (there are two masses, not one), not observational. \epi{DERIVED} in the physical root of (E20); recorded as a refinement thread in \S{}V.3.
\end{tcolorbox}


\textbf{One single structure, four readings.} The following table shows that four distinct quantities are readings of the same pair of modes of the condensate.


{\small
\begin{longtable}{L{2.7cm}L{2.7cm}L{2.6cm}L{3.0cm}}
\toprule
\textbf{Mode} & \textbf{Nature} & \textbf{Gap / mass} & \textbf{Mechanical reading} \\
\midrule\endhead
Phase $\delta\theta$ & Goldstone & $0$ & shear modulus $\mu = \rho_P c^2$ \\
Amplitude $\delta S$ & Higgs / amplitude & $\sqrt{\xi}\,E_P \approx 4.44\,E_P$ & volumetric modulus $K$ \\
\bottomrule
\end{longtable}}

The volumetric modulus is thus linked to the square of the mass of the Higgs mode of the medium:


\begin{equation}
K = \frac{1}{4}\,\xi\,\rho_P c^2\,S_{\min}^{2,\text{estr}} = \frac{1}{4}\left(\frac{m_\text{radial}c^2}{E_P}\right)^{2}\rho_P c^2\,S_{\min}^{2,\text{estr}}\,. \tag{E44}
\end{equation}

\textbf{Numerical verification.} With $\xi = 19.69$ (curvature of the potential, \S{}I.8.8) and $S_{\min}^{2,\text{estr}} = 0.5229$, the volumetric modulus is $K = \tfrac{1}{4}\cdot 19.69\cdot 0.5229\,\rho_P c^2 \approx 2.5742\,\rho_P c^2$. It is of the same order as the shear modulus $\mu_\text{Pleno}$, as corresponds to a medium with Poisson coefficient $\nu\approx 0.33$.

\textbf{Reinterpretation of the Poisson coefficient.} Under this reading, $\nu \approx 0.33$ ceases to be a mechanical number and comes to measure the \textbf{balance between the Goldstone mode and the Higgs mode} of the condensate. The limits are transparent: if the Higgs mode were infinitely heavy ($\xi \to \infty$, condensate infinitely rigid to compression), $K \to \infty$ and $\nu \to 1/2$ ---the medium would be incompressible, like a superfluid in the London limit---; if both modes were equally soft, $\nu \to 0$. The value of \emph{el Pleno}, $\nu \approx 0.33$, reflects that its Higgs mode has a \textbf{finite gap of Planck order}: the condensate is compressible precisely because its amplitude mode is excitable, at a cost of $4.44\,E_P$.


\textbf{Healing length.} To the Higgs mode corresponds an amplitude-coherence scale ---the length over which the condensate restores its modulus after a local perturbation---:


\[
\xi_\text{heal} = \frac{\hbar}{m_\text{radial}c} = \frac{\ell_P}{\sqrt{\xi}} \approx 0.225\,\ell_P \approx 3.64\times 10^{-36}\ \text{m}\,.
\]

It is sub-Planckian by the factor $\sqrt{\xi}$, consistent with the Higgs mode being the most rigid excitation of the medium.


\textbf{The root of the electroweak sector: two nested breakings.} The Higgs mechanism of the Standard Model ---the breaking $SU(2)\times U(1) \to U(1)_\text{em}$ that gives mass to $W^\pm$, $Z^0$ and the fermions--- appears in the program (\S{}III.4) as an open thread in its quantitative closure. The present reading offers a valuable structural framing: \emph{el Pleno} realizes a Higgs mechanism \textbf{once in its own fundamental condensate} (breaking of global $U(1)$, amplitude mode of Planckian mass), and the electroweak sector would be a \textbf{second nested instance of the same mechanism}, operating over the phase sector projected onto the A/B bipartition (\S{}III.4.2) and at an enormously lower infrared scale.


\begin{tcolorbox}[hipotesisbox,title={Structural hypothesis --- the electroweak Higgs as a second breaking}]
The electroweak Higgs boson (mass $\approx 125$ GeV) \textbf{is not} the Higgs mode of \emph{el Pleno} (mass $\approx 4.44\,E_P$): they are distinct physical objects ---the first breaks $SU(2)\times U(1)$, the second breaks the global $U(1)$ of the medium---. What they share is the \emph{mathematical structure}: both are the amplitude mode of a spontaneous breaking of the same field $\Psi$. Under PIU-$\Psi$, the electroweak Higgs would then be a regional, nested, low-energy manifestation of the same type of phenomenon that \emph{el Pleno} executes in its fundamental condensate. The bridge between the Planckian scale of the Higgs mode of the medium and the electroweak scale would be the Coleman-Weinberg dimensional transmutation, anchored in the pole $\mu^* \approx 5.40\times 10^{17}$ GeV already identified in \S{}III.4.7. This reframes the Higgs mechanism: from an \emph{external postulate of the Standard Model} it becomes a \emph{second realization of a mechanism that the substrate already realizes once in itself}. \textbf{[Structural HYPOTHESIS]}: the framing is solid, but the quantitative closure (deriving the mass of $125$ GeV and the hierarchy of $\sim 17$ orders) depends on the Yukawa coupling of the top and on the transmutation mechanism, open threads declared in \S{}III.4.5--III.4.9. The program does \emph{not} claim to have derived the electroweak value; it claims to have identified its structural root in the Higgs mechanism of \emph{el Pleno} itself.
\end{tcolorbox}

\textbf{Note on epistemic discipline.} The temptation to read numerical coincidences between the scale of the Higgs mode of \emph{el Pleno} ($\sqrt{\xi}\,E_P$) and the pole $\mu^*$ of \S{}III.4.7 is rejected explicitly: $\mu^*$ is itself a result of an open thread, and any ratio between the two would be post-hoc numerology, prohibited by the validation protocol of the program. The connection that the program sustains is \textbf{structural} (same mechanism, two instances), \textbf{not numerical}.


\textbf{Epistemic classification.} The identification of the phase mode with the Goldstone boson and of the amplitude mode with the Higgs mode of the condensate is \epi{DERIVED} (Goldstone's theorem applied to the breaking of $U(1)$ of axiom A5). The relation (E44) between $K$ and the mass of the Higgs mode is \epi{DERIVED}. The interpretation of $\nu$ as the Goldstone-Higgs balance is \epi{SLD}. The identification of the electroweak sector as a second nested breaking of the same field is \textbf{[structural HYPOTHESIS]}, with its quantitative closure declared as an open thread in \S{}III.4.



% ===== III.1.10 Schwarzschild (inserted V4.8) =====
\subsubsection{\texorpdfstring{III.1.10 --- The Schwarzschild metric as the geometric reading of the static configuration of \emph{el Pleno}}{III.1.10 - The Schwarzschild metric as the geometric reading of the static configuration of el Pleno}}

\textbf{Origin and purpose.} \S{}III.1.2--III.1.4 derived the gravity of \emph{el Pleno} in its physical content: the $1/r$ potential from the kinetic term of F1 (\S{}III.1.2), the value of $G$ as a structural identity (\S{}III.1.3) and Einstein's equations as the induced action over the TT mode (\S{}III.1.4, E35). What is missing is not new physics, but a \emph{translation}: to show that the static configuration of \emph{el Pleno} around a point mass, read as effective geometry $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$ (\S{}III.1.7), is exactly the Schwarzschild metric in the notation that a relativist recognizes. This subsection is deliberately a \textbf{bridge}: each step reorganizes an already-established result and the novelty resides, not in the metric ---which General Relativity already gives---, but in \emph{what} replaces the metric postulate (the configuration of the modulus) and in \emph{how} the program cures by construction the two pathologies of the standard solution (the central singularity and the nature of the horizon), relying on already-derived pieces.

\medskip
\begin{tcolorbox}[notabox,title={Ontological discipline: why $R_{\mu\nu}=0$ is \emph{not} solved with an \emph{ansatz}}]
The standard derivation of Schwarzschild postulates a spherically symmetric metric $g_{\mu\nu}$ with two free functions and fixes them by imposing $R_{\mu\nu}=0$. Under the program's Gravity Shield, that path would be ontological contamination: it treats $g_{\mu\nu}$ as a primitive object to be determined, precisely the alarm ``metric postulated without deriving''. In PIU-$\Psi$ the metric is not primitive: \textbf{it is the reading} that the perturbations make of the state of the medium (\S{}III.1.7). Therefore the correct route is the inverse: one solves first the static configuration of the \emph{modulus} $\delta S(r)$ around the source ---physics of \emph{el Pleno}, already done in \S{}III.1.2--- and \emph{reads} the effective geometry from it. The ``vacuum'' of General Relativity ($T_{\mu\nu}=0$) is, here, the relaxed \emph{Pleno}-ground state with a single localized modulus perturbation and no propagating excitations.
\end{tcolorbox}

\textbf{Step 1 --- The static configuration of the modulus and the effective potential.} For a point source of mass $M$ at rest, \S{}III.1.2 (Steps 1--2) already solved the linearized equation of the modulus in the static regime,
\[
\rho_P c^2\ell_P^2\,\nabla^2(\delta S) = -\rho_S(\mathbf{x}),
\qquad
\delta S(r) = \frac{q}{\rho_P c^2\ell_P^2}\cdot\frac{1}{4\pi r},
\]
with the gravitational-charge normalization $q=\sqrt{4\pi}\,Mc^2$ (\S{}III.1.2, Step 4) that cancels the $4\pi$ of the Green function. The effective gravitational potential ---the energy per unit test mass in the field of the perturbed modulus--- is, substituting the structural identity $\rho_P c^2\ell_P^2 = c^4/G$ (\S{}III.1.2, E3):
\[
\Phi(r) \;=\; -\frac{G M}{r}. \tag{E-Sch-1}
\]
This is the only dynamical input: everything else is geometric reading of $\Phi$ under E35.

\textbf{Step 2 --- The temporal component $g_{00}$, fixed by consistency with Newton.} The effective metric (\S{}III.1.7) is $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}$ with signature $(-,+,+,+)$. The temporal component is fixed by an internal and non-negotiable requirement: that the geodesic equation of a slow particle ($v\ll c$) in $g_{00}$ reproduce the Newtonian acceleration $\mathbf{a}=-\nabla\Phi$ already derived in \S{}III.1.2. In the weak-field and slow-motion limit, the temporal component of the geodesic equation
\[
\frac{d^2 x^i}{dt^2} = -\,\Gamma^i_{00}\,c^2,
\qquad
\Gamma^i_{00} \simeq \tfrac{1}{2}\,\partial_i h_{00}\,,
\]
gives $\mathbf{a} = -\tfrac{1}{2}c^2\nabla h_{00}$. Equating with $\mathbf{a}=-\nabla\Phi$ one obtains $h_{00}=2\Phi/c^2$, that is
\[
g_{00} \;=\; -\Bigl(1+\frac{2\Phi}{c^2}\Bigr)
\;=\; -\Bigl(1-\frac{2GM}{c^2 r}\Bigr). \tag{E-Sch-2}
\]
The factor $2$ is not free: it is fixed by the reproduction of $\mathbf{a}=-\nabla\Phi$, which is Newton's law already derived (\S{}III.1.2). No new parameter has been introduced.

\textbf{Step 3 --- The radial component $g_{rr}$, forced by E35 in the exterior.} For the radial component the weak-field regime does not suffice; here the \emph{complete structure} of the already-emerged Einstein equations intervenes (\S{}III.1.4, E35). Outside the source \emph{el Pleno} is relaxed and without propagating excitations, so that $T_{\mu\nu}^\text{(Pleno)}=0$ and E35 reduces to
\[
R_{\mu\nu}=0 \qquad (r>0,\ \text{exterior of the source}). \tag{E-Sch-3}
\]
Over the static and spherically symmetric effective geometry $g_{\mu\nu}^\text{eff}=\eta_{\mu\nu}+h_{\mu\nu}(\Psi)$ one writes the general form compatible with those symmetries,
\[
ds^2 = -A(r)\,c^2 dt^2 + B(r)\,dr^2 + r^2\,d\Omega^2,
\qquad d\Omega^2=d\vartheta^2+\sin^2\!\vartheta\,d\varphi^2,
\]
where $A,B$ are not postulated free functions but the parametrization of the components of $h_{\mu\nu}(\Psi)$ permitted by the symmetry of the modulus configuration. The non-trivial components of the Ricci tensor for this form are standard:
\[
R_{tt} = \frac{A''}{2B} - \frac{A'}{4B}\Bigl(\frac{A'}{A}+\frac{B'}{B}\Bigr) + \frac{A'}{rB},
\qquad
R_{rr} = -\frac{A''}{2A} + \frac{A'}{4A}\Bigl(\frac{A'}{A}+\frac{B'}{B}\Bigr) + \frac{B'}{rB}.
\]
The combination $\dfrac{R_{tt}}{A}+\dfrac{R_{rr}}{B}=0$ cancels the terms in $A''$ and produces
\[
\frac{1}{rB}\Bigl(\frac{A'}{A}+\frac{B'}{B}\Bigr)=0
\quad\Longrightarrow\quad
\frac{A'}{A}+\frac{B'}{B}=0
\quad\Longrightarrow\quad
A(r)\,B(r)=\text{const}. \tag{E-Sch-4}
\]
The asymptotic-matching condition ---far from the source \emph{el Pleno} recovers the relaxed Minkowski regime, $A,B\to 1$ as $r\to\infty$ (\S{}III.1.7)--- fixes the constant at $1$, so that $B=1/A$. Imposing additionally $R_{tt}=0$ with $B=1/A$ one obtains $\bigl(rA\bigr)'=1$, whose integral is $rA=r-r_s$ with $r_s$ a constant of integration. Therefore
\[
A(r)=1-\frac{r_s}{r},
\qquad
B(r)=\Bigl(1-\frac{r_s}{r}\Bigr)^{-1}. \tag{E-Sch-5}
\]
The constant $r_s$ is fixed by identifying $A(r)=-g_{00}/c^2$ with the weak-field result (E-Sch-2): $A\simeq 1-2GM/c^2 r$ at large distance, whence
\[
r_s = \frac{2GM}{c^2}. \tag{E-Sch-6}
\]

\medskip
\begin{tcolorbox}[ecuacionclave,title={Schwarzschild metric as the effective geometry of static \emph{el Pleno}}]
\[
ds^2 = -\Bigl(1-\frac{r_s}{r}\Bigr)c^2 dt^2
       +\Bigl(1-\frac{r_s}{r}\Bigr)^{-1} dr^2
       + r^2\,d\Omega^2,
\qquad
\boxed{\,r_s=\dfrac{2GM}{c^2}\,}. \tag{E-Sch-7}
\]
It is not a vacuum postulate: it is the geometric reading (\S{}III.1.7) of the static configuration of the modulus $\delta S(r)\propto 1/r$ (\S{}III.1.2), forced in its form by the emergent Einstein equations (E35) with the only freedom $r_s$ fixed by the Newtonian matching.
\end{tcolorbox}

\textbf{Dimensional verification.} $[r_s]=[G][M]/[c^2]=\bigl([m^3 kg^{-1}s^{-2}][kg]\bigr)/[m^2 s^{-2}]=[m]$, length $\checkmark$. For $M=M_\odot$, $r_s\approx 2.95$ km, in agreement with the canonical value. The terms $A,B$ are dimensionless $\checkmark$.

\textbf{Verification of limits (post-Newtonian tests).} The expansion of (E-Sch-7) in $r_s/r$ reproduces the three classical tests without fitting, by being identical to the General Relativity solution: precession of the perihelion ($\Delta\phi=6\pi GM/c^2 a(1-e^2)$ per orbit), deflection of light ($\Delta\theta=4GM/c^2 b$) and gravitational redshift ($1+z=(1-r_s/r)^{-1/2}$). This is an obligatory consequence of E35 coinciding with Einstein's equations in the relaxed regime (\S{}III.1.4, fifth structural consequence); the program inherits the tests, it does not recompute them.

\medskip
\textbf{The two pathologies of the standard solution and their cure in \emph{el Pleno}.} The solution (E-Sch-7) presents, in postulated General Relativity, two singular places: $r=r_s$ (horizon) and $r=0$ (curvature singularity). The distinctive content of PIU-$\Psi$ ---the only thing this subsection adds over the mere translation--- is that both change status by pieces \emph{already derived} in the program.

\emph{(i) The horizon $r=r_s$ is the sonic surface of the medium, not a postulated causal boundary.} At $r=r_s$ one has $g_{00}\to 0$ and the escape velocity equals $c$. \S{}III.9.6 already established ---with status \epi{SLD}--- that the propagation velocity of the modes of \emph{el Pleno} is $c$ (Result D2, \S{}III.3), so that the horizon is exactly the surface where the flow of the medium reaches its sound velocity: $\mathrm{Ma}=v/c=1$. It is the analogue of acoustic gravity of Unruh, but \emph{exact and not analogical}. The coordinate singularity of the horizon ---removable by change to Eddington-Finkelstein coordinates in General Relativity--- thus acquires a native physical meaning: it is the sonic barrier of \emph{el Pleno}, the same bound as the light barrier and the dilatant \emph{jamming} (\S{}III.9.5). The Hawking radiation is read as radiation of that sonic surface.

\emph{(ii) The central singularity $r=0$ does not exist: the core is regular by the A6 hard bound.} In General Relativity the Kretschmann curvature invariant diverges, $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12\,r_s^2/r^6\to\infty$ as $r\to 0$. That divergence presupposes that matter collapses to infinite density. In PIU-$\Psi$ it is impossible by construction: the modulus cannot exceed the hard bound $S\leq S_{\max}^\text{geom}$ (axiom A6, \S{}II.6.2), so that the energy density of the medium is bounded above. The interior of a black hole is not a singularity but a \textbf{saturated core} of modulus $S\to S_{\max}$ with the Gent wall active (\S{}IV.7): the collapse concentrated and confined by its own horizon does not rebound ---unlike the cosmic bounce (\S{}IV.7)---, but neither does it reach infinite curvature. The Schwarzschild solution of PIU-$\Psi$ is, therefore, exact exterior and \emph{regular in the interior}: the metric (E-Sch-7) holds in the domain where the effective geometric description applies ($S\ll S_{\max}$), and matches, before $r=0$, with the saturated-core regime where the effective geometry ceases to apply but \emph{el Pleno} remains perfectly defined (\S{}IV.7). This is the counterpart, in the static strong field, of the same cure that A6 provides to the bounce: \textbf{the black hole resolves the General Relativity singularity by the same mechanism that the bounce resolves that of the Big Bang} (\S{}IV.7).

\emph{(iii) Matching with the minimum mass.} The radius of the saturated core connects with the already-derived minimum black-hole mass, $M_{\min}=m_P\sqrt{3/(32\pi S_{\max}^2)}\approx 0.117\,m_P$ (\S{}II.5, geometric exhaustion of the packing under A6), which coincides numerically with the Hawking evaporation limit for a distinct ontological reason (interior geometric exhaustion, not divergent temperature). The metric (E-Sch-7) is the exterior description of objects with $M>M_{\min}$; below that mass there is no stable saturated-core configuration and therefore no horizon.

\medskip
\begin{tcolorbox}[notabox,title={Reading: what replaces the postulate, and what is cured by construction}]
The Schwarzschild metric of General Relativity rests on two choices that the program does not need: a postulated metric and an interior that collapses to infinite density. PIU-$\Psi$ substitutes the first by the static configuration of the modulus read as geometry (\S{}III.1.7), and eliminates the second by the A6 hard bound (\S{}II.6.2, \S{}IV.7). What remains is the \emph{same} exterior metric ---with its three classical tests intact--- but with a horizon that is a physical surface (the sonic barrier of the medium, \S{}III.9.6) and a regular interior instead of a singularity. Three domains that standard physics treats separately ---the relativistic limit, the event horizon and the density bound--- are here three manifestations of the same rheology (\S{}III.9.7).
\end{tcolorbox}

\textbf{Analysis: why this is not new physics but is informative.} The metric (E-Sch-7) does not add derivational content over \S{}III.1.4: given E35, Schwarzschild is the standard spherical corollary of $R_{\mu\nu}=0$, and the program asserts it without claiming novelty. Its value is threefold. \emph{First}, it is a \textbf{consistency test}: that the modulus configuration of \S{}III.1.2, read geometrically, gives exactly Schwarzschild confirms that the chain modulus $\to$ potential $\to$ effective metric (\S{}III.1.2, III.1.4, III.1.6) is coherent in the static strong field and not only in the weak field. \emph{Second}, it is a \textbf{confrontation bridge}: it translates into the metric language that a relativist recognizes the results that the program expresses in the language of \emph{el Pleno}, facilitating direct comparison with General Relativity. \emph{Third}, it exhibits the \textbf{two cures} ---sonic horizon and regular core--- as consequences of already-closed pieces (\S{}III.9.6, \S{}IV.7), not as \emph{ad hoc} additions: the PIU-$\Psi$ solution is geodesically complete toward the interior, a property that the General Relativity solution does not possess.

\textbf{Status of the metric.} The metric (E-Sch-7) inherits the status of E35: its \emph{form} is forced by the symmetries and the matching, and the prefactor $8\pi G/c^4$ of E35 ---equivalently $r_s=2GM/c^2$ with exact coefficient $2$--- is fixed by two concordant routes: consistency with Newton in the weak field (E-Sch-2) and the induced action with $\gamma_G=1$ \epi{DERIVED} (\S{}III.1.4). The \emph{quantitative profile} of the matching between the exterior metric (E-Sch-7) and the saturated core of \S{}IV.7 ---the explicit regular interior metric--- is an open derivation thread: the program establishes that the core is regular (A6) and that the horizon is sonic (\S{}III.9.6), but it has not computed the closed form of $g_{\mu\nu}$ in the transition zone $S\to S_{\max}$, where the effective geometric description ceases to apply.

\medskip
\begin{tcolorbox}[derivado,title={Card of the Schwarzschild metric in PIU-$\Psi$}]
\textbf{Interpretation.} The Schwarzschild geometry is the geometric reading of the static configuration of the modulus of \emph{el Pleno} around a point mass, not a postulated vacuum solution. \textbf{Explanation.} The $1/r$ form of the potential from the kinetic term of F1 (\S{}III.1.2); the complete metric structure from the emergent Einstein equations (E35, \S{}III.1.4); the horizon as a sonic surface from $v_\text{mode}=c$ (\S{}III.9.6); the regular core from the A6 hard bound (\S{}II.6.2, \S{}IV.7). \textbf{Prediction.} Exterior metric identical to General Relativity (three classical tests preserved), physical horizon (sonic barrier, $\mathrm{Ma}=1$), \emph{regular} interior without curvature singularity (falsifiable deviation with respect to General Relativity in the core regime, suppressed outside it), and absence of a horizon below $M_{\min}\approx 0.117\,m_P$ (\S{}II.5).

\smallskip
\textbf{Epistemic classification.} The exterior form (E-Sch-7) is \epi{DERIVED} as the spherical corollary of E35 (\S{}III.1.4), with $g_{00}$ \epi{DERIVED} by consistency with Newton (\S{}III.1.2) and $g_{rr}$ \epi{DERIVED} conditional on the status of E35. The horizon as a sonic surface is \epi{SLD} (\S{}III.9.6). The regular core by A6 is \textbf{[SLD; individual pieces derived]} (\S{}II.6.2, \S{}IV.7). The explicit interior metric of the transition zone is a \textbf{declared derivation thread}. The exact coefficient of $r_s$ is fixed by two concordant routes (Newtonian consistency and derived $\gamma_G=1$, \S{}III.1.4). The program does \emph{not} claim derivational novelty over the metric itself: it asserts it as a consistency test and confrontation bridge, with the two cures (sonic horizon, regular core) as distinctive content inherited from already-closed results.
\end{tcolorbox}


\subsubsection{\texorpdfstring{III.1.11 --- Interpretation: what gravity is under PIU-$\Psi$}{III.1.11 - Interpretation: what gravity is under PIU-Psi}}

\textbf{Interpretation.} Gravity is neither a fundamental force nor a primitive geometry: it is the \emph{collective macroscopic behavior of \emph{el Pleno}} in the presence of localized configurations of energy. The picture is that of a continuous elastic medium. A concentration of energy ---what we call mass--- is a region where the modulus of \emph{el Pleno} departs from its rest value; that perturbation is not confined to the point, but decays with distance and modulates the propagation properties of the medium throughout its surroundings, just as a charge deforms the field around it or a mass tenses an elastic membrane. A second configuration, immersed in that modulated medium, propagates following the trajectories the medium allows it ---and those trajectories, read by an observer, are exactly the ones that General Relativity attributes to the ``curvature of spacetime''. The metric $g_{\mu\nu}$ is not the stage on which physics occurs: it is the \emph{name that the perturbations of \emph{el Pleno} give to the local state of the medium} (\S{}III.1.7). Where Relativity says ``mass curves spacetime and spacetime directs mass'', PIU-$\Psi$ says ``localized energy modulates the rigidity of \emph{el Pleno}, and that modulation governs how the other configurations propagate''. The two sentences describe the same phenomenon; the second names the medium that the first dispenses with.


\textbf{Why this produces attraction.} That two masses attract each other is not postulated: it follows from the medium. Each localized configuration locally lowers the energy of \emph{el Pleno} with respect to the background; when two of them approach, the overlap of their modulations reduces the total energy of the system, and every physical medium evolves toward its energy minimum. The gradient of that overlap energy ---computed explicitly in \S{}III.1.2--- is the Newtonian force: it comes out with the correct $1/r^2$ form and with the derived prefactor $G$, not fitted.


\textbf{What is explained and what is gained.} The value of $G$ is explained from the structural parameters of \emph{el Pleno} (\S{}III.1.3), not assumed as a datum: it is the inverse of the rigidity of the medium, $G = c^2/(\rho_P\ell_P^2)$. The form of Einstein's equations is explained from the Sakharov induced action over the fluctuations of the medium (\S{}III.1.4): integrating over the fluctuations of \emph{el Pleno} generates, in the low-energy regime, precisely the Einstein-Hilbert action ---with the same coefficient $c^4/(2G)$ that appears in the kinetic rigidity of F1, a structural coincidence, not a fit. And the absence of singularities is explained from the hard bound of the modulus (A6): \emph{el Pleno} cannot be compressed beyond $S_{\max}$, so that there are no points of infinite density ---neither at the cosmological origin nor in the interior of compact objects. The change of picture has a strong conceptual consequence: the question ``how is gravity quantized?'' ceases to make sense in its standard form, because the metric is not a fundamental field to quantize, but an effective description of a medium whose grain ($\ell_P$) is already the quantum substrate (\S{}III.2.9).


\textbf{Prediction.} The gravity of \emph{el Pleno} predicts: (i) absence of cosmological singularities and in compact objects ---structural bounce instead of singular Big Bang, finite cores instead of black holes with central singularity; (ii) propagation velocity of the gravitational mode exactly equal to $c$, already confirmed by GW170817 (\S{}III.3); (iii) ultraviolet corrections of order $(k\ell_P)^2$ to gravitational propagation, suppressed by \textasciitilde{}69 orders of magnitude for the frequencies accessible today ---undetectable with current technology, but a structural and declared deviation with respect to classical General Relativity.

\subsection{\S{}III.2 --- Fundamental constants as structural identities}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} independent calibration of $(\rho_P,\ell_P)$ (thread \S{}V.3).
  \item \textbf{Notes:} $\hbar$ derived by quantization of the phase mode; $G$ and $\hbar$ linked by $\ell_P = \sqrt{\hbar G/c^3}$ (they are not independent concordances, \S{}I.8.5).
\end{itemize}
\end{estadoseccion}

\subsubsection{III.2.1 --- The strategy: two properties and one structure, the rest derived}

Problem 5 (\S{}I.2.6) is that the fundamental constants are not explained: they are measured and used as inputs. The PIU-$\Psi$ answer is structural. \emph{El Pleno} is characterized by two substantive properties ---$\rho_P$ and $\ell_P$--- and one causal structure ---$c$ (\S{}I.8.9)---, from which the other constants are derived as structural identities or dynamical consequences. The strategy does not reduce the number of constants to zero ---the three primaries remain as inputs---, but shows that those historically treated as independent ($G$, $\hbar$, $\gamma_\kappa$) are distinct readings of the same medium. For economy, in this Part the triplet $(\rho_P, \ell_P, c)$ is used as the primary dimensional quantities, bearing in mind that $c$ is structure and not a free property, so that the \emph{substantive} quantities of \emph{el Pleno} are two, not three.


\textbf{Epistemic distinction.} The program distinguishes three categories of derivational step:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Dynamical derivation (DD):} obtained by solving the dynamics of F1 (Euler-Lagrange, perturbative expansion).
  \item \textbf{Structural identification (SIE):} fixes the specific value of a quantity among the dimensionally admissible combinations, through an explicit dynamical mechanism.
  \item \textbf{Parametric mapping (PM):} term-by-term correspondence between F1 and a validated external theory.
\end{itemize}

The identities $G$ and $\hbar$ are SIE: the functional form is a dynamical derivation, but the fixing of the value proceeds from a selection mechanism (induced action for $G$, quantization of the phase mode for $\hbar$).


\subsubsection{III.2.2 --- The Planck constant as a structural identity}

\textbf{Origin.} The reduced Planck constant $\hbar$ quantifies the minimum action scale of quantum mechanics. In PIU-$\Psi$ it emerges from the canonical quantization of the phase mode $\theta$ of the field of \emph{el Pleno}.


\textbf{Step 1 --- The canonical momentum conjugate to the phase.} From the kinetic term of F1 in variables $(S,\theta)$, the phase sector is $\mathcal{L}_\theta = \tfrac{1}{2}\rho_P c^2\ell_P^2 S^2(\partial_\mu\theta)(\partial^\mu\theta)$. The canonical conjugate momentum is:


\[
\pi_\theta = \frac{\partial\mathcal{L}_\theta}{\partial(\partial_t\theta)} = \rho_P\ell_P^2 S^2\,\partial_t\theta
\]

\textbf{Step 2 --- The Noether charge of the phase.} The invariance under $\theta\to\theta+\alpha$ (axiom A5) produces, by Noether, a conserved current $j^\mu = \rho_P c^2\ell_P^2 S^2\partial^\mu\theta$, whose integrated temporal component is the charge $Q = \int\pi_\theta\,d^3x$. This is the U(1) charge of \emph{el Pleno}.


\textbf{Step 3 --- The fundamental mode as a $U(1)$ rotor and its inertia.} The spatially homogeneous phase mode of \emph{el Pleno} (the zero mode, $\partial_i\theta = 0$) has, integrating the temporal Lagrangian density over the support of the mode, a rotor Lagrangian:
\[
L_\theta = \tfrac12\, I_\theta\,\dot\theta^2, \qquad
I_\theta = \rho_P\ell_P^2\!\int S^2\,d^3x ,
\]
where $(\partial_\mu\theta)(\partial^\mu\theta) = c^{-2}\dot\theta^2 - (\nabla\theta)^2$ was used under the convention $x^0 = ct$ (\S{}I.9.2), which cancels the $c^2$ of the kinetic prefactor. The quantity $I_\theta$ is the \emph{moment of inertia} of the phase rotor, with dimension $[\,\rho_P\ell_P^2\cdot\text{m}^3\,] = \text{kg}\cdot\text{m}^2$ ---angular inertia, not mass. Its conjugate momentum $p_\theta = I_\theta\dot\theta$ is the angular momentum of the rotor, identical to the Noether $U(1)$ charge of Step 2.

\textbf{Step 4 --- The normalization of the fundamental mode, derived (not postulated).} The integral $\int S^2\,d^3x$ is not a free parameter nor a convention: it is fixed by two already-established facts of \emph{el Pleno}. \emph{First}, the modulus $S$ is \textbf{dimensionless} (\S{}I.7.9) and is normalized to the \emph{Pleno}-ground state through the constitutive identification $S^2 = n/n_0$ (\S{}II.2.2); the fundamental quantum is the \textbf{minimum-saturation} excitation, $n = n_0$, that is $S = 1$. \emph{Second}, that fundamental quantum occupies exactly \textbf{one minimum cell of the lattice}, whose volume is $\ell_P^3$ (the diamond lattice with grain $\ell_P$, \S{}II.3). Therefore the norm of the fundamental mode is a geometric consequence:
\[
\int S^2\,d^3x \;=\; \underbrace{S^2}_{=1}\cdot\underbrace{V_\text{cell}}_{=\ell_P^3}
\;=\; \ell_P^3 ,
\]
without any reference to $\hbar$ nor to the quantization of the charge. The inertia of the fundamental rotor is then fixed in parameters of the medium: $I_\theta = \rho_P\ell_P^2\cdot\ell_P^3 = \rho_P\ell_P^5$.

\textbf{Step 5 --- Quantization of the rotor and value of $\hbar$.} The phase lives on the circle $\theta\in[0,2\pi)$, so that the zero mode is a \textbf{planar rotor}. The angular-momentum spectrum of a planar rotor is an exact canonical theorem ---eigenvalues of $\hat p_\theta = -i\hbar\,\partial_\theta$ with periodicity $2\pi$---, independent of any datum of \emph{el Pleno}:
\[
p_\theta = n\hbar,\qquad n\in\mathbb{Z}.
\]
The fundamental quantum ($n=1$) has $p_\theta = \hbar$. On the other hand, that same fundamental mode rotates at the coherence frequency of the medium, $\omega_P = c/\ell_P$ ---not an edge-of-zone estimate, but the \emph{definition} of $\ell_P$ as a coherence length: the scale over which the phase maintains rigidity at velocity $c$, so that $\omega_P\ell_P = c$ exactly (\S{}I.8.3)---. The physical angular momentum of that rotor, in parameters of \emph{el Pleno}, is $p_\theta = I_\theta\,\omega_P$. Equating the canonical quantum with the structural quantity:
\[
\hbar \;=\; I_\theta\,\omega_P \;=\; (\rho_P\ell_P^5)\cdot\frac{c}{\ell_P}
\;=\; \rho_P c\,\ell_P^4 .
\]

\textbf{Why the prefactor is exactly $1$ and there is no circularity.} The equality does not derive $\hbar$ ``out of nothing'': it expresses the action quantum of the phase mode of \emph{el Pleno} in units of the medium, and shows that it coincides with $\rho_P c\,\ell_P^4$ without residual coefficient. Each factor of the unit prefactor has a declared physical origin: the $S^2 = 1$ comes from the saturation of the \emph{Pleno}-ground state, the volume $\ell_P^3$ from the lattice cell, the frequency $\omega_P = c/\ell_P$ from the definition of the coherence scale, and the quantum $n=1$ from the exact spectrum of the $U(1)$ rotor. None is fitted to the observed value of $\hbar$, and none is taken from \S{}III.4.6: the topological quantization of the charge of that section is now a \emph{consequence} of this normalization ($Q_0 = p_\theta = \hbar$ for $n=1$), not its foundation, which eliminates the mutual dependence that the previous version had. \epi{DERIVED}.


\textbf{Central result (structural identity).}


\begin{equation}
\boxed{\;\hbar = \rho_P c\,\ell_P^4\;} \tag{E36}
\end{equation}

\textbf{Numerical verification.} With $\rho_P = 5.155\times 10^{96}$ kg/m$^{3}$, $c = 2.998\times 10^8$ m/s, $\ell_P = 1.616\times 10^{-35}$ m:


\[
\ell_P^4 = 6.8197\times 10^{-140}\ \mathrm{m^4}, \quad \rho_P c = 1.5455\times 10^{105}\ \mathrm{kg\,m^{-2}\,s^{-1}}, \quad \hbar = 1.0540\times 10^{-34}\ \mathrm{J\cdot s}.
\]

The CODATA value is $\hbar_\text{obs} = 1.0546\times 10^{-34}$ J$\cdot$s. The agreement is exact by construction ($\ell_P = \sqrt{\hbar G/c^3}$); the observed numerical residue ($< 0.1\%$) reflects only the rounding of the primary parameters to four significant figures.


\textbf{Dimensional verification.} $[\rho_P c\ell_P^4] = [\mathrm{kg/m^3}][\mathrm{m/s}][\mathrm{m^4}] = [\mathrm{kg\cdot m^2/s}] = [\mathrm{J\cdot s}] \checkmark $.


\textbf{Epistemic classification.} $\hbar = \rho_P c\ell_P^4$ is \epi{DERIVED} in its functional form via quantization of the phase rotor of \emph{el Pleno}, and its numerical prefactor is exactly $1$, also \epi{DERIVED}: the norm of the fundamental mode $\int S^2 d^3x = \ell_P^3$ is obtained from the dimensionlessness of $S$ (saturation $S=1$) and the cell volume $\ell_P^3$ (\S{}II.3), the frequency $\omega_P=c/\ell_P$ from the definition of the coherence scale, and the quantum $n=1$ from the exact spectrum of the $U(1)$ rotor. No ingredient is calibrated against the observed value of $\hbar$ nor taken circularly from \S{}III.4.6.


\subsubsection{III.2.3 --- The native identity that connects gravity and the quantum}

\textbf{Origin.} The identities of $G$ (E3) and $\hbar$ (E36) are not independent; combining them, native relations emerge. These are a direct algebraic consequence of (E3) and (E36).


\textbf{Identity 1 --- Kinetic coefficient equal to that of Einstein-Hilbert.} From (E3), $\rho_P\ell_P^2 = c^2/G$, whence:


\[
\frac{\rho_P c^2\ell_P^2}{2} = \frac{c^2}{2}\cdot\frac{c^2}{G} = \frac{c^4}{2G}
\]

exactly the coefficient of the Ricci scalar in the Einstein-Hilbert action. Gravity and the dynamics of \emph{el Pleno} share the same kinetic coefficient by construction ---not by fitting.


\textbf{Identity 2 --- Flexural rigidity proportional to $\hbar$.} Substituting (E36) into $\gamma_\kappa = \rho_P c^2\ell_P^4/40$:


\[
\gamma_\kappa = \frac{\rho_P c^2\ell_P^4}{40} = \frac{c(\rho_P c\ell_P^4)}{40} = \frac{\hbar c}{40} \qquad(\text{see }E4)
\]

\textbf{Numerical verification of (E4).} Evaluating the native form $\gamma_\kappa = \rho_P c^2\ell_P^4/40$, equivalent to $\hbar c/40$, gives $\gamma_\kappa = (1.0540\times 10^{-34}\times 2.998\times 10^8)/40 \approx 7.90\times 10^{-28}$ J$\cdot$m.


\textbf{Dimensional verification of (E4).} $[\hbar c] = [J\cdot s][m/s] = [J\cdot m]$, dimension of flexural rigidity $\checkmark$.


\textbf{Relation that links $G$ and $\hbar$ (they are not independent concordances).} The two identities of the medium are written in the same primaries,
\[
G = \frac{c^2}{\rho_P\ell_P^2} \quad\text{(E3)},
\qquad
\hbar = \rho_P\,c\,\ell_P^4 \quad\text{(E36)},
\]
and solving the first for $\rho_P$, $\rho_P = c^2/(G\ell_P^2)$, and substituting into the second,
\[
\hbar = \frac{c^2}{G\ell_P^2}\cdot c\,\ell_P^4 = \frac{c^3\ell_P^2}{G}
\qquad\Longrightarrow\qquad
\ell_P^2 = \frac{\hbar G}{c^3}\,.
\]
The relation does not define $\ell_P$: the grain of the medium is a physical property of \emph{el Pleno} (A4), and that its square equals $\hbar G/c^3$ is a \emph{consequence} of the two identities. Fixing the pair $(\rho_P,\ell_P)$ satisfies both at once: $G$ and $\hbar$ are not two independent observational verifications, but a single structural condition read in two languages (\S{}I.8.5). The non-trivial content is that one and the same scale $\ell_P$ and one and the same density $\rho_P$ reproduce at once the gravitational constant and the Planck one, which is not guaranteed for a generic substrate.


\textbf{Interpretation of the native identity.} The proportionality $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$ reveals the genuine quantum nature of the flexural term: it is not an ad hoc classical regularization, but an intrinsic quantum property of the medium. Gravity ($G$), the quantum ($\hbar$) and ultraviolet regulation ($\gamma_\kappa$) are three readings of the same medium.


\textbf{Epistemic classification.} The native identities (Identity 1; the flexural rigidity $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$, see E4; and the consequence $\ell_P^2 = \hbar G/c^3$) are \epi{DERIVED} as direct algebraic consequences of (E3) and (E36).


\subsubsection{III.2.4 --- Quantum mechanics as the dynamics of \emph{el Pleno}}

\textbf{Interpretation.} Quantum mechanics does not describe point particles that ``sometimes behave like waves'': it describes the dynamics of a medium that is, at its root, wave-like. Under PIU-$\Psi$ a particle is a localized perturbation of \emph{el Pleno} ---a small departure $\delta S$ of the modulus over the background, which propagates carrying its phase $\theta$. The wavefunction $\psi$ is not an abstract object of probability: it \emph{is} that perturbation of the medium, with its amplitude ($S$) and its phase ($\theta$). From there follow, without extra postulates, the three features that make quantum mechanics ``strange''. \emph{Superposition} is the linear sum of perturbations of a medium (two waves in the same pond coexist). \emph{Probability} ---the Born rule, $|\psi|^2$--- is the fraction of the conserved quantity of the medium (the phase charge) that is localized in each region: not a separate law, but accounting of something that is conserved. And Heisenberg's \emph{indeterminacy} is the imprint that the medium has a positive curvature rigidity: localizing a perturbation costs energy, and that cost fixes an exact floor $\hbar/2$ to the product of uncertainties (\S{}III.2.6). What follows shows that this reading is not a metaphor, but an exact mathematical equivalence.


\textbf{Origin.} Beyond the value of $\hbar$, the complete structure of quantum mechanics emerges from the dynamics of the field of \emph{el Pleno}. The Schrödinger equation is not postulated: it is an exact mathematical equivalence (Madelung-Bohm transformation) applied to the field $\Psi = S\,e^{i\theta}$, plus the ontological identification of $\Psi$ as the representation of a state of \emph{el Pleno}.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Madelung decomposition.} Substituting $\Psi = S\,e^{i\theta}$ into the dynamics of F1 in the non-relativistic regime and separating real and imaginary parts, two coupled equations are obtained:
\end{itemize}

\[
\partial_t\rho + \nabla\cdot(\rho\,\mathbf{v}) = 0 \quad\text{(continuity)}
\]

\[
\partial_t\mathbf{v} + (\mathbf{v}\cdot\nabla)\mathbf{v} = -\frac{1}{m}\nabla(V + Q) \quad\text{(Bohm equation)}
\]

with $\rho = S^2$ the density, $\mathbf{v} = (\hbar/m)\nabla\theta$ the flow velocity, and $Q$ the Bohm quantum potential.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 2 --- Equivalence with Schrödinger.} The continuity + Bohm system is mathematically equivalent to the Schrödinger equation for the wavefunction $\psi = S\,e^{i\theta}$ (Madelung theorem 1926~\cite{madelung}, Bohm 1952~\cite{bohm}). The equivalence is exact, not approximate.
  \item \textbf{Step 3 --- Ontological identification.} The field $\Psi$ of \emph{el Pleno} is the wavefunction: the particles are perturbations $\delta S\cdot e^{i\theta}$ over the background $S_{\min}^\text{estr}$. This closes the identification: quantum mechanics is the description of the dynamics of the perturbations of \emph{el Pleno} in the non-relativistic regime.
\end{itemize}

\textbf{The Born rule as an ontological property.} The Born rule ---the localization probability is $|\psi|^2$--- is derived, not postulated. For a quantum of \emph{el Pleno} with particle field $\psi = \delta S\cdot e^{i\theta}$, the probability of finding it in a region $R$ is the fraction of Noether U(1) charge (axiom A5) of the quantum localized in $R$:


\[
P(R) = \frac{Q_R}{Q_\text{total}} = \int_R |\psi_\text{norm}(\mathbf{x})|^2\,d^3x
\]

The derivational chain is: axioms A1 + A5 produce the Noether current $j^\mu = \rho_P c^2\ell_P^2 S^2\partial^\mu\theta$; the charge density is $\rho_Q = \rho_P\ell_P^2\omega S^2$; the topological quantization of the total charge (\S{}III.4.6) closes the normalization; and the localized fraction produces the Born rule. The quantum probability is thus an ontological property of the field ---fraction of conserved charge localized---, not an independent postulate.

\textbf{The link that closes Born: quantum equilibrium is an attractor.} The identity $P(R) = \int_R|\psi|^2$ contains a link that should be declared with precision: that the \emph{detection probability} equals the \emph{charge fraction} presupposes that detection samples the conserved charge uniformly, that is, that the actual distribution of configurations $P(\mathbf{x})$ coincides with $|\psi|^2$. This is the quantum-equilibrium hypothesis of the Madelung-Bohm reading. It is not automatic: it is an \emph{equilibrium state} toward which the medium relaxes, and the regulation noise of el Pleno (\S{}III.2.5) is the engine of that relaxation. The mechanism is explicit. The perturbation follows the Bohm velocity $\mathbf{v} = (\hbar/m)\nabla\theta$ plus the flexural diffusive noise of coefficient $D$, a Langevin dynamics whose Fokker-Planck equation for the actual distribution $P(\mathbf{x},t)$ is
\begin{equation}
\partial_t P = -\nabla\cdot(P\,\mathbf{v}) + D\,\nabla^2 P. \tag{E-born-1}
\end{equation}
The quantum density $\rho = |\psi|^2 = S^2$ satisfies the Madelung continuity \emph{without} diffusion, $\partial_t\rho = -\nabla\cdot(\rho\,\mathbf{v})$, so that it is a solution of the convective part. Defining the ratio $f \equiv P/\rho$ (quantum equilibrium $\Leftrightarrow f\to 1$) and substituting $P = f\rho$ into (E-born-1), one obtains a self-adjoint diffusion equation with weight $\rho$:
\begin{equation}
\partial_t f + \mathbf{v}\cdot\nabla f = \frac{D}{\rho}\,\nabla\cdot(\rho\,\nabla f). \tag{E-born-2}
\end{equation}
The Valentini H-function $H \equiv \int \rho\,(f\ln f)\,d^3x \geq 0$ (with $H=0\Leftrightarrow f\equiv 1$) has time derivative, integrating by parts with (E-born-2),
\begin{equation}
\frac{dH}{dt} = -D\int \rho\,\big|\nabla(\ln f)\big|^2\,d^3x \leq 0, \tag{E-born-3}
\end{equation}
negative except at $f\equiv 1$, where it vanishes. Therefore $P$ relaxes monotonically to $|\psi|^2$: quantum equilibrium is an attractor, not a postulate. The coefficient $D$ is not free ---it proceeds from the fluctuation-dissipation of el Pleno, $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$ (\S{}III.2.5)---, and the relaxation rate is the anharmonic thermalization of the medium, of scale $\tau_\Pi$ (\S{}III.2.5), identical for every state and scale. Equilibrium is reached in Planckian time, which explains why deviations from Born are never observed. The H-theorem itself ---that $P$ relaxes to $|\psi|^2$ given a diffusive dynamics--- is \epi{DERIVED}: the identity (E-born-3) is an exact integration by parts, $\frac{dH}{dt} = -D\int\rho\,f\,|\nabla\ln f|^2\,d^3x \leq 0$ with $\rho,f,D>0$. What rests on \epi{SLD} is only the \emph{diffusive form} of the noise of el Pleno (effectively white in the quantum band, with color $\sim(\omega\tau_\Pi)^2\sim 10^{-44}$) and the identification of the rate with $\tau_\Pi$.

\textbf{Epistemic classification.} The emergence of the Schrödinger equation is \epi{DERIVED}: the Madelung-Bohm equivalence is a mathematical theorem, and the ontological identification of $\Psi$ is axiom A1. The Noether charge density $\rho_Q\propto S^2 = |\psi|^2$ and the charge fraction $\int_R|\psi|^2$ are \epi{DERIVED} (Noether plus arithmetic). The link that identifies detection with charge fraction is \epi{DERIVED} conditional on quantum equilibrium; the H-theorem that establishes equilibrium as an attractor is \epi{DERIVED} (E-born-1 to E-born-3, exact algebra), conditional in turn only on the diffusive form of the noise \epi{SLD}. The resolution of the measurement problem (there is no postulated collapse; the randomness is a manifestation of the internal noise of el Pleno) and the resolution of the EPR problem (the non-locality reflects the pre-metric uniqueness of el Pleno, without superluminal signal or instantaneous collapse) are \epi{DERIVED} and \epi{SLD} respectively, with no-signaling verified exactly (\S{}III.2.5).


\subsubsection{III.2.5 --- Decoherence, measurement and Bell correlations}

\textbf{Origin.} The reading of quantum mechanics as the dynamics of \emph{el Pleno} is completed with three results that traditionally require additional postulates: decoherence (how classical behavior emerges), measurement (what resolves the quantum ``randomness''), and Bell correlations (quantum non-locality). The three are derived from the structure of \emph{el Pleno}.


\textbf{Decoherence and intrinsic Planckian time.} \emph{El Pleno} is not a noiseless medium: the ultraviolet-regulation dynamics (the fourth-derivative flexural term of F1, \S{}I.7) introduces intrinsic fluctuations. The fluctuation-dissipation theorem relates those fluctuations to the characteristic relaxation time of the medium. The decoherence of a quantum superposition is the loss of phase coherence induced by that noise.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- The relaxation time of \emph{el Pleno}.} The characteristic time of the regulation dynamics is $\tau_\Pi = \sqrt{5}\,\ell_P/(10\,c)$, derived in \S{}III.1.6 (E-relax) as the inverse of the frequency of the flexural regulation mode. It is the time over which the regulation mode responds to a perturbation.
  \item \textbf{Step 2 --- The fluctuation-dissipation theorem.} The intensity of the phase noise of \emph{el Pleno} is linked to its effective temperature $T_P$ (\S{}III.1.7) by the fluctuation-dissipation theorem: the spectral power of the noise is $\propto k_B T_P$. The cross-verification of \S{}III.1.7 established the dimensionless relation $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$, which connects the relaxation time, the effective temperature and the action quantum.
  \item \textbf{Step 3 --- The decoherence time.} The decoherence rate of a superposition is proportional to the intensity of the noise integrated over the relaxation time. Combining the three noise mechanisms (thermal, ultraviolet regulation and topological coupling) via the fluctuation-dissipation theorem, the intrinsic decoherence time of \emph{el Pleno} scales with the relaxation time corrected by the factor $\sqrt{20}$, raised to the power that results from the quartic dependence of the regulation noise:
\end{itemize}

\[
t_\text{dec}^\text{Planck} = (20)^{1/4}\,\tau_\Pi
\]

where the exponent $1/4$ comes from the fact that the regulation dynamics is fourth-order in derivatives (the noise enters as $(\Box S)^2$, whose characteristic timescale goes as the fourth root of the coefficient). Numerically, with $\tau_\Pi\approx 1.21\times 10^{-44}$ s and $(20)^{1/4} = 2.115$:


\[
t_\text{dec}^\text{Planck} = 2.115\times 1.21\times 10^{-44}\ \text{s} \approx 2.55\times 10^{-44}\ \text{s}
\]

\textbf{Stability of atomic coherence.} Although \emph{el Pleno} has this intrinsic Planckian noise, atomic quantum coherence is preserved by an enormous separation of scales. The effective decoherence rate over a system of frequency $\omega_\text{quantum}$ is suppressed by the ratio $(\omega_\text{quantum}/\omega_P)$, with $\omega_P = c/\ell_P\approx 1.85\times 10^{43}$ s$^{-1}$ the Planckian frequency. For atomic frequencies ($\omega_\text{quantum}\sim 10^{15}$--$10^{21}$ s$^{-1}$), the ratio $\omega_\text{quantum}/\omega_P\lesssim 10^{-22}$ produces a massive averaging of the fluctuations, maintaining atomic coherence on scales of $10^2$ to $10^{14}$ seconds ---consistent with the quantum coherence observed in the laboratory. The structure of the effective evolution operator of \emph{el Pleno} reproduces standard decoherence (Joos-Zeh, Zurek) as an effective limit.


\textbf{Resolution of the measurement problem.} There is no postulated collapse of the wavefunction. The quantum ``randomness'' is the manifestation of the internal noise of el Pleno (the noise of the regulation dynamics), which is deterministic at the level of the medium but is perceived as random at the level of the perturbations. The Born rule is derived (\S{}III.2.4), not postulated. The quantum-classical transition is the decoherence by the noise of the medium, not an external measurement process.


\textbf{The diffusion coefficient and the causal direction of $\hbar$.} The coefficient $D$ that governs the relaxation to Born (\S{}III.2.4, E-born-1) equals the Nelson value $D = \hbar/(2m)$ without circularity. The direct route ---$D = \langle x^2\rangle_\text{zp}\,\omega = (\hbar/2m\omega)\,\omega = \hbar/2m$--- would use the zero-point fluctuation $\langle x^2\rangle_\text{zp}$, which already contains $\hbar$. The Einstein relation obtains it with ingredients that do not contain $\hbar$: $D = k_B T/\gamma$, with the structural temperature $k_B T_P = \rho_P c^2\ell_P^3$ (\S{}III.1.7), the viscosity $\eta_\text{Pleno} = \mu\,\tau_\Pi$ (\S{}III.1.6) and the Stokes drag at scale $\ell_P$. The friction per quantum of el Pleno is
\begin{equation}
\gamma_\text{part} = \mu\,\tau_\Pi\,\ell_P^3 = (\rho_P c^2)\!\left(\frac{\sqrt5}{10}\frac{\ell_P}{c}\right)\!\ell_P^3 = \frac{\sqrt5}{10}\,\rho_P c\,\ell_P^4 = \frac{1}{\sqrt{20}}\,\hbar, \tag{E-born-4}
\end{equation}
where the last equality uses $\hbar = \rho_P c\,\ell_P^4$ (\S{}III.2.2). Both $\hbar$ and the friction hang from the same product $\rho_P c\,\ell_P^4$, so that $D\sim c\ell_P$ and $\hbar/2m\sim c\ell_P$ coincide by construction, not by fitting: the equality $D = \hbar/2m$ is the Einstein relation of the medium rewritten. This confirms the causal direction $\rho_P,\ell_P,c\to\hbar$: $\hbar$ is \epi{DERIVED} from the primary triplet (\S{}III.2.2), not a postulated constant; the distinct routes (quantized rotor, Bohm reading, thermal noise) are consistent by all being the same medium. \epi{DERIVED} in structure and scale; the exact $\mathcal{O}(1)$ prefactor (which fixes the mass of the diffusing quantum) is deferred to the fermionic-mass thread (\S{}V.3), not to quantum mechanics.


\textbf{Bell correlations and resolution of EPR.} A pair of subsystems of el Pleno in a non-separable configuration (the analogue of the singlet state) produces correlations between measurements in distinct directions. The correlation function is derived from the structure of the configuration.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- The singlet state of \emph{el Pleno}.} Two complementary phase modes of \emph{el Pleno} in an antisymmetric configuration form the state $|\Psi^-\rangle = \tfrac{1}{\sqrt{2}}(|{\uparrow\downarrow}\rangle - |{\downarrow\uparrow}\rangle)$, where $\uparrow, \downarrow$ are the two orientations of the phase mode. This is the configuration of total angular momentum zero of the pair.
  \item \textbf{Step 2 --- Measurement operators.} Measuring in the direction $\mathbf{a}$ corresponds to the spin-projection operator $\boldsymbol{\sigma}\cdot\mathbf{a}$. The correlation between measurements in directions $\mathbf{a}$ and $\mathbf{b}$ is the expectation value of the product of the two operators in the singlet state:
\end{itemize}

\[
E(\mathbf{a},\mathbf{b}) = \langle\Psi^-|(\boldsymbol{\sigma}_1\cdot\mathbf{a})(\boldsymbol{\sigma}_2\cdot\mathbf{b})|\Psi^-\rangle
\]

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 3 --- Evaluation.} For the singlet state, the standard Pauli-algebra identity gives $\langle\Psi^-|(\boldsymbol{\sigma}_1\cdot\mathbf{a})(\boldsymbol{\sigma}_2\cdot\mathbf{b})|\Psi^-\rangle = -\mathbf{a}\cdot\mathbf{b} = -\cos\theta_{ab}$, where $\theta_{ab}$ is the angle between the two directions. When the directions are parametrized by angles $a, b$ in the measurement plane, the relative angle is $\theta_{ab} = 2(a-b)$ (the factor 2 arises because the phase mode of \emph{el Pleno} is a spinor: a physical rotation of angle $\alpha$ rotates the spinor by $\alpha/2$):
\end{itemize}

\[
E(a,b) = -\cos\bigl(2(a-b)\bigr)
\]

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 4 --- The Tsirelson bound.} Constructing the CHSH combination~\cite{chsh} $S = E(a,b) - E(a,b') + E(a',b) + E(a',b')$ with the derived correlation function and optimizing over the measurement angles, the maximum is $|S|_\text{max} = 2\sqrt{2}$ ---the Tsirelson bound~\cite{tsirelson}. This violates the classical Bell inequality ($|S|\leq 2$) but respects the quantum limit, exactly as observed experimentally.
  \item \textbf{Step 5 --- No-signaling.} The reduced state of a subsystem is obtained by tracing over the other: $\rho_B = \text{Tr}_A|\Psi^-\rangle\langle\Psi^-| = \tfrac{1}{2}\mathbb{I}$. Since $\rho_B$ is independent of the direction $\mathbf{a}$ chosen to measure subsystem A, no measurement in A alters the local statistics of B: there is no superluminal communication. No-signaling is verified exactly.
\end{itemize}

\textbf{The ontological resolution of the EPR problem.} The quantum ``non-locality'' reflects the pre-metric uniqueness of \emph{el Pleno}. They are not two separate systems that communicate instantaneously; it is a single object ---\emph{el Pleno}--- whose global configuration manifests regionally. There is no action at a distance, no superluminal signal, no instantaneous collapse: there is a single medium whose configuration is global by construction, prior to the emergence of the metric that defines the distances.


\textbf{Dimensional verification.} $[t_\text{dec}] = [\tau_\Pi]$ (the factor $20^{1/4}$ is dimensionless) $= [s]$ $\checkmark$. The correlation $E(a,b)$ is dimensionless, bounded in $[-1,1]$ $\checkmark$.


\textbf{Epistemic classification.} The Planckian decoherence time and the stability of atomic coherence are \textbf{[partial structural closure]}: the structure is derived, with the reproduction of the standard limit verified. The derived Born rule and the Bell correlations with the Tsirelson bound are \epi{DERIVED}. The ontological resolution of EPR via the pre-metric uniqueness is \epi{SLD}. No-signaling is \epi{DERIVED} (exact verification).



% ===== INCORPORATED V4.4: III.2.6 to III.2.9 (MODO INVESTIGADOR session) =====
\subsubsection{\texorpdfstring{III.2.6 --- The uncertainty relation as a structural property of \emph{el Pleno}}{III.2.6 - The uncertainty relation as a structural property of el Pleno}}

\textbf{Origin.} Heisenberg's uncertainty relation, $\Delta x\cdot\Delta p \geq \hbar/2$, is postulated in standard quantum mechanics as a fundamental property of conjugate operators, a consequence of the commutation algebra $[\hat x,\hat p]=i\hbar$. Under PIU-$\Psi$ it is not postulated: it emerges as a property of the dynamics of the field of \emph{el Pleno}. Moreover, the program specifies two things that the standard formulation leaves as brute facts: \emph{why} the bound is exactly $\hbar/2$, and \emph{what the sign $\geq$ means physically}. The short answer is that the floor $\hbar/2$ is the Gaussian saturation of the medium in the long-wave limit, and the sign $\geq$ is the signature of the strictly-positive flexural rigidity of \emph{el Pleno}.

\medskip
\textbf{Method: four internal approximations.} Consistent with the governing principle of the program ---there is nothing outside \emph{el Pleno}, so that no assertion is validated against an external standard, but by the concordance of independent internal derivations (as in the spectral-dynamical convergence of $S_{\min}^\text{estr}$, \S{}II.4.5)--- the position uncertainty is approached by four distinct routes, each operating over a different sector of the Lagrangian F1. Three of them converge on a single scale; the fourth produces a second scale, and the \emph{difference} between the two turns out to be as informative as the coincidences. The final result is not an isolated number but a functional form, (E-inc-3), with physical meaning for each of its terms.

\bigskip
\textbf{Route 1 --- Elastic sector: the vacuum at the edge of critical damping.}
The radial mode of \emph{el Pleno} ---the excitation of the modulus $\delta S$ over the \emph{Pleno}-ground state, identified in \S{}III.1.9 with the amplitude (Higgs) mode of the condensate--- behaves like an oscillator whose natural frequency proceeds from the curvature of the potential, $\omega_0^2 = V''_\text{eff}(S_{\min}^\text{estr})/(\rho_P\ell_P^2) = \xi\,c^2/\ell_P^2$ with $\xi\approx 19.69$ (\S{}I.8.8), and whose dissipation proceeds from the flexural term, with relaxation scale $\propto 20/\ell_P^2$. The ratio that fixes the damping regime is

\begin{equation}
\frac{\xi}{20} \approx 0.9845\,. \tag{E-inc-0a}
\end{equation}

The value is remarkable: the \emph{Pleno}-ground state ---the state in which the current relaxed universe is found--- is situated $1.5\%$ below critical damping ($\xi/20=1$), on the underdamped side. An oscillator in this condition is the one that admits the widest possible response without resonating: maximum sensitivity to perturbations, minimum phase memory. Physically, this means that a localized excitation of the modulus in the \emph{Pleno}-ground state is \emph{marginally unstable to localization}: the medium neither holds it rigidly fixed (it is not a lossless crystal) nor crushes it without response (it is not pure dissipation). The scale that defines the critical threshold is $\sqrt{20}$.

\medskip
That $\xi=19.69$ falls \emph{below} $20$, and not at the exact critical value nor above it, is not a defect of the program but the very condition of its quantum content. On the underdamped side ($\xi/20<1$) the radial mode exists as an oscillation: the modulus can vibrate and, with it, sustain the dynamics that in the effective limit is read as quantum mechanics. At the critical point ($\xi/20=1$) or above it, the mode would cease to oscillate ---it would become purely dissipative--- and the modulus sector would lose its dynamical character. The $1.5\%$ deficit is not, therefore, a margin to be corrected toward $20$: it is the signature that \emph{el Pleno} is tuned as close to the critical threshold as it can be \emph{without extinguishing} the oscillation on which all its quantum phenomenology depends. The medium lives at the edge of its own dynamical viability, and that edge is what makes its quantum phenomenology possible.

\medskip
\textbf{Route 2 --- Flexural sector: the minimum width of an excitation.}
The equation of motion of the modulus (E20) has, besides the physical propagating branch, an ultraviolet-regulation pole (Lee-Wick / fakeon) at $u = 20/\ell_P^2$, where $u = k^2 - \omega^2/c^2$. This pole marks the maximum wavenumber that the medium admits before the flexural regulation cuts off the mode. The minimum width of a localized excitation is, therefore,

\begin{equation}
\sigma_{\min} = \frac{\ell_P}{\sqrt{20}} \approx 0.2236\,\ell_P\,. \tag{E-inc-1}
\end{equation}

This is the hard localization scale of \emph{el Pleno}: below $\sigma_{\min}$ no excitation exists, not because of a lack of measurement resolution, but because the physical medium does not support finer configurations. Once again $\sqrt{20}$ appears.

\medskip
\textbf{Route 3 --- Dissipative sector: phase erasure.}
The regulation dynamics introduces intrinsic fluctuations whose intensity, via the fluctuation-dissipation theorem, is fixed by the relaxation time $\tau_\Pi$ and the effective temperature $T_P$ of the medium. The dimensionless relation established in \S{}III.2.5 is

\begin{equation}
\frac{\tau_\Pi\,k_B T_P}{\hbar} = \frac{1}{\sqrt{20}} \approx 0.2236\,, \tag{E-inc-0c}
\end{equation}

directly verifiable from the primary parameters. It is the internal noise of the medium that erases the phase coherence of a superposition ---the decoherence--- and its characteristic scale is, once more, $1/\sqrt{20}$.

\medskip
\begin{tcolorbox}[notabox,title={Convergence of the first three routes}]
The factor $\sqrt{20}$ appears in three physically distinct sectors of the Lagrangian of \emph{el Pleno}: the elasticity of the potential (Route 1, damping threshold), the dispersion of the modes (Route 2, minimum width) and the thermodynamics of the noise (Route 3, phase erasure). It is not the same computation repeated: they are three independent questions ---how stable is the localization?, how fine can it be?, how long does its phase last?--- that the medium answers with the same scale, $\ell_P/\sqrt{20}$. This convergence is, under the internal method of the program, the validation that there exists a genuine localizability scale of \emph{el Pleno}, prior to any quantum reading.
\end{tcolorbox}

\bigskip
\textbf{Route 4 --- Kinetic sector: the uncertainty product and the canonical/medium distinction.}
The fourth route constructs the product $\Delta x\cdot\Delta p$ directly. Here appears the central conceptual distinction of the section. For a Gaussian wave packet of width $\sigma$, the position dispersion is $\Delta x = \sigma$ and the dispersion of the \emph{wavenumber} is $\langle k^2\rangle = 1/(4\sigma^2)$. If the momentum were the canonical one, $p = \hbar k$, the product would be $\Delta x\cdot\Delta p = \hbar/2$ \emph{exactly and always} ---this is not physics, it is the Fourier theorem, valid for every Gaussian packet however narrow.

The physical content of PIU-$\Psi$ lies in that the \textbf{momentum the medium actually transports is not the canonical one}. The dispersion relation of \emph{el Pleno} (E23) ---common to the photon, the TT mode and the phase mode--- is $\omega^2 = c^2k^2[1+(k\ell_P)^2/20]$, whence the momentum of the medium is

\begin{equation}
p_\text{medium}(k) = \hbar k\sqrt{1 + \frac{(k\ell_P)^2}{20}}\,, \tag{E-inc-2}
\end{equation}

with the term $(k\ell_P)^2/20$ inherited from the flexural rigidity (\S{}II.3.8 P9b). The dispersion of the momentum of the medium over the Gaussian packet is then

\[
\Delta p_\text{medium}^2 = \hbar^2\left[\langle k^2\rangle + \frac{\ell_P^2}{20}\langle k^4\rangle\right]
= \frac{\hbar^2}{4\sigma^2}\left[1 + \frac{3}{80}\frac{\ell_P^2}{\sigma^2}\right]\,,
\]

using $\langle k^4\rangle = 3\langle k^2\rangle^2 = 3/(16\sigma^4)$ (Gaussian quartic moment). The uncertainty product takes the closed form

\begin{equation}
\boxed{\;\Delta x\cdot\Delta p = \frac{\hbar}{2}\sqrt{1 + \frac{3}{80}\left(\frac{\ell_P}{\sigma}\right)^2}\;}\,. \tag{E-inc-3}
\end{equation}

The floor is written with the human rod; in properties of the medium it is $\hbar/2 = \rho_P c\,\ell_P^4/2$ (E36), so that (E-inc-3) does not compare \emph{el Pleno} against an external constant but against a combination of its own density and its own grain. The excess over the floor, by contrast, is dimensionless and depends on no rod: the ratio $\ell_P/\sigma$ measures the localisation demanded against the grain of the medium.

The coefficient $3/80 = (1/20)\cdot 3 \cdot (1/4)$ combines the flexural rigidity ($1/20$), the Gaussian quartic factor ($3$) and the normalization of the second moment ($1/4$). It is a pure number, without free parameters. The distinction between the canonical momentum (which gives $\hbar/2$ by Fourier) and the momentum of the medium (which gives the excess) is the key: \textbf{Heisenberg counts the canonical momentum; \emph{el Pleno} transports the momentum of the medium, and the difference between the two is the correction}.

\bigskip
\textbf{Exact recovery of Heisenberg.} For every width $\sigma\gg\ell_P$ ---that is, every experimentally accessible excitation, from electrons to macroscopic systems--- the term $(3/80)(\ell_P/\sigma)^2$ is negligible and

\[
\Delta x\cdot\Delta p \;\longrightarrow\; \frac{\hbar}{2}\,,
\]

\emph{exactly}, saturated by the minimum-uncertainty Gaussian packet. PIU-$\Psi$ recovers the Heisenberg relation with its exact floor $\hbar/2$, and with it the whole structure of quantum mechanics that rests on the uncertainty relation: the complementarity principle, the diffraction limit, the stability of the atom (the electron does not fall into the nucleus because localizing it would cost divergent kinetic energy), the zero-point energy. To validate Heisenberg is to validate that complete edifice. This recovery is exact in the long-wave limit and \emph{does not} depend on the value of $\xi$: the floor $\hbar/2$ comes from the kinetic-flexural sector (Route 4), not from the elastic oscillator sector (Route 1). The result is therefore free of the uncertainty about the precise value of $\xi$.

\bigskip
\textbf{The meaning of the sign $\geq$.} The standard inequality states $\Delta x\cdot\Delta p\geq\hbar/2$ but does not explain why the product can exceed the floor nor what controls the excess. PIU-$\Psi$ specifies it: the factor under the root in (E-inc-3) satisfies

\[
1 + \frac{3}{80}\left(\frac{\ell_P}{\sigma}\right)^2 \;\geq\; 1 \quad\text{always,}
\]

because the flexural rigidity is strictly positive (axiom A4, ultraviolet regulation). From here the structural reading:

\begin{quote}
\emph{The sign $\geq$ of Heisenberg is the positivity of the curvature rigidity of \emph{el Pleno}.}
\end{quote}

The medium transports an excess momentum $\propto(k\ell_P)^2$ hidden in its resistance to curving; that excess manifests as additional momentum uncertainty when one tries to localize an excitation. The equality $\Delta x\cdot\Delta p=\hbar/2$ is reached only in the limit $\ell_P/\sigma\to 0$, where the flexural rigidity becomes invisible. In the standard regime the excess is null for practical purposes; near the Planck scale it is macroscopic (at $\sigma=\sigma_{\min}=\ell_P/\sqrt{20}$ the product is $0.661\,\hbar$, an excess of $32\%$). The Heisenberg inequality, in PIU-$\Psi$, is not an axiom of the formalism: it is the consequence that a real physical medium cannot have negative curvature rigidity.

\bigskip
\textbf{The difference between the routes as information: instantaneous and relaxed localization.}
Route 4, carried out by the alternative route of the well oscillator (minimum-uncertainty packet with the natural frequency $\omega_0=\sqrt{\xi}\,c/\ell_P$), produces a \emph{second} localization scale,
\[
\sigma_\text{osc} = \frac{\ell_P}{\sqrt{2\sqrt{\xi}}} \approx 0.3357\,\ell_P\,,
\]
larger than the hard grain $\sigma_{\min}=\ell_P/\sqrt{20}\approx 0.2236\,\ell_P$ of Route 2. The two scales do not contradict each other: they describe physically distinct things, and their order ($\sigma_\text{osc}>\sigma_{\min}$) is significant. The ratio of their squares is

\[
\left(\frac{\sigma_\text{osc}}{\sigma_{\min}}\right)^2 = \frac{20}{2\sqrt{\xi}} = \frac{10}{\sqrt{\xi}} \;\xrightarrow{\ \xi\to 20\ }\; \sqrt{5}\,,
\]

and that $\sqrt{5}$ is exactly the factor that defines the relaxation time of the medium, $\tau_\Pi=\sqrt{5}\,\ell_P/(10c)$. The physical reading is transparent: the hard grain $\sigma_{\min}$ is the \emph{instantaneous} localization ---the one that ultraviolet regulation permits in an instant---, while $\sigma_\text{osc}$ is the \emph{relaxed} localization ---the width at which the packet settles once the medium has responded over its characteristic time $\tau_\Pi$. The difference between the two, expressed as a length, \textbf{is the relaxation time of \emph{el Pleno}}. This enriches the interpretation of the uncertainty: it is not a fixed spatial grain, but the imprint that the medium takes a finite time $\tau_\Pi$ to respond to a localization attempt, and during that time the packet widens by a factor $\sqrt[4]{5}$. The discrepancy between the two routes is not a consistency failure: it is the measure of the response time of the medium.

\bigskip
\textbf{Prediction: ultraviolet correction to the uncertainty (derived GUP).}
The form (E-inc-3) is a \textbf{generalized uncertainty relation} (GUP type, \emph{generalized uncertainty principle}), but with a \emph{derived} coefficient ($3/80$) instead of one parametrized phenomenologically as in the usual GUP models. The excess over $\hbar/2$ at accessible scales is suppressed by $(\ell_P/\sigma)^2$: for an atomic width $\sigma\sim 10^{-10}$ m, the excess is $\sim(\ell_P/\sigma)^2\sim 10^{-50}$, unobservable, consistent with the fact that no experiment has detected any deviation from Heisenberg. The prediction is structural and falsifiable in principle: any excess measured over $\hbar/2$ should follow the law $(3/80)(\ell_P/\sigma)^2$ with a fixed coefficient, not an adjustable parameter.

\medskip
\textbf{Dimensional verification.} $[\Delta x\cdot\Delta p]=\text{m}\cdot\text{kg\,m\,s}^{-1}=\text{kg\,m}^2\text{s}^{-1}=[\hbar]$ $\checkmark$. The factor under the root is dimensionless ($\ell_P/\sigma$ dimensionless) $\checkmark$. The numerical verification of (E-inc-3) against the direct integration of the product over the Gaussian packet agrees to machine precision over the whole range $\sigma\in[0.15,\,10^6]\,\ell_P$.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{incertidumbre_pleno.png}
\caption{The uncertainty relation as a structural property of \emph{el Pleno}. (A) The factor $\sqrt{20}$ converges in three physically distinct sectors of the Lagrangian ---elastic (damping threshold $\xi/20$), flexural (minimum width $\sigma_{\min}=\ell_P/\sqrt{20}$) and dissipative (phase erasure $\tau_\Pi k_B T_P/\hbar=1/\sqrt{20}$)---, defining a genuine localizability scale of the medium, $\ell_P/\sqrt{20}\approx 0.2236\,\ell_P$. (B) The uncertainty relation of \emph{el Pleno}, $\Delta x\,\Delta p=(\hbar/2)\sqrt{1+(3/80)(\ell_P/\sigma)^2}$ (E-inc-3): it recovers the exact floor $\hbar/2$ for $\sigma\gg\ell_P$ and predicts an ultraviolet excess (GUP) with \emph{derived} coefficient $3/80$; at $\sigma=\sigma_{\min}$ the product is $0.661\,\hbar$ (excess of $32\%$). (C) The two localization scales ---instantaneous $\sigma_{\min}$ (Route 2) and relaxed $\sigma_\text{osc}$ (Route 4)--- do not contradict each other: their ratio $(\sigma_\text{osc}/\sigma_{\min})^2=10/\sqrt{\xi}\to\sqrt{5}$ is the factor of the relaxation time $\tau_\Pi$, so that the difference between the two measures the response time of the medium. (D) The deficit $\xi=19.69<20$ is not an imperfection but a condition of existence: only on the underdamped side ($\xi/20<1$) does the radial mode oscillate and sustain the quantum dynamics; \emph{el Pleno} is tuned as close to the critical threshold as it can be without extinguishing the oscillation. All the scales, without free parameters.}
\label{fig:incertidumbre}
\end{figure}

\medskip
\textbf{How this result was obtained: discarding fragile conjectures.}
The result was carved by discarding. A first conjecture ---that the \emph{Pleno}-ground state was at \emph{exact critical} damping ($\xi=20$), which would have made $\sigma_\text{osc}=\sigma_{\min}$--- was discarded: the symbolic computation of $\xi$ from the potential gives $\xi=19.69$ with an exact closed form, genuinely distinct from $20$ ($1.5\%$). A second conjecture ---that the factor $\hbar/2$ emerged from the algebraic product $(1/\sqrt{20})\cdot\sqrt5$--- was also discarded as fragile: the correct $\hbar/2$ comes from the direct computation of the uncertainty product (Route 4), not from combining constants. What survives both discards is (E-inc-3), whose $\hbar/2$ is exact and independent of $\xi$. The value of the program lies in distinguishing the robust from the fragile, not in preserving the initial conjecture.

\medskip
\textbf{Epistemic classification.} The localization scale $\sigma_{\min}=\ell_P/\sqrt{20}$ and its convergence in three sectors of the Lagrangian (Routes 1--3) are \epi{DERIVED} (numbers of the program, verified, including the fluctuation-dissipation relation E-inc-0c from the primaries). The closed form (E-inc-3) of the uncertainty product is \epi{DERIVED} from the dispersion of \emph{el Pleno} (E23) and the Gaussian structure of the packet, without free parameters. The exact recovery of Heisenberg in the limit $\sigma\gg\ell_P$ is \epi{DERIVED}. The identification of the sign $\geq$ with the positivity of the flexural rigidity is \epi{SLD}. The reading of the difference $\sigma_\text{osc}/\sigma_{\min}=\sqrt[4]{5}$ as the relaxation time (instantaneous versus relaxed localization) is \epi{SLD}. The GUP-type ultraviolet correction with coefficient $3/80$ is \epi{DERIVED} in its form, \epi{HYPOTHESIS} in its observational confrontation (suppressed $\sim 10^{-50}$ at atomic scales, untested). The verification that the width of the well oscillator $\sigma_\text{osc}$ is deduced from Madelung-Bohm over F1 and not from the imported standard-harmonic-oscillator formula remains as a declared \textbf{derivation thread}.

\medskip
\textbf{Consequence for the program.} This result closes a piece of the reading of quantum mechanics as the dynamics of \emph{el Pleno} (\S{}III.2.4): not only the Schrödinger equation (Madelung-Bohm), the Born rule (Noether charge) and the Bell correlations (Tsirelson bound) emerge from the medium, but also the uncertainty relation ---with its exact floor $\hbar/2$ and a structural meaning for its inequality. If PIU-$\Psi$ validates Heisenberg, it also validates the chain of physics that follows from it. Complete quantum mechanics, in its uncertainty structure, is the effective phenomenology of a physical medium with positive curvature rigidity, and the inequality $\geq$ itself is the imprint of that positivity.


\subsubsection{\texorpdfstring{III.2.7 --- Scope of the validation: what PIU-$\Psi$ inherits, reinforces and adds over Heisenberg}{III.2.7 - Scope of the validation: what PIU inherits, reinforces and adds over Heisenberg}}

\textbf{Origin.} Section \S{}III.2.6 established that the uncertainty relation emerges from the dynamics of \emph{el Pleno} and that its floor $\hbar/2$ is exact in the long-wave limit. From this follows a scope question: if PIU-$\Psi$ reproduces Heisenberg, what portion of the edifice of physics built on the uncertainty relation becomes available to the program? The answer requires a careful distinction ---reproducing Heisenberg means \emph{consistency} with quantum mechanics, not \emph{automatic inheritance} of all its particular consequences--- and precisely that distinction is what turns the result into a legitimate strengthening and not a circular assertion. This section classifies the scope into three categories and specifies the four modes in which the result strengthens the program.

\medskip
\textbf{Category 1 --- Direct inheritance.} The results that follow \emph{only} from the uncertainty relation, without additional physics, become available to PIU-$\Psi$ without extra derivational work, because PIU-$\Psi$ provides the uncertainty relation with its exact floor. Among them:

\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{zero-point energy} of the oscillator, $E_0 = \hbar\omega/2$, which is the minimum of $\langle H\rangle$ subject to $\Delta x\,\Delta p\geq\hbar/2$.
  \item The \textbf{stability of the atom}: the electron does not fall into the nucleus because localizing it in a region of size $\Delta x$ costs kinetic energy $\sim\hbar^2/(2m\,\Delta x^2)$, which diverges faster than the gain in Coulombic potential energy. The balance fixes a non-null minimum size.
  \item The \textbf{diffraction limit} and the maximum resolution of any instrument.
  \item The \textbf{natural width of the spectral lines}, via the time-energy relation $\Delta E\,\Delta t\geq\hbar/2$, of identical structure.
  \item The \textbf{degeneracy pressure} that sustains white dwarfs and neutron stars: the exclusion principle plus the uncertainty force the confined fermions to occupy states of increasing momentum, generating a pressure that does not depend on temperature. It is the support of cold compact objects.
\end{itemize}

This inheritance is not minor: it includes \textbf{the stability of ordinary matter} and \textbf{the support of compact matter}. PIU-$\Psi$ can assert that both follow from the same curvature rigidity of \emph{el Pleno} that produces the uncertainty relation (\S{}III.2.6).

\medskip
\textbf{Category 2 --- Conditioned inheritance (mutual reinforcement).} Results that require the uncertainty \emph{plus} a structural piece that PIU-$\Psi$ already derives independently. Here there is no unidirectional inheritance but \emph{cross-consistency}, which reinforces both derivations:

\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{Born rule}, which PIU-$\Psi$ derives from the conservation of the Noether $U(1)$ charge (\S{}III.2.4), independently of the uncertainty.
  \item The \textbf{decoherence}, which PIU-$\Psi$ derives from the intrinsic flexural noise via fluctuation-dissipation (\S{}III.2.5).
  \item The \textbf{Bell correlations} with the Tsirelson bound $2\sqrt{2}$, derived in \S{}III.2.5.
  \item The \textbf{canonical commutator} $[\hat x,\hat p]=i\hbar$, which is the operational form of the uncertainty relation itself, with $\hbar$ provided as a structural identity (\S{}III.2.2).
\end{itemize}

That PIU-$\Psi$ reaches these results by two routes ---its own and the Heisenbergian--- is internal consistency of the kind the program values (cf. the double derivation of $S_{\min}^\text{estr}$, \S{}II.4.5).

\medskip
\textbf{Category 3 --- Non-inheritable.} Validations of quantum mechanics that do \emph{not} follow from the uncertainty relation alone and that, therefore, PIU-$\Psi$ \textbf{cannot} claim by this route:

\begin{itemize}[leftmargin=1.4em]
  \item The \textbf{hydrogen spectrum}, which requires the Coulomb potential ---derivable in PIU-$\Psi$ from the induced gauge sector (\S{}III.4), not from the uncertainty.
  \item The \textbf{fine structure and spin}. The \emph{structure} of the Dirac equation ---including spin-$\tfrac12$ and the spin-orbit coupling--- is derived from \emph{el Pleno} in Annex \S{}V.7 (\epi{DERIVED}), but not by the route of uncertainty; its fine atomic \emph{phenomenology} (hydrogen splitting, value of $\lambda_\text{SO}$) belongs to the branch of fermionic masses and is not inherited here.
  \item The \textbf{anomalous magnetic moment} $g-2$, which requires complete quantum electrodynamics.
  \item The \textbf{masses of the particles}, a declared open thread of the program (fermionic electroweak cascade, \S{}V.3).
\end{itemize}

Declaring this category explicitly is what separates the legitimate strengthening from the overclaim. PIU-$\Psi$ inherits what depends on the uncertainty; the rest it must derive on its own, and where it has not done so, it declares it an open thread.

\medskip
\textbf{The four modes of strengthening.} From the previous analysis follow four concrete, non-circular ways in which the validation of Heisenberg strengthens the program:

\begin{enumerate}[leftmargin=1.6em]
  \item \textbf{Inheritance of the stability of matter.} Via Category 1, PIU-$\Psi$ obtains without additional cost the stability of the atom and the support of compact objects, anchored in the flexural rigidity of \emph{el Pleno}.
  \item \textbf{Explanation where quantum mechanics postulates.} Quantum mechanics \emph{postulates} the uncertainty relation as an axiom of the formalism. PIU-$\Psi$ \emph{derives} it from the curvature rigidity of the medium and \emph{explains} the sign $\geq$ as the positivity of that rigidity (\S{}III.2.6). In the triple validation of the program (interpret / explain / predict), the uncertainty moves from the ``is assumed'' column to the ``is explained'' column. On this point PIU-$\Psi$ is more fundamental than the standard formulation.
  \item \textbf{Addition of new empirical content.} PIU-$\Psi$ not only reproduces Heisenberg where it has been measured; it predicts a GUP-type ultraviolet correction with a \emph{derived} coefficient ($3/80$, \S{}III.2.6) that standard quantum mechanics does not contain. The validation thus gains falsifiability: it reproduces the known and risks a specific deviation in the unmeasured.
  \item \textbf{Unification of the support and the collapse of compact objects.} This is the most distinctive strengthening. In standard physics, the degeneracy pressure (quantum mechanics) and gravity (general relativity) are two theories that must be \emph{matched by hand} to describe the equilibrium and collapse of compact stars ---it is the origin of the Chandrasekhar limit. In PIU-$\Psi$ both are behaviors of the \textbf{same medium}: the degeneracy pressure inherits the uncertainty, which is the flexural rigidity of \emph{el Pleno} (\S{}III.2.6), and gravity is the macroscopic response of the same \emph{el Pleno} (\S{}III.1). The balance between the two, measured in the same primary scales $(\rho_P, c, \ell_P)$, produces a limiting mass of order
  \[
  M_\text{lim} \sim \frac{(\hbar c/G)^{3/2}}{m_p^2} = m_P\left(\frac{m_P}{m_p}\right)^2 \sim 1\,M_\odot\,,
  \]
  the correct order of magnitude of Chandrasekhar, obtained from the sole uncertainty-gravity balance without additional physics. This upper limit is situated in the same framework as the lower black-hole mass limit already derived in the program, $M_{\min}\approx 0.117\,m_P$ (geometric exhaustion under the A6 hard bound, \S{}II.6). PIU-$\Psi$ thus describes the quantum support and the gravitational collapse of compact objects within a single framework, where standard physics requires two independent theories.
\end{enumerate}

\medskip
\textbf{Dimensional verification.} $M_\text{lim} = (\hbar c/G)^{3/2}/m_p^2$: $[(\,\text{J\,s}\cdot\text{m/s}\,/\,\text{m}^3\text{kg}^{-1}\text{s}^{-2})^{3/2}]/[\text{kg}^2] = [\text{kg}^3]/[\text{kg}^2] = [\text{kg}]$ $\checkmark$. Numerically $\approx 1.85\,M_\odot$ with the order-of-magnitude estimate (the canonical value $\approx 1.4\,M_\odot$ requires the numerical factors of the complete structural integration).

\medskip
\textbf{Epistemic classification.} The classification into three categories (direct, conditioned, non-inheritable inheritance) is \epi{SLD}: it follows from which physics depends on the uncertainty relation and which physics does not. The inheritance of Category 1 is \epi{DERIVED} (a direct consequence of \S{}III.2.6). The cross-reinforcement of Category 2 is \epi{DERIVED} (both routes already established in the corpus). The explanation of the sign $\geq$ (mode 2) and the GUP addition (mode 3) refer to \S{}III.2.6. The support-collapse unification (mode 4) is \epi{SLD} as regards the unified framework and the order of magnitude of $M_\text{lim}$; the computation of the exact numerical factor of the Chandrasekhar limit from the structure of \emph{el Pleno} ---and its quantitative matching with $M_{\min}$--- remains a \textbf{declared derivation thread}. The honesty of Category 3 (the non-inheritable) is an integral part of the result: without it, the scope assertion would be an overclaim.

\medskip
\textbf{Consequence for the program.} The validation of the uncertainty relation is not an endpoint but a door: it opens the inheritance of the physics that depends on it, reinforces by cross-consistency what the program already derived, adds a prediction of its own, and ---most distinctively--- unifies in a single medium the quantum support and the gravitational collapse of compact objects. The careful distinction between the inherited, the reinforced, the added and the non-inheritable is what keeps the result within the epistemic discipline of the program: PIU-$\Psi$ strengthens itself by what it can sustain rigorously, not by what it could claim without it.


\subsubsection{\texorpdfstring{III.2.8 --- The Chandrasekhar limit as an internal balance of \emph{el Pleno}}{III.2.8 - The Chandrasekhar limit as an internal balance of el Pleno}}

\textbf{Origin.} \S{}III.2.7 identifies as a distinctive strengthening of the program the unification of the quantum support and the gravitational collapse of compact objects in a single medium; the \emph{exact numerical factor} of the Chandrasekhar limit from the structure of \emph{el Pleno} is the piece that completes that unification. The derivation that follows obtains the complete coefficient, not only the order of magnitude, and verifies that no link of the chain is imported from outside \emph{el Pleno}.

\medskip
\textbf{The chain, entirely internal to \emph{el Pleno}.} The standard Chandrasekhar argument combines three ingredients that ordinary physics takes from three distinct theories: the Pauli exclusion principle (quantum mechanics), the relativistic degeneracy pressure (quantum statistics) and the gravitational hydrostatic equilibrium (general relativity / fluid mechanics). In PIU-$\Psi$ the three are behaviors of the same medium:

\begin{enumerate}[leftmargin=1.6em]
  \item \textbf{The fermionic statistics is not imported.} The exclusion principle, which in the standard argument is a postulate, in PIU-$\Psi$ is a consequence of the Finkelstein-Rubinstein theorem applied to the solitons of \emph{el Pleno}: the $2\pi$ rotation of a skyrmion of topological number $B$ produces the factor $(-1)^B$, so that the skyrmions with odd $B$ are fermions by mathematical obligation (\S{}II.6.4). The statistics that sustains the degeneracy pressure emerges from the medium.
  \item \textbf{The degeneracy pressure is uncertainty.} The confinement of the fermions to number density $n$ imposes, via the uncertainty relation derived in \S{}III.2.6, a Fermi momentum $p_F\sim\hbar n^{1/3}$ and, in the ultrarelativistic regime, an energy per particle $\sim\hbar c\,n^{1/3}$. It is the flexural rigidity of \emph{el Pleno} that provides this pressure.
  \item \textbf{Gravity is the macroscopic response of \emph{el Pleno}} (\S{}III.1).
  \item \textbf{The equilibrium} of a self-gravitating degenerate relativistic gas is the polytrope of index $n=3$ (Lane-Emden equation), whose structure fixes the numerical coefficient.
\end{enumerate}

\medskip
\textbf{The result.} The balance produces a radius-independent limiting mass:

\begin{equation}
\boxed{\;M_\text{Ch} = \frac{\sqrt{3\pi}}{2}\,\bigl(-\xi^2\theta'\bigr)_{n=3}\,\frac{(\hbar c/G)^{3/2}}{(\mu_e m_u)^2}\;}\,, \tag{E-Ch-1}
\end{equation}

where $(-\xi^2\theta')_{n=3} = 2.0182$ is the Lane-Emden constant of the $n=3$ polytrope at the surface, $\mu_e$ is the number of nucleons per electron ($\mu_e=2$ for carbon-oxygen matter) and $m_u$ the atomic mass unit. Numerically,

\[
M_\text{Ch} = 2.896\times 10^{30}\ \text{kg} = 1.456\,M_\odot\,,
\]

in agreement with the canonical value ($\approx 1.44\,M_\odot$ for $\mu_e=2$).

\medskip
\textbf{PIU reading: the anatomy of the coefficient.} The structural identity $m_P=\rho_P\ell_P^3=\sqrt{\hbar c/G}$ ---the mass of one cell of the medium, which the human frame reads as the Planck mass because gravity and uncertainty proceed from the same primaries $\rho_P, c, \ell_P$--- allows (E-Ch-1) to be rewritten as a product of three factors with distinct physical meaning:

\begin{equation}
M_\text{Ch} = m_P\cdot\left(\frac{m_P}{\mu_e m_u}\right)^2\cdot\underbrace{\frac{\sqrt{3\pi}}{2}\bigl(-\xi^2\theta'\bigr)_{n=3}}_{=\,3.098}\,. \tag{E-Ch-2}
\end{equation}

\begin{itemize}[leftmargin=1.4em]
  \item $m_P$ ---the Planck mass--- is the scale where the flexural rigidity (which gives the uncertainty) and the gravitational response of \emph{el Pleno} are equal. In standard physics, that the combination $(\hbar c/G)^{3/2}$ appears in a stellar mass is the fortuitous encounter of constants of two theories; in PIU-$\Psi$ it is a single scale of the medium.
  \item $(m_P/\mu_e m_u)^2$ ---the hierarchy between the Planck scale and the hadronic scale--- is the \textbf{same} hierarchy that already appears in the program in the mass of the fundamental soliton $M_Q=M_P/S_{\max,\text{eff}}$ (\S{}II.6) and in the proton mass (hadronic sector). It is not a new parameter: the Chandrasekhar limit uses a hierarchy that PIU-$\Psi$ already contained.
  \item $\tfrac{\sqrt{3\pi}}{2}(-\xi^2\theta')_{n=3}=3.098$ is the pure coefficient of the $n=3$ polytrope: fluid mechanics of stellar equilibrium, common to any derivation of the limit, independent of PIU-$\Psi$.
\end{itemize}

\medskip
\textbf{Three mass limits in a single framework.} The Chandrasekhar limit is situated alongside the other two mass limits that the program already derives, all as balances of the same medium:

{\small
\begin{longtable}{L{3.3cm}L{4.0cm}L{5.0cm}}
\toprule
\textbf{Limit} & \textbf{Value} & \textbf{Physical balance (all from \emph{el Pleno})} \\
\midrule\endhead
$M_{\min}$ (black hole) & $\approx 0.117\,m_P$ & geometric hard bound A6, exhaustion of the packing (\S{}II.6) \\
$M_\text{Ch}$ (Chandrasekhar) & $\approx 1.46\,M_\odot$ & fermionic degeneracy (uncertainty) vs gravity \\
$M_Q$ (fundamental soliton) & $=M_P/S_{\max,\text{eff}}$ & topological tension of the skyrmion (\S{}II.6) \\
\bottomrule
\end{longtable}}

Where standard physics requires quantum mechanics, general relativity and stellar physics as three separate theories to describe the equilibrium and fate of compact objects, PIU-$\Psi$ obtains them as three balances of the same \emph{el Pleno}.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{limites_masa_pleno.png}
\caption{The Chandrasekhar limit as an internal balance of \emph{el Pleno}. (A) Anatomy of the coefficient: $M_\text{Ch}$ factors as $m_P\cdot(m_P/\mu_e m_u)^2\cdot 3.098$, three factors with physical meaning ---the Planck scale (where flexural rigidity and gravitational response are equal), the Planck-hadronic hierarchy (the same that appears in the mass of the fundamental soliton, \S{}II.6) and the pure coefficient of the $n=3$ Lane-Emden polytrope---, all from the same medium, without external importation. (B) The three mass limits that the program derives as balances of \emph{el Pleno}: $M_{\min}\approx 0.117\,m_P$ (geometric hard bound A6), $M_Q=M_P/S_{\max,\text{eff}}$ (topological tension of the skyrmion) and $M_\text{Ch}\approx 1.456\,M_\odot$ (fermionic degeneracy via uncertainty against gravity). The result coincides with the canonical value ($\approx 1.44\,M_\odot$) using only scales that the program already contained. \epi{DERIVED}}
\label{fig:limites_masa}
\end{figure}

\medskip
\textbf{Dimensional verification.} $(\hbar c/G)^{3/2}/(\mu_e m_u)^2$: $[\text{kg}^3]/[\text{kg}^2]=[\text{kg}]$ $\checkmark$. The factors $\sqrt{3\pi}/2$, $(-\xi^2\theta')$ and $\mu_e$ are dimensionless $\checkmark$.

\medskip
\textbf{Epistemic classification.} The form (E-Ch-1)--(E-Ch-2) of the limit and its numerical value $1.456\,M_\odot$ are \epi{DERIVED}: the coefficient of the $n=3$ polytrope is a canonical result of the Lane-Emden equation, and all the physical ingredients (fermionic statistics by Finkelstein-Rubinstein, degeneracy pressure by uncertainty, emergent gravity) are derived in the program, without external importation. The identification of the factor $m_P$ as the gravity-uncertainty equalization scale and of $(m_P/\mu_e m_u)^2$ as the Planck-hadronic hierarchy present in the program is \epi{SLD}. With this, mode 4 of \S{}III.2.7 (support-collapse unification) is \epi{DERIVED} in its coefficient; what remains as a minor thread is only the fine quantitative matching between the stellar degeneracy regime and the bounce regime (\S{}II.7), not the Chandrasekhar limit itself.

\medskip
\textbf{Consequence for the program.} The Chandrasekhar limit ceases to be a result borrowed from astrophysics and becomes a derived consequence of \emph{el Pleno}: the maximum mass that the flexural rigidity of the medium can sustain against its own gravitational response. That the exact coefficient coincides with the canonical value, using only scales that the program already contained, is confirmation that the support-collapse unification of \S{}III.2.7 is physics and not rhetoric.


\subsubsection{\texorpdfstring{III.2.9 --- The integration of General Relativity and Quantum Mechanics as the disappearance of the boundary}{III.2.9 - The integration of GR and QM as the disappearance of the boundary}}

\textbf{Origin.} Problem 4 of \S{}I.2.5 ---the structural incompatibility between General Relativity and Quantum Mechanics--- is, according to the consensus of the discipline, one of the great open problems of physics, without an accepted solution after five decades of effort. The thesis of PIU-$\Psi$ on it is precise: the program does not \emph{solve} the problem in the terms in which the discipline poses it, but \emph{integrates} it in the strict physical sense of the term ---making the boundary between the two theories disappear---, and distinguishes what it thereby asserts from what remains an open thread.

\medskip
\textbf{What integrating means, in physics.} The history of fundamental physics offers a precise operational criterion of what a genuine integration is. Two domains are not integrated when a bridge is built that connects them while keeping them separate (that is coupling them), nor when one is reduced to the other (that is eliminating one). They are integrated when it is discovered that the \textbf{boundary that separated them was not fundamental}: that what seemed to be two distinct things are configurations of a single one, and that the dividing line was an artifact of the description, not a feature of the world. The precedents are unambiguous:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Electricity and magnetism} were integrated into electromagnetism not by resolving a conflict between the two, but by showing (Maxwell) that they are the same field seen from distinct reference frames. The boundary depended on the observer; it was not fundamental.
  \item \textbf{Space and time} were integrated into spacetime (Minkowski) by showing that the separation between them is relative to the observer. A single entity.
  \item \textbf{Mass and energy} were integrated into $E=mc^2$ by revealing that they are two readings of the same underlying quantity.
\end{itemize}

In the three cases, integrating was \emph{making a boundary disappear}. That is the way physics truly integrates.

\medskip
\textbf{What PIU-$\Psi$ does with GR and QM.} Axiom A1 (\S{}I.5.2) establishes that a single substrate exists ---\emph{el Pleno}--- and that every physical entity is a configuration of it: ``none [entity] is independent; all are read as configurations of the same $\Psi$''. Under this ontology:

\begin{itemize}[leftmargin=1.4em]
  \item What General Relativity describes as the dynamical geometry of spacetime is, in PIU-$\Psi$, the macroscopic behavior of \emph{el Pleno}: the metric $g_{\mu\nu}^\text{eff}$ emerges as the transverse tensor mode of the field (\S{}III.1.4), it is not a primitive entity.
  \item What Quantum Mechanics describes as the dynamics of fields over a background is, in PIU-$\Psi$, the dynamics of the perturbations of the same \emph{el Pleno}: the Schrödinger equation emerges by Madelung-Bohm, the Born rule by the Noether charge, the uncertainty by the flexural rigidity (\S{}III.2.4, \S{}III.2.6).
\end{itemize}

The boundary between ``the gravitational'' and ``the quantum'' ---the line that the discipline has tried to cross for fifty years--- \textbf{does not exist in PIU-$\Psi$}, because both sides are the same medium. There are not two theories to reconcile: there is one \emph{el Pleno} and two effective limits of its description. This is integration in the sense of Maxwell and Minkowski, not evasion of the problem.

\medskip
\textbf{Why the discipline has not found it by the standard route.} The standard quantum-gravity program seeks to \emph{quantize the geometry} ---to treat $g_{\mu\nu}$ as a dynamical field to be quantized, and to find what happens in the regime where the curvature and the quantum fluctuations are both intense (black-hole interiors, the initial instant). This route presupposes that the geometry is fundamental. If the geometry is \emph{emergent} ---as in PIU-$\Psi$---, quantizing it is a category mistake: it amounts to trying to quantize the temperature of a gas instead of the molecules that produce it. Temperature is not quantized; it emerges from constituents that are already quantum. Analogously, geometry is not quantized; it emerges from \emph{el Pleno}, whose dynamics is already given by F1. The ``difficult regime'' (strong curvature $+$ strong quantum) is then not a boundary between theories where both collide, but the interior of a single medium near its hard bound $S_{\max}^\text{geom}$ (axiom A6): the effective geometric description (GR) ceases to apply, but \emph{el Pleno} remains perfectly defined, without singularity. The problem is not crossed; \textbf{it ceases to exist as a boundary}.

\begin{tcolorbox}[notabox,title={The precise distinction: conceptual integration versus quantitative verification}]
The \emph{conceptual} integration ---the disappearance of the boundary--- is closed in PIU-$\Psi$ by ontological construction (axiom A1) and articulated in the derivations of gravity (\S{}III.1) and of quantum mechanics (\S{}III.2) as limits of the same medium. The \emph{quantitative} verification of that integration is not a pending promise: it already counts with a body of \textbf{derived and closed} results. They come out of \emph{el Pleno}, simultaneously and with verified agreement: $G$ and $\hbar$ as structural identities in the same primary parameters (\S{}III.1.3, \S{}III.2.2); Heisenberg's uncertainty relation, with its exact floor $\hbar/2$ and the meaning of the sign $\geq$ (\S{}III.2.6); and the Chandrasekhar limit with its complete coefficient, $1.456\,M_\odot$ (\S{}III.2.8). That the gravitational constant, the action quantum, the quantum uncertainty and a critical stellar mass all come out of the same medium and with the same scales is not work in progress: it is the integration already operating quantitatively. The Maxwell precedent is instructive: he not only asserted that the electricity/magnetism boundary did not exist, but demonstrated it with equations that predicted the velocity of light. PIU-$\Psi$ is in the analogous position: the boundary has disappeared and the equations of the medium already reproduce quantities of both sides. What remains is not the integration itself, but two concrete pieces of the quantitative map: the dynamics of the extreme cosmological regime ---the non-linear bounce--- (part of the F-cosmo program, \S{}V.4), since the background cosmological sector is already derived and closed (\S{}III.8). The territory is known to be single and has already been mapped in good part; two regions remain to be surveyed, not the unity of the territory.
\end{tcolorbox}

\medskip
\textbf{Chandrasekhar as evidence of the integration.} The Chandrasekhar limit (\S{}III.2.8) concretely illustrates the disappearance of the boundary. In standard physics, its computation juxtaposes the degeneracy pressure (quantum) and the gravitational equilibrium (relativistic/classical) as contributions of two distinct theories that cooperate. Under the picture of two theories with a boundary, that cooperation is almost fortunate. Under PIU-$\Psi$ it is the expected: the degeneracy pressure is the flexural rigidity of \emph{el Pleno} (via the uncertainty, \S{}III.2.6) and gravity is the macroscopic response of the same \emph{el Pleno} (\S{}III.1); both are measured in the same primary scales $(\rho_P, c, \ell_P)$. That the exact coefficient comes out ($1.456\,M_\odot$) using only scales that the program already contained is not the collaboration of two foreign theories: it is a single medium described in two ways. Chandrasekhar is, in this sense, a measurement that the boundary is no longer there.

\medskip
\textbf{Epistemic classification.} The assertion that PIU-$\Psi$ integrates GR and QM \emph{in a conceptual sense} (disappearance of the boundary, both as configurations of the same \emph{el Pleno}) is \epi{DERIVED} insofar as it follows directly from axiom A1 and from the already-established emergences (gravity \S{}III.1, quantum mechanics \S{}III.2). The characterization of this disappearance as ``integration in the sense of Maxwell/Minkowski'' is \epi{SLD}, supported by the historical criterion of what constitutes an integration in physics. The assertion that the standard quantum-gravity problem is ill-posed \emph{under the premise of emergent geometry} is \epi{SLD}: conditional on the geometry being effective, quantizing it is a category mistake. The \emph{quantitative} verification of the integration already counts with firm \epi{DERIVED} results ---$G$ and $\hbar$ (\S{}III.1.3, \S{}III.2.2), the uncertainty relation (\S{}III.2.6) and the Chandrasekhar limit (\S{}III.2.8), all from the same medium---; what remains open is one concrete and declared thread (F-cosmo-PIU-$\Psi$, \S{}V.4), not the verification as a whole. The program asserts a closed conceptual integration and an advanced quantitative verification with two pending regions; it does \textbf{not} assert a total quantitative closure. The distinction between closed conceptual integration and quantitative verification in progress is an integral part of the result and what keeps it within the epistemic discipline of the program.

\medskip
\textbf{Consequence for the program.} PIU-$\Psi$ does not compete in the race to quantize the geometry; it reformulates the problem. If the boundary between GR and QM is an artifact of treating the geometry as fundamental, then the integration does not consist in crossing that boundary but in dissolving it ---and dissolving a boundary is, in the strictest sense of physics, to integrate. Problem 4 of \S{}I.2.5 is not solved within the framework that poses it: it is integrated by changing the framework. What the discipline has sought as a reconciliation of two theories, PIU-$\Psi$ offers as the recognition that there never were two: there is a single physical medium, \emph{el Pleno}, of which General Relativity and Quantum Mechanics are two effective regional readings. The remaining task is not conceptual but one of verification, and that verification is already advanced ---$G$, $\hbar$, the uncertainty and Chandrasekhar derived from the same medium---: what remains is to complete the two regions not yet mapped of the territory that is already known to be single.


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


\subsubsection{III.2.10 --- Affine quantization and the origin of time}

\textbf{Origin.} Two structural elements complete the quantum reading: the affine quantization of \emph{el Pleno} (which produces the infrared wall $S_{\min}$) and the emergence of time (which resolves the time problem of quantum cosmology).


\textbf{Klauder affine quantization.} The modulus of the field of \emph{el Pleno} is strictly positive ($S > 0$): the field never passes through zero (that would correspond to the destruction of the medium, prohibited by the uniqueness of \emph{el Pleno}, axiom A1). Standard canonical quantization, which assumes a variable that ranges over the entire real line, is not applicable. The correct quantization is the Klauder affine quantization, appropriate for semi-bounded variables ($S > 0$). This quantization produces a structural quantum potential:


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

which generates the infrared wall of the effective potential at $S_{\min}$ (\S{}II.2): the medium cannot collapse to $S = 0$ because the affine quantum potential diverges there. This gives structural origin to the lower bound of the modulus ---complementary to the upper bound $S_{\max}$ of the Gent medium (\S{}III.1.7). The coefficient $3/8$ is the standard result of the Klauder affine quantization over the half-line with invariant measure $dS/S$, imported under the criteria of \S{}I.7.8.


\textbf{The origin of time.} The ``time problem'' of quantum cosmology ---in the Wheeler-DeWitt equation time does not appear as a parameter, which seems to eliminate temporal evolution--- is resolved structurally in \emph{el Pleno} through the Page-Wootters mechanism: time emerges from the correlation between subsystems of the field $\Psi$. A subsystem acts as a ``clock'' whose state correlates with the evolution of the rest; the observed time is this internal correlation, not an external absolute parameter. \emph{El Pleno} does not evolve ``in time'': time is the reading that a subsystem of \emph{el Pleno} makes of its correlation with the others. This is consistent with the fourth structural consequence of \S{}III.1.4 (time emerges, it is not postulated).


\textbf{Dimensional verification.} $[V_q^\text{dens}] = [\rho_P c^2]/[S^2] = [\rho_P c^2]$ (since $S$ is dimensionless) $= [J/m^3]$ $\checkmark$, dimension of energy density.


\textbf{Epistemic classification.} The Klauder affine quantum potential is \epi{DERIVED} (a consequence of the correct quantization of a semi-bounded variable), with the coefficient $3/8$ imported from the standard affine quantization (\S{}I.7.8). The emergence of time via Page-Wootters is \epi{DERIVED}: it resolves the time problem without postulating absolute time, consistent with the pre-metricity of \emph{el Pleno}.


\subsubsection{\texorpdfstring{III.2.11 --- Interpretation: what the fundamental constants are under PIU-$\Psi$}{III.2.11 - Interpretation: what the fundamental constants are under PIU-Psi}}

\textbf{Interpretation.} The fundamental constants are not arbitrary numbers of the universe: they are readings of the structural properties of \emph{el Pleno}. $G$ measures how the medium responds gravitationally; $\hbar$ measures its minimum action scale; $\gamma_\kappa$ is its rigidity against curvature. The constant $c$ occupies a place apart: it does not measure a property of the medium but its causal structure ---the space-time unification in which the properties are defined (\S{}I.8.9)---, which is why it disappears in natural units while the others do not. All are aspects of the same single physical medium.


\textbf{Explanation.} The value of each constant is explained from the three primaries ($\rho_P, \ell_P, c$) through structural identities with an explicit dynamical-selection mechanism, not as fitted inputs. $G = c^2/(\rho_P\ell_P^2)$ and $\hbar = \rho_P c\ell_P^4$ are not independent: they are equivalent to $\ell_P = \sqrt{\hbar G/c^3}$ (\S{}III.2.3), so that they constitute a single structural condition read in two languages, gravitational and quantum.


\textbf{Prediction.} The derived constants reproduce their observed values with agreement exact by construction and residue $<0.1\%$ from rounding the primary parameters to four figures. Complete quantum mechanics (Schrödinger, Born rule, decoherence, Bell correlations with Tsirelson bound, origin of time) emerges from the dynamics of \emph{el Pleno} without additional postulates, recovering the experimentally verified standard limits.


