% !TeX root = main.tex
\subsection{\S{}III.3 --- Transverse tensor modes: gravitational waves}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own; P15 (suppression of the primordial tensor spectrum) is a falsifiable hypothesis.
  \item \textbf{Notes:} gravitational wave as a TT mode of \emph{el Pleno}, not an independent geometric entity. Formal retirement of the old exponential attenuation formula.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{III.3.1 --- The question: what is a gravitational wave under PIU-$\Psi$}{III.3.1 - The question: what is a gravitational wave under PIU-Psi}}

The direct detection of gravitational waves by LIGO (2015), and in particular GW170817 (neutron-star merger with an electromagnetic counterpart), confirmed that they are real perturbations that propagate with a finite velocity equal to that of light ---one of the six arguments of \S{}I.3.4 in favor of \emph{el Pleno}. The phenomenological question is which configuration of \emph{el Pleno} corresponds to the observed wave, and the structural answer is already in \S{}II.5.5: the transverse-traceless tensor mode (TT mode).


\textbf{An ontological distinction.} It would be a mistake to treat the gravitational wave as something that travels \textit{through} \emph{el Pleno} ---as if there were a gravitational medium separable from the wave, on which an attenuation of the type $A(f,L) = \exp(-\kappa L f^2)$ acted. That violates the uniqueness of \emph{el Pleno}: the wave is a configuration of the medium ---its TT mode---, not an object that propagates against an external background, and therefore there is no attenuation against that background. The propagation of the TT mode is that of the medium itself.


\subsubsection{III.3.2 --- Rigorous identification of the TT mode}

\textbf{Origin.} The TT mode is constructed as a correlated modulation of the perturbations of the field of \emph{el Pleno} that satisfies three geometric conditions: it preserves the local amplitude, is transverse, and is traceless.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Condition 1 --- Amplitude preservation.} The TT mode does not alter the modulus: $\delta S^\text{TT} = 0$. The TT configuration preserves $|\Psi|$, modulating only the tensor structure of the phase.
  \item \textbf{Condition 2 --- Transversality.} $\partial^i h^\theta_{ij} = 0$.
  \item \textbf{Condition 3 --- Tracelessness.} $\delta^{ij}h^\theta_{ij} = 0$.
\end{itemize}

\textbf{Identification with the metric perturbation.} Via the induced-action chain (\S{}III.1.4), the microscopic TT modulation of \emph{el Pleno} is identical to the transverse-traceless metric perturbation $h^\text{TT}_{\mu\nu}$ of General Relativity. They are not two distinct objects: the observed macroscopic gravitational wave is the macroscopic TT mode of the field $\Psi$.


\textbf{Effective Lagrangian of the TT mode.} It is obtained by substituting the TT configuration into F1 and retaining the quadratic order.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- The TT perturbation over the background.} $S = S_0$ (constant background), tensor modulation encoded in the phase sector $\theta \to \theta_0 + h^\theta_{ij}$, with $h^\theta_{ij}$ transverse, traceless, amplitude-preserving. The polarization $h_+$ is one of the two independent components.
  \item \textbf{Step 2 --- Kinetic term.} With $S = S_0$ and the phase modulated by $h_+$, the quadratic part is $\mathcal{L}^\text{cin}_\text{TT} = \tfrac{1}{2}\rho_P c^2\ell_P^2 S_0^2(\partial h_+)^2$.
  \item \textbf{Step 3 --- Flexural term.} The flexural term of F1 acts over the curvature of the complete field $\Psi$ (\S{}I.7.7, form E4b--E4c), and its phase component ---derived from the diamond-lattice operator in \S{}II.3.8 (P9b)--- is $-\tfrac{\rho_P c^2\ell_P^4}{40}S_0^2(\Box h_+)^2$ at quadratic order. It is not a postulated generalization: it is the phase component of the same lattice regulator that produces the flexural term of the modulus, with coefficient $1/40$ identical because both sectors share the bond-difference operator $|\Psi_n-\Psi_m|^2$ (\S{}II.3.8, equation E13b). The weighting $S_0^2$ is the same as that of the phase kinetic term.
  \item \textbf{Step 4 --- Effective Lagrangian:}
\end{itemize}

\[
\mathcal{L}_\text{TT} = \tfrac{1}{2}\rho_P c^2\ell_P^2 S_0^2(\partial h_+)^2 - \gamma_\kappa S_0^2(\Box h_+)^2
\]

with $S_0 = S_{\min}^\text{estr}$. This is a Pais-Uhlenbeck structure (standard kinetic plus fourth order in derivatives), whose high-energy pole is regularized at the scale $M_\text{LW} = \sqrt{20}/\ell_P$ (\S{}III.3.4).


\textbf{Dimensional verification.} The kinetic term has $[\rho_P c^2\ell_P^2 S_0^2(\partial h_+)^2] = [J/m^3]$ (energy density, $h_+$ dimensionless). The flexural one has $[\gamma_\kappa S_0^2(\Box h_+)^2] = [J\cdot m][m^{-2}]^2 = [J/m^3]$ $\checkmark$.


\textbf{Epistemic classification.} The identification of the TT mode as a transverse-traceless configuration of \emph{el Pleno} is \epi{DERIVED} from F1. The identification with $h^\text{TT}_{\mu\nu}$ via the induced action is \epi{DERIVED} (\S{}III.1.4). The TT mode is not a quantum entity additional to the single scalar field.


\subsubsection{III.3.3 --- Four derivational results on the propagation}

\textbf{Result D1 --- Decoupling of the potential.} Since $V(S)$ depends only on the modulus $|S|$ and the TT configuration preserves $|\Psi|$, the potential does not affect the TT mode at linear order. This excludes a mass for the TT mode and its coupling to the walls $V_q$ (IR) and $V_\text{topo}$ (UV). \epi{DERIVED}.


\textbf{Result D2 --- Wave equation and absence of mass.} In the infrared regime, with the potential decoupled (D1), the field equation of the TT mode is the free wave equation $\Box h_+ = 0$, without a mass term or a zeroth-order correction. What is non-trivial here is the \textbf{absence of mass and dispersion at zeroth order}: the TT mode could have been massive ---as the radial mode is (\S{}III.1.9)---, and it is not, because the TT configuration preserves the modulus and the potential does not affect it (D1). That the resulting velocity is $c$ is not an independent measurement of $c$ but the \textbf{signature of Lorentz covariance} (axiom A3): $c$ is already in the operator $\Box = (1/c^2)\partial_t^2 - \nabla^2$ of F1, so that $\Box h_+ = 0$ gives $v = c$ by structural consistency, not by derivation of the value of $c$ (cf. the circularity warning in \S{}III.1.8). \epi{DERIVED}: the absence of mass of the gravitational mode. That the propagation velocity of the modes of the medium is $c$ has a direct rheological consequence: the event horizon of a black hole ---where the escape velocity equals $c$--- is exactly the sonic surface of \emph{el Pleno} (Mach number unity), as developed in \S{}III.9.6.


\textbf{Result D3 --- Absence of dissipative attenuation.} The TT mode propagates without dissipation for three independent reasons: (1) there is no environment external to \emph{el Pleno} to which energy can be transferred; (2) the Noether U(1) current associated with the tensor phase is exactly conserved (axiom A5); (3) the first-order dissipative terms in temporal derivative are prohibited by the effective Lorentz invariance and the parity of the Lagrangian (\S{}I.7). The three converge: the TT mode is not attenuated. \epi{DERIVED} from the structure of F1.


\textbf{Result D4 --- Planckian ultraviolet correction.} The dispersion relation of the TT mode is derived in \S{}II.5.4 (E23): the quadratic $\Box F(1 - \tfrac{\ell_P^2}{20}\Box F) = 0$ has two branches ---the infrared physical one ($\Box F = 0$, luminal propagation) and the ultraviolet-regulation one ($\Box F = 20/\ell_P^2$, Lee-Wick pole, \S{}III.3.4)---. The dispersion relation is:


\[
\omega^2 = c^2 k^2\left[1 + \frac{(k\ell_P)^2}{20}\right] \qquad(\text{see }E23)
\]

identical to (E23). The coefficient $1/20$ has a structural origin (\S{}II.5.4 and \S{}II.3.8 P9b): it is the phase flexural coefficient $1/40$ of F1 ---derived from the diamond-lattice operator acting on the phase sector, not by analogy with the radial mode--- multiplied by the factor 2 of differentiating $(\Box h_+)^2$ with respect to $\Box h_+$. Since the TT mode and the photon are both modes of the phase sector, the ultraviolet correction is identical for the two (ratio between coefficients equal to unity, \S{}III.4.3).


\textbf{Phenomenological reading.} The correction $(k\ell_P)^2/20$ is the only deviation with respect to luminal propagation. For accessible frequencies ($k\ell_P \sim 10^{-43}$ at \textasciitilde{}100 Hz), the term $(k\ell_P)^2/20 \sim 10^{-87}$, against the observational bound $\sim 10^{-17}$: suppressed by \textasciitilde{}70 orders of magnitude.


\textbf{Dimensional verification of (E23).} $[c^2 k^2] = [m^2/s^2][m^{-2}] = [s^{-2}] = [\omega^2]$ $\checkmark$; $(k\ell_P)^2$ dimensionless $\checkmark$.


\textbf{Epistemic classification.} The dispersion relation (E23) is \epi{DERIVED} in \S{}II.4.4. D1, D2, D3 are \epi{DERIVED}. The contribution of this subsection is the quantification of the observational suppression, which shows consistency with GW170817.


\textbf{Result D4-bis --- The imaginary part of the dispersion and its scale.} The relation (E23) is the \emph{reactive} (real) part of the wave number: it fixes the phase velocity and is conservative. The reading of the Lee-Wick pole as a finite-memory mode $\tau_\Pi$ adds the \emph{dissipative} (imaginary) part, which completes the complex wave number $k = k_0 + i\alpha$ of the TT mode. For a transverse shear wave ---and the TT mode is one--- in a medium with kinematic viscosity $\nu_\text{TT} = \eta_\text{TT}/\rho_P$, the damped wave equation gives the viscous dispersion relation $k^2 = (\omega^2/c^2)\,(1 - i\,\nu_\text{TT}\,\omega/c^2)^{-1}$. For $\nu_\text{TT}\omega/c^2 \ll 1$ (every sub-Planckian regime), the root expands to first order:
\begin{equation}
k = \frac{\omega}{c}\left(1 - i\,\frac{\nu_\text{TT}\,\omega}{c^2}\right)^{-1/2}
= \frac{\omega}{c} + i\,\frac{\nu_\text{TT}\,\omega^2}{2c^3} + \mathcal{O}(\nu_\text{TT}^2), \tag{E23b}
\end{equation}
so that the spatial attenuation is the imaginary part of the wave number,
\begin{equation}
\alpha(\omega) = \operatorname{Im}(k) = \frac{\nu_\text{TT}}{2c^3}\,\omega^2 = \frac{1}{2\sqrt{20}}\,\frac{\ell_P}{c^2}\,\omega^2, \qquad \nu_\text{TT} = \frac{\eta_\text{TT}}{\rho_P} = \frac{1}{\sqrt{20}}\,c\,\ell_P, \tag{E23c}
\end{equation}
with pure coefficient $1/(2\sqrt{20})\approx 0.1118$, the same angular word $\sqrt{20}$ of the flexural that governs the real dispersion and the localizability floor $\sigma_\text{min}$. The same flexural term ($1/40\to 1/20$) thus provides the two coefficients of the complex wave number: the real one $(k\ell_P)^2/20$ (reactive) and the imaginary one $\alpha$ (dissipative, memory $\tau_\Pi$).

This component \emph{does not contradict} Result D3: D3 asserts that the TT mode is not attenuated by energy transfer to an external environment ---which does not exist, A1--- nor by first-order dissipative terms ---forbidden by the parity of the Lagrangian (\S{}I.7)---, and that still holds. The imaginary part of (E23c) is not dissipation against a background: it is the finite-memory response of the medium itself, of second order in $\omega$, suppressed by $\ell_P/c^2$. Its magnitude confirms it: the absorption length $1/\alpha$ is of enormous cosmological scale except at Planckian frequency.
\begin{center}
{\small
\begin{tabular}{llll}
\toprule
\textbf{Band} & $f$ (Hz) & absorption length $1/\alpha$ & in Hubble radii \\
\midrule
LISA & $10^{-3}$ & $\sim 1.3\times10^{57}$ m & $\sim 1\times10^{31}$ \\
LIGO & $10^{2}$ & $\sim 1.3\times10^{47}$ m & $\sim 1\times10^{21}$ \\
Planck ($1/\tau_\Pi$) & $\sim 8\times10^{43}$ & $\sim \ell_P$ & attenuation $\mathcal{O}(1)$ \\
\bottomrule
\end{tabular}}
\end{center}
At the relaxation frequency $1/\tau_\Pi = \sqrt{20}\,c/\ell_P \approx 8\times10^{43}$ Hz, the absorption length falls to the scale of $\ell_P$ (attenuation of order unity), confirming that the intrinsic dissipation is only appreciable at the Planck scale. \textbf{Prediction.} Gravitational waves propagate without observable absorption or dispersion, with deviations only at Planckian frequencies. The imaginary part of the dispersion is \epi{DERIVED} (coefficient $1/(2\sqrt{20})$ via $\tau_\Pi$ and $\eta_\text{TT}$), consistent with the gravitational waves observed without attenuation.


\subsubsection{III.3.4 --- The ultraviolet-regulation pole and unitarity}

\textbf{The high-energy pole.} The flexural term produces, in the propagator of the TT mode, a second pole at high energy in addition to the massless physical pole, located at:


\[
M_\text{LW} = \frac{\sqrt{20}}{\ell_P} \approx 2.77\times 10^{35}\ \text{m}^{-1} \quad\Longleftrightarrow\quad M_\text{LW}\,\hbar c \approx 5.46\times 10^{19}\ \text{GeV}
\]

where $\sqrt{20}$ is the relaxation invariant of \emph{el Pleno} (\S{}III.1.6). The pole is situated at $\sqrt{20}\approx 4.47$ times the Planck mass in natural units of wavenumber, but its energy scale $M_\text{LW}\,\hbar c\approx 5.46\times 10^{19}$ GeV remains above $E_P\approx 1.22\times 10^{19}$ GeV; the qualifier ``sub-Planckian'' refers to the associated \textbf{length} $\ell_P/\sqrt{20}$, which is smaller than $\ell_P$. It is the scale at which the fourth-order flexural term dominates and regularizes the ultraviolet behavior of the propagator.


\textbf{Resolution of unitarity.} In a naive treatment, this second pole would correspond to a negative-norm state that would violate unitarity. The program treats it through the benign-ghost prescription (the same that regularizes quantum gravity in the schemes of consistent quadratic gravity): the high-energy pole is not interpreted as an asymptotic particle of the spectrum, but as an auxiliary regulation mode that does not appear in the external states. With this prescription the theory is unitary and has better ultraviolet behavior than pure Einstein gravity.


\textbf{Unitarity as a native consequence of F1.} The benign-ghost prescription does not need to be imported: it emerges from the structure of \emph{el Pleno} itself. The flexural term $(\Box S)^2$ makes F1 a Pais-Uhlenbeck theory, in which the Ostrogradsky theorem predicts a Hamiltonian linear in one of the momenta and, therefore, a generic ghost. But the Ostrogradsky instability requires two conditions that \emph{el Pleno} blocks:
\begin{itemize}[leftmargin=1.6em]
  \item \textbf{Bounded phase space.} The two walls of the modulus ---the Klauder IR quantum wall (axiom A2, which repels $S\to 0$) and the Gent UV topological wall (axiom A6, which repels $S\to S_{\max}$)--- bound the coordinate $q_1=S$ on both sides, and the hard deformation bound $\dot\gamma\leq c/\ell_P$ (\S{}III.9.2) bounds its conjugate momentum. Both axes of the physical pair are bounded, which prevents the escape to infinity that the instability needs. The deformation bound acts as a momentum bounder, dual to the configuration bounder (A2 plus A6): the walls compactify the field, the deformation bound its rate of change.
  \item \textbf{The regulation mode does not oscillate.} The ultraviolet root of the propagator has $\omega^2<0$ (E20): it is \emph{overdamped}, not an oscillating mode of negative energy. The Ostrogradsky instability requires an oscillating mode that resonates with the physical mode and transfers energy secularly; an overdamped mode does not oscillate, does not resonate, and there is no secular transfer. There is no escape.
\end{itemize}
Thus, the unitarity of \emph{el Pleno} is reduced from an external recipe to a consequence of F1 plus the walls of the modulus (A2 in the IR, A6 in the UV) plus the deformation bound. The benign ghost of the TT mode and the ``fakeon'' of the radial mode are not two distinct recipes: they are the same root $\omega^2<0$ of the same flexural term in two sectors, just as the invariant $\sqrt{20}$ converges in three sectors of the Lagrangian (\S{}III.2.6). The rigorous formalization to all orders remains a declared technical thread (\S{}V.3), reduced from imported to native.


\textbf{Consistency with the absence of dissipation.} This prescription is coherent with D3: the regulation mode does not transfer energy from the physical mode to a dissipative channel, because it is not an asymptotic state.


\textbf{Three propagation regimes.} (i) \textbf{Infrared} ($k\ell_P \ll 1$): pure luminal propagation, the regime of all current observations. (ii) \textbf{Transition} (close to $M_\text{LW}$): the correction $(k\ell_P)^2/20$ becomes appreciable; inaccessible. (iii) \textbf{Ultraviolet} (above $M_\text{LW}$): the regulation mode dominates; Planckian scale.


\textbf{Dimensional verification.} $[M_\text{LW}] = [\sqrt{20}/\ell_P] = [m^{-1}]$, which in natural units is a mass $\checkmark$.


\textbf{Epistemic classification.} The location of the pole at $M_\text{LW} = \sqrt{20}/\ell_P$ is \epi{DERIVED}. Unitarity as a native consequence of F1 ---phase space bounded by A2 and the deformation bound, regulation mode with $\omega^2<0$ that does not resonate--- is \epi{SLD}, reduced from an imported prescription to a property of \emph{el Pleno} itself; its complete formalization to all orders remains a technical thread. The consistency with the absence of dissipation is \epi{DERIVED}.


\subsubsection{III.3.5 --- Numerical verification against observation}

For a gravitational wave of frequency $f$, $k \approx 2\pi f/c$, and the fractional correction to the velocity is of order $(k\ell_P)^2/20$.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{LIGO/Virgo at 100 Hz.} $k\ell_P \approx 3.4\times 10^{-41}$, $(k\ell_P)^2/20 \sim 10^{-83}$, against the observational bound $\sim 10^{-17}$: margin \textasciitilde{}66--69 orders.
  \item \textbf{GW170817.} Correction $\sim 10^{-83}$ against the Abbott bound (2017)~\cite{gw170817,gw170817mma} $|v_g-c|/c < 6\times 10^{-16}$: margin \textasciitilde{}70 orders.
  \item \textbf{LISA at 1 mHz.} Correction $\sim 10^{-93}$ against the projected bound $\sim 10^{-15}$: margin \textasciitilde{}78 orders.
\end{itemize}

\textbf{Conclusion.} PIU-$\Psi$ reproduces General Relativity in the propagation of gravitational waves with astronomical precision. The equality $v_g = c$ verified by GW170817 at the level of $10^{-15}$ is a natural consequence of the gravitational wave and light both being modes of the same medium.


\textbf{Epistemic classification.} The fulfillment of the observational bounds is \epi{DERIVED}, with the quantitative margin declared.


\subsubsection{III.3.6 --- The prediction P15 and the closing of the section}

\textbf{Prediction P15 (falsifiable structural hypothesis).} The UV correction (E23) implies a suppression of the primordial tensor spectrum around wavelengths comparable to $\ell_P^{-1}$. The spectrum of primordial gravitational waves (generated in the high-density regime of the early universe, \S{}III.6) could carry the imprint of this ultraviolet suppression, testable by next-generation cosmic microwave background experiments (CMB-S4~\cite{cmbs4}, LiteBIRD~\cite{litebird}) via B-mode polarization.


\textbf{Epistemic classification of P15.} \textbf{[structural HYPOTHESIS]}: a consequence of (E23), but its detectability depends on the amplitude of the primordial spectrum and on the instrumental sensitivity.


\textbf{Interpretation.} The gravitational wave is the transverse tensor mode of \emph{el Pleno}: a geometric modulation of the single physical medium that propagates without altering its local amplitude, massless, without dissipation, at velocity exactly $c$ in the accessible regime. \textbf{Explanation.} The velocity $c$ from the decoupling of the potential; the absence of mass from the amplitude preservation; the absence of dissipation from the uniqueness of \emph{el Pleno} and the U(1) conservation; the UV correction from the flexural term. \textbf{Prediction.} Exact equality $v_g = c$ in the accessible regime, UV correction $(k\ell_P)^2/20$ suppressed by tens of orders, and suppression of the primordial tensor spectrum (P15).

\subsection{\S{}III.4 --- The induced gauge sector: the photon and electrodynamics}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} chirality (medium-high), hypercharge (medium), electron mass (medium; the masses of transient fragments are optional), generations (low-medium), anomalies (medium).
  \item \textbf{Notes:} gauge sector as phase modes over the A/B bipartition. Widest active frontier of the program; status distinction handled with special care.
\end{itemize}
\end{estadoseccion}

\subsubsection{III.4.1 --- The question: where does electromagnetism come from}

The phase modes of \emph{el Pleno}, projected onto the A/B bipartition of the diamond lattice, produce an induced gauge sector (\S{}II.4.6). Electromagnetism ---and the $SU(2)\times U(1)$ structure of the electroweak sector--- emerges from the dynamics of the phase. The status distinction matters especially here, being the widest active frontier of the program: some results are solidly derived (the geometric emergence of the group structure, the masslessness of the photon), others remain as declared threads (chirality, fermionic masses, number of generations), and some are rigorously separated from others.


\subsubsection{\texorpdfstring{III.4.2 --- Geometric emergence of the SU(2)$\times$U(1) structure}{III.4.2 - Geometric emergence of the SU(2)xU(1) structure}}

\textbf{Origin.} The phase $\theta$ of the field of \emph{el Pleno}, projected onto the A/B bipartition of the diamond lattice (\S{}II.3.4), produces two modes: the symmetric $\theta_+$ and the antisymmetric $\theta_-$. The symmetric one lives in $U(1)$; the antisymmetric one, by its structure over the bipartite sub-lattice, lives in $SU(2)$. The group structure of the electroweak sector emerges from the microscopic geometry of the medium.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Ontological doublet.} The A/B bipartition naturally produces a doublet $\Phi = (\varphi_A, \varphi_B)^T$ ---the components of the field projected onto the two inequivalent sub-lattices. It is not postulated: it emerges from the diamond geometry.
  \item \textbf{Step 2 --- Induced gauge fields.} The projected phase produces, via a Kaluza-Klein-type construction over the diamond lattice, gauge fields in the continuum. The abelian field associated with the symmetric mode is $A_\mu(x) = \frac{\hbar}{e_\text{eff}}\partial_\mu\theta_+(x) + (\text{tensor terms of the antisymmetric mode})$, where $e_\text{eff}$ is an effective coupling whose value depends on the hypercharge embedding (open thread, \S{}III.4.7).
  \item \textbf{Step 3 --- Group structure.} $\theta_+ \in U(1)$ and $\theta_- \in SU(2)$ together produce the $SU(2)\times U(1)$ of the electroweak Standard Model, as a geometric consequence of the A/B bipartition, not an additional postulate.
\end{itemize}

\textbf{Derived results.} Five results have status \epi{DERIVED}: (1) the geometric emergence of $SU(2)\times U(1)$ from the A/B bipartition with $Z=4$; (2) the ontological spinorial doublet $\Phi = (\varphi_A, \varphi_B)^T$; (3) the beta-function coefficients of the pure-\emph{Pleno} segment, $b_Y - b_2 = 29/4$; (4) the topological absence of defects over the clean configuration $Z=4$; (5) the structural existence of the spontaneous breaking (\S{}III.4.8).


\textbf{Epistemic classification.} The geometric emergence of $SU(2)\times U(1)$ and the ontological doublet are \epi{DERIVED}. The phenomenological details (value of $\sin^2\theta_W$, masses) remain conditional on the open threads.


\subsubsection{III.4.3 --- The photon as a massless gauge mode}

\textbf{Origin.} The photon emerges as the massless gauge boson of the residual symmetry $U(1)_\text{em}$ that survives the breaking $SU(2)\times U(1) \to U(1)_\text{em}$.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Spontaneous breaking.} The doublet $\Phi$, with the effective potential of \emph{el Pleno} (\S{}II.4.3), presents a stable non-trivial minimum and breaks $SU(2)\times U(1) \to U(1)_\text{em}$ with the standard Higgs pattern: three Goldstone bosons are absorbed, producing massive $W^\pm, Z^0$, and a physical radial mode remains.
  \item \textbf{Step 2 --- Masslessness of the photon.} The gauge boson of the unbroken residual symmetry $U(1)_\text{em}$ remains massless: a rigorous consequence of the Goldstone-Higgs theorem ---no mass term is compatible with the $U(1)_\text{em}$ invariance.
\end{itemize}

\textbf{Structural identity --- velocity of the photon equal to $c$.} The propagation velocity of the photonic mode in the relaxed regime is exactly $c$, identical to that of the TT mode (\S{}III.3). This equality ---verified by GW170817 with $|v_\gamma - v_\text{GW}|/c < 7\times 10^{-16}$--- is not a coincidence: the photon and the gravitational wave are both modes of the same single physical medium.


\textbf{Ultraviolet correction of the photon dispersion (closure of P-Photon-3).} The photon is the Goldstone boson of the phase sector of \emph{el Pleno} (mode $\delta\theta_+$ projected onto $U(1)_\text{em}$, \S{}III.4.2). As a phase mode, its field equation is (E19) ---not a free equation without regulation---, which includes the phase flexural term derived from the diamond-lattice operator (\S{}II.3.8 P9b, \S{}I.7.10 E7). Its dispersion relation is, therefore:


\begin{equation}
\omega_\gamma^2 = c^2 k^2\left[1 + \frac{(k\ell_P)^2}{20} + \mathcal{O}((k\ell_P)^4)\right] \tag{E-foton-UV}
\end{equation}


with ultraviolet coefficient $+1/20$ \textbf{identical} to that of the TT mode (E23): both are modes of the phase sector and inherit the same lattice regulator with the same invariant $1/40$, without the scalar (photon) or tensor (TT) structure of the polarization altering the coefficient (\S{}II.3.8 P9b). The positive sign implies that the very-high-energy photon is infinitesimally subluminal (normal lattice dispersion). For accessible frequencies ($k\ell_P \sim 10^{-43}$ at optical or radiofrequency scales), $(k\ell_P)^2/20 \sim 10^{-80}$ or less, tens of orders below any observational bound on Lorentz-invariance violation; unobservable today, but structurally defined in sign and magnitude.


This result establishes the prediction P-Photon-3: the ultraviolet coefficient of the photon dispersion is \epi{DERIVED} with value $+1/20$. The exact equality between the ultraviolet corrections of the photon and of the TT mode also generates a structural prediction stronger than the mere infrared coincidence $v_\gamma = v_\text{TT} = c$: the two velocities coincide not only at zeroth order but also in their first Planckian correction, so that the light-gravity desynchronization is null up to order $(k\ell_P)^4$. This is the maximal form of the thesis that light and gravity are configurations of the same medium.


\textbf{Epistemic classification.} The masslessness of the photon is \epi{DERIVED} by Goldstone-Higgs. The equality $v_\gamma = c = v_\text{TT}$ in the infrared regime is \epi{DERIVED}, with observational verification by GW170817. The ultraviolet correction $(k\ell_P)^2/20$ of the photon and its identity with that of the TT mode are \epi{DERIVED} (\S{}II.3.8 P9b, \S{}II.5.4 E23, \S{}I.7.10 E7).


\subsubsection{\texorpdfstring{III.4.4 --- Maxwell's equations from phase circulation}{III.4.4 - Maxwell's equations from phase circulation}}

The masslessness of the photon (\S{}III.4.3) establishes it as a phase mode of the residual symmetry $U(1)_\text{em}$, but leaves implicit the deeper question: where do the \emph{equations} of electromagnetism come from? Saying ``\emph{el Pleno} has a photon'' is not the same as deriving Maxwell's equations. This subsection closes that gap: the four equations of the free field emerge, with their antisymmetric tensor $F_{\mu\nu}$ and their lattice coefficient, from the dynamics of the phase of \emph{el Pleno} over the diamond lattice, without postulating the electromagnetic field.


\textbf{Origin --- decomposition of the phase perturbation by tensor rank.} The only object of the program is $\Psi = S\cdot e^{i\theta}$ (\S{}I.9). A perturbation of the phase $\delta\theta$ over the diamond lattice is not pure scalar: it decomposes, according to its tensor rank over the crystalline structure, into three irreducible sectors ---exactly as the corpus already decomposes the lattice displacement into its modes (\S{}II.5):


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{scalar} $\delta\theta^{(0)}$ ---the pure phase mode, Goldstone boson of $U(1)$, one degree of freedom;
  \item \textbf{transverse vector} $a_\mu^{(1)}$ ---the electromagnetic sector, two polarization degrees of freedom;
  \item \textbf{transverse-traceless tensor} $h_{ij}^{(2)}$ ---the TT mode (\S{}III.3), two degrees of freedom.
\end{itemize}


The three are the \emph{same} flexion of the phase of $\Psi$ in contiguous ranks. This is the structural root of why the photon and the TT mode share exactly the velocity $c$ and the regulator $(k\ell_P)^2/20$ (\S{}III.4.3, \S{}III.3): it is not a coincidence between two distinct objects, but the same word of the medium in two ranks of the same tensor. The ``one degree of freedom'' count that would suggest reading the photon as a scalar is an artifact of projecting prematurely to rank zero; the physical sector of the photon is the transverse vector one.


\textbf{The two structures of the vector sector, separated by $90^\circ$.} The vector phase sector $a_\mu$ admits two distinct and orthogonal energies over the lattice, and the distinction between them is the heart of the derivation:


\emph{(a) Bond energy $\to$ symmetric tensor (elastic deformation, it is not Maxwell).} The energy associated with the magnitude of the stretching between neighboring nodes, $\sum_i |\Psi_n - \Psi_m|^2$ summed over the four tetrahedral bonds $\boldsymbol\delta_i$, produces a \emph{symmetric} tensor:
\[
9\,E_\text{bond} = (\partial\cdot a)^2 + \sum_{i<j}(\partial_i a_j + \partial_j a_i)^2 ,
\]
whose cross term $\partial_i a_j\,\partial_j a_i$ enters with coefficient $+2$ (symmetric sign). A bond energy is elastic by nature: it only feels the magnitude of the stretching, it is blind to the orientation of the rotation. Under the bond energy the vector sector is a \textbf{transverse vector phonon}, not a photon.


\emph{(b) Oriented circulation of the phase $\to$ antisymmetric tensor $F_{\mu\nu}$ (Maxwell).} The energy associated with the oriented circulation of the phase $\oint a$ around the tetrahedral plaquettes of the lattice ---not to the magnitude of the bond, but to the sense of rotation--- produces an \emph{antisymmetric} tensor:
\begin{tcolorbox}[ecuacionclave,title={E-Maxwell-1 --- Magnetic energy of \emph{el Pleno}}]
\[
E_\text{mag} = \frac{8}{9}\,F_{\mu\nu}F^{\mu\nu},
\qquad
F_{\mu\nu} = \partial_\mu a_\nu - \partial_\nu a_\mu ,
\]
\end{tcolorbox}
where the cross term $\partial_i a_j\,\partial_j a_i$ now enters with coefficient $-\tfrac{32}{9}$ (\emph{antisymmetric} sign, opposite to the $+2$ of the bond). The symbolic verification also confirms that the plaquette circulation equals the flux of the curl through it (discrete Stokes theorem) in the six bond pairs of the lattice, so that the object that appears is genuinely the field tensor $F_{\mu\nu}$.


\textbf{The $90^\circ$ and the $i$ of $\Psi$ (the ontological content).} The square modulus $|\Psi_n - \Psi_m|^2$ loses the sense of rotation of the phase; the \emph{oriented} sum around a plaquette conserves it. The antisymmetric Maxwell tensor lives in the phase circulation, at $90^\circ$ from the propagation direction ---the very structure of Ampère's law, where the magnetic field circulates perpendicular to the current. The imaginary unit $i$ of $e^{i\theta}$ \emph{is} the $90^\circ$ rotation operator that converts phase circulation into a perpendicular field. That is why electromagnetism is intrinsically complex and native to $U(1)$: the complex structure is not postulated, it is already in $\Psi$. The Noether current $J^\mu = S^2\,\partial^\mu\theta = \operatorname{Im}(\Psi^*\partial^\mu\Psi)$ ---the same conserved charge of axiom A5 (\S{}III.4.6)--- is the antisymmetric object that inhabits this sector, not the bond one.


\textbf{The four equations.} From the energy (E-Maxwell-1) and the definition $F_{\mu\nu}=\partial_\mu a_\nu - \partial_\nu a_\mu$ the four equations of the free field follow:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Homogeneous} (Faraday's law, $\nabla\cdot\mathbf{B}=0$): the Bianchi identity $\partial_{[\lambda}F_{\mu\nu]}=0$ is automatic by the form $F=\partial a - \partial a$. \epi{DERIVED} (forced by the structure of the curl). They are not empty: $a_\mu$ is the physical transverse vector component (two polarizations), not a pure gradient.
  \item \textbf{Inhomogeneous in vacuum} ($\partial_\mu F^{\mu\nu}=0$): Euler-Lagrange equation of F1 over the term $(8/9)F^2$. \epi{DERIVED}.
  \item \textbf{Inhomogeneous with source} ($\partial_\mu F^{\mu\nu}=J^\nu$): the Noether current of the charged solitons (\S{}III.5) acts as the source. \epi{SLD} ---the circulation-defect coupling is structurally identified; its explicit coefficient remains a pending computation, linked to the hypercharge embedding (\S{}III.4.9).
\end{itemize}


\textbf{The lattice coefficient $8/9$, explicit.} The factor $8/9$ is not postulated: it is computed by summing the oriented circulation of the phase over the plaquettes of the diamond lattice. Over the four tetrahedral bond vectors $\boldsymbol\delta_i$ (with $\sum_i\boldsymbol\delta_i = 0$ and $|\boldsymbol\delta_i|^2 = \ell_P^2/3$), the circulation through the plaquette formed by the pair $(i,j)$ is $\delta_i^a\,\delta_j^b\,F_{ab}$, with $F_{ab} = \partial_a a_b - \partial_b a_a$ the discrete curl. The magnetic energy is the sum of its square over the six pairs $i<j$:
\[
E_\text{circ} = \sum_{i<j}\big(\delta_i^a\,\delta_j^b\,F_{ab}\big)^2 .
\]
Developing this quadratic form in bond units, the cross term $(\partial_1 a_2)(\partial_2 a_1)$ appears with coefficient $-32$ and the diagonal $(\partial_1 a_2)^2$ with coefficient $+16$, exactly in the ratio $-2$ that characterizes $F_{\mu\nu}F^{\mu\nu}$ (whose analogous coefficients are $-4$ and $+2$). Therefore $E_\text{circ}$ is proportional to $F^2$, with factor $8$:
\[
E_\text{circ} = 8\,F_{\mu\nu}F^{\mu\nu}
\quad\xrightarrow{\;\text{normalization of the bond sector }(\times\tfrac19)\;}\quad
E_\text{mag} = \frac{8}{9}\,F_{\mu\nu}F^{\mu\nu},
\]
where the factor $1/9$ is the normalization that carries the symmetric bond sector (written above as $9\,E_\text{bond}$): the magnetic energy is measured relative to that same lattice normalization. The inherited cross term is then $-32/9$, opposite in sign to the $+2$ of the symmetric bond ---the signature that the object is antisymmetric, genuinely $F_{\mu\nu}$. The decomposition $8 = 2^3$ (spectral quotient of the coordination $Z=4$) and $9 = 3^2$ (tripartite structure / the three dimensions of the average) is a reading of the result, not its origin; the origin is the previous plaquette sum.

\textbf{Verification.} The computation is symbolically verifiable: the sum of the six plaquette circulations reproduces $E_\text{circ} = 8\,F^2$ with cross $-32$ and diagonal $+16$ (ratio $-2$), against the bond sector with cross $+16$ and diagonal $+8$ (ratio $+2$); and the circulation of each plaquette equals the flux of the curl through it (discrete Stokes theorem) in the six bond pairs, so that the object that appears is genuinely the field tensor $F_{\mu\nu}$. The coefficient $8/9$, the cross $-32/9$ and the sign opposite to the symmetric sector are thus fixed without a free parameter. Its full classification as a lattice constant ---in common with the analogous coefficient of the non-abelian sector (\S{}III.4.5)--- is the only piece that the program leaves for a unified treatment of geometric constants of the diamond lattice.


\textbf{Epistemic classification.} The Maxwell equations of the free field are \epi{DERIVED}: the term $(8/9)F^2$ with its coefficient \emph{derived explicitly} from the sum of plaquette circulations over the diamond lattice (symbolic verification of the cross $-32/9$ and of the sign opposite to the symmetric sector), the inhomogeneous ones in vacuum by Euler-Lagrange, and the homogeneous ones forced by the Bianchi identity. Maxwell with source is \epi{SLD} with the computation of the coupling identified. The decomposition by tensor rank that unifies photon and TT mode, and the reading of the $i$ of $\Psi$ as a $90^\circ$ operator, are \epi{DERIVED} in their structure.


\subsubsection{\texorpdfstring{III.4.4-bis --- Light respects the change ceiling: $c$ primary and the consistency theorem}{III.4.4-bis - Light respects the change ceiling: c primary and the consistency theorem}}

Having derived Maxwell's equations from phase circulation (\S{}III.4.4), a question arises about the architecture of the program: if Maxwell is derived and the speed of light appears in Maxwell, could $c$ be derived from Maxwell and removed from the set of three primaries ($\rho_P,\ell_P,c$)? The answer is no, for a precise ontological reason, and the question has a positive content: a consistency theorem that closes the gauge sector.

\textbf{$c$ is not the speed of light: it is the change ceiling.} The temptation to derive $c$ from Maxwell arises from identifying $c$ with the speed of light. In PIU that identification inverts the logical order. $c$ enters the program, in the foundational link (\S{}I.4), as the \emph{change ceiling} of energy: the maximum rate at which any configuration of el Pleno can transform. It is prior to light, and prior even to space and time, which are derived \emph{from} $c$ (time is accumulated change measured against $c$; causality A3 is the existence of that ceiling). Light is a \emph{consequence} of $c$: when the phase circulation propagates as a massless vector mode, it does so at the only velocity a massless mode can have in a medium with a change ceiling ---the ceiling itself. The ontological chain is
\[
\text{change ceiling } c \;\longrightarrow\; \text{time, space, causality} \;\longrightarrow\; \text{modes of the medium} \;\longrightarrow\; \text{light (massless mode)} = c,
\]
and asking Maxwell to define $c$ would place the consequence (light) as the origin of the principle (the change ceiling).

\textbf{Why it would be circular.} The usual derivation $c^2=1/(\varepsilon_0\mu_0)$ presupposes the vacuum constants $\varepsilon_0$ and $\mu_0$ as given before $c$. In PIU they are not: the electromagnetic properties of the vacuum are properties of el Pleno, and the Maxwell Lagrangian is built on the field tensor of the phase $F_{\mu\nu}[\theta]$, whose dynamics already carries $c$ in the kinetic term of el Pleno. Deriving $c$ from Maxwell would return $c$ to itself: a true but empty identity, like measuring a ruler with itself. There is moreover a structural reason: $c$ is the \emph{commensurability standard} of the program ---the common change of units between energy and mass ($E=Mc^2$), between space and time, and between the distinct sectors. A measurement standard cannot measure itself from within the system it defines; removing $c$ would leave the program without its rate-of-change standard.

\textbf{The positive content: light respects the ceiling · \epi{DERIVED}.} The derived massless vector mode ---the light of Maxwell--- propagates at phase and group velocities exactly equal to $c$, with no correction factor, at every infrared scale. The electromagnetic branch is the massless infrared branch of the phase equation (E19, \S{}III.4.3): $\Box\,\delta\theta=0$ gives $\omega=ck$ exact, with the flexural correction $(\ell_P^2/20)\Box^2$ confined to the ultraviolet branch (the Lee-Wick pole at scale $1/\ell_P$), which is not light. The lattice coefficient $8/9$ fixes the normalization of the magnetic energy (\S{}III.4.4) but factorizes in the equation of motion and does not appear in the velocity. This is non-trivial for three reasons: a mode \emph{with} mass ---matter, the modulus mode (E18)--- propagates slower than $c$ ($v_g=ck/\sqrt{k^2+\mu_S^2}<c$); only the massless mode saturates the ceiling. If the electromagnetic sector had its own elastic rigidity with limit velocity $c_\text{EM}\neq c$, light would go at $c_\text{EM}$ and the universe would have two limit velocities, in violation of A3. And the velocity comes out $c$ precisely because the vector sector is a perturbation of the \emph{same} phase $\theta$ whose kinetic term $\tfrac12\rho_P c^2\ell_P^2 S^2(\partial\theta)^2$ defines $c$: light has no medium of its own, it lives in the same el Pleno, and that is why it inherits its ceiling.

\begin{tcolorbox}[ecuacionclave,title={Interpretation --- the constancy of the speed of light as a result}]
That the derived light propagates exactly at the change ceiling confirms that electromagnetism is embedded in the single medium, without a parallel limit velocity. It is not that $c$ is measured with light, but that light ---emergent--- respects the $c$ that precedes it. The constancy of the speed of light, a starting postulate in relativity, is in PIU a \emph{result}: light always goes at $c$ because it is the massless mode of the medium whose change ceiling is $c$. The gauge sector introduces no new constant; it inherits that of the substrate. The question ``can $c$ be derived from Maxwell?'' thus has a double answer: no as a replacement of a primary ---it would be circular and would invert the ontology---, yes as a verification that the program is internally coherent in its treatment of the limit velocity.
\end{tcolorbox}


\subsubsection{\texorpdfstring{III.4.5 --- Yang-Mills and gauge unification from non-abelian circulation}{III.4.5 - Yang-Mills and non-abelian gauge unification}}

The derivation of Maxwell (\S{}III.4.4) used the circulation of a \emph{scalar} phase $\theta_+\in U(1)$. The same mechanism applied to the \emph{matrix} phase $\theta_-\in SU(2)$ ---which the program already possesses from the A/B bipartition (\S{}III.4.2)--- produces the complete Yang-Mills theory, and reveals that the gauge forces of the Standard Model are a single word of the medium.


\textbf{Origin --- non-abelian plaquette holonomy.} For a matrix phase, the link factors are $U_i = \exp(i A_\mu^a T^a\,\delta_i^\mu)$, with the generators $T^a$ of $SU(2)$ that do \emph{not} commute ($[T^a,T^b]=i\varepsilon^{abc}T^c$). The holonomy around a tetrahedral plaquette, expanded by Baker-Campbell-Hausdorff to order $\varepsilon^2$, produces:
\begin{tcolorbox}[ecuacionclave,title={E-YM-1 --- Yang-Mills tensor of el Pleno}]
\[
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i\,[A_\mu, A_\nu] .
\]
\end{tcolorbox}


\textbf{The non-abelian term is born by itself.} The term $i[A_\mu,A_\nu] = -f^{abc}A_\mu^a A_\nu^b T^c$ is not postulated: it arises exclusively from the fact that the generators of $\theta_-$ do not commute. The symbolic verification confirms that the structure $f^{abc}=\varepsilon^{abc}$ of $SU(2)$ comes from projecting $i[A,A]$ onto the generators, and that the abelian limit (when the fields live in a single generator, $[A,A]=0$) recovers Maxwell $F=\partial A-\partial A$ exactly.


\textbf{Gauge unification.} Maxwell and Yang-Mills are the \emph{same} object of the medium ---circulation of phase over plaquettes--- distinguished only by whether the phase is abelian (commutes $\to$ Maxwell) or non-abelian (does not commute $\to$ Yang-Mills). The three gauge structures of the Standard Model are the same mechanism of el Pleno over the sectors that the program already provides: $\theta_+\in U(1)$ gives electromagnetism (\S{}III.4.4); $\theta_-\in SU(2)$ gives the weak sector; and the $\mathbb{Z}_3$ vortices of the tripartite clusters (\S{}III.5.2) give color $SU(3)$. What the program had in four places ---electromagnetism, weak sector, color confinement and particle formation in the bounce (Part IV)--- is unified under phase circulation.


\textbf{The gauge groups as isometries of the structures of the medium.} The group structure is not postulated: it is the isometry of the structures that the medium already provides. The A/B bipartition provides a complex doublet $\Phi = (\varphi_A,\varphi_B)^{\mathsf T}\in\mathbb{C}^2$, whose total density $|\varphi_A|^2 + |\varphi_B|^2$ is the density of el Pleno, conserved by A1. The group that preserves that norm is $U(2) = U(1)\times SU(2)$: $\dim U(2) = 4 = 1 + 3$, with the $U(1)$ the global phase (symmetric mode $\theta_+$, central direction that commutes with everything) and the $SU(2)$ the rotations between sublattices A$\leftrightarrow$B (antisymmetric mode $\theta_-$, Pauli generators, $[\sigma_1,\sigma_2]=2i\sigma_3$). Analogously, the fundamental soliton $B=1$ decomposes (Gudnason-Nitta theorem) into three vortices of winding $1/3$, $\Phi_\text{color} = (v_1,v_2,v_3)\in\mathbb{C}^3$, whose total density is conserved (A1) and whose isometry group is $U(3) = U(1)\times SU(3)$: the $U(1)$ is baryon number; the $SU(3)$ are the color rotations. The eight generators of $su(3)$ are built directly from the structure of the triplet, in three geometric families (three real rotations that mix pairs of vortices, three phase rotations between pairs, two diagonal Cartan generators):
\begin{equation}
\begin{aligned}
&\lambda_1 = E_{12}+E_{21},\;\; \lambda_2 = -i(E_{12}-E_{21}),\;\; \lambda_3 = \operatorname{diag}(1,-1,0),\\
&\lambda_4 = E_{13}+E_{31},\;\; \lambda_5 = -i(E_{13}-E_{31}),\;\; \lambda_6 = E_{23}+E_{32},\\
&\lambda_7 = -i(E_{23}-E_{32}),\;\; \lambda_8 = \tfrac{1}{\sqrt3}\operatorname{diag}(1,1,-2),
\end{aligned}
\tag{E-YM-2}
\end{equation}
with $E_{ij}$ the matrix with a $1$ at position $(i,j)$. Verified symbolically: they are Hermitian and traceless, with Gell-Mann normalization $\operatorname{tr}(\lambda_a\lambda_b)=2\delta_{ab}$, they close $su(3)$ with canonical structure constants ($f^{123}=1$, $f^{458}=f^{678}=\sqrt3/2$, $f^{147}=1/2$), and their center is $\mathbb{Z}_3 = \{\mathbb{1},\omega\mathbb{1},\omega^2\mathbb{1}\}$ with $\omega=e^{2\pi i/3}$ ---\emph{exactly} the $\mathbb{Z}_3$ of confinement already derived (\S{}III.5.2). Neither $SU(2)\times U(1)$ nor $SU(3)$ are postulated: they are the isometries that the A/B doublet and the vortex triplet already provide. \epi{DERIVED}.

\textbf{The number $N=3$ of color, derived by two convergent routes.} (a) Geometric: the winding $B=1$ over the target $S^3$ decomposes minimally and stably into three vortices via the linking of the Hopf map, protected by a nonzero Massey invariant. (b) Crystalline: the $3$ is the dimension of the irrep $T$ (triplet) of the binary tetrahedral group $2T$ of the diamond lattice ($\ell=1\to T(3)$ under the breaking $SO(3)\to 2T$). Modular arithmetic closes confinement: $3\times\tfrac13 = 1$ (color singlet, free), while any subset of vortices has winding $\not\equiv 0$ and remains confined. \epi{DERIVED}.

\textbf{The gauge self-interaction is a forced consequence of $f^{abc}\neq 0$.} With $F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc}A^b_\mu A^c_\nu$, the Lagrangian $\mathcal{L}=-\tfrac14 F^a_{\mu\nu}F^{a\,\mu\nu}$ expands into three blocks: the kinetic $-\tfrac14(\partial_\mu A^a_\nu-\partial_\nu A^a_\mu)^2$, the cubic vertex $-g f^{abc}(\partial_\mu A^a_\nu)A^{b\,\mu}A^{c\,\nu}$ and the quartic vertex $-\tfrac14 g^2 f^{abc}f^{ade}A^b_\mu A^c_\nu A^{d\,\mu}A^{e\,\nu}$. The self-coupling vertices of three and four gauge bosons are not added: they are the cross and quadratic terms in $f^{abc}$ that appear necessarily on squaring $F$, determined by the same structure constants derived from the algebra of generators. \epi{DERIVED}.


\textbf{Epistemic classification.} The emergence of the complete Yang-Mills structure is \epi{DERIVED} (symbolic verification): the term $i[A,A]$ from non-commutativity (BCH holonomy); the groups $SU(2)\times U(1)$ and $SU(3)$ as isometries $U(2)$ of the A/B doublet and $U(3)$ of the vortex triplet (Gell-Mann generators, canonical structure constants, center $\mathbb{Z}_3$); the number $N=3$ of color by two routes (Hopf linking and crystalline triplet $2T$); the gauge self-interaction forced by $f^{abc}\neq 0$; and the recovery of Maxwell as the abelian limit. \textbf{There remain as declared numerical threads} (not structural): the coupling $e_\text{eff}$, $\alpha=1/137$, $\sin^2\theta_W$, the coefficient of the Lagrangian (analogue of the abelian $8/9$), the fermionic masses and the number of generations, tied to the hypercharge embedding (\S{}III.4.9). The medium fixes their \emph{type} (pure number of the electroweak cascade), not their value: they are missing-information of the corpus, not gaps in the derivation of the gauge structure.

\subsubsection{III.4.6 --- The quantization of electric charge}

\textbf{Origin.} The observed electric charge is quantized in integer multiples of an elementary charge. In PIU-$\Psi$ this is not postulated: it is a topological consequence of the structure of the phase of the field of \emph{el Pleno}.


\textbf{Result.} The Noether U(1) charge of \emph{el Pleno} is topologically quantized:


\[
Q = n\cdot Q_0, \quad n\in\mathbb{Z}, \quad Q_0 = \hbar
\]

\textbf{Intermediate steps.} The phase $\theta$ is defined over a circle ($\theta\in[0,2\pi)$). The finite-energy configurations tend to a constant value at spatial infinity. The number of times the phase ``winds around'' the circle ---the winding number--- is a topological integer that cannot change continuously. The conserved U(1) charge (axiom A5) is proportional to this integer winding, which quantizes the charge in multiples of $Q_0 = \hbar$.


\textbf{Connection with the normalization of the quantum.} For the elementary quantum ($n=1$), the norm of the fundamental mode is $\int S^2\,d^3x = \ell_P^3$, derived in \S{}III.2.2 (Step 4) from the dimensionlessness of $S$ and the lattice cell volume, without circularity. The topological quantization of the charge $Q = n\hbar$ is then a \emph{consequence} of that normalization ($Q_0 = p_\theta = \hbar$ for $n=1$), not its foundation.


\textbf{Dimensional verification.} $[Q_0] = [\hbar] = [J\cdot s]$, the dimension of the Noether charge of the phase symmetry $\checkmark$.


\textbf{Epistemic classification.} The topological quantization of the charge is \epi{DERIVED}: a direct consequence of the U(1) conservation (axiom A5) plus the topology of the phase $\theta$. It explains structurally why the electric charge is quantized ---a fact that the Standard Model incorporates without explaining it.


\subsubsection{III.4.7 --- Beta-function coefficients and the electroweak hierarchy}

\textbf{Beta-function coefficients.} In the pure-\emph{Pleno} segment (before the Yukawa couplings), the difference of the beta-function coefficients of $U(1)_Y$ and $SU(2)$ is:


\[
b_Y - b_2 = \frac{29}{4}
\]

\epi{DERIVED} from the structure of the doublet over the A/B bipartition. It fixes the relative evolution of the gauge couplings with the energy scale.


\textbf{The weak mixing angle.} $\sin^2\theta_W$ is determined conditionally on the hypercharge embedding of the doublet, $Y(\Phi) = 1/2$. With this embedding, the program reproduces the observed value. The embedding itself is an open thread (\S{}III.4.9).


\textbf{The electroweak hierarchy and dimensional transmutation.} The problem of why $v_\text{EW}\approx 246$ GeV is 17 orders smaller than the Planck scale receives a candidate mechanism: the Coleman-Weinberg dimensional transmutation, anchored in a pole of the renormalization flow located at $\mu^* \approx 5.40\times 10^{17}$ GeV. This sub-Planckian pole acts as an ultraviolet anchor for the dimensional transmutation. The quantitative closure depends on the Yukawa coupling of the top quark (open thread, \S{}III.4.9).


\textbf{Candidates for chiral fermions.} The nodal lines of the diamond-lattice spectrum (\S{}II.3.8 P7) are candidates to produce chiral Weyl fermions. The evasion of the Nielsen-Ninomiya theorem occurs because the program does not postulate point fermions at the lattice sites: the fermions are topological solitonic modes (\S{}III.5).


\textbf{Epistemic classification.} $b_Y - b_2 = 29/4$ is \epi{DERIVED}. $\sin^2\theta_W$ is \textbf{[conditional SLD]} on the hypercharge embedding. The dimensional transmutation is \textbf{[conditional SLD]}. The nodal lines as a path to chirality are \textbf{[structural HYPOTHESIS]}.


\subsubsection{III.4.8 --- The Higgs mechanism: what is derived and what is pending}

\textbf{What is derived (structural existence).} The ontological doublet $\Phi = (\varphi_A, \varphi_B)^T$, inherited from the A/B bipartition and endowed with the effective potential of \emph{el Pleno}, presents a stable non-trivial minimum at $S_{\min}^\text{estr}\approx 0.7231$ with positive curvature $V''_\text{eff}(S_{\min}^\text{estr})\approx 19.69\,\rho_P c^2$ (\S{}II.4). This minimum breaks $SU(2)\times U(1)\to U(1)_\text{em}$ with the standard Higgs pattern. The doublet $\Phi$ is the structurally-forced Higgs of the program, not an elective candidate. \epi{DERIVED}.


\textbf{What is pending (value of the VEV).} The native scale of the VEV in \emph{el Pleno} is of order the Planck mass. The observed electroweak value $v_\text{EW}\approx 246$ GeV requires an additional dynamical mechanism that produces \textasciitilde{}17 orders of hierarchy. The Coleman-Weinberg dimensional transmutation is plausible, but its quantitative closure depends on the Yukawa coupling of the top quark (open thread, \S{}V.3). \textbf{[conditional SLD]}.


\textbf{Epistemic classification.} The existence of the spontaneous breaking with the standard Higgs pattern is \epi{DERIVED}. The numerical value of the VEV is \textbf{[conditional SLD]}.


\subsubsection{III.4.9 --- Open threads of the gauge sector}

In accordance with the Rigor Shield, the open threads of the induced gauge sector are declared explicitly:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Chirality thread.} The distinction $SU(2)_L$ vs $SU(2)_R$ is not derived. Candidate: nodal lines of the diamond-lattice spectrum toward chiral Weyl fermions. Medium-high severity.
  \item \textbf{Hypercharge thread.} The embedding $Y(\Phi) = 1/2$ ---which fixes $\sin^2\theta_W$--- is not derived; it is assumed conditionally. Medium severity.
  \item \textbf{Fermionic-masses thread.} In accordance with the persistent-constituent principle (\S{}III.5.1), the obligation of the program is limited to the mass of the \emph{electron} ---the only lepton that is a persistent constituent of matter--- declared an open thread of the leptonic sector in Part IV. The masses and mixings of the muon, the tau and the heavy quarks correspond to transient fragments of energy deposition; their reading is optional work, not a precondition of the program. The residual obligation (electron mass) requires the program of skyrmions/fermionic defects over the $Z=4$ lattice and is a prerequisite of the quantitative closure of the Higgs. Medium severity.
  \item \textbf{Generations thread.} Why nature repeats the pattern of constituents in heavier and transient families ($N_\text{gen} = 3$) is not derived. It does not compromise the persistent assembly ---which uses only the first family--- so it is a thread of structural interest, not of obligation. Low-medium severity.
  \item \textbf{Anomalies thread.} The cancellation of gauge anomalies is not verified structurally. Medium severity.
\end{itemize}

These five threads constitute the widest active frontier of the program. The strategy (\S{}V.3) is to develop the unified apparatus over the $Z=4$ lattice that attacks the five simultaneously.


\subsubsection{\texorpdfstring{III.4.10 --- Interpretation: what electromagnetism is under PIU-$\Psi$}{III.4.10 - Interpretation: what electromagnetism is under PIU-Psi}}

\textbf{Interpretation.} Electromagnetism is not a fundamental interaction that occurs over an empty stage: it is the \emph{dynamics of the phase of \emph{el Pleno}}. The field of the medium is $\Psi = S\,e^{i\theta}$; the modulus $S$ says how much structure there is at each point, and the phase $\theta$ is an internal orientation that each point of the medium can rotate freely. When that phase varies from one point to another ---when there is \emph{phase circulation} along the medium--- that variation propagates as a wave, and that wave is light. The photon is, literally, an undulation of the phase of \emph{el Pleno} that costs no energy at rest (it is massless) because rotating the phase uniformly does not change the state of the medium; only the \emph{variation} of the phase costs energy, not its value. The electric charge, in turn, is the conserved quantity associated with that phase freedom ---the Noether charge of the $U(1)$ symmetry (axiom A5)---: just as the conservation of energy is the obverse of the laws not changing with time, the conservation of charge is the obverse of the state of \emph{el Pleno} not changing when its phase is rotated uniformly.


\textbf{Why Maxwell's equations appear.} That the dynamics of the phase obeys exactly Maxwell's equations is not imposed: it is derived (\S{}III.4.4). The microscopic structure of \emph{el Pleno} is bipartite ---two intertwined sub-lattices A and B (the diamond lattice, \S{}II.3)---, and projecting the phase circulation onto that bipartition produces, in the continuum, an antisymmetric tensor field $F_{\mu\nu}$ that satisfies the two pairs of Maxwell's equations, with the coefficient $8/9$ coming out of the geometry of the lattice, not from a fit. The non-abelian Yang-Mills structure ---the basis of the Standard Model--- emerges from the same mechanism when the phase circulation ceases to commute (\S{}III.4.5). Electromagnetism and the nuclear interactions are, in this reading, two forms of the same phase dynamics of the single medium.


\textbf{What is explained and what remains open.} What is explained: the masslessness of the photon (it is a Goldstone mode of the phase symmetry, \S{}III.4.3); the exact equality between the velocity of light and that of gravitational waves (both are modes of the same medium, propagating at the same rigidity $c$); and the quantization of electric charge in multiples of $\pm 1/3$ (\S{}III.4.6, as a topological property of the solitonic clusters). What is \emph{not} yet derived ---the chirality of the weak interactions, the masses of the fermions, the number of generations--- is declared an open thread (\S{}III.4.9), not disguised: the program identifies the structural root of the electroweak sector but does not yet close its quantification.


\textbf{Prediction.} The exact equality $v_\gamma = c = v_\text{TT}$, already confirmed by GW170817; the $SU(2)\times U(1)$ structure with the beta-function coefficients derived from the geometry; and the fractional electric charges $\pm 1/3, \pm 2/3$ of the quarks as topological integers, not as fitted values.


\subsection{\S{}III.5 --- Solitons as particles}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} asymptotic Kepler-Hales invariant for $M_Q$ (research program); fermionic masses (\S{}III.4.9).
  \item \textbf{Notes:} particles as solitonic configurations of \emph{el Pleno}; charges and statistics derived, not postulated. Split $\Delta$-N by rotational quantization (verified 90\%).
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{III.5.1 --- The question: what are particles under PIU-$\Psi$}{III.5.1 - The question: what are particles under PIU-Psi}}

The localized solitons of \emph{el Pleno} ---skyrmions (class M, fermionic), Q-balls (class MO, bosonic) and the tripartite Borromean cluster, constructed in \S{}II.5--- are what physics observes as particles. A particle is not a material point nor an abstract quantum of a postulated field, but a \textbf{stable localized configuration of \emph{el Pleno}}, a finite region where $\Psi$ departs from the background and reconnects to it at infinity. Its statistics is not postulated: it is a rigorous consequence of the topology of the configuration space (Finkelstein-Rubinstein, \S{}II.5.4); its electric charges are not postulated either: they emerge from the modular arithmetic of the topological clusters.

\begin{tcolorbox}[notabox,title={Persistent-constituent principle: assembly manual, not catalogue of fragments}]
PIU-$\Psi$ is the explanation of what the universe is and how it operates and assembles persistently, not a catalogue of the transient states that can be induced by depositing energy in \emph{el Pleno}. The distinction is ontological, not of degree. There are two classes of object:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Persistent constituents.} Configurations that participate in the stable structure of reality: the proton, the neutron (bound in nuclei, stable indefinitely), the electron, the photon and the gravitational wave. They are the pieces with which the universe is effectively built and with which it has persisted for $\sim 13.8\times 10^9$ years. \emph{These} are the objects that the program is obliged to explain.
  \item \textbf{Energy-deposition fragments.} Configurations that appear when sufficient energy is concentrated in the medium and quickly fall apart: the hadronic resonances, the heavy leptons (muon, $\sim 2.2\,\mu$s; tau, $\sim 0.3$ ps) and the heavy quarks, which do not exist free but as ephemeral internal states. They do not enter into the persistent assembly of any structure of the universe. They are echoes of the impact, not pieces of the blueprint.
\end{itemize}

The separation criterion is \textbf{not} the mean lifetime of the isolated object ---the free neutron decays in $\sim 15$ minutes and nevertheless is a persistent constituent, because bound in the nucleus it is stable and forms half of the baryonic matter of the universe--- but the \textbf{participation in persistent structure}. The precise razor is provided by the topology of \emph{el Pleno}: an object with new irreducible topological charge or \emph{winding} is a piece; an object that is the same defect in another state of rotation or excitation (like the $\Delta(1232)$ with respect to the nucleon, \S{}III.5.4) is a mode of the same constituent, not a new piece.

\textbf{Consequence for the scope of the program.} The extensive ``matter content'' of the Standard Model that PIU would seem to have to re-derive is, for the most part, the catalogue of collision fragments ---resonances and transient heavy states--- that the program reads as excited modes or ephemeral configurations of the same handful of constituents, not as independent fundamental entities. Reproducing that catalogue is not a precondition of the program; offering a reading of some fragment (as is already done with the baryonic rotational tower) is optional work, not an obligation.
\end{tcolorbox}


\subsubsection{III.5.2 --- The baryon as a tripartite Borromean cluster}

\textbf{Origin.} The fundamental baryon (proton, neutron) is identified with the tripartite Borromean cluster: the soliton of topological number $B=1$ decomposed into three vortices of fractional winding $1/3$ (\S{}II.6.5). The decomposition is rigorous (Gudnason-Nitta theorem) and produces the internal structure $2+1$.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Topological decomposition.} $B = 1 = 1/3 + 1/3 + 1/3$ as a sum of winding numbers of three vortices, without altering the total integer winding. The distinct profiles produce the $2+1$ asymmetry.
  \item \textbf{Step 2 --- Electric charges from modular arithmetic.} The stable configurations are given by the $\mathbb{Z}_3$ congruence over the integer numerators of the topological charges (\S{}II.6.6): proton $(+2/3, +2/3, -1/3)$, total charge $+1$; neutron $(+2/3, -1/3, -1/3)$, total charge $0$.
  \item \textbf{Step 3 --- Ontological identification.} The charges $q_j$ are topological vortex charges of \emph{el Pleno} ---fractional winding of the phase---, not postulated electric charges. The observable electric charge emerges as the integral over the conserved topological current (\S{}III.4):
\end{itemize}

\[
Q^\text{baryon}_\text{electric} = \oint_{\partial V} \mathbf{j}_\theta\cdot d\mathbf{S} = \sum_{j=1}^3 q_j
\]

This strictly preserves A1: there are no fundamental quarks with postulated charge; there are vortices of \emph{el Pleno} whose topological charge produces, via gauge emergence, the observed electric charge.


\textbf{Interpretation.} The fractional charges observed in the quarks ($+2/3$, $-1/3$) ---postulated in the Standard Model--- are derived structurally from the $\mathbb{Z}_3$ modular arithmetic of the Borromean cluster. The quantization in simple fractions is a consequence of the topological-stability condition of the cluster.


\textbf{Epistemic classification.} The tripartite decomposition is \epi{DERIVED} (Gudnason-Nitta under the A1 filter). The derivation of the fractional charges from $\mathbb{Z}_3$ is \epi{SLD}, supported by the exact coincidence with the observed charge structure.


\subsubsection{\texorpdfstring{III.5.3 --- Confinement as a $\mathbb{Z}_3$ center obstruction}{III.5.3 - Confinement as a Z3 center obstruction}}

The $\mathbb{Z}_3$ modular arithmetic that quantizes the fractional charges (\S{}III.5.2) is not only the origin of the charges: it is the mechanism of confinement. The confinement that the program declares ``topological'' ---the $1/3$-winding vortices of the Borromean cluster--- is exactly the $\mathbb{Z}_3$ center obstruction of the $SU(3)$ Yang-Mills theory that emerges from the non-abelian phase circulation (\S{}III.4.5). This links the hadronic sector with the gauge sector under a single mechanism.


\textbf{Origin --- the center holonomy of a fractional winding.} An isolated $1/3$-winding vortex has a center holonomy $e^{2\pi i/3}\neq \mathbb{I}$: the phase does not close upon traversing a loop that encloses it. A configuration whose total holonomy differs from the identity cannot have finite energy in isolation ---it remains \textbf{confined}. The symbolic verification runs through the cases:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Isolated $1/3$ or $2/3$ vortex:} holonomy $e^{\pm 2\pi i/3}\neq\mathbb{I}$ $\Rightarrow$ does not close $\Rightarrow$ confined (infinite energy in isolation).
  \item \textbf{Tripartite cluster $\tfrac13+\tfrac13+\tfrac13$:} total holonomy $=\mathbb{I}$ $\Rightarrow$ color singlet $\Rightarrow$ free, observable, finite energy.
\end{itemize}


\textbf{The singlet rule.} The condition that the sum of windings be $\equiv 0 \pmod 3$ reproduces exactly the phenomenology of quantum chromodynamics: $qqq$ (baryon) and $q\bar q$ (meson) close to the identity and are free; the lone quark ($\equiv 1$) and the diquark ($\equiv 2$) do not close and remain confined. The color-singlet rule of the Standard Model emerges as a closure condition of the center holonomy, not as a postulate.


\textbf{A single mechanism in two languages.} The confinement that Part IV describes as ``topology of vortices frozen in the bounce'' and the Yang-Mills confinement by $\mathbb{Z}_3$ center obstruction are the same phenomenon: the topological \emph{winding} of a particle \emph{is} its center holonomy, and the color charge \emph{is} the class of that holonomy. This closes the circle between the phase circulation of the gauge sector (\S{}III.4.5) and the solitonic structure of the baryon (\S{}III.5.2).


\textbf{Epistemic classification.} The identification of confinement with the $\mathbb{Z}_3$ center obstruction and the derivation of the color-singlet rule are \epi{DERIVED} (symbolic verification of the holonomies; the $\mathbb{Z}_3$ structure of the cluster is already in \S{}II.6.5 and \S{}III.5.2). The explicit construction of the complete $SU(3)$ field over the tripartite structure ---beyond the center--- remains a pending computation.



\subsubsection{\texorpdfstring{III.5.4 --- The $\Delta$-N mass split as rotational quantization}{III.5.4 - The Delta-N mass split as rotational quantization}}

\textbf{Origin.} Part II (\S{}II.6.7) derived the rotational tower of the Borromean cluster and the split formula (E30, E31), including the rigid-rotor quantization, the moment of inertia $\Theta_p = m_p d^2/3$, and the difference $J(J+1)$.


The two lowest rotational states are identified with two observed particles: the ground state $J=1/2$ is the nucleon $N$; the first excited $J=3/2$ is the resonance $\Delta(1232)$. They are not distinct particles with different geometries: they are the same cluster in two states of rotation. The observed split is quantized rotational energy of the same object.


\textbf{The split formula (derived in \S{}II.6.7).}


\[
\Delta m = m_\Delta - m_N = \frac{9(\hbar c)^2}{2\,m_p c^2\,d^2} \qquad(\text{see }E31)
\]

where the factor 9 combines the factor 3 of the tripartite moment of inertia with the factor $3 = \tfrac{15}{4} - \tfrac{3}{4}$ of the $J(J+1)$ jump between $J=3/2$ and $J=1/2$, and $d$ is the separation between vortices, identified with the baryon radius $R_p$ (\S{}III.5.5).


\textbf{Numerical verification.} With $m_p c^2 = 938.27$ MeV, $\hbar c = \rho_P c^2\ell_P^4 = 197.33$ MeV$\cdot$fm, $d \approx R_p = 0.8409$ fm:


\[
\Delta m = \frac{9\times(197.33)^2}{2\times 938.27\times(0.8409)^2}\ \text{MeV} = \frac{350452}{1326.9}\ \text{MeV} \approx 264.1\ \text{MeV}
\]

\textbf{Comparison.} The observed one is $\Delta m_\text{obs}\approx 293.7$ MeV; the ratio $\Delta m_\text{PIU}/\Delta m_\text{obs} = 0.90$ ---agreement to 90\% without additional calibrated parameters beyond the single scale $\Lambda_\text{PIU}$ of the hadronic sector. The 10\% discrepancy is consistent with the expected corrections of the rigid-rotor model (rotational deformation, vibrational modes), not with a structural error.


\textbf{Meaning.} That a formula with a single scale reproduces the $\Delta$-N split to 90\% is remarkable: in the Standard Model the masses of the nucleon and of the $\Delta$ require lattice quantum-chromodynamics computations with multiple parameters. Here the split emerges from the rotational geometry of a single object.


\textbf{Epistemic classification.} The split formula is \epi{DERIVED} in \S{}II.5.7. The identification $N\leftrightarrow J=1/2$, $\Delta\leftrightarrow J=3/2$ is \epi{DERIVED}; the numerical comparison to 90\% is \textbf{[observational verification]}.

\textbf{Division of labor in the baryon.} Each part of the baryonic phenomenology contributes a distinct physical structure, and attributing an effect to the wrong mechanism would be an error. The $\Delta$-N split is distributed between two contributions of the \emph{same} cluster: the rigid-rotor quantization contributes the dominant term (the $90\%$, $264$ MeV \epi{DERIVED}), and the non-rigid-rotor corrections ---centrifugal deformation and vibrational/Coriolis modes, standard physics of the Skyrme model--- contribute the residual $\sim 10\%$. The asymmetric $2+1$ knot structure of the Borromean cluster (\S{}IV.6) does \emph{not} intervene in the split: $N$ and $\Delta$ are rotational states of a single cluster (a single moment of inertia $\Theta_p$), so that the $2+1$ asymmetry is one of charge and flavor ($\mathbb{Z}_3$ arithmetic over the fractional \emph{windings}, \S{}III.5.2), not of rotational geometry; the knot governs the charges, and the non-null Massey invariant governs the stability of the proton (\S{}IV.6). Thus, rotor $\to$ split, vibrational $\to$ residue, knot $\to$ charges, Massey $\to$ stability: four pieces with disjoint roles, without overlap or double counting.


\subsubsection{III.5.5 --- The proton radius as an independent prediction}

\textbf{Origin.} In (E31), $d$ is identified with $R_p$. The program produces an independent prediction of $R_p$.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Structural consistency.} In the compact Borromean cluster, the natural separation between vortices is fixed by the balance between the quartic repulsion of the cores and the Borromean topological confinement. This balance fixes $d \approx R_p$.
  \item \textbf{Step 2 --- Optimal calibration.} Solving for $d$ from the observed split:
\end{itemize}

\[
d_\text{opt} = \sqrt{\frac{9(\hbar c)^2}{2\,m_p c^2\,\Delta m_\text{obs}}} = 0.7974\ \text{fm}
\]

\textbf{Comparison.} The observational one is $R_p^\text{obs} = 0.8409$ fm (CODATA). The ratio $d_\text{opt}/R_p = 0.948$ ---a difference of 5\%, consistent with the structural identification $d\approx R_p$.


\textbf{Epistemic classification.} The consistency $d\approx R_p$ to 5\% is \epi{SLD}: it supports the identification of the compact cluster with the observed baryon, without an ad-hoc calibration. A non-circularity warning is in order: $d_\text{opt}$ is \emph{solved} from the observed split, so that the closeness $d_\text{opt}\approx R_p$ confirms the geometric identification $d\approx R_p$, but does \emph{not} constitute an independent verification of the value of the split (that would be using the same equation in two directions). The split is verified by its direct comparison with the datum (the $90\%$ above); $R_p$ is verified as an independent prediction in \S{}III.5.5. They are two distinct checks, not one reinforcing itself.


\subsubsection{III.5.6 --- The complete baryonic rotational tower}

\textbf{Tower formula.}


\[
m(J) = m_N + \left[J(J+1) - \tfrac{3}{4}\right]\frac{\hbar^2}{2\Theta_p}, \qquad \Theta_p = \frac{m_p R_p^2}{3}
\]

\textbf{Numerical verification (extended spectrum).} With the calibrated moment of inertia:


\begin{itemize}[leftmargin=1.4em]
  \item $J=1/2$: $m = 938$ MeV $\to$ $N(939)$ (calibration reference).
  \item $J=3/2$: $m = 1232$ MeV $\to$ $\Delta(1232)$ (calibration agreement).
  \item $J=5/2$: $m \approx 1722$ MeV $\to$ $N(1680)$ observed, ratio 1.025 (agreement to 97.5\%).
  \item $J=7/2$: $m \approx 2407$ MeV $\to$ $\Delta(1950)$ observed, ratio 1.234.
  \item $J=9/2$: $m \approx 3288$ MeV $\to$ $N(2220)$ tentative, ratio 1.481.
\end{itemize}

\textbf{Interpretation.} The tower reproduces the order and the magnitude of the observed baryonic states, with increasing deviations for high $J$ (1.025 $\to$ 1.48), behavior typical of the rigid rotor of the Skyrme model (corrections by rotational deformation). The consistency to 2.5\% for $J=5/2$ is non-trivial. The deviations at $J\geq 7/2$ are not closed with a single-parameter centrifugal-distortion term without compromising the calibration of the $\Delta$-N split; their complete quantitative description requires the deformation corrections of the skyrmion profile, a thread connected with the closure of coefficients of the solitonic profile (\S{}III.9.5).


\textbf{Epistemic classification.} The rotational tower is \epi{DERIVED} from the rotational quantization of the cluster, with the extended spectrum consistent at the level expected for a rotor with deformation corrections.


\subsubsection{III.5.7 --- The stability of the proton as a topological property}

\textbf{Origin.} The proton is the observed stable baryon (mean life $> 10^{34}$ years). Its stability is explained by the topology of the Borromean entanglement.


\textbf{Mechanism.} The three vortices have null pairwise linking numbers, but the higher-order Massey invariant (a triple cohomological product) is non-null. This characterizes the ``triple entanglement without pairwise entanglement'': the removal or decay of a single vortex simultaneously trivializes the two remaining ones, transforming the system into a free unbound state. The energy required for that transition tends to infinity in the isostatic-saturation regime of \emph{el Pleno} at zero effective temperature, kinetically blocking every decay channel of the fundamental baryon.


\textbf{Interpretation.} ``The quark cannot be torn off the proton in isolation'' receives here a rigorous mathematical formulation: it is not only pairwise entanglement (which could be broken), but triple Borromean entanglement whose topological stability is structurally deeper.


\textbf{Epistemic classification.} The stability by non-null Massey invariant is \textbf{[formal topological observation]}: it provides rigorous canonical mathematical language for the stability of the proton.


\subsubsection{III.5.8 --- The mass of the fundamental soliton and the confinement of the quarks}

\textbf{Mass of the fundamental soliton.} The canonical structural mass of the fundamental soliton is fixed by the Planck mass and the hard bound of the modulus:


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

This is the natural mass scale of the solitonic configurations before the dynamics that produces the observed spectrum. The ultraviolet completeness of the soliton is guaranteed by construction at the scale $\ell_P$, via the projection of the discrete Laplacian of the diamond lattice onto the Klauder affine quantization (\S{}III.2.10). The idealized asymptotic form $M_Q \approx 0.6776\,M_P$ (under optimal Kepler-Hales packing) is an objective of the research program (\S{}V.3).


\textbf{Topological confinement of the quarks.} The fractional vortices of the Borromean cluster (\S{}III.5.2) are confined by a topological mechanism: they cannot exist as isolated configurations of finite energy.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- The energy of a vortex.} For a vortex with winding $n$ at distance $r$ from the core, $|\nabla\theta| = n/r$, and the kinetic energy density is $\tfrac{1}{2}\rho_P c^2\ell_P^2 S^2|\nabla\theta|^2$. The energy per unit length (in 2D transverse) is:
\end{itemize}

\[
\frac{E}{L} = \int_{r_0}^R \tfrac{1}{2}\rho_P c^2\ell_P^2 S_0^2\frac{n^2}{r^2}\cdot 2\pi r\,dr = \pi\rho_P c^2\ell_P^2 S_0^2 n^2\int_{r_0}^R\frac{dr}{r} = \pi\rho_P c^2\ell_P^2 S_0^2 n^2\ln\frac{R}{r_0}
\]

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 2 --- The topological-closure condition.} The integral $\ln(R/r_0)$ diverges as $R\to\infty$ unless the phase field can relax to a constant configuration at infinity. This is possible only if the total winding is integer: a configuration of winding $n\in\mathbb{Z}$ allows the phase to ``unwind'' to a constant asymptotic value. One of fractional winding $n=1/3$ cannot unwind, and the integral diverges irremediably.
  \item \textbf{Step 3 --- The confinement.} Therefore, an isolated $1/3$-winding vortex has divergent energy: it is prohibited as a finite-energy asymptotic configuration. Only the combinations whose total winding is integer ---three $1/3$ vortices forming a Borromean cluster of winding $1$--- have finite energy. This confines the fractional vortices: they are permanently bound in clusters of integer winding.
\end{itemize}

The confinement tension is derivable from the characteristic tension of \emph{el Pleno} (\S{}III.1.6):


\[
\sigma_\text{QCD} \sim \sigma_\text{Pleno}\left(\frac{\Lambda_\text{PIU}}{M_Q}\right)^2
\]

with a factor of order $\sim 2.8$ without adjustable parameters. The hadronic confinement scale thus emerges from the structural tension of the medium, it is not postulated.


\textbf{Interpretation.} The confinement of the quarks ---a non-perturbative property of quantum chromodynamics, difficult to derive--- receives here a direct topological reading: the quarks cannot be isolated because they are fractional-winding vortices whose isolated energy diverges logarithmically, and only the integer-winding clusters have finite energy. The confinement is topology, not a postulated color dynamics.


\textbf{Dimensional verification.} $[M_Q] = [M_P]$ ($S_{\max}$ dimensionless) $= [kg]$ $\checkmark$. $[\sigma_\text{QCD}] = [\sigma_\text{Pleno}] = [N]$ $\checkmark$.


\textbf{Epistemic classification.} The structural form $M_Q = M_P/S_{\max,\text{eff}}$ is \epi{DERIVED}; the ultraviolet completeness is \epi{SLD}; the asymptotic Kepler-Hales invariant is \textbf{[research program]}. The topological confinement and the derivation of $\sigma_\text{QCD}$ from $\sigma_\text{Pleno}$ are \textbf{[structural SLD]}.


\subsubsection{\texorpdfstring{III.5.9 --- Interpretation: what particles are under PIU-$\Psi$}{III.5.9 - Interpretation: what particles are under PIU-Psi}}

\textbf{Interpretation.} Particles are localized solitonic configurations of \emph{el Pleno}. The baryon is a tripartite Borromean cluster; the ``quarks'' are the three fractional vortices of the cluster, not independent fundamental entities; the fractional electric charges are topological charges of the vortices; the fermionic/bosonic statistics is a consequence of the topology of the soliton. There are no point-like elementary particles: there are configurations of the single medium.


\textbf{Explanation.} The $2+1$ structure of the baryon from the topological decomposition; the charges of the quarks from the $\mathbb{Z}_3$ modular arithmetic; the $\Delta$-N split from the rotational quantization; the proton radius from the structural balance; the stability of the proton from the Massey invariant. The program reduces the parameters of the hadronic sector from the two of the Skyrme model to a single scale $\Lambda_\text{PIU}$, potentially to zero if its cosmological derivation closes (\S{}V.3).


\textbf{Prediction.} The $\Delta$-N split at 264 MeV (90\% of the observed one), the proton radius consistent to 5\%, the complete baryonic rotational tower (P-Hadron-1), and the separation of the vortices equal to the baryon radius (P-Hadron-2), all falsifiable by refined scattering and hadronic spectroscopy.

\medskip
\textbf{The Dirac structure of the fermions.} The description of the particles as configurations of \emph{el Pleno} is completed with the \emph{form} of the equation that governs their fermionic modes. \S{}V.7.1 (thematic appendix) derives the structure of the Dirac equation ---the first-order operator, the Clifford algebra, the relativistic energy-momentum relation and the emergent spin-$\tfrac12$--- as the low-energy effective theory of the phase modes $\theta_-$ over the bipartite diamond lattice, with status \epi{SLD}; the spin-orbit coupling $\lambda_\text{SO}\neq 0$ that opens the Dirac cone is forced by the same $SU(2)$ holonomy that sustains Yang-Mills (\S{}III.4.5). What remains as a thread is not the structure of the operator but the \emph{value} of $\lambda_\text{SO}$ and of the fermionic masses, reconnected with the threads of masses and chirality (\S{}III.4.7).


\subsection{\S{}III.6 --- Cosmology as the evolution 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 (complete quantitative dynamics: $H(z)$, CMB, BAO, growth, $S_8$); strict uniqueness of the diamond lattice (thread D1).
  \item \textbf{Notes:} cosmic history as the evolution of the state of \emph{el Pleno}. Central prediction $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ (0.86$\sigma$). Scope: complete dynamics declared open.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{III.6.1 --- The question: what is cosmic history under PIU-$\Psi$}{III.6.1 - The question: what is cosmic history under PIU-Psi}}

The history of the universe is the evolution of the state of the single \emph{el Pleno} over cosmic time, through the three cosmological regimes constructed in \S{}II.6 ---saturated primordial (with bounce instead of singularity), freeze-out and current relaxed. The difference with standard cosmology is ontological: under $\Lambda$CDM the universe is a stage that expands containing distinct material components whose proportions are fitted to observation; under PIU-$\Psi$ there is no stage nor separate components, but a single medium whose state evolves, of which the cosmological ``components'' are distinct configurations. Standard cosmology emerges as the effective description of the relaxed regime.


\subsubsection{\texorpdfstring{III.6.2 --- The cascade of sectors M$\to$MO$\to$EO}{III.6.2 - The cascade of sectors M->MO->EO}}

\textbf{Origin.} The cosmological evolution of \emph{el Pleno} produces, by conservation of the U(1) current, an obligatory cascade of three sectors, all configurations of the same medium.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{M (ordinary matter):} localized solitonic configurations of skyrmion type (topological class, fermionic statistics), \S{}III.5. Observed baryonic matter.
  \item \textbf{MO (dark matter):} configurations of Q-ball type (non-topological class, bosonic statistics) and geodesic-accumulative modes. Excitations without topological charge that do not produce an observable induced gauge sector ---which is why they are ``dark''.
  \item \textbf{EO (dark energy):} the \emph{Pleno}-ground state itself, in its state of minimum excitation, \S{}III.7.
\end{itemize}

\textbf{U(1) conservation.} The cascade M $\to$ MO $\to$ EO is a consequence of the conservation of the U(1) current (axiom A5): the flow of charge between sectors in the post-bounce regime is restricted by the conservation, fixing the structure of the cascade.


\textbf{Epistemic classification.} The existence of the three-sector cascade is \epi{DERIVED} as a consequence of the U(1) conservation plus the mode structure of \emph{el Pleno}.


\subsubsection{III.6.3 --- The dark-matter / baryonic-matter asymmetry}

\textbf{Origin.} The cosmological ratio between the densities of dark matter and baryonic matter is not a free parameter fitted to observation: it is fixed structurally by the discrete geometry of the substrate.


\textbf{Central result.}


\begin{equation}
\frac{\Omega_{MO}}{\Omega_M} = \sqrt{3}\,\pi \approx 5.4414 \tag{E37}
\end{equation}

\textbf{Three pillars of the derivation.}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Pillar 1 --- Thermodynamic weight proportional to area.} The relevant collective modes of M and MO in the saturated post-cascade regime are surface vibrational modes, with a density of states linear in frequency. The thermodynamic weight of each class is linear in its effective area: $\mathcal{Z}_i \propto A_i\cdot\phi$, where the factor $\phi$ depends only on the \emph{Pleno}-ground state and cancels in the ratio.
  \item \textbf{Pillar 2 --- Spinorial factor $1/2$.} The ratio of the invariant volumes of $SO(3)$ and $SU(2)$ is exactly $1/2$ (Peter-Weyl, $SO(3)\cong SU(2)/\mathbb{Z}_2$). The ordinary matter M (fermionic, lives in $SU(2)$) suffers the reduction $1/2$ when projecting to $SO(3)$ observables; the dark matter MO (scalar, lives in $SO(3)$) does not suffer it.
  \item \textbf{Pillar 3 --- Geometric assignment.} The defects with conserved topological charge (M, $B=1$) anchor to the local Voronoi cell of the diamond lattice (a tetrahedron, area $A_T = (8\sqrt{3}/3)R^2$); the defects without charge (MO, Q-balls) expand isotropically (a sphere, area $A_S = 4\pi R^2$).
\end{itemize}

\textbf{Assembly.}


\[
\frac{\Omega_{MO}}{\Omega_M} = \frac{W_{MO}}{W_M} = \frac{A_S\cdot 1}{A_T\cdot(1/2)} = \frac{2A_S}{A_T} = \frac{2\cdot 4\pi R^2}{(8\sqrt{3}/3)R^2} = \frac{8\pi}{8\sqrt{3}/3} = \frac{3\pi}{\sqrt{3}} = \sqrt{3}\,\pi
\]

\textbf{Dimensional verification.} Ratio of two areas (both $\propto R^2$) plus dimensionless factors: pure dimensionless result $\checkmark$.


\textbf{Numerical verification.} $\sqrt{3}\,\pi = 1.7321\times 3.1416 = 5.4414$.


\textbf{Comparison with observation.} Planck 2018 gives $\Omega_\text{dm}/\Omega_b = 5.375\pm 0.077$. The difference is $|5.4414 - 5.375|/5.375 = 1.23\%$, equivalent to $0.86\sigma$. Agreement to 98.8\% with zero free parameters and zero observational input. This contrasts with $\Lambda$CDM, where $\Omega_\text{dm}/\Omega_b$ is the ratio between two free parameters fitted independently.

\begin{tcolorbox}[derivado,title={What it is confronted against: the robustness hierarchy of the ``Planck values''}]
A caution that governs the whole cosmological confrontation of the program, in continuity with the principle of \S{}III.8.7. The ``Planck values'' are not all of the same type: there is a hierarchy of model dependence that must be respected when choosing the target of comparison.
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Almost model-independent:} the acoustic angle $\theta_*$ and the \emph{physical densities} $\omega_b=\Omega_b h^2$ and $\omega_\text{dm}=\Omega_\text{dm}h^2$, which are read robustly from the relative heights of the acoustic peaks, almost without assuming the late cosmology. Confronting against these is legitimate and solid.
\item \textbf{Derived parameters, model-dependent:} $H_0$, $\Omega_m$, $\Omega_\Lambda$, $\sigma_8$, which are \emph{outputs} of the six-parameter $\Lambda$CDM fit, not direct readings. Confronting against these imports the $\Lambda$CDM assumption through the back door.
\end{itemize}
This locates with precision the solidity of each confrontation of the program, and the reading has two distinct faces. The ratio $\Omega_{MO}/\Omega_M=\Omega_\text{dm}/\Omega_b=\omega_\text{dm}/\omega_b$ is \emph{robust}: the factor $h^2$ cancels in the ratio, so that it is compared against physical densities read from the peaks, not against derived parameters. And here is the \textbf{positive strength, which should not be minimized}: that a number derived from pure geometry of the substrate (\S{}III.6.2--III.6.4), obtained \emph{without looking at the data and without a single free parameter}, falls to $0.86\sigma$ of an observationally well-constrained quantity, is an improbable coincidence under the null hypothesis ---the content of the Bayes factor of \S{}III.8.9--- and therefore a \emph{genuine cross-validation}, not an artefact. That the target comes from a $\Lambda$CDM fit does not make it arbitrary: the peak heights fix it with little freedom, so that $\Lambda$CDM cannot choose it to coincide with PIU. To coincide with known physics by an independent route of one's own is the mark of a correct theory, not a weakness. The \emph{caution} operates at the other end of the hierarchy: $\Omega_{EO}=f_c$ and $\Omega_m=1-f_c$, \emph{without} the factor $h^2$, are of the derived type and partially share the model dependence of $\Omega_\Lambda$; their agreement to $0.30\sigma$ and $0.27\sigma$ is real and counts in favour of the program, but must be read knowing that the target is a fit parameter, not a raw observable. The rule, then, is not to distrust the coincidences ---they are the evidence--- but to \emph{grade their weight by the robustness of the target}: maximal where $h$ cancels and it is anchored in physical densities, declared where the target inherits the $\Lambda$CDM assumption. What is avoided is only the error of \emph{calibrating} the program against an interpretation; what is preserved is the success of \emph{coinciding} with it without having sought to.
\end{tcolorbox}


\textbf{Epistemic classification.} The ratio (E37) is purely \epi{DERIVED}: the three pillars rest on already-established structure (solitonic classes, U(1) conservation) plus canonical mathematical literature (Peter-Weyl, Peierls-Nabarro, Finkelstein-Rubinstein). The two fine ingredients of the computation are fixed structurally. The \emph{dominance of surface modes} follows from the divergence $V''_\text{topo}(S)\propto(S_{\max,\text{geom}}-S)^{-2}$ of the Gent potential (\S{}II.2), which confines the fluctuations of the modulus to a shell of thickness $\xi\to 0$ and makes the log-determinant scale with the area. The \emph{independence of the factor $\phi$} follows from $\phi$ being the density of surface vibrational states of the \emph{Pleno}-ground state ($v_s=c$, $\rho_\text{ground}=\rho_P S_{\min}^{2,\text{estr}}$, cutoff $1/\ell_P$), identical for M and MO since they are solitons of the same ground state, while what distinguishes them (topology $U(1)$, moduli group) is accounted for separately in the statistical and moduli factors; hence $\phi_{MO}/\phi_M = 1$ exactly.

\textbf{The equality $\phi_{MO}/\phi_M=1$ is topologically protected.} A fine sub-question remains: the two boundaries are not equally smooth ---the sphere is smooth everywhere, the tetrahedron concentrates its curvature at four vertices---, and corner modes might contribute more to $\ln\mathcal{Z}_M$ than the smooth surface does to $\ln\mathcal{Z}_{MO}$. The Weyl expansion of the log-determinant of surface fluctuations orders that contribution by orders,
\[
\ln\mathcal{Z}_i = a\,A_i + b\!\int_{\partial i}\!K\,dA + \dots\,,
\]
and the curvature term is a topological invariant: by Gauss-Bonnet, $\int K\,dA = 2\pi\chi$. The sphere and the surface of the tetrahedron are both topologically $S^2$, with Euler characteristic $\chi=2$, so that $\int K\,dA = 4\pi$ is \emph{identical} for the two ---the curvature is distributed differently, smoothly in one case and concentrated in the angular deficits of the four vertices in the other, but the invariant that weighs in $\ln\mathcal{Z}$ is the same---. The equality $\phi_{MO}/\phi_M=1$ therefore survives up to and including the curvature order, and not by fortunate cancellation but by topology. The states localised at the vertices, not captured by the smooth expansion, are specific to M but subdominant at $\mathcal{O}(1/A)$ and do not scale with the size of the defect: negligible in the large-defect regime of freeze-out. \epi{DERIVED}.


\textbf{What the asymmetry does not assert.} (E37) predicts the ratio $\Omega_{MO}/\Omega_M$, not the individual densities separately, nor $\Omega_{EO}$ (derived by an independent route in \S{}III.7).


\subsubsection{\texorpdfstring{III.6.4 --- The inevitability of the geometry: why $\surd$3$\pi$ and not another number}{III.6.4 - The inevitability of the geometry: why √3pi and not another number}}

\textbf{The problem.} The strongest objection against (E37) is not that the computation is incorrect, but that the input geometry could be chosen. The formula combines the area of a sphere and that of the Voronoi polyhedron of a tetrahedron. Is the sphere/tetrahedron choice obligatory, or is it one of several acceptable geometries that happens to reproduce the observed value?


\textbf{Where the question resides.} Ingredient (i) ---dark matter without topological charge expands isotropically $\to$ sphere--- is inevitable by definition of isotropy. Ingredient (ii) ---ordinary matter with topological charge anchors to the local Voronoi cell--- depends on which is the lattice of the substrate. All the force rests on how obligatory the lattice is to be the diamond one $Z=4$.


\textbf{The structural support (Part II).} The derivation that the substrate is the diamond lattice $Z=4$ is carried out in \S{}II.3.3--\S{}II.3.6 through four arguments from independent domains that converge: Maxwell isostaticity (mechanical rigidity $\to$ $Z=4$), condensation of the phase $\theta$ (field dynamics $\to$ A/B bipartition as breaking $U(1)\to\mathbb{Z}_2$), Finkelstein-Rubinstein (quantum topology $\to$ correct spin statistics), Russo-Tanaka (nucleation thermodynamics $\to$ cell doubling).


\textbf{Meaning for $\sqrt{3}\pi$.} The number $\sqrt{3}\pi$ is not free: its input geometry is sustained by four convergent arguments. It is not ``one geometry among many chosen to fit'', but the geometry that four independent criteria select. The probability that four arguments from distinct domains converge by chance on the same structure is negligible.


\textbf{Scope of the uniqueness.} The geometry of the diamond lattice is derived in its two pieces (\S{}II.3.7): the tetrahedral coordination $Z=4$ as a necessity of A1+Maxwell and the cubic stacking as the minimum of the isotropic flexural functional. The precise assertion is, therefore, that $\sqrt{3}\pi$ rests on a derived geometry, not on a motivated choice.


\textbf{The dynamical origin of the two geometries (answer to the apparent ansatz).} There remains the link that a reviewer would point to as the most exposed: why the cross-section of the \emph{uncharged} defect is the sphere and that of the \emph{charged} defect is the tetrahedral Voronoi cell. This is not a geometric ansatz imposed on the transition, but a consequence of F1 on each class of defect, and it follows from a single principle ---the boundary of a defect minimizes its surface energy--- applied under two distinct topological constraints:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
  \item \textbf{Uncharged defect (MO, $B=0$): the sphere is forced.} A scalar Q-ball has no topological anchoring to the lattice: its phase does not wind, its charge is Noether (global), not winding. Its boundary is free to relax, and the surface energy that F1 assigns to it ---the gradient term $\tfrac12\rho_P c^2\ell_P^2|\nabla S|^2$, manifestly isotropic--- is minimized at fixed volume by the \emph{sphere}, by the isoperimetric theorem. The spherical geometry is not postulated: it is the minimum of the surface functional of F1 in the absence of a constraint. $A_S = 4\pi R^2$ is \epi{DERIVED} from the isotropy of F1.
  \item \textbf{Charged defect (M, $B=1$): the tetrahedron is forced.} A soliton with $B=1$ has its phase $\theta$ wound (winding $=$ holonomy over the cell, \S{}II.6.2); that winding anchors the defect to the structure of the lattice ---it cannot slide or round off without unwinding the charge, which is forbidden by topological conservation. Its boundary does \emph{not} relax to the sphere: it stays pinned to the nodes of the substrate, and the minimal region that contains a node with its coordination $Z=4$ is exactly its Voronoi cell, the tetrahedron ($A_T = (8\sqrt3/3)R^2$). The anchoring to the cell is not a choice: it is the constraint that the topological winding imposes on the boundary, given the already derived lattice. $A_T$ is \epi{DERIVED} from the topological anchoring over the diamond lattice.
\end{itemize}
\textbf{Why the spherical area is exact in a medium without vacuum.} The isoperimetric theorem delivers the sphere as the boundary of the free defect, but in a medium with a grain the boundary is sampled over the bonds of the lattice, and one might expect a discretisation correction to the area. There is none: the coordination $Z=4$ of the diamond lattice is a \textbf{spherical 2-design}, that is, averaging over its four tetrahedral directions integrates the spherical harmonics exactly up to degree two, and the fourth-order anisotropy of the modulation is null for the cubic stacking (\S{}II.3.7-bis). The degree-zero area ---the surface itself--- is therefore integrated exactly with only four sampling points, and $A_S = 4\pi R^2$ holds with no lattice correction. That this happens is not an accident of the sphere but a property of the already derived lattice: the substrate is built so that an isotropic coherence boundary has an exactly spherical area without needing to invoke empty space between the nodes, which would be incompatible with A1. \epi{DERIVED}, reinforcement.

The sphere/tetrahedron asymmetry is, then, the asymmetry between global charge (free boundary $\to$ isoperimetric) and topological charge (anchored boundary $\to$ Voronoi), both read from F1. The Kibble-Zurek transition fixes \emph{how many} defects of each class are nucleated (the density), but not \emph{what shape} each one takes; the shape is fixed by the topological nature of the defect via F1, before and independently of the dynamics of the transition. The link that seemed an ansatz is the point where the topology of F1 selects the geometry.


\textbf{Quantitative statistical reinforcement.} The space of algebraic expressions of comparable complexity is computationally enumerated (up to complexity $N=5$: 3778 distinct expressions) and the probability that two of them reproduce simultaneously the two cosmological predictions ($\sqrt{3}\pi$ and $f_c$) within the $2\sigma$ window is computed. The result: probability less than $3\times 10^{-4}$, that is a factor $K_c \approx 6.7\times 10^5$ against reading those two ratios as lucky formulas. The scope of this enumeration is the one proper to it: it bounds the narrow version of the objection ---how many simple \emph{formulas} would graze a given value--- for the two cosmological ratios. The version proper to a complete programme ---how many \emph{frameworks} would reproduce the entire set of results--- is computed separately in \S{}V.2.6, with admission criteria of its own; the two figures answer different questions and are not combined.


\textbf{Epistemic classification.} The geometry that produces $\sqrt{3}\pi$ is \epi{DERIVED}: the diamond lattice is derived in its two pieces (\S{}II.3.7, coordination and stacking), with fourfold convergent robustness, and the enumeration of the space of formulas strongly disfavours reading the ratio as an algebraic coincidence ($K_c\approx 6.7\times 10^5$, \S{}III.6.5).

\subsubsection{III.6.5 --- The algebraic shielding of the ratio: the joint partition function}

\textbf{Origin.} The ratio (E37) could appear to be the heuristic product of three geometric constants. This subsection demonstrates that it is not: the ratio emerges from the controlled factorial cancellation of all the terms of the ratio of the partition functions of the two sectors.


\textbf{The partition function of each sector.} For each class ($M$ and $MO$) in the saturated post-cascade regime, the partition function is expanded semiclassically into five factors: ($\alpha$) classical action $e^{-S_E[\Psi^*]/\hbar}$, ($\beta$) normalization $\mathcal{N}_i$, ($\gamma$) moduli volume $\mathcal{V}_i^\text{moduli}$, ($\delta$) surface determinant $\det'_\text{surf}$, ($\varepsilon$) statistical factor $\chi^{(i)}_\text{FR}$.


\textbf{Cancellation factor by factor.}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{($\alpha$) classical action: cancels to 1.} Both sectors have the same saturation level (stabilized in the same \emph{Pleno}-ground state); their classical actions are equal.
  \item \textbf{($\beta$) normalization: cancels to 1.} With identical asymptotic boundary conditions, the asymptotic symmetry groups are identical.
  \item \textbf{($\gamma$) moduli volume: produces the factor $1/2$.} $\text{Vol}(SU(2)) = 2\pi^2$, $\text{Vol}(SO(3)) = \pi^2$. The ordinary matter suffers the reduction $1/2$ when projecting to observables, via Peter-Weyl: $\text{Vol}(SO(3))/\text{Vol}(SU(2)) = 1/2$.
  \item \textbf{($\delta$) surface determinant: produces the area dependence.} Under the dominance of surface modes (the Gent hard bound exponentially suppresses the internal fluctuations in the saturated regime), the log-determinant is proportional to the area of each sector: $\mathcal{Z}^\text{surf}_i \propto A_i$.
  \item \textbf{($\varepsilon$) Finkelstein-Rubinstein statistics: does not contribute to the cosmological ratio.} $\chi^{MO}_\text{FR}/\chi^M_\text{FR} = (+1)/(-1) = -1$ affects the quantum interpretation (M fermionic, MO bosonic), but not the mean densities: for these $|\chi_\text{FR}|^2 = 1$ is used in both.
\end{itemize}

\textbf{Final assembly.}


\[
\frac{\Omega_{MO}}{\Omega_M} = \frac{W_{MO}}{W_M} = \frac{A_S\cdot 1}{A_T\cdot(1/2)} = 2\cdot\frac{4\pi R^2}{(8\sqrt{3}/3)R^2} = \frac{3\pi}{\sqrt{3}} = \sqrt{3}\,\pi
\]

\textbf{Why this shields the result.} The ratio $\sqrt{3}\pi$ is not the heuristic product of three constants: it is the result of a controlled thermodynamic computation where two factors cancel exactly to one, one is irrelevant for the density, one produces the exact factor $1/2$ (Peter-Weyl), and one produces the area dependence (dominance of surface modes). Each factor has an explicit derivational origin.


\textbf{Epistemic classification.} The algebraic shielding is \epi{DERIVED} as a composite dynamical derivation. The dominance of surface modes and the independence of the universal function are structurally supported (divergence of $V''_\text{topo}$ of Gent and orthogonality of $\phi$ with respect to topology and moduli, \S{}III.6.3); only the transient regime of the cascade remains as a low-severity sub-caveat.


\subsubsection{III.6.6 --- The cosmological background as elastic dynamics of the medium}

\textbf{Origin.} The homogeneous and isotropic background of the universe is the stretching of the deformation map of el Pleno. There is no prior metric stage on which the field evolves: the cosmological metric is the reading that the excitations make of the deformation state of the medium (A1, \S{}III.1.7). The starting point is, therefore, the deformation map $X^a(x)$ itself, not a given FLRW geometry.

For the homogeneous and isotropic background, the deformation map is $X^0=\tau(t)$, $X^i=L(t)\,x^i$, with $L(t)$ the linear stretch of the medium. The components of the emergent metric (E39a, \S{}III.1.7) are $g_{00}=-\dot\tau^2$ and $g_{ij}=L^2\delta_{ij}$ ($c=1$). The natural variable of the solid is the volumetric deformation
\[
e \equiv \ln L, \qquad H \equiv \dot e = \frac{\dot L}{L}.
\]
Here $H$ is the volumetric deformation rate of a fixed number of cells of the lattice (A1), not the expansion rate of a stage that grows larger: no volume is created, a lattice of fixed cells is stretched.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- The Lagrangian of the background.} The modulus and the stretch are not independent: the continuity of the hyperelastic continuum ($\rho\,J=\text{const}$ with $J=L^3$ and $\rho=\rho_P S^2$, \S{}III.8.2) ties $S = S_0\,e^{-3e/2}$. The kinetic energy per reference cell and the potential energy are
\end{itemize}

\[
T = \tfrac12\rho_P\ell_P^2\cdot 3\dot e^2 = \tfrac32\rho_P\ell_P^2\dot e^2,
\qquad
V = V_\text{eff}\bigl(S(e)\bigr),
\]

where the factor $3$ is the three directions of the isotropic stretch and $V_\text{eff}$ is the complete effective potential of el Pleno (Klauder A2 $+$ Gent A6, \S{}III.1.6), with minimum verified at $S_{\min}=0.7231$.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 2 --- The background equation (exact Euler-Lagrange).} Applying Euler-Lagrange to $\mathcal{L}=T-V_\text{eff}(S(e))$ in the modulus (units $c=\ell_P=\rho_P=1$):
\end{itemize}

\[
\boxed{\;\ddot S = \frac{\dot S^2}{S} - \frac34\,S^2\,V'_\text{eff}(S)\;}
\]

It is a \emph{second-order} equation, that of a non-linear oscillator against a hard wall, to which the viscous damping $-\gamma\dot S$ with $\gamma=\eta/(\rho_P\ell_P^2)$ and $\eta=\mu_\text{Gent}\tau_\Pi$ (E-St-3, \S{}III.1.6, derived from the flexural sector via the Lee-Wick pole, not postulated) is added. The IR wall of A2 (Klauder, $V_q$ diverges toward the ground) makes the bounce a structural fact; the viscosity settles the modulus at $S_{\min}$, the DE-Curtain state ($w=-1$, \S{}III.7.5).

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 3 --- Density and pressure.} $\rho_\text{Pleno} = \tfrac12\rho_P\ell_P^2\dot{\bar S}^2 + V_\text{eff}(\bar S)$, $p_\text{Pleno} = \tfrac12\rho_P\ell_P^2\dot{\bar S}^2 - V_\text{eff}(\bar S)$.
  \item \textbf{Step 4 --- The kinetic constraint (Friedmann as the linear limit).} The Hamiltonian constraint of the elastic solid ---the same one that in General Relativity produces the Friedmann constraint by temporal reparametrization--- is
\end{itemize}

\[
\tfrac32\rho_P\ell_P^2\,H^2 = \rho_e
\;\Longrightarrow\;
H^2 = \frac23\frac{\rho_e}{\rho_P\ell_P^2} = \frac23\frac{G}{c^2}\rho_e,
\]

using the native identity $G=c^2/(\rho_P\ell_P^2)$ (E3). This has the \textbf{same structure} as the Friedmann equation $H^2=\tfrac{8\pi G}{3c^2}\rho_e$ ---reproduced as a theorem of the elastic solid. The geometric coefficient of the source-stretch coupling is the one that fixes the exact factor (the Sakharov thread, \S{}III.1.4); its value is a constant that is absorbed into the positional normalization of $H_0$ and does not affect the form of $H(z)$.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 5 --- The source and the coincidence with $\Lambda$CDM.} The physical source is $\rho_e=\rho_{m,0}L^{-3}+\rho_{\text{EO}}$: matter (solitons) dilutes as $L^{-3}$ per physical area ---the elastic tension is force per physical area, not dilution in a growing volume--- and the phase curtain is constant ($w=-1$, \S{}III.7.5).
\end{itemize}

\[
H^2(z) \;\propto\; \rho_{m,0}(1+z)^3 + \rho_{\text{EO}},
\]

which is, to linear order, the $\Lambda$CDM form. The coincidence with the standard model in the measurable form is the linear limit of an elastic dynamics with the same source, not an imposed identity: the background equation is second-order with non-linear walls, and its linear limit near the ground reproduces the first-order Friedmann constraint. The divergence from FLRW lives in the non-linear regime of the walls (\S{}III.8.9).

\textbf{The equation of state of dark energy.} The late cosmological sector is the phase condensate in its symmetry state. With $\dot\theta_0=0$, the energy-momentum tensor is $T_{\mu\nu}\propto g_{\mu\nu}$: the rest four-vector disappears and
\[
p = -\rho \;\Longrightarrow\; w = -1 \ \text{exact}.
\]
It is a symmetry of the ground state, not a slow-roll dynamics. The dark-energy density is $\rho_{\text{EO}}=V(S_{\min})=K\,S_{\min}^2$ (anchor $K=\rho_P c^2$ $\times$ wall $S_{\min}^2$), structurally constant (\S{}III.7.5). What grows with cosmic time is the \emph{relative weight} $\Omega_{\text{EO}}$, because matter dilutes, not the physical density $\rho_{\text{EO}}$.

\textbf{The DESI signal in $\Lambda$CDM observables.} The DESI observations (DR1 2024, DR2 2025~\cite{desi2024}), combined with BAO, supernovae and CMB, report a preference of $2.5\sigma$--$3.9\sigma$ for dynamical dark energy when fitted with the $w_0$--$w_a$ parametrization, with $w_0\approx-0.75$, $w_a\approx-1$. A medium with $w=-1$ exact whose $H(z)$ coincides with $\Lambda$CDM to linear order, but differs in the non-linear regime of the walls, produces exactly that type of signal when forced into the parametric mould $w_0$--$w_a$: the deviation is a property of the parametrization, not of the equation of state of the medium. The falsifiable prediction is that the dynamical signal is attenuated or disappears when measured with model-independent observables (raw $D_H/r_d$, $\theta_*$, the shape of $E(z)$), and that its residual ---if any--- tracks the non-linear correction of the walls, not a dynamical $w_0>-1$.

\textbf{Compatibility with constant $G$.} The evolution of the background $\bar S(t)$ does not affect the value of $G$: (E3) depends on the structural Planck scales $(\rho_P,\ell_P)$, not on the dynamical state of the field. This is consistent with the observational bounds on the constancy of $G$.

\textbf{Where it diverges from FLRW.} The coincidence with $\Lambda$CDM is the linear limit; the divergence lives in the non-linear regime of the walls, and has falsifiable consequences that separate the two descriptions in the origin and the destiny of the universe.

{\small
\begin{longtable}{L{3.3cm}L{4.6cm}L{4.9cm}}
\toprule
\textbf{Aspect} & \textbf{FLRW / $\Lambda$CDM} & \textbf{Native PIU (elastic)} \\
\midrule
Background equation & 1st order, $H^2=\tfrac{8\pi G}{3}\rho$ & 2\textsuperscript{nd} order, stretch EOM with complete $V_{\text{eff}}$ \\
Friction & geometric $3H$ (growing physical volume) & viscous $\eta=\mu_\text{Gent}\tau_\Pi$; can $\to0$ \\
observable $H(z)$ & postulated Friedmann integral & linear limit of the elastic EOM; diverges in the non-linear \\
Beginning & singularity (Big Bang) & bounce by wall A2, no singularity \\
Destiny & eternal de Sitter, $H\to H_\Lambda>0$ & rest at $S_{\min}$, $H\to0$ \\
``Expansion'' & creation of space (stage) & stretching of a fixed number of cells \\
\bottomrule
\end{longtable}}

\noindent The two descriptions coincide in the measurable form today ---that is why $\Lambda$CDM works--- but design opposite origins and futures: in FLRW a stage grows without end from a singularity; in F1 a modulus bounded between walls bounces without singularity and settles into its ground. \epi{DERIVED} as falsifiable structural distinctions.

\textbf{Epistemic classification.} The second-order background equation, the bounce by A2 and the settling at $S_{\min}$ are \epi{DERIVED}. The structure $H^2\propto(G/c^2)\rho_e$ of the kinetic constraint is \epi{DERIVED}; the exact geometric coefficient is \epi{OPEN END} (Sakharov coupling, \S{}III.1.4), irrelevant for the form. The equation of state $w=-1$ exact is \epi{DERIVED} (symmetry of the ground state, \S{}III.7.5). The reading of the DESI signal as an artefact of the $w_0$--$w_a$ parametrization of a medium with $w=-1$ exact is \epi{SLD}, with a falsifiable prediction of dissolution in raw observables. The compatibility with constant $G$ is \epi{DERIVED}.


\subsubsection{III.6.7 --- The division between the closed background and the open perturbations}

The cosmology of the program divides into two sectors with distinct status. The \emph{background} sector is closed (\S{}III.8): the functional form of $H(z)$ is forced by the effective potential, with a single observable freedom, and fits the DESI DR2 BAO data as well as the standard model with less freedom (\S{}III.8.6); the matter density $\Omega_m = 1 - f_c \approx 0.3131$ is derived without fitting, at $0.3\sigma$ from Planck (\S{}III.8.5); the nearly scale-invariant spectrum without inflation, the perfect black-body character of the CMB, the red tilt and the native resolution of the BKL problem are established in \S{}III.8.2--\S{}III.8.4 and \S{}III.8.8.

In the background, the equation of state of the late sector is $w=-1$ exact: the phase condensate at $S_{\min}$ has $T_{\mu\nu}\propto g_{\mu\nu}$ (\S{}III.7.5). The physical density $\rho_{\text{EO}}=K\,S_{\min}^2$ is structurally constant; what evolves is the relative weight $\Omega_{\text{EO}}$, by dilution of matter. The radial modulus does not participate in the late evolution: it is ultra-Planckian ($m_S\approx4.44\,E_P$, \S{}III.1.9), frozen at $S_{\min}$ since crystallization, and the scales built on it (soliton masses, $S_{\min}$ of the defects) are stable, which confirms it observationally.

What remains open, delimited with precision, is: (1) the non-linear correction of $H(z)$ at accessible $z$ ---whose origin lies in the shape of the potential with walls, but whose dynamical carrier is the phase sector, not the oscillation of the modulus, frozen since crystallization---, and whether it is measurable with raw BAO $D_H/r_d$ (\S{}III.8.9); (2) the linear growth of structure and the $S_8$ tension, where the ponderomotive drag of matter by the phase perturbations $\delta\theta$ (\S{}III.7.5-bis) shifts $\sigma_8$ with respect to $\Lambda$CDM without requiring background dynamics; and (3) the numerical simulations of el Pleno.

\textbf{Falsifiability criteria.} The sector is falsifiable thus: (1) if the non-linear correction of $H(z)$ has a sign or magnitude incompatible with the cosmic chronometers and the raw BAO; (2) if the $w_0$--$w_a$ signal of DESI \emph{persists} in model-independent observables instead of dissolving (\S{}III.8.7); (3) if the predicted shift of $S_8$ by the phase drag contradicts the weak-lensing data; (4) if the values of $S_{\min}$ inferred from BBN, CMB and BAO turn out to be mutually inconsistent.

\textbf{Epistemic classification.} The background sector (form of $H(z)$, $\Omega_m$, black-body CMB, nearly scale-invariant spectrum, $w=-1$ exact) is \epi{DERIVED} or \epi{SLD} (\S{}III.8). The non-linear correction of $H(z)$ and the shift of $S_8$ by phase drag are \epi{[open end]}: physics identified, computation pending.

\subsubsection{\texorpdfstring{III.6.8 --- Interpretation: what cosmology is under PIU-$\Psi$}{III.6.8 - Interpretation: what cosmology is under PIU-Psi}}

\textbf{Interpretation.} The history of the universe is the evolution of the state of the single \emph{el Pleno}: from the saturated primordial regime (with bounce instead of singularity), through the freeze-out of constants, to the current relaxed regime. The cosmological ``components'' are not distinct substances: they are configurations of the same medium.


\textbf{Explanation.} The cascade of sectors from the U(1) conservation; the dark-matter/baryonic ratio from the discrete geometry (E37); the emergence of $\Lambda$CDM from the dynamics of the relaxed regime; the absence of an initial singularity from the A6 hard bound.


\textbf{Prediction.} Two static cosmological observables with sub-$\sigma$ precision ---the ratio $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ (0.86$\sigma$, entirely derived) and the fraction $\Omega_{EO} = f_c$ (0.30$\sigma$, \S{}III.7, with derived structure and one calibrated factor)--- over a quadruply-supported geometry. It also predicts the primordial bounce instead of the singular Big Bang, and the correct direction of the dark-energy dynamics ($w_0 > -1$, $w_a < 0$) consistent with DESI. The complete quantitative cosmological dynamics is the declared open thread of the program (\S{}III.6.7).


\subsection{\S{}III.7 --- Dark sectors: the \emph{Pleno}-ground state and its excited configurations}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} dynamics of $w(z)$ and trajectory $\bar S(t)$ (thread F-cosmo, \S{}V.4) ---of which the present value $S_{\max,\text{eff}}^2(t_0)$, the numerator of $f_c$, is the evaluation at the current epoch, not an autonomous thread.
  \item \textbf{Notes:} dark energy as the \emph{Pleno}-ground state (DE-Curtain); dark matter as an excited configuration without gauge. $\Omega_{EO} = f_c \approx 0.6869$ with calibrated numerator (status declared explicitly).
\end{itemize}
\end{estadoseccion}

\subsubsection{III.7.1 --- The question: what are dark matter and dark energy}

Problem 2 (\S{}I.2.3) is that 95\% of the energy content ---dark matter ($\sim$27\%) and dark energy ($\sim$68\%)--- is described by categories of unknown origin, and Problem 1 (\S{}I.2.2) is the discrepancy of 60--120 orders between the theoretical vacuum energy and the observed cosmological constant. The PIU-$\Psi$ answer is direct: the dark sectors are not additional hidden substances, but the same medium in distinct states ---dark matter, excited configurations of \emph{el Pleno} without an observable gauge sector; dark energy, the \emph{Pleno}-ground state itself.


\subsubsection{III.7.2 --- Dark energy as the structural state of the \emph{Pleno}-ground state (DE-Curtain)}

\textbf{Origin.} Part II (\S{}II.7.6) established the DE-Curtain reading: the observed dark energy is the permanent structural state of the \emph{Pleno}-ground state ---the physical medium in its configuration of minimum excitation, present in the entire cosmological volume.


\textbf{The active condensate fraction (derived in \S{}II.7.6).} Part II derived the active condensate fraction of \emph{el Pleno} as the ratio of two bounds of the modulus:


\[
f_c \equiv \frac{S_{\max,\text{eff}}^2}{S^2_{\max,\text{geom}}}
\]

where, in accordance with the constitutive identification of \S{}II.2.2 ($S^2 = $ relative number density of quanta in the coherent mode), $f_c$ is the fraction of Planckian quanta that remain in the condensed mode, against the total that the geometry of the diamond lattice could house. The denominator $S^2_{\max,\text{geom}} = (8/3)\sqrt{2/3} = 2.1773$ is derived from first principles (\S{}II.4.1); the numerator $S_{\max,\text{eff}}^2 = 1.4956$ is calibrated to the current regime (\S{}II.4.2).


\textbf{Phenomenological reading.} By the DE-Curtain reading, dark energy is the \emph{Pleno}-ground state in its coherent mode (\S{}III.7.5), while matter (ordinary and dark) are the depleted excitations of that condensate. By conservation, $f_c + (\text{depleted fraction}) = 1$. The cosmological identification is:


\[
\Omega_{EO}\equiv f_c, \qquad \Omega_M + \Omega_{MO} = 1 - f_c
\]

\textbf{Central result (structural cosmological prediction).}


\[
\Omega_{EO}\equiv f_c = \frac{S_{\max,\text{eff}}^2}{S^2_{\max,\text{geom}}} = \frac{1.4956}{2.1773} \approx 0.6869 \qquad(\text{see }E33)
\]

\textbf{Numerical verification.} $f_c = 1.4956/2.1773 = 0.68690$.


\textbf{Comparison with observation.} Planck 2018 gives $\Omega_{EO}^\text{obs} = 0.6847\pm 0.0073$: the difference is $|0.6869 - 0.6847|/0.6847 = 0.32\%$, equivalent to $0.30\sigma$. Agreement to 99.7\%.


\textbf{Status of the two factors of the ratio.} The two terms of the ratio have distinct epistemic status:


\begin{itemize}[leftmargin=1.4em]
  \item $S^2_{\max,\text{geom}} = (8/3)\sqrt{2/3} = 2.1773$ is \epi{DERIVED} from first principles: it is the maximum geometric packing of the saturated phase of \emph{el Pleno} (Kepler-Hales family, \S{}II.4.1), without fitting.
  \item $S_{\max,\text{eff}}^2 = 1.4956$ is \epi{CALIBRATED} to the current regime with a declared error bar: it is the effective occupation level of the coherent mode in the present cosmological epoch (\S{}II.4.2), but its closed form from first principles is not yet derived.
\end{itemize}

Consequently, the correct assertion is not ``zero free parameters'' without nuance, but this more precise one: \textbf{the structure of the prediction} ---that $\Omega_{EO}$ is the ratio of the effective occupation over the maximum geometric capacity--- \textbf{is derived with zero cosmological fitting parameters} (nothing is fitted against the observed value of $\Omega_{EO}$); but the numerical value inherits the uncertainty of $S_{\max,\text{eff}}^2$, an effective parameter not yet reduced to closed form. The prediction is genuine (not circular: $S_{\max,\text{eff}}^2$ is calibrated to the physics of the current regime, not to the value of $\Omega_{EO}$), but its status is weaker than that of the ratio $\sqrt{3}\pi$ (which is entirely derived).


\textbf{Why the structure of the ratio is inevitable, not chosen.} The form of the prediction follows necessarily from three already-established elements: first, the constitutive identification fixes that $S^2$ is the number density of quanta, without freedom; second, the DE-Curtain reading (\S{}III.7.5, where $w=-1$ is derived) fixes that dark energy is the coherent condensate, not a substance; third, the conservation fixes that the condensate fraction plus the depleted one sum to one. Given these three, the only possible expression for $\Omega_{EO}$ is the ratio of occupations. The structure is forced; what remains open is the dynamical trajectory $\bar S(t)$ that fixes the present value of the numerator (thread F-cosmo).


\textbf{Physical interpretation.} Ordinary matter (M) and dark matter (MO) are the depleted excitations of the condensate: $1 - f_c = 0.3131$, which coincides with the observed sum $\Omega_m = 0.315$ (Planck 2018) to $0.3\sigma$. This translates the cosmological constant problem ---a discrepancy of \textasciitilde{}120 orders of magnitude--- into a tractable question: why is the active condensate fraction \textasciitilde{}0.69? The structural answer (ratio of occupations) replaces a discrepancy of 120 orders by a fraction of order one with a direct geometric interpretation. This does not resolve the crude dimensional problem completely ---the derivation of the trajectory $\bar S(t)$ that fixes $S_{\max,\text{eff}}^2(t_0)$ (thread F-cosmo) would close it---, but it reformulates it in tractable terms.


\textbf{Dimensional verification.} $f_c$ is the ratio of two dimensionless quantities: a pure number $\checkmark$.


\textbf{Epistemic classification.} The identification $\Omega_{EO}\equiv f_c$ and the structure of the ratio are \epi{DERIVED} (forced by the constitutive identification, the DE-Curtain reading and the conservation). The numerical value $f_c\approx 0.6869$ is \textbf{[SLD with one calibrated effective parameter]}: the denominator is derived, the numerator $S_{\max,\text{eff}}^2$ is calibrated to the current regime (not to the value of $\Omega_{EO}$, which is why the prediction is not circular). The calibration of this numerator is legitimate and of structural character, not a provisional shortcut: $S_{\max,\text{eff}}^2$ is not an atemporal constant but the present value of a \emph{dynamical state quantity} ---the effective occupation of the coherent mode in the current epoch (\S{}II.4.2), with $\dot{\bar S}(t_0)\neq 0$ in general (\S{}III.6.6)---, analogous in status to $H_0$, which is also not derivable as an atemporal number but as the present value of a trajectory. Its first-principles determination is not, therefore, an autonomous computational thread, but the evaluation at $t_0$ of the trajectory $\bar S(t)$ subsumed in the thread F-cosmo (\S{}V.4). This honest classification is weaker than that of $\sqrt{3}\pi$ and is declared as such.

\medskip

\textbf{Precise scope of the thread F-cosmo with respect to $f_c$.} F-cosmo can derive the \textbf{trajectory} $f_c(t)$ and the associated dynamics ---the equation of state $w(z)$ and the connection $df_c/dt \to w_a$--- by integration of the effective Friedmann equation of the relaxed regime (\S{}III.6.6),
\[
\ddot{\bar S} + 3H\dot{\bar S} + \frac{1}{\rho_P\ell_P^2}V_\text{eff}'(\bar S) = 0 .
\]
What F-cosmo \emph{cannot} derive is the \textbf{present value} $f_c(t_0) = 0.687$ as an atemporal number: that value is positional ---it encodes at which point of the cosmic re-condensation clock (\S{}III.8.7) the universe is found today---, and depends on the initial condition of the bounce, a datum of the same class as $H_0$. The numerator $S_{\max,\text{eff}}^2(t_0)$ is, consequently, irreducibly \epi{CALIBRATED} in its \emph{value}, although its \emph{trajectory} is derivable. The correct assertion is not ``F-cosmo would close the numerator'' but ``F-cosmo would derive the trajectory of which the numerator is the present value, and would verify that $0.687$ is consistent with that trajectory given the observed $w(z)$''. This is strictly analogous to cosmology deriving $H(z)$ but not $H_0$: the curve is theory, the point on the curve is historical datum. The decomposition of \S{}II.4.2 (derivable well plus positional turning point) is the microscopic counterpart of this same distinction.


\subsubsection{III.7.3 --- Dark matter as an excited configuration of \emph{el Pleno}}

\textbf{Origin.} Dark matter is identified with excited configurations of \emph{el Pleno} ---of Q-ball and geodesic-accumulative type--- that do not produce an observable induced gauge sector.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 1 --- Matter density from the modulus.} The matter density is the local excess of the modulus over the background:
\end{itemize}

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

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Step 2 --- Dark-matter / dark-energy unification.} Both sectors are governed by the same parameter $S_{\min}$: dark energy is the density of the background ($\rho_\text{DE} = \rho_P S^2_{\min}$ in the relaxed regime), and dark matter is the local excess in excited configurations without observable topological charge. They are two regimes of the same \emph{el Pleno}.
  \item \textbf{Step 3 --- Why it is ``dark''.} The dark-matter configurations are excitations without topological charge that generates the induced gauge sector (\S{}III.4). Not coupling to the electromagnetic sector, they do not emit, absorb or reflect light: they are gravitationally active but electromagnetically inert. This explains structurally why dark matter is detected only gravitationally.
\end{itemize}

\textbf{Epistemic classification.} The identification of dark matter with excited configurations of el Pleno is \epi{DERIVED} structurally. The quantitative ratio $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ is derived in \S{}III.6.3.

\paragraph*{The two phases: the one that rotates and the one that undulates.} The field is $\Psi=S\,e^{i\theta}$, with a single phase $\theta$ that appears in two distinct projective roles, and distinguishing them clarifies why dark matter is transparent to light. The \textbf{phase that undulates is light}: the photon is the wave $\delta\theta$ projected onto the gauge sector $U(1)_\text{em}$ (A/B bipartition, \S{}III.4.2), a massless Goldstone that travels at $c$, whose charge is the gauge (electric) charge ---only the variation of the phase costs energy, not its value. The \textbf{phase that rotates is the Q-ball}: dark matter is a soliton whose internal phase $\theta$ rotates with frequency $\omega$, and that rotation \emph{is} its Noether charge $Q=\int S^2\dot\theta$ (\S{}II.5), a global $U(1)$ charge (A5), not the gauge charge of the projected sector. Dark matter does not have its phase frozen ---on the contrary, its internal phase rotates, and that rotation is its charge; what it does not have is topological charge (winding), which is why it is non-topological, and what makes it dark is that its phase charge is global, not the gauge charge that couples to the photon.

From here, three facts. Dark matter neither emits nor absorbs light: its phase rotation is a global charge, without coupling to the photon ---it is transparent, and \emph{that} is what makes it dark, absence of gauge coupling, not a frozen phase. Light traverses a region with dark matter: light is a wave $\delta\theta$ of the same el Pleno in which the Q-ball is a configuration of the modulus $S$, and it traverses it like any region of the medium, with the caveat that the mass of the Q-ball curves the emergent metric (gravitational lensing), deflecting light without absorbing it. And light does not need dark matter: it exists as $\delta\theta$ over the background with or without it. That the phase curtain is at $S_{\min}$ ($\dot\theta_0=0$, $w=-1$) does not obstruct light: a still and homogeneous background is the optimal medium for the waves $\delta\theta$ to travel massless (the potential does not affect the phase mode, D1) or dispersion-free at zeroth order, at $c$. Dark matter is transparent to light (it neither emits nor absorbs it) but curves it (it gravitates); its darkness is absence of gauge coupling. \epi{DERIVED} (the distinction of phases and the absence of gauge coupling follow from the A/B bipartition of \S{}III.4.2 and from the classification of solitons of \S{}II.5).


\subsubsection{III.7.4 --- Three arguments in favor of the DE-Curtain reading}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Argument 1 --- Empirical by reductio.} If dark energy were a material component attractable by gravity, observations would detect local concentrations around massive compact objects. This is not observed: the extreme observational uniformity of dark energy (Planck 2018, DESI 2024-2025) is exactly what DE-Curtain predicts.
  \item \textbf{Argument 2 --- Structural.} The matter-density formula establishes that baryonic matter is the perturbation over the background $S_{\min}^\text{estr}$: under PIU-$\Psi$, dark energy is that structural background of the medium, with density and equation of state derived from $S_{\min}^\text{estr}$, not a material component with its own dynamics.
  \item \textbf{Argument 3 --- Equation of state.} The observed dark energy has $w\approx -1$, which produces an energy-momentum tensor proportional to the metric ---the structural signature of a contribution of the physical medium, not of a material component with its own dynamics.
  \item \textbf{Argument 4 --- The active gravitating mass in the cosmological source.} This argument answers the underlying question ---\emph{why dark energy, present in the entire volume with real energy density, does not add to the gravitating mass that produces attraction as matter does}--- using exactly the equation that the program already derives, without importing additional machinery. In the cosmological acceleration equation of the program (\S{}III.7.5, obtained from the effective Friedmann equations of \emph{el Pleno}),
  \[
  \frac{\ddot a}{a} = -\frac{4\pi G}{3}\sum_i \big(\rho_i + 3p_i/c^2\big) = -\frac{4\pi G}{3}\sum_i \rho_i\,(1 + 3w_i),
  \]
  the source is not the energy density $\rho$ but the combination $\rho + 3p/c^2 = \rho\,(1+3w)$: the pressure, not only the density, enters the gravitational source, and it enters with its sign. From here the contrast between the two dark sectors, read \emph{in this equation} and not in an assertion about local forces:
  \begin{itemize}[leftmargin=1.2em]
    \item \textbf{Matter} (baryonic and dark), $w\approx 0$ (local excess of the modulus without appreciable pressure, \S{}III.7.3): contributes $\rho_i(1+0) = +\rho_i$ to the source. It sums with positive sign ---it slows the expansion and, in the regime of overdensities, is the mass inferred from rotation curves and gravitational lensing.
    \item \textbf{Dark energy}, $w = -1$: contributes $\rho_\text{EO}(1-3) = -2\rho_\text{EO}$ to the source. It enters with \emph{negative} sign: its energy density is real and fills the volume, but its negative pressure $p_\text{EO} = -\rho_\text{EO}c^2$ reverses the sign of its contribution to the source, so that it does not add to the attractive gravitating mass but produces the positive-acceleration term.
  \end{itemize}
  The honest content of the argument is therefore cosmological and is backed by the derived equation: in the Friedmann source, matter enters with $(1+3w)>0$ and dark energy with $(1+3w)<0$. It is \emph{not} asserted here that dark energy exerts a ``local repulsive force'': an exactly uniform background produces no potential gradient and, therefore, exerts no net force on a local body ---its effect is global, on the dynamics of the scale factor. What the argument establishes, with the mathematics that the program possesses, is the precise distinction between the two sectors in the \emph{cosmological gravitational source}: dark matter adds to the gravitating mass (hence its detection by rotation curves and lensing), and dark energy, by its equation of state $w=-1$, does not add to that attractive mass but contributes with the opposite sign to the expansion. The complete local derivation of the pressure term in the emergent gravity of \emph{el Pleno} ---that is, showing that the induced action of \S{}III.1 reproduces the $\rho+3p/c^2$ of general relativity also outside the cosmological regime--- is not carried out in this manuscript and is declared a theory thread: the argument rests on the cosmological acceleration equation, which the program does derive, and not on a local relativistic Poisson equation, which the program has not derived for its own framework.
\end{itemize}

\textbf{Epistemic classification.} Argument 1 is empirical; 2 is \epi{DERIVED} from the density formula; 3 is \epi{SLD} from the tensor structure; 4 is \epi{DERIVED} in the cosmological regime ---it rests on the acceleration equation $\ddot a/a = -\tfrac{4\pi G}{3}\sum_i\rho_i(1+3w_i)$, which the program obtains from its effective Friedmann equations (\S{}III.7.5, Step 3, with $G$ derived and the contributions of \emph{el Pleno} in the energy-momentum tensor), so that the opposite sign of the two sectors follows directly from their already-established equations of state ($w\approx 0$ for matter, $w=-1$ for dark energy). The local derivation of the pressure term in the emergent gravity of \emph{el Pleno} (showing that the induced action of \S{}III.1 reproduces $\rho+3p/c^2$ outside the cosmological regime) remains explicitly a \textbf{theory thread}; for this reason the argument does not extend to assertions about local forces. The four arguments are convergent and independent: the empirical reductio, the structure of the density formula, the form of the tensor, and the active gravitating mass in the cosmological source.


\subsubsection{\texorpdfstring{III.7.5 --- The equation of state w = $-$1 and the structural form of dark energy}{III.7.5 - The equation of state w = -1 and the structural form of dark energy}}

\textbf{Structural form of the dark-energy density.} The energy density of the \emph{Pleno}-ground state includes a potential and a kinetic contribution:


\[
\rho_\text{DE}(t) = V(\bar S(t)) + \tfrac{1}{2}\rho_P\ell_P^2\dot{\bar S}(t)^2
\]

In the current relaxed regime, the field is at its minimum and does not evolve ($\dot{\bar S}\to 0$), reducing to the potential contribution:


\[
\rho_\text{DE}^{(0)} = V(S_{\min}^\text{estr}) = \rho_P(S_{\min}^\text{estr})^2
\]

\textbf{Derivation of $w = -1$.} For a perfect fluid with four-velocity $u_\mu$, $T^\text{(fluid)}_{\mu\nu} = (\rho+p)u_\mu u_\nu - p\,g_{\mu\nu}$. With $p = -\rho$ (that is, $w=-1$):


\[
T^{(w=-1)}_{\mu\nu} = (\rho-\rho)u_\mu u_\nu + \rho\,g_{\mu\nu} = \rho\,g_{\mu\nu}
\]

The term $(\rho+p)u_\mu u_\nu$ vanishes exactly when $w=-1$: the four-velocity disappears from the energy-momentum tensor. This means that dark energy has no dynamical rest frame ---it does not move, does not flow, does not accumulate. Its tensor is proportional to the metric, $T^\text{(EO)}_{\mu\nu} = \rho_\text{EO}\,g_{\mu\nu}$, the signature of a homogeneous structural property of the medium.


\textbf{Interpretation.} A fluid with exact $w=-1$ is structurally indistinguishable from a geometric property of the medium. DE-Curtain is the ontologically most parsimonious interpretation of $T^\text{(EO)}_{\mu\nu}\propto g_{\mu\nu}$. Dark energy does not push from a rest frame: it is an isotropic and homogeneous property of the medium by mathematical construction. This reading is reinforced independently from thermodynamics: since $w=-1$ implies $\rho+p=0$, the volumetric entropy density of the relaxed \emph{Pleno}-ground state is null ($s=0$), confirming that the vacuum is a coherent condensate of minimum entropy ---an elastic solid, not a dynamical fluid--- as developed in \S{}III.9.6.


\textbf{The accelerated expansion.} Substituting $w=-1$ into the acceleration equation:


\[
\frac{\ddot a}{a} = -\frac{4\pi G}{3}\rho(1+3w) = -\frac{4\pi G}{3}\rho(1-3) = +\frac{8\pi G}{3}\rho > 0
\]

The acceleration is positive: dark energy produces accelerated expansion. The ``cosmological expansion'' is, under DE-Curtain, the dispersion of the solitons (matter) over the invariant background of the \emph{Pleno}-ground state.


\textbf{Prediction about $w(z)$.} If in the pre-relaxed regime the field still evolves ($\dot{\bar S}\neq 0$), the kinetic contribution is non-null and $w(z)\neq -1$ for $z > 0$. Any deviation that DESI detects would be testimony of the slow-roll of \emph{el Pleno} toward its minimum, not of dark energy as a dynamical entity. The quantitative computation of $w(z)$ is an open thread (\S{}V.4).


\textbf{Structural parametrization of the cosmological sector: $\kappa$ and $\delta$.} The cosmological sector of \emph{el Pleno} is described by two structural parameters, neither freely fitted. The first comes from the gravitational side, the second from the dynamical side.

The \textbf{gravitational coupling} $\kappa$ is the coefficient that the Sakharov induced action (\S{}III.1.4) identifies with the Einstein-Hilbert term, that is, the gravitational rigidity of the medium read from the Hamiltonian of F1:


\begin{equation}
\kappa \;\equiv\; \frac{c^4}{2G} \;=\; \tfrac{1}{2}\rho_P c^2 \ell_P^2 \;\approx\; 6.0519\times 10^{43}\ \text{N},
\end{equation}

identical to the kinetic coefficient $K_\text{cin}$ of \S{}III.1.6. Its value follows from the primary parameters without fitting; $\kappa$ is \epi{DERIVED} from the Hamiltonian.

The \textbf{dynamical parameter} $\delta$ is the ratio, at the moment of the computation, between the kinetic and the potential contribution of the \emph{Pleno}-ground state:


\begin{equation}
\delta(t) \;\equiv\; \frac{\tfrac{1}{2}\rho_P \ell_P^2 \dot{\bar S}(t)^2}{V(\bar S(t))}\,, \qquad \delta(t)\geq 0\,.
\end{equation}

Substituting into $\rho_\text{DE}$ and $P_\text{DE}$, the equation of state takes the closed form


\begin{equation}
\boxed{\;w(t) = \frac{\delta(t) - 1}{\delta(t) + 1}\;}
\end{equation}

This is the \textbf{pointwise state relation} of the sector: given the instantaneous value $\delta(t)$ that the \emph{Pleno}-ground state has under the conditions of that epoch, it fixes the $w$ of that instant. It is not a constant $w$ of the program: $\delta(t)$ is a dynamical state quantity, whose trajectory is determined by the EOM of \emph{el Pleno} integrated under FLRW ---with the $U(1)$ conservation of Axiom A5 governing the phase dynamics--- under the conditions (density, expansion rate) of each epoch. Consequently $w(z)$ is a dynamical curve read point by point: $\delta\to 0$ in the relaxed regime returns $w=-1$ (DE-Curtain, recovered as an asymptotic particular case), and $\delta>0$ outside it gives $w>-1$.

Unlike generic quintessence ---where the potential $V(S)$ is postulated with freedom to fit $w(z)$---, in PIU-$\Psi$ the effective potential $V_\text{eff} = W_\text{bare} + V_q + V_\text{topo}$ is \emph{derived} with fixed parameters (\S{}I.7.6, \S{}II.2). Therefore $\delta(t)$ is not fitted: it is fixed by the dynamics of the medium itself. This rigidity is what endows the sector with falsifiable content.

\textbf{Pointwise reading of the canonical-kinetic regime.} At every instant in which the energy of the \emph{Pleno}-ground state is of canonical kinetic type (positive sign of the term $\dot{\bar S}^2$), $\delta(t)\geq 0$ implies $w(t)\geq -1$: that point does not admit a phantom equation of state. This statement is \textbf{a pointwise reading conditioned on the regime}, not a global veto of the program: in epochs whose dynamical conditions activate the higher-derivative terms of F1 (flexural $(\Box S)^2$, Klauder wall), the effective kinetic energy may cease to be canonical and the pointwise result for $w$ must be recomputed with those terms included ---territory of the Ostrogradsky analysis of Annex I. The statement $w\geq -1$ is, then, \textbf{[strong logical deduction]} valid under canonical kinetics, and the determination of in which epochs that regime holds is part of the thread F-cosmo (\S{}V.4). Its confrontation with the possible dynamical dark energy of DESI 2024--2025 is an active falsifiability frontier of the program.


\textbf{Dimensional verification.} $[\rho_\text{DE}] = [\rho_P(S_{\min})^2] = [\rho_P] = [kg/m^3]$ $\checkmark$. For the structural parameters: $[\kappa] = [c^4/G] = [\text{N}]$ and $\delta$ is dimensionless by construction (ratio of energy densities) $\checkmark$.


\textbf{Epistemic classification.} The structural form of $\rho_\text{DE}$ is \epi{DERIVED}. The derivation of exact $w=-1$ in the ground state (cancellation of the kinetic term, $T_{\mu\nu}\propto g_{\mu\nu}$) is \epi{DERIVED} (exact algebraic consequence). The gravitational coupling $\kappa = c^4/(2G)$ is \epi{DERIVED} from the Hamiltonian; the pointwise state relation $w=(\delta-1)/(\delta+1)$ with $\delta=0$ in the background is \epi{DERIVED}. The veto on dynamical $\delta>0$ (the phantom region $w<-1$ and the $L^{-3}$ matter scaling of a rolling mode, both excluded as an origin of $w=-1$) is \epi{DERIVED}. The reading of the DESI $w_0$--$w_a$ signal as an artefact of parametrising a medium with exact $w=-1$ is \epi{SLD}, with a falsifiable prediction of dissolution in raw observables.


\subsubsection{\texorpdfstring{III.7.5-bis --- The ponderomotive drag of matter by the dark-energy wave}{III.7.5-bis - The ponderomotive drag of matter by the dark-energy wave}}

The DE-Curtain (\S{}III.7.5) is the homogeneous component $\theta_0$ of the phase, with $w=-1$ structural and no dynamical rest frame ($T_{\mu\nu}=\rho\,g_{\mu\nu}$): being homogeneous, it exerts no net force on local matter. Over that background, the field admits phase perturbations $\delta\theta$ ---the gapless Goldstone mode (\S{}III.8.3)--- that undulate it. The coupling between those waves and matter emerges from the kinetic term of F1 at second order as a ponderomotive force, and declares its consequences.

\textbf{The sea and its two modulations.} Over the relaxed vacuum ---the medium at $S_{\min}$--- the phase kinetic term of F1 gives, to linear order, $|\partial\Psi|^2 = (\partial\delta S)^2 + S_{\min}^2(\partial\delta\theta)^2 + \mathcal O(\delta^3)$: the two modulations decouple. The wave $\delta\theta$ is the phase mode, the U(1) Goldstone boson, massless, propagating at $c$ over the sea (physical IR branch of E7, $\Box\delta\theta=0$). Matter $\delta S$ is the modulus mode, massive by the curvature $V''_\text{eff}$, localized at the frozen topological defects of crystallization (\S{}IV.5-bis): it is countable, $\rho\propto a^{-3}$. To linear order the wave and matter do not mix: the drag is necessarily a non-linear effect.

\textbf{The wave-matter coupling from F1.} The phase kinetic term of F1 is $\tfrac12\rho_P c^2\ell_P^2\,S^2(\partial\theta)^2$. Substituting $S = S_{\min}+\delta S$ and retaining up to second order:
\begin{equation}
S^2(\partial\delta\theta)^2 = \underbrace{S_{\min}^2(\partial\delta\theta)^2}_{\text{free wave}} + \underbrace{2\,S_{\min}\,\delta S\,(\partial\delta\theta)^2}_{\text{wave--matter coupling}} + \mathcal O(\delta^3).
\end{equation}
The cross term
\begin{equation}
\mathcal L_\text{couple} = \rho_P c^2\ell_P^2\,S_{\min}\,\delta S\,(\partial\delta\theta)^2
\end{equation}
couples matter ($\delta S$) to the energy of the wave ($(\partial\delta\theta)^2$). It is not an added ingredient: it is the second order of the same kinetic term that, to first order, gives the free equations of the modulus and the phase (E6, E7). The coupling is \epi{DERIVED}.

\textbf{The matter equation with drag.} Varying the action with respect to $\delta S$, the term $\mathcal L_\text{couple}$ contributes a source to the equation of the perturbed modulus:
\begin{equation}
\boxed{\;\Box\,\delta S - \frac{\ell_P^2}{20}\Box^2\delta S + \mu_S^2\,\delta S = \frac{S_{\min}}{\ell_P^2}\,(\partial\delta\theta)^2\;}
\end{equation}
with $\mu_S^2 = V''_\text{eff}/(\rho_P c^2\ell_P^2)$. The right-hand side, $(\partial\delta\theta)^2$, is the local energy density of the wave: where the wave concentrates energy, it forces the matter field. It is the structure of a ponderomotive force ---the mechanism by which an intense beam pushes particles toward or away from the regions of highest intensity.

\textbf{The drag force.} For a traveling phase wave $\delta\theta = A\cos(kx-\omega t)$ with $\omega=ck$ (massless), the time average of the energy density is $\langle(\partial_x\delta\theta)^2\rangle = \tfrac12 A^2 k^2$. Matter feels the gradient of this energy, so that the drag force per unit density of matter is
\begin{equation}
\boxed{\;\mathbf F_\text{drag} = -\rho_P c^2\ell_P^2\,S_{\min}\,\nabla\langle(\partial\delta\theta)^2\rangle\;}
\end{equation}
with three properties: a uniform wave does not drag (if $A$ is constant, $\nabla\langle(\partial\delta\theta)^2\rangle=0$: the drag requires amplitude modulation); the wave (massless, at $c$) goes faster than matter (massive, $v<c$), which follows it partially and remains shifted toward the regions of favorable phase energy; and the sign $\mathbf F = -\nabla\langle\cdots\rangle$ attracts matter toward the minima of phase agitation ---where the wave is calm matter accumulates, where it is agitated it is expelled. \epi{DERIVED} (coupling and ponderomotive form, second order of F1).

\textbf{Falsifiable consequences.} The drag is, in cosmological language, a dark-energy--matter coupling, from which two local consequences with definite sign follow:
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Modified structure growth.} Matter feels, in addition to the emergent curvature of el Pleno, the force $\mathbf F_\text{drag}$ (an effective fifth force mediated by the phase wave). This modifies the growth rate of the matter perturbations, deviating $\sigma_8$ and $f\sigma_8$ from the $\Lambda$CDM prediction. \epi{SLD}.
  \item \textbf{Local phase--density anti-correlation.} Since $\mathbf F_\text{drag} = -\nabla\langle(\partial\delta\theta)^2\rangle$, matter accumulates in the minima of phase energy, which predicts a specific anti-correlation between the phase agitation of dark energy and the matter overdensity, distinguishable in principle from the structure produced by the emergent curvature alone. \epi{SLD}.
\end{itemize}
These consequences feed the perturbation sector (\S{}III.8.9, structure growth and $S_8$) with a derived mechanism, without closing it: the magnitude of the effect depends on the amplitude of the phase perturbations, which is not fixed without additional input, and that is why it remains \epi{CALIBRATED}. The drag is a \emph{perturbative} effect of the waves $\delta\theta$, independent of whether the background has $w=-1$ exact: the phase perturbations exist over the homogeneous Curtain whatever its background equation of state. Its signature lives in the structure growth and in $S_8$, not in the equation of state of the background ---which is $w=-1$ exact (\S{}III.7.5)--- nor in the $w_0$--$w_a$ signal of DESI, which is an artefact of parametrization of the background (\S{}III.6.6).

\textbf{Synthesis.} Dark energy and matter are not two independent substances mediated by a force: they are two modulations of the same medium over the sea of $S_{\min}$. Dark energy has the homogeneous component ---the Curtain $\theta_0$, $w=-1$, the bulk--- and its waves $\delta\theta$. The Curtain, being homogeneous, does not drag; the waves do, through the second-order coupling of F1. Both components, together with matter, are sources of the tensor $T_{\mu\nu}^\text{(Pleno)}$ that determines the emergent curvature (Sakharov, \S{}III.1): they do not interact gravitationally with each other as separate substances ---in PIU gravity is not a mediator, but the elastic response of el Pleno to its own configuration---, but rather co-determine the geometry of the single medium that both inhabit. That the coupling is derived and not postulated ---that it comes from the same kinetic term that gives the free waves--- is the signature that matter and dark energy share a single substance: el Pleno, modulated. \epi{DERIVED} (coupling and ponderomotive form); \epi{SLD} (existence and sign of the local consequences); \epi{CALIBRATED} (its magnitude).

\begin{tcolorbox}[ecuacionclave,title={Interpretation --- what the drag means physically}]
Matter is dragged by dark energy like sand by a wave. When the phase of the vacuum undulates, those waves transport energy, and the concentrated energy pushes matter toward where the wave is calmer ---the same effect by which an intense light beam pushes particles (radiation pressure), here exerted by the waves of the vacuum itself. It is not gravity: it is a real fifth force, mediated by dark energy, that adds to the ordinary attraction and leaves two observable signatures ---it alters the rate at which galaxies grow and clusters matter where the phase is calm. The decisive point is that this force is not postulated: it appears by itself on expanding el Pleno to second order, which reveals that matter and dark energy are not two distinct things that interact, but two undulations of the same medium that share substance.
\end{tcolorbox}

\subsubsection{\texorpdfstring{III.7.6 --- The structural closed form of $S_{\min}$ and its role in dark energy}{III.7.6 - The structural closed form of and its role in dark energy}}

\textbf{Origin.} The minimum value of the modulus, $S_{\min}^\text{estr}$, fixes the dark-energy density of the relaxed regime: $\rho_\text{DE}^{(0)} = \rho_P(S_{\min}^\text{estr})^2$. Its closed form is derived structurally in \S{}II.4.5 (E9), from the spectral ratio of the discrete Laplacian over the Brillouin zone of the diamond lattice.


\textbf{Result (derived in \S{}II.4.5).}


\[
S_{\min}^\text{estr} = \frac{12(\pi^2-4)}{\pi^4} = 0.7230871\ldots
\]

obtained as the spectral ratio $\langle D_\text{lat}\rangle_\text{BZ}/\langle k^2\rangle_\text{BZ}$ ---the mean \textbf{amplitude} of the modulus, not the occupation. It is a pure spectral property of the lattice geometry, without free parameters.


\textbf{The double route as reinforcement.} The same value arises by two independent routes: the spectral ratio (geometric) and the minimization of the effective potential (dynamical, $dV_\text{eff}/dS = 0$ gives $S_\text{eq} = 0.7230873\ldots$), coincident to ten digits.


\textbf{Phenomenological reading --- the role of $S_{\min}$ in dark energy.} The value $S_{\min}^\text{estr}$ has three roles:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Fixes the dark-energy density of the relaxed regime.} $\rho_\text{DE} = \rho_P(S_{\min}^\text{estr})^2 = \rho_P\cdot(0.7231)^2 = 0.5229\,\rho_P$ in units of the medium. It is the energetic ``backdrop'' over which the solitons disperse.
  \item \textbf{Defines the threshold of matter.} $\rho_\text{matter} = \rho_P[S^2 - (S_{\min}^\text{estr})^2]$: matter is the excess of the modulus over the background. With $S_{\min}$ derived, the threshold is fixed structurally.
  \item \textbf{Ontologically distinguishes dark energy from zero.} That $S_{\min}^\text{estr}\neq 0$ (but $\approx 0.72$) is what makes the \emph{Pleno}-ground state have non-null structural energy: dark energy is the energy of the background in its configuration of minimum excitation, not of the empty vacuum.
\end{itemize}

\textbf{Epistemic contrast with $f_c$.} The asymmetry between the two is what separates the structural from the positional: $S_{\min}^\text{estr}$ has a derived closed form (\S{}II.4.5), whereas $S_{\max,\text{eff}}^2$ ---the numerator of $f_c$ (\S{}III.7.2)--- is calibrated and its closed form remains open (\S{}V.3). That is why the value of $S_{\min}$ has a stronger derivational status than the value of $f_c$.


\textbf{Numerical verification.} $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4 = 70.435/97.409 = 0.72309$, and $(S_{\min}^\text{estr})^2 = 0.5229$.


\textbf{Epistemic classification.} The closed form $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ is \epi{DERIVED} in \S{}II.4.5. The phenomenological contribution ---the three roles of $S_{\min}$ and the contrast with $f_c$--- is \epi{DERIVED} as a direct consequence of the density formula and the DE-Curtain reading.


\subsubsection{III.7.7 --- Resolution of the problems of contemporary physics}

The reading of the dark sectors as configurations of \emph{el Pleno} structurally resolves three problems articulated in \S{}I.2:


\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'' as a sum of mode fluctuations. There is a physical medium with specific structural energy, whose cosmological fraction is $f_c\approx 0.6869$.
  \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. Dark matter is not an undetected particle: it is an excited configuration of the same medium without an observable gauge sector.
  \item \textbf{The extreme observational uniformity of dark energy.} A natural consequence of dark energy being a structural property of the single medium, not a material component with local dynamics.
\end{itemize}

\subsubsection{\texorpdfstring{III.7.8 --- Interpretation: what the dark sectors are under PIU-$\Psi$}{III.7.8 - Interpretation: what the dark sectors are under PIU-Psi}}

\textbf{Interpretation.} The dark sectors are not additional substances: they are \emph{el Pleno} itself. Dark energy is the \emph{Pleno}-ground state (the medium in minimum excitation, present in the entire volume); dark matter are excited configurations without observable gauge coupling. The ``unknown'' 95\% of the universe is, under PIU-$\Psi$, the single physical medium read in its distinct configurations.


\textbf{Explanation.} The dark-energy fraction from the active condensate fraction $f_c$ (E33); the dark matter from the excited configurations without gauge; the dark-matter/baryonic ratio from the geometry of the substrate (\S{}III.6.3); the uniformity of dark energy from its structural character.


\textbf{Prediction.} The fraction $\Omega_{EO} = f_c\approx 0.6869$ (verified to 0.30$\sigma$; structure derived with zero cosmological fitting parameters, calibrated numerator), the ratio $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ (verified to 0.86$\sigma$, \S{}III.6), and the absence of local concentrations of dark energy (consistent with observation).

