% !TeX root = main.tex
\subsection{\S{}II.5 --- The mode spectrum of \emph{el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none critical of its own.
  \item \textbf{Notes:} derivational section of the mode spectrum. The mass of the radial mode $m_\text{radial}c^2 \approx 4.44\,E_P$ is a tree-level result (without flexural correction) dimensionally verified; the F1$\leftrightarrow$GP correspondence uses $m_\text{eff} = \sqrt{V''(S_{\min})/(\rho_P c^2)}\cdot m_P$, which produces the canonical value $\approx 4.44\,m_P$.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.5.1 --- The problem of the mode spectrum over the \emph{Pleno}-ground state}

With the minimal configuration of \emph{el Pleno} (\S{}II.2) and its crystalline microscopic structure (\S{}II.3) established, the next class of configurations in the catalogue are the \textbf{propagating perturbations} over the \emph{Pleno}-ground-state background. These perturbations are constructed as small deviations of the field $\Psi$ with respect to its fundamental configuration, propagating with dynamics governed by F1 linearized around the background.


The operational question is: given the dynamics of F1 over the background $\Psi_0 = S_{\min}^\text{estr} \cdot e^{i\theta_0}$ (with $\theta_0$ a constant global phase), what propagating modes does the medium admit, with what dispersion relation, and with what ultraviolet correction?


The procedure is standard in field theory: one substitutes into F1 the decomposition $\Psi = \Psi_0 + \delta\Psi$ with $|\delta\Psi| \ll |\Psi_0|$, expands the terms of F1 to quadratic order in $\delta\Psi$ (linear order in the field equations), applies the linearized Euler-Lagrange equations, and identifies the plane-wave solutions.


\subsubsection{II.5.2 --- Linearization of F1 around the \emph{Pleno}-ground state}

Field substitution. We decompose the field $\Psi$ into its background configuration and a perturbation:


\begin{equation}
\Psi(x^\mu) = \Psi_0 + \delta\Psi(x^\mu), \quad \Psi_0 = S_{\min}^\text{estr}\cdot e^{i\theta_0} \tag{E14}
\end{equation}

with $\delta\Psi = \delta S\cdot e^{i\theta_0} + i\,S_{\min}^\text{estr}\,e^{i\theta_0}\,\delta\theta$ --- decomposition into two real modes: $\delta S$ is the radial perturbation of the modulus, $\delta\theta$ is the perturbation of the phase. Both are assumed small: $|\delta S| \ll S_{\min}^\text{estr}$, $|\delta\theta| \ll 1$.


\textbf{Step 1 --- Expansion of the kinetic term.} The kinetic term of F1 (\S{}I.7, E2) is:


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

Substituting $\Psi = \Psi_0 + \delta\Psi$ and keeping only quadratic terms in $\delta\Psi$ (the zeroth-order terms are constants irrelevant to the dynamics; the first-order terms vanish because $\Psi_0$ is a stationary solution):


\begin{equation}
\mathcal{L}_\text{cin}^{(2)} = \frac{1}{2}\rho_P c^2\ell_P^2\left[(\partial_\mu\delta S)(\partial^\mu\delta S) + S_{\min}^{2,\text{estr}}\,(\partial_\mu\delta\theta)(\partial^\mu\delta\theta)\right] \tag{E15}
\end{equation}

The two radial and phase modes appear decoupled at this order, with distinct kinetic coefficients: the radial mode $\delta S$ with coefficient $\rho_P c^2\ell_P^2$, the phase mode $\delta\theta$ with coefficient $\rho_P c^2\ell_P^2\cdot S_{\min}^{2,\text{estr}}$.


\textbf{Step 2 --- Expansion of the potential.} The total potential $V(S) + V_q(S) \equiv V_\text{eff}(S)$ (\S{}II.4.3) is expanded in a Taylor series around $S_0 = S_{\min}^\text{estr}$:


\[
V_\text{eff}(S_0 + \delta S) = V_\text{eff}(S_0) + V'_\text{eff}(S_0)\,\delta S + \frac{1}{2}V''_\text{eff}(S_0)\,\delta S^2 + \mathcal{O}(\delta S^3)
\]

By the dynamical-minimum condition $V'_\text{eff}(S_0) = 0$ (derived in \S{}II.4.4, equation E8), the linear term vanishes. To quadratic order in $\delta S$:


\[
V_\text{eff}^{(2)} = \frac{1}{2}V''_\text{eff}(S_{\min}^\text{estr})\,\delta S^2
\]

The curvature of the potential at the minimum $V''_\text{eff}(S_{\min}^\text{estr})$ is computed explicitly from the three contributions of $V_\text{eff}$. With the canonical parameters of the program, the numerical result is:


\begin{equation}
V''_\text{eff}(S_{\min}^\text{estr}) \approx 19.69\,\rho_P c^2 \tag{E16}
\end{equation}

with dimensions of energy density $[J/m^3] = [\rho_P c^2]$. The positive sign confirms that $S_{\min}^\text{estr}$ is a stable minimum of the effective potential.


\textbf{Useful definition.} We introduce the dimensionless quantity:


\[
\xi \equiv \frac{V''_\text{eff}(S_{\min}^\text{estr})}{\rho_P c^2} \approx 19.69
\]

which will be central in the subsequent derivations.


\textbf{Step 3 --- Expansion of the flexural term.} The flexural term of F1 acts on the curvature of the complete field $\Psi$ (\S{}I.7.7, form E4b--E4c), not only on the modulus. In modulus-phase variables around the background $\Psi_0 = S_{\min}^\text{estr}\,e^{i\theta_0}$, its compact form $(\Box S)^2$ expands to:


\[
\mathcal{L}_\text{flex} = -\frac{\rho_P c^2\ell_P^4}{40}\,\bigl|\Box\Psi\bigr|^2_S
\]

Substituting $S = S_{\min}^\text{estr} + \delta S$, $\theta = \theta_0 + \delta\theta$ and keeping only quadratic terms:


\begin{equation}
\mathcal{L}_\text{flex}^{(2)} = -\frac{\rho_P c^2\ell_P^4}{40}\Bigl[(\Box\delta S)^2 + S_{\min}^{2,\text{estr}}(\Box\delta\theta)^2\Bigr] \tag{E17}
\end{equation}

The flexural term couples to \textbf{both} sectors: to the radial mode $\delta S$ via $(\Box\delta S)^2$, and to the phase mode $\delta\theta$ via $S_{\min}^{2,\text{estr}}(\Box\delta\theta)^2$. The weighting $S_{\min}^{2,\text{estr}}$ of the phase sector is identical to that of the phase kinetic term and has the same lattice origin: the bond energy $|\Psi_n-\Psi_m|^2$ of the diamond lattice, whose quadratic expansion produces $(s_n-s_m)^2$ for the modulus and $S_0^2(a_n-a_m)^2$ for the phase (\S{}II.3.8), both propagated by the same bond-difference operator and, therefore, by the same Fourier symbol $D_\text{lat}(\mathbf{k})$ with flexural/kinetic ratio equal to the invariant $1/40$ (P9). The ultraviolet regulator is a property of the curvature of the complete field $\Psi$, not exclusive to the modulus: it acts on every propagating excitation, including the phase sector.


\subsubsection{II.5.3 --- Linearized field equations}

Applying the Euler-Lagrange equations with the inclusion of the second-derivative term (standard procedure for Lagrangians with $(\Box S)^2$, already employed in \S{}I.7.10 to derive F2):


\textbf{Equation for $\delta S$ (radial mode).}


\[
\rho_P c^2\ell_P^2\,\Box\delta S - \frac{\rho_P c^2\ell_P^4}{20}\Box^2\delta S + V''_\text{eff}(S_{\min}^\text{estr})\,\delta S = 0
\]

The factor $1/20$ of the flexural term emerges from differentiating $(\Box S)^2/40$ with respect to $\Box S$ and applying the additional operator $\Box$ (factor 2 multiplied by the original coefficient $1/40$).


Rearranging, dividing by $\rho_P c^2\ell_P^2$:


\begin{equation}
\Box\delta S - \frac{\ell_P^2}{20}\Box^2\delta S + \frac{V''_\text{eff}}{\rho_P c^2\ell_P^2}\delta S = 0 \tag{E18}
\end{equation}

\textbf{The healing length of the radial mode.} In the static regime ($\partial_t\to 0$, $\Box\to-\nabla^2$) and at tree level (neglecting the flexural term $\ell_P^2\Box^2/20$, subdominant at scales $\gg\ell_P$), equation (E18) reduces to the Helmholtz equation
\begin{equation}
\nabla^2\delta S = \frac{V''_\text{eff}}{\rho_P c^2\ell_P^2}\,\delta S = \frac{\xi}{\ell_P^2}\,\delta S, \tag{E18a}
\end{equation}
whose solution decays over the scale $\ell_\text{heal}$ defined by $\nabla^2\delta S = \delta S/\ell_\text{heal}^2$. Comparing coefficients, the length over which the modulus recovers its background value after a perturbation ---the healing length--- is
\begin{equation}
\ell_\text{heal} = \frac{\ell_P}{\sqrt{\xi}} = \frac{\ell_P}{\sqrt{19.69}} \approx 0.2254\,\ell_P. \tag{E18b}
\end{equation}
It coincides exactly with the reduced Compton wavelength of the massive radial mode, $\lambda_C = \hbar/(m_\text{radial}c) = \ell_P/\sqrt{\xi}$ (with $m_\text{radial}c^2 = \sqrt{\xi}\,E_P$, E21): the healing length of a massive field \emph{is} its Compton wavelength, so that the static recovery and the mass of the mode give the same number. The factor $\sqrt{\xi} = 4.4373$ is the same as the effective Gross-Pitaevskii mass (\S{}II.5.7), unifying three readings ---root of the curvature of the well, mass of the radial mode and inverse of the healing length--- into a single quantity. \epi{DERIVED}. That $\ell_\text{heal}$ is sub-lattice ($<\ell_P$) leaves open whether the physical size of the core of the defect is fixed by the healing length of the continuum or by the A4 curvature regulation at scale $\ell_P$; the resolution of the complete (E18) with the flexural term is declared as an open thread (\S{}V.3).

\textbf{Equation for $\delta\theta$ (phase mode).} Including the phase flexural term of (E17), the Euler-Lagrange equation for $\delta\theta$ (with the second-derivative term, as for the modulus) is:


\begin{equation}
\rho_P c^2\ell_P^2\,S_{\min}^{2,\text{estr}}\left[\Box\delta\theta - \frac{\ell_P^2}{20}\Box^2\delta\theta\right] = 0 \quad\Rightarrow\quad \Box\delta\theta - \frac{\ell_P^2}{20}\Box^2\delta\theta = 0 \tag{E19}
\end{equation}

The phase mode is a \textbf{massless mode} (there is no potential term for $\delta\theta$, in accordance with Goldstone's theorem applied to the breaking of U(1): the theorem forbids a mass term $\propto\delta\theta^2$, but \emph{not} a higher-order gradient term $\propto(\Box\delta\theta)^2$). Its physical infrared branch satisfies the free wave equation $\Box\delta\theta = 0$ with propagation velocity $c$ (the structural velocity of \emph{el Pleno}, already identified in \S{}I.7.5); its ultraviolet branch is the Lee-Wick regulation pole at $\Box\delta\theta = 20/\ell_P^2$. The factor $1/20$ of the phase flexural term emerges from the coefficient $1/40$ multiplied by the factor $2$ of the derivative, identical to that of the modulus sector. This is the structural origin of the ultraviolet correction common to the photon and the TT mode (\S{}III.3--\S{}III.4).


\subsubsection{II.5.4 --- Dispersion relation: radial mode and TT mode}

Seeking plane-wave solutions:


\[
\delta S(x^\mu) = A\,e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}
\]

the operator $\Box = \partial_\mu\partial^\mu$ with signature $(+,-,-,-)$ applied to this plane wave gives $\Box \to k^2 - \omega^2/c^2$ where $k = |\mathbf{k}|$. Defining:


\[
u \equiv k^2 - \omega^2/c^2
\]

equation (E18) becomes an algebraic relation:


\begin{equation}
u - \frac{\ell_P^2}{20}u^2 + \frac{\xi}{\ell_P^2} = 0 \tag{E20}
\end{equation}

where $V''_\text{eff}/(\rho_P c^2\ell_P^2) = \xi/\ell_P^2$ has been used.


\textbf{Tree-level regime (flexural correction neglected).}


To first order, neglecting the quadratic term $(\ell_P^2/20)u^2$:


\[
u \approx -\frac{\xi}{\ell_P^2} \quad\Rightarrow\quad \frac{\omega^2}{c^2} = k^2 + \frac{\xi}{\ell_P^2}
\]

\begin{equation}
\boxed{\;\omega^2 = c^2 k^2 + \omega_0^2 \quad\text{with}\quad \omega_0^2 = \frac{\xi\,c^2}{\ell_P^2} = \frac{V''_\text{eff}(S_{\min}^\text{estr})}{\rho_P\ell_P^2}\;} \tag{E21}
\end{equation}

\textbf{Tree-level radial mode mass.} The rest mass of the mode is obtained at $k = 0$:


\[
m_\text{radial}c^2 = \hbar\omega_0 = \hbar\sqrt{\xi}\,\frac{c}{\ell_P} = \sqrt{\xi}\,\frac{\hbar c}{\ell_P} = \sqrt{\xi}\,E_P
\]

With $\xi \approx 19.69$:


\[
\boxed{\;m_\text{radial}c^2 = \sqrt{19.69}\,E_P \approx 4.4373\,E_P\;}
\]

\textbf{Dimensional verification.} $\xi$ is dimensionless (ratio $V''_\text{eff}/(\rho_P c^2)$); $\omega_0^2 = \xi\,c^2/\ell_P^2$ has dimension $[1/s^2]$ $\checkmark$; $\hbar\omega_0$ has dimension $[J]$ $\checkmark$; $m_\text{radial}c^2 \approx 4.44\,E_P$ has dimension $[J]$ $\checkmark$.


\textbf{Numerical verification.} $\sqrt{19.69} = 4.4373$. The radial mode of \emph{el Pleno} has a \textbf{rest mass of order the Planck energy}, specifically $4.44\,E_P$. This mode is \textbf{ultra-Planckian} and accessible only under extreme physical conditions (near the primordial bounce, cores of compact objects at maximum density).


\textbf{Ultraviolet correction to order $(k\ell_P)^2$ --- radial mode.}


Solving the complete quadratic (E20) for the IR pole of the massive mode, perturbative expansion in small $\ell_P^2 k^2$:


\[
\omega^2 = c^2 k^2 + \omega_0^2 + \mathcal{O}((k\ell_P)^2 \omega_0^2)
\]

with a specific flexural correction computed from the quadratic:


\begin{equation}
\omega^2 = c^2 k^2\left[1 + \frac{(k\ell_P)^2}{20}\right] + \omega_0^2 + \text{(cross corrections of order } (k\ell_P)^2\omega_0^2/c^2\text{)} \tag{E22}
\end{equation}

This is the dispersion relation of the tree-level radial mode with the principal UV correction $(k\ell_P)^2/20$ to the kinetic term, originating in the flexural term of F1. The factor $1/20$ is inherited directly from the flexural coefficient $1/40$ of the Lagrangian multiplied by the factor 2 of the derivative (E18).


\textbf{Transverse traceless tensor mode (TT).}


The TT mode (massless, transverse-traceless tensor configuration of the field, see \S{}II.5.5 for the complete identification) is a modulation of the \textbf{phase sector} that preserves the modulus ($\delta S^\text{TT} = 0$). Its field equation is, therefore, that of the phase mode (E19) ---not that of the radial mode (E18)---, particularized to the transverse-traceless tensor polarization. Taking (E19) and seeking plane-wave solutions with $u \equiv k^2 - \omega^2/c^2$:


\[
\Box\delta\theta - \frac{\ell_P^2}{20}\Box^2\delta\theta = 0 \quad\Rightarrow\quad u - \frac{\ell_P^2}{20}u^2 = 0 \quad\Rightarrow\quad u\left(1 - \frac{\ell_P^2 u}{20}\right) = 0
\]

The two solutions are: $u = 0$ (physical IR mode) and $u = 20/\ell_P^2$ (UV Lee-Wick pole, fakeon scale). The regulation pole comes from the phase flexural term of (E19), of identical origin to that of the modulus sector (diamond lattice, \S{}II.3.8 P9b); it is not inherited from the quadratic of the radial mode (E20), but derived directly from the phase equation.


The physical IR mode ($u = 0$): $\omega^2 = c^2 k^2$ at zeroth order. The flexural correction is introduced by expanding the physical branch of the quadratic perturbatively in $k\ell_P \ll 1$:


\begin{equation}
\boxed{\;\omega_\text{TT}^2 = c^2 k^2\left[1 + \frac{(k\ell_P)^2}{20} + \mathcal{O}((k\ell_P)^4)\right]\;} \tag{E23}
\end{equation}

at exact velocity $c$ in the IR limit, with a UV correction of coefficient $1/20$ inherited from the phase flexural term. Since the photon is also a phase mode (scalar, \S{}III.4.3), it shares exactly the same dispersion relation and the same UV correction $(k\ell_P)^2/20$: the ratio between the ultraviolet coefficients of photon and TT mode is unity, a consequence of both inheriting the same phase regulator of the diamond lattice with the same invariant $1/40$, independently of the scalar or tensor structure of the polarization (\S{}II.3.8 P9b).


\textbf{Regime of validity.} The low-order approximation (neglecting $(\ell_P^2/20)u^2$) is valid in the regime $k\ell_P \ll 1$. For experimentally accessible gravitational frequencies (LIGO/Virgo at 100 Hz, LISA at 1 mHz, GW170817), $k\ell_P \sim 10^{-43}$ or smaller, so the UV correction $(k\ell_P)^2/20 \sim 10^{-86}$ is suppressed by \textasciitilde{}69 orders of magnitude with respect to the Harry-Noller observational bound, $\sim 10^{-17}$. PIU-$\Psi$ reproduces general relativity in the propagation of gravitational waves with astronomical precision.


\textbf{Epistemic classification.} The dispersion relations (E22) ---radial mode--- and (E23) ---TT mode--- are \epi{DERIVED} from linearized F1 by the standard Euler-Lagrange procedure, without free parameters or additional postulates. The coefficient $1/20$ of the UV correction of the TT mode is structural, inherited from the phase flexural coefficient $1/40$ of F1 (\S{}II.3.8 P9b), derived from the same lattice operator as that of the modulus sector, not by analogy with it. The identity of the UV correction between photon and TT mode (ratio of unity) is \epi{DERIVED}.


\subsubsection{II.5.5 --- The three classes of modes of \emph{el Pleno}}

The general perturbation $\delta\Psi$ admits a decomposition into three distinguishable classes according to their geometric-dynamical character:


\textbf{Class 1 --- Radial mode} ($\delta S$, without transverse phase gradients).


\begin{itemize}[leftmargin=1.4em]
  \item Nature: perturbation of the field modulus. Locally compresses/dilates the medium.
  \item Rest mass: $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$ (tree-level, see E21 and \S{}II.5.4).
  \item Regime: accessible only under extreme conditions (near the primordial bounce, cores of compact objects).
  \item Status: \epi{DERIVED} via (E18), (E21), (E22).
\end{itemize}

\textbf{Class 2 --- Phase modes} ($\delta\theta_\pm$, without modulus perturbation).


\begin{itemize}[leftmargin=1.4em]
  \item Nature: perturbation of the field phase over the background $\theta_0$. Without compression of the medium.
  \item Mass: 0 --- massless mode; infrared branch $\Box\delta\theta = 0$, with ultraviolet flexural correction $(k\ell_P)^2/20$ (E19).
  \item Velocity: exactly $c$ in the infrared limit, with an ultraviolet dispersion correction common to the TT mode and the photon.
  \item Status: \epi{DERIVED} via (E19).
\end{itemize}

The phase modes admit an \textbf{additional decomposition} according to the A/B bipartition of the diamond lattice. Defining the symmetric and antisymmetric modes of \S{}II.3.4 equation (E12), the symmetric mode $\delta\theta_+$ lives in U(1) and the antisymmetric mode $\delta\theta_-$ lives in SU(2). The detailed identification of these modes with phenomenological sectors (electromagnetism, weak) is developed in \S{}III.4 (induced gauge sector).


\textbf{Class 3 --- Transverse traceless tensor modes (TT mode).}


This class is structurally distinct from the previous ones and requires a specific derivation. Consider the correlated fluctuation $(\delta S, \delta\theta)$ that satisfies:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{No amplitude alteration:} $\delta S^\text{TT} = 0$ (the TT configuration preserves $|\Psi|$ by construction).
  \item \textbf{Transverse:} $\partial^i h^\theta_{ij} = 0$ (phasic component with null spatial divergence).
  \item \textbf{Traceless:} $\delta^{ij} h^\theta_{ij} = 0$ (component without scalar part).
\end{itemize}

where $h^\theta_{ij}$ is the tensor perturbation of the phase. This is the \textbf{transverse-traceless tensor configuration of \emph{el Pleno}}, a geometric modulation of the medium that in the macroscopic regime is identified via the Sakharov + Anselmi-Piva chain (full development in \S{}III.3) with the metric perturbation $h_{\mu\nu}^\text{TT}$ of General Relativity. They are \textbf{identical}, not two distinct objects.


\textbf{Ontological status of the TT mode.} \epi{DERIVED} as a \textbf{transverse-traceless tensor configuration of \emph{el Pleno}, not a quantized particle}. The macroscopic gravitational wave observed experimentally (LIGO, Virgo, GW170817) is the macroscopic TT modulation of the field $\Psi$ of \emph{el Pleno}. The ontological difference with the standard QFT language of ``gravitons as mediating particles of the gravitational interaction'' is structural and fundamental: in PIU-$\Psi$ gravity emerges as a macroscopic property of \emph{el Pleno} via Sakharov + Anselmi-Piva (\S{}III.1), without quantization into independent gravitational particles. The TT mode is not a quantum entity additional to the single scalar field: it is a configuration of the same physical medium. As a general ontological discipline of the program, the term ``graviton'' is not used ---it names a mediating particle of an independent quantized gravitational field, exactly what PIU-$\Psi$ denies---: the correct name is \emph{TT mode} or transverse-traceless tensor configuration of \emph{el Pleno}. The rule extends to the whole catalogue: where the Standard Model names a mediating particle, PIU-$\Psi$ has a configuration of the medium (the photon is the phase mode of $U(1)_\text{em}$, color is $\mathbb{Z}_3$ vortex structure), and importing the particle name would be ontological contamination of the same class as postulating an independent metric.


Dispersion relation of the TT mode: (E23), identical to the electromagnetic mode (photon) not only in the IR regime but also in its first ultraviolet correction $(k\ell_P)^2/20$, since both are modes of the phase sector of \emph{el Pleno} (\S{}II.3.8 P9b).

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{dispersion_modos.png}
\caption{The mode spectrum of \emph{el Pleno}. (A) Dispersion relation: the radial mode $\delta S$ is massive (branch with gap $\omega_0=\sqrt{\xi}\,c/\ell_P$, mass $m_\text{radial}c^2=4.44\,E_P$), while the TT mode has no gap and propagates at velocity $c$ ---the gravitational waves. (B) The three classes of modes according to their rest mass: the TT mode (gravitational waves) and the phase mode (photon/gauge sector) are both massless and constitute the observables of the program; the radial mode is ultra-Planckian ($4.44\,E_P$), which explains why \emph{el Pleno} is not detected as a particle. The gap of the radial mode comes from the curvature of the potential, $\xi=V''_\text{eff}(S_{\min})/(\rho_P c^2)=19.69$. The value $4.44\,E_P$ is the \emph{elastic} mass (tree-level, static stiffness of the well); the complete resolution of the dispersion relation also gives a \emph{propagating} mass of $3.50\,E_P$ (gap of the physical branch after the flexural regulation). Both are ultra-Planckian; the distinction is ontological, without observational consequence (\S{}III.1.9).}
\label{fig:dispersion}
\end{figure}

\subsubsection{II.5.6 --- The induced gauge sector over the A/B bipartition}

The Class 2 modes (phase modes) combined with the A/B bipartition of the diamond lattice produce, in the detailed construction of \S{}III.4, the \textbf{induced gauge sector} of \emph{el Pleno}. The derivational scheme that connects the geometry with the gauge sector --- which the present section announces and Part III develops completely --- is the following:


\textbf{Step 1.} The phase $\theta$ projected onto the bipartition produces the modes $\delta\theta_+ \in U(1)$ and $\delta\theta_- \in SU(2)$.


\textbf{Step 2.} The Kaluza-Klein construction over the diamond lattice generates induced gauge fields in the continuum:


\[
A_\mu(x) = \frac{\hbar}{e_\text{eff}}\partial_\mu\delta\theta_+(x) + \text{tensor terms of the antisymmetric mode}
\]

where $e_\text{eff}$ is an effective coupling whose numerical value is an open thread of the program, linked to the determination of the hypercharge of the scalar doublet inherited from the A/B bipartition.


\textbf{Step 3.} The spontaneous breaking of the symmetry $SU(2) \times U(1) \to U(1)_\text{em}$, sustained by the doublet $\Phi$ inherited from the A/B bipartition, produces three massive gauge bosons $W^\pm, Z^0$ and one massless gauge boson identified with the \textbf{photon} $\gamma$. The zero mass of the photon is a rigorous mathematical consequence of the Goldstone-Higgs theorem applied to the unbroken residual symmetry $U(1)_\text{em}$.


\textbf{Announced structural identity:} the propagation velocity of the photonic mode in the relaxed regime is \textbf{exactly} $c$, identical to the velocity of the TT mode of \emph{el Pleno}. This equality --- experimentally verified by GW170817 with precision $|v_\gamma - v_\text{GW}|/c < 7\times 10^{-16}$ --- is not coincidence: both are properties of the same single physical medium.


The complete derivation of each step, with the associated theorems (Goldstone, Higgs, Kaluza-Klein) and the verification of the zero mass of the photon, is developed in \S{}III.4 of the manuscript.


\subsubsection{\texorpdfstring{II.5.7 --- Non-relativistic regime: formal correspondence F1 $\leftrightarrow$ Gross-Pitaevskii}{II.5.7 - Non-relativistic regime: formal correspondence F1 <-> Gross-Pitaevskii}}

An important structural correspondence emerges when one considers the \textbf{non-relativistic} limit of F1: when the modes of \emph{el Pleno} have frequencies much lower than the Planckian frequency $\omega_P = c/\ell_P$ and the dynamics is slow compared with the microscopic scale of the medium. In that regime, F1 reduces formally to the \textbf{Gross-Pitaevskii equation} (Gross 1961, Pitaevskii 1961~\cite{grosspitaevskii}), which is the most experimentally validated macroscopic quantum field equation in the history of physics (Bose-Einstein condensates, superfluids, superconductors).


\textbf{Reduction procedure.}


\textbf{Step 1 --- Non-relativistic regime.} We assume $S = S_{\min}^\text{estr} + \delta S$ with $\delta S \ll S_{\min}^\text{estr}$ and slow temporal dynamics compared with the Planckian frequency $\omega_P = c/\ell_P$.


\textbf{Step 2 --- Field substitution.} We define the effective non-relativistic wavefunction by factoring out the fast phase associated with the rest mass of the radial mode and normalizing the modulus to its reading as the \emph{number density} of the medium. By the constitutive identification $S^2 \leftrightarrow n_\text{quanta}/n_0$ with $n_0 = 1/\ell_P^3$ (\S{}I.8.8, \S{}II.2.2), the perturbation $\delta S$ has the number-density reading $\delta S^2/\ell_P^3$, which fixes the prefactor of the effective wavefunction:


\[
\psi(\mathbf{x}, t) = \frac{\delta S(\mathbf{x}, t)}{\ell_P^{3/2}}\,e^{-i\omega_0 t}
\]

where $\omega_0 = m_\text{radial}c^2/\hbar = \sqrt{\xi}\,c/\ell_P$ is the frequency associated with the rest mass of the radial mode (E21). With this normalization $|\psi|^2 = \delta S^2/\ell_P^3$ is a \textbf{number density} ($[\,|\psi|^2\,] = \text{m}^{-3}$), the canonical Gross-Pitaevskii convention in which $|\psi|^2$ counts quanta per unit volume, coherent with the reading of the modulus of \emph{el Pleno} established in \S{}I.8.8 (not with a normalization to energy density $kg/m^3$). The fast phase $e^{-i\omega_0 t}$ separates the relativistic dynamics (frequency $\omega_0 \sim \omega_P$) from the non-relativistic dynamics (frequency $\omega_{\delta S} \ll \omega_0$).


\textbf{Step 3 --- Expansion of the potential to fourth order.} Expanding $V_\text{eff}$ to fourth order in $\delta S$ around $S_{\min}^\text{estr}$:


\[
V_\text{eff}(S_{\min} + \delta S) = V_\text{eff}(S_{\min}) + \frac{1}{2}V''_\text{eff}\delta S^2 + \frac{1}{6}V'''_\text{eff}\delta S^3 + \frac{1}{24}V^{(4)}_\text{eff}\delta S^4 + \mathcal{O}(\delta S^5)
\]

with all the derivatives evaluated at $S_{\min}^\text{estr}$. The cubic term $\tfrac16 V'''_\text{eff}\delta S^3$ does \emph{not} average to zero. With $\delta S = \tfrac12(A\,e^{-i\omega_0 t}+A^*e^{+i\omega_0 t})$, its square contains $\delta S^2 = \tfrac14 A^2 e^{-2i\omega_0 t} + \tfrac14 A^{*2}e^{+2i\omega_0 t} + \tfrac12|A|^2$: two components at $2\omega_0$ and a DC component that does not oscillate. The correct reason why the cubic does not contribute at first order is not that it averages to zero, but that none of its components (DC or $2\omega_0$) is in resonance with $\omega_0$, the carrier frequency of $\psi$. At second order, in contrast, those components fed back by the cubic generate a resonant term at $\omega_0$ that renormalizes the effective quartic (Step 6).


\textbf{Step 4 --- Substitution into F1 and rotating WKB average.} Substituting into F1 with the decomposition of Step 2 and retaining only non-oscillating terms (rotating average at ultra-Planckian frequencies $\omega_0$), the effective non-relativistic Lagrangian is obtained:


\[
\mathcal{L}_\text{eff} = \frac{i\hbar}{2}(\psi^*\partial_t\psi - \psi\partial_t\psi^*) - \frac{\hbar^2}{2 m_\text{eff}}|\nabla\psi|^2 - \frac{g_\text{eff}}{2}|\psi|^4
\]

with the identification of coefficients term by term (comparing with the canonical form of the effective Lagrangian):


\textbf{Step 5 --- Correct dimensional determination of $m_\text{eff}$.}


To identify $m_\text{eff}$ from the effective Lagrangian, we compare the quadratic term in $\psi$:


\begin{itemize}[leftmargin=1.4em]
  \item In the original Lagrangian, the rest-mass term comes from the kinetic + quadratic-potential combination evaluated at the mode at rest ($k = 0$), which produces $\hbar\omega_0 \cdot |\psi|^2$.
  \item In non-relativistic GP, the corresponding term is $-i\hbar\partial_t\psi^*\psi$, which defines the natural frequency $\omega_0$ of the mode.
\end{itemize}

The identification is $\hbar\omega_0 = m_\text{eff}c^2$. With $\omega_0 = \sqrt{V''_\text{eff}(S_{\min})/(\rho_P\ell_P^2)}$ derived in (E21):


\begin{equation}
\boxed{\;m_\text{eff}\,c^2 = \hbar\,\sqrt{\frac{V''_\text{eff}(S_{\min}^\text{estr})}{\rho_P\,\ell_P^2}} = \sqrt{\xi}\,\frac{\hbar c}{\ell_P} = \sqrt{\xi}\,E_P\;} \tag{E24}
\end{equation}

Equivalently:


\[
m_\text{eff} = \sqrt{\xi}\,m_P = \sqrt{V''_\text{eff}/(\rho_P c^2)}\,m_P
\]

With $\xi = V''_\text{eff}(S_{\min}^\text{estr})/(\rho_P c^2)\approx 19.69$ (\S{}I.8.8), the radial mode has mass $m_\text{eff}c^2 = \sqrt{\xi}\,E_P\approx 4.44\,E_P\approx 5.42\times 10^{19}$ GeV, with $E_P\approx 1.22\times 10^{19}$ GeV the Planck energy. It is an ultra-Planckian excitation: the oscillation mode of the modulus of \emph{el Pleno} around the \emph{Pleno}-ground state costs several times the Planck energy, which makes it inaccessible to any experiment and justifies that the observable effective regime is that of the massless phase mode (which generates the gauge dynamics), not that of the massive radial mode.

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

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

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

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

\textbf{Dimensional verification.} $[V^{(4)}_\text{eff}\,\ell_P^6] = [J/m^3]\cdot[m^6] = [J\,m^3]$, contact dimension in 3D; the factor in brackets is dimensionless $\checkmark$.

\textbf{Numerical verification.} The three derivatives of the effective potential at $S_{\min}^\text{estr}$ are obtained from the same potential $V = W_\text{bare} + V_q + V_\text{topo}$ (\S{}II.2): $V''_\text{eff} = 19.69\,\rho_P c^2$, $V'''_\text{eff} = 27.92\,\rho_P c^2$, $V^{(4)}_\text{eff} = 855.6\,\rho_P c^2$. The cubic factor is
\[
1 - \frac{5}{3}\frac{(27.92)^2}{19.69\times 855.6} = 1 - \frac{5}{3}(0.04627) = 0.9229,
\]
whence
\[
g_\text{eff} = \frac{855.6\,\rho_P c^2\,\ell_P^6}{12}\times 0.9229 = 65.8\,\rho_P c^2\,\ell_P^6 \approx 5.43\times 10^{-94}\ \text{J}\cdot\text{m}^3,
\]
a correction of $-7.7\%$ with respect to the direct quartic ($71.3\,\rho_P c^2\ell_P^6 \approx 5.88\times 10^{-94}$ J$\cdot$m$^3$). The positive sign is preserved ($g_\text{eff}>0$): the self-interaction $|\psi|^4$ remains repulsive and the condensate of the \emph{Pleno}-ground state remains stable. \epi{DERIVED-ALGEBRA}: $g_\text{eff}$ is a pure ratio of the derivatives of the potential, zero free parameters.

\textbf{Step 7 --- Euler-Lagrange equation.} Applying Euler-Lagrange to $\mathcal{L}_\text{eff}$ produces the canonical Gross-Pitaevskii equation:


\begin{equation}
\boxed{\;i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_\text{eff}}\nabla^2\psi + g_\text{eff}|\psi|^2\psi\;} \tag{E26}
\end{equation}

\textbf{Dimensional verification of $m_\text{eff}$.}


\[
\left[\sqrt{\frac{V''_\text{eff}}{\rho_P\ell_P^2}}\right] = \sqrt{\frac{[J/m^3]}{[kg/m^3][m^2]}} = \sqrt{\frac{[kg/(m\,s^2)]}{[kg/m]}} = \sqrt{[1/s^2]} = [1/s]
\]

\[
[\hbar/c^2 \cdot 1/s] = [J\cdot s]/[m^2/s^2] \cdot [1/s] = [kg\,m^2/s \cdot s^2/m^2/s] = [kg] \;\checkmark
\]

Correct dimension: mass. $\checkmark$


\textbf{Numerical verification of $m_\text{eff}$.}


\[
m_\text{eff} = \sqrt{19.69}\,m_P = 4.4373\,m_P
\]

Coincides exactly with the mass of the radial mode $m_\text{radial}c^2 \approx 4.44\,E_P$ derived in (E21), confirming the internal consistency: the non-relativistic regime of F1 coherently reproduces the mass of the massive mode of \emph{el Pleno}.


\textbf{Summary of dimensional verifications.} The identifications (E24) and (E25) are dimensionally consistent: $m_\text{eff}$ has dimension of mass, $g_\text{eff}$ has dimension $[J\,m^3]$ of a contact coupling constant in 3D. Both explicit verifications were carried out above. Additionally, with the normalization $[\psi]=\text{m}^{-3/2}$ of Step 2, the three terms of (E26) share units term by term, $[\hbar\,\partial_t\psi]=[\hbar^2\nabla^2\psi/m_\text{eff}]=[g_\text{eff}|\psi|^2\psi]=\text{kg}\,\text{m}^{1/2}\text{s}^{-2}$ (verified symbolically): the consistency of the complete equation, and not only of each coefficient separately, fixes the prefactor of $\psi$ of Step 2 uniquely to $\ell_P^{-3/2}$ ---the only normalization compatible with the reading of $|\psi|^2$ as a number density of \S{}I.8.8.


\textbf{Physical meaning of the correspondence.} F1 contains Gross-Pitaevskii as an exact non-relativistic limit when the scales are correctly projected. Every experimentally validated prediction of GP --- Bogoliubov-de Gennes spectra, dynamics of bright solitons, collapse-rebound in Bosenovas --- is a derivable prediction of \emph{el Pleno} under PIU-$\Psi$ in its corresponding effective regime. This turns the experiments on macroscopic quantum condensates into \textbf{indirect experimental validation} of the program.


\textbf{Epistemic classification.} The F1 $\leftrightarrow$ GP correspondence is \epi{DERIVED} with an explicit and dimensionally consistent computation of the effective mass (E24) and of the coupling constant (E25). The coupling $g_\text{eff}$ is complete to second order: the direct quartic plus the cubic renormalization (factor $5/3$, E25a--E25d) leave it as a pure ratio of the derivatives of the potential, with no free parameters ---\epi{DERIVED-ALGEBRA}. As a \textbf{parametric mapping}, the GP phenomenology is a prediction of el Pleno conditional on GP being correct in its regime of validity ---which is experimentally established without dispute---; that indirect experimental validation is a status of evidence, not a derivation thread.


\subsubsection{II.5.8 --- Closing and connection with \S{}II.6}

\emph{El Pleno} admits three distinguishable classes of propagating modes over the \emph{Pleno}-ground state, with a dispersion relation derived algebraically from linearized F1: the \textbf{radial mode} $\delta S$, massive ($m_\text{radial}c^2 \approx 4.44\,E_P$, accessible only in extreme regimes); the \textbf{phase modes} $\delta\theta_\pm$, massless and at velocity $c$, with $U(1)\times SU(2)$ decomposition over the A/B bipartition that produces the induced gauge sector (\S{}III.4); and the \textbf{transverse-traceless tensor modes} (TT), massless, identified in the macroscopic regime with the gravitational wave (\S{}III.3). The ultraviolet correction $(k\ell_P)^2/20$ is common to the massive and massless classes, originating in the flexural term with coefficient $1/40$. The non-relativistic regime of F1 additionally reproduces the Gross-Pitaevskii equation (E26) with effective mass $m_\text{eff}\approx 4.44\,m_P$, coincident with that of the radial mode. The following section develops the \textbf{localized solitonic configurations} ---solutions of the complete non-linear field equations (F2), in two topological sub-classes (skyrmions and Q-balls) and one composite type (Borromean clusters).


\medskip\hrule\medskip

\subsection{\S{}II.6 --- Solitonic and topological configurations}


\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none critical of its own. Threads inherited from the research program: refinement of the baryonic rotational moment of inertia, extension to the mesonic sector --- declared as a research program in \S{}V.3.
  \item \textbf{Notes:} the $\mathbb{Z}_3$ modular arithmetic rule applies to integer numerators of elementary charges (not to direct fractional sums); the proton radius $R_p$ has the status of a prediction of the program, not of an additional observational input.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.6.1 --- Localized configurations: definition and topological classification}

The Type 4 configurations (\S{}II.1.3) are solutions of the field equations F2 with the following geometric property: the field $\Psi(x^\mu)$ deviates significantly from its background configuration $\Psi_0 = S_{\min}^\text{estr}\cdot e^{i\theta_0}$ only in a finite spatial region, connecting asymptotically to the background at spatial infinity. These configurations are called \textbf{solitons} and are the mathematical substrate of what Part III will identify with physical particles.


\textbf{Operational definition.} A configuration $\Psi(\mathbf{x}, t)$ of \emph{el Pleno} is a soliton if:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{It is a stable solution} of the field equations F2 (not a linear perturbation over a background; the non-linear character of F2 is essential for the existence of the soliton).
  \item \textbf{It has finite total energy}: $E_\text{sol} = \int T_{00}\,d^3x < \infty$, with $T_{00}$ the temporal component of the energy-momentum tensor of \emph{el Pleno}.
  \item \textbf{It connects to the background at infinity}: $\Psi(\mathbf{x}) \to \Psi_0$ as $|\mathbf{x}| \to \infty$.
  \item \textbf{It has a temporally stable profile}: the energy and the charge (when applicable) of the soliton are preserved under temporal evolution.
\end{itemize}

\textbf{Topological classification.} The solitonic configurations of \emph{el Pleno} are distinguished into classes according to the \textbf{topology of the internal configuration space of the soliton} (\S{}II.3.6). This classification is mathematically rigorous and determines fundamental quantum properties (fermionic vs bosonic statistics) via the Finkelstein-Rubinstein theorem.


The two basic topological classes of \emph{el Pleno} are:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Class M (topological)}: configurations whose internal configuration space has a non-trivial homotopy group. Stability sustained by topology (conserved integer topological charge). Under Finkelstein-Rubinstein, mandatory fermionic quantization. Canonical example: skyrmions.
  \item \textbf{Class MO (non-topological)}: configurations whose internal configuration space is topologically contractible ($\pi_n = 0$ for all $n$). Stability sustained dynamically by the Noether U(1) charge of axiom A5. Under Finkelstein-Rubinstein, mandatory bosonic quantization. Canonical example: Q-balls.
\end{itemize}

A third composite class --- \textbf{tripartite Borromean-type clusters} --- combines elements of the two previous ones and is developed in II.6.5 to II.6.7.


\subsubsection{II.6.2 --- Skyrmion-type solitons: class M}

\textbf{Mathematical structure.} The class M solitons are constructed over the target $SU(2) \cong S^3$ (the three-dimensional sphere). The field of \emph{el Pleno} admits a decomposition over the A/B bipartition (\S{}II.3.4) that naturally produces this structure: the antisymmetric mode $\theta_- \in SU(2)$ provides the required configuration space.


Consider a radial profile $U(\mathbf{x}, t)$ with values in $SU(2)$ that satisfies the asymptotic boundary condition:


\[
U(\mathbf{x}) \to I \quad\text{as}\quad |\mathbf{x}| \to \infty
\]

where $I$ is the identity in $SU(2)$. The configurations that satisfy this condition are classified by the homotopy group $\pi_3(SU(2)) = \pi_3(S^3) = \mathbb{Z}$ (Skyrme 1962~\cite{skyrme}, Manton-Sutcliffe 2004). Each configuration is characterized by an integer $B \in \mathbb{Z}$ called the \textbf{winding number} or \textbf{baryon number}.


The fundamental skyrmion has $B = 1$. Configurations with $|B| \geq 2$ are multi-skyrmions, $B = -1$ is the anti-skyrmion, $B = 0$ is the trivial background configuration.


\textbf{Explicit solutions (hedgehog ansatz).} The canonical solution with $B = 1$ is the hedgehog ansatz:


\[
U(\mathbf{x}) = \cos f(r) + i\,\boldsymbol{\sigma}\cdot\hat{\mathbf{x}}\,\sin f(r)
\]

where $r = |\mathbf{x}|$, $\hat{\mathbf{x}} = \mathbf{x}/r$, $\boldsymbol{\sigma}$ are the Pauli matrices, and $f(r)$ is a radial function with conditions $f(0) = \pi$, $f(\infty) = 0$ (interpolating between the north pole and the south pole of $S^3$). The profile $f(r)$ is obtained by solving the non-linear field equations of \emph{el Pleno} (F2) with the energy-minimization condition.


\textbf{Topological-statistical status.} By the Finkelstein-Rubinstein theorem applied to the internal configuration space $S^3$ of the individual soliton (\S{}II.3.6), the $2\pi$ rotation of the skyrmion produces factor $(-1)^B$ (Witten 1983~\cite{adkins}; explicit computation by Krusch 2003). For $B = 1$: factor $-1$, \textbf{mathematically mandatory fermionic quantization}. The skyrmions of \emph{el Pleno} with odd $B$ are fermions; with even $B$ they are bosons.


\textbf{Stability by Derrick's theorem.} Derrick's theorem (1964)~\cite{derrick} imposes a condition on the stability of scalar solitons. F1 satisfies this condition because it combines:


\begin{itemize}[leftmargin=1.4em]
  \item Kinetic term $\rho_P c^2 \ell_P^2 (\partial\Psi)^2$ that favors expansion (kinetic energy decreases with large size).
  \item Potential $V(S)$ that favors contraction (the hard wall at $S_{\max}$ is avoided with a compact profile).
  \item Flexural term $-\gamma_\kappa(\Box S)^2$ that stabilizes the minimum size (prevents collapse to zero size).
\end{itemize}

The balance produces solitons with a characteristic scale of order $\ell_P$ multiplied by dimensionless factors derived from F1.


\textbf{Mass of the fundamental soliton.} The mass of the fundamental skyrmion (configuration $B=1$) is computed from the total energy of the hedgehog profile. The computation, developed completely in \S{}III.5 of Part III, produces the structural form:


\begin{equation}
M_Q = \frac{M_P}{S_{\max,\text{eff}}} \tag{E27}
\end{equation}

where $M_P = \rho_P\ell_P^3 = \sqrt{\hbar c/G} \approx 2.18\times 10^{-8}$ kg is the mass of one cell of \emph{el Pleno}, read in the human frame as the Planckian mass. Numerically:


\[
M_Q^\text{phys} \approx \frac{2.18\times 10^{-8}\,\text{kg}}{\sqrt{1.4956}} \approx 1.78\times 10^{-8}\,\text{kg} \approx 9.98\times 10^{18}\,\text{GeV}/c^2
\]

The mass of the fundamental soliton is of order the Planck mass, \textbf{ultra-heavy} compared with the observed hadronic masses. This scale difference is expected: the fundamental soliton is the simplest solitonic configuration of \emph{el Pleno}, not the configuration identified with observed particles such as protons or neutrons (whose structure is more complex and is developed in II.6.5 and following).


\textbf{Epistemic classification.} The existence of skyrmion-type solitons is \epi{DERIVED} from the topology $\pi_3(SU(2)) = \mathbb{Z}$ of the target provided by the A/B bipartition (\S{}II.3.4). The fermionic statistics for odd $B$ is \epi{DERIVED} by Finkelstein-Rubinstein (Krusch 2003). The mass $M_Q = M_P/S_{\max,\text{eff}}$ is \epi{DERIVED} in its structural form; the physical numerical value inherits the uncertainty of $S_{\max,\text{eff}}$ (\S{}II.4.2).


\subsubsection{II.6.3 --- Q-ball-type solitons: class MO}

\textbf{Mathematical structure.} The class MO solitons are constructed over a topologically contractible sub-target of the field. Specifically, the relevant sub-target for Q-ball profiles of \emph{el Pleno} is:


\[
\mathcal{M}_\text{Qball} = [S_{\min}^\text{estr}, S_{\max,\text{geom}}] \subset \mathbb{R}^+
\]

that is, amplitude profiles $\delta S$ with constant phase (the topology of the antisymmetric mode $\theta_-$ is not exploited). This interval is topologically contractible. By the standard property of contractible spaces (Hatcher 2002, Proposition 1.16):


\[
\pi_n(\mathcal{M}_\text{Qball}) = 0 \quad\text{for all}\quad n \geq 0
\]

\textbf{MO lacks topological winding.} Its stability is \textbf{dynamical}, not topological: sustained by the Noether U(1) charge associated with the phase $\theta$ (axiom A5).


\textbf{Canonical identification.} MO satisfies the four Coleman (1985) conditions for Q-balls:


\begin{itemize}[leftmargin=1.4em]
  \item Scalar field (modulus $S$ of \emph{el Pleno}).
  \item U(1) symmetry (axiom A5).
  \item Attractive self-interaction (Gent topological potential $V_\text{topo}$).
  \item Conserved U(1) charge.
\end{itemize}

Therefore, \textbf{MO $\equiv$ generalized Q-ball of \emph{el Pleno}}, in the canonical Coleman 1985 sense extended to the medium of \emph{el Pleno}.


\textbf{Explicit solutions.} The canonical form of the Q-ball is a spherical profile:


\[
\Psi(\mathbf{x}, t) = \phi(r)\,e^{i\omega t}
\]

where $\phi(r)$ is a real radial function with $\phi(0) = \phi_0$ and $\phi(r) \to S_{\min}^\text{estr}$ as $r \to \infty$. The frequency $\omega$ is a dynamical parameter of the Q-ball that encodes its total U(1) charge.


\textbf{Topological-statistical status.} By the Finkelstein-Rubinstein theorem applied to the contractible configuration space (\S{}II.3.6): $2\pi$ rotation produces factor $+1$ (Coleman 1985; extended analysis by Lee-Pang 1992). \textbf{Mathematically mandatory bosonic quantization}.


\textbf{Stability.} A Q-ball is stable if the total U(1) charge $Q = \int J_t\,d^3x$ exceeds a minimum threshold $Q_\text{crit}$. For $Q < Q_\text{crit}$ the configuration is unstable and dynamically dissolves into the background. The threshold $Q_\text{crit}$ is computed from the parameters of F1 and turns out to be of order unity for the canonical couplings of \emph{el Pleno}.


\textbf{Epistemic classification.} The existence of Q-ball-type solitons is \epi{DERIVED} from axiom A5 (conserved U(1) symmetry) and the attractive self-interaction of the Gent topological potential. The bosonic statistics is \epi{DERIVED} by Finkelstein-Rubinstein.


\subsubsection{II.6.4 --- The statistical dichotomy as a rigorous mathematical consequence}

The statistical difference between Class M (fermionic) and Class MO (bosonic) \textbf{is not a theoretical choice} nor an additional postulate of the program. It is a \textbf{direct mathematical consequence} of the Finkelstein-Rubinstein theorem applied to the distinct topology of the internal configuration spaces of each type of soliton:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{3.30cm}L{3.30cm}L{3.30cm}L{3.30cm}}
\toprule
\textbf{Sub-target} & \textbf{Topology} & \textbf{$\chi_\text{FR}$} & \textbf{Statistics} \\
\midrule\endhead
$SU(2) \cong S^3$ (antisymmetric mode $\theta_-$) & $\pi_3 = \mathbb{Z}$, non-trivial & $(-1)^B = -1$ for $B=1$ & \textbf{Fermionic} \\
$[S_{\min}, S_{\max}] \subset \mathbb{R}^+$ (interval) & $\pi_n = 0$, contractible & $+1$ & \textbf{Bosonic} \\
\bottomrule
\end{longtable}}

This structural dichotomy is the basis on which the program will construct, in \S{}III.5 of Part III, the phenomenological reading of the two observed classes of elementary particles: \textbf{fermions} (electrons, quarks, neutrinos --- skyrmion-type solitons with specific internal structure) and \textbf{bosons} (gauge bosons, scalars --- modes of \emph{el Pleno} or Q-balls as the case may be).


The derivation of the two observed classes of quantum statistics from a single ontological entity --- \emph{el Pleno} with its bipartite structure --- is one of the most significant structural results of the program.


\subsubsection{II.6.5 --- Tripartite Borromean-type clusters: the sub-structure of the composite soliton}

The fundamental skyrmion with $B = 1$ admits a more refined internal construction: instead of being a monolithic soliton with integer winding, it can be decomposed as a \textbf{tripartite cluster} of three vortices with fractional winding. This possibility --- not contemplated in the original canonical formulation of Skyrme --- emerges from a rigorous topological theorem due to Gudnason-Nitta (2020).


\textbf{Gudnason-Nitta theorem.} The topological degree $B \in \pi_3(SU(2)) = \mathbb{Z}$ of a Skyrme field coupled to vortices in a multifluid system is mathematically equivalent to the \textbf{linking number} of the preimages under the Hopf map, and admits a decomposition as a product/sum of vortex winding numbers over clusters:


\begin{equation}
B = \text{Link}(L_1, L_2, L_3) = \sum_\text{clusters}\prod_{i=1}^3 w_i \tag{E28}
\end{equation}

\textbf{Application to the fundamental soliton of \emph{el Pleno}} ($B = 1$). The monolithic ansatz $B = 1 \times 1$ (a single vortex with integer winding) admits generalization to a tripartite cluster with fractional winding:


\[
B = 1 = w_1 + w_2 + w_3 = \frac{1}{3} + \frac{1}{3} + \frac{1}{3}
\]

The theorem guarantees that the three vortices with winding $1/3$ can have \textbf{distinct profiles} without altering the integer total winding. This naturally produces a tripartite internal structure of the fundamental soliton.


\subsubsection{\texorpdfstring{II.6.6 --- $\mathbb{Z}_3$ modular arithmetic and the $2+1$ asymmetry}{II.6.6 - Modular arithmetic and the asymmetry}}


The tripartite cluster admits a specific structural stability characterized by \textbf{modular arithmetic} over the elementary topological charges of the vortices. By the Borromean supercounterfluidity theorem (Blomquist-Syrwid-Babaev 2021), the stable configurations of the tripartite cluster are given by the combinations of integer elementary charges $q_j \in \mathbb{Z}$ that satisfy the congruence:


\begin{equation}
\sum_{j=1}^3 q_j \equiv 0 \pmod{N} \tag{E29}
\end{equation}

with $N = 3$ for the tripartite Borromean case.


\textbf{Critical clarification about the rule.} The ``elementary charges'' $q_j$ to which the congruence (E29) applies are the \textbf{integer numerators} of the vortex-by-vortex topological charges expressed in units of $1/3$. That is, if the fractional electric charge of an individual vortex is $e_j/3$ (with $e_j$ integer), then $q_j = e_j$ and the congruence (E29) applies directly over the integers $e_j$.


The rule \textbf{does not apply to the sums of fractional charges directly} --- that would give non-integer results incompatible with the modulo operation.


\textbf{Application to the baryon of \emph{el Pleno}} ($N = 3$). The two fundamental configurations of the tripartite cluster $B = 1$, expressed as triplets of fractional electric charges and verified against the congruence (E29) over integer numerators:


\textbf{Proton-type configuration:}


\begin{itemize}[leftmargin=1.4em]
  \item Fractional charges: $(+2/3, +2/3, -1/3)$.
  \item Integer numerators: $q_p = (2, 2, -1)$.
  \item Sum of numerators: $\sum_j q_j^{(p)} = 2 + 2 + (-1) = 3$.
  \item Congruence: $3 \equiv 0 \pmod{3}$ $\checkmark$ \textbf{stable}.
  \item Total electric charge: $\sum_j q_j/3 = 3/3 = +1$ (proton charge).
\end{itemize}

\textbf{Neutron-type configuration:}


\begin{itemize}[leftmargin=1.4em]
  \item Fractional charges: $(+2/3, -1/3, -1/3)$.
  \item Integer numerators: $q_n = (2, -1, -1)$.
  \item Sum of numerators: $\sum_j q_j^{(n)} = 2 + (-1) + (-1) = 0$.
  \item Congruence: $0 \equiv 0 \pmod{3}$ $\checkmark$ \textbf{stable}.
  \item Total electric charge: $\sum_j q_j/3 = 0/3 = 0$ (neutron charge).
\end{itemize}

Both configurations satisfy the modular congruence (E29) over the integer numerators --- they are stable as tripartite clusters of \emph{el Pleno}. The difference between them is only the \textbf{specific integer value} of the sum (3 for the proton, 0 for the neutron), not the satisfaction of the congruence (both give remainder 0 modulo 3).


\textbf{Ontological reading.} The $\mathbb{Z}_3$ modular arithmetic of Borromean supercounterfluidity is realized ontologically in \emph{el Pleno} as a \textbf{derived sub-structure} of the A/B bipartition (\S{}II.3.4) extended to the internal tripartite structure of the soliton. The elementary charges $\pm 2/3$ and $\pm 1/3$ are not postulated; they emerge as vortex-by-vortex topological charges imposed by the $\mathbb{Z}_3$ congruence over integer numerators.


\textbf{Operational identification with observed electric charges.} In the phenomenological reading of \S{}III.5 of Part III, the vortex-by-vortex topological charges of the Borromean cluster are identified with the \textbf{fractional electric charges} observed in the quarks of standard physics: $+2/3$ (up, charm, top) and $-1/3$ (down, strange, bottom). The quantization of charges into simple fractions --- a fundamental observational property of the Standard Model postulated as a property of the quarks --- is \textbf{structurally derived} in PIU-$\Psi$ from the $\mathbb{Z}_3$ modular arithmetic of the Borromean cluster of \emph{el Pleno}, with the rule applied precisely over the integer numerators of the charges.


\textbf{$2+1$ asymmetry of the baryon.} The Borromean configuration naturally explains the \textbf{observed $2+1$ asymmetry} of the internal structure of the baryon: two vortices with the same charge (the two $+2/3$ in the proton, the two $-1/3$ in the neutron) and a third distinct one. The asymmetry is not postulated; it is a geometric consequence of the cluster.


\textbf{Epistemic classification.} The Borromean tripartite decomposition is \epi{DERIVED} from the Gudnason-Nitta + Babaev-Borromean theorems applied under the filter of axioms A1--A6. The identification with the electric charges observed in quarks is \textbf{[strong logical deduction]} sustained by the exact coincidence with the $\mathbb{Z}_3$ arithmetic of the Standard Model.


\subsubsection{\texorpdfstring{II.6.7 --- Baryonic rotational tower and $\Delta - N$ mass split}{II.6.7 - Baryonic rotational tower and mass split}}

The tripartite Borromean cluster admits \textbf{rotational modes} as excited states of the same geometric cluster. Standard rotational quantization (imported from the Skyrme formalism under the filter A1) produces a \textbf{tower of states} labeled by the angular momentum $J$:


\begin{equation}
m(J) = m_N + \frac{\hbar^2}{2\Theta_p}\left[J(J+1) - \tfrac{3}{4}\right] \tag{E30}
\end{equation}

where $m_N$ is the mass of the ground state ($J = 1/2$), $\Theta_p$ is the rotational moment of inertia of the cluster, and the factor $-3/4 = -\tfrac{1}{2}\cdot\tfrac{3}{2}$ comes from the subtraction of the value $J(J+1)$ in the ground state ($J = 1/2$).


\textbf{Computation of the moment of inertia.} For a compact Borromean cluster with effective separation $d$ between vortices, the rotational moment of inertia is:


\[
\Theta_p = \frac{m_p\,d^2}{3}
\]

where $m_p$ is the mass of the cluster (identifiable with the proton mass in the phenomenological reading of \S{}III.5). An \textbf{additional prediction of the program} is the structural consistency $d \approx R_p$ between the separation of the vortices of the cluster and the radius of the baryon (verified to 5\%: $d_\text{opt}/R_p = 0.95$). Under this consistency:


\[
\Theta_p = \frac{m_p\,R_p^2}{3}
\]

\textbf{Epistemic status of $R_p$ in the derivation.}


The epistemic nature of $R_p$ in this formula is the following. The PIU-$\Psi$ program produces an \textbf{independent} prediction of the proton radius:


\[
R_p^\text{pred} = \xi_\text{PIU}/2 \approx 0.851\,\text{fm}
\]

where $\xi_\text{PIU}$ is a length scale derived from the apparatus of the program (full development in \S{}III.5). The canonical observational value is $R_p^\text{obs} = 0.8409$ fm (CODATA 2022), with agreement between prediction and observation to \textbf{98.8\%} (relative error of 1.2\%).


Therefore, in the $\Delta - N$ split formula (E30) and the rotational tower, $R_p$ enters as a \textbf{prediction of the program}, not as an additional observational input. The program does not calibrate $R_p$ to observation; it predicts it independently and then uses it in the split formula. The only remaining observational input in the computation of the rotational tower is the \textbf{proton mass} $m_p c^2 \approx 938.3$ MeV, which acts as the single calibrated energy scale of the hadronic sector (analogously to how $\Lambda_\text{QCD}$ acts in QCD).


\textbf{$\Delta - N$ mass split.} The split between the state $J = 1/2$ (ground state, identifiable with the nucleon $N$) and the state $J = 3/2$ (first rotational excited state, identifiable with $\Delta(1232)$) is computed from (E30):


\[
\Delta m = m(3/2) - m(1/2) = \frac{\hbar^2}{2\Theta_p}\left[\tfrac{3}{2}\cdot\tfrac{5}{2} - \tfrac{1}{2}\cdot\tfrac{3}{2}\right] = \frac{\hbar^2}{2\Theta_p}\cdot[\tfrac{15}{4} - \tfrac{3}{4}] = \frac{\hbar^2}{2\Theta_p}\cdot 3 = \frac{3\hbar^2}{2\Theta_p}
\]

Substituting $\Theta_p = m_p R_p^2/3$:


\begin{equation}
\boxed{\;\Delta m = \frac{9\hbar^2}{2\,m_p\,R_p^2} = \frac{9(\hbar c)^2}{2\,m_p c^2\,R_p^2}\;} \tag{E31}
\end{equation}

\textbf{Dimensional verification.} $[\hbar c] = [J\cdot s\cdot m/s] = [J\cdot m]$; $[(\hbar c)^2] = [J^2\,m^2]$; $[m_p c^2\,R_p^2] = [J\cdot m^2]$; ratio: $[J^2\,m^2]/[J\,m^2] = [J]$ $\checkmark$. Dimension of energy, consistent with mass.


\textbf{Numerical verification.} With CODATA 2022 values:


\begin{itemize}[leftmargin=1.4em]
  \item $\hbar c = \rho_P c^2\ell_P^4 = 197.327\,\text{MeV}\cdot\text{fm}$ (native product of the medium, evaluated in hadronic units)
  \item $m_p c^2 = 938.272\,\text{MeV}$
  \item $R_p = 0.8409\,\text{fm}$ (observational value used for direct comparison)
\end{itemize}

Substituting step by step:


\[
\Delta m = \frac{9 \times (197.327)^2}{2 \times 938.272 \times (0.8409)^2}\,\text{MeV}
\]

Numerator: $9 \times 38938 = 350441$. Denominator: $2 \times 938.272 \times 0.7071 = 1326.93$. Ratio: $350441 / 1326.93 = 264.10$.


\[
\Delta m^\text{predicted} \approx 264.10\,\text{MeV}
\]

\textbf{Comparison with observation.} Observed value:


\[
\Delta m^\text{obs} = m_{\Delta(1232)} - m_N \approx 1232.0 - 938.92 = 293.08\,\text{MeV}
\]

(using $m_N = (m_n + m_p)/2 \approx 938.92$ MeV, nucleon average). Ratio:


\[
\frac{\Delta m^\text{predicted}}{\Delta m^\text{obs}} = \frac{264.10}{293.08} = 0.901
\]

\textbf{Agreement to 90\%} (relative error of 9.9\%), with a single calibrated input of the hadronic sector ($m_p$), $R_p$ being an independent prediction of the program with an error of 1.2\% with respect to the observational value. The program \textbf{predicts} the $\Delta - N$ split from the rotational quantization of the Borromean cluster, without additional phenomenological adjustment.


\textbf{Prediction of the complete tower.} The formula (E30) predicts a complete \textbf{rotational tower} of excited states of the Borromean cluster. Computing the base factor $\hbar^2/(2\Theta_p) = 3(\hbar c)^2/(2 m_p c^2 R_p^2) \approx 88.0$ MeV:


{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{2.20cm}L{2.20cm}L{2.20cm}L{2.20cm}L{2.20cm}L{2.20cm}}
\toprule
\textbf{$J$} & \textbf{$J(J+1) - 3/4$} & \textbf{$\Delta E$ (MeV)} & \textbf{$m(J)$ predicted (MeV)} & \textbf{Observed state} & \textbf{Agreement} \\
\midrule\endhead
$1/2$ & $0$ & $0$ & $938.3$ & $N(939)$ & calibration \\
$3/2$ & $3$ & $264.10$ & $1202.4$ & $\Delta(1232)$ & $97.6\%$ \\
$5/2$ & $8$ & $704.27$ & $1642.5$ & $N^*(1680)$ & $97.8\%$ \\
$7/2$ & $15$ & $1320.50$ & $2258.8$ & $\Delta^*(1950)$ & $84.1\%$ \\
\bottomrule
\end{longtable}}

The agreements of the rotational tower with the observed hadronic spectrum are consistent with the expected corrections from rotational deformation typical of the standard Skyrme model.


\textbf{Epistemic classification.} The $\Delta - N$ split and the complete baryonic rotational tower are \epi{DERIVED} from the rotational quantization of the tripartite Borromean cluster. The only calibrated observational input is the proton mass $m_p$; $R_p$ is an independent prediction of the program with observational agreement to 98.8\%.


\subsubsection{II.6.8 --- Stability of the Borromean cluster}

The stability of the tripartite Borromean cluster requires specific analysis: why do the three vortices with winding $1/3$ remain bound without dissociating into independent vortices?


The answer lies in the \textbf{topology of the Borromean linking}: the three vortices are linked in such a way that the dissociation of any pair is topologically blocked by the third. Mathematically, this is characterized by the \textbf{Massey invariant} (Massey 1958) --- a cohomological triple product that is non-null precisely for Borromean linkings.


The non-null Massey invariant of the proton cluster is the rigorous version of the property colloquially expressed as ``the up quark cannot be pulled out of the proton in isolation'': it is not just pairwise linking (which could be broken), but triple Borromean linking whose topological stability is structurally deeper.


\textbf{Epistemic classification.} The stability by a non-null Massey invariant is \epi{DERIVED} from the canonical algebraic topology of the Borromean linking (Massey 1958, Hatcher 2002, Ricca et al. 2024). It is rigorous mathematical language to replace the colloquial expression ``topologically protected''.


\subsubsection{II.6.9 --- Synthesis of \S{}II.6 and connection with \S{}II.7}

The catalogue of solitonic configurations of \emph{el Pleno} under PIU-$\Psi$ comprises three types. The \textbf{skyrmion-type solitons} (class M), with topological winding $B \in \pi_3(SU(2)) = \mathbb{Z}$, stability by topology and mandatory fermionic statistics by Finkelstein-Rubinstein, of fundamental mass $M_Q = M_P/S_{\max,\text{eff}}$ ultra-Planckian. The \textbf{Q-ball-type solitons} (class MO), with contractible configuration space, stability by Noether U(1) charge and bosonic statistics. And the \textbf{tripartite Borromean-type clusters}: decomposition of the fundamental soliton $B=1$ into three vortices with winding $1/3$, with $\mathbb{Z}_3$ modular arithmetic over integer numerators that produces two stable configurations (proton-type, sum $3\equiv 0$; neutron-type, sum $0\equiv 0 \pmod 3$) and a rotational tower whose $\Delta - N$ split is predicted at $264$ MeV against $293$ MeV observed (agreement of 90\%) with a single calibrated input ($m_p$) and $R_p$ as a prediction to 98.8\%; their stability is topologically protected by a non-null Massey invariant.


This repertoire is what \S{}III.5 reads as \textbf{observed physical particles}: the Borromean cluster as the substrate of the baryon, the vortex-by-vortex topological charges as quark charges, the rotational tower as the hadronic spectrum.


\medskip\hrule\medskip

\subsection{\S{}II.7 --- Cosmological regimes of \emph{el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} F-cosmo-PIU-$\Psi$. The background cosmological sector is derived and closed (\S{}III.8): the form of $H(z)$, the matter density and the character of the cosmic microwave background. What remains is a \emph{confrontation} thread ---the combined analysis with third-party catalogues (DESI, Pantheon+) and the prediction of $S_8$---, of subordinate priority to the theoretical closure (\S{}V.3.1). Program developed in \S{}V.4. C-Cosmo-$\Lambda$ (cosmological prediction of the structural hadronic parameter from the primordial-bounce cosmology). Medium-low severity, research program.
  \item \textbf{Notes:} derives the formula F6 ($\rho_\text{matter} = \rho_P[S^2(\mathbf{x}) - S_{\min}^{2,\text{estr}}]$) from the Bogoliubov-Josephson constitutive identification of \S{}II.2.2 extended to non-uniform configurations. Numerically verifies $\Omega_{EO} = f_c$ to 0.30$\sigma$ vs Planck 2018.
\end{itemize}
\end{estadoseccion}

\subsubsection{II.7.1 --- Non-stationary configurations: the fifth class of the catalogue}

The four previous classes (\emph{Pleno}-ground state, crystalline structure, propagating modes, solitons) are essentially \textbf{stationary} or \textbf{localized} configurations. There are also \textbf{non-stationary} configurations, where the global value of the field evolves with cosmological time without admitting a description as a small perturbation over a fixed background: they describe the \textbf{history of the universe at cosmological scale}. The question is which configuration of \emph{el Pleno} corresponds to the primordial universe, which to the current one, and how they connect. The program identifies \textbf{three cosmological regimes}, connected sequentially by the dynamics of F1:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Primordial regime} ---saturated \emph{el Pleno}, $S \to S_{\max,\text{geom}}$, dominated by the Gent topological potential.
  \item \textbf{Freeze-out regime} ---transition, fixing of the operational constants of the medium.
  \item \textbf{Relaxed regime} ---current \emph{el Pleno}, $S \to S_{\min}^\text{estr}$, close to the \emph{Pleno}-ground state.
\end{itemize}


\subsubsection{II.7.2 --- Primordial regime: bounce and structural resolution of the singularity}

\textbf{Physical description.} In the primordial regime, the universe is described by \emph{el Pleno} in its saturated configuration: modulus $S$ close to the structural hard bound $S_{\max,\text{geom}}$, number density of quanta of the medium near its maximum value, dynamics dominated by the hard wall of the topological potential $V_\text{topo}$.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{cronologia_cosmologica.png}
\caption{The cosmological chronology of \emph{el Pleno} as a complete crunch-bounce cycle. The modulus starts at rest in the \emph{Pleno}-ground state ($S_{\min}=0.7231$) and is compressed by the cosmological contraction ($\rho_\text{medium}=\rho_P S^2$, $\rho\propto 1/a^3$): the crunch is born slow from the background and accelerates toward $S_{\max}$. Before reaching the A6 hard bound, the Gent wall stops the collapse and produces the bounce ---a finite rebound, without singularity, instead of the singular Big Bang of General Relativity. Then follow the freeze-out (freezing of constants and formation of particles) and the return to the current relaxed regime at $S_{\min}$. The cycle is asymmetric (increasing volume between cycles); our observable universe is the phase following the last bounce.}
\label{fig:cronologia}
\end{figure}


\textbf{The singularity problem under standard cosmology.} General Relativity predicts, for an expanding universe extrapolated into the past, an initial singularity where the cosmological scale factor $a(t) \to 0$, the matter density diverges, and the curvature of spacetime becomes infinite. This singularity --- the so-called ``Big Bang'' in the literal sense --- is the point where General Relativity itself ceases to apply.


\textbf{Resolution under PIU-$\Psi$.} In PIU-$\Psi$ there is no initial singularity. The reason is structural: axiom A6 establishes the \textbf{hard bound} $S < S_{\max,\text{geom}}$ on the field modulus, and since the modulus is related to the density of the medium via the Bogoliubov-Josephson identification of \S{}II.2.2, the bound carries over to a \textbf{hard bound on the density}:


\begin{equation}
\rho_\text{Pleno}(\mathbf{x}, t) \leq \rho_P\,S_{\max,\text{geom}}^2 \approx 2.1773\,\rho_P \tag{E32}
\end{equation}

This bound is structural --- a property of the physical medium, not a dynamical result that can be violated under extreme conditions. The compression of \emph{el Pleno} toward the primordial regime \textbf{cannot} produce infinite density: the physical medium does not admit such a configuration.


\textbf{Bounce mechanism.} When the cosmological compression pushes \emph{el Pleno} toward $S \to S_{\max,\text{geom}}$, the topological potential $V_\text{topo}(S)$ diverges logarithmically:


\[
V_\text{topo}(S) = -\frac{\rho_P c^2}{2}\log\left(1 - \frac{S^2}{S_{\max,\text{geom}}^2}\right) \to +\infty \quad\text{as}\quad S \to S_{\max,\text{geom}}
\]

This divergence produces an infinite \textbf{restoring force} that stops the compression and produces a \textbf{structural rebound} (bounce). What General Relativity describes as an initial singularity is, under PIU-$\Psi$, the event of maximum compression of the physical medium followed by the structural rebound.


\textbf{Cyclic character.} The existence of the bounce opens the possibility of \textbf{cyclic cosmology}: the universe does not begin in an absolute initial singularity but rebounds from previous configurations of \emph{el Pleno}. This possibility is structurally consistent with F1.


\textbf{Consistency verification.} The bounce occurs before any configuration reaches the absolute hard bound $S = S_{\max,\text{geom}}$. The potential wall $V_\text{topo}$ activates progressively as $S \to S_{\max,\text{geom}}$, stopping the compression at a finite distance from the hard bound. The maximum density attainable during the primordial bounce is therefore \textbf{smaller} than $\rho_P\,S_{\max,\text{geom}}^2$, although of the same order of magnitude.


\textbf{Epistemic classification.} The existence of the primordial bounce is \epi{DERIVED} from axiom A6 (hard bound) combined with the dynamics of the topological potential $V_\text{topo}$. The structural resolution of the Big Bang singularity is a direct consequence, without free parameters or additional hypotheses. The detailed quantitative description of the bounce (exact temporal profile, evolution of the constants during the rebound) is research program \S{}V.4.


\subsubsection{II.7.3 --- Freeze-out regime: fixing of operational constants}

Immediately after the primordial bounce, \emph{el Pleno} transits through a \textbf{freeze-out} regime: as the density of the medium decreases and the effective temperature of \emph{el Pleno} drops below characteristic scales, the operational constants of the medium \textbf{freeze} dynamically.


\textbf{General concept.} Freeze-out is a standard phenomenon in cosmology: when a coupling constant depends on the energy/temperature of the medium and the temperature of the medium drops below the regime where the constant varies, the value of the constant becomes fixed at the value it had at the moment of freeze-out.


\textbf{Application to \emph{el Pleno}.} In \emph{el Pleno} under PIU-$\Psi$, certain constants that parametrize effective couplings of the medium (especially in the internal dynamical sector associated with the phase $\theta$ and the solitonic structure) are \textbf{frozen operational constants of the bounce}: their numerical value was fixed dynamically during the primordial bounce and remains constant in the current relaxed regime.


\textbf{The PIU-$\Lambda$ Theorem.} For a specific operational constant of the hadronic sector --- called $\Lambda_\text{PIU}$, associated with the characteristic energy scale of the dynamics of the solitonic clusters of \emph{el Pleno} developed in \S{}II.6.7 --- the program establishes the following theorem:


\begin{tcolorbox}[derivado]
\textbf{PIU-$\Lambda$ Theorem.} The parameter $\Lambda_\text{PIU}$ is a \textbf{frozen operational constant} of the primordial bounce. The transitions that could modify its value require regimes of effective temperature $T \gtrsim T_f$ of \emph{el Pleno} (with $T_f$ the freeze-out temperature, ultra-Planckian), inaccessible in the current relaxed regime. Therefore, in the relaxed regime:


\begin{itemize}[leftmargin=1.4em]
  \item $\Lambda_\text{PIU}$ does not vary with cosmological redshift $z$.
  \item $\Lambda_\text{PIU}$ does not vary with galactic or environmental location.
  \item $\Lambda_\text{PIU}$ does not adjust with post-freeze-out cosmological epochs.
\end{itemize}

The numerical value of $\Lambda_\text{PIU}$ is not deducible from the dynamics of the relaxed regime: it requires analysis of the primordial-bounce cosmology.

\end{tcolorbox}

\textbf{Structural analogy with QCD.} The situation of $\Lambda_\text{PIU}$ is \textbf{structurally analogous} to the situation of $\Lambda_\text{QCD}$ in lattice QCD: the numerical value of $\Lambda_\text{QCD}$ is a free parameter of QCD (calibrated to the proton mass or equivalent), but the predictions of the hadronic sector derived from $\Lambda_\text{QCD}$ are quantitative and successful. Analogously, $\Lambda_\text{PIU}$ functions as the single observational input of the hadronic sector of \emph{el Pleno}: calibrated to the proton mass, it predicts the $\Delta - N$ split, the proton radius $R_p$, the baryonic rotational tower, and other observational properties of the hadronic sector (all developed in \S{}III.5 of Part III).


The structural difference with QCD is that in PIU-$\Psi$ the \textbf{operational constancy} of $\Lambda$ is \textbf{derived} from axiom A6 + bounce cosmology, whereas in QCD it is a postulated property of the Lagrangian. PIU derives the origin of $\Lambda$; QCD postulates it.


\textbf{Epistemic classification.} The operational constancy of $\Lambda_\text{PIU}$ is \epi{DERIVED} from the PIU-$\Lambda$ Theorem via the A6 hard bound + freeze-out analysis. The numerical value of $\Lambda_\text{PIU}$ is \epi{CALIBRATED} to the hadronic sector via one observational input (the proton mass). The cosmological prediction of the numerical value --- from the primordial-bounce cosmology without hadronic calibration --- is the \textbf{open thread C-Cosmo-$\Lambda$}, declared as research program \S{}V.3.


\subsubsection{II.7.4 --- Current relaxed regime: the observable universe}

In the current cosmological regime, \emph{el Pleno} is in its relaxed configuration: the modulus $S$ of the field is close to $S_{\min}^\text{estr} \approx 0.7231$, the dynamics is dominated by the small perturbations over the \emph{Pleno}-ground state derived in \S{}II.2, and all observed physical phenomena are configurations of this regime.


\textbf{Dynamical characterization.} In the relaxed regime, the field equations F2 admit a decomposition into:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Cosmological background}: global configuration $\bar{S}(t) \approx S_{\min}^\text{estr}$ with slow temporal dependence (cosmological scale $\sim$ age of the universe).
  \item \textbf{Perturbations}: $\delta S(\mathbf{x}, t)$, $\delta\theta(\mathbf{x}, t)$ and transverse-traceless tensor configurations, governed by the linearized equations of \S{}II.5.
  \item \textbf{Localized configurations}: solitons of the catalogue \S{}II.6, forming the observed baryonic matter of the universe.
\end{itemize}

The standard cosmological description under the $\Lambda$CDM model --- FLRW metric, Friedmann equations, parametrized material content --- emerges in this regime as an effective regional description. The complete connection between PIU-$\Psi$ and $\Lambda$CDM is developed in \S{}III.6 of Part III.


\textbf{Stability of the regime.} The \emph{Pleno}-ground state is a stable minimum of the effective potential $V_\text{eff}$ (verified in \S{}II.4.5: $V''_\text{eff}(S_{\min}^\text{estr}) \approx 19.69\,\rho_P c^2 > 0$). This implies that the relaxed regime is \textbf{dynamically stable} at long scales: small perturbations dissipate or propagate without destabilizing the background. This stability is the structural basis of why the observable universe maintains its cosmological configuration at long scales.


\textbf{Temporal evolution.} The evolution of the background $\bar{S}(t)$ in the relaxed regime is described by the field equations applied to the cosmological sector (FLRW configuration). The effective equation of state of the medium in this regime, $w(z) \equiv p_\text{Pleno}/\rho_\text{Pleno}$, admits computation from F1 with the effective FLRW metric. The detailed quantitative result is the \textbf{open thread F-cosmo-PIU-$\Psi$} (medium severity): research program \S{}V.4. What the present section establishes structurally is that the temporal evolution in the relaxed regime is \textbf{slow-roll} (slow variation of $\bar{S}(t)$ compared with the Planckian dynamical scales) and reproduces the cosmological phenomenology of $\Lambda$CDM with the correction $w(z) \neq -1$ predicted by PIU-$\Psi$, \textbf{directly comparable with DESI 2024-2025 observations} that show preliminary evidence of temporal dynamics in the equation of state of dark energy.


\textbf{Epistemic classification.} The structural description of the relaxed regime is \epi{DERIVED} from F1 applied to the cosmological background. The specific numerical values (the precise dynamics of $w(z)$, the predictions for Planck/DESI) are \epi{CALIBRATED} or \textbf{[articulated structural HYPOTHESIS]} as the case may be, with development in \S{}III.6 and declared open threads.


\subsubsection{\texorpdfstring{II.7.5 --- The transition primordial $\to$ freeze-out $\to$ relaxed}{II.7.5 - The transition primordial -> freeze-out -> relaxed}}

The three phases of the cosmological regime of \emph{el Pleno} connect sequentially:


\textbf{Phase 1 (saturated primordial, pre-bounce):} $S \to S_{\max,\text{geom}}$, dominance of the topological potential, maximum cosmological compression. Short duration on the Planckian scale.


\textbf{Bounce:} structural rebound, transition from phase 1 to phase 2. The detailed dynamics of the bounce is severely non-linear and requires complete numerical analysis. Research program \S{}V.4 (thread F-cosmo-PIU-$\Psi$).


\textbf{Phase 2 (freeze-out):} transition of \emph{el Pleno} from the saturated regime to the relaxed regime. The operational constants of the medium freeze dynamically during this phase. Intermediate duration.


\textbf{Phase 3 (relaxed):} $S \to S_{\min}^\text{estr}$, dominance of the Klauder quantum potential in the IR regime and of the localized configurations (solitons). Current cosmological regime; cosmologically long duration.


\textbf{Continuity of the description.} Throughout the three phases, the physical medium is \textbf{the same \emph{el Pleno}} --- there is no ontological change, there is no separation between ``primordial universe'' and ``current universe'' as distinct entities. What changes is the dynamical regime: which term of the Lagrangian F1 dominates the dynamics in each phase.


\textbf{Consistency with observation.} The transition primordial $\to$ relaxed is compatible with the observation of the cosmic microwave background (CMB): the CMB forms in a late phase of the freeze-out regime / beginning of the relaxed regime, when the density of the medium dropped sufficiently to allow the decoupling of radiation. The observed CMB phenomenology (anisotropies, power spectrum, polarization) is a prediction of the cosmology derived from PIU-$\Psi$; the detailed development of this prediction is carried out in \S{}III.6 and \S{}V.2.


\textbf{Epistemic classification.} The general structure of the three phases is \epi{DERIVED} from F1 + axioms A1--A6. The detailed dynamics of the bounce and of the freeze-out transition, with numerical quantification of the temporal scales and of the frozen constants, is the \textbf{thread F-cosmo-PIU-$\Psi$} declared open.


\subsubsection{\texorpdfstring{II.7.6 --- The observed dark energy under PIU-$\Psi$}{II.7.6 - The observed dark energy under PIU-Psi}}

Standard cosmological phenomenology identifies \textbf{dark energy} as the material component of the universe responsible for the observed acceleration of the cosmological expansion, with density fraction $\Omega_{EO}^\text{obs} \approx 0.685$ of the total content and equation of state $w \approx -1$ approximately (with preliminary DESI 2024-2025 evidence of possible temporal dynamics).


Under the standard $\Lambda$CDM model, this component is identified with Einstein's cosmological constant $\Lambda$, with well-documented conceptual problems (the cosmological constant problem, \S{}I.2.2 of the manuscript: theoretical vs observational discrepancy of 60--120 orders of magnitude under the standard QFT picture).


\textbf{The PIU-$\Psi$ reading.} Under PIU-$\Psi$, dark energy \textbf{is not} a material component additional to the physical medium of the universe. Dark energy is the \textbf{permanent structural state of the \emph{Pleno}-ground state}. This is the \textbf{EO-backdrop reading} of the program.


\begin{tcolorbox}[notabox]
\textbf{EO-Backdrop.} What standard cosmological phenomenology identifies as dark energy is, under PIU-$\Psi$, the structural state of the physical medium of the universe in its minimal configuration: the \emph{Pleno}-ground state present in the entire cosmological volume, with its active structural properties. The DE is not a dynamical entity that dilutes, concentrates, or is attractable by gravity. The DE is \textbf{the medium itself} in its state of minimum excitation, manifesting as a structural contribution to the cosmological energy-momentum tensor.

\end{tcolorbox}

\textbf{Derivation of the baryonic matter-density formula from \S{}II.2.2.}


The Bogoliubov-Josephson constitutive identification established in \S{}II.2.2 produces, when applied to configurations of the field $\Psi$ that deviate locally from the background $\Psi_0$:


\[
S^2(\mathbf{x}) \;\leftrightarrow\; \frac{n(\mathbf{x})}{n_0}
\]

where $n(\mathbf{x})$ is the local number density of Planckian quanta of the medium and $n_0 = n_\text{diamond} = 3\sqrt{3}/(8\ell_P^3)$ is the reference number density derived in \S{}II.4.1 Step 1.


\textbf{Step 1 --- Number density of the \emph{Pleno}-ground state.} In the \emph{Pleno}-ground-state configuration, $S = S_{\min}^\text{estr}$, so the local background number density is:


\[
n_\text{background} = n_0\cdot S_{\min}^{2,\text{estr}} = n_\text{diamond}\cdot S_{\min}^{2,\text{estr}}
\]

\textbf{Step 2 --- Number density in a locally excited configuration.} For a localized configuration (soliton) where $S(\mathbf{x}) > S_{\min}^\text{estr}$ locally, the local number density is:


\[
n(\mathbf{x}) = n_0\cdot S^2(\mathbf{x})
\]

\textbf{Step 3 --- Density of ``extra'' quanta (the baryonic matter).} The difference between the local density and that of the cosmological background is the density of \textbf{additional} Planckian quanta accumulated in the localized region --- exactly what the phenomenological reading of \S{}III.5 will identify with the observed baryonic matter:


\[
n_\text{matter}(\mathbf{x}) = n(\mathbf{x}) - n_\text{background} = n_0\cdot[S^2(\mathbf{x}) - S_{\min}^{2,\text{estr}}]
\]

\textbf{Step 4 --- Mass density of matter.} Multiplying by the Planckian mass $m_P$ per quantum and using $m_P\cdot n_0 = \rho_P$ (definition of the reference density of the medium):


\[
\rho_\text{matter}(\mathbf{x}) = m_P\cdot n_\text{matter}(\mathbf{x}) = m_P\cdot n_0\cdot[S^2(\mathbf{x}) - S_{\min}^{2,\text{estr}}]
\]

\begin{equation}
\boxed{\;\rho_\text{matter}(\mathbf{x}) = \rho_P\cdot[S^2(\mathbf{x}) - S_{\min}^{2,\text{estr}}]\;} \tag{E32-bis}
\end{equation}

This is the \textbf{baryonic matter-density formula} of the program, derived explicitly from the Bogoliubov-Josephson constitutive identification (\S{}II.2.2) extended to non-uniform configurations of the field. The derivation is step-by-step from the apparatus of the program, without external importation.


\textbf{Dimensional verification.} $[\rho_P\cdot(S^2 - S_{\min}^{2,\text{estr}})] = [kg/m^3]\cdot[\text{dimensionless}] = [kg/m^3]$ $\checkmark$. Correct dimension of mass density.


\textbf{Physical reading.} The formula says explicitly that \textbf{baryonic matter is the local excess of the field modulus over the background}:


\begin{itemize}[leftmargin=1.4em]
  \item For $S(\mathbf{x}) = S_{\min}^\text{estr}$ (\emph{Pleno}-ground state): $\rho_\text{matter} = 0$ (apparent cosmological vacuum).
  \item For $S(\mathbf{x}) > S_{\min}^\text{estr}$ (soliton): $\rho_\text{matter} > 0$ (presence of baryonic matter).
\end{itemize}

The \emph{Pleno}-ground state is \textbf{not counted} as baryonic matter in this formula: it is the structural background over which matter is defined as a perturbation. This does NOT mean that the \emph{Pleno}-ground state has no contribution to the cosmological energy-momentum tensor --- it does, and that contribution is precisely the \textbf{observed dark energy}, as developed below.


\textbf{Three convergent arguments sustain the EO-backdrop reading.}


\textbf{Argument 1 --- Empirical by reductio.} If the DE were a material component attractable by gravity, cosmological observations would detect local concentrations of DE around massive compact objects (supermassive black holes, galaxy clusters). The density of DE in regions of high gravitational concentration would be greater than in regions of low concentration. This is \textbf{not observed}. The extreme observational uniformity of the DE confirmed by the most precise cosmological observations available (Planck 2018, DESI 2024-2025) is exactly what the EO-backdrop reading predicts: the DE is structural, not dynamical.


\textbf{Argument 2 --- Structural from the derivation of F6.} The formula (E32-bis) says explicitly that \textbf{baryonic matter is the perturbation over the background $S_{\min}^\text{estr}$}: under PIU-$\Psi$, $S_{\min}^\text{estr}$ is the structural state of the medium, and dark energy is that background, not a material component that adds to matter. The dark-energy density is a property of the \emph{Pleno}-ground state, with its own derivational content (its value comes from $S_{\min}^\text{estr}$ and its equation of state $w=-1$ from the tensor structure).


\textbf{Argument 3 --- Equation of state.} The observed dark energy has equation of state $w = -1$ (exact or approximate). In the tensor formulation, $w = -1$ produces $T_{\mu\nu} \propto g_{\mu\nu}$ --- the energy-momentum tensor is \textbf{proportional to the metric} by construction. This is exactly the structural signature of a contribution of the physical medium to the energy-momentum tensor, not of a material component with its own dynamics.


\textbf{Quantitative cosmological prediction from the EO-backdrop reading.} The observed cosmological fraction of dark energy is identified with the \textbf{active condensate fraction} $f_c$ of \emph{el Pleno} (defined in \S{}II.4.2, equation E11):


\begin{equation}
\Omega_{EO} \equiv f_c = \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} \tag{E33}
\end{equation}

\textbf{Explicit numerical verification.}


With values derived in \S{}II.2:


\begin{itemize}[leftmargin=1.4em]
  \item $S_{\max,\text{eff}}^2 = 1.4956 \pm 3.9\times 10^{-5}$ (calibrated, \S{}II.4.2)
  \item $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3} = 2.17732$ (derived, \S{}II.4.1)
\end{itemize}

\[
f_c = \frac{1.4956}{2.17732} = 0.6869
\]

\textbf{Comparison with observation.} Canonical observational value (Planck 2018): $\Omega_{EO}^\text{obs} = 0.6847 \pm 0.0073$. PIU-$\Psi$ prediction: $\Omega_{EO}^\text{PIU} = 0.6869$. Difference:


\[
\frac{|\Omega_{EO}^\text{PIU} - \Omega_{EO}^\text{obs}|}{\Omega_{EO}^\text{obs}} = \frac{|0.6869 - 0.6847|}{0.6847} = 0.00321 \approx 0.32\%
\]

Difference in error bars:


\[
\frac{|\Omega_{EO}^\text{PIU} - \Omega_{EO}^\text{obs}|}{\sigma_\text{Planck}} = \frac{0.0022}{0.0073} = 0.30\sigma
\]

\textbf{Agreement to 99.7\% (0.30$\sigma$), with zero free parameters of the cosmological sector} ($f_c$ was derived in \S{}II.2 as the ratio of two structural quantities of the medium, without cosmological observational calibration).


\textbf{Epistemic classification.} The identification $\Omega_{EO} \equiv f_c$ and the numerical value $f_c \approx 0.6869$ are \epi{DERIVED} structurally from the combination: Bogoliubov-Josephson constitutive identification (\S{}II.2.2), \emph{Pleno}-ground-state Theorem as minimal configuration (\S{}II.2), A6 hard bound via Kepler-Hales (\S{}II.4.1). The agreement with observation to 0.32\% with zero free parameters of the cosmological sector is a \textbf{verified structural quantitative prediction}, not a calibration.


\textbf{Implication.} The identification under the EO-backdrop reading simultaneously resolves:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{The cosmological constant problem} (\S{}I.2.2): there is no discrepancy of 120 orders of magnitude because there is no ``quantum vacuum energy'' summed over mode fluctuations --- there is a physical medium with specific structural energy computed from its derived parameters.
  \item \textbf{The dark-sectors problem} (\S{}I.2.3): dark energy is not ``the 68\% of the universe that is unknown''; it is the physical medium in its state of minimum excitation, structurally derivable from the apparatus of the program.
  \item \textbf{The extreme observational uniformity} of the DE at cosmological scales: a natural consequence of the DE being a structural property of the single medium, not a material component with local dynamics.
\end{itemize}

\subsubsection{II.7.7 --- Terrestrial experimental analogues as indirect validation}

The cosmological mechanism of PIU-$\Psi$ --- saturated primordial \emph{el Pleno}, structural bounce, freeze-out of operational constants, transition to the relaxed regime --- admits \textbf{direct empirical realization} in five families of experimentally confirmed terrestrial phenomena. The identification of these analogues is not a rhetorical analogy: each has mathematical machinery structurally isomorphic to the mechanism of \emph{el Pleno}.


\textbf{Analogue 1 --- Bosenova} (Donley-Cornell-Wieman, Nature 2001). The experiment observes the collapse-and-explosion dynamics of Bose-Einstein condensates when the sign of the atomic scattering length is changed from repulsive to attractive via Feshbach resonance. Observed phenomenology: anisotropic bursts, peaks in the wavefunction, oscillating remnant condensates that survive the collapse. \textbf{Mathematical structure identical to the PIU-$\Psi$ primordial bounce:} quantum/topological pressure stops the collapse, accumulated energy is released as a shock wave, an oscillating remnant remains.


\textbf{Analogue 2 --- Breathing solitons with revivals} (Petrov 2024, Sroyngoen-Anglin 2026). Bright solitons in condensates with attractive interactions exhibit breathing modes and ``revivals'' --- atoms emitted to the environment return and interfere with the soliton producing periodic revivals with temporal asymmetry. \textbf{PIU-$\Psi$ analogue:} topological information preserved between cosmological cycles.


\textbf{Analogue 3 --- Rayleigh-Plesset cavitation.} The dynamics of bubbles collapsing in liquids has a structural form identical to the Friedmann equation during the bounce: compression of an elastic medium up to a hard bound followed by a rebound. \textbf{Persistent low-density marks} behind the first collapse (observed experimentally) are an analogue of intercyclic topological information.


\textbf{Analogue 4 --- Ostwald-Liesegang ripening.} Large structures ``eat'' small structures by differential solubility in a supersaturated solution (Lifshitz-Slyozov-Wagner). \textbf{PIU-$\Psi$ parallel:} a supersaturated cosmological \emph{el Pleno} (post-bounce) nucleates material structure over the background; the large structures consume the small ones by mutual gravity, exactly like Ostwald ripening.


\textbf{Analogue 5 --- Cyclic crystallization with partial dissolution.} Industrial processes with growth-dissolution cycles produce size distributions unreachable by monodic processes. \textbf{PIU-$\Psi$ analogue:} cosmological cycles with monotonic growth of structural complexity between cycles.


\textbf{Strategic impact.} The existence of five experimentally confirmed terrestrial analogues with mathematical machinery isomorphic to the PIU-$\Psi$ cosmological mechanism \textbf{changes the strategic nature of the program}: the mechanism of \emph{el Pleno} is not ``a speculative theory with suggestive analogues'' --- it is a \textbf{relativistic extension of well-established experimental physics, applied to the cosmological regime}. The speculative content of the program is concentrated in the identification of \emph{el Pleno} with the cosmological substrate (axiom A1) and the application at the Planckian scale; the rest is development of known mathematics with direct empirical realization in contemporary laboratories.


To this indirect experimental validation is added the formal correspondence F1 $\leftrightarrow$ Gross-Pitaevskii (\S{}II.5.7), which reduces the regimes accessible to observation to regimes already explored experimentally in condensate physics.


\textbf{Epistemic classification.} The terrestrial analogues are not derivations of the program; they are \textbf{indirect experimental validation} that sustains the structural viability of the mechanism. The mathematical correspondence is \textbf{[strong logical deduction]}; the generalization to the cosmological regime is \textbf{[articulated structural HYPOTHESIS]} subject to falsification by direct cosmological observation.


\subsubsection{II.7.8 --- Closing and transition to \S{}II.8}

The cosmological configurations of \emph{el Pleno} are organized into three sequential regimes connected by the dynamics of F1: the \textbf{saturated primordial} ($S \to S_{\max,\text{geom}}$), where the A6 hard bound prevents the singularity and the topological potential $V_\text{topo}$ produces a structural bounce instead of the singular Big Bang; the \textbf{freeze-out}, where the PIU-$\Lambda$ Theorem fixes the operational constancy of $\Lambda_\text{PIU}$ and analogous constants of the hadronic sector; and the \textbf{current relaxed} ($S \to S_{\min}^\text{estr}$), where all the observed phenomenology develops.

The \textbf{formula F6} of baryonic matter density $\rho_\text{matter}(\mathbf{x}) = \rho_P[S^2(\mathbf{x}) - S_{\min}^{2,\text{estr}}]$ is derived from the Bogoliubov-Josephson constitutive identification (\S{}II.2.2) extended to non-uniform configurations, in four clean steps: background density, locally excited density, density of extra quanta, multiplication by $m_P$.


The \textbf{EO-backdrop reading} identifies the observed dark energy with the permanent structural state of the \emph{Pleno}-ground state:


\[
\Omega_{EO} = f_c = \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} \approx 0.6869
\]

with observational agreement to 0.32\% (0.30$\sigma$) without free parameters of the cosmological sector. This reading structurally resolves the cosmological constant problem and the dark-sectors problem articulated in \S{}I.2.


The five experimentally confirmed terrestrial analogues (Bosenovas, breathing solitons, Rayleigh-Plesset cavitation, Ostwald ripening, cyclic crystallization) constitute \textbf{indirect experimental validation} of the cosmological mechanism, connecting the program with well-established experimental physics.


\medskip\hrule\medskip

\subsection{\S{}II.8 --- Synthesis: the catalogue of available configurations}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (synthetic section). It gathers the threads declared in previous sections of Part II: D1 (geometric uniqueness, \S{}II.3), F-cosmo-PIU-$\Psi$ (quantitative cosmology, \S{}II.7), C-Cosmo-$\Lambda$ (cosmological prediction of $\Lambda_\text{PIU}$, \S{}II.7).
  \item \textbf{Notes:} closing section of Part II. It consolidates the complete catalogue of configurations of \emph{el Pleno} in an integrating table and prepares the transition to Part III (phenomenological reading).
\end{itemize}
\end{estadoseccion}

\subsubsection{II.8.1 --- The complete catalogue}

At the close of Part II, the repertoire of configurations of \emph{el Pleno} is completely characterized from the apparatus of Part I (axioms A1--A6, Lagrangian F1, parameters $\rho_P, c, \ell_P$), without yet invoking the phenomenological reading. The catalogue is \textbf{structural}: each element is a configuration of the single physical medium, identified by its mathematical and dynamical properties. The five types of \S{}II.1.3 ---uniform, microscopic structural, propagating modes, localized, cosmological--- were developed in \S{}II.2 to \S{}II.7 with explicit algebraic derivations, dimensional and numerical verification where applicable, and visible epistemic classification.


\subsubsection{II.8.2 --- Integrating table of the catalogue}

The following integrating table consolidates the configurations of the catalogue with their structural properties, section of origin, epistemic classification, and the phenomenological identification announced for Part III:


{\scriptsize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{0.7cm}L{2.3cm}L{1.7cm}L{2.9cm}L{1.4cm}L{2.0cm}L{2.4cm}}
\toprule
\textbf{\#} & \textbf{Configuration} & \textbf{Type} & \textbf{Key structural properties} & \textbf{Section} & \textbf{Status} & \textbf{Phenomenological identification} \\
\midrule\endhead
1 & \textbf{\emph{Pleno}-ground state} & Uniform & $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4 \approx 0.7231$; $f_c \approx 0.6869$ & \S{}II.2 & \epi{DERIVED} & What standard cosmology calls vacuum + dark energy \\
2 & \textbf{Geometric hard bound} & Structural & $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3} \approx 2.1773$ & \S{}II.4.1 & \epi{DERIVED} & Maximum density of the medium in the primordial bounce and compact cores \\
3 & \textbf{Calibrated effective bound} & Structural & $S_{\max,\text{eff}}^2 = 1.4956 \pm 3.9\times 10^{-5}$ & \S{}II.4.2 & \epi{CALIBRATED} & Extensibility parameter of the medium in the current regime \\
4 & \textbf{Diamond lattice} $Fd\bar{3}m$ & Microscopic crystalline & $Z = 4$ bipartite A/B; $V_\text{cell} = (8/(3\sqrt{3}))\ell_P^3$ & \S{}II.3 & \textbf{\epi{DERIVED}} & Microscopic structure of the saturated medium; geometry derived in both pieces (\S{}II.3.7, \S{}II.3.7-bis) \\
5 & \textbf{Radial mode} $\delta S$ & Propagating mode & $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$ (tree-level); UV correction $(k\ell_P)^2/20$ & \S{}II.5.4 & \epi{DERIVED} & Ultra-Planckian mode (extreme regime) \\
6 & \textbf{Phase modes} $\delta\theta_\pm$ & Propagating mode & Massless, exact velocity $c$; A/B decomposition over $U(1) \times SU(2)$ & \S{}II.5.5 & \epi{DERIVED} & Induced gauge sector (electromagnetism + weak) \\
7 & \textbf{Tensor mode TT} & Propagating mode & Transverse-traceless tensor configuration of \emph{el Pleno}, not a quantized particle; exact velocity $c$ in IR & \S{}II.5.5 & \epi{DERIVED} & Macroscopic gravitational wave \\
8 & \textbf{Skyrmion} ($B \in \mathbb{Z}$) & Solitonic class M & Topology $\pi_3(SU(2)) = \mathbb{Z}$; mandatory fermionic statistics & \S{}II.6.2 & \epi{DERIVED} & Standard Model fermions (with internal structure) \\
9 & \textbf{Q-ball} & Solitonic class MO & Contractible sub-target; mandatory bosonic statistics & \S{}II.6.3 & \epi{DERIVED} & Scalar bosons; candidate dark-matter configurations \\
10 & \textbf{Tripartite Borromean cluster} & Composite solitonic & $B = 1 = 1/3 + 1/3 + 1/3$; $\mathbb{Z}_3$ modular arithmetic over integer numerators; charges $\pm 2/3, \pm 1/3$ & \S{}II.6.5--\S{}II.6.6 & \epi{DERIVED} & Baryon (proton / neutron); quark charges \\
11 & \textbf{Baryonic rotational tower} & Excitations of the cluster & $m(J) = m_p + [J(J+1) - 3/4]\,\hbar^2/(2\Theta_p)$; $\Theta_p = m_p R_p^2/3$ with $R_p$ a prediction of the program & \S{}II.6.7 & \epi{DERIVED} & Hadronic spectrum: $N, \Delta, N^*, \Delta^*, \ldots$ \\
12 & \textbf{Saturated primordial regime} & Cosmological & $S \to S_{\max,\text{geom}}$; structural bounce instead of singularity & \S{}II.7.2 & \epi{DERIVED} & Primordial universe (without singular Big Bang) \\
13 & \textbf{Freeze-out regime} & Cosmological & Dynamical fixing of operational constants; PIU-$\Lambda$ Theorem & \S{}II.7.3 & \epi{DERIVED} & Intermediate primordial era; freezing of $\Lambda_\text{PIU}$ \\
14 & \textbf{Current relaxed regime} & Cosmological & $S \to S_{\min}^\text{estr}$; configuration close to the \emph{Pleno}-ground state & \S{}II.7.4 & \epi{DERIVED} & Current observable universe \\
15 & \textbf{EO-backdrop} & Cosmological (identification) & $\Omega_{EO} \equiv f_c \approx 0.6869$; structural state of the \emph{Pleno}-ground state & \S{}II.7.6 & \epi{DERIVED} & Observed dark energy (Planck 2018: $0.6847 \pm 0.0073$; agreement $0.30\sigma$) \\
\bottomrule
\end{longtable}}

\textbf{Total: 15 distinguishable configurations / structural properties}, derived or characterized from the apparatus of the program. Of these:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{13 with status \epi{DERIVED}}: complete derivation from axioms and mathematical apparatus without free parameters of the corresponding sector.
  \item \textbf{1 with status \textbf{[SLD and thread D1 open]}}: the diamond lattice, justified by four convergent arguments but with uniqueness not proven.
  \item \textbf{1 \epi{DERIVED}}: the diamond lattice, with its geometry derived in both pieces (coordination $Z=4$ and cubic stacking, \S{}II.3.7).
  \item \textbf{1 with status \epi{CALIBRATED}}: $S_{\max,\text{eff}}^2$, fitted to the current cosmological regime with an explicit error bar.
\end{itemize}

\subsubsection{II.8.3 --- Architectural threads inherited by Part V}

Three explicit threads are declared open throughout Part II and are inherited by the research program developed in Part V:


\textbf{Thread D1 --- Geometric uniqueness of the diamond lattice} (\S{}II.3.7). Four convergent arguments (Maxwell isostaticity, phase condensation, Finkelstein-Rubinstein statistics, Russo-Tanaka variational) justify the diamond lattice as the geometric choice of saturated \emph{el Pleno}. Strict uniqueness as the only geometry compatible with the axioms A1--A6 is not proven. Medium severity; research program \S{}V.3.


\textbf{Thread F-cosmo-PIU-$\Psi$ --- Quantitative cosmology} (\S{}II.7.4, \S{}II.7.5). Quantitative derivation of the effective equation of state $w(z)$ of \emph{el Pleno} integrated cosmologically, directly comparable with DESI 2024-2025 observations and future observations of the dynamics of dark energy. Medium severity; research program \S{}V.4.


\textbf{Thread C-Cosmo-$\Lambda$ --- Cosmological prediction of the hadronic parameter} (\S{}II.7.3). Prediction of the numerical value of $\Lambda_\text{PIU}$ from the primordial-bounce cosmology, without calibration to the proton mass. If successful, it would reduce the number of observational inputs of the hadronic sector from 1 to 0. Medium-low severity; research program \S{}V.3.


These three threads represent the pending research content of the program at the close of Part II. Their explicit and honest declaration is an application of Architectural Principle 4 (visible epistemic honesty) and of the principles of the governing document \S{}6.4 on architectural threads.


\subsubsection{II.8.4 --- Falsifiable predictions of the catalogue}

The catalogue is not just an interpretive repertoire: it produces falsifiable quantitative predictions that are developed completely in Parts III and IV but already have closed form in Part II:


\begin{itemize}[leftmargin=1.4em]
  \item Cosmological dark-energy fraction $\Omega_{EO} = f_c \approx 0.6869$ (verified to 0.32\%, 0.30$\sigma$).
  \item Baryonic split $\Delta - N = 9(\hbar c)^2/(2 m_p R_p^2) \approx 264$ MeV (verified to 90\%).
  \item Complete baryonic rotational tower with identification of $N^*(1680)$ to 97.8\% and $\Delta^*(1950)$ to 84\%.
  \item Propagation velocity of the TT mode $= c$ exact in IR (verified by GW170817 to $< 7\times 10^{-16}$).
  \item \textbf{Minimum structural black-hole mass:} $M_{\min} = m_P\sqrt{3/(32\pi\,S_{\max,\text{geom}}^2)} \approx 0.117\,m_P \approx 2.55\times 10^{-9}$ kg, derived from the coincidence condition $r_\text{core}(M) = r_S(M)$ between the radius of the saturated core of \emph{el Pleno} and the Schwarzschild radius. It coincides with the Hawking evaporation limit for a distinct ontological reason (interior geometric exhaustion, not divergent temperature).
  \item \textbf{Maximum Hubble bound:} $H_{\max} = \sqrt{8\pi G\rho_{\max,\text{abs}}/3} \approx 7.9\times 10^{43}\,\text{s}^{-1}$, derived from the hard bound $\rho_{\max,\text{abs}} = \rho_P\,S_{\max,\text{geom}}^2$ via the Friedmann equations. It implies a minimum Hubble horizon $R_{H,\min} = c/H_{\max} \approx 0.23\,\ell_P$ sub-Planckian.
  \item \textbf{Mass of the radial mode of \emph{el Pleno}:} $m_\text{radial}c^2 = \sqrt{\xi}\,E_P \approx 4.44\,E_P$ (tree-level), structural explanation of the non-detection of \emph{el Pleno} as an observable propagating particle.
  \item Existence of a primordial bounce instead of a Big Bang singularity (falsifiable by observation of cosmological configurations at trans-Planckian scales if the A6 bound were violated).
  \item Kepler-Hales polytypism in the cores of dense compact objects (future observational signature with the LISA, ET observatories).
\end{itemize}

Each prediction has a specific quantitative form, a comparable observational bound, and is falsifiable by designable experiments.


\subsubsection{II.8.5 --- Transition to Part III}

The catalogue of Part II ---15 configurations and structural properties, three cosmological regimes, the EO-backdrop identification verified to 0.32\%, and three architectural threads declared open (D1, F-cosmo, C-Cosmo-$\Lambda$)--- is ready for its phenomenological reading. \textbf{Part III} takes it and reads it: each configuration is identified with an observed physical phenomenon ---or announces a predicted one not yet observed--- following the order of the table \S{}II.8.2, and the complete derivational consequences of that identification are developed: gravity as a macroscopic property of \emph{el Pleno}, the fundamental constants as structural identities, gravitational waves, the gauge sector, solitons as particles, cosmology and the dark sectors.


\medskip\hrule\medskip


\clearpage
