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% PIU-Psi V5.0 -- Mathematical Compendium -- PART 3: Matter, forces and cosmology (sec. 9-12)
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\section{The gauge sector: Maxwell, Yang-Mills and the charge}
\label{sec:gauge}
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\subsection{The Maxwell equations from phase circulation --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\textbf{The two energies of the vector sector.} A perturbation of the phase
$\delta\theta$ over the lattice decomposes by tensor rank into scalar (Goldstone),
transverse vector (electromagnetism) and transverse tensor (TT mode). The vector
sector $a_\mu$ admits two orthogonal energies:

\emph{(a) Bond energy} (magnitude of the stretching $\sum_i|\Psi_n-\Psi_m|^2$):
produces a \emph{symmetric} tensor, with the cross term $\partial_i a_j\,\partial_j a_i$
of coefficient $+2$. It is a vector phonon, not a photon.

\emph{(b) Oriented circulation of the phase} over the plaquettes: produces an
\emph{antisymmetric} tensor $F_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu$.

\textbf{The coefficient $8/9$ (sum of plaquettes, verified symbolically).} The
circulation energy is the sum of the square of the circulation
$\delta_i^a\delta_j^b F_{ab}$ over the six pairs $i<j$ of tetrahedral bonds, with
the normalized vectors $\boldsymbol{\delta}_i=\tfrac{1}{\sqrt3}(\pm1,\pm1,\pm1)$
(each component therefore carries the factor $1/\sqrt3$, and the product
$\delta_i^a\delta_j^b$ the factor $1/3\cdot1/3=1/9$):
\[
E_\text{circ}=\sum_{i<j}\big(\delta_i^a\,\delta_j^b\,F_{ab}\big)^2 .
\]
Expanding the quadratic form with $F_{ab}$ antisymmetric (sympy), the contributions
over the six pairs sum, for each independent component,
\[
E_\text{circ}=\frac{16}{9}\big(F_{01}^2+F_{02}^2+F_{12}^2\big)
=\frac{16}{9}\cdot\tfrac12\,F_{\mu\nu}F^{\mu\nu}=\frac{8}{9}\,F_{\mu\nu}F^{\mu\nu},
\]
where $F_{\mu\nu}F^{\mu\nu}=2(F_{01}^2+F_{02}^2+F_{12}^2)$ was used for an antisymmetric
tensor. The $16/9$ already includes the $1/9$ of the bond normalization; the crude $16$
is $Z^2=4^2$ (square of the coordination). In terms of derivatives, each
$F_{12}^2=(\partial_1 a_2-\partial_2 a_1)^2$ contributes the diagonal $(\partial_1 a_2)^2$ with
coefficient $+16/9$ and the cross term $(\partial_1 a_2)(\partial_2 a_1)$ with $-32/9$,
exactly in the ratio $-2$ that characterizes $F_{\mu\nu}F^{\mu\nu}$:
\begin{equation}
\boxed{\;E_\text{mag}=\frac{8}{9}\,F_{\mu\nu}F^{\mu\nu},\qquad F_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu\;} \tag{E-Maxwell-1}
\end{equation}
The cross term $-32/9$, opposite to the $+2$ of the symmetric bond sector, is the signature that
the object is antisymmetric, genuinely $F_{\mu\nu}$.

\textbf{Verification by discrete Stokes.} That the object is the field tensor and not an
artefact is confirmed independently: the oriented circulation of the phase over
each tetrahedral plaquette equals the flux of the curl through it ---the discrete
Stokes theorem--- over the six bond pairs of the lattice (verified symbolically).
The sum of the six circulations reproduces $E_\text{circ}=8\,F_{\mu\nu}F^{\mu\nu}$ with
cross term $-32$ and diagonal $+16$ (ratio $-2$), against the bond sector with cross term $+16$
and diagonal $+8$ (ratio $+2$): the two sectors are orthogonal, and only the circulation one
carries the antisymmetry of $F_{\mu\nu}$. The decomposition $8=2^3$ (spectral quotient of the
coordination $Z=4$) and $9=3^2$ (the three dimensions of the average) is a reading of the
result, not its origin; the origin is the sum of plaquettes.

\textbf{The four Maxwell equations.} With the Lagrangian
$\mathcal{L}_\text{EM}=-\tfrac{8}{9}F_{\mu\nu}F^{\mu\nu}$ (E-Maxwell-1) the four
equations come out, they are not postulated:
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Homogeneous} (Faraday $\nabla\times\mathbf{E}=-\partial_t\mathbf{B}$ and
$\nabla\cdot\mathbf{B}=0$). Since $F_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu$ is a
curl, it identically satisfies the Bianchi identity
\[
\partial_{[\lambda}F_{\mu\nu]}=\partial_\lambda F_{\mu\nu}+\partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}=0,
\]
verifiable by substituting $F=\partial a-\partial a$: each term cancels by the
commutativity of the partial derivatives. It is not a dynamical law but a geometric
identity ---that is why there are no magnetic monopoles in the U(1) sector.
\item \textbf{Inhomogeneous in vacuum} ($\nabla\cdot\mathbf{E}=0$,
$\nabla\times\mathbf{B}=\partial_t\mathbf{E}$). The Euler-Lagrange equation of
$\mathcal{L}_\text{EM}$ with respect to $a_\nu$,
\[
\partial_\mu\frac{\partial\mathcal{L}_\text{EM}}{\partial(\partial_\mu a_\nu)}=0
\;\Longrightarrow\;
\partial_\mu F^{\mu\nu}=0,
\]
where the global coefficient $8/9$ cancels upon equating to zero, so that the form of
the equation is independent of it (the $8/9$ fixes the \emph{normalization} of the field, not
the dynamics).
\item \textbf{With source} ($\partial_\mu F^{\mu\nu}=J^\nu$). The current
$J^\nu=\rhoP c^2\lP^2 S^2\partial^\nu\theta$ is the same U(1) Noether current (E7c)
of the charged solitons; coupled to the vector sector, it appears as a source term. The
coefficient remains tied to the hypercharge embedding (\epi{SLD}, open end §\ref{sec:cabos}).
\end{itemize}

\textbf{Why electromagnetism is complex: the origin of $i$.} This is the conceptual
piece of the result. The phase lives in $U(1)$: $\Psi=Se^{i\theta}$, and the imaginary
unit $i$ is the generator of rotations of the phase circle, that is, a rotation
of $90^\circ$ in the plane $(\operatorname{Re}\Psi,\operatorname{Im}\Psi)$. When the phase
circulates over a plaquette, that $90^\circ$ turn is what converts the
\emph{circulation} of phase (an oriented gradient) into a \emph{perpendicular field}
---exactly the relation $\mathbf{E}\perp\mathbf{B}$, both perpendicular to the
propagation. Electromagnetism is not complex by mathematical convention: it is so because
it \emph{is} the dynamics of a $U(1)$ phase field, and the complex structure is intrinsic
to the circle on which the phase lives. Light is, literally, a wave of phase circulation
of el Pleno propagating at $c$.

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- electromagnetism, derived}]
The four Maxwell equations come out of el Pleno, they are not postulated. The field tensor
$F_{\mu\nu}$ is the oriented circulation of the phase over the plaquettes of the lattice, with
coefficient $8/9$ \emph{computed} ---sum of the six tetrahedral circulations,
verified by the discrete Stokes theorem, without a free parameter. The homogeneous ones
(Faraday, $\nabla\cdot\mathbf{B}=0$) are the Bianchi identity that $F=\mathrm{d}a$
satisfies identically ---that is why there are no monopoles in the U(1) sector; the inhomogeneous ones
come out of Euler-Lagrange; the source is the U(1) Noether current of the charged
solitons. And the complex structure of electromagnetism has an origin: the $i$ of
$\Psi=Se^{i\theta}$ is the $90^\circ$ turn that converts phase circulation into a
perpendicular field, the relation $\mathbf{E}\perp\mathbf{B}$. The only open coefficient is the
coupling of the source, tied to the hypercharge embedding (declared, §\ref{sec:cabos}).
\end{tcolorbox}

\subsection{Consistency theorem: light propagates at the change ceiling --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Once Maxwell is derived, a question arises about the architecture of primaries: if Maxwell is
derived and $c$ appears in Maxwell, could $c$ be derived from Maxwell and cease to be primary?
The answer is negative by circularity; the positive content is the theorem that follows.

\begin{tcolorbox}[clave,title={Theorem --- speed of light $=$ change ceiling}]
The massless transverse vector mode of the gauge sector, with Lagrangian
$\mathcal{L}_\text{EM}=-\tfrac{8}{9}F_{\mu\nu}F^{\mu\nu}$,
$F_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu$ (E-Maxwell-1), has phase and
group velocity exactly equal to $c$, the primary change ceiling, at every infrared scale.
\epi{DERIVED}
\end{tcolorbox}

\textbf{Proof. (i) The wave equation.} The Euler-Lagrange equation of
$\mathcal{L}_\text{EM}$ is $\tfrac{8}{9}\partial_\mu F^{\mu\nu}=0\Rightarrow\partial_\mu F^{\mu\nu}=0$:
the global coefficient $8/9$ factors out and cancels ---it does not affect the propagation dynamics,
only the normalization of the energy. In the Lorenz gauge ($\partial_\mu a^\mu=0$),
\[
\Box\,a^\nu=0,\qquad \Box=\frac{1}{c^2}\partial_t^2-\nabla^2,
\]
with the d'Alembertian of the medium, built with the same $c$ of the kinetic term of el Pleno.

\textbf{(ii) Dispersion relation.} For a plane wave $a^\nu\sim\exp[i(kx-\omega t)]$:
\[
\Box\,a^\nu=0\;\Longrightarrow\;-\frac{\omega^2}{c^2}+k^2=0\;\Longrightarrow\;\omega=ck,
\]
whence $v_\text{phase}=\omega/k=c$ and $v_\text{group}=d\omega/dk=c$, both exactly $c$,
without a factor. The coefficient $8/9$, which does fix the normalization of the magnetic energy
(E-Maxwell-1), does not appear in the velocity.

\textbf{(iii) Absence of dispersion in the light branch.} The complete phase equation
(E19) includes the flexural correction:
\[
\Box\,\delta\theta-\frac{\ell_P^2}{20}\Box^2\delta\theta=0
\;\Longrightarrow\;
\Box\Big(1-\frac{\ell_P^2}{20}\Box\Big)\delta\theta=0,
\]
which factors into two branches: (a) $\Box\,\delta\theta=0$, the massless infrared mode (light),
with $\omega=ck$ exact and without dispersion; (b) $(1-\tfrac{\ell_P^2}{20}\Box)\delta\theta=0$, a
massive mode at the ultraviolet scale $\sim1/\ell_P$ (Lee-Wick pole), which is not light. The
electromagnetic branch is (a): $v=c$ at every infrared scale. $\blacksquare$

\textbf{Non-triviality.} The theorem is not a tautology. (i) A massive mode does not saturate the ceiling:
the modulus mode (matter, E18) obeys $\Box\,\delta S+\mu_S^2\,\delta S=0$, with
$\omega^2/c^2=k^2+\mu_S^2$ and $v_g=ck/\sqrt{k^2+\mu_S^2}<c$; only the massless mode propagates
at $c$, and that light does so depends on $F_{\mu\nu}$ being a curl and the U(1) symmetry
protecting the zero mass of the photon. (ii) A sector with its own rigidity $c_\text{EM}\neq c$ would give
two limiting velocities, in violation of A3. (iii) The velocity is $c$ because the vector
sector $a_\mu$ 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 inherits $c$ from the term that defines it.

\textbf{Corollary: $c$ cannot be derived from Maxwell.} For two independent reasons.
\emph{Circularity}: the standard derivation $c^2=1/(\varepsilon_0\mu_0)$ presupposes
$\varepsilon_0,\mu_0$ prior to $c$; in PIU the vacuum constants are properties of el
Pleno that already contain $c$ (the Lagrangian is built over $F_{\mu\nu}[\theta]$, and the
dynamics of $\theta$ carries $c$ in its kinetic term), so that deriving $c$ from Maxwell would return
$c$ to itself ---a true but empty identity, like measuring a ruler with itself.
\emph{Ontological inversion}: $c$ is not the speed of light but the change ceiling of the
energy, prior to space, to time and to light (from $c$ emerge time as accumulated
change and causality A3); light is the consequence ---the massless mode that saturates the
ceiling---, and deriving $c$ from Maxwell would put the consequence as the origin. Moreover $c$ is the standard
of commensurability ($E=Mc^2$, spacetime, sectors); a measurement standard does not measure itself.
The theorem is the correct exploitation of the question: $c$ is not derived, it is verified
that light ---derived--- respects the ceiling that $c$ embodies. The constancy of the speed of
light, a postulate in relativity, is here a result.

\subsection{Yang-Mills from the non-abelian circulation --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

Applying the same mechanism to the matrix phase $\theta_-\in SU(2)$, the bond
factors are $U_i=\exp(iA_\mu^a T^a\delta_i^\mu)$ with non-commuting generators
$[T^a,T^b]=i\varepsilon^{abc}T^c$. The difference from $U(1)$ is that the transport over a
closed plaquette no longer cancels: the holonomy $W=U_1 U_2 U_1^{-1}U_2^{-1}$, expanded
by Baker-Campbell-Hausdorff, leaves a residue proportional to the commutator,
\[
\ln W=-[A_1,A_2]+\mathcal{O}(\varepsilon^3),
\]
which does not vanish because the $A_\mu$ do not commute. That residue is the extra term of the field
tensor:
\begin{equation}
\boxed{\;F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+i[A_\mu,A_\nu]\;} \tag{E-YM-1}
\end{equation}
The term $i[A_\mu,A_\nu]=-f^{abc}A_\mu^a A_\nu^b T^c$ is born from the non-commutativity; the
abelian limit ($[A,A]=0$) recovers Maxwell exactly ---Yang-Mills is Maxwell when the
phase ceases to commute. The three gauge structures of the Standard Model are the same
phase-circulation mechanism: $\theta_+\in U(1)$ (electromagnetism),
$\theta_-\in SU(2)$ (weak), $\mathbb{Z}_3$ vortices (color $SU(3)$).

\subsection{Quantization of the charge and beta function --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The phase lives on a circle ($\theta\in[0,2\pi)$); the finite-energy
configurations tend to a constant value at infinity, and the integer winding (number of
turns) is a topological invariant. The conserved U(1) charge is proportional to it:
\[
Q=n\,Q_0,\qquad Q_0=\hbar,\qquad n\in\mathbb{Z}.
\]
The difference of beta-function coefficients of $U(1)_Y$ and $SU(2)$ is
$b_Y-b_2=29/4$ \epi{DERIVED}. The mixing angle $\sin^2\theta_W$ is
\epi{SLD} conditional on the hypercharge embedding $Y(\Phi)=1/2$ (open end,
§\ref{sec:cabos}).

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest: the gauge sector}]
The conserved charge of the previous link unfolds: Maxwell and Yang-Mills are the
same word of the medium ---phase circulation over plaquettes--- distinguished only
by whether the phase commutes or not. The quantization of the charge is topological (integer
winding), which explains why the electric charge comes in integer multiples, a fact
that the Standard Model incorporates without explaining.

\emph{What it means physically: the fundamental forces are not separate entities
added to the universe. Electromagnetism is the way in which el Pleno propagates the
twists of its own phase; light is a wave of that circulation. The factor $8/9$, the
complex structure of electrodynamics, the quantization of the charge and the unification
of the three gauge forces in a single mechanism are not coincidences of the formalism: they are
consequences of there being a single phase field over a single lattice. What remain open are
the value of the coupling $g$ and the hypercharge embedding, but the \textbf{form} of the
gauge sector ---why there are forces of the Maxwell-Yang-Mills type--- is derived.}
\end{tcolorbox}

\medskip\hrule\medskip

% =====================================================================

\section{Solitons as particles}
\label{sec:solitones}
% =====================================================================

\subsection{Topological classification and statistics --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

A material particle is, in this program, a \emph{localized and stable configuration
of el Pleno}: a region where the field $\Psi$ departs from the background $\Psi_0=\Smin^\text{estr}e^{i\theta_0}$
and returns to it at infinity. That such configurations exist, be stable and have
a definite statistics is not postulated: it follows from the topology of the field and the form of F1.

\textbf{Two topological classes.} According to the topology of the internal configuration
space of the soliton, there are two families, and the distinction \emph{fixes the statistics}:
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Topological class (skyrmions).} Over the target $SU(2)\cong S^3$ (antisymmetric
mode $\theta_-$), the finite-energy configurations with $U\to\mathbb{1}$ at
infinity are classified by $\pi_3(SU(2))=\pi_3(S^3)=\mathbb{Z}$. The integer is the
\textbf{baryon number} $B$, a conserved topological charge: it cannot change
continuously, which gives stability without need of an external confining force.
\item \textbf{Non-topological class (Q-balls).} Over the contractible sub-target
$[\Smin,\Smax]$ ($\pi_n=0$), there is no topological charge; the stability is sustained
dynamically by the U(1) Noether charge (Coleman conditions).
\end{itemize}

\textbf{The hedgehog ansatz ($B=1$).} The fundamental configuration with baryon number
unity is the ``hedgehog'', where the internal orientation of the field points radially:
\[
U(\mathbf{x})=\cos f(r)+i\,\boldsymbol{\sigma}\cdot\hat{\mathbf{x}}\,\sin f(r),
\qquad f(0)=\pi,\ f(\infty)=0,
\]
with $f(r)$ the radial profile that interpolates between the north pole ($f=\pi$ at the
center) and the south pole ($f=0$ at infinity) of $S^3$, covering it \emph{exactly once}: that
unique covering \emph{is} $B=1$. The profile $f(r)$ is obtained by minimizing the energy
over the non-linear field equations (F2).

\textbf{The baryon number as a topological integral.} $B$ is not a label but the
degree of the map $\mathbb{R}^3\cup\{\infty\}\cong S^3\to SU(2)\cong S^3$, given by the
integral of the baryon density with $L_i=U^{-1}\partial_i U$:
\[
B=-\frac{1}{24\pi^2}\int \varepsilon^{ijk}\operatorname{Tr}(L_i L_j L_k)\,d^3x .
\]
For the hedgehog ansatz this integral reduces to a single dimension,
\[
B=-\frac{2}{\pi}\int_0^\infty \sin^2 f\,\frac{df}{dr}\,dr
=\frac{2}{\pi}\int_0^\pi \sin^2 f\,df
=\frac{2}{\pi}\cdot\frac{\pi}{2}=1,
\]
where the change of variable uses $f(0)=\pi,\,f(\infty)=0$ (verified symbolically). The
result is an \emph{integer} independent of the concrete profile $f(r)$: any
continuous deformation of the field leaves it at $1$ ---that invariance is the topological
stability.

\textbf{Stability: the Derrick~\cite{derrick} balance.} Derrick's theorem (1964) warns that a
pure scalar soliton in 3D is unstable ---either it collapses to zero size or it dilutes. F1
evades it because it combines \emph{three} terms with opposite tendencies that balance at a
finite size:
\begin{itemize}[leftmargin=1.4em]
\item the \textbf{kinetic} $\rhoP c^2\lP^2(\partial\Psi)^2$ favors \emph{expanding} (its
energy decreases as the size grows);
\item the \textbf{potential} $V(S)$, with its Gent hard wall at $\Smax$, favors
\emph{contracting} (a compact profile avoids saturation);
\item the \textbf{flexural} $-\gamma_\kappa(\Box S)^2$ prevents the \emph{collapse} to zero
size (it penalizes infinite curvature).
\end{itemize}
The balance of the three fixes a characteristic scale of the order of $\lP$ by
dimensionless factors of F1: the soliton has its own size, neither pointlike nor diffuse. This is what
makes matter have finite extension.

\textbf{The profile equation and the mass.} The profile $f(r)$ minimizes the energy
functional of the hedgehog. With the dimensionless variable $\tilde r=r/\lP$ and the structure of F1
(kinetic + Skyrme~\cite{skyrme}-type flexural + potential), the radial energy density has the form
\[
E[f]=4\pi\,\rhoP c^2\lP^3\!\int_0^\infty\!\Big[\underbrace{\tfrac12\tilde r^2 f'^2+\sin^2 f}_{\text{kinetic}}
+\underbrace{\sin^2 f\Big(f'^2+\tfrac{\sin^2 f}{2\tilde r^2}\Big)}_{\text{flexural (stabilizes)}}
+\underbrace{\tilde r^2\,\mathcal{V}(f)}_{\text{potential}}\Big]\,d\tilde r,
\]
and the condition $\delta E/\delta f=0$ gives the profile equation, a non-linear second-order
ODE:
\[
\big(\tilde r^2+2\sin^2 f\big)f''+2\tilde r f'+\sin 2f\,\Big(f'^2-1-\tfrac{\sin^2 f}{\tilde r^2}\Big)
-\tilde r^2\,\mathcal{V}'(f)=0,
\]
with $f(0)=\pi$, $f(\infty)=0$. Its numerical solution, substituted into $E[f]$, gives a pure number
$\mathcal{E}_0=\mathcal{O}(1)$ of order unity, so that the mass is
$M_Q=\mathcal{E}_0\,M_P/S_{\max,\text{eff}}$. With the normalization of the program
($\mathcal{E}_0\to 1$ by the reading of $S_{\max,\text{eff}}$):
\begin{equation}
M_Q=\frac{M_P}{S_{\max,\text{eff}}}\approx\frac{2.18\times 10^{-8}\,\text{kg}}{\sqrt{1.4956}}
\approx 9.98\times 10^{18}\ \text{GeV}/c^2, \tag{E27}
\end{equation}
with $M_P=\sqrt{\hbar c/G}$ the Planck mass. The fundamental soliton is ultra-heavy
($\sim M_P$): it is the \emph{simplest} configuration, not the one identified with the proton
---that is a composite cluster, developed below, whose \emph{structure} (three vortices,
fractional charge, confinement) emerges from the topology. Its absolute hadronic mass, in
contrast, is not derived: $m_p c^2\approx 938.3$ MeV is the \textbf{only calibrated energy
scale} of the hadronic sector, analogous to $\Lambda_\text{QCD}$ in QCD (status
\epi{CALIBRATED}). What the program derives is the \emph{structure} and the \emph{ratios}
(split $\Delta$-$N$, radius, charges), not the absolute scale.

\textbf{Spin-statistics, derived (Finkelstein-Rubinstein).} The
spin-statistics theorem is not postulated: it \emph{emerges} from the topology. The idea of
Finkelstein-Rubinstein is that rotating a soliton by $2\pi$ is not the
identity if its configuration space has non-contractible loops. For skyrmions of
charge $B$, the configuration space $\mathcal{Q}_B$ (maps of degree $B$ with boundary
condition) has fundamental group
\[
\pi_1(\mathcal{Q}_B)=\mathbb{Z}_2\quad(B\neq 0),
\]
so that the closed loop that describes the physical rotation of $2\pi$ of the soliton is the
non-trivial generator of that $\mathbb{Z}_2$: it cannot be contracted to a point. The quantization
over a space with $\pi_1=\mathbb{Z}_2$ admits two sectors ---even or
odd wave function under the loop--- and the physical sector is fixed by the degree: the rotation of $2\pi$ produces
on the wave function of the soliton the factor
\[
\chi_\text{FR}=(-1)^B:\qquad B\ \text{odd}\to -1\ (\text{fermion}),\quad
B\ \text{even}\to +1\ (\text{boson}).
\]
For the fundamental baryon $B=1$ the factor is $-1$: \textbf{mathematically obligatory
fermionic quantization} (Witten 1983; explicit computation by Krusch 2003). A Q-ball,
with contractible configuration space ($\pi_1=0$), only admits the trivial sector, factor
$+1$: boson. The statistics is not a choice of the model nor an added axiom ---it is
a consequence of the distinct topology of each class.

\begin{center}
\begin{tabular}{L{3.2cm}L{3.0cm}L{2.4cm}L{3.0cm}}
\toprule
\textbf{Sub-target} & \textbf{Topology} & \textbf{$\chi_\text{FR}$} & \textbf{Statistics}\\
\midrule
$SU(2)\cong S^3$ & $\pi_3=\mathbb{Z}$ & $(-1)^B=-1$ & Fermionic\\
$[\Smin,\Smax]$ & $\pi_n=0$ & $+1$ & Bosonic\\
\bottomrule
\end{tabular}
\end{center}
\emph{Reading: that ordinary matter be fermionic ---the Pauli exclusion principle,
on which all of chemistry rests--- ceases to be a postulate. It is the imprint of
the matter particles being topological knots ($B$ odd) in the field of el Pleno, and the
topology of the knot forces the statistics.}

\subsection{The Borromean baryon and the \texorpdfstring{$\mathbb{Z}_3$}{Z3} charge --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The observed baryon (proton, neutron) is not the fundamental monolithic soliton, but a
\emph{richer sub-structure} of the same $B=1$. By the Gudnason-Nitta theorem (2020), the
topological degree $B\in\pi_3(SU(2))$ is equivalent to the linking number (linking
number) of the preimages under the Hopf map, and admits a decomposition into vortex winding:
\begin{equation}
B=\text{Link}(L_1,L_2,L_3)=\sum_\text{clusters}\prod_{i=1}^3 w_i. \tag{E28}
\end{equation}
Applied to the baryon of el Pleno, the integer winding $B=1$ splits into \textbf{three vortices of
fractional winding} $1/3$, which may have distinct profiles without altering the total:
\[
B=1=w_1+w_2+w_3=\tfrac{1}{3}+\tfrac{1}{3}+\tfrac{1}{3}.
\]
Those three vortices of $1/3$ \emph{are} the three quarks: neither their number nor their
fractional charge is postulated, they come from the fact that the unit linking number factorizes into three thirds.

\textbf{The $\mathbb{Z}_3$ arithmetic and the $2+1$ asymmetry.} By the Borromean
supercounterfluidity theorem (Blomquist-Syrwid-Babaev 2021), the \emph{stable}
configurations of the cluster satisfy a congruence modulo $N=3$ over the \textbf{integer numerators}
of the charges (in units of $1/3$), not over the fractions:
\begin{equation}
\sum_{j=1}^3 q_j\equiv 0\pmod 3 . \tag{E29}
\end{equation}
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Proton} $(+\tfrac23,+\tfrac23,-\tfrac13)\to q=(2,2,-1)$: $\sum=3\equiv 0$
\checkmark stable; total charge $\sum q_j/3=+1$.
\item \textbf{Neutron} $(+\tfrac23,-\tfrac13,-\tfrac13)\to q=(2,-1,-1)$: $\sum=0\equiv 0$
\checkmark stable; total charge $0$.
\end{itemize}
The complete structure of the nucleon emerges without postulates: the fractional charges
$\pm\tfrac23,\pm\tfrac13$, their number (three), the $2+1$ asymmetry between quarks, and the
confinement as a $\mathbb{Z}_3$ center obstruction ---why the quarks do not separate
is the same topological reason that the Borromean rings do not unlink pairwise.

\emph{Reading: the proton is not a bag of three point particles put in by hand. It is a
single topological knot of el Pleno whose unit linking is realized, in the most
stable way, as three sub-vortices knotted à la Borromeo. The fractional charge of the quark and its
confinement ---two of the strangest facts of the Standard Model--- are consequences of
that linking geometry, not axioms.}

\subsection{Rotational tower and the \texorpdfstring{$\Delta$-$N$}{Delta-N} split --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\textbf{Collective-coordinate quantization.} The classical skyrmion is invariant under
internal rotations; upon quantizing that freedom (collective coordinate $A(t)\in SU(2)$), the
rotational Lagrangian is that of a rigid rotor,
\[
L_\text{rot}=\tfrac12\Theta_p\,\omega^2,\qquad
H_\text{rot}=\frac{\mathbf{J}^2}{2\Theta_p},
\]
with $\Theta_p$ the moment of inertia of the soliton and $\mathbf{J}$ the spin. The eigenvalues of
$\mathbf{J}^2$ are $\hbar^2 J(J+1)$, so that each rotational level has energy
$\hbar^2 J(J+1)/(2\Theta_p)$. Taking the nucleon ($J=\tfrac12$) as reference and subtracting its
value $J(J{+}1)=\tfrac12\cdot\tfrac32=\tfrac34$:
\begin{equation}
m(J)=m_N+\frac{\hbar^2}{2\Theta_p}\big[J(J+1)-\tfrac{3}{4}\big]. \tag{E30}
\end{equation}
The half-integer spin is \emph{forced} by the fermionic quantization of
Finkelstein-Rubinstein ($B=1$): the skyrmion only admits $J=\tfrac12,\tfrac32,\dots$, not
integer $J$. The tower is, therefore, a consequence of the same topology that gives the statistics.

\textbf{Moment of inertia.} For an approximately uniform baryon-charge distribution
in the radius $R_p$, the moment of inertia is $\Theta_p=m_p R_p^2/3$ (imported Skyrme
framework). The split between $J=\tfrac32$ ($\Delta$) and $J=\tfrac12$ ($N$):
\[
\Delta m=\frac{\hbar^2}{2\Theta_p}\Big[\tfrac{3}{2}\cdot\tfrac{5}{2}-\tfrac{1}{2}\cdot\tfrac{3}{2}\Big]=\frac{\hbar^2}{2\Theta_p}\cdot 3,
\]
and substituting $\Theta_p=m_p R_p^2/3$:
\begin{equation}
\boxed{\;\Delta m=\frac{9\hbar^2}{2 m_p R_p^2}=\frac{9(\hbar c)^2}{2 m_p c^2 R_p^2}\;} \tag{E31}
\end{equation}
\textbf{Numerical verification.} With $\hbar c=197.327$ MeV$\cdot$fm,
$m_p c^2=938.272$ MeV, $R_p=0.8409$ fm:
\[
\Delta m=\frac{9\times(197.327)^2}{2\times 938.272\times(0.8409)^2}=264.1\ \text{MeV}.
\]
Observed $m_\Delta-m_N\approx 293.08$ MeV: agreement to $90\%$, with $m_p$ as the only
calibrated input. The residual $10\%$ is not a structural error: it is the expected correction
of the non-rigid-rotor model ---centrifugal deformation and vibrational/Coriolis modes,
standard physics of the Skyrme model--- over the rigid-rotor approximation that gives the dominant
term. The proton radius is an independent prediction
$R_p^\text{pred}=\xi_\text{PIU}/2=0.851$ fm against $0.8409$ fm observed
($98.8\%$). The complete tower (base factor $88.0$ MeV, verified):

\begin{center}
\begin{tabular}{L{1.4cm}L{2.6cm}L{2.8cm}L{2.6cm}L{1.6cm}}
\toprule
$J$ & $J(J+1)-3/4$ & $m(J)$ predicted & Observed & Agreement\\
\midrule
$1/2$ & $0$ & $938.3$ & $N(939)$ & calibr.\\
$3/2$ & $3$ & $1202.4$ & $\Delta(1232)$ & $97.6\%$\\
$5/2$ & $8$ & $1642.6$ & $N^*(1680)$ & $97.8\%$\\
$7/2$ & $15$ & $2258.8$ & $\Delta^*(1950)$ & $84.1\%$\\
\bottomrule
\end{tabular}
\end{center}

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Matter is not an ingredient added to the universe: it is the knotted form of el Pleno itself.
A particle is a localized and stable configuration of the field, and everything that
characterizes it emerges from its topology and the form of F1, without postulates: its \textbf{existence
and stability} (Derrick balance between kinetic, potential and flexural, which gives it finite
size); its \textbf{statistics} (Finkelstein-Rubinstein: $B$ odd $\to$ fermion, whence
the Pauli principle and, with it, the structure of chemistry); its \textbf{charge} (topological
winding, quantized in integers); and, for the baryon, its \textbf{three-quark
sub-structure} with fractional charges, $2+1$ asymmetry and confinement, all coming from the
Borromean $\mathbb{Z}_3$ linking. The hadronic masses and the rotational tower are
predicted with a single calibrated scale ($m_p$), to $90$--$98\%$. The spin-$\tfrac12$, the
fractional charge and the confinement ---three of the facts that the Standard Model incorporates
without explaining--- are here geometric consequences. What remains as an open end is the
\emph{value} of the individual masses and the rotational framework, imported from Skyrme; the
\textbf{form} of matter is derived.
\end{tcolorbox}


\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest: the particles}]
Particles are not added to the programme: they are the stable configurations F1 admits over the
already derived lattice. The topological classification sorts the catalogue into classes by
winding number and charge, and from each class come properties that are not fitted but
computed: Finkelstein-Rubinstein statistics without postulating spin-statistics, the
$\mathbb{Z}_3$ charge of the Borromean baryon, the rotational tower with its $\Delta$-$N$
split, and the ratio $\sqrt3\pi$ between the weights of the two classes. That ratio deserves
emphasis for its origin: it follows from the tetrahedral geometry of the lattice site combined
with the invariant measure of $SU(2)$; the spherical area entering it is exact with no lattice
correction because the four tetrahedral directions form a spherical 2-design, and its weight
quotient is topologically protected by Gauss-Bonnet ---sphere and tetrahedron are both $S^2$ with $\chi=2$, so that the curvature term
of the Weyl expansion is identical for the two and the equality survives to that order by
topology and not by fortunate cancellation.

What remains open is bounded and declared: the values of the fermionic masses and the
spin-orbit coupling hang on the electroweak cascade, and the $\Delta$-$N$ split closes to
ninety per cent with the pending rotor correction. Neither affects the structure ---the
classification and the geometric ratios are derived---, and both appear in the map of open ends
with their severity.
\end{tcolorbox}

\medskip\hrule\medskip

% =====================================================================

\section{Quantitative cosmology}
\label{sec:cosmologia}
% =====================================================================

\subsection{The dark matter / baryonic ratio \texorpdfstring{$\sqrt{3}\pi$}{sqrt3 pi} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

\begin{equation}
\boxed{\;\frac{\Omega_{MO}}{\Omega_M}=\sqrt{3}\,\pi\approx 5.4414\;} \tag{E37}
\end{equation}
The ratio is computed as the quotient of partition functions $\mathcal{Z}_{MO}/\mathcal{Z}_M$
at the freeze-out temperature, integrating over the topological configurations of each
sector. Three pillars fix it; the important thing is that none of the three is a free
assumption ---each follows from already-established structure.

\textbf{Pillar 1 --- the weight is proportional to the area (derived, not postulated).} The
key claim ---that $\mathcal{Z}_i\propto A_i$, linear in the area--- is not postulated: it follows from the
Gent wall. The potential $V''_\text{topo}(S)\propto(S_{\max,\text{geom}}-S)^{-2}$ diverges
as the hard bound is approached (§\ref{sec:smax}), which \emph{confines} the modulus
fluctuations to a surface shell of thickness $\xi\to 0$ around the defect. In
consequence the log-determinant of the fluctuations (the contribution to $\ln\mathcal{Z}$)
scales with the \emph{area} of that shell, not with the volume: $\mathcal{Z}_i\propto A_i\phi$.
The dominance of surface modes is, then, a consequence of the Gent divergence of
link 6, not a hypothesis. The prefactor $\phi$ is the vibrational density of states of the
ground-Pleno ($v_s=c$, cutoff $1/\lP$), \emph{identical} for M and MO because they are solitons of the
\emph{same} ground; what distinguishes them (topology, moduli) is accounted for separately, so
that $\phi_{MO}/\phi_M=1$ exact and $\phi$ cancels in the quotient. \epi{SLD}.

\textbf{Pillar 2 --- spinor factor $1/2$.} The quotient of invariant volumes
$\text{Vol}(SO(3))/\text{Vol}(SU(2))=\pi^2/(2\pi^2)=1/2$ (Peter-Weyl, $SO(3)\cong SU(2)/\mathbb{Z}_2$):
ordinary matter (fermionic, lives in $SU(2)$) undergoes the reduction $1/2$ upon projecting to
$SO(3)$ observables; the dark one (scalar, in $SO(3)$) does not. \epi{DERIVED}.

\textbf{Pillar 3 --- geometric assignment.} The defects with conserved topological charge
(M, $B=1$) anchor to the local Voronoi cell of the diamond lattice (tetrahedron,
$A_T=(8\sqrt{3}/3)R^2$); the chargeless ones (MO, Q-balls) expand isotropically (sphere,
$A_S=4\pi R^2$). The two forms \emph{emanate from} $\mathcal{L}_{F1}$, they are not postulated: the chargeless Q-ball ($B=0$, global Noether charge, unwound phase) has a free boundary and minimizes the isotropic gradient term $\tfrac12\rho_P c^2\ell_P^2|\nabla S|^2$ at fixed volume $\to$ sphere (isoperimetric theorem); the charged soliton ($B=1$, wound phase, winding $=$ holonomy) has its boundary \emph{pinned} to the nodes of the substrate ---it cannot be rounded off without unwinding the charge, forbidden by topological conservation--- and remains confined to the Voronoi cell of the $Z=4$ node $\to$ tetrahedron. The Kibble-Zurek transition fixes \emph{how many} defects are nucleated, not \emph{what shape} each takes; the shape is selected by the topology of F1. \epi{DERIVED}.


\textbf{Why $A_S=4\pi R^2$ 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, and the reason is an already derived property of the
substrate.

\emph{The coordination $Z=4$ is a spherical 2-design.} A set of points
$\{\hat n_i\}_{i=1}^N$ on $S^2$ is a $t$-design if the discrete average over it reproduces
exactly the continuous average for every polynomial of degree $\leq t$:
\[
\frac1N\sum_{i=1}^{N} p(\hat n_i)\;=\;\frac{1}{4\pi}\int_{S^2} p(\hat n)\,d\Omega ,
\qquad \deg p\leq t .
\]
The four tetrahedral directions satisfy that condition with $t=2$. Indeed, for degree one the
sum vanishes, $\sum_i\hat n_i=0$, which is the equilibrium condition already imposed; and for
degree two,
\[
\frac14\sum_{i=1}^{4}\hat n_i^a\hat n_i^b=\frac13\delta^{ab}
=\frac{1}{4\pi}\int_{S^2}\hat n^a\hat n^b\,d\Omega ,
\]
which is exactly the second-order isotropy from which $\cos\theta=-1/3$ was deduced: the four
vectors reproduce the normalised identity tensor with no anisotropic residue.

\emph{Consequence for the area.} The area of an isotropic surface is a degree-zero observable
in the sampling directions, and degree zero lies trivially within the reach of a 2-design. The
surface integral is therefore \emph{exact} with only four sampling points, with no lattice
correction term:
\[
A_S=4\pi R^2 \quad\text{with no correction},
\]
and the first anisotropy that could spoil it is the fourth-order one, which the cubic stacking
annuls ---precisely the discriminant $0$ against $4/81$ that selects cubic diamond over
wurtzite.

\emph{Why this matters here.} The result does not rest on assuming empty space between the
nodes, which would violate A1: the area comes out exact \emph{because} the already derived
geometry of the substrate is a 2-design, not because the defect is treated as a bubble in a
vacuum. The ratio $\sqrt3\pi$ inherits that exactness. \T{DERIVED}, reinforcement.

\textbf{Assembly} (verified symbolically):
\[
\frac{\Omega_{MO}}{\Omega_M}=\frac{A_S\cdot 1}{A_T\cdot(1/2)}=\frac{2A_S}{A_T}
=\frac{2\cdot 4\pi R^2}{(8\sqrt{3}/3)R^2}=\frac{8\pi}{8\sqrt{3}/3}=\frac{3\pi}{\sqrt{3}}=\sqrt{3}\,\pi .
\]
\textbf{Confrontation.} Planck 2018: $\Omega_\text{dm}/\Omega_b=5.375\pm 0.077$;
difference $1.23\%$, equivalent to $0.86\sigma$, without free parameters.

\begin{tcolorbox}[apunte,title={Against what it is confronted: robustness hierarchy of the ``Planck values''}]
Caution that governs the cosmological confrontation (continues the principle of §11.9). The ``Planck values'' are not of the same type: the acoustic angle $\theta_*$ and the \emph{physical densities} $\omega_b=\Omega_b h^2$, $\omega_\text{dm}=\Omega_\text{dm}h^2$ are read in a nearly model-independent way from the peak heights; in contrast $H_0$, $\Omega_m$, $\Omega_\Lambda$, $\sigma_8$ are \emph{outputs} of the six-parameter $\Lambda$CDM fit. The reading has two sides. The \textbf{strength, which must not be minimized:} the ratio $\Omega_{MO}/\Omega_M=\omega_\text{dm}/\omega_b$ is the \emph{most robust} confrontation of the sector ---the factor $h^2$ cancels and it is compared against physical densities--- and that a pure-geometry number fall to $0.86\sigma$ of it \emph{without a free parameter and without looking at the datum} is an improbable coincidence under the null hypothesis (Bayes factor, §15), genuine cross-validation, not an artefact; that the target come out of a fit does not make it arbitrary, since the peaks fix it with little freedom. The \textbf{caution}, at the other extreme: $\Omega_{EO}=f_c$ and $\Omega_m=1-f_c$ are of the derived type (they share the model dependence of $\Omega_\Lambda$); their agreement is real and counts in favor, but their target is a fit parameter, not a raw datum. The rule is not to distrust coincidences ---they are the evidence--- but to grade their weight by the robustness of the target. One avoids \emph{calibrating} against an interpretation; one preserves the success of \emph{coinciding} with it without seeking it.
\end{tcolorbox}

\textbf{The objection: is the geometry forced?} The strongest criticism against
(E37) is not that the computation fails ---the algebra is exact--- but that the \emph{input
geometry} could be chosen: the formula combines the area of a sphere and that of the Voronoi of a
tetrahedron; is the sphere/tetrahedron choice forced, or is it one of several that coincidentally
reproduces the datum? The sphere is inevitable (isotropy $\Rightarrow$ sphere, by definition).
All the force falls on whether the lattice of the substrate \emph{must} be the diamond $Z=4$. The
distinction D1-a/D1-b (link 5) situates the weight of the argument: the \textbf{tetrahedral
coordination $Z=4$ is a theorem}, not a conjecture ---the only arrangement of four
nearest neighbors compatible with the isotropy of A1 (D1-a, via spherical
2-design)--- and the \textbf{cubic stacking is derived} (D1-b, minimum of the
isotropic flexural functional: fourth-order uniaxial anisotropy $0$ cubic vs $4/81$
wurtzite). $\sqrt{3}\pi$ depends on the tetrahedral coordination and the A/B bipartition
---both fixed--- and the geometry of the lattice is derived in its two pieces. The precise
statement is, then, ``$\sqrt{3}\pi$ rests on a derived tetrahedral coordination (D1-a),
a derived cubic stacking (D1-b) and an axiomatic bipartition''. The status of (E37) is
\epi{DERIVED} in its algebra and \epi{SLD} in its input geometry: the weight of the argument
falls on an entirely derived lattice and an axiomatic bipartition, without a pending geometry open
end.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the harvest: cosmology}]
F1 taken to the whole universe: the proportion of dark to baryonic matter ---in
$\Lambda$CDM, two free parameters fitted separately--- is here a pure number
of geometry, two areas (sphere vs tetrahedron) and a spinor factor $1/2$, with the surface
weight derived from the Gent wall. The agreement at $0.86\sigma$ with zero fitting is one of
the two firm cosmological predictions of the program, both inherited from the diamond lattice of
link 5, whose geometry is derived in both pieces ---coordination $Z=4$ and cubic stacking---,
so that neither hangs on an assumption. The partition between the two populations further has
an exact point on its trajectory: they weigh the same when the total modulus equals
$\sqrt2\,\Smin$, a closed form independent of the mechanism of the transition. And the local
value of $H_0$ is excluded as a background by a margin that does not rest on the exact value of
$\Omega_m$: the age ceiling moves $1.8\hspace{0.08em}\%$ over the whole plausible range,
against the $8.1\hspace{0.08em}\%$ by which that value lies from it.
\end{tcolorbox}

\subsection{Effective Friedmann equations and \texorpdfstring{$w(z)$}{w(z)} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

On FLRW, the background $\bar S(t)$ obeys (restriction of F2 to the homogeneous background):
\[
\ddot{\bar S}+3H\dot{\bar S}+\frac{1}{\rhoP\lP^2}V'_\text{eff}(\bar S)=0,
\]
with $\rho_\text{Pleno}=\tfrac{1}{2}\rhoP\lP^2\dot{\bar S}^2+V_\text{eff}$,
$p_\text{Pleno}=\tfrac{1}{2}\rhoP\lP^2\dot{\bar S}^2-V_\text{eff}$. Substituting into
the emergent Einstein equations (E35) with $G$ from (E3):
\[
H^2=\frac{8\pi G}{3}(\rho_\text{Pleno}+\rho_M+\rho_{MO}),
\qquad
\frac{\ddot a}{a}=-\frac{4\pi G}{3}\sum_i(\rho_i+3p_i),
\]
identical in form to the standard ones (the ``components'' are configurations of the same
medium, $G$ derived). The equation of state is the quotient $w=p_\text{Pleno}/\rho_\text{Pleno}$;
writing the kinetic term as $K\equiv\tfrac12\rhoP\lP^2\dot{\bar S}^2$ and defining
$\delta(t)\equiv K/V_\text{eff}(\bar S)\geq 0$,
\[
w=\frac{p_\text{Pleno}}{\rho_\text{Pleno}}=\frac{K-V_\text{eff}}{K+V_\text{eff}}
=\frac{\delta V_\text{eff}-V_\text{eff}}{\delta V_\text{eff}+V_\text{eff}},
\]
whence, cancelling $V_\text{eff}$,
\begin{equation}
\boxed{\;w(t)=\frac{\delta(t)-1}{\delta(t)+1}\;}
\end{equation}
and the gravitational coupling $\kappa=c^4/(2G)$. In the relaxed limit
($\dot{\bar S}\to 0$, $\delta\to 0$), $w\to -1$ exact. The canonical kinetics
guarantees $w\geq -1$, since $w+1=2K/(K+V_\text{eff})=\dot{\bar S}^2/(\tfrac12\dot{\bar S}^2+V_\text{eff}/(\rhoP\lP^2))\geq 0$:
the phantom region ($w<-1$) is structurally vetoed, without need to impose it.

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
The entire history of the universe ---its expansion, its content, its acceleration--- is the
history of a single medium changing configuration. What cosmology calls
``matter'', ``radiation'' and ``dark energy'' are not separate ingredients thrown into
an empty container, but distinct states of the same Pleno: how much its modulus moves
($\dot{\bar S}$) against how much tension it stores ($V_\text{eff}$) fixes the equation of
state at each epoch. From this comes, without fitting, that the universe can never behave
like phantom energy ($w<-1$): that region is physically forbidden because it would require
negative kinetic energy, something a real medium does not have. The Friedmann expansion is,
literally, the medium breathing between motion and tension.
\end{tcolorbox}

\subsection{Dark energy \texorpdfstring{$w=-1$}{w=-1} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The equation of state of the ground-Pleno is not $w=-1$ by postulate, but the result
of the derived relation $w(t)=(\delta-1)/(\delta+1)$ (§11.2), with
$\delta=K/V_\text{eff}\geq 0$. The late cosmological sector is the phase condensate at
$\Smin$ with $\dot\theta_0=0$ and the radial modulus frozen (§11.9): $\delta=0$
\emph{structurally}, so that $w=-1$ \emph{exact} is the present value, not only
an asymptotic attractor. The positivity of the canonical kinetics also vetoes the phantom
region ($w+1\geq0$), and no internal mechanism displaces $w$ from $-1$ (the two candidates
are excluded by algebra, below). The value $w=-1$ is geometric ---the symmetry
$T_{\mu\nu}\propto g_{\mu\nu}$ of the ground state---, not a calibrated number.

In the relaxed limit, with $p=-\rho$, the energy-momentum tensor of the fluid
$T_{\mu\nu}=(\rho+p)u_\mu u_\nu-p\,g_{\mu\nu}$ reduces to
\[
T^{(w=-1)}_{\mu\nu}=(\rho-\rho)u_\mu u_\nu+\rho\,g_{\mu\nu}=\rho\,g_{\mu\nu}:
\]
the four-velocity disappears, the dark energy has no dynamical rest frame
---it is a homogeneous structural property (DE-Curtain). The acceleration:
\[
\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho(1+3w)=-\frac{4\pi G}{3}\rho(1-3)=+\frac{8\pi G}{3}\rho>0 .
\]
Density: $\rho_\text{DE}^{(0)}=V(\Smin^\text{estr})=\rhoP(\Smin^\text{estr})^2=0.5229\,\rhoP$.
Matter is the excess $\rho_\text{matter}=\rhoP[S^2-(\Smin^\text{estr})^2]$.
Thermodynamically, $\rho+p=0$ implies entropy $s=0$: the relaxed vacuum is an elastic
solid, not a fluid.

\textbf{The $w_0$--$w_a$ signal of DESI as a parametrization artefact.} The late cosmological sector is the phase condensate at $\Smin$ with $\dot\theta_0=0$: $T_{\mu\nu}\propto g_{\mu\nu}$ gives $w=-1$ \emph{exact}, without a kinetic contribution to displace it (the radial modulus is ultra-Planckian and frozen, §11.9). The two mechanisms that could produce a dynamic deviation are vetoed by algebra: a condensate that gained physical density would give $w<-1$ phantom (forbidden by the positivity of the canonical kinetics, $w+1\geq0$), and a phase mode that \emph{rolled} ($\dot\theta_0\neq0$) does not give $w=-1$ constant but dilutes like matter ---from the conserved Noether charge $Q=S^2\dot\theta\,L^3=\text{const}$ and the continuity $S^2L^3=\text{const}$ it follows $\dot\theta=\text{const}$ and $\rho_{\text{kin}}=\tfrac12\rhoP\lP^2 S^2\dot\theta^2\propto S^2\propto L^{-3}$, dust scaling incompatible with a $w=-1$ component. The ground state has $\dot\theta_0=0$ exact (minimum of the phase potential), so that only $w=-1$ remains. The DESI observations (2024--2025), combined with BAO/SNe/CMB, prefer at $2.5$--$3.9\sigma$ dynamical dark energy with $w_0\approx -0.75$, $w_a\approx -1$ when fitted with the $w_0$--$w_a$ parametrization. A medium with $w=-1$ exact whose $H(z)$ coincides with $\Lambda$CDM at linear order, but differs in the non-linear regime of the walls, produces exactly that type of signal when forced within the parametric mold: 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 attenuates or disappears when measured with model-independent observables (raw $D_H/r_d$, the shape of $E(z)$), and that its residual ---if any--- tracks the non-linear correction of the walls, not a dynamical $w_0>-1$. \epi{SLD}.

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Dark energy is not a substance added to the universe: it is the tension of the relaxed
vacuum itself. When el Pleno rests in its configuration of minimal rigidity ($\Smin$), its
phase is frozen in coherence and its pressure equals in magnitude its density with opposite
sign ($\rho+p=0$): it is an elastic solid under uniform tension, not a gas. That tension
does not dilute as space expands ---they are not particles that separate, but a
property of the medium at each point---, and that is why it dominates more and more as matter
rarefies, accelerating the expansion. Its equation of state is $w=-1$ exact: the signal of
dynamical dark energy that DESI hints at when fitting with the $w_0$--$w_a$ parametrization is not
a physical deviation of the vacuum, but the artefact of forcing a medium with $w=-1$ exact
---whose $H(z)$ differs from $\Lambda$CDM only in the non-linear regime of the walls--- within
that mold. The program predicts that the signal dissolves in raw observables.
\end{tcolorbox}


\subsection{The ponderomotive drag of matter by the dark-energy wave --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The DE-Curtain is the homogeneous component $\theta_0$ of the phase, with $w=-1$ and without a rest
frame ($T_{\mu\nu}=\rho\,g_{\mu\nu}$): being homogeneous, it exerts no net local force. On that
background, the phase perturbations $\delta\theta$ ---the gapless Goldstone mode--- ripple it,
and their coupling to matter emerges from the second order of the kinetic term of F1.

\textbf{The two modulations.} On the relaxed vacuum at $\Smin$, the phase kinetic term
gives, at linear order, $|\partial\Psi|^2=(\partial\delta S)^2+\Smin^2(\partial\delta\theta)^2+\mathcal O(\delta^3)$:
the two modulations decouple. The wave $\delta\theta$ (Goldstone, massless, $\Box\delta\theta=0$,
at $c$) and the matter $\delta S$ (massive by $V''_\text{eff}$, localized in defects) do not mix
at linear order: the drag is necessarily non-linear.

\textbf{The coupling from F1.} The phase kinetic term is
$\tfrac12\rhoP c^2\lP^2 S^2(\partial\theta)^2$. With $S=\Smin+\delta S$, at second order:
\[
S^2(\partial\delta\theta)^2=\underbrace{\Smin^2(\partial\delta\theta)^2}_{\text{free wave}}+\underbrace{2\,\Smin\,\delta S\,(\partial\delta\theta)^2}_{\text{coupling}}+\mathcal O(\delta^3).
\]
The cross term $\mathcal L_\text{couple}=\rhoP c^2\lP^2\,\Smin\,\delta S\,(\partial\delta\theta)^2$
couples the matter to the energy of the wave. It is not an added ingredient: it is the second order of the
same term that at first order gives the free equations E6 and E7. \epi{DERIVED}.

\textbf{The matter equation with drag.} Varying with respect to $\delta S$:
\begin{equation}
\boxed{\;\Box\,\delta S-\frac{\lP^2}{20}\Box^2\delta S+\mu_S^2\,\delta S=\frac{\Smin}{\lP^2}\,(\partial\delta\theta)^2\;}
\end{equation}
with $\mu_S^2=V''_\text{eff}/(\rhoP c^2\lP^2)$. The right side $(\partial\delta\theta)^2$ is the
local energy density of the wave: where it concentrates energy, it forces the matter field ---the
structure of a ponderomotive force.

\textbf{The drag force.} For a traveling wave $\delta\theta=A\cos(kx-\omega t)$ with
$\omega=ck$, the time average $\langle(\partial_x\delta\theta)^2\rangle=\tfrac12 A^2k^2$. The
matter feels the gradient of this energy:
\begin{equation}
\boxed{\;\mathbf F_\text{drag}=-\rhoP c^2\lP^2\,\Smin\,\nabla\langle(\partial\delta\theta)^2\rangle\;}
\end{equation}
A uniform wave does not drag ($\nabla\langle\cdots\rangle=0$); the wave (massless, at $c$) goes
faster than the matter (massive, $v<c$), which follows it partially; and the sign
$\mathbf F=-\nabla\langle\cdots\rangle$ attracts the matter toward the minima of phase agitation.
\epi{DERIVED} (coupling and ponderomotive form, second order of F1).

\textbf{Falsifiable consequences.} The drag is a dark-energy--matter coupling with two
local consequences of definite sign, \epi{SLD}: (i) modified structure growth ---the
matter feels, besides the emergent curvature, $\mathbf F_\text{drag}$ (effective fifth force
mediated by the wave), deviating $\sigma_8$ and $f\sigma_8$ from $\Lambda$CDM---; (ii)
local phase-density anti-correlation ---the matter accumulates in the minima of phase energy,
distinguishable from the structure by curvature alone. These feed the open end F-cosmo (structure
growth, $S_8$) with a derived mechanism, without closing it: the magnitude depends on the amplitude of
the phase perturbations, not fixed without additional input, and remains \epi{CALIBRATED}. The
global deviation of $w=-1$ that DESI observes does not come from this mechanism (it is the non-relaxed
background, zeroth order), but the drag is a separate perturbative effect on the growth
of structure.

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Matter is dragged by dark energy like sand by a wave. When the phase
of the vacuum ripples, those waves transport energy, and the concentrated energy pushes the
matter toward where the wave is calmest ---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 speed at which
galaxies grow and clusters matter where the phase is calm. The decisive thing is that
this force is not postulated: it appears by itself upon expanding el Pleno to second order, which
reveals that matter and dark energy are not two distinct things that interact, but two
ripples of the same medium that share substance.
\end{tcolorbox}


\subsection{Phase freezing: the origin of the partition between dark energy and matter (Kibble-Zurek) --- \texorpdfstring{\T{SLD}}{[SLD]} / \texorpdfstring{\T{CALIBRATED}}{[CALIBRATED]}}

\textbf{Angular ontology of $\Smax$.} The saturation bound $S_{\max,\text{geom}}^2=8\sqrt6/9$
admits an angular reading: the factor $8/9$ is $\sin^2\theta_\text{tet}$, with
$\theta_\text{tet}=\arccos(-1/3)=109.47^\circ$ the tetrahedral bond angle,
\[
\sin^2\theta_\text{tet}=1-\cos^2\theta_\text{tet}=1-\tfrac19=\tfrac89 .
\]
At $\Smax$ el Pleno is continuous mass without vacuum, compressed to saturate the geometry of the bond
---not packed spheres with gaps; the Kepler-Hales packing is the contact anchor, the
angular reading is the ontology of the continuous medium.

\textbf{The path of the continuum.} The post-bounce expansion traverses the modulus from $\Smax$ to $\Smin$
(densities $\rho\propto S^2$, in $\rhoP$): from $\Smax$ ($\rho=2.177$, amorphous continuous mass, U(1)
restored) to $S^*$ ($\rho=1.067$, crystallization to diamond lattice and phase separation) to $\Smin$
($\rho=0.523$, relaxed diamond lattice, the present sea). The total compression is
$\Smax^2/\Smin^2\approx4.16$.

\textbf{The freezing.} At $\Smax$ the phase $\theta$ is thermally disordered (U(1)
restored, without condensate). Upon crossing the transition at $S^*$, U(1) breaks (A5): causally
disconnected regions choose independent phases, forming domains of coherent phase
(Kibble-Zurek). Upon the completion of crystallization, the phase remains frozen in two complementary
structures:
\begin{itemize}[leftmargin=1.4em]
\item Within each domain: uniform coherent phase, $\Box\theta=0$, propagating mode. It is the
dark energy (the DE-Curtain, ground-Pleno in its coherent mode).
\item At the boundaries: trapped phase circulation, topological defects of \emph{winding}. They are
the particles (the depleted matter, the vortices of the hadronic sector).
\end{itemize}
The volume fraction in coherent domains is $f_c$; the fraction in defects is $1-f_c$.

\textbf{Derived properties of dark energy.} As intra-domain coherent phase over the sea
$\Smin$: it has $w=-1$ ($\Box\theta=0$, $S=\Smin$ uniform, $\rho+p=0$, \derivado); it does not dilute (the
coherent phase is not countable, does not go as $a^{-3}$, \derivado); it propagates over the medium (satisfies
E7, $\partial_\mu(S^2\partial^\mu\theta)=0$, with coefficient $S^2$: the phase does not exist without the modulus).
The matter (defects) is trapped circulation, with mass, countable, $\rho\propto a^{-3}$. The IR wall
of Klauder guarantees $S\ge\Smin$: the sea never empties.


\textbf{Corollary: 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: the point where the two populations weigh the same has a closed form and does
not depend on the mechanism of the transition.

\emph{Construction of the variable.} Measure the total field amplitude relative to the floor
with
\[
x \;\equiv\; \frac{S_\text{tot}-\Smin^\text{estr}}{\Smin^\text{estr}}
\qquad\Longleftrightarrow\qquad
S_\text{tot}=(1+x)\,\Smin^\text{estr},
\]
a dimensionless variable that equals $0$ when the medium is exactly at the floor and grows with
the excess. Under the constitutive identification $\rho\propto S^2$ of \S{}2, the total density
and that of the coherent floor are
\[
\rho_\text{tot}=K\,S_\text{tot}^2=K(1+x)^2(\Smin^\text{estr})^2,
\qquad
\rho_\text{DE}=K\,(\Smin^\text{estr})^2,
\qquad K=\rhoP c^2 .
\]

\emph{The partition.} The relative weight of the coherent floor is the quotient of those two
densities,
\[
\Omega_\text{DE}(x)=\frac{\rho_\text{DE}}{\rho_\text{tot}}
=\frac{K(\Smin^\text{estr})^2}{K(1+x)^2(\Smin^\text{estr})^2},
\]
and here both the anchor $K$ and the square of the floor $(\Smin^\text{estr})^2$ cancel at
once, leaving
\begin{equation}
\Omega_\text{DE}(x)=\frac{1}{(1+x)^2},
\qquad
\Omega_m(x)=1-\frac{1}{(1+x)^2}=\frac{x(2+x)}{(1+x)^2},
\tag{E62}
\end{equation}
where the second form follows from placing the $1$ over a common denominator:
$[(1+x)^2-1]/(1+x)^2=[1+2x+x^2-1]/(1+x)^2=x(2+x)/(1+x)^2$. Note that
$\Omega_\text{DE}+\Omega_m=1$ identically, as it must be under flatness.

\emph{The equality point.} The two populations weigh the same when $\Omega_\text{DE}=\tfrac12$,
that is
\[
\frac{1}{(1+x)^2}=\frac12
\;\Longrightarrow\;
(1+x)^2=2
\;\Longrightarrow\;
1+x=\sqrt2
\;\Longrightarrow\;
x=\sqrt2-1=0.4142136,
\]
taking the positive root because $S_\text{tot}\geq\Smin^\text{estr}$ forces $x\geq0$. In terms
of the field,
\[
S_\text{tot}=\sqrt2\;\Smin^\text{estr}=1.4142136\times0.7230871=1.0225995 ,
\]
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. \T{DERIVED}.

\emph{Scope, declared.} (E62) 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 it contributes is not the value of $\Omega_\text{DE}$ ---which remains positional,
with its calibrated ingredient--- but that the equality point has a closed form independent of
the mechanism of the transition: any history that carries the medium through that value of the
modulus passes through the half-and-half partition.

\emph{A false friend of this sector.} The value $\sqrt2\,\Smin^\text{estr}=1.0226$ falls within
$1\hspace{0.08em}\%$ of the self-dual radius
$S^*=\sqrt{\Smin^\text{estr}S_{\max,\text{geom}}}=1.0329$, and both lie near the density
$\rho\sim\rhoP$. They are not the same quantity: their closed forms differ ---$\sqrt2\cdot
12(\pi^2-4)/\pi^4$ against the geometric mean of the two walls--- 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.

\textbf{Status of the partition $f_c$.} The partition between coherent dark energy ($f_c\approx0.6869$) and
depleted matter ($1-f_c\approx0.3131=\Omega_m$) is not derived geometrically: although its \emph{present value} is of type A ---it is read from the instantaneous value of the numerator $S_{\max,\text{eff}}^2(t_0)$, as $H_0$ is read from $\Theta_0$ (§4.4)---, that value was \emph{imprinted} by the rate of the last phase transition: the fraction of
defects in Kibble-Zurek depends on the \emph{rate} of the transition (the velocity of the bounce). This situates $f_c$ as a measure of the freezing velocity of the
last phase transition of the universe: physical but trajectory-dependent, on equal footing
with $H_0$ and with the numerator $S_{\max,\text{eff}}^2(t_0)$ of $f_c$ itself. The mechanism (freezing
at $S^*$) is \epi{SLD}; the value remains \epi{CALIBRATED}, the same positional character that governs
the tilt $n_s$ ---the three quantities are imprinted in the same bounce transition, and all are state readings of the single trajectory $\bar S(t)$.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- a single substance, three modulations}]
One and the same continuous medium, modulated in three ways over $\Smin$, gives dark energy (coherent phase
wave), matter (trapped vortices) and space (the diamond lattice of the sea). One and the same event
---the crystallization at $S^*$ during the bounce--- gives the spectral tilt, the
matter/dark-energy separation and the origin of coherence. That the wave-matter coupling be derived and
not postulated ---that it come out of the same kinetic term that gives the free waves--- is the signature that
matter and dark energy share a single substance: el Pleno, modulated.
\end{tcolorbox}


\subsection{The perfect blackbody and the primordial spectrum --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

\textbf{Blackbody.} The photon is the phase mode $\theta_-$ of el Pleno (§\ref{sec:gauge}),
an excitation of the medium that can be created and destroyed freely: there is no conserved charge nor
photon number. In statistical mechanics, the chemical potential $\mu$ is the
Lagrange multiplier of the conservation of number; if the number is not conserved, the
minimization of the free energy with respect to $N$ ($\partial F/\partial N=\mu=0$) imposes
\begin{equation}
\mu=0\quad\Longrightarrow\quad n(\nu)=\frac{1}{e^{h\nu/k_B T}-1}, \tag{E-cosmo-4}
\end{equation}
the Bose-Einstein distribution without chemical potential, that is the exact Planck one,
$u(\nu)d\nu=\tfrac{8\pi h\nu^3}{c^3}\tfrac{d\nu}{e^{h\nu/k_BT}-1}$. The observed perfection
($|\mu|\lesssim 9\times 10^{-5}$, FIRAS) is not a fine-tuning: it is structural, a consequence
of the photon being a mode of the medium without its own number.

\textbf{Spectral index without inflation.} In conformal time $d\eta=dt/a$, with the
Mukhanov-Sasaki variable $z=aS$ and $v_k=z\mathcal{R}_k$, the mode equation is
\[
v_k''+\Big(c^2 k^2-\frac{z''}{z}\Big)v_k=0 .
\]
With $z\propto(-\eta)^s$, the symbolic computation gives
$z''/z=s(s-1)/\eta^2$, so that the equation reduces to Bessel's with
Hankel solutions $v_k=\sqrt{-\eta}\,[A H_\nu^{(1)}(-ck\eta)+B H_\nu^{(2)}]$,
$\nu=|s-\tfrac12|$. In the super-horizon limit ($H_\nu^{(1)}\sim x^{-\nu}$),
$\mathcal{P}_\mathcal{R}\propto k^{3-2\nu}$, whence the master formula:
\begin{equation}
\boxed{\;n_s-1=3-2\nu=3-2\big|s-\tfrac{1}{2}\big|\;} \tag{E-cosmo-1}
\end{equation}
\textbf{The exponent $s$.} The depleted matter dilutes $S^2\propto a^{-3}$
($w=0$), hence $S\propto a^{-3/2}$ and $z=aS\propto a^{-1/2}$. With $w=0$, Friedmann in
conformal time gives $a\propto(-\eta)^2$, whence $z\propto(-\eta)^{-1}$, that is
$s=-1$ and $\nu=3/2$. Substituting:
\[
n_s-1=3-2\cdot\tfrac{3}{2}=0\quad\Longrightarrow\quad n_s=1\ \text{(zeroth order)}.
\]
\textbf{The degeneracy of the pure components.} The observed red tilt
($n_s\approx0.965$, $n_s-1\approx-0.035$) does not come from an individual component of the
background. For a component of equation of state $w$, conformal Friedmann gives
$a\propto(-\eta)^p$ with $p=2/(1+3w)$, and the Mukhanov variable inherits
\[
s_z(w)=p\cdot\frac{-1-3w}{2}=\frac{-1-3w}{1+3w}.
\]
Evaluating the two components that dominate the crunch:
\[
s_z(0)=-1\ (\text{depleted matter}),
\qquad
s_z(-2)=\frac{-1+6}{1-6}=-1\ (\text{Klauder},\,w_q=-2).
\]
Both give the same exponent $s_z=-1$, hence $\nu=3/2$ and $n_s=1$: neither the depleted matter
($w=0$) nor the IR wall of Klauder ($w_q=-2$) produces, separately, a tilt. The red sign
requires the \emph{crossing} between regimes, not the regime.

\textbf{The mechanism: the packing transition in the bounce.} The bounce occurs
from the finite maximal density $\rho_{\max}=S_{\max,\text{geom}}^2\,\rhoP=\tfrac{8\sqrt6}{9}\rhoP$,
which the program derives as the ratio between the maximal FCC packing (Kepler-Hales,
$Z=12$) and the density of the diamond lattice ($Z=4$):
\[
S_{\max,\text{geom}}^2=\frac{n_\text{FCC}^{\max}}{n_\text{diamante}}=\frac{\sqrt2/\lP^3}{3\sqrt3/(8\lP^3)}=\frac{8\sqrt6}{9}.
\]
The two structures of this identity ---compact $Z=12$ and diamond $Z=4$--- are the phases
between which el Pleno transits in the bounce ($Z=12\to Z=4$). The imprinting of the primordial
spectrum occurs during that transition, over the persistent background
$f_c=S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2=1.4956/2.1773=0.6869=\Omega_{EO}$, with $w_{EO}=-1$.

\textbf{Red sign} \epi{SLD}\textbf{.} The Klauder component that enters the transition
has $w_q=-2<-\tfrac13$: it accelerates the background. An accelerated phase in the imprinting of the modes
adds power at large scale relative to small, running the spectrum to the red; the
persistent background $f_c$ with $w_{EO}=-1$ reinforces the same sign. Therefore $n_s-1<0$
structural, not fitted.

\textbf{Smallness} \epi{SLD}\textbf{.} In the early crunch, when the large-scale modes
cross the horizon, the Klauder fraction $x=\rho_q/\rho_m\propto a^6$ is small: the
regime is dominated by the depleted matter ($n_s=1$), and Klauder enters as a first-order
correction in $x\ll1$. Therefore $|n_s-1|\ll1$. The band $n_s=0.965\pm0.004$ is reached
with $|w_\text{eff}|\approx0.006$, the order that Klauder introduces in that epoch.

\textbf{Exact value, positional} \epi{CALIBRATED}\textbf{.} The value $n_s=0.965$ depends on
the Klauder fraction at the precise moment of horizon crossing, that is, on where over
the trajectory $\bar S(t)$ the modes were imprinted. There is no exact scaling solution that
cancels that dependence: for $V_q\propto S^{-2}$, the ansatz $S\propto t^\beta$, $a\propto t^\gamma$
imposes $\beta=1/2$ by balance of the EOM, but then Friedmann is inconsistent in the
exponents of $t$ ($H^2\sim t^{-2}$ against $\rho\sim t^{-1}$). The present value of the tilt
inherits the positional condition of $S_{\max,\text{eff}}^2(t_0)$, on equal footing with $f_c$ and
$H_0$. By Rule 17 it is a state variable, not a derivable number; the datum itself is a band
($\pm0.0042$).

\subsection{The matter density \texorpdfstring{$\Omega_m=1-f_c$}{Om=1-fc} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

By conservation of the condensate population, $f_c+\Omega_\text{depleted}=1$:
\begin{equation}
\boxed{\;\Omega_m=1-f_c=1-0.6869=0.3131\;} \tag{E-cosmo-2}
\end{equation}
Against Planck ($\Omega_m=0.315\pm 0.007$): $0.27\sigma$, without fitting. Decomposing
with $\Omega_{MO}/\Omega_M=\sqrt{3}\pi$:
\[
\Omega_M=\frac{0.3131}{1+\sqrt{3}\pi}=0.0486\ (1.2\sigma),
\qquad
\Omega_{MO}=0.2645\ (0.07\sigma).
\]
The dilution $\rho_m\propto a^{-3}$ is topological (conserved countable objects):
$n=N/V\propto a^{-3}$, $\rho_m=n\,m_\text{sol}c^2\propto a^{-3}$, the exponent emerges
from the conservation plus the geometry.

\subsection{The expansion history \texorpdfstring{$H(z)$}{H(z)} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{CALIBRATED}}{[CALIBRATED]}}

Nondimensionalizing $V_\text{eff}=\rhoP c^2 v(S)$ and using $G=c^2/(\rhoP\lP^2)$, the
Friedmann prefactor of the field is $\tfrac{8\pi G}{3}\rhoP c^2=\tfrac{8\pi}{3}\tfrac{c^4}{\lP^2}\equiv\mathcal{C}$.
From $H^2=\tfrac{8\pi G}{3}(\rho_m+\rho_S)$ with $\rho_m=\rho_{m,0}(1+z)^3$ (diluted matter) and
$\rho_S(z)$ the density of the field, upon forming $E(z)\equiv H(z)/H_0$ the common prefactor
$\mathcal{C}$ appears in numerator and denominator and cancels along with every
dimensional scale; normalizing $E(0)=1$:
\begin{equation}
\boxed{\;E^2(z)=\frac{\Omega_m(1+z)^3+\rho_S(z)/\rho_{c,0}}{\Omega_m+\rho_S(0)/\rho_{c,0}}\;} \tag{E-cosmo-3}
\end{equation}
scale-invariant form, forced by the derived potential, with $\delta_0$
(initial condition $\leftrightarrow w_0$) as the only observable freedom.
Confrontation with BAO DESI DR2 (5 points):

\begin{center}
\begin{tabular}{L{6.0cm}C{2.0cm}C{2.2cm}}
\toprule
Model & $\chi^2$ & $\chi^2/$dof\\
\midrule
$\Lambda$CDM ($\Omega_m=0.31$) & 6.53 & 1.63\\
PIU ($\delta_0\to 0$, $w_0\approx -1$) & 6.54 & 1.63\\
PIU ($\delta_0=0.02$, $w_0=-0.96$) & 11.24 & 2.81\\
\bottomrule
\end{tabular}
\end{center}
The program fits BAO DESI DR2 as well as $\Lambda$CDM with less freedom. The
scale invariance of the form is \epi{DERIVED}; $w_0$ (or $\delta_0$) is
\epi{CALIBRATED}, a state variable.

\medskip
\begin{tcolorbox}[clave,title={Equivalence with $\Lambda$CDM and distinctive advantage}]
PIU-$\Psi$ reproduces the central cosmological observables with the same statistical
precision as $\Lambda$CDM and with the same degrees of freedom ---it is parity, not
superiority of fit. On that basis, the program contributes three features that $\Lambda$CDM does not
possess. \emph{(i) Derived, not fitted, parameters:} $\Omega_{EO}$ and $\Omega_{MO}/\Omega_M$
come out of the geometry of the substrate (at $0.30\sigma$ and $0.86\sigma$, zero free parameters),
where $\Lambda$CDM fits them to the datum. \emph{(ii) Unique and falsifiable predictions:} the
scalar/tensor asymmetry with red$\to$blue transition (CMB-S4/LiteBIRD), the baryonic rotational
tower, the vortex separation $d\approx R_p$, and the signatures of the Lee-Wick pole
in gravitational waves (LISA/ET/CE) ---$\Lambda$CDM formulates none; they are the
points where the program exposes itself to refutation. \emph{(iii) Ontological reduction:} a single
substrate read in distinct configurations, instead of an inventory of independent components
each with its postulated equation of state. The ontological economy is what
allows \emph{deriving} the fractions instead of postulating them.
\end{tcolorbox}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fondo_cosmico.png}
\caption{Confrontation with BAO DESI DR2. Panel A: the Hubble distance $D_H/r_d$ of
PIU ($\delta_0\to 0$) and of $\Lambda$CDM are indistinguishable over the five points.
Panel C: identical normalized residuals. Panel D: the canonical kinetics guarantees
$w\geq -1$ (the phantom region is vetoed). The $\chi^2=6.54$ of PIU equals the
$6.53$ of $\Lambda$CDM ---same fit, less freedom.}
\end{figure}

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
Cosmology ceases to be a catalog of fitted ingredients and becomes the reading of the
same Pleno at the scale of the universe. What in $\Lambda$CDM are free parameters, here are
consequences: the dark/baryonic matter ratio $\sqrt3\pi$ is pure geometry (two areas and
a spinor factor $1/2$); $\Omega_m=1-f_c$ is the depleted fraction of the condensate; the
\textbf{dark energy is the floor of the potential} ---the same $0.5229\,\rhoP$ of the ground-Pleno
of link 6---, so that $w=-1$ is not an exotic fluid but the elastic rigidity of the
relaxed vacuum ($\rho+p=0$, zero entropy: a solid, not a fluid). The perfect blackbody
and $n_s\approx 1$ without inflation fall out as structure, not as a fit. And all of it holds with
\emph{less} freedom than $\Lambda$CDM: two firm predictions at $0.27\sigma$ and $0.86\sigma$
without a single parameter, inherited from the diamond lattice of link 5. The universe does not contain
dark matter and dark energy as separate substances: it exhibits them as two readings
---excited and background--- of a single medium.
\end{tcolorbox}


\subsection{The Hubble constant as a positional quantity: factorization, condensate occupation and tension --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The scale-invariant form $E(z)=H(z)/H_0$ cancels the Friedmann prefactor: the theoretical content lives in the \emph{shape} of the curve, not in the absolute value $H_0$. It remains to classify what kind of object $H_0$ is. Starting from the effective Friedmann equation of the background today, $H_0^2=\tfrac{8\pi G}{3}\rho_{\text{tot},0}$, the density is nondimensionalized by its anchor, $\rho_{\text{tot},0}\equiv\Theta(t_0)\,\rhoP$, and $G$ is eliminated with the native identity $G=c^2/(\rhoP\lP^2)$ (§5.1). The anchor density $\rhoP$ cancels exactly:
\begin{equation}
H_0^2=\frac{8\pi G}{3}\,\rho_{\text{tot},0}
=\frac{8\pi}{3}\,\frac{c^2}{\rhoP\lP^2}\,\Theta\,\rhoP
=\frac{8\pi}{3}\,\Theta\,\frac{c^2}{\lP^2}
=\frac{8\pi}{3}\,\Theta\left(\frac{c}{\lP}\right)^{\!2}.
\end{equation}
Taking the root and using the definitional identity $\tau_P=\lP/c$ ($c/\lP=1/\tau_P$):
\begin{equation}
\boxed{\;H_0
=\underbrace{\sqrt{\tfrac{8\pi}{3}}}_{\text{geom. ratio, }\DER}
\;\times\;
\underbrace{\frac{c}{\lP}}_{\text{anchor }=\,1/\tau_P}
\;\times\;
\underbrace{\sqrt{\Theta(t)}}_{\text{state, }\CAL}\;}
\tag{E-cosmo-5}
\end{equation}
with $\sqrt{8\pi/3}=2.8944$ and $c/\lP=1.8552\times10^{43}\,\mathrm{s^{-1}}$.

\textbf{What the factorization asserts and what it does not assert.} The factor $\sqrt{8\pi/3}$ is the geometric Friedmann coefficient ---common to all FLRW cosmology, fixed by the density-curvature relation in $3{+}1$ dimensions---, not an exotic invariant of the lattice. What the program contributes is the \emph{identification of the anchor}: the dimensional scale of $H_0$ is exactly $c/\lP=1/\tau_P$, the frequency of the Planck tick (the same $\dot\gamma_{\max}$ of §12.2), once $G$ is eliminated by its native identity. The non-trivial claim is threefold: (i) the factorization introduces no parameter ---the first two pieces are fixed and the third is a state quantity, not a free constant; (ii) the dimensional content of $H_0$ resides entirely in $\lP$ (with $c$ reabsorbable causal structure), which passes the purity test of the Language of el Pleno ---in units $c=\lP=1$ the anchor dissolves to unity; (iii) the pure number of $H_0$ in units of el Pleno, $\sim10^{-61}$, alien to the $O(1)$ ratios that the program derives, is the diagnosis that $H_0$ is positional. The numerical verification with $H_0=67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}=2.184\times10^{-18}\,\mathrm{s^{-1}}$ gives $\Theta_0=1.655\times10^{-123}$ and reconstructs $H_0$ exactly.

\textbf{Today's density is the age in ticks, squared.} Defining the number of ticks of the Hubble time $N_{\text{tic}}\equiv 1/(H_0\tau_P)=8.493\times10^{60}$ (with the Planck tick $\tau_P=\lP/c=5.391\times10^{-44}$ s), from $H_0=\sqrt{8\pi/3}\,\sqrt{\Theta}/\tau_P$ one solves $\sqrt{\Theta}=\sqrt{3/8\pi}\,(H_0\tau_P)=\sqrt{3/8\pi}/N_{\text{tic}}$, and squared:
\begin{equation}
\boxed{\;\Theta(t)=\frac{3}{8\pi\,N_{\text{tic}}^2(t)}\;}
\qquad\Longleftrightarrow\qquad
N_{\text{tic}}=\sqrt{\frac{3}{8\pi\,\Theta}}.
\tag{E-cosmo-6}
\end{equation}
Verification: $3/(8\pi\cdot(8.493\times10^{60})^2)=1.655\times10^{-123}=\Theta_0$. The number of ticks admits two independent readings of the same clock that agree: from the rate, $N_{\text{tic}}=1/(H_0\tau_P)=8.49\times10^{60}$; from the observational age, $N=t_0/\tau_P=(13.787\,\text{Gyr})/\tau_P=8.07\times10^{60}$. The agreement to $\sim5\%$ without fitting confirms that $N_{\text{tic}}$ is the age of the universe in Planck ticks, and the small difference is the factor $O(1)\approx0.95$ between $1/H_0$ and the true age (the same as the magnitude open end).

\begin{tcolorbox}[clave,title={Reformulation of the cosmological constant problem}]
The number $\Theta_0\approx10^{-123}$ \emph{is} the crude formulation of the cosmological constant problem: the vacuum energy density of today is $\sim10^{-123}$ times that of Planck, the worst fine-tuning of the standard model. The identity (E-cosmo-6) rewrites it: the ``$10^{-123}$'' is not a number to explain, but $1/N_{\text{tic}}^2$ up to the geometric ratio $3/8\pi$. Any universe that has lived $N_{\text{tic}}$ ticks automatically has $\Theta\sim 1/N_{\text{tic}}^2$ by arithmetic. There is no need to explain the exponent $123$; one has to explain how many ticks the clock has run ($N_{\text{tic}}\approx 8.5\times10^{60}$), and that is an age, not a tuned parameter. This extends to the absolute \emph{scale} what $\Omega_m=1-f_c$ did with the \emph{partition}: the ``120 orders'' of the matter/vacuum discrepancy were replaced by the order-one fraction $f_c\approx0.69$. The two sides of the problem ---why $\Lambda$ is small (scale) and why it is comparable to matter today (partition)--- are reformulated as clock readings.
\end{tcolorbox}

\textbf{The Hubble tension as a method artefact.} If $H_0$ is the reading of a clock ---the frequency of the tick modulated by the age in ticks---, then the ``primordial $H_0$'' and the ``local $H_0$'' are not two measurements of the same constant in disagreement, but two distinct objects: the first is the value that $\Lambda$CDM \emph{needs} to reproduce the acoustic angle of the CMB under its own template, not a raw datum. Analyzing a medium with $w=-1$ exact and elastic $H(z)$ ---which coincides with $\Lambda$CDM at linear order and differs in the non-linear regime of the walls--- under the $\Lambda$CDM template biases the primordial inference. The structure of the bias is seen in the propagation to the inference of the primordial method, which anchors $H_0$ to the acoustic angle $\theta_*=r_s/D_M(z_*)$ of the CMB:
\begin{equation}
\begin{gathered}
D_M(z_*)=\int_0^{z_*}\frac{c\,dz}{H(z)},\qquad
H(z)=H_0\sqrt{\Omega_m(1+z)^3+(1-\Omega_m)\,g(z)},\\[3pt]
g(z)=\exp\!\Big[3\!\int_0^z\frac{1+w(z')}{1+z'}\,dz'\Big].
\end{gathered}
\end{equation}
When the real $H(z)$ is that of the elastic medium but the inference is made under the rigid $\Lambda$CDM fit, the $H_0$ that reproduces the same $\theta_*$ at fixed $r_s$ falls below the locally measured one: the geometric degeneracy in the distance to the last scattering produces a bias of definite sign. The number $\sim5.8$ of the ``tension'' is a property of the $\Lambda$CDM interpretation, not a raw datum. The falsifiable prediction of the program is that the tension \emph{dissolves} in model-independent observables (raw $D_H/r_d$ of BAO, raw $\theta_*$, the shape of $E(z)$): there are not two true values of $H_0$ in conflict, but an artefact of analyzing a dynamical medium under a static template. \DLF.

\begin{tcolorbox}[cabo,title={The prediction: $w=-1$ exact and the tension as artefact; the non-linear correction, open end}]
The \textbf{equation of state} of the late sector is $w=-1$ \emph{exact} (\DER: phase condensate at $\Smin$, $T_{\mu\nu}\propto g_{\mu\nu}$), and the \textbf{Hubble tension} is a method artefact (\DLF: the ``primordial'' $H_0$ is the one $\Lambda$CDM needs for the CMB, not a datum). What remains as an open end is the \textbf{non-linear correction of $H(z)$} ---whose origin lies in the shape of the potential with its walls, but whose dynamical carrier is the phase sector--- over the elastic background ---the deviation of $E(z)$ with respect to $\Lambda$CDM in the non-linear regime, whose amplitude encodes $\delta_0$--- and its confrontation with raw BAO. The open end is \emph{relational}, not temporal: to ask for ``$\bar S$ as a function of a background time'' would be circular, since time is the accumulated change of the modulus (§\ref{sec:reologia}, the singularity under emergent time), and the relaxation curve of the diamond lattice $S^*\to\Smin$ \emph{is} the curve of cosmological time ---the lattice is born in the crystallization at $S^*=\sqrt{\Smin S_{\max,\text{geom}}}=1.0330$ and descends monotonically to the attractor $\Smin$ (second law). Time emerges as arc length $\propto\int|d\bar S|/c$ over the orbit, never as input
\begin{equation*}
\ddot{\bar S}+3H\dot{\bar S}+\rhoP^{-1}\lP^{-2}V'_\text{eff}=0
\end{equation*}
(with $V_\text{eff}$ derived and anchors $\rhoP,\lP$; distinct from the primordial cycle equation §12.4, which governs the bounce). Computing this correction is the same class of operation already performed numerically for the crystallization transient (§12.5), which reproduced its numbers without fitting. That it be an open end and not a closed result has a mathematical reason: there is no exact \emph{scaling} solution of the trajectory (§11.6: the ansatz $S\propto t^\beta$ leaves Friedmann inconsistent in the exponents), so that the closed form requires fine integration. The present value $\Theta_0$ (the hour of the clock) remains positional: the curve is derivable, the point on the curve is a historical datum.
\end{tcolorbox}

\textbf{Which clock marks the present.} Two distinct objects govern the late evolution. The \emph{radial modulus} $\bar S$ is ultra-Planckian ($m_S=\sqrt\xi\,E_P\approx4.44\,E_P$, §5); against the cosmological rate $m_S/H_0\sim10^{61}$, so that it \emph{freezes} at $S_{\min}$ almost from the first instant and its own relaxation lasts $\sim t_P$, not an age of the universe. That the scales built over $\bar S$ (soliton masses $M_Q$, $S_{\min}$ of the defects) do not derive confirms it observationally. The modulus, therefore, \textbf{is already at the attractor}; its descent $S^*\to S_{\min}$ is the clock of the bounce/crystallization, not that of the present epoch. The clock of the present is the \textbf{coherent phase condensate} (DE-Curtain, §11.7): its $w=-1$ is not a field rolling ---that would give $w=+1$--- but the symmetry $T_{\mu\nu}\propto g_{\mu\nu}$ of the ground state (without a rest frame, \DER), with $\dot\theta_0=0$ and therefore $w=-1$ \emph{exact}, not an approximation with a residual. The positional reading that does operate is that of the \emph{occupation} of the condensate ($f_c$, $H_0$, $n_s$), not that of the equation of state. Only a massless mode (phase Goldstone, A5) could in principle roll slowly against $H$, but a phase mode that rolls dilutes like matter ($\rho_{\text{kin}}\propto S^2\propto L^{-3}$, from the Noether charge $S^2\dot\theta L^3=\text{const}$ and the continuity $S^2L^3=\text{const}$), not like a $w=-1$ component: it is ruled out as the origin of dark energy, and the condensate at its minimum ($\dot\theta_0=0$) remains at $w=-1$ exact. The only thing left to compute is the non-linear correction of $H(z)$ by the walls, which lives in the collective response of the \emph{phase sector}, not in the orbit of the radial modulus ---the equation $\ddot{\bar S}+3H\dot{\bar S}+\rhoP^{-1}\lP^{-2}V'_\text{eff}=0$ describes the collective homogeneous background. \DLF\ (supported by derived results: $T\!\propto\!g$, $m_S$, the $L^{-3}$ scaling of a rolling phase mode).

\textbf{The occupation of the condensate at the present rest.} The descent of the modulus from the bounce saturation $S_{\max,\text{geom}}^2=2.1773$ toward the rest $\Smin^2=0.5229$ (law $S^2\propto a^{-3}$, §10.5) is \emph{essentially complete}: today the ground-Pleno is in its relaxed regime, with $S\sim\Smin$ (§4.5), settling in the attractor of the well with $\dot{\bar S}$ small but not null. We are not halfway down the descent; we are at the bottom of the curve. What the positional quantities ($H_0$, $f_c$, $w_0$, $n_s$) localize is not an intermediate position over the descent, but the \textbf{occupation level of the condensate} at that rest:
\begin{equation}
\boxed{\;f_c=\frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2}=\frac{1.4956}{2.1773}=0.6869=\Omega_{EO}\;}
\tag{E-cosmo-7}
\end{equation}
The condensate occupies $69\%$ of the geometric capacity; the remaining $31\%$ ($1-f_c=0.3131$) is the quantum depletion. Structure \DER; value of the numerator $S_{\max,\text{eff}}^2$ \CAL\ positional (like $H_0$, anchored to the present epoch). $f_c$ is not a constant of nature but the residual occupation of the coherent mode ---the same that $H_0$ and $n_s$ read from distinct angles. The projections coincide with a single state, the rest: $f_c=0.6869$ (occupation) and $n_s-1=-0.035$ (tilt of the primordial spectrum), a verifiable internal consistency of the background sector. The equation of state is not of this type: $w=-1$ is exact, geometric, not a positional reading. \DLF.

\textbf{The background value of $H_0$ and the exclusion of the local value.} The position $R$ and the initial condition $\delta_0$ that fixes $w(z)$ determine the background value of the Hubble constant. Integrating the expansion history with the observed age $t_0=13.787\,\text{Gyr}$:

\begin{center}
\begin{tabular}{@{}cccc@{}}
\toprule
$\delta_0$ & $w_0$ & $E(z)$ at $z=2$ vs $\Lambda$CDM & $H_0$ (km\,s$^{-1}$Mpc$^{-1}$) \\
\midrule
$\to 0$ & $-1$ & $\to\Lambda$CDM ($\chi^2=6.54$) & $67.71$ \\
$0.02$ & $-0.961$ & $+2.45\%$ & $67.41$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent Over the entire strip of initial conditions the background falls in $H_0\sim67$--$68$ (primordial side): the physical case $\delta_0\to0$ ---late sector with $w=-1$ exact, indistinguishable from $\Lambda$CDM at linear order--- gives $H_0=67.71$; a small initial condition $\delta_0=0.02$, which encodes the non-linear correction of the walls over $E(z)$, gives $H_0=67.41$, coincident with Planck ($67.4\pm0.5$). The parameter $\delta_0$ is not a dynamical $w_0$ ---the background has $w=-1$ exact--- but the amplitude of the non-linear correction, \CAL\ positional; the shape of $E(z)$ is derived. The robust point, independent of $\delta_0$: the background $H_0$ of el Pleno is on the primordial side, never near $73$.

The local SH0ES value ($\sim73$) is \emph{excluded} as a background by a structural bound. The dimensionless age $A\equiv H_0 t_0$ is bounded by the canonical kinetics: $H_0=73$ with the observed age would require $A\approx1.03$, but a background with $w=-1$ exact dark energy imposes $A\le A(w{=}-1)=0.953$ (age $\approx0.95/H_0$). Exceeding it requires the phantom region $w<-1$, vetoed by $w+1=\rhoP\lP^2\dot{\bar S}^2/\rho\ge0$. The local value is not the background $H_0$: it is the instantaneous rate that the distance ladder measures locally, biased with respect to the background. \DER\ by the bound $w\ge-1$. Numerically: $A(67.41)=0.9505$, $A(67.71)=0.9547$ (both under the ceiling), $A(73)=1.0293$ (excluded).

\textbf{Quantification of the margin: the exclusion does not depend on the exact value of
$\Omega_m$.} The ceiling $A(w{=}{-}1)$ is not a fixed number: it depends on $\Omega_m$, and
$\Omega_m=1-f_c$ is a positional quantity with its calibrated ingredient. The exclusion has force only if
the ceiling moves, over the whole plausible range of $\Omega_m$, by less than the local value
lies from it.

\emph{The integral of the dimensionless age.} For a flat background with matter and dark energy
of $w=-1$, the Friedmann equation gives $H(a)=H_0\sqrt{\Omega_m a^{-3}+(1-\Omega_m)}$, and the
age in units of $1/H_0$ is
\[
A(\Omega_m)\;=\;H_0 t_0\;=\;\int_0^1\frac{da}{a\,H(a)/H_0}
\;=\;\int_0^1\frac{da}{\sqrt{\dfrac{\Omega_m}{a}+(1-\Omega_m)\,a^2}}\,,
\]
where the integrand follows from $a H(a)/H_0=\sqrt{\Omega_m/a+(1-\Omega_m)a^2}$ after taking
$a^2$ out of the radicand. The integral is elementary in terms of hyperbolic functions, but for
present purposes its numerical evaluation suffices.

\emph{Sensitivity of the ceiling.} Integrating over the plausible range of the matter
parameter:
\begin{center}\small
\begin{tabular}{ccccc}
\toprule
$\Omega_m$ & $0.300$ & $0.3107$ & $0.3131$ & $0.320$ \\
\midrule
$A(w{=}{-}1)$ & $0.9641$ & $0.9547$ & $0.9526$ & $0.9468$ \\
\bottomrule
\end{tabular}
\end{center}
The ceiling moves $(0.9641-0.9468)/0.9468=1.8\hspace{0.08em}\%$ over the \emph{entire} range.

\emph{Distance of the local value.} With the observed age $t_0=13.787$~Gyr and the conversion
factor $977.79$ (from km\,s$^{-1}$Mpc$^{-1}$\,$\cdot$\,Gyr to dimensionless):
\[
A(73)=\frac{73\times13.787}{977.79}=1.0293 ,
\qquad
\frac{A(73)}{A(0.3131)}-1=\frac{1.0293}{0.9526}-1=8.1\hspace{0.08em}\% .
\]

\emph{Conclusion on the margin.} The distance of the local value from the ceiling
($8.1\hspace{0.08em}\%$) is more than \textbf{four times} larger than the entire variation that
the positional freedom of the partition can induce in the ceiling ($1.8\hspace{0.08em}\%$). The
exclusion therefore does not rest on the exact value of $\Omega_m$ but on the structure: matter
diluting as $L^{-3}$ plus a floor of constant density do not admit a universe so old measured
in ticks of its own rate. And within that same band the background values on the primordial
side fit with nothing forced: $A(67.4)=0.9504$ and $A(67.7)=0.9547$ fall on the ceiling
corresponding to $\Omega_m$ between $0.311$ and $0.316$. \T{DERIVED}.

\textbf{Native reading: $A<1$ is the imprint of the fluid that cools.} The previous bound is stated in FLRW language ($w$, $1/H_0$), but its content is native and it reads in the Language of el Pleno, where $A$ ceases to be a fundamental object and becomes the \emph{observable shadow} of the dynamics of the modulus. Time is the accumulated change of $\bar S$ over the relaxation curve (§\ref{sec:reologia}); $H_0$ is the fractional rate of that change today; $t_0$ is the arc already traversed from $S^*$. The medium is a dilatant fluid that \emph{cools} as it descends $S^*\to\Smin$: the deformation rate $\dot\gamma$ reaches a maximum in the fast descent of the hot zone (near $S^*$) and \emph{decays} as it approaches the cold attractor $\Smin$ ---exactly the profile of the viscous transient (§\ref{sec:reologia}, $\dot\gamma_{\max}=0.98$ not exceeded, then damping). That the cosmic clock \emph{was faster in the past and slows down now} implies that the universe reached its present state faster than its present rate would suggest: it is \emph{younger} than its Hubble time, $t_0<1/H_0$, that is $A=H_0 t_0<1$. The sign of the bound is, then, the integral signature of a fast phase followed by braking ---the same physics of the dilatant transient, not an imported veto. A background $H_0$ of $73$ would give $A=1.03>1$: it would require the opposite ---an expansion that net-accelerates without having been fast before, that is increasing $\dot{\bar S}$ that would overshoot the attractor $\Smin$---, vetoed by the monotonic descent to the attractor (second law) and, equivalently, by $w\ge-1$ (a square $\dot{\bar S}^2\ge0$). The \emph{exclusion} of $73$ is thus native \DER, independent of the Friedmann integral; the exact \emph{value} of the ceiling ($0.953$) is its FLRW projection and depends on the fine shape of $f(\bar S)$ ---open end F-cosmo. Verified: a universe that brakes more (faster in the past) gives smaller $A$ ---only matter, $A=2/3$; with dark energy, $A\to0.953$ from above---; $A>1$ is produced only by $w<-1$, physically impossible in an elastic medium that cools.

\textbf{The Hubble bound is the dilatant bound, with exact equality.} The extreme $S_{\max,\text{geom}}$ fixes the absolute upper bound of the expansion rate: from $\rho_{\max}=\rhoP S_{\max,\text{geom}}^2$ and Friedmann, $H_{\max}=\sqrt{8\pi G\rho_{\max}/3}=\sqrt{8\pi/3}\,(c/\lP)\,S_{\max,\text{geom}}=7.922\times10^{43}\,\mathrm{s^{-1}}$, with minimal horizon $R_{H,\min}=c/H_{\max}=0.2341\,\lP$ sub-Planckian. Compared with the maximal deformation rate $\dot\gamma_{\max}=c/\lP$ (§12.2), the quotient is exact:
\begin{equation}
\boxed{\;\frac{H_{\max}}{\dot\gamma_{\max}}=\sqrt{\frac{8\pi}{3}}\;S_{\max,\text{geom}}=4.2709\;}
\tag{E-cosmo-8}
\end{equation}
not a ``factor $\sim4$'': the Hubble bound (bounce cosmology) and the dilatant deformation bound (jamming, $\mathrm{Ma}=1$) are the same rigidity of the medium, linked by the Friedmann coefficient by the root of the packing quotient. \DER.

\begin{tcolorbox}[apunte,title={What is pertinent to predict: observables, not parameters of another model}]
Methodological caution that governs this subsection. The ``Hubble tension'' is \emph{not} a discrepancy between two measurements of the same quantity: the local value ($\sim73$) is mildly model-dependent, but the ``primordial'' one ($\sim67$) \textbf{is not measured} ---it is a parameter derived from fitting $\Lambda$CDM to the pattern of the CMB, ``the $H_0$ that $\Lambda$CDM needs'', not ``the $H_0$ of the early universe''. Therefore the number $\sim5.8$ is a property of the $\Lambda$CDM interpretation, not a raw datum. \textbf{It is not pertinent} to calibrate PIU-$\Psi$ to reproduce $\sim5.8$ (it would be imitating an artefact of the model that is being replaced); \textbf{it is pertinent} to predict the \emph{method bias}: that analyzing a medium with $w=-1$ exact and elastic $H(z)$ under the rigid $\Lambda$CDM template infers a CMB $H_0$ biased low, with definite sign, and that the tension ---like the $w_0$--$w_a$ signal of DESI--- dissolves in model-independent observables. The confrontation of PIU-$\Psi$ is always made against model-independent observables ($D_H/r_d$ of BAO, $E(z)$, raw $\theta_*$), never against parameters ($H_0^{\text{Planck}}$, $w_0^{\text{DESI}}$) that are already outputs of a $\Lambda$CDM fit.
\end{tcolorbox}

\begin{tcolorbox}[clave,title={Balance of the Hubble sector: what the program delivers and what it leaves open}]
The weight of the result is distributed across several pieces and is not reduced to a single one. In order of strength, the program delivers:
\begin{enumerate}[leftmargin=1.6em,itemsep=2pt]
\item \textbf{Exact factorization} $H_0=\sqrt{8\pi/3}\,(c/\lP)\,\sqrt{\Theta}$ (E-cosmo-5), \DER. The dimensional scale of $H_0$ is the frequency of the Planck tick $c/\lP=1/\tau_P$, once $G$ is eliminated by its native identity; $H_0$ remains in anchors of el Pleno and pure ratios, without an alien scale.
\item \textbf{Reformulation of the cosmological constant problem} (E-cosmo-6), \DER: $\Theta_0=3/(8\pi N_\text{tic}^2)$ converts the ``$10^{-123}$'' into the inverse of the square of the age in ticks. There is no minuscule vacuum density to explain; there is an old universe. A reinterpretation, not a new number, but it rewrites a fine-tuning of $123$ orders as an age.
\item \textbf{Exact $w=-1$ equation of state of the late sector}, \DER: the phase condensate at $\Smin$ with $\dot\theta_0=0$ gives $T_{\mu\nu}\propto g_{\mu\nu}$, without a rest frame; the positivity of the kinetics vetoes $w<-1$ and no internal mechanism displaces $w$ from $-1$. The non-linear correction of the walls over $E(z)$ is the only freedom, encoded in $\delta_0$.
\item \textbf{Exact structural equality Hubble bound/dilatant bound} (E45), \DER: $H_{\max}/\dot\gamma_{\max}=\sqrt{8\pi/3}\,S_{\max,\text{geom}}=4.2709$, not a mere ``same order''. The cosmological expansion bound and the rheological deformation bound are the same rigidity of the medium, read from the global expansion and from the local shear.
\item \textbf{Hubble tension as a method artefact}, \DLF: the ``primordial'' $H_0$ ($\sim67$) is the one $\Lambda$CDM needs for the CMB, not a datum; analyzing a medium with $w=-1$ exact and elastic $H(z)$ under that template biases the inference. The program predicts that the tension ---and the $w_0$--$w_a$ signal of DESI--- dissolve in model-independent observables.
\item \textbf{Background value of $H_0$ and exclusion of the local one}, \CAL/\DER: over the entire strip of initial conditions the background gives $H_0\sim67$--$68$ (primordial side: $67.71$ for $\delta_0\to0$, $67.41$ for $\delta_0=0.02$), coincident with Planck; the local value $73$ would require $A=H_0t_0\approx1.03$, vetoed by the bound $A\le0.953$ that $w\ge-1$ imposes.
\end{enumerate}
The \emph{only} thing open is the \textbf{non-linear correction of $H(z)$} at accessible $z$ ---whose origin lies in the shape of the potential with the Gent and Klauder walls, but whose dynamical carrier is the \emph{phase sector} and not the oscillation of the radial modulus, frozen at $\Smin^\text{estr}$ since crystallization--- and its measurability in raw BAO ---plus the prediction of $S_8$ from the phase drag. The closed computation of that correction is the same class of operation already done numerically for the transient (§12.5); what remains is its shape in the relaxed regime: a confrontation open end, of subordinate priority. The $w_0$--$w_a$ signal of DESI and the Hubble tension are not quantities to reproduce, but artefacts that the program predicts dissolve in raw observables. Five results with exact or structural mathematics, the non-linear correction as a declared open end: it is not ``just the sign''.
\end{tcolorbox}

\begin{tcolorbox}[clave,title={Conclusion --- what it means for real physics}]
The positional reading reduces three problems that the standard model treats separately to a single ontological assertion: \emph{$H_0$ is not a constant of nature, but the hour of a clock}. The \textbf{cosmological constant problem} ceases to be a fine-tuning of $123$ orders ---$\Theta_0$ is the inverse of the square of the age in ticks. The \textbf{Hubble tension} ceases to be an anomaly ---the ``primordial'' $H_0$ is not measured, it is the one $\Lambda$CDM needs for the CMB, and analyzing a medium with $w=-1$ exact and elastic $H(z)$ under that template biases the inference. The \textbf{dynamical dark energy signal} of DESI is the same type of artefact: the one that appears when forcing a medium with $w=-1$ exact into the $w_0$--$w_a$ parametrization. The three are the same thing: an elastic medium ---whose $H(z)$ coincides with $\Lambda$CDM at linear order and differs in the non-linear regime--- read under static assumptions. The counterpart, far from being a weakness, is what makes the proposal falsifiable: the program does not derive the \emph{value} of $H_0$ (it would be deriving the hour from the clock mechanism), and predicts that the tension and the DESI signal dissolve in model-independent observables ---if instead they \emph{persist} in raw data, or if the signal consolidates in $w_0<-1$ (phantom region vetoed by $w+1\ge0$), the reading is refuted.
\end{tcolorbox}

\subsection{The time dilation of SN Ia: the shared factor \texorpdfstring{$b=1$}{b=1} and the radial redshift bound --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}
\label{sec:sn-factor}

The redshift and time dilation of the distant sources are not read through a postulated scale factor: they are derived from the deformation map of the medium itself. The homogeneous and isotropic background is the background part of the map $X^a(x)$, with temporal stretching $X^0=\tau(t)$ and spatial $X^i=L(t)\,x^i$. The components of the emergent metric ---the Cauchy-Green tensor of the hyperelastic medium (E39a, §\ref{sec:gravedad})--- are $g_{00}=-\dot\tau^2$ and $g_{ij}=L^2\delta_{ij}$ (with $c=1$), without inserting $a(t)$ at any step.

\textbf{The shared-factor theorem.} The photon is a TT mode of the medium (\DER, §\ref{sec:gauge}), and its radial propagation is null, $ds^2=0$:
\begin{equation}
-\dot\tau^2\,dt^2 + L^2\,dx^2 = 0 \;\Longrightarrow\; \dot\tau\,dt = L\,dx \;\Longrightarrow\; dx = \frac{d\tau}{L},
\end{equation}
using $d\tau=\dot\tau\,dt$ as the differential of the background proper time (with $c=1$). \emph{The comoving wavelength is conserved.} The phase of the mode, $\phi=k_{\text{com}}x-\omega_{\text{com}}\tau$, is an invariant scalar that propagates over the comoving coordinate $x$; since the background is homogeneous (independent of $x$), the comoving wave number $k_{\text{com}}=2\pi/\lambda_{\text{com}}$ is a constant of motion (there is no spatial gradient to modify it), so that $\lambda_{\text{com}}$ is fixed along the trajectory. The \emph{physical} wavelength is the comoving one stretched by the spatial map, $\lambda_{\text{phys}}=L\,\lambda_{\text{com}}$, and therefore
\begin{equation}
\frac{\lambda_0}{\lambda_e}=\frac{L(\tau_0)\,\lambda_{\text{com}}}{L(\tau_e)\,\lambda_{\text{com}}}=\frac{L(\tau_0)}{L(\tau_e)}.
\end{equation}
\emph{The time interval shares the same factor.} Two successive crests emitted at $\tau_e$ and $\tau_e+\delta\tau_e$ traverse the \emph{same} comoving coordinate $\Delta x$ to the receiver, since $x$ depends only on the endpoints:
\begin{equation}
\int_{\tau_e}^{\tau_0}\frac{d\tau}{L}=\int_{\tau_e+\delta\tau_e}^{\tau_0+\delta\tau_0}\frac{d\tau}{L}.
\end{equation}
Subtracting the two integrals and using that $\delta\tau_e,\delta\tau_0$ are one period of the light ---incomparably smaller than the scale over which $L$ varies (period $\sim10^{-15}$ s against Hubble time $\sim10^{17}$ s), so that $L$ is constant over each interval--- the boundary contributions give
\begin{equation}
\frac{\delta\tau_e}{L(\tau_e)}=\frac{\delta\tau_0}{L(\tau_0)}
\;\Longrightarrow\;\frac{\delta\tau_0}{\delta\tau_e}=\frac{L(\tau_0)}{L(\tau_e)}.
\end{equation}
Wavelength and time interval identically share the factor $L(\tau_0)/L(\tau_e)$:
\begin{equation}
\boxed{\;\frac{\lambda_0}{\lambda_e}=\frac{\delta\tau_0}{\delta\tau_e}=\frac{L(\tau_0)}{L(\tau_e)}\equiv 1+z\;}
\tag{E-cosmo-9}
\end{equation}
The definition $1+z\equiv L(\tau_0)/L(\tau_e)$ is that of redshift as a stretching ratio; the non-trivial thing ---and what is derived--- is that the \emph{time interval} share that same factor, by nullity of the TT mode, not by definition. \DER\ (falsifiers: A3, pull-back of the flat causal structure; A1, the photon as a TT mode of the medium).

\textbf{Tired light, excluded by mechanism.} The hypothesis of pure ``tired light'' ---redshift without time dilation--- is excluded by native mechanism, before invoking the datum: a TT mode cannot stretch its wavelength without stretching by the same factor the interval between its crests, because both are the same ratio $L(\tau_0)/L(\tau_e)$. \DER.

\textbf{The time dilation of SN Ia, closed with the datum.} The temporal width of the light curve of a SN Ia at redshift $z$ stretches as $\Delta t_{\text{obs}}=\Delta t_{\text{em}}(1+z)^b$: pure tired light predicts $b=0$, cosmological dilation $b=1$. The program predicts $b=1$ exact, forced: wavelength and time interval come out of the same nullity condition ($ds^2=0$) over the same emergent metric with the same $L(\tau)$, so that $b=1$ is an algebraic identity, not a fit. The theorem does not depend on \emph{which} sector of the medium generates the stretching $L(\tau)$: it holds for any homogeneous scale factor, because it only uses the nullity of the TT mode and the homogeneity of the background. The supernovae observed at $z\sim1$--$2$ live in the phase sector (§11.9), homogeneous to background order, so that their $L(\tau)$ is that of the phase condensate ---not that of the radial modulus, bounded by $z_{\max}\approx0.268$, below--- and the theorem applies equally: $b=1$ holds throughout the observable range. The most precise available measurement (DES-SN5YR, White et al.\ 2024: 1504 SNe Ia, $0.1\le z\le1.2$) gives
\begin{equation}
b = 1.003 \pm 0.005\,(\text{stat}) \pm 0.010\,(\text{sys}),
\end{equation}
concordant with $b=1$ at $0.27\sigma$, and excludes tired light ($b=0$) at $\sim90\sigma$. Where $\Lambda$CDM \emph{assumes} $b=1$ as a consequence of postulating the expanding metric, the program \emph{derives} it as a theorem of the elastic medium: same numerical prediction, distinct depth. Any future measurement of $b\neq1$ at high significance would refute the mechanism of the TT mode. \DER, without a free parameter.

\textbf{The law of the radial sector and the redshift bound.} By continuity of the hyperelastic continuum (mass conservation, $\rho\,J=\text{const}$ with $J=L^3$ and $\rho=\rhoP S^2$):
\begin{equation}
\rhoP S^2\,L^3=\text{const}\;\Longrightarrow\; S^2 L^3=\text{const}\;\Longrightarrow\;
\boxed{\;L(S)\propto S^{-2/3}\;}
\tag{E-cosmo-10}
\end{equation}
\DER\ from continuity and $\rho=\rhoP S^2$ (amplitude $S$ against occupation $\rhoP S^2$). With the radial modulus frozen at $\Smin$, the redshift accumulated over the segment $S^*\to\Smin$ is bounded:
\begin{equation}
(1+z)_{\max}=\left(\frac{S^*}{\Smin}\right)^{2/3}
=\left(\frac{S_{\max,\text{geom}}}{\Smin}\right)^{1/3}
=\left(\frac{1.4756}{0.7231}\right)^{1/3}=1.2684
\;\Longrightarrow\; z_{\max}\approx0.268.
\end{equation}
\DER, structural (depends only on the fixed walls $\{\Smin,S_{\max,\text{geom}}\}$). The second equality is an exact algebraic identity: since the critical point is the geometric mean of the walls, $S^*=\sqrt{\Smin\,S_{\max,\text{geom}}}$ (§\ref{sec:reologia}, bilateral rheological structure), one has $(S^*/\Smin)^{2/3}=\big((\Smin S_{\max,\text{geom}})^{1/2}/\Smin\big)^{2/3}=(S_{\max,\text{geom}}/\Smin)^{1/3}$, without approximation. Numerically, $(1.4756/0.7231)^{1/3}=1.2684$, hence $z_{\max}=0.2684$. This bound forces the \textbf{two-clock structure}: the radial sector, with its redshift bounded to $z_{\max}\approx0.268$, cannot produce the redshift of the supernovae ($z\sim1$--$2$); therefore the clock of the observable epoch is the phase sector, not the radial one (§11.9), and it is its scale factor $L(\tau)$ that enters the shared-factor theorem above.

This same number arises, by a second independent route, from the \emph{background dynamics} of the modulus. The background equation (§11.2) is a non-linear oscillator, $\ddot{\bar S}+3H\dot{\bar S}+\rhoP^{-1}\lP^{-2}V'_\text{eff}(\bar S)=0$, which starts from the crystallization at $S^*$ and descends to the attractor $\Smin$. Combined with the continuity law $L\propto S^{-2/3}$ just obtained (E-cosmo-10), the total accumulated stretching over that segment ---normalizing $L=1$ at the crystallization $S=S^*$--- is
\begin{equation}
\frac{L(\Smin)}{L(S^*)}=\left(\frac{S^*}{\Smin}\right)^{2/3}=1.2684 .
\end{equation}
That the \emph{kinematic bound} of the radial redshift (from the law $L\propto S^{-2/3}$ read over the fixed walls) and the \emph{dynamic stretching} of the background (the same law read over the orbit of the oscillator of §11.2, from $S^*$ to $\Smin$) give the same $1.2684$ is not a fitted numerical coincidence: both evaluate the same law $L\propto S^{-2/3}$ over the same segment $S^*\!\to\!\Smin$ ---the first as a bound of light phase accumulation, the second as an integrated stretching of the homogeneous background. Two readings of the same segment, the same number, without a free parameter.

\begin{tcolorbox}[clave,title={Conclusion --- redshift and time dilation, an identity of the medium}]
The redshift and the time dilation of distant sources are not two effects that must be postulated separately: they are the \emph{same} stretching factor $L(\tau_0)/L(\tau_e)$ of the deformation map, forced to coincide by the nullity of the TT mode ($ds^2=0$). Hence the dilation exponent of the light curves is $b=1$ \emph{exact}, without a fit ---confirmed at $0.27\sigma$ by DES-SN5YR and excluding tired light at $\sim90\sigma$. And from the mass conservation of the continuum comes $L(S)\propto S^{-2/3}$, whose bound $z_{\max}\approx0.268$ for the frozen radial sector forces the clock of the observable epoch to be the phase sector: the same two-clock structure that governs the positional reading of $H_0$. The value of that bound, $(S^*/\Smin)^{2/3}=1.2684$, comes out by two converging routes ---the kinematics of the redshift and the dynamics of the background oscillator, both over the segment $S^*\!\to\!\Smin$---, without a single fitted parameter.
\end{tcolorbox}

\medskip\hrule\medskip
% =====================================================================

\section{Rheology of el Pleno and the cosmic cycle}
\label{sec:reologia}
% =====================================================================

\subsection{The flow index and the Herschel-Bulkley reading --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

El Pleno behaves as a dilatant non-Newtonian fluid with yield
(Herschel-Bulkley). Defining the normalized packing fraction
$x\equiv\rho/\rho_{\max}=S^2/S_{\max,\text{geom}}^2\in[0,1)$, identified with the
relative deformation rate $x=\dot\gamma/\dot\gamma_{\max}$ (E46) ---both measure how close
the medium is to its hard bound---, the reduced stress that measures the departure from
rest (with $\tau\to 0$ when $x\to 0$) is $\tau_\text{red}(x)=x/(1-x)$, where the
$(1-x)$ in the denominator is the signature of geometric jamming (A6). The
\emph{flow index} is defined as the local logarithmic slope of the flow
curve, $n\equiv d\ln\tau/d\ln\dot\gamma=d\ln\tau_\text{red}/d\ln x$ (since
$d\ln x=d\ln\dot\gamma$). With $\ln\tau_\text{red}=\ln x-\ln(1-x)$,
\[
n(x)=x\,\frac{d}{dx}\big[\ln x-\ln(1-x)\big]
=x\Big(\frac{1}{x}+\frac{1}{1-x}\Big)
=1+\frac{x}{1-x},
\]
whence
\begin{equation}
\boxed{\;n(x)=\frac{1}{1-x}\;}\,,\qquad x=\frac{\dot\gamma}{\dot\gamma_{\max}}. \tag{E47}
\end{equation}
$n\to 1$ (Newtonian) in the diluted vacuum and $n\to\infty$ (stiffening by
jamming) upon approaching the maximal packing. \emph{At the present rest} the packing is $x_\text{rest}=\Smin^2/S_{\max,\text{geom}}^2=0.5229/2.1773=0.2401$, so that the medium today is slightly dilatant, almost Newtonian: $n_\text{rest}=1/(1-0.2401)=1.316$. This value is \emph{geometric} (depends only on the fixed extremes), not positional: it does not change with the relaxation of the lattice, unlike $R$, $f_c$ or $w_0$. The limit $n\to 1$ at $x\to 0$ is
\epi{DERIVED} (model-independent); the dilatant character $n>1$ and the divergence
at the bound are \epi{SLD}, and the closed form $1/(1-x)$ depends on the constitutive
packing model $x=S^2/S_{\max,\text{geom}}^2$, so that it is \epi{SLD}. The yield
stress comes from the IR wall of Klauder: to deform the modulus from
rest one must overcome the restoring response $-dV_q/dS$ of the quantum
potential $V_q^\text{dens}=3\hbar^2/(8\rhoP\lP^8 S^2)$ (E5). Differentiating and using
$\hbar=\rhoP c\lP^4$,
\[
-\frac{dV_q^\text{dens}}{dS}=\frac{3\hbar^2}{4\rhoP\lP^8 S^3}
=\frac{3\rhoP c^2}{4S^3},
\]
and evaluating at the ground-Pleno $S=\Smin^\text{estr}$:
\begin{equation}
\boxed{\;\tau_0=\frac{3}{4\,(\Smin^\text{estr})^3}\,\rhoP c^2\approx 1.9838\,\rhoP c^2\;} \tag{E48}
\end{equation}
\textbf{Verification} (sympy): with $\Smin^\text{estr}=12(\pi^2-4)/\pi^4$,
$3/[4(\Smin^\text{estr})^3]=1.98376$. The complete constitutive law is
$\tau=\tau_0+\kappa_\text{HB}\dot\gamma^{\,n(x)}$, with $\tau_0$ \epi{DERIVED} (IR wall
of Klauder) and the exponent $n(x)$ classified above. The medium has, then, the three
features of a Herschel-Bulkley fluid ---yield threshold $\tau_0$, power law,
and variable index $n(x)$--- all derived from F1.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{curva_reologica.png}
\caption{El Pleno as a dilatant Herschel-Bulkley fluid. The flow index
$n(x)=1/(1-x)$ passes from Newtonian ($n\to 1$) in the diluted vacuum to divergent
($n\to\infty$) at the hard bound. The present universe ($x=0.240$, $n=1.32$) is almost in
the Newtonian regime; the strong dilatancy only activates near the saturation. The
yield stress $\tau_0=1.9838\,\rhoP c^2$ is the IR wall of Klauder.}
\end{figure}

\subsection{The maximal deformation rate \texorpdfstring{$\dot\gamma_{\max}=c/\lP$}{gamma_max=c/lP} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

In rheology the independent variable is the \textbf{deformation rate} $\dot\gamma$
($[\text{s}^{-1}]$). Axiom A4 (UV regulation) makes the propagation operator
of F1 carry the correction $\Box\to\Box(1+\lP^2\Box/20)$ (dispersion E23), so that
every excitation has a wave number bounded by $k_{\max}\sim\lP^{-1}$ and, with velocity
$c$, frequency bounded by $\omega_{\max}=c\,k_{\max}\sim c/\lP$. A shear of the medium
is an excitation of its modulus, hence its maximal rate is that cutoff frequency:
\begin{equation}
\dot\gamma_{\max}=\frac{c}{\lP}=\frac{1}{t_P}\approx 1.8552\times10^{43}\ \text{s}^{-1},\tag{E45}
\end{equation}
the \textbf{Planck frequency}, with $t_P=\lP/c$ (verified: $c/\lP=1.8552\times10^{43}$).
It is not a dimensional translation of $c$: it is the cutoff frequency that the UV regulation of
F1 imposes, read as a deformation rate. The dimensionless \textbf{relative rate}:
\begin{equation}
x\equiv\frac{\dot\gamma}{\dot\gamma_{\max}}=\dot\gamma\,\frac{\lP}{c}\in[0,1)\tag{E46}
\end{equation}
coincides with the \textbf{Deborah number} $\mathrm{De}=\tau_\Pi^\star\dot\gamma$ with
$\tau_\Pi^\star=\lP/c=t_P$: it measures how close the deformation is to the limit rate. The
viscoelastic fluid$\leftrightarrow$solid transition occurs at $\mathrm{De}\sim1$,
which here coincides with the hard bound $x\to1$. \epi{DERIVED} from A4 and E23.

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The packing fraction $x=\rho/\rho_{\max}=S^2/S_{\max,\text{geom}}^2$ of the flow
index (E47) has thus a twin kinetic reading: $x=\dot\gamma/\dot\gamma_{\max}$.
The same $x$ that governs the dilatancy of the medium (jamming upon saturating) governs its
response to the deformation rate, anchoring the Herschel-Bulkley reading to the hard bound
of F1. The equality of the two readings of $x$ is a \emph{physical identification}
(density vs.\ rate), not a derived equation.
\end{tcolorbox}

\subsection{The Lorentz factor as a kinetic flow index --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The \emph{same} dilatant flow index that stiffens the medium upon saturating its density,
evaluated over the \emph{kinetic} fraction used of the ceiling $c$, reproduces special
relativity.

The analogy is structural. In the rheological sector, $x=\dot\gamma/\dot\gamma_{\max}$ measures
how much of the limit deformation rate is used, and the medium stiffens as
$n(x)=1/(1-x)$. In the kinetic sector, a configuration that moves at velocity $v$ uses
the fraction $u\equiv v^2/c^2$ of its capacity for change with respect to the ceiling $c$ ---the same
structure of ``fraction of the limit used''. Evaluating the flow index over this
fraction:
\begin{equation}
n(u)=\frac{1}{1-u}=\frac{1}{1-v^2/c^2}=\gamma^2 . \tag{E49--E50}
\end{equation}
The Lorentz factor squared \emph{is} the flow index of el Pleno with respect to the kinetic rate
of change (verified symbolically). This is not a formal coincidence: in
§\ref{sec:F1} the mass-shell constraint gave $E=\gamma Mc^2$, so that
$E^2=\gamma^2(Mc^2)^2=n(u)\,(Mc^2)^2$; the relativistic energy \emph{is} the structural
energy amplified by the flow index. The \textbf{time dilation} is,
literally, the dilatant stiffening of the medium when the rate of change approaches its
ceiling: the time of a fast object ``thickens'' by the same reason that el Pleno
thickens upon saturating its density. This re-obtains, by the rheological route, the result of
§\ref{sec:F1} (mass shell), closing the consistency between the two readings ---the
relativity that \emph{opened} the chain (link 1) reappears as a particular case of the
rheology that \emph{closes} it.

\subsection{The cycle equation: damped Gent oscillator --- \texorpdfstring{\T{SLD}}{[SLD]}}

Here the rheology becomes cosmology. The homogeneous background $\bar S(t)$ ---the mean density
of the universe--- obeys a non-linear oscillator equation, where the Gent potential
provides the restoring force with its hard wall at $\Smax$, and the dilatant viscosity
provides the damping:
\begin{equation}
\rhoP\lP^2\ddot{\bar S}+\underbrace{\eta(\bar S)\,\dot{\bar S}}_{\text{dilatant dissipation}}+V'_\text{eff}(\bar S)=0, \tag{E57}
\end{equation}
with effective viscosity $\eta(\bar S)\propto\dot\gamma^{\,n(x)-1}$ that \emph{grows} toward the
packing (the medium thickens upon collapsing). The structure is that of a pendulum in a
well with an infinitely hard wall at one end: the \textbf{hard bound $\Smax$ of Gent
prevents the collapse from reaching infinite density}. When the universe contracts and $\bar S$
grows toward $\Smax$, the term $V'_\text{eff}$ diverges (the Gent wall,
$\propto 1/(\Smax^2-\bar S^2)^3$) and \emph{brakes and reverses} the collapse before the
singularity: the crunch \textbf{bounces}. There is no singular Big Bang ---there is a bounce from the
finite maximal density $\rho_{\max}=S_{\max,\text{geom}}^2\rhoP\approx 2.1773\,\rhoP$.

\textbf{Three destinies according to the collapse energy.}
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Subcritical}: the energy is not enough to reach $\Smax$; damped oscillation
that relaxes to the ground-Pleno $\Smin^\text{estr}$ (does not bounce to a new universe).
\item \textbf{Critical}: the collapse reaches $\Smax$ and bounces completely (crunch-bounce),
initiating a new cosmic cycle.
\item \textbf{Supercritical (local)}: incomplete crystallization of a region $\to$ primordial
black hole, whose core remains frozen at the universal density $2.1773\,\rhoP$.
\end{itemize}
The \textbf{minimal collapse mass} is the scale below which the flexural rigidity prevents the
trapping:
\[
M_{\min}=\sqrt{\frac{3}{32\pi\,S_{\max,\text{geom}}^2}}\;m_P\approx 0.117\,m_P,
\]
(the same scale that regulates the regular Schwarzschild core, §\ref{sec:schwarzschild}):
below $M_{\min}$ there is no horizon, the curvature that the flexural penalizes is
too large for the trapping. It is a falsifiable prediction: a spectrum of primordial black
holes with a lower cutoff at $\sim 0.1\,m_P$ and cores at universal density.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{integracion_ciclo.png}
\caption{Numerical integration of the cycle equation (E57). (A) Trajectory of the modulus
$\bar S(t)$ with the dilatant viscosity: the collapse starts near $\Smax$, the Gent wall
reverses the crunch \emph{before} the hard bound (finite bounce, without singularity) and the
oscillation is damped relaxing to the ground-Pleno $\Smin$. (B) The dilatancy keeps the deformation
rate under its ceiling $\dot\gamma\leq c/\lP$ dynamically: with dilatancy (blue) the
medium never crosses $x=1$, whereas without it (red) the rate would shoot to
superluminal values ($x\sim 10^3$). The dilatant thickening is what regulates the bounce.}
\end{figure}

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- a single rheology}]
A single flow index, $n(x)=1/(1-x)$, unifies three phenomena: the stiffening by
jamming of the medium (late cosmology), the Lorentz dilation and the viscosity that
damps the cosmic cycle. The Lorentz factor $\gamma^2=n(v^2/c^2)$ is a particular
case of that same rheology: relativity and the cycle of the universe are two readings
of the same dilatant thickening of el Pleno. The physical consequence is that the universe does not
arise from a singularity nor dies in one: the hard bound of the modulus converts the Big Bang into
a bounce from finite density and forbids the infinite everywhere ---at the cosmic
origin and at the center of each black hole, whose core remains at the universal density
$2.1773\,\rhoP$. The singularity of general relativity does not occur: the medium is too
rigid to allow it. The core at $2.1773\,\rhoP$ and the cutoff $M_{\min}\approx 0.117\,m_P$
are falsifiable predictions.
\end{tcolorbox}


\subsection{The viscous crystallization transient: how the partition is imprinted --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The matter\,/\,dark-energy partition is fixed by the phase freezing at $S^*$, and the crystallization point $S^*$ by the rheological route. Between the two is the dynamical link: how the modulus relaxes from $S^*$ to its minimum $\Smin$, in how much time, and why the partition survives intact to today. The cycle equation (E57) with the derived effective potential (E52) and the dilatant Gent viscosity closes that link \emph{without a single free parameter}:
\begin{equation}
\rhoP\lP^2\,\ddot S+\eta(\dot\gamma)\,\dot S+V'_\text{eff}(S)=0,
\qquad
\eta(\dot\gamma)=\frac{\eta_0}{1-\dot\gamma/\dot\gamma_{\max}},
\qquad
\dot\gamma=2\frac{|\dot S|}{S},
\end{equation}
with $\dot\gamma_{\max}=c/\lP=1/\tau_P$ the hard bound (A6) 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{\rhoP c^2}{2}\ln\!\Big(1-\frac{S^2}{S_{\max,\text{geom}}^2}\Big)
+\frac{3\rhoP c^2}{8S^2}.
\end{equation}

\textbf{The coefficients are all derived.} In units of el Pleno ($\rhoP=c=\lP=1$, $\dot\gamma_{\max}=1$), the bounds are pure ratios of the lattice: $\Smin=12(\pi^2-4)/\pi^4=0.7231$, $S_{\max,\text{geom}}^2=8\sqrt6/9=2.1773$, $S^*=\sqrt{\Smin\,S_{\max,\text{geom}}}=1.0329$. The viscosity is the Maxwell relation of the shear sector with its two derived factors, $\mu_\text{Gent}=\rhoP c^2/S_{\max,\text{geom}}^2$ and $\tau_\Pi=\lP/c=\tau_P$ (Deborah number unity), 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 $V'_\text{eff}(\Smin)=0$, and the curvature of the minimum reproduces the structural Higgs, $V''_\text{eff}(\Smin)=\xi=19.69\,\rhoP c^2$. Three cross verifications confirm the consistency: the Klauder coefficient comes out $c_k=0.3750=3/8$ (the value that the affine quantization fixes independently); 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 $\rho(S^*)=1.067\,\rhoP\approx\rhoP$ (Planck, where the space recrystallizes) and $\rho(\Smin)=0.523\,\rhoP$ (the present relaxed sea). \epi{DERIVED}, without a free parameter.

\textbf{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 in free oscillation would reach $\dot\gamma\approx2\omega_S A/\Smin\approx3.8\gg1$, impossible: the Gent viscosity diverges when $\dot\gamma\to\dot\gamma_{\max}$, braking the motion before reaching the bound. The numerical integration of the EOM with the derived coefficients confirms that the rate never reaches its bound ---the maximum is $0.98<1$. The dilatant rheology ---the same one that gives special relativity as a stiffening upon approaching $c$ (§12.3)--- imposes that the fall respect the deformation limit: the A6 bound acts here on the \emph{velocity} of the radial mode, not on its amplitude.

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{fig_transitorio.png}
\caption{Trajectory of the modulus during the crystallization transient, integrating the viscous equation of motion with all the derived coefficients. The modulus $S(t)$ (violet) descends monotonically from $S^*=1.033$ and overshoots $\Smin$ by inertia to a minimum of $0.672$ at $t\approx1.1\,t_P$, after which it oscillates with decreasing amplitude and settles at $\Smin=0.7231$. The deformation rate $\dot\gamma(t)$ (orange) reaches a maximum of $0.98$ ---grazing the Planck bound $\dot\gamma_{\max}=c/\lP=1$ without exceeding it--- during the fast descent, and pulses toward zero at each turning point (where $\dot S=0$). The decreasing peaks of $\dot\gamma$ are the signature of viscous damping. Time is in Planck ticks.}
\end{figure}

\textbf{The transient is very brief; the rest, lasting.} With the self-limited fall, the modulus settles at $\Smin$ in a few $t_P$; the permanent entry into the $1\%$ of $\Smin$ 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 fixes 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 (§11.3). There is no coherent radial oscillation that persists and dilutes over eons ---the viscosity damps it at the Planck scale. Matter does not come from that oscillation, but from the topological defects formed by the intense shear of the hot transient, persistent like dust, $\rho\propto a^{-3}$.

\begin{tcolorbox}[apunte,title={Simulation consistency, not proof}]
The numerical integration of the EOM reproduces independently all the numbers of the transient ($E_0=4.2546$, $\xi=19.69$, $\eta_0=0.4593$, $\dot\gamma_{\max}$ not exceeded with maximum $0.98$, minimum $S=0.672$ at $t\approx1.1\,t_P$, settling in a few $t_P$). This is simulation \emph{consistency} with the derived coefficients, not a closed analytical proof of the trajectory: the closed form of $\bar S(t)$ integrated under FLRW remains as open end F-cosmo. What is established without fitting is the mechanism ---transient $\sim t_P$, rest for the remainder of the history--- by which the geometric partition $f_c$ is imprinted and survives. \epi{SLD}.
\end{tcolorbox}


\subsection{Incompressible Navier-Stokes as the hydrodynamic limit of F1 --- \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{Madelung dictionary.} The decomposition $\Psi=S\,e^{i\theta}$ translates F1 to
fluid variables with identifications fixed by the conserved quantities:
\[
\rho\equiv\rhoP\,S^2\ \text{(density)},\qquad \mathbf v\equiv c^2\lP^2\,\nabla\theta\ \text{(velocity)}.
\]
The coefficient $c^2\lP^2$ is fixed by the U(1) Noether current (E7c): its spatial
component $J^i=\rhoP c^2\lP^2 S^2\,\partial^i\theta$ equals the mass flux
$\rho v^i=\rhoP S^2 v^i$, whence $v^i=c^2\lP^2\partial^i\theta$. The flow is irrotational
in the pure-phase sector ($\mathbf v=\nabla\phi$, $\phi=c^2\lP^2\theta$); the vorticity
resides in the topological defects of $\theta$.

\textbf{Continuity.} In the IR limit ($L\gg\lP$) the flexural term of the phase equation
E7 is suppressed by $\lP^2k^2\to0$ and remains $\partial_\mu(S^2\partial^\mu\theta)=0$.
Separating in time and space and multiplying by $\rhoP c^2\lP^2$:
\begin{equation}
\boxed{\;\partial_t\rho+\nabla\!\cdot(\rho\,\mathbf v)=0\;}
\end{equation}
The mass conservation of the fluid is the conservation of the U(1) Noether current.
\epi{DERIVED}.

\textbf{Euler.} The IR phase dynamics is a Hamilton-Jacobi equation for $\phi$:
\[
\partial_t\phi+\tfrac12|\nabla\phi|^2+\frac{\mu(\rho)}{\rhoP}+Q=0,
\qquad Q=-\frac{c^2\lP^2}{2}\frac{\nabla^2\sqrt\rho}{\sqrt\rho}\ \text{(Bohm potential)}.
\]
Taking the gradient and using $(\nabla\phi\!\cdot\!\nabla)\nabla\phi=(\mathbf v\!\cdot\!\nabla)\mathbf v$:
\begin{equation}
\boxed{\;\partial_t\mathbf v+(\mathbf v\!\cdot\!\nabla)\mathbf v=-\frac{1}{\rho}\nabla p-\nabla Q\;}
\end{equation}
with barotropic pressure $p(\rho)$ from $V+V_q$ and the Bohm quantum correction. \epi{DERIVED}.

\textbf{The origin of the viscous term.} The Madelung~\cite{madelung} derivation is conservative; the viscous
term is not added by hand, but comes from the flexural correction of the Noether
current (E7c). In the non-relativistic IR limit $\Box\theta\approx\nabla^2\theta$, and the
exact identity is verified
\[
\partial^i(\nabla^2\theta)=\nabla^2(\partial^i\theta)=\frac{1}{c^2\lP^2}\nabla^2 v^i,
\]
so that the current of el Pleno has the diffusive form
$\mathbf J=\rho\mathbf v+(\text{coef})\nabla^2\mathbf v$. A current $\propto\nabla^2\mathbf v$
in the transport of momentum is a viscous term. Closing the coefficient with $\tau_\Pi$
(Maxwell-Kubo), the viscosity is the already-derived $\eta_\text{Pleno}=\mu_\text{Gent}\tau_\Pi$.
The flexural term $(\Box S)^2$ ---the ultraviolet regulator--- is the physical source of the
infrared viscosity.

\textbf{The diluted limit.} Three reductions controlled by $x=S^2/S_{\max,\text{geom}}^2$:
(a) $Q\to0$ at $L\gg\lP$; (b) $n(x)\to1$ at $x\to0$ (Newtonian), which contributes $\eta\nabla^2\mathbf v$;
(c) Mach $\to0$, $\rho\approx$ const, $\nabla\!\cdot\!\mathbf v=0$. The result:
\begin{equation}
\boxed{\;\partial_t\mathbf v+(\mathbf v\!\cdot\!\nabla)\mathbf v=-\frac{1}{\rho}\nabla p+\nu\,\nabla^2\mathbf v,\qquad \nabla\!\cdot\!\mathbf v=0\;}
\end{equation}
with native kinematic viscosity
$\nu=\eta_\text{Pleno}/(\rhoP S^2)=\sqrt5\,c\,\lP/(10\,S^2\,S_{\max,\text{geom}}^2)$,
consistent with $\nu_\text{TT}=(1/\sqrt{20})c\lP$ in the pure-shear sector. \epi{SLD}:
continuity, Euler, Bohm and the Newtonian limit are \derivado; the identification of the
viscous term with the flexural (verified symbolically) is \epi{SLD}; by Rule 19 the set
is \epi{SLD}. Navier-Stokes appears as an effective limit of F1, not as origin.

\subsection{Global regularity of the flow of el Pleno --- \texorpdfstring{\T{SLD}}{[SLD]}}

\begin{tcolorbox}[apunte,title={Scope: regularity of el Pleno, not the Clay problem}]
It is not the proof of the Clay problem (pure Navier-Stokes: constant viscosity, without
kinematic bound). The flow of el Pleno possesses the A6 bound and divergent viscosity; the
regularity \emph{of el Pleno} is established, whose regularization comes from its departing from
Navier-Stokes where the latter could blow up.
\end{tcolorbox}

\textbf{Kinematic bound.} The maximal deformation rate is finite (E45):
$\dot\gamma=\sqrt{2S_{ij}S_{ij}}\le c/\lP$, with $S_{ij}=\tfrac12(\partial_i v_j+\partial_j v_i)$
and $x=\dot\gamma/\dot\gamma_{\max}\in[0,1)$ always.

\textbf{Beale-Kato-Majda criterion.} There is a singularity at $T^*$ if and only if
$\int_0^{T^*}\|\boldsymbol\omega\|_\infty\,dt=\infty$. The A6 bound limits the deformation
$S_{ij}$, not directly the vorticity (a rigid rotation has $S_{ij}=0$,
$\boldsymbol\omega\neq0$). The only route to blow-up is vortex stretching, of rate
\[
\tfrac12\frac{D|\boldsymbol\omega|^2}{Dt}\Big|_\text{stretch}=\omega_i\,S_{ij}\,\omega_j,
\]
so that only the deformation $S_{ij}$ stretches the vortices, and the A6 bound controls the
growth of $\boldsymbol\omega$ by the physically correct route.

\textbf{At most exponential growth.} From $\omega_i S_{ij}\omega_j\le\|S\|_\text{op}|\boldsymbol\omega|^2$
and $\|S\|_\text{op}\le\dot\gamma_{\max}/\sqrt2$:
\[
\frac{d|\boldsymbol\omega|}{dt}\le\frac{\dot\gamma_{\max}}{\sqrt2}|\boldsymbol\omega|
\ \Longrightarrow\
|\boldsymbol\omega(t)|\le|\boldsymbol\omega(0)|\,e^{\dot\gamma_{\max}t/\sqrt2},
\]
and the BKM integral converges for every finite $T$:
\[
\int_0^T\|\boldsymbol\omega\|_\infty\,dt\le\|\boldsymbol\omega(0)\|_\infty\frac{\sqrt2}{\dot\gamma_{\max}}\big(e^{\dot\gamma_{\max}T/\sqrt2}-1\big)<\infty.
\]
There is no finite-time singularity: the flow is globally regular. The dynamical reinforcement
is given by the Herschel-Bulkley viscosity, which diverges upon approaching the bound
($\eta_\text{eff}\propto 1/(1-x)^2\to\infty$): negative feedback that makes the
growth sub-exponential. \epi{SLD}.

\subsection{The singularity under emergent time --- \texorpdfstring{\T{SLD}}{[SLD]}}

\begin{tcolorbox}[apunte,title={Scope: the singularity dissolves through the definition of time}]
Reinterpretation of the terms of the problem: under the definition of time of PIU (time
as accumulated change), the notion of finite-time singularity dissolves.
\end{tcolorbox}

PIU derives time, does not postulate it: the internal change $C\in[0,c]$ has ceiling $c$, and the
physical time is the accumulated change (canonical definition in corpus §I.8.9-bis:
$\tau=(1/c)\int_\gamma C\,d\lambda$), $\tau_\text{phys}\propto\int\|\nabla\mathbf v\|\,dt$,
which is identically the integrand of the BKM criterion. From this, exactly:
\begin{equation}
\boxed{\;\text{blow-up (BKM)}\iff\int\|\nabla\mathbf v\|\,dt=\infty\iff\tau_\text{phys}=\infty\;}
\end{equation}
A blow-up is, by its own mathematical definition, an event at infinite physical time. For
the self-similar blow-up $\|\nabla\mathbf v\|=A/(T^*-t)$:
\[
\tau_\text{phys}(T)\propto\int_0^T\frac{A\,dt}{T^*-t}=A\ln\frac{T^*}{T^*-T}\xrightarrow{T\to T^*}\infty.
\]
The gradient diverges in finite coordinate time $T^*$, but the physical time diverges
logarithmically: the singularity is never reached in finite physical time. A blow-up
would require infinite local change (forbidden by A1, $C\le c$; the A6 bound is A1 plus the grain,
$\dot\gamma_{\max}=c/\lP$) in finite physical time (impossible, since $\tau_\text{phys}$ diverges):
infinite change is not accumulated in finite accumulated change. \epi{SLD}, internal to PIU.

\subsection{Bilateral rheological structure and the critical point \texorpdfstring{$S^*$}{S*} --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

The modulus lives between two walls: the IR of Klauder ($S\to\Smin$, A2) and the hard of Gent
($S\to\Smax$, A6). The flow index (E47) measures the proximity to Gent, and its mirror toward the
IR wall ---by the same procedure of logarithmic slope of the restoring stress---
is $m$:
\[
n(x)=\frac{1}{1-x}=\frac{\Smax^2}{\Smax^2-S^2},\quad x=\frac{S^2}{\Smax^2};
\qquad
m(\bar y)=\frac{1}{1-\bar y}=\frac{S^2}{S^2-\Smin^2},\quad \bar y=\frac{\Smin^2}{S^2}.
\]
The bilateral index $N(S)=n+m$ diverges at both walls and has a minimum (minimal rigidity) at
\begin{equation}
\boxed{\;S^*=\sqrt{\Smin\,\Smax}\approx1.0330\;}
\end{equation}
the exact geometric mean of the walls (with $\Smin=0.7231$ and
$\Smax=S_{\max,\text{geom}}=\sqrt{8\sqrt6/9}\approx1.4756$). At $S^*$ the packing
fractions are identical, $x^*=\bar y^*=\Smin/\Smax\approx0.4900$ (exact). \epi{DERIVED}.

\textbf{Robustness.} Five definitions of the bilateral index (sum $n+m$, product $nm$,
log-sum, excess, inverse) give the same $S^*$: the root is the inversion symmetry
$S\to\Smin\Smax/S$, which exchanges the two walls ($x\leftrightarrow\bar y$,
Gent$\leftrightarrow$Klauder) and of which $S^*$ is the only fixed point. Every symmetric
combination of $n$ and $m$ is extremized at the self-symmetric point. \epi{DERIVED}.

\textbf{$S^*$ as a crystallization point.} $S^*$ is not a force equilibrium (the minimum
of the potential is $\Smin$) but the minimal-rigidity one, where the elastic barrier to reorganize
the lattice is minimal; therefore the packing transition ($Z=12\leftrightarrow Z=4$) occurs
at $S^*$, at density
\[
\rho(S^*)=x^*\,\rho_{\max}=\frac{\Smin}{\Smax}\,S_{\max,\text{geom}}^2\,\rhoP\approx1.067\,\rhoP\approx\rhoP,
\]
that is, the recrystallization of space occurs upon crossing the Planck density. \epi{SLD}.

\medskip
\begin{tcolorbox}[nota,title={Conclusion --- the hydrodynamics of el Pleno}]
Incompressible Navier-Stokes is the IR limit of F1, with the viscous term as an infrared
imprint of the ultraviolet flexural. The flow of el Pleno is globally regular (A6 bound plus
divergent viscosity), and under the emergent time the singularity lives at
$\tau_\text{phys}=\infty$. The modulus has a self-symmetric point $S^*=\sqrt{\Smin\Smax}$ of
minimal rigidity, where it crystallizes upon crossing $\rho\approx\rhoP$.
\end{tcolorbox}

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
The universe flows like an ordinary viscous fluid, and cannot blow up. In its
smooth and slow regime, el Pleno obeys the same Navier-Stokes equations as water or
air ---the viscosity that governs them is not added: it is the same grain rigidity that at
fine scale regulates the medium, seen from afar as friction. And since every real material
has a finite rigidity and a maximal deformation velocity, the flow of el Pleno never
develops the infinities (``blow-up'') that torment ideal hydrodynamics: a
whirlpool can only sharpen up to the limit that the grain allows, and the viscosity grows
without ceiling just when it comes closest, braking it. The point $S^*$ ---halfway
between the two walls, where the medium is most fluid--- is where the vacuum recrystallizes upon
compressing to the Planck density. El Pleno behaves, in sum, like a genuine physical
fluid: viscous, regular, and with a definite phase-transition point.
\end{tcolorbox}


\subsection{The inversion self-duality: generating operator and conformal inversion of the field --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The inversion symmetry $I:S\to\Smin\Smax/S$ of the critical point admits two deeper readings: that the operator $I$ not only \emph{detects} the self-duality but \emph{constructs} it from a single wall, and that over the complete field $\Psi=S\,e^{i\theta}$ the inversion of the modulus is the conformal inversion of the complex plane. Both follow from F1 without a new parameter.

\textbf{The operator generates the mirror wall.} In the logarithmic variable $u=\ln(S/S^*)$, the inversion $I$ is the exact reflection $u\to-u$, with $I(S^*)=S^*$ and $n(u)=m(-u)$. Applied to the only geometric wall of F1, that of Gent,
\begin{equation}
V_\text{topo}(S)=-\frac{\rhoP c^2}{2}\ln\!\Big(1-\frac{S^2}{S_{\max,\text{geom}}^2}\Big),
\end{equation}
the operator generates its mirror image toward the lower wall:
\begin{equation}
I(V_\text{topo})(S)=V_\text{topo}\!\Big(\frac{\Smin\Smax}{S}\Big)
=-\frac{\rhoP c^2}{2}\ln\!\Big(1-\frac{\Smin^2}{S^2}\Big),
\end{equation}
an IR wall of Gent form, divergent at $S\to\Smin$. The exact self-dual potential is the canonical symmetrization of the real wall under the idempotent projector $P_D=\tfrac12(1+I)$:
\begin{equation}
\boxed{\;V_D=(1+I)\,V_\text{topo}=V_\text{topo}+I(V_\text{topo}),\qquad P_D^2=P_D\;}
\end{equation}
exactly even in $u$ by construction. The upper Gent wall contains its own lower wall under inversion: the self-duality is not postulated, the operator $I$ constructs it from a single geometric wall. \epi{DERIVED} (via the Language of el Pleno).

\textbf{The measured breaking: Klauder is not Gent.} The \emph{physical} potential of F1 is not $V_D$: its lower wall is the quantum one of Klauder ($V_q=3\rhoP c^2/8S^2$, §2.3), not the geometric IR wall that $I$ generates. For exact self-duality, the coefficient of the real IR wall (Klauder, $3/8=0.375$) would have to equal that of the generated one (Gent-IR, divergence coefficient $\Smin^2=0.5229$). They do not coincide:
\begin{equation}
\frac{\Smin^2}{3/8}=\frac{0.5229}{0.375}=1.394,\qquad
\Smin^2-\frac{3}{8}=0.148.
\end{equation}
The ratio $1.394$ does not correspond to any known invariant of the program (were discarded $\sqrt2$, $\pi^2/7$, $1/f_c$ as false friends). It is not a failure but physical information: $0.148$ measures how \emph{quantum} (Klauder, from A2) and not \emph{geometric} (Gent, from A6) the IR wall is. The self-duality would be exact if the IR wall were purely geometric; that it be of Klauder is what breaks it, by a computable quantity. The generalized force $Q(u)=S\,V'(S)$ of the physical potential $V=V_q+V_\text{topo}$ is not odd under $u\to-u$, confirming that $u\to-u$ is not a symmetry of the dynamical potential. \epi{DERIVED}.

\textbf{The phase is not a spectator.} The kinetic term of F1 is $\tfrac12\rhoP c^2\lP^2[(\partial S)^2+S^2(\partial\theta)^2]$. The factor $S^2$ that weights the phase makes it rigid where $S$ is large (Gent) and soft where $S$ is small (Klauder). Under $I$ that weight is inverted: $S^2\to S^{*4}/S^2$. The complete symmetry that leaves F1 invariant is not only $u\to-u$, but an inversion combined with a dilation of the phase gradient by $e^{2u}$. \epi{DERIVED}.

\textbf{The inversion of the modulus is the conformal inversion of the field.} Over $\Psi=S\,e^{i\theta}$, the inversion of the modulus is exactly the conformal inversion of the complex plane with respect to the circle of radius $S^*$:
\begin{equation}
\boxed{\;\Psi\;\longrightarrow\;\frac{S^{*2}}{\overline{\Psi}}\;}
\end{equation}
Direct verification: with $\overline\Psi=S\,e^{-i\theta}$,
\begin{equation}
\frac{S^{*2}}{\overline\Psi}=\frac{S^{*2}}{S\,e^{-i\theta}}=\frac{S^{*2}}{S}\,e^{i\theta},
\end{equation}
which inverts the modulus ($S\to S^{*2}/S$, with $S^{*2}=\Smin\Smax$) and \emph{preserves} the phase $\theta$. It is the classical operation of spherical mirror ---circle inversion--- in the plane $\Psi$, the same that in the theory of complex functions generates the Droste effect via $z\to\log z$. The self-duality of el Pleno is not an analogy with the Droste: it is literally the conformal inversion of the field. \epi{DERIVED}.

\textbf{The invariant set is a circle, not a point.} Over the isolated modulus, $S^*\approx1.0330$ was a fixed point. Over the complete plane $\Psi$, the set of fixed points of the conformal inversion is the \emph{entire} circumference $|\Psi|=S^*$: the self-duality has no center but an invariant circle. The plane $\Psi$ is genuinely circular (radius $=$ modulus $S$, angle $=$ phase $\theta$); the inversion reflects each interior point ($S<S^*$, toward Klauder) to an exterior one ($S>S^*$, toward Gent) over the same radius, conserving the phase. The central hole $|\Psi|<\Smin$ is the region forbidden by the Klauder wall, where the phase would remain undefined at $\Psi=0$ ---exactly what the strict positivity of A2 prevents. The modulus-only reading is the shadow, in the modulus, of this operation over the complete field. \epi{DERIVED}.

\begin{tcolorbox}[clave,title={What this self-duality fixes and what it does not}]
The geometric structure of el Pleno under inversion is exact: the operator $I$ generates the mirror wall from a single wall (projector $P_D$), the fixed point $S^*$ and the invariant circle $|\Psi|=S^*$ are exact, and the inversion of the modulus is the conformal inversion $\Psi\to S^{*2}/\overline\Psi$ of the field. What is broken is the self-duality of the \emph{dynamical} potential: the physical IR wall is of Klauder (quantum), not the Gent-IR that $I$ would generate, and the difference ---$\Smin^2-3/8=0.148$, ratio $1.394$--- measures how quantum and not geometric that wall is. The self-similarity of el Pleno is bounded by construction: the minimal scale $\lP$ of axiom A4 cuts the zoom from below and the coherence scale from above, so that F1 forbids the strict fractal and admits only self-similarity in a finite band ---the class to which every real physical fractal belongs. \epi{DERIVED}.
\end{tcolorbox}



\subsection{The thermal statistics of el Pleno by configuration counting --- \texorpdfstring{\T{DERIVED}}{[DERIVED]}}

The entropy density of the mode gas is not taken from standard statistical mechanics: it is
obtained by counting the configurations of the eigenmodes of F1. The count returns the equilibrium occupation without
assuming it, and with it the coefficient $2\pi^2/45$ of the Stefan-Boltzmann law.

\textbf{Step 1 --- each phase mode of F1 is a harmonic oscillator.} The phase sector of the
quadratic expansion of F1 around the \emph{Pleno}-ground state has kinetic term
$\tfrac12\rhoP c^2\lP^2 (\Smin^\text{estr})^2(\partial\theta)^2$ and flexural term
$-(\rhoP c^2\lP^4/40)(\Smin^\text{estr})^2(\Box\delta\theta)^2$. Decomposing the perturbation
into Fourier modes,
\[
\delta\theta(\mathbf{x},t)=\sum_{\mathbf{k}} q_\mathbf{k}(t)\,e^{i\mathbf{k}\cdot\mathbf{x}},
\]
and substituting into the Lagrangian, the cross terms vanish by orthogonality of the
exponentials over the volume, so that each $\mathbf{k}$ gives an independent oscillator
Lagrangian $L_\mathbf{k}=\tfrac12 M_k\dot q_\mathbf{k}^2-\tfrac12 K_k q_\mathbf{k}^2$ with
inertia and stiffness
\begin{equation}
M_k=\rhoP\lP^2 (\Smin^\text{estr})^2,
\qquad
K_k=\rhoP c^2\lP^2 (\Smin^\text{estr})^2\left(k^2+\frac{\lP^2 k^4}{20}\right).
\tag{E-BE-1}
\end{equation}
The $k^2$ comes from the spatial kinetic term and the $k^4$ from the flexural one, with its
coefficient $1/40$ and a factor $2$ from the second derivative of the square, which leaves
$1/20$. Hence
\[
\omega_k^2=\frac{K_k}{M_k}
=\frac{\rhoP c^2\lP^2 (\Smin^\text{estr})^2\big(k^2+\lP^2k^4/20\big)}
       {\rhoP\lP^2 (\Smin^\text{estr})^2}
=c^2k^2\left[1+\frac{(k\lP)^2}{20}\right],
\]
where $\rhoP$, $\lP^2$ and $(\Smin^\text{estr})^2$ all cancel: the dispersion relation does
not depend on the amplitude of the ground state. It is exactly the one already derived for the
phase sector, with its ultraviolet correction of coefficient $1/20$ and its luminal limit
$\omega_k\to ck$ in the infrared. The levels of each mode are $E_m=m\,\hbar\omega_k$ with
$m=0,1,2,\dots$

\textbf{Verification of the units of (E-BE-1).} $[M_k]=[\rhoP][\lP^2]=(M/L^3)(L^2)=M/L$;
$[K_k]=[\rhoP][c^2][\lP^2][k^2]=(M/L^3)(L^2/T^2)(L^2)(1/L^2)=M/(L\,T^2)$; and
$[K_k/M_k]=1/T^2=[\omega^2]$ $\checkmark$.

\textbf{Step 2 --- microcanonical counting, with no assumed distribution.} Consider a region
with $N$ modes of frequency $\omega$ among which $q$ indistinguishable quanta are distributed.
The number of distributions is the stars-and-bars coefficient,
\[
\Omega(q,N)=\binom{q+N-1}{q}=\frac{(q+N-1)!}{q!\,(N-1)!},
\]
and the entropy of \emph{el Pleno} is its logarithm, $\sigma_\text{Pleno}=\ln\Omega$
(dimensionless; $k_B$ enters only when the count is dressed with the kelvin rod). Applying
Stirling $\ln n!\simeq n\ln n-n$ in the regime $q,N\gg1$:
\[
\ln\Omega\simeq (q+N)\ln(q+N)-(q+N)-\big[q\ln q-q\big]-\big[N\ln N-N\big]
=(q+N)\ln(q+N)-q\ln q-N\ln N,
\]
where the linear terms cancel exactly: $-(q+N)+q+N=0$. Writing the mean occupation
$n\equiv q/N$ and factoring out $N$,
\[
\ln\Omega = N\Big[(1+n)\ln\big(N(1+n)\big)-n\ln(Nn)-\ln N\Big]
= N\big[(1+n)\ln(1+n)-n\ln n\big],
\]
because the $\ln N$ terms cancel: $(1+n)\ln N-n\ln N-\ln N=0$. The entropy per oscillator is
then
\begin{equation}
\frac{\sigma(n)}{k_B}=(1+n)\ln(1+n)-n\ln n .
\tag{E-BE-2}
\end{equation}
The step is pure combinatorics over an equally spaced spectrum: no statistics has been chosen.

\textbf{Step 3 --- equilibrium forces the occupation.} With the thermodynamic definition of
temperature, $1/T=\partial S/\partial E$, the energy of the region is
$E=q\,\hbar\omega=N n\hbar\omega$ and its entropy $S=N k_B\,\sigma(n)/k_B$, so that
\[
\frac{\partial S}{\partial E}
=\frac{\partial S/\partial n}{\partial E/\partial n}
=\frac{N k_B\,\partial_n\big[(1+n)\ln(1+n)-n\ln n\big]}{N\hbar\omega}
=\frac{k_B}{\hbar\omega}\,\partial_n\big[(1+n)\ln(1+n)-n\ln n\big].
\]
Differentiating the bracket term by term,
\[
\partial_n\big[(1+n)\ln(1+n)\big]=\ln(1+n)+1,
\qquad
\partial_n\big[n\ln n\big]=\ln n+1,
\]
and subtracting, the ones cancel and $\ln\frac{1+n}{n}$ remains. The equilibrium condition
$1/T=(k_B/\hbar\omega)\ln[(1+n)/n]$ rearranges to $\ln[(1+n)/n]=\hbar\omega/k_BT$, and
exponentiating and inverting:
\[
\frac{1+n}{n}=e^{\hbar\omega/k_BT}
\qquad\Longrightarrow\qquad
\frac1n=e^{\hbar\omega/k_BT}-1 ,
\]
whence
\begin{equation}
\boxed{\;\langle n\rangle=\frac{1}{e^{\hbar\omega/k_BT}-1}\;}
\tag{E59}
\end{equation}
the Bose-Einstein occupation, \emph{derived} from the count and not postulated.

\textbf{Step 4 --- entropy density and the Stefan-Boltzmann coefficient.} Integrating (E-BE-2)
with the occupation (E59) over the three-dimensional mode measure, with $g_*$ polarisations
and in the infrared regime $\omega=ck$ where the $1/20$ correction is negligible,
\[
\frac{s}{k_B}=\frac{g_*}{2\pi^2}\int_0^\infty k^2\,\sigma_k\,dk ,
\qquad
\sigma_k=(1+n_k)\ln(1+n_k)-n_k\ln n_k .
\]
With the change of variable $x=\hbar ck/k_BT$, whence $k=(k_BT/\hbar c)x$ and
$dk=(k_BT/\hbar c)dx$, the factor $k^2dk$ contributes $(k_BT/\hbar c)^3x^2dx$ and comes out of
the integral:
\[
\frac{s}{k_B}=\frac{g_*}{2\pi^2}\left(\frac{k_BT}{\hbar c}\right)^{3}
\int_0^\infty x^2\Big[(1+n)\ln(1+n)-n\ln n\Big]dx ,
\qquad n=\frac{1}{e^x-1}.
\]
The dimensionless integral equals $4\pi^4/45$, and $(1/2\pi^2)(4\pi^4/45)=2\pi^2/45$, whence
\begin{equation}
s=\frac{2\pi^2}{45}\,g_*\,k_B\left(\frac{k_BT}{\hbar c}\right)^{3},
\tag{E-BE-3}
\end{equation}
which is the Stefan-Boltzmann law with its coefficient.

\textbf{Numerical check built so that it can fail.} The occupation is obtained by solving
$\ln[(1+n)/n]=x$ by bisection ---without using (E59)--- and with it $\int k^2\sigma_k\,dk$ is
integrated numerically. The result, $0.877298168985721$ for $g_*=2$, agrees with
$(2\pi^2/45)g_*=0.877298168985721$ to all fifteen digits of double precision. A check that
re-evaluated the closed formula could not fail and would not be a check.

\textbf{What enters the count and what does not.} Three ingredients, and none of them is
imported quantum statistics. \emph{First}, that the mode $\mathbf{k}$ is a harmonic
oscillator: this is the quadratic expansion of F1 (Step 1), not an added model. \emph{Second},
that the quanta are indistinguishable: this is not a statistical choice but what it means for
the level $|m\rangle$ of a mode to be $m$ excitations of the same degree of freedom ---there
are no labels to attach---. The choice between statistics arises when \emph{particles} are
distributed among \emph{levels}; here there are no particles, there is a field with modes.
\emph{Third}, the bosonic character ---occupation with no upper bound--- which the programme
itself fixes: the phase mode is a spin-$0$ Goldstone boson and its angular momentum is
$p_\theta=n\hbar$ with $n\in\mathbb{Z}$ without a ceiling; a fermionic occupation would
require truncating at $m=1$, and nothing in the spectrum does so.

The only element taken from outside is the equally spaced spectrum of the quantised harmonic
oscillator, which the programme already uses in the quantisation of the $U(1)$ rotor from
which $\hbar$ follows: it is the quantisation of an eigenmode of F1, not foreign physics, and
it rests on the grain (A4) and the $U(1)$ phase (A5).

\subsection{Consequences: viscosity bound and sonic horizon --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]}}

\textbf{The cold vacuum is an elastic solid, not a fluid.} In the relaxed regime the
dark energy has $w=-1$ (§11.3), hence $\rho+p=0$ and the entropy density of the
ground-Pleno is null, $s=(\rho+p)/T=0$: a pure coherent condensate of minimal entropy,
which reinforces the DE-Curtain reading from thermodynamics. \epi{DERIVED}
(consequence of $w=-1$).

\textbf{The thermal Pleno and the KSS bound.} In high excitation (bounce, interior of a black
hole, early universe) the TT modes ($g_*=2$, two polarizations) form a radiation-type
gas with $s=(2\pi^2/45)\,g_*\,k_B\,(k_B T/\hbar c)^3$. For any fluid, the
shear-viscosity / entropy ratio is bounded by the
Kovtun-Son-Starinets limit:
\begin{equation}
\frac{\eta}{s}\;\geq\;\frac{\hbar}{4\pi k_B}\,,\tag{E51}
\end{equation}
saturated in conformal theories with a gravitational dual and approximated in the
quark-gluon plasma ($\sim2$--$2.5$ times the bound). The ratio of the TT gas is bounded from two
sides without the complete collisional kinetics.

\textbf{The ultraviolet sub-bound is a pure number: complete derivation.} The value of $R$ at
the most extreme point of the TT regime depends on no measuring rod: the three primaries cancel
by dimension and $k_B$ enters twice and annihilates itself.

\emph{Step 1 --- the saturation temperature.} The A6 bound fixes the maximum temperature as
the one at which the energy density of the radiation equals the saturation density of the
medium. With the density derived from the same count as in the previous subsection,
\[
\frac{\pi^2}{30}\,g_*\,\frac{(k_BT)^4}{(\hbar c)^3}\;=\;S_{\max,\text{geom}}^2\,\rhoP c^2 ,
\qquad g_*=2 .
\]
Substituting the native form $\hbar=\rhoP c\,\lP^4$ (structural identity), the cube in the
denominator is
\[
(\hbar c)^3=\big(\rhoP c\,\lP^4\cdot c\big)^3=\rhoP^3\,c^6\,\lP^{12},
\]
whence
\[
(k_BT_{\max})^4=\frac{30\,S_{\max,\text{geom}}^2}{\pi^2 g_*}\;\rhoP c^2\cdot\rhoP^3 c^6\lP^{12}
=\frac{30\,S_{\max,\text{geom}}^2}{\pi^2 g_*}\;\rhoP^4\,c^8\,\lP^{12},
\]
and taking the fourth root ---where $\rhoP^4\to\rhoP$, $c^8\to c^2$ and $\lP^{12}\to\lP^3$---:
\[
k_BT_{\max}=\left(\frac{30\,S_{\max,\text{geom}}^2}{\pi^2 g_*}\right)^{1/4}\rhoP c^2\lP^3 .
\]
With $g_*=2$ and $S_{\max,\text{geom}}^2=8\sqrt6/9$ the bracket equals
$30(8\sqrt6/9)/(2\pi^2)=40\sqrt6/(3\pi^2)$, and its fourth root admits the closed form
\begin{equation}
k_BT_{\max}=\left(\frac{40\sqrt6}{3\pi^2}\right)^{1/4}\rhoP c^2\lP^3
=\frac{5^{1/4}\,6^{7/8}}{3\sqrt\pi}\;\rhoP c^2\lP^3
=1.348741293\;\rhoP c^2\lP^3 .
\tag{E-R-1}
\end{equation}

\emph{Step 2 --- the dimensionless ratio.} The dissipation window is measured against the
universal bound $\hbar/(4\pi k_B)$, so that the relevant ratio is
\[
R\;\equiv\;\frac{\eta/s}{\hbar/(4\pi k_B)}
=\frac{4\pi k_B}{\hbar}\cdot\frac{\eta}{s}
=\frac{4\pi k_B}{\hbar}\cdot\frac{\tau_\text{col}T}{5}
=\frac{4\pi}{5}\,\frac{\tau_\text{col}\,(k_BT)}{\hbar},
\]
where in the last step $k_B$ and $T$ group into the product $k_BT$, which is an energy. Here
is the first half of the self-annihilation: $k_B$ enters once in the denominator of the bound
and once in writing the temperature as energy per degree of freedom, and the two occurrences
combine leaving no trace.

\emph{Step 3 --- evaluation at the extreme point.} The relaxation time cannot fall below the
ultraviolet regulation time, $\tau_\Pi=\lP/(\sqrt{20}\,c)$, and the temperature cannot rise
above (E-R-1). Substituting both, together with $\hbar=\rhoP c\lP^4$:
\[
R(\tau_\Pi,T_{\max})
=\frac{4\pi}{5}\cdot\frac{\lP}{\sqrt{20}\,c}
\cdot\frac{1.348741293\,\rhoP c^2\lP^3}{\rhoP c\,\lP^4}
=\frac{4\pi}{5\sqrt{20}}\cdot 1.348741293 ,
\]
because the central quotient simplifies term by term: $\rhoP/\rhoP=1$, $c^2/(c\cdot c)=1$ and
$\lP\cdot\lP^3/\lP^4=1$. With the closed forms instead of the decimals,
\begin{equation}
R(\tau_\Pi,T_{\max})=\frac{2\cdot 5^{3/4}\,6^{7/8}\sqrt\pi}{75}=0.757973 .
\tag{E61}
\end{equation}

\emph{Verification that no free symbol remains.} In (E61) neither $\rhoP$, nor $\lP$, nor $c$,
nor $k_B$, nor $\hbar$ appears. The three primaries cancel by dimensional analysis ---the ratio
$R$ is dimensionless by construction--- and $k_B$ self-annihilates by the mechanism of Step 2:
it enters twice and measures nothing of \emph{el Pleno} that is not already in the energy.
Checked symbolically: the set of free symbols of the resulting expression is empty.

\emph{Reading of the factors.} The numbers that remain are words of the medium: the $5$ and
the $\sqrt\pi$ of the angular family that appears in the average over directions, and the
$\sqrt6$ ---here as $6^{7/8}$--- of the tetrahedral bond that fixes
$S_{\max,\text{geom}}^2=8\sqrt6/9$. That $R<1$ at that point means that the physical window of
the TT gas lies above the sub-bound, not that the universal bound is violated: the value
$0.757973$ is the extreme of a minimal-relaxation estimate, not a value attained.

\textbf{The phase of the ratio.} Pairing the viscosity of the saturated solid
$\eta_\text{Pleno}=\mu_\text{Gent}\,\tau_\Pi$ with the entropy of the radiation gas is
inconsistent: they are distinct phases of the medium (elastic solid vs TT gas). The ratio so
formed, evaluated at the saturation temperature, gives $\eta/s\approx0.60\,\hbar/4\pi k_B$,
below the universal bound ---physically impossible, which betrays the pairing.
The correct computation uses the viscosity of the TT-mode gas itself.

\textbf{The TT gas ratio in the relaxation-time approximation.} For an
ultrarelativistic gas of quasiparticles (RTA, Boltzmann),
\[
\eta=\frac{1}{5}(\varepsilon+P)\,\tau_\text{col},
\qquad
s=\frac{\varepsilon+P}{T},
\]
with $\varepsilon=\rho_\text{rad}$ and $P=\rho_\text{rad}/3$. The ratio cancels the density:
\begin{equation}
\boxed{\;\frac{\eta}{s}=\frac{1}{5}\,\tau_\text{col}\,T\;}
\end{equation}
independent of $\rho_\text{rad}$: it depends only on the product $\tau_\text{col}\,T$.

\textbf{Maximal temperature by A6 saturation.} The maximal physical temperature of the TT gas is
that at which its energy density reaches the maximal density of el Pleno (saturation A6).
With Stefan-Boltzmann ($g_*=2$) and the native identity $\hbar=\rhoP c\,\lP^4$,
\[
\rho_\text{rad}(T)=\frac{\pi^2}{30}\,g_*\,\frac{(k_B T)^4}{(\hbar c)^3},
\qquad
\rho_\text{rad}(T_{\max})=\rho_{\max}=S_{\max,\text{geom}}^2\,\rhoP c^2,
\]
\[
k_B T_{\max}=\Big(30\,S_{\max,\text{geom}}^2\,\rhoP c^2\,(\hbar c)^3/\pi^2 g_*\Big)^{1/4}
=\frac{\sqrt[4]{5}\;6^{7/8}}{3\sqrt\pi}\,\rhoP c^2\,\lP^3.
\]

\textbf{Two-sided bound.} The lower bound is the universal E51,
$R\equiv(\eta/s)/(\hbar/4\pi k_B)\geq1$. The upper bound is structural: no process of el
Pleno relaxes below the grain $\lP$, so that $\tau_\text{col}\geq\tau_\Pi=\lP/\sqrt{20}\,c$.
Evaluating $\eta/s=\tfrac15\tau_\Pi T$ at $T_{\max}$:
\[
R(\tau_\Pi,T_{\max})=\frac{2\cdot5^{3/4}\,6^{7/8}\sqrt\pi}{75}\approx0.758<1.
\]
That the UV floor $\tau_\Pi$ fall below E51 is informative: the universal bound forces
$\tau_\text{col}\geq\tau_\Pi/0.758\approx1.319\,\tau_\Pi$. The upper extreme is fixed by the
natural thermal time $\tau_T=\hbar/k_B T_{\max}$, with $\tau_T/\tau_\Pi\approx3.32$,
corresponding to $\eta/s\approx2.5\,$E51. The ratio remains in a narrow window:
\begin{equation}
\boxed{\;1\;\lesssim\;\frac{\eta/s}{\hbar/4\pi k_B}\;\lesssim\;2.5\;}
\end{equation}
the strip of the quark-gluon plasma, the most perfect fluid known experimentally.
\epi{SLD}: universal KSS lower bound, upper bound by the native UV scale $\tau_\Pi$
evaluated at $T_{\max}$; the exact value within the window requires the precise
collisional relaxation time and remains as a bounded open end.

\textbf{The event horizon is the sonic surface of el Pleno.} The velocity of the
modes is $c$ (TT mode). At the horizon the escape velocity equals $c$, hence the
horizon is the surface where the flow of the medium reaches its sound velocity
---analog of Unruh, here exact. The Mach number is $\mathrm{Ma}=v/c$, and the
impossibility of free supersonic flow is the same bound as the light barrier and the
dilatant jamming; the Hawking radiation is read as radiation from that sonic
surface. \textbf{[Strong logical deduction]}: the identity
$v_\text{sound}=v_\text{escape}=c$ is exact; the acoustic reading is an interpretation
consistent with the black-hole physics of the program.

\medskip
\begin{tcolorbox}[nota,title={Conclusion}]
The rheology of el Pleno delivers two closing readings: the cold vacuum is a solid
($s=0$, derived) and the thermal Pleno is a near-perfect fluid whose ratio $\eta/s$
is bounded from two sides, $1\lesssim(\eta/s)/(\hbar/4\pi k_B)\lesssim2.5$ ---the strip
of the near-perfect fluid, by the universal KSS bound below and the native UV scale $\tau_\Pi$
at the saturation temperature above. The horizon as a sonic surface unifies the
light barrier, the dilatant jamming and the escape velocity in a single bound
$\mathrm{Ma}\le1$.

Two pieces close the chapter with material of its own. The \emph{thermal statistics} of that gas is not imported: the equilibrium occupation (E59) and the coefficient $2\pi^2/45$ of the Stefan-Boltzmann law (E-BE-3) follow from counting the configurations of the eigenmodes of F1, so that the entropy entering the ratio belongs to the medium and not to an auxiliary theory. And the \emph{ultraviolet sub-bound} $R=2\cdot5^{3/4}6^{7/8}\sqrt\pi/75=0.757973$ (E61) is a pure number: the three primaries cancel by dimension and $k_B$ self-annihilates by entering twice ---in the denominator of the bound and in writing the temperature as energy---, so that the position of the window relative to the universal bound depends on no measuring rod.
\end{tcolorbox}

\begin{tcolorbox}[clave,title={Conclusion --- what it means physically}]
The vacuum changes state according to its temperature, like water. Cold and relaxed, el Pleno
is a perfect solid: it does not flow, has no internal friction, generates no entropy ---it is the
vacuum at rest. Heated to the extreme (in the bounce, inside a black hole, in the
newborn universe), it becomes a near-perfect liquid, with the minimal friction that
the quantum laws allow: exactly the regime of the quark-gluon plasma that is
created in heavy-ion colliders, the most perfect fluid ever measured in the
laboratory. That the hot vacuum of the universe and that drop of ultradense matter of the
laboratory share the same fluidity strip is not a coincidence: both are the same
fundamental medium in its thermal phase. And the horizon of a black hole is, in this
reading, simply the surface where that fluid reaches the sound velocity ---it
cannot flow faster, just as air cannot in a nozzle.
\end{tcolorbox}

\medskip\hrule\medskip

% =====================================================================
