% !TeX root = main.tex
\clearpage
\section*{Part IV --- The dilatant crunch-bounce cycle and particle formation}
\phantomsection
\addcontentsline{toc}{section}{Part IV --- The dilatant crunch-bounce cycle and particle formation}
% ============================================================
\begin{tcolorbox}[notabox,title={Purpose of Part IV}]
\textbf{To show that the dilatant rheology of \emph{el Pleno} (\S{}III.9) is not only a static characterization of the medium, but the dynamical engine that governs the crunch-bounce cosmological cycle and the asymmetric formation of the particles.} Parts I--III established \emph{el Pleno} as a medium, its catalogue of configurations and its phenomenological reading; Part V confronts them with observation. Part IV articulates a dynamical narrative that connects four results already derived in the program ---the Gent elasticity (\S{}II.2), the diamond-lattice crystallization (\S{}II.3), the emergent gravity (\S{}III.1) and the non-linear rheology (\S{}III.9)--- under a single mechanism, without introducing any new free parameter. Each link carries its explicit epistemic classification; the bulk rests on quantities derived in Parts I--III, and what remains a strong deduction or hypothesis is declared as such.
\end{tcolorbox}

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} individual mass of the electron ($m_e$, thread of the leptonic sector); formal homotopy demonstration of the $-1$ \emph{winding} of the interstitial over $Fd\bar{3}m$; absolute temporal duration of the crystallization plateau.
  \item \textbf{Notes:} the cycle is built entirely on already-derived pieces (tripartite potential and $E_0$ of \S{}II.2; dilatant rheology of \S{}III.9; gravity of \S{}III.1; Russo-Tanaka crystallization of \S{}II.3; Borromean cluster of the hadronic sector). Part IV is a \emph{dynamical synthesis}: it organizes those pieces into a coherent physical chronology and establishes the methodological principle of the interpretation-independent observable.
\end{itemize}
\end{estadoseccion}

\subsection{\S{}IV.1 --- The three velocities of \emph{el Pleno}}

The dynamics of the cycle rests on three characteristic velocities of \emph{el Pleno} that, confused, produce apparent contradictions and, distinguished, order all the physics of the bounce. The three already appear in Parts I--III; their explicit contrast is the following.

\begin{enumerate}[leftmargin=1.6em]
  \item \textbf{Signal velocity} $v_T = c$. It is the propagation velocity of the transverse-traceless mode (TT mode, Result D2, \S{}III.3). It is the real causal bound of the program, the signature of Lorentz covariance (axiom A3).
  \item \textbf{Compression phase velocity} $v_L = \sqrt{(K+\tfrac{4}{3}\mu)/\rho_P} \approx 1.98\,c$ (\S{}III.1.8). It is a \emph{phase} velocity of a massive mode (the radial mode has gap $\omega_0$), not a signal velocity: it does not transport information, just as in a waveguide or a plasma. There is no causal conflict.
  \item \textbf{Maximum deformation rate} $\dot\gamma_{\max} = c/\ell_P = \omega_P \approx 1.855\times 10^{43}\ \text{s}^{-1}$ (E45, \S{}III.9.2). It is the bound on the velocity at which the medium can reorganize its density ---the deformation rate, not a propagation velocity.
\end{enumerate}

\textbf{The bounce is governed by the third}, $\dot\gamma_{\max}$, not by $v_L$. This distinction is what makes the whole narrative of Part IV consistent with Part III, and resolves in advance the objection that the bounce ``would travel'' at $v_L>c$: the bounce is not a wave that propagates, but the global temporal evolution of the modulus, bounded by the reorganization rate of the medium. \epi{DERIVED} (the three velocities, in \S{}III).

\subsection{\S{}IV.2 --- The scenario: two populations of \emph{el Pleno}}

\emph{El Pleno} does not participate in the cycle as a single homogeneous fluid. By the EO-Backdrop reading (\S{}III.7.2), there are two physically distinct populations:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{EO backdrop} (fraction $f_c = 0.6869$): the \emph{Pleno}-ground state in its coherent mode, with equation of state $w=-1$, homogeneous. It is \textbf{gravitationally inert}: it does not attract itself, does not agglomerate, does not participate in the contraction. It is the permanent background ---the ``ocean''.
  \item \textbf{Depleted matter} (fraction $1-f_c = 0.3131$): the excitations of the condensate (ordinary and dark matter). It is the only thing that gravity concentrates and the only thing that cycles ---the ``waves''. The coincidence with the observed $\Omega_m=0.315$ (Planck 2018, at $0.3\sigma$) is the verification of \S{}III.8.5.
\end{itemize}

\textbf{Immediate consequence:} the universe does not collapse entirely because the 69\% (EO) is gravitationally inert. Only 31\% cycles, and the matter budget per cycle is bounded by the available depleted fraction. This dynamically realizes the principle that a particle is a modulation $\delta S$ over the background, not a portion of the medium (``wave, not water''): the ocean is the EO backdrop; the waves are the local crunch-bounces where matter forms. \epi{DERIVED} (the two populations, from \S{}III.7.2).

\subsection{\texorpdfstring{\S{}IV.3 --- The effective potential of the cycle (exact, without free parameters)}{Sec.IV.3 - The effective potential of the cycle}}

The potential energy of the modulus $S$ of \emph{el Pleno} is the tripartite potential derived in \S{}I.7 and \S{}II.2, which is rewritten here in its operative form for the dynamics of the cycle:

\begin{equation}
V_\text{eff}(S) = \underbrace{\frac{E_0}{2}\frac{S^4}{(S_{\max}^2-S^2)^2}}_{W_\text{bare}\ (\text{Gent, hard wall})} \;-\; \underbrace{\frac{\rho_P c^2}{2}\log\!\left(1-\frac{S^2}{S_{\max}^2}\right)}_{V_\text{topo}} \;+\; \underbrace{\frac{3\rho_P c^2}{8 S^2}}_{V_q\ (\text{IR wall})}\,. \tag{E52}
\end{equation}

All the constants are derived in Parts I--II, without any fitting parameter:
\begin{itemize}[leftmargin=1.4em]
  \item $E_0 = 4.2546\,\rho_P c^2$ ---Gent modulus, derived by the dynamical-balance condition $dV_\text{eff}/dS=0$ at $S_{\min}$ (\S{}II.4.4), verified symbolically to 10 digits.
  \item $S_{\max}^2 = S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3} = 2.1773$ ---hard bound of the packing (\S{}II.4.1).
  \item $S_{\min} = 12(\pi^2-4)/\pi^4 = 0.7231$ ---\emph{Pleno}-ground state (\S{}II.2).
\end{itemize}

\textbf{Verification.} It was confirmed symbolically that $dV_\text{eff}/dS = 0$ exactly at $S_{\min}=0.7231$ (minimum of the potential). The restoring force of the medium grows catastrophically upon approaching the hard bound:

\begin{center}
\begin{tabular}{cccc}
\toprule
\textbf{$S$} & \textbf{$V_\text{eff}$ [$\rho_P c^2$]} & \textbf{$-dV_\text{eff}/dS$} & \textbf{$S_{\max}^2-S^2$} \\
\midrule
0.7231 ($S_{\min}$) & 1.067 & 0.00 & 1.654 \\
0.900 & 1.442 & $-4.91$ & 1.367 \\
1.100 & 4.044 & $-27.82$ & 0.967 \\
1.300 & 26.554 & $-354.0$ & 0.487 \\
1.400 & 174.375 & $-4\,959$ & 0.217 \\
1.450 & 1\,681.5 & $-134\,850$ & 0.075 \\
1.470 & 36\,826 & $-1.33\times 10^{7}$ & 0.030 \\
\bottomrule
\end{tabular}
\end{center}

\textbf{The Gent wall and the divergence of the force.} The restoring force comes from the hyperelastic term; its derivative (the force) is:
\begin{equation}
-\frac{dW_\text{bare}}{dS} = -\frac{2\,E_0\,S^3\,S_{\max}^2}{(S_{\max}^2-S^2)^3}\,. \tag{E53}
\end{equation}
The key is in the denominator $(S_{\max}^2-S^2)^3$ ---the ``gap'' between the current compression and the hard bound, raised to the cube. When $S$ rises from 1.300 to 1.400, the gap is reduced to less than half (from 0.487 to 0.217), but since it enters cubed in the denominator, its effect is amplified: $\text{gap}^3$ falls by a factor $\approx 11$, which fires the force by a factor $\approx 14$ in an interval of only $\Delta S=0.1$. The decomposition confirms that \emph{only} the Gent term fires: $V_\text{topo}$ and $V_q$ remain orders of magnitude below and hardly vary in that region (Figure~\ref{fig:pared_gent}).

This divergence is the mathematical signature of a medium with \textbf{limited extensibility} (axiom A6): \emph{el Pleno} cannot be compressed indefinitely, and upon approaching its hard bound it becomes catastrophically more rigid. Physically, it is what stops the compression and produces the structural rebound (\S{}II.7.2) ---the rebound occurs before touching $S_{\max}$ because the force diverges first, so that the maximum density of the bounce is less than $\rho_P S_{\max}^2$. \epi{DERIVED} (potential and constants, from \S{}II.2; direct numerical verification).

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{pared_de_gent.png}
\caption{The Gent wall of \emph{el Pleno}. (A) The potential $V_\text{eff}(S)$: well at $S_{\min}$, hard wall at $S_{\max}$. (B) The restoring force $|dV_\text{eff}/dS|$ on a logarithmic scale, exploding toward $S_{\max}$; the values of the discussed zone are marked. (C) Decomposition into the three terms: only the Gent wall (red) fires; $V_\text{topo}$ and $V_q$ remain well below. (D) The root cause: the gap $(S_{\max}^2-S^2)^3$ in the denominator, which falls $\times 11$ between $S=1.30$ and $S=1.40$ and fires the force $\times 14$.}
\label{fig:pared_gent}
\end{figure}

\subsection{\texorpdfstring{\S{}IV.4 --- The assembled dilatant viscosity (without free parameters)}{Sec.IV.4 - The assembled dilatant viscosity}}

The non-linear response of \emph{el Pleno} is that of a Herschel-Bulkley dilatant fluid (\S{}III.9), with flow index (E47)
\begin{equation}
n(x) = \frac{1}{1-x}\,, \qquad x = \frac{\dot\gamma}{\dot\gamma_{\max}}\,, \qquad \dot\gamma_{\max} = \frac{c}{\ell_P}\,,
\end{equation}
with $n>1$ always: the medium thickens the faster it is deformed, diverging at the bound. The rate-dependent effective viscosity is composed of already-derived quantities ---it is not a free parameter:
\begin{equation}
\eta(\dot\gamma) = \frac{\mu_\text{Gent}\,\tau_\Pi}{1-\dot\gamma/\dot\gamma_{\max}}\,, \tag{E54}
\end{equation}
with $\mu_\text{Gent} = \rho_P c^2/S_{\max}^2$ \epi{DERIVED} and $\tau_\Pi = \sqrt{5}\,\ell_P/(10c)$ \epi{DERIVED} (inverse of the frequency of the Lee-Wick regulation mode, \S{}III.1.6, E-relax). In dimensionless form (time in units of $t_P=\ell_P/c$, $V_\text{eff}$ in units of $\rho_P c^2$):
\begin{equation}
\tilde\eta(x) = \frac{\sqrt{5}/(10\,S_{\max}^2)}{1-x} \approx \frac{0.1027}{1-x}\,, \tag{E55}
\end{equation}
where the coefficient $\tilde\eta_\text{lin} = \sqrt{5}/(10\,S_{\max}^2)$ is the exact composition of the two derived quantities, made dimensionless: $\tilde\eta_\text{lin} = (\mu_\text{Gent}\,\tau_\Pi)/(\rho_P c^2\cdot t_P) = (1/S_{\max}^2)\cdot(\sqrt{5}/10) = \sqrt{5}/(10\cdot 2.1773) = 0.1027$. It is the same factor $\sqrt{5}/10$ that defines $\tau_\Pi$ (\S{}III.1.6), divided by $S_{\max}^2$; in physical units it reproduces the Maxwell viscosity $\eta_\text{Pleno} = \mu_\text{Gent}\tau_\Pi \approx 2.56\times 10^{69}$ Pa$\cdot$s of \S{}III.1.6, without any fitting coefficient.

\textbf{Key physical link.} The physical deformation rate that enters the dilatancy is not $\dot S$ directly, but the \textbf{volumetric deformation}, derived from the constitutive identification $\rho_\text{medium} = \rho_P S^2$ (\S{}II.2.2): since the volume goes as $V\propto 1/\rho \propto 1/S^2$,
\begin{equation}
\dot\gamma = \left|\frac{d\ln V}{dt}\right| = 2\,\frac{|\dot S|}{S}\,, \tag{E56}
\end{equation}
with the bound $\dot\gamma \leq \dot\gamma_{\max} = c/\ell_P$. \epi{DERIVED} (the relation $\rho\sim S^2$ is from the canon; the Maxwell viscosity $\eta_\text{Pleno}=\mu_\text{Gent}\tau_\Pi$ is \epi{DERIVED} as the product of two derived quantities via Maxwell-Kubo, \S{}III.1.8; the dilatant viscosity (E54) extends that linear viscosity to the non-linear regime by the factor $1/(1-x)$ of \S{}III.9).

\subsection{\texorpdfstring{\S{}IV.5 --- The equation of the cycle and its chronology}{Sec.IV.5 - The equation of the cycle and its chronology}}

Bringing together the potential (E52), the dilatant viscosity (E54--E55) and the volumetric deformation (E56), the dynamics of the modulus of \emph{el Pleno} in the homogeneous cosmological mode is a non-linear oscillator ---a \textbf{Gent oscillator with dilatant damping}:
\begin{equation}
\rho_P\ell_P^2\,\ddot S \;+\; \eta(\dot\gamma)\,\dot S \;+\; \frac{dV_\text{eff}}{dS} \;=\; F_\text{ext}\,, \qquad \dot\gamma = \frac{2|\dot S|}{S}\,, \tag{E57}
\end{equation}
with $F_\text{ext}$ the external forcing (the cosmological contraction; see Phase 1). The chronology of the cycle follows from this equation in four phases (Figure~\ref{fig:formacion}).

\subsubsection{Phase 1 --- Crunch (slow compression)}

\textbf{Engine: the global cosmological contraction.} The crunch that carries \emph{el Pleno} toward $S_{\max}$ is the Friedmann dynamics of the universe in a collapse phase, not the local self-gravity of matter. This distinction is a result of Part IV itself. Upon deriving the gravitational force of an overdensity of depleted matter from the excess mass-energy over the background, $\Delta\varepsilon(S) = V_\text{eff}(S)-V_\text{eff}(S_{\min})$, the integration of (E57) with that force shows that the local self-gravity does \emph{not} reach $S_{\max}$: the restoring force $dV_\text{eff}/dS$ defeats gravity near $S_{\min}$ and the system stabilizes just above the background (at $S\approx 0.743$). The energetic argument is explicit: the excess energy that the local self-gravity can contribute is $\Delta\varepsilon(0.74)-\Delta\varepsilon(S_{\min}) \approx 2.8\times 10^{-3}\,\rho_P c^2$, while carrying the modulus to the wall zone ($S\approx 1.45$) demands $\Delta\varepsilon(1.45) \approx 1.68\times 10^{3}\,\rho_P c^2$ ---a factor $\sim 6\times 10^{5}$ greater. The excess of the depleted matter over the backdrop is, therefore, insufficient by five orders of magnitude to compress \emph{el Pleno} to the Gent wall on its own.

The compression to $S_{\max}$ therefore demands the global cosmological forcing. By the identification $\rho_\text{medium}=\rho_P S^2$ (\S{}II.2.2) and $\rho\propto 1/a^3$ during the contraction, it follows that $S^2 \propto 1/a^3$: the contraction of the scale factor $a(t)$ compresses the modulus of \emph{el Pleno} toward $S_{\max}$.

\textbf{Why it is slow:} the crunch goes \emph{against} the restoring force of Gent, and the dilatancy makes the resistance grow upon approaching $S_{\max}$ ($n\to\infty$). It is quasi-static, which gives the medium time to homogenize and order itself.

\textbf{Deep connection: the Hubble bound is the dilatant bound.} The maximum Hubble bound of the program $H_{\max} = \sqrt{8\pi G\rho_{\max}/3} \approx 7.9\times 10^{43}$ s$^{-1}$ (\S{}II, from the A6 hard bound) and the maximum deformation rate $\dot\gamma_{\max} = c/\ell_P \approx 1.9\times 10^{43}$ s$^{-1}$ (\S{}III.9) are of the same order (factor $\sim 4$). The maximum cosmological contraction rate is bounded by the maximum reorganization rate of the medium: the Hubble bound (cosmology, Part II) and the dilatant bound (rheology, \S{}III.9) are the same physical bound seen from two sides, tied by the Friedmann coefficient through the square root of the packing quotient. \epi{DERIVED} for the equality, which is exact algebra; the unifying reading of the two bounds as one is \epi{SLD} (cosmological engine, from \S{}II.7.2; connection $H_{\max}\sim\dot\gamma_{\max}$, Part IV).

\subsubsection{Phase 2 --- Bounce (rapid decompression, regulated)}

\textbf{Engine:} the elastic energy of Gent stored in the compression (which explodes near $S_{\max}$, table of \S{}IV.3) is released. The rebound is structural and occurs before reaching $S_{\max}$ (\S{}II.7.2).

\textbf{The dilatant regulation (central link).} The pure mechanical oscillator, without viscosity, would give a bounce with a highly superluminal deformation rate: the numerical integration of (E57) without the dilatant term (constant minimal damping), releasing the system from the compression $S=1.45$, produces deformation rates that scale up to $\sim 1.1\times 10^{3}\,\dot\gamma_{\max}$ (Figure~\ref{fig:integracion}, red curve) ---three orders of magnitude over the bound, physically impossible. \textbf{The dilatancy of the program is exactly the regulator that prevents it:} when the deformation rate $\dot\gamma=2|\dot S|/S$ approaches $\dot\gamma_{\max}=c/\ell_P$ (that is $x\to 1$), the viscosity (E55) diverges, creating an infinite dissipation barrier that the bounce cannot cross. The integration of (E57) with the dilatant viscosity and $\dot\gamma=2|\dot S|/S$ confirms this effect robustly: sweeping the starting point of the bounce from $S_0=0.9$ to $S_0=1.45$, the deformation rate always remains bounded by the bound ($x_{\max}$ grows monotonically from $0.989$ to $1.000$ without ever crossing it), with convergence verification under refinement of the integration step. The leftover Gent energy is dissipated as heat near the bound, not as an impossible deformation velocity.

This dynamical regulation is also consistent with the structural bound: $\dot\gamma_{\max}=c/\ell_P$ is of the same range as the A6 hard bound on the modulus. The divergence of the viscosity at $x=1$ is the rheological manifestation that the medium cannot reorganize its density faster than the Planck frequency ---kinematic bound and dynamical divergence are the same physics, and the numerical integration realizes them simultaneously.

\begin{tcolorbox}[notabox]
\textbf{Central result of Part IV.} The dilatant rheology is not a descriptive ornament of \emph{el Pleno}: it is what makes the bounce physically consistent (it prevents the superluminal reorganization of the medium) and, at the same time, the source of the temporal asymmetry slow-crunch/fast-bounce that will distinguish protons from electrons. The same property ---the shear-thickening with a hard deformation bound--- fulfills the two functions.
\end{tcolorbox}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{integracion_ciclo.png}
\caption{Numerical integration of the equation of the cycle (E57). (A) Trajectory of the modulus $S(t)$ with the dilatant viscosity: the bounce is regulated and the modulus oscillates between $S_{\min}$ and the initial compression without reaching $S_{\max}$. (B) Deformation rate $x=\dot\gamma/\dot\gamma_{\max}$: with dilatancy (blue) it remains bounded by the bound $x\leq 1$ ---the viscosity diverges at $x=1$ and brakes the bounce before crossing it; without the dilatant mechanism (red, constant damping) the rate scales up to $\sim 1.1\times 10^3\,\dot\gamma_{\max}$, three orders of magnitude over the bound, physically impossible. The dilatancy is the regulator that keeps the bounce subluminal. Integration with \texttt{scipy} (Radau method, tolerances $10^{-9}/10^{-11}$, convergence verified under refinement of the step); all parameters derived ($E_0=4.2546$, $S_{\max}=1.4756$, $S_{\min}=0.7231$, $\eta_0=0.1027$).}
\label{fig:integracion}
\end{figure}

\subsubsection{Phase 3 --- Crystallization plateau (particle formation)}

Upon expanding, the medium cools and crosses the crystallization point: the homogeneous fluid \emph{el Pleno} transforms into the crystalline diamond lattice with defects (the particles). This transition is already in the program (\S{}II.3.6, Russo-Tanaka argument): it is a Landau-Ginzburg transition with coupled order parameters (scalar density $\rho$, tetranematic tensor $Q_{ijk}$, third gradient $\nabla^3\rho$), where the quadrupolar coupling selects the coordination $Z=4$ with cell doubling, verified by the structure factor $F(\mathbf{G}) = 1 + e^{i\mathbf{G}\cdot\boldsymbol{\delta}_0}$, $\boldsymbol{\delta}_0=(a/4)(1,1,1)$, which is exactly that of the diamond lattice.

\textbf{Why there is a plateau:} a first-order transition is not instantaneous. Fluid and crystal coexist while the latent heat is released; the system stagnates in a plateau of quasi-constant conditions until the crystallization is completed. What \emph{sustains} that plateau is a structural thermodynamic bottleneck: during the crunch the medium packs toward the geometry of maximum compactness ($Z=12$, Kepler-Hales packing, the geometry that fixes $S_{\max,\text{geom}}$, \S{}II.4.1), and to crystallize to the diamond lattice of coordination $Z=4$ it must yield the latent heat of the structural reorganization. Unlike the quasi-degenerate polytypes of the ceiling (whose lattice energies differ by $\sim 10^{-4}\,\rho_P c^2$ per site, \S{}II.3), the geometries of $Z=12$ and $Z=4$ have substantially distinct lattice energies by their sole difference of coordination (twelve against four neighbors), so that the latent heat of the transition is of order $\mathcal{O}(1)\,\rho_P c^2$ per site. The medium \emph{cannot} crystallize to diamond until that heat is evacuated; that finite evacuation is what maintains the quasi-constant conditions and gives the plateau its duration. The \emph{scale} of the latent heat (of order one, not degenerate) is \epi{SLD}; its numerical \emph{value} requires the computation with the Gent potential of the program, $W_\text{bare}(S)=(E_0/2)\,S^4/(S_{\max}^2-S^2)^2$, and remains a declared computational thread (\S{}IV.8). From this plateau three physical consequences follow:
\begin{enumerate}[leftmargin=1.6em]
  \item \textbf{The size of the particles} is decided by the plateau: if it lasts long enough, the defect relaxes to its equilibrium size; if the growth is truncated, it remains smaller (see \S{}IV.6).
  \item \textbf{The identity of the particles} (all identical) comes from the latent heat that keeps $T$ quasi-constant during the plateau: homogeneous conditions $\Rightarrow$ all the defects grow under the same conditions.
  \item \textbf{The nucleation spacing} (how many nuclei crystallize per volume) fixes the separation between particles.
\end{enumerate}
\epi{SLD} (the Landau-Ginzburg structure of the crystallization is \S{}II.3.6; the plateau reading as a formation window is proper to Part IV).

\subsubsection{Phase 4 --- Freeze-out (freezing)}

Once the crystallization is completed, the medium continues cooling and the operational constants of \emph{el Pleno} freeze dynamically ($\Lambda_\text{PIU}$ as a frozen constant of the bounce). The configurations are fixed. \textbf{[DERIVED, \S{}II.7.3]}.

\subsection{\texorpdfstring{\S{}IV.5-bis --- The phase freeze-out: the origin of the partition between dark energy and matter}{Sec.IV.5-bis - The phase freeze-out: the origin of the partition between dark energy and matter}}

The crystallization plateau (\S{}IV.5, Phase 3) does not only form the particles: it is the event in which the two populations of el Pleno separate ---the DE-Curtain ($f_c$, coherent mode $w=-1$) and the depleted matter ($1-f_c$). What the transition decides is the \emph{partition}, and the two populations do not arrive at the event in the same condition.

\emph{What dark energy is} ---its density $\rho_\text{DE}=V(S_{\min}^\text{estr})=K\,S_{\min}^{2,\text{estr}}$ and its equation of state $w=-1$--- is not produced at the transition: it is the ground state of the medium, the floor of the effective potential. That floor exists for as long as F1 governs \emph{el Pleno}, with or without a bounce, and its modulus is obtained by two routes that do not touch cosmic history ---the spectral one (\S{}II.4.5) and the dynamical one (\S{}II.4.4)---, neither of which needs the transition.

\emph{The defects, by contrast, are nucleated there}: before the crossing of $S^*$ there are no solitons, and the Kibble-Zurek-type phase freeze-out is the event that produces them. But this introduces no second substance. By A1 there is a single medium, and a defect is that same medium in another configuration: what crystallization creates is not matter \emph{ex novo} but a configuration in which part of the volume becomes trapped.

Hence the precise statement, and the one that gives this subsection its title: the transition originates neither dark energy ---which was already there--- nor a new substance ---which does not exist---, but the \emph{partition}: which fraction of the volume came to rest on the coherent floor and which fraction was trapped in defects.

That separation has a hard consequence for the stability of $w$. If $\rho_\text{DE}$ is the value of the potential at its minimum, then a dark-energy sector with $w\neq-1$ would require the minimum itself to move in time, that is, that F1 not be the Lagrangian of the medium. The constancy of $w=-1$ ceases to be a fact about how the universe ended up and becomes a structural impossibility: a minimum does not evolve. This is a third lock, independent of the two that \S{}III.7.5 already establishes ---the positivity of the canonical kinetic term, which vetoes the phantom region, and the $\propto L^{-3}$ dilution of a rolling phase mode---.

The partition is produced as a Kibble-Zurek-type phase freeze-out over the critical point $S^*$.

\subsubsection{\texorpdfstring{IV.5-bis.1 --- Ontology of $S_{\max}$: continuous mass, not spheres}{Sec.IV.5-bis.1 - Ontology of S_max: continuous mass}}

The saturation bound $S_{\max,\text{geom}}^2 = 8\sqrt6/9$ (\S{}II.4.1) admits an angular reading in addition to the packing one. The factor $8/9$ is $\sin^2\theta_\text{tet}$, the squared sine of the tetrahedral bond angle: with $\theta_\text{tet} = \arccos(-1/3) = 109.47^\circ$ (\S{}II.3.7),
\begin{equation}
\sin^2\theta_\text{tet} = 1 - \cos^2\theta_\text{tet} = 1 - \tfrac19 = \tfrac89 .
\end{equation}
At $S_{\max}$ el Pleno is not packed spheres with gaps: it is a continuous mass, without void, compressed until it saturates the geometry of the tetrahedral bond. The Kepler-Hales packing (\S{}II.4.1) is the anchor of contact with standard physics; the angular reading is the ontology of the continuous medium, consistent with the diamond lattice being the modulation of a continuous scalar field (\S{}II.3.7-bis), not a scaffolding of spheres.

\subsubsection{IV.5-bis.2 --- The path of the continuum}

The post-bounce expansion traverses the modulus from $S_{\max}$ to $S_{\min}$, with densities $\rho\propto S^2$ (in units of $\rho_P c^2$):

\begin{center}
{\small
\begin{tabular}{lcL{6.2cm}}
\toprule
\textbf{Point} & $\rho/\rho_P$ & \textbf{Physical state} \\
\midrule
$S_{\max}$ & $2.177$ & amorphous continuous mass, no lattice, disordered phase (U(1) restored) \\
$S^*$ & $1.067$ & crystallization to diamond lattice; phase separation \\
$S_{\min}$ & $0.523$ & relaxed diamond lattice; the present ``sea'' \\
\bottomrule
\end{tabular}}
\end{center}

The total compression is $S_{\max}^2/S_{\min}^2\approx 4.16$. The midpoint $S^* = \sqrt{S_{\min}S_{\max}}\approx 1.0330$ (\S{}III.9.11) is the point of minimum rigidity of the modulus, where the elastic barrier to reorganize the lattice is minimal and, therefore, where the packing transition occurs ($Z=12\to Z=4$, the same as the plateau of \S{}IV.5) at density $\rho(S^*)\approx\rho_P$.

\subsubsection{IV.5-bis.3 --- The phase freeze-out}

At $S_{\max}$ the phase $\theta$ is thermally disordered: U(1) symmetry restored, without long-range coherence, without condensate. On expanding and crossing the transition at $S^*$, the U(1) symmetry breaks (A5): the phase chooses a value, but causally disconnected regions choose independent phases, forming domains of coherent phase (Kibble-Zurek mechanism). On completing crystallization, the phase is frozen in two complementary structures:
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Within each domain:} uniform coherent phase, $\Box\theta = 0$, gapless propagating mode. It is dark energy ---the DE-Curtain of \S{}IV.2, el Pleno-ground state in its coherent mode.
  \item \textbf{At the boundaries between domains:} trapped phase circulation, topological defects of \emph{winding}. They are the particles ---the depleted matter, the vortices of the hadronic sector (\S{}IV.6).
\end{itemize}
Both are born from the same freeze-out at $S^*$. The volume fraction in coherent domains is $f_c$; the fraction in defects is $1-f_c$. The matter/dark-energy separation is not the meeting of two substances, but the two complementary structures ---coherent interior and boundary with circulation--- of the same phase freeze-out.

\subsubsection{IV.5-bis.4 --- Derived properties of dark energy}

Dark energy, as a coherent phase within a domain over the relaxed sea $S_{\min}$:
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Has $w=-1$:} with $\Box\theta = 0$ and $S = S_{\min}$ uniform, the energy-momentum tensor of the phase sector gives $\rho+p = 0$ (\S{}III.7.5, where $w=-1$ is derived as the vanishing of the velocity term). \epi{DERIVED}.
  \item \textbf{Does not dilute:} the coherent phase is not countable; it does not scale as $a^{-3}$. It is persistent vacuum energy, gravitationally inert (\S{}IV.2). \epi{DERIVED}.
  \item \textbf{Propagates over the medium:} the phase wave satisfies the phase equation (E7), $\partial_\mu(S^2\partial^\mu\theta)=0$, with coefficient $S^2$. The phase does not exist without the modulus: it propagates over the medium at $S_{\min}$, which is the substrate.
\end{itemize}
Matter (defects) is trapped circulation, with mass, countable, $\rho\propto a^{-3}$: perturbations of the medium that do dilute. The Klauder IR wall ($V_q\propto 1/S^2$, diverges as $S\to 0$, \S{}I.7.8) guarantees $S\ge S_{\min}$: the sea never empties, maintaining the substrate over which the wave propagates and the defects exist as trapped vortices.

\subsubsection{IV.5-bis.5 --- Status of the partition $f_c$}

The exact partition between coherent dark energy ($f_c\approx 0.6869$) and depleted matter ($1-f_c\approx 0.3131 = \Omega_m$, already in the corpus by population conservation, \S{}III.8.5) is not derived geometrically: the fraction of defects in a Kibble-Zurek freeze-out depends on the \emph{rate} of the transition ---the speed of the bounce---, which is positional over the trajectory $\bar S(t)$. This locates $f_c$ as a quantity that measures the speed of the freeze-out of the last phase transition of the universe: a physical quantity but trajectory-dependent, on equal footing with $H_0$ and with the numerator $S_{\max,\text{eff}}^2(t_0)$ (\S{}III.7.2). The mechanism (freeze-out at $S^*$) is \epi{SLD}; the value remains \epi{CALIBRATED}, in full consistency with the classification of $f_c$ already declared in \S{}III.7.2 and with the positional status of the tilt $n_s$ (\S{}III.8.4): the same positional character governs the three quantities, because the three are imprinted in the same bounce transition.


\textbf{An exact point on the trajectory of the partition.} That the present value is positional does not prevent the \emph{shape} of the trajectory and its notable points from being geometric. Measuring the total field amplitude relative to the floor with the dimensionless variable $x \equiv (S_\text{tot}-S_{\min}^\text{estr})/S_{\min}^\text{estr}$, the partition between the coherent floor and what exceeds it is written without $S_{\min}^\text{estr}$ appearing ---it cancels---:
\begin{equation}
\Omega_\text{DE}(x) = \frac{1}{(1+x)^2}\,,
\qquad
\Omega_m(x) = 1-\frac{1}{(1+x)^2}\,.
\tag{E62}
\end{equation}
Equality between the two populations, $\Omega_\text{DE}=\tfrac12$, then occurs at
\[
(1+x)^2 = 2
\quad\Longrightarrow\quad
x = \sqrt2-1 = 0.414214
\quad\Longrightarrow\quad
S_\text{tot} = \sqrt2\,S_{\min}^\text{estr} = 1.022600\,,
\]
an exact, uncalibrated value: the geometry does not say \emph{when} the crossing occurs ---the corresponding $z$ is positional, like $H_0$--- but it does say in \emph{which field configuration} it occurs. \epi{DERIVED}.

The scope of (E62) is stated with the same precision: it is an algebraic identity of the partition, and $x$ is a relabelling of $\sqrt{\rho_\text{tot}/\rho_\text{DE}}-1$, not an independent reading of the field. What (E62) contributes is not the value of $\Omega_\text{DE}$ ---which remains positional--- but that the equality point has a closed form independent of the mechanism of the transition.

\textbf{A false friend of this sector.} The value $\sqrt2\,S_{\min}^\text{estr}=1.0226$ falls within $1\,\%$ of the self-dual radius $S^*=\sqrt{S_{\min}^\text{estr}S_{\max,\text{geom}}}=1.0329$ (\S{}II.4.7), and both lie near the density $\rho\sim\rho_P$. They are not the same quantity: their closed forms differ ---$\sqrt2\cdot12(\pi^2-4)/\pi^4$ against $\sqrt{S_{\min}^\text{estr}S_{\max,\text{geom}}}$--- and so do their origins, the first from the late partition and the second from the geometry of the annulus. The numerical proximity is not read as an identity.

A single continuous medium, modulated in three ways over $S_{\min}$, gives dark energy (coherent phase wave), matter (trapped vortices) and space (the diamond lattice of the sea). A single event ---the crystallization at $S^*$ during the bounce--- gives the spectral tilt (\S{}III.8.4), the matter/dark-energy separation and the origin of coherence. \epi{SLD} (mechanism of separation by phase freeze-out, anchored in $S^*$, the U(1) breaking of A5 and the crystallization $Z=12\to Z=4$ already derived); \epi{CALIBRATED} (the value of $f_c$).

\begin{tcolorbox}[ecuacionclave,title={Interpretation --- where matter, dark energy and space come from}]
Matter, dark energy and space are born from the same act: the vacuum freezing. When the universe expands after the bounce and crosses the crystallization point $S^*$, the phase of el Pleno freezes ---like water turning to ice--- and, just as ice forms domains separated by boundaries, the medium is divided in two. Within each domain, the phase is uniform and coherent: that is dark energy, the persistent tension of the vacuum. At the boundaries between domains phase circulation is trapped: those knotted whirls are the particles of matter. And the crystalline pattern of the freeze-out itself is space. They are not three distinct substances put together, but three faces of a single medium that solidifies. How much matter and how much dark energy remain ($f_c$) depends on the speed with which the universe crossed that transition ---that is why that number measures the speed of the last freeze-out of the cosmos, not a derivable constant.
\end{tcolorbox}

\subsection{\texorpdfstring{\S{}IV.5-ter --- The viscous crystallization transient: how the dynamics imprints the partition $f_c$}{Sec.IV.5-ter - The viscous crystallization transient}}

The matter\,/\,dark-energy partition was fixed geometrically by the phase freeze-out at $S^*$ (\S{}IV.5-bis) and the crystallization point $S^*$ by the rheological route (\S{}III.9.11). Between the two, the dynamical link was missing: how the modulus relaxes from $S^*$ to its minimum $S_{\min}$, in how much time, and why the imprinted partition survives intact to today. The viscous equation of motion of the cycle (E54--E57), with the derived effective potential (E52) and the dilatant Gent viscosity, closes that link \emph{without a single free parameter}:
\begin{equation}
\rho_P\ell_P^2\,\ddot S + \eta(\dot\gamma)\,\dot S + \frac{dV_{\text{eff}}}{dS} = 0,
\qquad
\eta(\dot\gamma)=\frac{\mu_{\text{Gent}}\,\tau_\Pi}{1-\dot\gamma/\dot\gamma_{\max}},
\qquad
\dot\gamma=\frac{2|\dot S|}{S},
\end{equation}
with $\dot\gamma_{\max}=c/\ell_P=1/\tau_P$ the hard deformation bound (A6, \S{}III.9.2) and the effective potential (E52)
\begin{equation}
V_{\text{eff}}(S)=\frac{E_0}{2}\frac{S^4}{(S_{\max,\text{geom}}^2-S^2)^2}
-\frac{\rho_P c^2}{2}\ln\!\Big(1-\frac{S^2}{S_{\max,\text{geom}}^2}\Big)
+\frac{3\rho_P c^2}{8\,S^2},
\end{equation}
whose three terms are the hyperelastic body of the well, the Gent UV topological wall (A6) and the Klauder IR quantum wall (A2).

\paragraph*{The coefficients are all derived.} In units of el Pleno ($\rho_P=c=\ell_P=1$, $\dot\gamma_{\max}=1$), the modulus bounds are pure ratios of the lattice: $S_{\min}=12(\pi^2-4)/\pi^4=0.7231$, $S_{\max,\text{geom}}^2=8\sqrt6/9=2.1773$, $S^*=\sqrt{S_{\min}\,S_{\max,\text{geom}}}=1.0329$. The viscosity is the Maxwell relation of the shear sector with its two derived factors, $\mu_{\text{Gent}}=\rho_P c^2/S_{\max,\text{geom}}^2$ (\S{}III.1.6) and $\tau_\Pi=\ell_P/c=\tau_P$ (unit Deborah number, \S{}III.9.4), so that $\eta_0=\mu_{\text{Gent}}\tau_\Pi=1/S_{\max,\text{geom}}^2=0.4593$. The coefficient $E_0=4.2546$ is fixed by $V_{\text{eff}}'(S_{\min})=0$, and the curvature of the minimum reproduces the structural Higgs of the program, $V_{\text{eff}}''(S_{\min})=\xi=19.69\,\rho_P c^2$ (\S{}II.4). Three cross-checks confirm the internal consistency of the potential: on imposing minimum and curvature, the coefficient of the Klauder term comes out $c_k=0.3750=3/8$, exactly the value that the Klauder affine quantization fixes independently (E5); the three walls sum the curvature, $V_q''(8.230)+V_{\text{topo}}''(0.986)+W_{\text{bare}}''(10.473)=19.69=\xi$; and the densities of the critical points reproduce the readings of the program, $\rho(S^*)=1.067\,\rho_P\approx\rho_P$ (Planck density, where space recrystallizes), $\rho(S_{\min})=0.523\,\rho_P$ (the present relaxed sea). \epi{DERIVED}, with no free parameter.

\paragraph*{The deformation bound self-limits the fall.} The natural frequency of the well is $\omega_S=\sqrt{\xi}=4.44/\tau_P$, far above $\dot\gamma_{\max}=1/\tau_P$. A modulus oscillating freely would reach a deformation rate $\dot\gamma\approx 2\omega_S A/S_{\min}\approx 3.8\gg 1$, impossible: the Gent viscosity diverges when $\dot\gamma\to\dot\gamma_{\max}$, braking the motion before the bound is reached. The integration of the EOM with the derived coefficients confirms that the deformation rate never reaches its bound ---the maximum recorded is $0.98<1$. The dilatant rheology ---the same one that gives special relativity as stiffening on approaching $c$ (\S{}III.9.5)--- imposes that the fall respects the deformation limit of el Pleno: the hard bound of A6 acts here on the \emph{velocity} of the radial mode, not on its amplitude. The medium does not flow faster than the Planck tick.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.96\textwidth]{fig_transitorio.png}
  \caption{\textbf{Trajectory of the modulus during the crystallization transient}, integrating the viscous equation of motion with all coefficients derived. The modulus $S(t)$ (violet, left axis) descends monotonically from $S^*=1.033$ and overshoots $S_{\min}$ by inertia to an absolute minimum of $0.672$ at $t\approx 1.1\,t_P$, after which it oscillates with decreasing amplitude and settles at $S_{\min}=0.7231$. The deformation rate $\dot\gamma(t)$ (orange, right axis) reaches a maximum of $0.98$ ---grazing the Planck bound $\dot\gamma_{\max}=c/\ell_P=1$ without exceeding it--- during the fast descent, and pulses toward zero at each turning point of the modulus (where $\dot S=0$). The decreasing peaks of $\dot\gamma$ are the signature of the viscous damping. Time is in Planck ticks.}
  \label{fig:transitorio}
\end{figure}

\paragraph*{The transient is very brief; the rest, durable.} With the self-limited fall, the modulus settles at $S_{\min}$ in a time of the order of a few $t_P$; the permanent entry into the $1\%$ of $S_{\min}$ occurs at $t_{\text{settle}}\sim t_P=5.4\times10^{-44}$ s. Over the age of the universe ($\sim4.4\times10^{17}$ s), the transit occupies a fraction $\sim10^{-61}$. This resolves what dark energy is dynamically: during $99.99\ldots\%$ of the history, the radial mode is still at its minimum, where its contribution is that of the DE-Curtain with $w=-1$ structural (\S{}III.7.5). There is no coherent radial oscillation that persists and dilutes over eons ---the viscosity damps it on the Planck scale. Matter does not come from that oscillation, but from the topological defects formed by the intense shear of the hot transient, stable by the Massey invariant (\S{}IV.6) and persistent as dust, $\rho\propto a^{-3}$.

\begin{tcolorbox}[ecuacionclave,title={Dynamics and geometry are the same physics}]
The matter\,/\,dark-energy partition admits two equivalent readings. \textbf{Geometric:} $f_c$ is the ratio of lattice occupations, fixed by the endpoints $\{S_{\min},S_{\max,\text{geom}}\}$ and the turning point $S_{\max,\text{eff}}$ (\S{}III.7.2). \textbf{Dynamical:} $f_c$ is the fraction of coherent domains that the Kibble--Zurek freeze-out leaves in the viscous transient at $S^*$. They do not compete: the viscous transient \emph{produces} the $f_c$ that the geometry predicts,
\[
f_c=\frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2}
=\underbrace{\frac{S_{\max,\text{eff}}^2}{S_{\min}^2}}_{\text{state}=2.860}
\times
\underbrace{\frac{S_{\min}^2}{S_{\max,\text{geom}}^2}}_{\text{geom}=0.2401}
=0.6869,
\qquad \Omega_m=1-f_c=0.3131,
\]
at $0.27\sigma$ from the observed value. The dynamics imprints the partition near the Planck density; the subsequent cosmology is standard ---matter $\propto a^{-3}$, dark energy of the Curtain with $w=-1$--- without an anomalous factor. There is no order-of-magnitude problem: the component that would dilute by a factor $\sim10^{90}$ ---the coherent radial oscillation--- does not exist beyond the transient, because the viscosity damps it in $\sim t_P$. What persists are the topological defects and the DE-Curtain, both with densities fixed at the freeze-out.
\end{tcolorbox}

\paragraph*{Which part is confrontable.} The transient occurs at Planck density, $\sim10^{-43}$ s after crystallization; between it and the earliest observation ---nucleosynthesis at $\sim1$ s--- there are $\sim43$ orders of magnitude in time, and the Planck epoch is opaque to every probe. The transient is not directly observable, nor can it be: it is the imprint mechanism, and what is confrontable are the seals it engraves and that survive frozen. There are three. The \textbf{spectral index}: the packing crossing $Z=12\to Z=4$ on passing through $S^*$ imprints the tilt of the primordial spectrum (\S{}III.8.4); observed $n_s=0.9649\pm0.0042$ (Planck 2018). The \textbf{matter\,/\,dark-energy partition}: $\Omega_m=1-f_c=0.3131$; observed $\Omega_m=0.315\pm0.007$, agreement to $0.27\sigma$. The \textbf{absence of an initial singularity}: that the trajectory begins at $S^*$ with a finite value, not at a divergence, is the assertion that there was a bounce and not a singular Big Bang, with a nearly scale-invariant spectrum \emph{without inflation} ---the dilatant bounce does the work that inflation does in the standard model (\S{}III.8.3). Nucleosynthesis ($t\sim1$ s) occurs much later, with the modulus already at $S_{\min}$ and standard physics emerging as a limit; the cosmology from BBN onward is the standard one. The distinctive observable difference of PIU-$\Psi$ is not in the early universe ---where it confronts already-imprinted seals, in agreement with the data--- but in the late one: the trajectory of $w(z)$, where the dark energy of el Pleno evolves instead of being constant (\S{}III.8.7).

\begin{tcolorbox}[notabox,title={Epistemic classification --- the viscous transient}]
The integration of the EOM with derived coefficients (potential E52, Gent viscosity, A6 bound) is \epi{DERIVED} with no free parameter, with three cross-checks of internal consistency (the emergent $3/8$ of Klauder, the sum of the three walls to $\xi=19.69$, the densities of the critical points). The self-limitation of the fall by $\dot\gamma_{\max}=c/\ell_P$ and the duration $\sim t_P$ of the transient are \epi{DERIVED}. That this transient imprints the geometric partition $f_c$ is \epi{SLD}; the present value of the numerator $S_{\max,\text{eff}}^2$ remains \epi{CALIBRATED} positional, of the class already declared for $H_0$ and $f_c(t_0)$ (\S{}III.7.2, \S{}III.8.7). No new thread is opened: the structure is closed without a free parameter.
\end{tcolorbox}

\subsection{\S{}IV.6 --- The origin of the three real fundamental particles}

The temporal asymmetry slow-crunch/fast-bounce, over the diamond lattice that crystallizes, produces the three real stable particles. Each one with its moment of the cycle and its conditions, all consistent with the hadronic sector of the program (Block K, tripartite Borromean decomposition of the baryon).

\paragraph*{Proton --- $(u,u,d)$, $B=1$, charge $+1$, permanent.}
It is a tripartite Borromean cluster $B=1$: three vortices of fractional \emph{winding} $1/3$, entangled in a knot of six crossings, with $2+1$ asymmetry selected by the $\mathbb{Z}_3$ modular rule of the Borromean supercounterfluidity. It forms during the post-bounce crystallization (the plateau), anchored in the diamond lattice in quasi-equilibrium (the slow crunch permits the complete formation). It is stable by the non-null Massey invariant: removing a vortex trivializes the other two, which demands infinite energy at $T\ll T_P$. Mass $m_p = 8.10\,\Lambda_\text{PIU}$ (excess of tension over the background). \textbf{[Hadronic sector, Block K]}.

\paragraph*{Neutron --- $(u,d,d)$, $B=1$, charge $0$, almost-stable.}
It is the \emph{same} Borromean cluster $B=1$, with a quark $u$ changed for a $d$ (variant of the proton, not a distinct object; they differ by $\sim 1$ part in a thousand). It forms in the same crystallization as the proton. \textbf{It does not contain an electron}: upon decaying, one of the $d$ quarks converts into a $u$ emitting an electron, $n(u,d,d)\to p(u,u,d)+e^- +\bar\nu$ ---the electron is \emph{created} in the decay, it was not inside. The mass balance verifies the mechanism: $m_n - m_p - m_e = 939.565 - 938.272 - 0.511 = 0.782$ MeV, exactly the $Q$-value of the beta decay. It decays free in $\sim 880$ s. \textbf{[Hadronic sector + verification in Part IV]}.

\paragraph*{Electron --- lepton, charge $-1$, permanent.}
It is a \emph{single} defect (not a tripartite cluster), with direct integer \emph{winding} $-1$. It lives in an interstitial hole of the diamond lattice. It forms in the bounce/expansion, when the diamond lattice emerges and opens the interstitial holes. It is light because it is the minimum defect that carries integer charge: it does not pay the self-energy of the tripartite cluster. Its mass is by Compton quantization ($m_e\,c\,r_e \sim \hbar$), a mechanism \emph{distinct} from the proton's; the individual value of $m_e$ belongs to the leptonic sector, an open thread of the program. \textbf{[Leptonic sector; individual mass open]}.

\textbf{The hierarchy $m_p/m_e$ comes from distinct mechanisms} (tripartite cluster with self-energy versus single defect with Compton scale), not from a single ratio ---consistent with the conclusion of the hadronic sector of the program.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{formacion_particulas_ciclo.png}
\caption{Particle formation in the crunch-bounce cycle of \emph{el Pleno}. Top: the trajectory $S(t)$ of the cycle, which starts from \emph{rest} at the \emph{Pleno}-ground state ($S_{\min}$) and is compressed by the cosmological contraction ($\rho\propto 1/a^3$, $S^2\propto 1/a^3$) ---the crunch is born slow from the background and accelerates toward $S_{\max}$---, followed by the fast bounce regulated by dilatancy, the crystallization plateau where the particles form, and the return to the relaxed regime. Bottom: the origin, moment of the cycle, conditions and mass of the three real fundamental particles, consistent with the hadronic sector of the program.}
\label{fig:formacion}
\end{figure}

\paragraph*{The holes and what the electron is in the hole.}
The interstitial holes that house the electron emerge from the volumetric expansion of the bounce: the lattice dilates by a factor $(S_{\max}/S_{\min})^2 = 4.16$ (from $\rho\sim S^2$), creating a 76\% volume of holes and a 24\% of anchored lattice. The hole that opens in the bounce does \emph{not} release ``material'' (that would violate the principle that a particle is a modulation of the field, not a portion of the medium): the hole \textbf{creates the topological condition} for a defect of \emph{winding} $-1$ ---it gives the \emph{site} where the topology $-1$ can exist (the interstitial of the lattice), while the topology gives the \emph{nature} (the single defect of \emph{winding} $-1$). The electron is the defect that the hole makes possible, not the substance of the hole.

This is anchored in the geometry of the diamond lattice $Fd\bar{3}m$: its two sub-lattices are related by inversion, which inverts the sign of the \emph{winding} (nodes: \emph{winding} $+$; interstitials: \emph{winding} $-$). The diamond occupies half of the tetrahedral sites, leaving exactly one empty interstitial per node ($1:1$). From there the \textbf{cosmic neutrality} emerges: one node $(+1)$ for each interstitial $(-1)$ $\Rightarrow$ net charge zero, each proton with its electron. The charge $+1$ of the proton and the $-1$ of the electron have the same origin (integer \emph{winding} of the defect), with the sign given by the sub-lattice. \textbf{[ontological SLD; the individual mass $m_e$ and the formal homotopy of the \emph{winding} $-1$ over $Fd\bar{3}m$ remain open]}.

\paragraph*{Particles and gauge bosons: two regimes of the same phase circulation.}
The formation of the particles in the bounce and the propagating gauge sector (\S{}III.4.4, \S{}III.4.5) are not two phenomena: they are two regimes of the phase circulation of \emph{el Pleno}. The topological \emph{winding} that defines a particle and the non-abelian holonomy that defines a gauge field are the same object ---the homotopy class of the phase loop of $\Psi$. The distinction is one of regime:
\begin{itemize}[leftmargin=1.6em]
  \item \textbf{Propagating regime} (relaxed medium, today): weak circulation $\Rightarrow$ gauge bosons ---the photon ($U(1)$), the modes of the weak sector ($SU(2)$), the TT mode--- as oscillations of the phase.
  \item \textbf{Trapped regime} (bounce, crystallization): strong and \emph{frozen} circulation $\Rightarrow$ particles as \emph{winding} defects, whose charge is the trapped holonomy class.
\end{itemize}
During the crunch the compression toward $S_{\max}$ forces the phase; in the crystallization plateau the non-trivial holonomies are trapped as \emph{winding} defects ---the particles. The confinement of the clusters by $\mathbb{Z}_3$ center obstruction (\S{}III.5.3) is the frozen version of the same $\mathbb{Z}_3$ structure that the non-abelian circulation produces in the propagating regime. Thus, the bounce is the perturbative$\to$topological transition of the phase circulation: what in the relaxed medium propagates as a boson, in the compressed medium freezes as a particle. \textbf{[SLD; the circulation mechanism is derived in \S{}III.4.5 and \S{}III.5.3, the reading of the two regimes is the dynamical synthesis of this Part]}.



\subsubsection{\texorpdfstring{\S{}IV.6.1 --- The size of the particles and the principle of the clean observable}{Sec.IV.6.1 - The size of the particles and the clean observable}}

The proton radius is already derived in the hadronic sector of the program: $R_p = \xi_\text{PIU}/2 = 0.851$ fm against an observed value of $0.841$ fm ---prediction to $98.9\%$ with a single input ($\Lambda_\text{PIU}$ calibrated to $m_p$, just as quantum chromodynamics uses an input). The size is the extension of the Borromean cluster, \textbf{without gluons}.

\begin{tcolorbox}[notabox,title={Principle of the interpretation-independent observable}]
The ``observed proton size'' is not a neutral datum. Under the Standard Model it is interpreted with the ontology of the gluons (the interior filled with color field, where $\sim 99\%$ of the mass resides). PIU does not postulate gluons, so the comparison must be made against the \textbf{interpretation-independent} observable: the charge radius, measured by how the proton deflects electrons (\emph{scattering}), not the gluon confinement radius. The corpus compares against that clean observable ($0.841$ fm) and reproduces it to $1.1\%$ without gluons.

\textbf{As a general principle:} when confronting PIU with the Standard Model or with $\Lambda$CDM, interpretation-independent observables must be used ---what is effectively \emph{measured} (charge radii, cross sections, energies, shifts)--- and not quantities that incorporate the ontology of the other framework (confinement radii, gluon contributions, parametrized components). PIU does not need to reproduce the interpretive artifacts of the other framework; it needs to reproduce the observed. Confusing the two would lead to the error of declaring ``failed'' a prediction that in reality hits the measured datum. This principle protects every observational confrontation of the program.
\end{tcolorbox}

\textbf{How the crystallization plateau fits.} The equilibrium size of the cluster ($\xi_\text{PIU}$, which gives $R_p=0.851$ fm) is a static property, already derived. The plateau does not redefine it: it decides \emph{whether} the cluster reaches its equilibrium or remains truncated. The proton (slow crunch, long plateau) relaxes to $\xi_\text{PIU}$; the electron (fast bounce, truncated) does not reach the equilibrium of the cluster and remains a much smaller single defect. The two pieces fit together ---the hadronic sector gives the equilibrium size; the plateau and the dilatancy give whether it is reached or truncated, which is the origin of the proton/electron asymmetry. Scale verification: $R_p \approx 4\times$ the Compton length of the proton ($\hbar/m_p c = 0.210$ fm), coherent with a cluster of hadronic Compton scale. \textbf{[Equilibrium size derived in the corpus; role of the plateau, strong logical deduction]}.

\subsection{\S{}IV.7 --- Self-verification: three destinies of the gravitational collapse}

The same Gent potential and the A6 hard bound produce, without any fitted parameter, the three destinies of the gravitational collapse that are observed in nature. The distinction is made by the concentration of mass-energy and the confinement, not new parameters:

\begin{center}
{\scriptsize
\begin{tabular}{p{3.0cm}p{2.4cm}p{2.4cm}p{4.4cm}}
\toprule
\textbf{Regime} & \textbf{Concentration} & \textbf{Reaches $S_{\max}$?} & \textbf{Destiny} \\
\midrule
Diffuse overdensity & low & no & structure (galaxies, gas) \\
Concentrated collapse & high (over threshold) & yes, saturated core & black hole: does not rebound (confined), without singularity (A6) \\
Global cosmological crunch & all of space & yes, simultaneous & Big Bounce: rebounds (without confinement) \\
\bottomrule
\end{tabular}}
\end{center}

\textbf{The key physical distinction.} The core of a black hole and the cosmic bounce reach the \emph{same} state ($S\to S_{\max}$, Gent wall active), but have opposite destinies by the confinement conditions. The black hole is confined by its own horizon: the external gravity retains it, it does not rebound, it remains stable. The cosmic bounce has no ``outside'' to confine it: the Gent wall pushes without retention and it rebounds. Same potential, same mechanism, distinct result by the boundary conditions.

\textbf{Why it is a genuine self-verification.} PIU reproduces the observed phenomenology of gravity with a single mechanism ---diffuse collapse forms structure, concentrated collapse forms stable black holes without singularity, cosmic origin rebounds--- without fitting anything. The program already derives the individual pieces: the minimum black-hole mass $M_{\min} = m_P\sqrt{3/(32\pi S_{\max}^2)} \approx 0.117\,m_P$ (coincides with the Hawking evaporation limit by a distinct ontological reason: interior geometric exhaustion, not divergent temperature), the absence of singularities by the A6 bound (\S{}II.7.2, \S{}III.1), and the primordial bounce (\S{}II.7.2). Part IV unifies them under the Gent rheology: \textbf{the black hole resolves the singularity of General Relativity by the same mechanism that the bounce resolves that of the Big Bang} ---the Gent wall that prevents $S>S_{\max}$. \textbf{[SLD; individual pieces derived in the program]}.

\textbf{Primordial black holes as incomplete crystallization.} The same thermodynamic bottleneck that sustains the plateau (\S{}IV.5, Phase 3) has a consequence for the black holes: the crunch is not perfectly homogeneous, and where the evacuation of the latent heat does not manage to complete the crystallization, the destiny changes. The crystallization $Z=12\to Z=4$ advances as a \emph{front} from the cold interfaces toward the interior of the hot regions, while the expansion of the bounce dilutes the medium; both velocities are bounded by the same hard bound $\dot\gamma_{\max}=c/\ell_P$ (\S{}III.9.2). The destiny is decided by the competition:
\begin{itemize}[leftmargin=1.6em]
  \item where the crystallization velocity exceeds that of expansion, the front catches up and the medium crystallizes to diamond $\Rightarrow$ ordinary matter;
  \item where the expansion exceeds the crystallization, the material escapes before crystallizing and remains in trapped saturated $Z=12$ packing $\Rightarrow$ primordial black hole.
\end{itemize}
This refines the black hole of the central row of the table: the primordial black hole is the population that the bounce left in a saturated state ($S\to S_{\max}$, maximum-packing core retained by its own horizon), not a later stellar collapse. It adds the thermodynamic balance ---insufficiency of the heat sink--- to the confinement that the program already had.

\textbf{Prediction: universal core density.} If the primordial black holes are trapped saturated maximum packing, their core density is $\rho_\text{core} = S_{\max,\text{geom}}^2\,\rho_P \approx 2.1773\,\rho_P$, \emph{universal} ---independent of the mass of the black hole---, finite and non-singular, in contrast with the singularity of General Relativity. It connects with the minimum mass $M_{\min}\approx 0.117\,m_P$, derived from the coincidence between the radius of the saturated core and the Schwarzschild radius with $S_{\max,\text{geom}}$. \textbf{[SLD; falsifiable]}.

\textbf{Connection with the early black holes.} A region that becomes trapped as a primordial black hole, while it does so, gravitationally drags ---by the emergent gravity of the medium itself (\S{}III.1), not by a postulated force--- the neighboring material that does decompress and crystallize. The result is co-natal: a massive central black hole with matter assembling around it from the origin, which offers a reading of the enigma of the massive black holes in epochs too early and of their correlation with the host galaxies already present in the young universe. The quantitative dynamics of the dragging is not computed and is declared a thread (\S{}IV.8). \textbf{[strong HYPOTHESIS; the mechanism is anchored in the emergent gravity, the computation remains open]}.



\subsection{\S{}IV.8 --- Synthesis and epistemic classification}

Part IV articulates a single causal chain, anchored end-to-end in the program, that joins four sections under the dilatant rheology:

\begin{center}
\begin{tabular}{p{7.2cm}p{2.6cm}p{2.4cm}}
\toprule
\textbf{Link} & \textbf{Status} & \textbf{Support} \\
\midrule
Two populations (inert EO / depleted matter) & derived & \S{}III.7.2 \\
Potential $V_\text{eff}$, $E_0$, $S_{\max}$, $S_{\min}$ & derived & \S{}I.7, \S{}II.2 \\
Dilatant flow index $n(x)=1/(1-x)$ & SLD & \S{}III.9.3 \\
Volumetric deformation $\dot\gamma=2|\dot S|/S$ & derived & \S{}II.2.2 \\
Maxwell viscosity $\eta_\text{Pleno}=\mu_\text{Gent}\tau_\Pi$ & derived & \S{}III.1.8 \\
Dilatant viscosity $\tilde\eta = \eta_\text{Pleno}/(1-x)$ & SLD & dilatant factor \\
Structural bounce before $S_{\max}$ & derived & \S{}II.7.2 \\
Crunch engine = cosmological contraction ($S^2\propto 1/a^3$) & derived & \S{}II.7.2 \\
Hubble bound $=$ dilatant bound ($H_{\max}\sim\dot\gamma_{\max}$) & SLD & \S{}II + \S{}III.9 \\
Dilatancy regulates the bounce to $\dot\gamma\leq c/\ell_P$ & SLD & Part IV \\
Slow-crunch / fast-bounce asymmetry & SLD & Part IV \\
Plateau $=$ Landau-Ginzburg crystallization & SLD & \S{}II.3.6 \\
Freeze-out of constants & derived & \S{}II.7.3 \\
Three destinies of collapse (structure / BH / bounce) & SLD & \S{}II.7.2 + Part IV \\
Proton size $R_p=0.851$ fm (without gluons) & derived & hadronic sector \\
Origin of the three particles in the cycle & SLD / hypothesis & hadronic sector + Part IV \\
\bottomrule
\end{tabular}
\end{center}

\textbf{What Part IV contributes to the program:}
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Unifies} four sections (\S{}II.2, \S{}II.3, \S{}III.1, \S{}III.9) under a single dynamical mechanism: the dilatant rheology.
  \item \textbf{Converts a postulate into a consequence:} the slow-crunch/fast-bounce asymmetry ceases to be a postulate and comes to be derived from the Gent wall plus the dilatancy.
  \item \textbf{Gives a physical mechanism} to the asymmetric formation of particles (complete proton / truncated electron), coherent with the hadronic sector.
  \item \textbf{Explains why the universe does not collapse entirely} (inert EO; only 31\% cycles).
  \item \textbf{Self-verification:} the same Gent potential and the A6 bound give the three destinies of the collapse without fitting anything; black hole and bounce resolve their respective singularities by the same mechanism.
  \item \textbf{Demonstrates that the dilatant rheology is necessary, not decorative:} without it the bounce would be superluminal. It is the regulator that makes the cycle physically possible.
  \item \textbf{Establishes the principle of the interpretation-independent observable} (\S{}IV.6.1), which protects every observational confrontation of the program.
\end{itemize}

\textbf{Open threads of Part IV.} (i) The individual mass of the electron $m_e$ (leptonic sector). (ii) The formal homotopy demonstration of the $-1$ \emph{winding} of the interstitial over $Fd\bar{3}m$. (iii) The absolute temporal duration of the crystallization plateau (the equilibrium size is already derived; what would be missing is the computation of the duration to also fix the truncation scale of the electron in numbers). (iv) The exact form of $F_\text{ext}$ coupling the Friedmann contraction to (E57) from first principles.

\medskip

With Part IV closed, Part V quantitatively confronts with observation the set of predictions generated throughout the five Parts and declares the threads that remain open.

\medskip\hrule\medskip

