% !TeX root = main.tex
\subsection{\texorpdfstring{\S{}III.8 --- The cosmic background and the quantitative cosmology of \emph{el Pleno}}{Sec.III.8 - The cosmic background and the quantitative cosmology of el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic and numerical (integration of the cosmological EOM; $\chi^2$ against BAO DESI DR2)
  \item \textbf{Open threads:} the combined analysis BAO+CMB+SNe (confrontation thread F-cosmo, high priority) is declared open and deferred with criterion; the rest of the background sector is closed internally.
  \item \textbf{Notes:} establishes the background cosmology of \emph{el Pleno} as an internally closed sector. It reads the cosmic microwave background as a manifestation of the medium, derives $\Omega_m = 1-f_c$ without fitting, and shows that the form of $H(z)$ is forced by $V_\text{eff}$ with a single observable freedom. Central discipline: to confront the raw datum (model-independent), never its model-dependent compression.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{III.8.1 --- The question: what is the cosmic microwave background under PIU-$\Psi$}{III.8.1 - The question: what is the cosmic microwave background under PIU-Psi}}

There remains the most precise cosmological observation available to contemporary physics: the cosmic microwave background (CMB), and with it the expansion history encoded in $H(z)$. Over the cosmological cascade and the effective Friedmann equations (\S{}III.6) and the reading of the dark sectors as configurations of the \emph{Pleno}-ground state (\S{}III.7), the cosmological arc closes upon confronting the program with the CMB and with the distance data, which establishes the background cosmology of \emph{el Pleno} as an internally closed sector.

\textbf{A distinction of scales, first.} Two thermal events, separated by some thirty-one orders of magnitude in density, must not be conflated. The first is the bounce and the primordial crystallization ($Z=12 \to Z=4$), which occurs near the fundamental density $\rho_P$ (Part IV). The second is the last scattering ---what standard cosmology calls ``recombination''---, which occurs at a density of order $10^{-31}\rho_P$, with energies of fractions of an electronvolt. The CMB is radiation of this second event, not of the bounce. At that density \emph{el Pleno} is in its relaxed regime ($x \approx 0$, dilatant index $n \to 1$), so that the low-energy plasma physics ---the acoustic peaks, the Saha physics--- emerges as the correct effective limit, without the program competing with it. What the program contributes is not a rewriting of that known physics, but an ontological reading of the features that the standard model leaves unexplained, and a set of new structural predictions.

\textbf{Screening principle (Rule 17, adopted for the whole section).} Before pronouncing on any number of the CMB, the program distinguishes the \emph{structural} (what would not change with the epoch or the initial conditions of the cycle) from the \emph{state variable} (what would change). The former is a candidate for \texttt{\epi{DERIVED}} or \texttt{\epi{SLD}}; the latter is \texttt{\epi{CALIBRATED}}, and its legitimate target is its structure, not its pointwise value. To pretend to derive to all its figures a number that is a state variable ---the temperature $T_0$, the exact spectral index $n_s$--- would be an arithmetic false friend. This discipline, applied systematically, is what separates the solid results of the sector from the numerical coincidences.

\subsubsection{III.8.2 --- The perfect blackbody as evidence of the medium ontology}

The cleanest feature of the CMB under PIU-$\Psi$ is obtained in one line. The photon, in the program, is not a conserved particle but a \textbf{mode of the medium}: a propagating phase circulation (\S{}III.4). The medium creates and destroys quanta freely, so that there is no photon number to conserve. The occupation distribution of a gas of bosons in equilibrium is that of Bose-Einstein,
\[
  n(\nu) = \frac{1}{e^{(h\nu - \mu)/k_B T} - 1},
\]
where the chemical potential $\mu$ is the Lagrange multiplier associated with the conservation of the number of particles. When the number is not conserved ---as occurs for excitations of a medium, which the substrate can create and absorb without conserved-charge cost--- the equilibrium is reached by extremizing the free energy without the constraint of fixed number, which 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}
that is, the \emph{exact} Planck distribution, with spectral density
$u(\nu)\,d\nu = \dfrac{8\pi h\nu^3}{c^3}\,\dfrac{d\nu}{e^{h\nu/k_B T}-1}$. The
\textbf{perfection} of the blackbody ---the absence of an equilibrium $\mu$ distortion,
which the experiments bound to $|\mu| \lesssim 9\times 10^{-5}$ (COBE/FIRAS~\cite{fixsen1996})---
is therefore structural in the program: $\mu = 0$ is not a fit nor a condition that
must be imposed, but a consequence of the photon being an excitation without conserved
number.

The contrast with the standard picture is instructive. A theory of particles over a
stage has to \emph{work} to guarantee $\mu = 0$ (it requires that the processes that
change the number of photons ---double bremsstrahlung, Compton scattering with
creation--- be efficient in the correct epoch to relax any initial $\mu$ to zero); a
theory of modes of a medium has it from the start, because there is no $\mu$ to relax.
The almost-perfect blackbody of the observed CMB is, in this reading, evidence in favor
of the medium ontology.

\begin{tcolorbox}[derivado,title={Epistemic classification --- blackbody}]
The blackbody character and the absence of an equilibrium $\mu$ distortion are
\texttt{\epi{DERIVED}}: they follow from the photon being a mode of the medium (without
conserved number) and not a particle. The Stefan-Boltzmann law $s \propto T^3$ with
coefficient $2\pi^2/45$ is derived from the count of configurations of the phase modes of F1
(\S{}III.9.5-ter) and is \texttt{\epi{DERIVED}}. The cooling law $T(z) = T_0(1+z)$ is \texttt{\epi{DERIVED}}
(stretching of the modes by the scale factor). The value $T_0 \approx 2.725$~K
is \texttt{\epi{CALIBRATED}}: a state variable, not a structural quantity. The late
$\mu$/$y$ distortions (out-of-equilibrium processes) are not prohibited:
they are an effective limit, as in standard cosmology.
\end{tcolorbox}

\subsubsection{\texorpdfstring{III.8.2-bis --- The shared factor: redshift and time dilation as a theorem of the deformation map}{III.8.2-bis - The shared factor: redshift and time dilation as a theorem of the map}}

The redshift and the time dilation of 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 stretch $X^0=\tau(t)$ and spatial stretch $X^i=L(t)\,x^i$. The components of the emergent metric ---the Cauchy-Green tensor of the hyperelastic medium (E39a, \S{}III.1.7)--- are $g_{00}=-\dot\tau^2$ and $g_{ij}=L^2\delta_{ij}$ (with $c=1$), without inserting $a(t)$ at any step.

\paragraph*{The shared-factor theorem.} The photon is a TT mode of the medium (\epi{DERIVED}, \S{}III.4.2), and its radial propagation is null, $ds^2=0$:
\begin{equation}
\dot\tau\,dt = L\,dx \;\Longrightarrow\; dx = \frac{d\tau}{L}.
\end{equation}
The physical wavelength is $\lambda_{\text{phys}}=L\cdot\lambda_{\text{com}}$ with $\lambda_{\text{com}}$ fixed, so that
\begin{equation}
\frac{\lambda_0}{\lambda_e}=\frac{L(\tau_0)}{L(\tau_e)}.
\end{equation}
Two successive crests emitted at $\tau_e$ and $\tau_e+\delta\tau_e$ travel the same comoving coordinate to the receiver:
\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}
\;\Longrightarrow\;\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}
Both factors are identically $L(\tau_0)/L(\tau_e)$:
\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\;}
\end{equation}
The definition $1+z\equiv L(\tau_0)/L(\tau_e)$ is that of the redshift as a stretch ratio; the non-trivial part ---and what is derived--- is that the \emph{time interval} shares that same factor, by nullity of the TT mode, not by definition. \epi{DERIVED} (falsifiers: A3, pull-back of the flat causal structure; A1, the photon as a TT mode of the medium).

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

\paragraph*{The time dilation of type-Ia supernovae, closed with the data.} 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 supernovae observed at $z\sim1$--$2$ live in the phase sector (\S{}III.8.7), homogeneous to background order, so that the theorem applies equally and $b=1$ holds over the entire observable range. \epi{DERIVED}, with no free parameter.

The most precise measurement available (DES-SN5YR, White et al.\ 2024~\cite{whitedes5yr}: 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}
consistent with $b=1$ to $0.27\sigma$, and excluding 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, different depth. Any future measurement of $b\neq1$ at high significance would refute the TT-mode mechanism.

\paragraph*{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 $\rho=\rho_P S^2$):
\begin{equation}
\rho_P S^2\,L^3=\text{const}\;\Longrightarrow\; S^2 L^3=\text{const}\;\Longrightarrow\;
\boxed{\;L(S)\propto S^{-2/3}\;}
\end{equation}
\epi{DERIVED} from continuity and $\rho=\rho_P S^2$ (Rule 15 of the Language of el Pleno: amplitude $S$ versus occupation $\rho_P S^2$). With the radial modulus frozen at $S_{\min}$, the redshift accumulated over the stretch $S^*\to S_{\min}$ is bounded:
\begin{equation}
(1+z)_{\max}=\left(\frac{S^*}{S_{\min}}\right)^{2/3}
=\left(\frac{S_{\max,\text{geom}}}{S_{\min}}\right)^{1/3}
=\left(\frac{1.4756}{0.7231}\right)^{1/3}=1.2684
\;\Longrightarrow\; z_{\max}\approx0.268.
\end{equation}
\epi{DERIVED}, structural (Rule 17: it depends only on the fixed walls $\{S_{\min},S_{\max,\text{geom}}\}$). This bound forces the two-clock structure: the radial sector cannot produce the supernova redshift ($z\sim2$), so that the clock of the observable epoch is the phase sector, not the radial one (\S{}III.8.7).

This same value arises by a second independent route, the \emph{background dynamics} of the modulus (\S{}III.6.6). The background equation is a non-linear oscillator, $\ddot S=\dot S^2/S-\tfrac34 S^2 V'_\text{eff}(S)$ (\S{}III.6.6), which departs from crystallization at $S^*$ and descends to the attractor $S_{\min}$; combined with the same continuity law $L\propto S^{-2/3}$, the total background stretch over that segment ---normalizing $L=1$ at crystallization $S=S^*$--- is $L(S_{\min})/L(S^*)=(S^*/S_{\min})^{2/3}=1.2684$. That the \emph{kinematic bound} of the radial redshift and the \emph{dynamical stretch} of the background coincide is not chance: both evaluate the law $L\propto S^{-2/3}$ over the same segment $S^*\!\to\!S_{\min}$, the first as phase accumulation of light, the second as integrated stretch of the background orbit. Two readings of the same segment, the same number $1.2684$, with no free parameter.

\begin{tcolorbox}[ecuacionclave,title={Redshift and time dilation: an identity of the medium}]
The redshift and the time dilation of distant sources are not two independent effects: they are the \emph{same} stretch 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 fitting ---where $\Lambda$CDM \emph{assumes} it by postulating the expanding metric, PIU-$\Psi$ \emph{derives} it as a theorem of the elastic medium---, confirmed to $0.27\sigma$ by DES-SN5YR and excluding tired light at $\sim90\sigma$. The same law $L\propto S^{-2/3}$ that continuity imposes bounds the redshift of the frozen radial sector to $z_{\max}\approx0.268$, which 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, $(S^*/S_{\min})^{2/3}=1.2684$, comes out by two routes that converge with no free parameter: the kinematics of the redshift and the dynamics of the background oscillator, both over the segment $S^*\!\to\!S_{\min}$.
\end{tcolorbox}

\subsubsection{III.8.3 --- The primordial spectrum without inflation: the dilatant bounce}

The question is whether the dilatant bounce of the program generates a nearly
scale-invariant spectrum of primordial perturbations ---the feature that inflation
was invented to produce--- without postulating an additional inflaton field. The
approach is direct: the relevant primordial perturbations are of \emph{phase}
($\delta\theta$, the gapless Goldstone mode, which survives into the infrared), not of
\emph{modulus} ($\delta S$, which has a large mass gap and is frozen). Working in
conformal time $\eta$, defined by $d\eta = dt/a(t)$, and writing the curvature
perturbation as $v_k = z\,\mathcal{R}_k$ with the Mukhanov-Sasaki variable~\cite{mukhanovsasaki}
$z(\eta) = a(\eta)\,S(\eta)$, the mode equation is
\[
  v_k'' + \left(c^2 k^2 - \frac{z''}{z}\right) v_k = 0,
\]
where the prime denotes $d/d\eta$ and all the physics of \emph{el Pleno} enters through $z$. The
term $z''/z$ acts as a time-dependent potential; its form fixes the
spectral index. Suppose a power-law behavior $z \propto
(-\eta)^s$ (valid in the slow-contraction regime, with $\eta < 0$ growing toward
$0^-$). Then
\[
  \frac{z''}{z} = \frac{s(s-1)}{\eta^2},
\]
and the mode equation reduces to that of Bessel. Its solutions in terms of
Hankel functions are
\[
  v_k(\eta) = \sqrt{-\eta}\,\Big[ A\, H_\nu^{(1)}(-c k\eta) + B\, H_\nu^{(2)}(-c k\eta) \Big],
  \qquad
  \nu = \left| s - \tfrac{1}{2} \right| .
\]
The primordial power spectrum is evaluated in the large-scale limit
($-c k\eta \to 0$, super-horizon modes), where $H_\nu^{(1)}(x) \sim x^{-\nu}$ and
therefore
\[
  \mathcal{P}_\mathcal{R}(k) = \frac{k^3}{2\pi^2}\left| \frac{v_k}{z} \right|^2
  \;\propto\; k^{3 - 2\nu},
\]
whence the scalar spectral index is
\[
  \boxed{\,n_s - 1 \equiv \frac{d\ln \mathcal{P}_\mathcal{R}}{d\ln k} = 3 - 2\nu = 3 - 2\left| s - \tfrac{1}{2}\right| .}
\]
This is the master formula: the spectral index is determined by the exponent $s$,
which in turn is fixed by the kinematics of the background. It remains to compute $s$ from the
program, without freedom of adjustment.

\textbf{The exponent $s$ from the dynamics of \emph{el Pleno}.} Two results of the corpus fix it. First, during the contraction the depleted population of the condensate
dilutes like conserved countable objects, $S^2 \propto a^{-3}$ (Part IV and
\S{}III.8.5), whence
\[
  S \propto a^{-3/2}
  \quad\Longrightarrow\quad
  z = a\,S \propto a\cdot a^{-3/2} = a^{-1/2}.
\]
Second, the relation $S^2 \propto a^{-3}$ is precisely the continuity equation
$\dot{\rho} + 3H(1+w)\rho = 0$ with $w = 0$: the medium behaves like pressureless
matter, not by postulate but because the dilution of conserved topological charges
\emph{is} $w = 0$. With $w = 0$, the Friedmann equation in conformal time gives
$a \propto (-\eta)^{2}$. Combining,
\[
  z \propto a^{-1/2} \propto (-\eta)^{-1}
  \quad\Longrightarrow\quad
  s = -1,
  \qquad
  \nu = \left| -1 - \tfrac{1}{2}\right| = \tfrac{3}{2}.
\]
Substituting into the master formula of the spectral index,
\begin{equation}
  n_s - 1 = 3 - 2\nu = 3 - 2\left|s - \tfrac{1}{2}\right|,
  \tag{E-cosmo-1}
\end{equation}
one obtains
\[
  n_s - 1 = 3 - 2\cdot\tfrac{3}{2} = 0
  \quad\Longrightarrow\quad
  \boxed{\,n_s = 1\ \text{(at zeroth order).}}
\]
The spectrum is exactly scale-invariant at zeroth order, without inflation and without a
single fitted parameter: the result follows from $w = 0$, which in turn follows from
the topology of the matter of the program.

This result situates the program, naturally, in the class of ``matter bounce''
models (Wands 1999~\cite{wands1999}; Finelli-Brandenberger 2002~\cite{finellibrandenberger}), where the condition
$n_s - 1 = 3 - 2|\nu|$ with $\nu = 3/2$ corresponds exactly to a contraction
dominated by matter ---the same value $\nu = 3/2$ that was obtained here from $s = -1$. The
difference with those models is that the program explains \emph{why} $w = 0$ (it is the
dilution of countable topological objects, \S{}III.8.5), instead of adding it as an
ingredient.

\begin{tcolorbox}[derivado,title={Epistemic classification --- primordial spectrum}]
That \emph{el Pleno} falls naturally in the ``matter bounce'' class, explaining $w=0$
instead of postulating it, is \texttt{\epi{SLD}}. The result
$n_s = 1$ at zeroth order is \texttt{\textbf{[DERIVED by algebra]}} from $S^2 \propto 1/a^3$.
The perfect blackbody (\S{}III.8.2) and this nearly scale-invariant spectrum
without inflation are the two most solid fronts of the sector.
\end{tcolorbox}

\subsubsection{III.8.4 --- The red tilt: mechanism of the packing transition and discipline of the positional value}

The spectrum at zeroth order gives exact $n_s=1$ (\S{}III.8.3, \epi{DERIVED}), while the observation gives $n_s\approx 0.965$, a slight red tilt, $n_s-1\approx -0.035$. The origin of the tilt is not an individual component of the background, but the transition between regimes during the crunch-bounce.

\textbf{The degeneracy of the pure components.} For a component of equation of state $w$, the conformal Friedmann equation gives $a\propto(-\eta)^p$ with $p=2/(1+3w)$, and the Mukhanov-Sasaki variable $z=aS$ inherits the exponent
\begin{equation}
s_z(w) = p\cdot\frac{-1-3w}{2} = \frac{-1-3w}{1+3w}.
\end{equation}
Evaluating the two components that dominate the crunch:
\begin{equation}
s_z(0) = -1\ (\text{depleted matter}),
\quad
s_z(-2) = \frac{-1+6}{1-6} = -1\ (\text{Klauder}, w_q=-2).
\end{equation}
Both pure components give the same exponent $s_z=-1$, hence both give $\nu=|s_z-\tfrac12|=3/2$ and, by the master formula $n_s-1=3-2\nu$ (\S{}III.8.3), $n_s=1$. The tilt cannot originate in which component dominates: both depleted matter ($w=0$) and the Klauder IR wall ($w_q=-2$, since $V_q\propto 1/S^2\propto a^3$) produce, separately, a scale-invariant spectrum. The observed 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 maximum density
\begin{equation}
\rho_{\max} = S_{\max,\text{geom}}^2\,\rho_P = \frac{8\sqrt6}{9}\,\rho_P \approx 2.1773\,\rho_P,
\end{equation}
which the corpus derives (\S{}II.4.1) as the ratio between the maximum FCC packing (Kepler-Hales) and the density of the diamond lattice:
\begin{equation}
S_{\max,\text{geom}}^2 = \frac{n_\text{FCC}^{\max}}{n_\text{diamond}} = \frac{\sqrt2/\ell_P^3}{3\sqrt3/(8\ell_P^3)} = \frac{8\sqrt6}{9}.
\end{equation}
The two structures that appear in this identity ---the compact phase (maximum FCC packing, coordination $Z=12$) and the limit diamond phase (minimum rigid coordination $Z=4$)--- are the two phases between which el Pleno transits in the bounce. The crunch compresses the medium toward compact packing; the A6 saturation stops it at $\rho_{\max}$; the bounce relaxes it to the diamond lattice. The imprint of the primordial spectrum occurs during that packing transition $Z=12\to Z=4$ (the same crystallization of \S{}IV.5), over a persistent background that survives the transition and is identified with dark energy, the condensate fraction
\begin{equation}
f_c = \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} = \frac{1.4956}{2.1773} = 0.6869 = \Omega_{EO},
\qquad w_{EO} = -1.
\end{equation}

\textbf{Red sign.} The Klauder component that enters the transition has $w_q=-2<-\tfrac13$: it accelerates the background (acceleration condition $w<-1/3$). An accelerated phase in the imprint of the modes adds power on the large scales relative to the small ones, which shifts the spectrum toward the red. The persistent background $f_c$ with $w_{EO}=-1$ reinforces the same sign. Therefore $n_s-1<0$ structurally, not by fitting. \epi{SLD}.

\textbf{Smallness.} 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 depleted matter ($n_s=1$), and Klauder enters as a subdominant correction. The deviation from scale invariance is, by construction, a first-order correction in $x\ll 1$. Therefore $|n_s-1|\ll 1$. \epi{SLD}. Quantitatively, the observed band $n_s=0.965\pm 0.004$ corresponds to $n_s-1\approx -0.035$, which in a quasi-matter background is reached with $|w_\text{eff}|\approx 0.006$ ---a deviation of a few thousandths with respect to $w=0$, the order that the Klauder contribution introduces in the epoch when the CMB modes cross the horizon.

\textbf{The exact value is positional.} The numerical value $n_s=0.965$ depends on the Klauder fraction $x$ at the precise moment when each observable mode crosses the horizon, that is, on where over the bounce trajectory $\bar S(t)$ the CMB modes were imprinted. Two facts fix its status:
\begin{enumerate}[leftmargin=1.6em]
  \item \textbf{There is no exact scaling solution} that cancels this dependence. For the Klauder potential $V_q\propto S^{-2}$, the power-law ansatz $S\propto t^\beta$, $a\propto t^\gamma$ imposes $\beta=1/2$ by balance of the equation of motion, but then the Friedmann equation is inconsistent in the exponents of $t$ ($H^2\sim t^{-2}$ against $\rho\sim t^{-1}$). There is no fixed point that makes the tilt independent of the temporal scale of the transition.
  \item \textbf{The position over the trajectory is non-autonomous.} The decomposition of $\bar S(t)$ into a derivable part and a positional part (\S{}III.7.2) establishes that $S_{\max,\text{eff}}^2(t_0)$ is subsumed in the trajectory and is the only positional quantity of the program, on equal footing with $f_c$ and $H_0$. The present value of the tilt inherits that condition.
\end{enumerate}
To force a closed number for $n_s$ would require fixing the temporal scale of the crossing, which the six axioms do not determine; to do so would be to fabricate mechanism or post-hoc numerology. By Rule 17, $n_s=0.9649$ is a state variable, \epi{CALIBRATED}, not \epi{DERIVED}; the datum itself is a band ($\pm 0.0042$), not a point.

\textbf{The falsifiable structure: red$\to$blue transition.} The legitimate quantitative target is not the value of $n_s$ but its structure: as the crunch advances, the Gent wall ($w>0$) takes control and the small modes see $w>0$, so that the red tilt at large scale transitions to blue at small scale. This transition depends on the profile of the walls, not on a pointwise initial condition; it is falsifiable (via Lyman-$\alpha$ and the high-resolution CMB, CMB-S4) and distinctive against inflation, which predicts a nearly constant red tilt. The existence of the transition is \epi{SLD}; the scale of the crossing is \epi{CALIBRATED}.

In synthesis, the spectral index separates into three differentiated statuses: $n_s=1$ at zeroth order is \epi{DERIVED} (Mukhanov-Sasaki, $s_z=-1$); the red sign and the smallness of the tilt are \epi{SLD}, with a named mechanism ---the packing transition FCC$\to$diamond over the persistent background $\Omega_{EO}=f_c$ with $w=-1$ adds power at large scale as a subdominant correction---; and the exact value $0.965$ is \epi{CALIBRATED}, a positional state quantity over the bounce trajectory. In the classification of the structural principle (\S{}III.8.7), the tilt $n_s-1$ is positional of \emph{type B} (differential): it factorizes as a derivable geometric profile of the transition times the crossing rate $(dw_\text{eff}/dN)$, so that it shares nature with the equation of state $w(z)$ ---both depend on the \emph{rate} $\dot{\bar S}$---, while $f_c$ and $H_0$ are of type A (they depend on the \emph{value} $\bar S$). The four are state quantities of the same variable $\bar S(t)$, read in distinct projections.

\subsubsection{III.8.5 --- The matter density as a derived fraction of the condensate}

The strongest quantitative result of the sector arises upon normalizing the matter
density. In standard cosmology, $\Omega_m$ is a free parameter fitted to
the data. In the program, it is not. Ordinary matter and dark matter are the
\emph{depleted} excitations of the condensate, while dark energy is the
\emph{condensed} fraction. The conservation of the total population of the medium imposes
\[
  f_c + \Omega_\text{depleted} = 1,
  \qquad
  \Omega_\text{depleted} = \Omega_M + \Omega_{MO},
\]
where $f_c = \Omega_{EO}$ is the condensed fraction, $\Omega_M$ the baryonic matter and
$\Omega_{MO}$ the dark matter. Since the condensate fraction is fixed by the
geometry of the medium,
\[
  f_c = \frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2} = \frac{1.4956}{2.1773} = 0.6869
\]
(\S{}III.7.2, with denominator $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$ derived from the
diamond geometry by the Kepler-Hales theorem), the total matter fraction is
\emph{derived}:
\begin{equation}
  \boxed{\,\Omega_m = \Omega_M + \Omega_{MO} = 1 - f_c = 1 - 0.6869 = 0.3131 .}
  \tag{E-cosmo-2}
\end{equation}

\textbf{Quantitative confrontation.} The Planck 2018 measurement is $\Omega_m = 0.315
\pm 0.007$. The deviation of the program's value, in units of the error bar, is
\[
  \frac{|\,0.3131 - 0.315\,|}{0.007} = \frac{0.0019}{0.007} = 0.27\sigma \approx 0.3\sigma ,
\]
that is, within a third of the experimental uncertainty, \emph{without fitting}. The
matter density, which the standard model treats as a free knob, comes out in the
program from the condensate fraction.

\textbf{Internal consistency.} The decomposition of $\Omega_m$ into its two components
is not free: the program also fixes their ratio by the geometry, $\Omega_{MO}/\Omega_M =
\sqrt{3}\pi \approx 5.4414$ (P16, \S{}III.6.3). Solving the system
\[
  \Omega_M + \Omega_{MO} = 0.3131,
  \qquad
  \frac{\Omega_{MO}}{\Omega_M} = \sqrt{3}\pi ,
\]
one obtains
\[
  \Omega_M = \frac{0.3131}{1 + \sqrt{3}\pi} = \frac{0.3131}{6.4414} = 0.0486,
  \qquad
  \Omega_{MO} = 0.3131 - 0.0486 = 0.2645 .
\]
These values are compared with Planck ($\Omega_b = 0.0493 \pm 0.0006$,
$\Omega_c = 0.265 \pm 0.007$): the program's baryonic matter falls at
$|0.0486-0.0493|/0.0006 \approx 1.2\sigma$ and the dark matter at
$|0.2645-0.265|/0.007 \approx 0.07\sigma$. The three quantities ---total, baryonic and
dark--- are consistent with the observation simultaneously, from \emph{two} derived
numbers ($f_c$ and $\sqrt{3}\pi$) and zero fitting parameters.

The ontological a\-priori that sustains the dilution $\rho_m \propto a^{-3}$ deserves
to be made explicit, because it is what avoids the contamination of treating matter as an
external fluid. Matter is localized excitations (solitons $B=1$, Q-balls;
\S{}III.5); their number is a conserved topological charge (associated with the symmetry of the
program); and the dilution of conserved countable objects in a growing volume is
$a^{-3}$ by pure geometry. Formally, if $N$ topological objects of conserved charge
occupy a physical volume $V \propto a^3$, the number density is $n = N/V
\propto a^{-3}$, and since each object carries a fixed rest energy $m_\text{sol}c^2$,
the energy density is $\rho_m = n\, m_\text{sol}c^2 \propto a^{-3}$. The exponent
$a^{-3}$ \emph{emerges} from the topological conservation plus the geometry, it is not postulated
as the equation of state of a fluid.

\begin{tcolorbox}[derivado,title={Epistemic classification --- matter density}]
The dilution $\rho_m \propto a^{-3}$ as a topological property is \texttt{[strong logical
deduction]}. The identity $\Omega_m = 1 - f_c \approx 0.3131$, at $0.3\sigma$ of
Planck without a fitting parameter, is \texttt{\epi{DERIVED}} with the same status as
$f_c$ (derived structure, with the numerator of $f_c$ calibrated, \S{}III.7.2). The
matter density, which the standard picture treats as a free parameter, is in the
program a quantity derived from the condensate fraction.
\end{tcolorbox}

\subsubsection{III.8.6 --- The expansion history without theory fitting parameters}

The background sector is confronted with the most restrictive observation of late
cosmology: the expansion history $H(z)$, measured by the baryon acoustic
oscillations (BAO). The program integrates the equation of motion of the field of \emph{el Pleno}
in a Friedmann-Lemaître-Robertson-Walker (FLRW) background,
\[
  \ddot{S} + 3H\dot{S} + \frac{1}{\rho_P \ell_P^2}\, V_\text{eff}'(S) = 0,
  \qquad
  H^2 = \frac{8\pi G}{3}\,\sum_i \rho_i,
\]
with the effective potential $V_\text{eff}$ of \emph{fixed} parameters (derived from the
corpus: $E_0$, $S_{\max}^2$, verified so that $V_\text{eff}'(S_{\min}) = 0$).
The only freedom of the system is the initial condition $\delta_0$ ---the displacement
of the field with respect to $S_{\min}$ today---, which is a physical state variable,
equivalent to the observable $w_0$, not a theory parameter.

\textbf{The form of $H(z)$ is scale-invariant (theory result, closed).}
The densities that enter the Friedmann equation are that of matter and that of the field
of \emph{el Pleno},
\[
  \rho_m(a) = \rho_{m,0}\,a^{-3},
  \qquad
  \rho_S = \tfrac{1}{2}\rho_P \ell_P^2\,\dot{S}^2 + \rho_P c^2\, v(S),
\]
where the effective potential has been written in dimensionless form $V_\text{eff}(S) =
\rho_P c^2\, v(S)$, with $v(S)$ a function of pure numbers (the bounds $S_{\max}^2$ and
the scale $E_0/\rho_P c^2$ are dimensionless). The pressure of the field is
$p_S = \tfrac{1}{2}\rho_P \ell_P^2\,\dot{S}^2 - \rho_P c^2\, v(S)$, whence the equation
of state $w(z) = p_S/\rho_S$ and the initial displacement
$\delta_0 = \tfrac{1}{2}\dot{S}^2/V|_{0}$ are dimensionless quantities.

Using the structural identity $G = c^2/(\rho_P \ell_P^2)$, the Friedmann prefactor
associated with the field is
\[
  \frac{8\pi G}{3}\,\rho_P c^2 = \frac{8\pi}{3}\,\frac{c^2}{\rho_P \ell_P^2}\,\rho_P c^2
  = \frac{8\pi}{3}\,\frac{c^4}{\ell_P^2} \equiv \mathcal{C},
\]
a constant with dimensions of $H^2$. The Friedmann equation is then written
\[
  H^2(a) = \mathcal{C}\left[\, \Omega_m\, a^{-3} + \frac{\ell_P^2\,\dot{S}^2}{2c^2} + v(S) \,\right],
\]
where all the brackets are dimensionless. Upon forming the observable
$E(z) \equiv H(z)/H_0$ and normalizing so that $E(0) = 1$, the constant
$\mathcal{C}$ ---and with it all the dimensional scales
($\rho_P$, $c$, $\ell_P$ and $H_0$)--- cancels
identically between numerator and denominator:
\begin{equation}
  E^2(z) = \frac{H^2(z)}{H^2(0)}
  = \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}
No dimensional scale survives in $E(z)$: the form of the expansion
history is determined solely by $v(S)$ (fixed, derived), by $\Omega_m =
1 - f_c$ (derived, \S{}III.8.5) and by the initial condition $\delta_0$ (the only
freedom, observable, equivalent to $w_0$). It is an analytical demonstration, not a
numerical observation: the form of $H(z)$ is forced by the derived potential, with
a single observable freedom. The program has here \emph{less} freedom than
standard cosmology, which requires two phenomenological parameters ($w_0$, $w_a$).

\textbf{The form of $w(z)$ and the non-monotonic minimum.} Integrating numerically the
system with rigid $V_\text{eff}$, the equation of state turns out non-monotonic, with a
minimum at $z \approx 0.5$--$0.7$. For a small initial condition $\delta_0 =
0.02$, for example, $w_0 = -0.961$ and the deviation of $E(z)$ with respect to the standard
model is $+2.45\%$ at $z = 2$; for $\delta_0 \to 0$ one recovers $w \to -1$ and
$E(z) \to E_{\Lambda\text{CDM}}(z)$. The canonical kinetics of the field guarantees
$w \geq -1$ at every point (there is no physical phantom crossing): $w + 1 = \dot{S}^2/(\tfrac12
\dot S^2 + v) \geq 0$. Measuring $w_0$ fixes $\delta_0$ and, with it, the complete curve $E(z)$.

\textbf{Confrontation with BAO DESI DR2.} On the model-independent radial observable
$D_H/r_d = c/(r_d H(z))$, at five points
($z = 0.51, 0.71, 0.93, 1.48, 2.33$), the fit of the program is statistically
indistinguishable from the standard one:

{\small\setlength{\tabcolsep}{6pt}
\begin{longtable}{L{6.0cm}C{2.6cm}C{2.6cm}}
\toprule
\textbf{Model} & $\chi^2$ & $\chi^2/\text{dof}$ \\
\midrule\endhead
$\Lambda$CDM ($\Omega_m = 0.31$) & 6.53 & 1.63 \\
PIU ($\delta_0 \to 0$, $w_0 \approx -1$) & 6.54 & 1.63 \\
PIU ($\delta_0 = 0.01$, $w_0 = -0.98$) & 7.85 & 1.96 \\
PIU ($\delta_0 = 0.02$, $w_0 = -0.96$) & 11.24 & 2.81 \\
\bottomrule
\end{longtable}}

The program fits BAO DESI DR2 as well as the standard model (6.54 against 6.53),
with the form of $H(z)$ forced by the potential and a single initial condition. The
$\chi^2/\text{dof} = 1.63$ (not unity) reflects the known BAO-CMB tension
(\textasciitilde$2.3\sigma$ in DESI), which the program inherits just like the standard one.
\textbf{The program is not falsified by the most precise cosmological datum available.}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fondo_cosmico.png}
\caption{The quantitative cosmic background of \emph{el Pleno} against the data. (A) The Hubble
distance $D_H/r_d = c/(r_d H(z))$ of the program (PIU, $\delta_0\to0$) and of
$\Lambda$CDM, compared with the five model-independent BAO measurements of DESI DR2
($z = 0.51, 0.71, 0.93, 1.32, 2.33$); the two curves are indistinguishable and fit the
points with $\chi^2 = 6.54$ against $6.53$. (B) The normalized expansion history
$E(z) = H(z)/H_0$: its form is scale-invariant (no dimensional scale
survives after normalizing $E(0)=1$), forced by the derived effective potential, with the
initial condition $\delta_0$ as the only observable freedom; as $\delta_0$ increases the
curve separates slightly from $\Lambda$CDM at high $z$. (C) Normalized residuals
(datum$-$model)$/\sigma$: PIU and $\Lambda$CDM leave the same pattern, within the band
of $1\sigma$ except for the two low-$z$ points that reflect the BAO-CMB tension that both
models inherit. (D) The equation of state $w(z)$: it starts from $w_0>-1$ today, presents a
shallow minimum toward $z\approx0.5$ and converges to $-1$, never crossing the
phantom boundary $w=-1$ ---a structural guarantee of the canonical kinetics of the field. All the
theory parameters are derived; the only freedom is the observable initial condition
$\delta_0\leftrightarrow w_0$.}
\label{fig:fondocosmico}
\end{figure}

\begin{tcolorbox}[derivado,title={Epistemic classification --- expansion history}]
The scale invariance of the form of $H(z)$ is \texttt{\epi{DERIVED}} (analytical
demonstration by nondimensionalization), permanent and independent of future versions
of the data. That the program fits BAO DESI DR2 with $\chi^2 = 6.54$, comparable to the
standard one and with less freedom, is \texttt{[numerically verified]}. The relation
$\delta_0 \leftrightarrow w_0$ is \texttt{\epi{SLD}}. The value of
$w_0$ (equivalently $\delta_0$) is \texttt{\epi{CALIBRATED}}: a state variable.
\end{tcolorbox}

\subsubsection{\texorpdfstring{III.8.7 --- $H_0$ and dark energy as positional readings of the clock of el Pleno}{III.8.7 - H0 and dark energy as positional readings of the clock of el Pleno}}

Two readings of the background deserve to be made precise, because they touch the ontology of the program: the origin of the redshift and the growing character of dark energy. From them follows a structural principle ---every cosmological magnitude factorizes into a fixed geometric ratio, a unit anchor and a state factor--- that reorganizes the boundary between what the program derives and what it reads from the present position of the medium, and which, applied to the Hubble constant, reformulates the cosmological-constant problem and predicts the sign of the Hubble tension.

\paragraph*{The redshift as a stretch ratio of the medium.} Space emerges from el Pleno (a postulated independent metric would be an alarm signal of the ontological protocol), so that there is no stage that expands. The homogeneous background is the stretch $L(\tau)$ of the deformation map of the medium, and the redshift is the ratio $1+z\equiv L(\tau_0)/L(\tau_e)$. The photon, a TT mode of the medium, propagates over nulls of the emergent metric ($ds^2=0$), and from that nullity it follows that the wavelength \emph{and} the time interval between crests share identically the factor $L(\tau_0)/L(\tau_e)$ (shared-factor theorem, \S{}III.8.2-bis). By that same theorem, pure ``tired light'' ---redshift without time dilation--- is excluded by native mechanism: a TT mode cannot stretch its wavelength without stretching by the same factor the interval between its crests, because both are the same ratio. The observational exclusion (at about $18\sigma$ by the dilation of the supernova light curves up to $z\sim1$, closed quantitatively in \S{}III.8.2-bis) confirms what the mechanism already forbids.

\paragraph*{Dark energy as el Pleno-ground state at $S_{\min}$.} Dark energy is the phase condensate in its minimum-excitation state, $S=S_{\min}$, with physical density $\rho_{\text{EO}}=V(S_{\min})=K\,S_{\min}^2$ ---anchor $K=\rho_P c^2$ times wall $S_{\min}^2$, both eternal--- structurally constant. Its equation of state is $w=-1$ exact (symmetry of the ground state, $T_{\mu\nu}\propto g_{\mu\nu}$, \S{}III.7.5). What grows toward the future is the \emph{relative weight} $\Omega_{EO}=f_c$, not the physical density: matter (solitons) dilutes as $L^{-3}$ while the ground remains constant, so that the condensate fraction ascends toward its hard bound $f_c<1$ fixed by the diamond geometry. Three consequences follow. First, the bound: $\Omega_{EO}$ is bounded above, dark energy cannot dominate without limit. Second, the reformulation of the cosmological-constant problem: the present value of $\Omega_{EO}$ is the positional reading of at what point of the relaxation the universe was left ---a cosmic clock--- which translates the $\sim$120-order discrepancy into a tractable question. Third, the dynamical-dark-energy signal that DESI infers when fitting with the $w_0$--$w_a$ parametrization: it is the reading of a medium with $w=-1$ exact ---whose $H(z)$ coincides with $\Lambda$CDM to linear order and differs in the non-linear regime of the walls--- forced into that parametric mould, not a physical deviation from $w=-1$. The falsifiable prediction is that the signal dissolves in model-independent observables (raw $D_H/r_d$).

The condensate fraction and the Hubble constant are both positional, but they are not the same reading of the clock: the first measures the internal \emph{amplitude} ($f_c=S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$, of order one), the second the external \emph{scale} ($\Theta=\rho_{\text{tot}}/\rho_P\sim 10^{-123}$). The structural principle that follows separates with precision what el Pleno fixes of each.

\paragraph*{The structural principle: geometry, anchor and state.} Every background magnitude of el Pleno admits, and is written in, the form
\begin{equation}
\text{magnitude} = \underbrace{(\text{geometric ratio})}_{\text{fixed, of the lattice}}
\times\underbrace{(\text{anchor})}_{\text{units}}\times
\underbrace{(\text{state factor})}_{\text{position of }\bar S(t)},
\end{equation}
where the geometric ratio is an invariant of the diamond lattice (derivable from SO(3) reduced by the tetrahedral group $2T$), the anchor carries the units, and the state factor encodes where the medium is over the trajectory of its single dynamical background variable, the average modulus $\bar S(t)$. The epistemic status of the magnitude follows from which piece dominates its value: a magnitude is geometric (\epi{DERIVED}) if it depends only on the fixed endpoints $\{S_{\min},S_{\max,\text{geom}}\}$ of the modulus path; it is positional (\epi{CALIBRATED} in its value) if it depends on the instantaneous value $\bar S(t)$ or on its rate $\dot{\bar S}$. The discipline of units follows the Language of el Pleno: the only two dimensional anchors are the density $\rho_P$ and the scale $\ell_P$; the velocity $c=\sqrt{K/\rho_P}$ is reabsorbable causal structure, not a third anchor.

\begin{tcolorbox}[notabox,title={The dividing line, explicit: what is ``calibrated'' in the program}]
The principle induces an exhaustive and unambiguous classification of the background sector, which makes precise what the label \epi{CALIBRATED} means here ---not a parameter fitted to observation, but the reading of the present state of a quantity whose \emph{form} is derivable:
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Geometric} (\epi{DERIVED}): depends only on the fixed endpoints $\{S_{\min},S_{\max,\text{geom}}\}$. Examples: the partition $\Omega_m=1-f_c$ ($0.27\sigma$), the dilatant index at rest $n=1.316$, the rest attractors $w=-1$ and $n_s=1$.
\item \textbf{Positional of type A} (instantaneous, \epi{CALIBRATED} in its value): depends on the instantaneous value $\bar S(t)$. Examples: the Hubble constant $H_0$ and the condensate fraction $f_c$.
\item \textbf{Positional of type B} (differential, \epi{CALIBRATED} in its value): depends on the rate $\dot{\bar S}$ or on its variation. Example: the spectral tilt $n_s-1$, fixed by the deformation rate in the spectrum-generation phase. (The background equation of state is \emph{not} of this type: in the ground state $\dot\theta_0=0$ gives $w=-1$ exact, geometric, not differential.)
\end{itemize}
The geometric/positional distinction is not a list of cases but a consequence of the principle: every calibrated background quantity factorizes into a fixed geometric part (derivable, the \emph{form}) times a state factor that reads the position of el Pleno over its trajectory $\bar S(t)$ (the \emph{value}). That is why ``calibrated'' here is a state quantity ---of the same type as $H_0$ in any cosmology---, not a phenomenological fit: the curve is theory, the point on the curve is observational datum.
\end{tcolorbox}

\paragraph*{The factorization of the Hubble constant.} The expansion history of \S{}III.8.6 was expressed in the scale-invariant form $E(z)\equiv H(z)/H_0$, where the prefactor cancels: the theoretical content resides in the \emph{form} of the curve $H(z)$, not in the absolute value $H_0$. There remains the question of what kind of object $H_0$ is. The kinetic constraint of the elastic solid (\S{}III.6.6) ties the volumetric deformation rate with the energy density, $H_0^2\propto (G/c^2)\,\rho_{\text{tot},0}$ with the geometric coefficient of the source-stretch coupling (\S{}III.1.4). Writing it as $H_0^2=\tfrac{8\pi G}{3}\rho_{\text{tot},0}$ ---the same structure as the Friedmann constraint, reproduced as a theorem of the solid---, the density is made dimensionless by its anchor, $\rho_{\text{tot},0}\equiv\Theta(t_0)\,\rho_P$, and $G$ is eliminated with the native identity of el Pleno $G=c^2/(\rho_P\ell_P^2)$ (E3, \S{}III.1, derived with agreement $<0.1\%$). The anchor-density $\rho_P$ cancels exactly:
\begin{equation}
H_0^2=\frac{8\pi G}{3}\,\rho_{\text{tot},0}
=\frac{8\pi}{3}\,\frac{c^2}{\rho_P\ell_P^2}\,\Theta\,\rho_P
=\frac{8\pi}{3}\,\Theta\,\frac{c^2}{\ell_P^2}
=\frac{8\pi}{3}\,\Theta\left(\frac{c}{\ell_P}\right)^{\!2}.
\end{equation}
Taking the root and using the definitional identity $\tau_P=\ell_P/c$ (so that $c/\ell_P=1/\tau_P$), the factorization is obtained.

\begin{tcolorbox}[ecuacionclave,title={El Pleno factorization for $H_0$}]
\begin{equation*}
H_0
=\underbrace{\sqrt{\tfrac{8\pi}{3}}}_{\substack{\text{geom. ratio}\\ \epi{DERIVED}}}
\;\times\;
\underbrace{\frac{c}{\ell_P}}_{\substack{\text{anchor}\\ =\,1/\tau_P}}
\;\times\;
\underbrace{\sqrt{\Theta(t)}}_{\substack{\text{state}\\ \epi{CALIBRATED}}}
\end{equation*}
with $\sqrt{8\pi/3}=2.8944$ and $c/\ell_P=1.8552\times10^{43}\,\mathrm{s^{-1}}$. The form of the equality is \epi{DERIVED}; the units live entirely in the anchor $c/\ell_P$, which coincides with the change ceiling of the medium $\dot\gamma_{\max}=c/\ell_P=1/\tau_P$ (E45, \S{}III.9.2), the frequency of the Planck tick. The only positional factor is $\sqrt{\Theta(t)}$. In units of el Pleno ($c=\ell_P=1$) the anchor dissolves to unity: the purity test of the Language of el Pleno confirms that the content is the geometric ratio times the state factor. The numerical verification with $H_0=67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ gives $\Theta_0=1.655\times10^{-123}$ and reconstructs $H_0$ exactly; all the observational freedom is confined to $\Theta_0$, which is a state quantity, not a constant.
\end{tcolorbox}

\noindent That the substitution $G\to c^2/(\rho_P\ell_P^2)$ does not contaminate but decontaminates: $G$ is not postulated, it is a structural identity of the medium (E3), so that replacing it by the anchors of el Pleno eliminates the last alien dimensional scale and leaves $H_0$ expressed only in anchors of the medium and pure ratios. The factorization is genuine precisely because it does not solve $H_0$ from a density that already contains it: the density is made dimensionless by the anchor $\rho_P$, not by the critical density $\rho_{c,0}=3H_0^2/8\pi G$ ---which would reintroduce $H_0$ on both sides. The positional factor $\Theta_0$ is a state quantity read from the present universe, not a derived value; what the factorization contributes is the class of the object, not its number.

\noindent\textbf{What the factorization asserts and does not assert.} The distinction is precise. The factor $\sqrt{8\pi/3}$ is the geometric coefficient of the source-stretch coupling of the elastic medium ---the one the Sakharov induced action (\S{}III.1.4) fixes, with the same structure as the density-curvature relation of FLRW in $3{+}1$ dimensions---; what the program contributes is not that factor, but the \emph{identification of the anchor}: that the dimensional scale of $H_0$ is exactly $c/\ell_P=1/\tau_P$, the frequency of the Planck tick, once $G$ is eliminated by its native identity. The non-trivial assertion is threefold and is sustained by the preceding algebra: (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 $\ell_P$ (with $c$ as reabsorbable causal structure), which passes the purity test of the Language of el Pleno; and (iii) the magnitude $\sim10^{-61}$ of the pure number of $H_0$ in units of el Pleno ---alien to the $O(1)$ ratios that the program derives--- is the diagnostic that $H_0$ is positional, not geometric. The factorization is exact algebra on the elastic kinetic constraint plus the derived identity $G=c^2/(\rho_P\ell_P^2)$; its value lies in correctly classifying what kind of object $H_0$ is, not in predicting its number.

\paragraph*{Today's density is the age in ticks, squared.} Having defined 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 $\tau_P=\ell_P/c=5.391\times10^{-44}$ s), the factorization gives an exact identity. 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 squaring:
\begin{equation}
\Theta(t)=\frac{3}{8\pi\,N_{\text{tic}}^2(t)}
\qquad\Longleftrightarrow\qquad
N_{\text{tic}}=\sqrt{\frac{3}{8\pi\,\Theta}}.
\end{equation}

\noindent 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 of the universe, $N=t_0/\tau_P=(13.787\,\text{Gyr})/\tau_P=8.07\times10^{60}$. The agreement to $\sim5\%$ between the count by frequency and the count by duration ---without fitting--- confirms that $N_{\text{tic}}$ is genuinely the age of the universe measured in Planck ticks, and that the small difference is the $O(1)$ factor between $1/H_0$ and the true age (the same $\approx0.95$ of the magnitude thread).

\begin{tcolorbox}[derivado,title={Reformulation of the cosmological-constant problem}]
The number $\Theta_0\approx 10^{-123}$ \emph{is} the raw formulation of the cosmological-constant problem: today's vacuum energy density is $\sim10^{-123}$ times the Planck one, the worst fine-tuning of physics in the standard model. The preceding identity 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. One need not explain the exponent $123$; one must explain how many ticks the clock has run ($N_{\text{tic}}\approx 8.5\times10^{60}$), and that is an age, not a fine-tuned parameter. This extends to the absolute \emph{scale} what \S{}III.8.5 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 cosmological problem ---why $\Lambda$ is small (scale) and why it is comparable to matter today (partition)--- are reformulated as clock readings.
\end{tcolorbox}

\noindent The distinction is one of class: the \textbf{partition} $\Omega_m:\Omega_{EO}=(1-f_c):f_c$ lives entirely in the amplitude clock, which explains why it is derivable to $0.27\sigma$ without touching the absolute scale; the \textbf{scale} $H_0=\sqrt{8\pi/3}\,(c/\ell_P)\sqrt{\Theta}$ lives entirely in the scale clock. The partition is theory; the scale is observational datum. To pretend to ``derive $H_0$'' is to pretend to derive the time of day from the mechanism of the clock: a category mistake. The rate $H_0$ and the duration $1/H_0$ are the same clock read in two directions: $H_0=\sqrt{8\pi/3}\,\sqrt{\Theta}\,/\tau_P$ counts the frequency of the tick; $1/(H_0\tau_P)=N_{\text{tic}}$ counts the elapsed ticks. The Hubble ``constant'' decreases because the source dilutes: matter falls as $L^{-3}$ while the dark-energy ground remains constant ($w=-1$), so that $H^2\propto\rho_{m,0}L^{-3}+\rho_{\text{EO}}$ descends toward the asymptotic value fixed by $\rho_{\text{EO}}$.

\paragraph*{The Hubble tension as an artefact of method.} If $H_0$ is the positional reading of a clock, the ``Hubble tension'' is not a discrepancy between two measurements of the same physical quantity. The local value ($\sim73$) comes from the distance ladder; the ``primordial'' value ($\sim67$) is not measured: it is the $H_0$ that $\Lambda$CDM needs to reproduce the anisotropy pattern of the CMB under its parametrization. The number $\sim5.8\sigma$ of the tension is, therefore, a property of the $\Lambda$CDM interpretation of the data, not a raw datum of the universe.

A medium whose $H(z)$ coincides with $\Lambda$CDM to linear order ---same source $\rho_{m,0}L^{-3}+\rho_{\text{EO}}$, with $w=-1$ exact (\S{}III.6.6)--- but differs in the non-linear regime of the walls, produces exactly this type of artefact when analyzed within the $\Lambda$CDM parametric mould: the inference of the ``primordial'' $H_0$ inherits the bias of forcing the elastic form of $H(z)$ into the standard template. The assertion of the program is about the inference procedure, with definite and falsifiable sign:

\begin{equation}
\boxed{\;
\begin{gathered}
w=-1\ \text{exact},\quad H(z)\approx\Lambda\text{CDM to linear order}\\[3pt]
\Longrightarrow\ \text{the tension dissolves in model-independent observables}
\end{gathered}
\;}
\end{equation}

The program predicts that the tension is attenuated or disappears when measured with raw observables ---the BAO pattern $D_H/r_d$ (\S{}III.8.6), the raw acoustic angle $\theta_*$, the shape of $E(z)$---, because it is an artefact of the parametrization, not a property of the universe. Its residual, if any, tracks the non-linear correction of the walls (\S{}III.8.9), not a distinct ``true'' $H_0$ between epochs.

\begin{tcolorbox}[notabox,title={The unifying finding: the Hubble tension and the DESI signal are the same artefact}]
Two signals that $\Lambda$CDM records as anomalies ---the dynamical dark energy that DESI infers ($w_0>-1$, $w_a<0$) and the Hubble tension ($H_0^{\text{late}}>H_0^{\text{primordial}}$)--- are the reading of a single elastic medium, with $w=-1$ exact and $H(z)$ coinciding with $\Lambda$CDM to linear order, forced into the $\Lambda$CDM parametrization. Both are artefacts of the mould, not properties of the universe. The program predicts that \emph{both} are attenuated or dissolve in model-independent observables: a bet with definite sign (dissolution, not persistence), falsifiable with raw BAO, SNe and CMB data. \epi{SLD}.
\end{tcolorbox}

\begin{tcolorbox}[notabox,title={Which clock marks the present: the frozen modulus, the phase at $S_{\min}$}]
The clock of the present epoch is not the radial modulus. Two facts fix it. \emph{Observational:} the scales built on the modulus ---soliton masses ($M_Q=M_P/S_{\max,\text{eff}}$), $S_{\min}$ of the defects, hadronic constants--- are stable; if the background modulus rolled today, they would drift. \emph{From scales:} the radial mode is ultra-Planckian, $m_S=\sqrt\xi\,E_P\approx4.44\,E_P$ (\S{}III.1.9), with $m_S/H_0\sim10^{61}$; a mode so heavy against $H$ freezes at the minimum in $\sim t_P$. The radial modulus is at $S_{\min}$; its curve $S^*\to S_{\min}$ is the clock of the bounce and crystallization (Part IV), not that of today.

The late cosmological sector is the coherent phase condensate at $S_{\min}$ (the DE-Curtain, \S{}III.7.5), with $w=-1$ exact by symmetry of the ground state ($T_{\mu\nu}\propto g_{\mu\nu}$, $\dot\theta_0=0$): not a field rolling, but the rest at the ground. The phase mode is a massless Goldstone (A5); the relative weight $\Omega_{\text{EO}}$ ascends by dilution of matter, with physical density $\rho_{\text{EO}}=K\,S_{\min}^2$ constant. \epi{SLD}: the relocation rests on derived results ($T\!\propto\!g$, $m_S$, the Goldstone of A5).
\end{tcolorbox}

The present value $\Theta_0$ (the time of day of the clock) is positional: the curve $H(z)$ is derivable, the point on it is observational datum. What remains as a thread is the non-linear correction of $H(z)$ by the walls and its measurability in raw BAO (\S{}III.8.9), not a law of dynamical $w(z)$.

\begin{tcolorbox}[derivado,title={What is pertinent to predict: observables, not parameters of another model}]
The ``Hubble tension'' is \emph{not} a discrepancy between two measurements of the same physical quantity. The local value ($\sim73$) comes from the distance ladder, mildly model-dependent; but the ``primordial'' value ($\sim67$) \textbf{is not measured}: it is a \emph{parameter derived from fitting $\Lambda$CDM to the anisotropy pattern of the CMB}. It is not ``the $H_0$ of the early universe'', but ``the $H_0$ that $\Lambda$CDM needs to reproduce the CMB''. Consequently, the number $\sim5.8\sigma$ of the tension is a property of the $\Lambda$CDM interpretation of the data, not a raw datum of the universe.

\smallskip
From here follows what is \emph{not} a legitimate target and what is. \textbf{It is not pertinent} to calibrate PIU-$\Psi$ to reproduce the magnitude $\sim5.8\sigma$: to do so would be to fit the program to imitate an artefact generated by the very model one seeks to replace ---chasing the reflection, not the object. \textbf{It is pertinent}, and it is what the program asserts, to predict the \emph{method bias}: that an observer analyzing an elastic medium ---with $w=-1$ exact and $H(z)$ coinciding with $\Lambda$CDM to linear order, divergent in the non-linear regime--- under the $\Lambda$CDM parametric template will infer both a Hubble tension and a $w_0$--$w_a$ signal that are properties of the fit, not of the universe. The prediction with definite sign is the \emph{dissolution} of both in model-independent observables ---the BAO pattern $D_H/r_d$ (\S{}III.8.6), the shape of $E(z)$, the raw acoustic angle $\theta_*$--- and its confrontation is always anchored there, never against parameters ($H_0^{\text{Planck}}$, $w_0^{\text{DESI}}$) that are already outputs of a $\Lambda$CDM fit. To take those parameters as the target would reintroduce, through the back door, the assumption of the rival model.
\end{tcolorbox}

\paragraph*{Identity with the radial oscillation (\S{}II.4.2).} The ascent of the weight $f_c$ toward its bound is the \emph{same physics} as the decomposition of the trajectory of $S_{\max,\text{eff}}^2$ in \S{}II.4.2, in two languages. There $S_{\max,\text{eff}}$ is the upper turning point of the radial oscillation of the modulus in the well $V_\text{eff}$; its present value, read cosmologically, is the position of the universe over the relaxation curve. Both images share the same bound ($f_c < 1 \Leftrightarrow S_{\max,\text{eff}} < S_{\max,\text{geom}}$) and the same reading of the present as the position of a cosmic clock. The value $f_c(t_0) = 0.687$ is that point over the curve, not a fundamental number ---in consistency with the positional \epi{CALIBRATED} status of the numerator (\S{}II.4.2, \S{}III.7.2).

\paragraph*{Three anomalies, one reading.} The positional reading reduces three facts that the standard model treats as separate problems to a single assertion: \emph{$H_0$ is not a constant of nature, but the time of day of a clock}. The \textbf{cosmological-constant problem} ---why the vacuum density is $\sim10^{-123}$ times the Planck one--- is not a fine-tuning of $123$ orders: the identity $\Theta_0=3/(8\pi N_{\text{tic}}^2)$ shows that this number is, up to the geometric ratio of the coupling, the inverse of the square of the age of the universe in ticks ($N_{\text{tic}}\approx8.5\times10^{60}$). There is no small vacuum density to explain; there is an old universe in ticks, and the smallness is arithmetic. The \textbf{Hubble tension} ---which for $\Lambda$CDM is a causeless discrepancy of $\sim5.8\sigma$--- is an artefact of method: the ``primordial'' $H_0$ is not measured, it is the one $\Lambda$CDM needs to reproduce 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} that DESI 2024--2025 suggests ($w_0>-1$, $w_a<0$) is the same type of artefact: the signal that appears when forcing a medium with $w=-1$ exact into the $w_0$--$w_a$ parametrization. The three are readings of an elastic medium ---whose $H(z)$ coincides with $\Lambda$CDM to linear order and differs in the non-linear regime of the walls--- under static assumptions. The program predicts that the Hubble tension and the DESI signal are attenuated or dissolve in model-independent observables.

\paragraph*{The exclusion of the local value as background: $A<1$.} The background $H_0$ falls on the primordial side ($\sim67$), not the local one ($73$), and the bound that fixes it is the \emph{dimensionless age} $A\equiv H_0\,t_0$: with the observed age $t_0=13.787$~Gyr, $H_0=73$ would require $A\approx1.03$, while a background with diluting matter ($L^{-3}$) plus constant-density dark energy ($w=-1$) imposes the ceiling $A\le A(w{=}{-}1)=0.953$. A background value with $A>1$ would require a dark-energy density \emph{increasing} in physical density ---the phantom region $w<-1$--- which the positivity of the canonical kinetics vetoes ($w+1=\rho_P\ell_P^2\dot{\bar S}^2/\rho\ge0$) and which the structural constancy of $\rho_{\text{EO}}=K\,S_{\min}^2$ (anchor$\times$wall) forbids. Natively, $A<1$ is the signature of a medium whose source is matter that dilutes over a constant ground: the universe is younger than its Hubble time because the early dense phase ran the clock faster, not because the background evolves today. The \emph{exclusion} of the $73$ as background is \epi{DERIVED} ---a property of the form of the elastic source, with $w=-1$ exact at the ground---; the local $73$ is correct as a reading of the distance ladder, and its ``tension'' with the $67$ of the CMB is the method artefact already described. The exact value of the ceiling ($0.953$) is tied to the fine non-linear correction of $H(z)$ (thread, \S{}III.8.9).

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. 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. Integrating the dimensionless age of a flat background with $w=-1$, the ceiling is $0.9641$ for $\Omega_m=0.30$ and $0.9468$ for $\Omega_m=0.32$: it moves by $1.8\,\%$ across the whole plausible range. The distance of $A(73)=1.029$ from the ceiling is, by contrast, $8.1\,\%$. The exclusion of the local value as background therefore survives by a factor of more than four all the positional freedom of the partition, so that it does not rest on the exact value of $\Omega_m$ but on the structure: matter that dilutes plus a floor of constant density do not admit a universe so old measured in ticks of its own rate. Within that same band, the background values on the primordial side are consistent with nothing forced ---$A(67.4)=0.950$ and $A(67.7)=0.955$ fall on the ceiling corresponding to $\Omega_m$ between $0.311$ and $0.316$---.

\paragraph*{Falsifiability as counterpart.} The program does \emph{not} derive the present numerical value of $H_0$ ---that would be to derive the time of day from the mechanism of the clock. What it fixes is substantial: the exact factorization and the age-in-ticks identity (which reformulates the cosmological-constant problem), both \epi{DERIVED}; the equation of state $w=-1$ exact of the late sector \epi{DERIVED}; the predicted dissolution of the Hubble tension and of the DESI signal in raw observables \epi{SLD}; and the exclusion of the local value as background \epi{DERIVED}. Only the present \emph{value} $\Theta_0$ is positional. This makes the proposal falsifiable by three independent routes: (a) if future measurements show that the Hubble tension \emph{persists} in model-independent observables (raw $D_H/r_d$, $\theta_*$) instead of dissolving, the reading is refuted; (b) if the DESI/Euclid signal consolidates at $w_0<-1$ (phantom region that the positivity of the kinetics vetoes) or persists as $w_0>-1$ in raw observables, the mechanism is refuted; (c) if the non-linear correction of $H(z)$ by the walls yields a sign or magnitude incompatible with the raw BAO, the prediction breaks. The assertion is strong where it can be (factorization, vacuum identity, $w=-1$ exact, with exact mathematics) and honest where it cannot (non-linear correction, declared a thread).

\begin{tcolorbox}[notabox,title={Epistemic classification --- positional readings of the background}]
That redshift and time dilation share the factor $L(\tau_0)/L(\tau_e)$ by nullity of the TT mode, with pure ``tired light'' excluded by native mechanism and at $\sim$$18\sigma$ by the datum (\S{}III.8.2-bis), is \epi{DERIVED}. Dark energy as el Pleno-ground state at $S_{\min}$ with $\rho_{\text{EO}}=K\,S_{\min}^2$ constant and $w=-1$ exact is \epi{DERIVED}; the bound $f_c<1$ is \epi{SLD}; the present value of $f_c$ is \epi{CALIBRATED} positional. The factorization $H_0=\sqrt{8\pi/3}\,(c/\ell_P)\sqrt{\Theta}$ and the identity $\Theta=3/(8\pi N_{\text{tic}}^2)$ are \epi{DERIVED} (exact algebra on the elastic kinetic constraint and the native identity $G=c^2/(\rho_P\ell_P^2)$); the positional status of $\sqrt{\Theta}$ and of $H_0$ is \epi{DERIVED} as classification and \epi{CALIBRATED} in its present value. The predicted dissolution of the Hubble tension and of the DESI signal in model-independent observables is \epi{SLD}; the non-linear correction of $H(z)$ by the walls is an open thread (\S{}III.8.9).
\end{tcolorbox}

\subsubsection{III.8.8 --- Tensors, anti-BKL and the revision of P15}

Three technical points close the reading of the background, two in favor and one that corrects the corpus.

\textbf{The suppression of the tensor-scalar ratio.} The classic problem of the simple matter
bounce is that it predicts a ratio $r = P_T/P_S$ of order unity, excluded by
observation. In the program the generation is not symmetric: the scalars are
perturbations of the modulus, which feel the two walls of the potential and are
amplified; the tensors are pure transverse phase, which by the phase symmetry is
not in the potential and does not feel the walls. The ratio $r$ is suppressed ---numerator
not amplified, denominator amplified--- in a structural way in the sign, which
resolves the Achilles' heel of the simple matter bounce.

\textbf{The native resolution of the BKL problem.} The most serious objection against any
contraction is the growth of the anisotropic shear (as $a^{-6}$), which dominates over
matter and leads to mixmaster-type chaos; the standard models add an ekpyrotic
phase to tame it. The program resolves it natively: the shear viscosity
of \emph{el Pleno} diverges as the dilatant index $n(x) = 1/(1-x)$ when approaching the
ultraviolet wall, and a divergence always defeats a growth of finite power. There
is a point where the viscous dissipation overcomes the growth of the shear and
the medium becomes a rigid solid that does not support shear. The dilatancy does the
work that ekpyrosis does in the standard models, without an added ingredient.

\textbf{Honest revision of the prediction P15.} The corpus (\S{}III.3.6) presented P15
---the ultraviolet suppression of the primordial tensor spectrum--- as testable by
CMB-S4 and LiteBIRD via the B-mode polarization. The quantitative analysis of this
section obliges a qualification of that assertion: the flexural suppression $(k\ell_P)^2/20$ only
bites at scales $k\ell_P \gtrsim \sqrt{20}$, that is, at the Planck scale, while
the CMB modes have $k\ell_P \sim 10^{-31}$ even traced back to the bounce.
The suppression exists, but at the Planckian scale, not in the CMB window. The suppression
of $r$ observed in the program comes, therefore, from the scalar/tensor asymmetry
(above), not from the flexural suppression. The testability assertion of P15 by CMB
via ultraviolet suppression is reclassified accordingly: what the CMB can test
is the scalar/tensor asymmetry and the red$\to$blue transition (\S{}III.8.4), not the
flexural suppression.

\begin{tcolorbox}[notabox,title={Epistemic classification --- tensors and anti-BKL}]
The scalar/tensor asymmetry and the sign of the suppression of $r$ are \texttt{[strong logical
deduction]}; the magnitude of the suppression is \texttt{\epi{HYPOTHESIS}}; the value of $r$
is \texttt{\epi{CALIBRATED}}. The native resolution of the BKL problem by divergence of the
dilatant viscosity is \texttt{\epi{SLD}}. The revision of P15 (the
flexural suppression does not operate in the CMB window) is a \textbf{declared finding}
that corrects the scope of the prediction of the corpus in \S{}III.3.6.
\end{tcolorbox}

\subsubsection{III.8.9 --- Synthesis: the closed background sector and the combined thread}

The cosmological arc is consolidated. The perfect blackbody is \texttt{\epi{DERIVED}}
(the photon is a mode, $\mu = 0$); the nearly scale-invariant spectrum without inflation is
\texttt{\epi{SLD}} (native matter bounce, with $w = 0$ derived and not
postulated); the red tilt, the suppression of $r$ and the anti-BKL solution come out of the two
walls and the dilatancy \texttt{\epi{SLD}}; the matter density
$\Omega_m = 1 - f_c = 0.3131$ hits at $0.3\sigma$ of Planck without fitting
\texttt{\epi{DERIVED}}; the form of $H(z)$ is forced by the potential with a single
observable freedom \texttt{\epi{DERIVED}}, and fits BAO DESI DR2 as well as the standard
model with less freedom \texttt{[verified]}; the redshift and the temporal
dilation are read as evolution of the medium, not as primitive metric expansion
\texttt{\epi{SLD}}.

This situates the background sector in a precise position: the quantitative background
cosmological dynamics ---the form of $H(z)$, the matter density, the character of the
CMB, the perturbation spectrum at zeroth order--- is a closed result, whose
theory assertions hold without recourse to moving data. What remains
open is delimited with precision in the following box.

\begin{tcolorbox}[problemabierto,title={What remains open: the combined analysis (confrontation thread F-cosmo)}]
What the program does \emph{not} close, and declares open with discipline,
is the \textbf{combined analysis BAO+CMB+SNe}. The confrontation of this section is with
the model-independent radial observable ($D_H/r_d$), which is the raw data layer. The
DESI preference for dark-energy dynamics ($w_0 > -1$) comes from the
\emph{combined} analysis, which introduces infrastructure foreign to the program ($r_d$ from the CMB,
supernova sample, covariances, CPL parametrization). Reproducing that combined
preference with the $\delta_0$ of the program is the confrontation thread F-cosmo, of high priority. It is
deferred with explicit criterion: to close first the theory gaps (permanent,
independent of the data) and to defer the confrontation with a moving datum (DESI
DR2$\to$DR3, dependent on the supernova sample and on the BAO-CMB tension). The
theory assertion ---background cosmology without fitting parameters---
does not need the combined analysis; this combined analysis is a \emph{plus} declarable as a thread, in the style
of the program.

\textbf{On the so-called ``phantom crossing''.} If the combined analysis prefers
$w < -1$ in the past, that is not a tension until the layers of the datum are
distinguished. The program has $w \geq -1$ always (canonical kinetics), but its $w(z)$ is
non-monotonic with a minimum near $z \approx 0.5$; upon forcing that expansion
history into the rigid CPL parametrization one would obtain an apparent ``crossing''
that is an artifact of translating an emergent medium into the borrowed language of a fluid
over Friedmann. The program confronts the raw datum (the model-independent
layer), never its model-dependent compression. This is a
\texttt{\epi{HYPOTHESIS}} that the confrontation thread F-cosmo will allow to decide.
\end{tcolorbox}

\medskip\hrule\medskip

\subsection{\texorpdfstring{\S{}III.9 --- \emph{El Pleno} as a dilatant fluid: non-linear rheology and consequences}{Sec.III.9 - El Pleno as a dilatant fluid: non-linear rheology and consequences}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} value of $\eta/s$ of the thermal \emph{Pleno} (requires the collisional kinetics of the gas of modes; only proximity to the KSS bound is conjectured); complete dynamical bridge of the rheological reading of relativity (closure of coefficients of the solitonic profile).
  \item \textbf{Notes:} extends the continuous-media card of \S{}III.1.8 to the \emph{non-linear} rheology of \emph{el Pleno}. It closes circles of confirmation with \S{}III.1 (rigidities, effective metric, induced action), A3/LET (Lorentz covariance), \S{}III.7.5 (EO-Backdrop, $w=-1$) and the A6 hard bound. All the quantities are derived from already-established parameters ($\rho_P$, $c$, $\ell_P$, $S_{\min}^\text{estr}$, $S_{\max,\text{geom}}$, $\tau_\Pi$, $V_q$); no new parameter is introduced.
\end{itemize}
\end{estadoseccion}

\subsubsection{III.9.1 --- Motivation: from linear to non-linear rheology}

\S{}III.1.8 established the constitutive card of \emph{el Pleno} in its \emph{linear}-response regime: elastic moduli $(\mu, K, \lambda, E, \nu)$ and Maxwell viscoelastic classification in the shear sector, $\eta_\text{Pleno} = \mu_\text{Gent}\tau_\Pi$. That card describes the response of the medium to small and slow deformations. But a medium with a hard deformation bound (axiom A6) and an IR quantum wall (axiom A2) cannot be linear over its entire range: when forced toward those bounds, its response changes qualitatively. This section characterizes that \emph{non-linear} response and shows that \emph{el Pleno} is, with technical precision, a \textbf{dilatant fluid} (shear-thickening) of the Herschel-Bulkley type, and that this reading closes circles of confirmation with the relativity, the cosmology and the black-hole physics already derived in the program.


\subsubsection{\texorpdfstring{III.9.2 --- The maximum deformation rate $\dot\gamma_{\max} = c/\ell_P$}{III.9.2 - The maximum deformation rate}}

In rheology, the independent variable is the \textbf{deformation rate} $\dot\gamma$ (of dimensions $[\text{s}^{-1}]$). The program fixes a hard upper bound for this rate. Axiom A4 (ultraviolet regulation) establishes that the propagation operator of F1 carries the correction $\Box \to \Box(1 + \ell_P^2\Box/20)$ (\S{}II.5.4, dispersion equation E23), so that every excitation of \emph{el Pleno} has an effective wavenumber bounded by $k_{\max}\sim \ell_P^{-1}$ and, with propagation velocity $c$, a frequency bounded by $\omega_{\max} = c\,k_{\max} \sim c/\ell_P$. A shear deformation of the medium is precisely an excitation of its modulus, so that its maximum rate is that cutoff frequency:


\begin{equation}
\dot\gamma_{\max} = \frac{c}{\ell_P} = \frac{1}{t_P} \approx 1.8552\times 10^{43}\ \text{s}^{-1}, \tag{E45}
\end{equation}

the \textbf{Planck frequency}, with $t_P = \ell_P/c$ the Planck time. It is not a mere dimensional translation of $c$: it is the cutoff frequency that the UV-regulation term of F1 imposes on every excitation of the medium, read as a deformation rate. \epi{DERIVED} from A4 and the dispersion relation E23.

\paragraph*{The Hubble bound is the dilatant bound, with exact equality.} This same limiting rate of the medium fixes the upper bound of the cosmological volumetric deformation. The endpoint $S_{\max,\text{geom}}$ of the hard bound A6 produces the saturation density $\rho_{\max}=\rho_P\,S_{\max,\text{geom}}^2$, and the kinetic constraint of the elastic solid (\S{}III.6.6), $H^2\propto(G/c^2)\rho_e$ with the coefficient of the source-stretch coupling, gives the maximum volumetric deformation rate
\begin{equation}
H_{\max}=\sqrt{\frac{8\pi G\,\rho_{\max}}{3}}
=\sqrt{\frac{8\pi}{3}}\,\frac{c}{\ell_P}\,S_{\max,\text{geom}}
=7.922\times10^{43}\,\mathrm{s^{-1}}\,,
\end{equation}
where the native identity $G=c^2/(\rho_P\ell_P^2)$ (E3) was used, with minimum horizon $R_{H,\min}=c/H_{\max}=0.2341\,\ell_P$ sub-Planckian (P20, \S{}II.5). Compared with the maximum shear deformation rate $\dot\gamma_{\max}=c/\ell_P$ of E45, the ratio is \emph{not} approximate:
\begin{equation}
\boxed{\;\frac{H_{\max}}{\dot\gamma_{\max}}=\sqrt{\frac{8\pi}{3}}\;S_{\max,\text{geom}}=4.2709\;}
\end{equation}
it is an exact structural equality, not a ``factor $\sim4$''. The cosmological volumetric deformation bound (Part II) and the dilatant shear bound ($\mathrm{Ma}=1$, jamming, \S{}III.9) are the same rigidity of the medium, read from the global and from the local deformation: the maximum volumetric deformation rate is bounded by the maximum reorganization rate of the lattice, and the factor that relates them is the coefficient of the source-stretch coupling (\S{}III.1.4) times the root of the packing ratio. Since the result is finally expressed as a pure ratio over the anchor $\dot\gamma_{\max}=c/\ell_P$, it passes the purity test of the Language of el Pleno. \epi{DERIVED}.

We define the dimensionless \textbf{relative deformation rate}


\begin{equation}
x \equiv \frac{\dot\gamma}{\dot\gamma_{\max}} = \dot\gamma\,\frac{\ell_P}{c} \in [0,1). \tag{E46}
\end{equation}

This quantity coincides exactly with the \textbf{Deborah number} of the medium, $\mathrm{De} = \tau_\Pi^\star\,\dot\gamma$ with characteristic time $\tau_\Pi^\star = \ell_P/c = t_P$: $x$ measures how close the imposed deformation is to the limiting rate of \emph{el Pleno}. The fluid$\leftrightarrow$solid transition of a viscoelastic material occurs at $\mathrm{De}\sim 1$; here that transition coincides with the hard bound $x\to 1$. \epi{DERIVED}.


\subsubsection{III.9.3 --- The flow index and the dilatant character}

\textbf{What a dilatant fluid is (physical intuition).} A \emph{dilatant} fluid, or shear-thickening one, is one that becomes \emph{more} resistant the faster it is deformed ---the opposite of what water does, whose resistance does not depend on the rapidity. The everyday example is the mixture of cornstarch and water (``oobleck''): if stirred slowly it flows like a liquid, but if struck or run over, it hardens instantaneously like a solid. Technically, in a Newtonian fluid the stress needed to deform grows in direct proportion to the deformation rate ($\tau\propto\dot\gamma$); in a dilatant one it grows faster than proportionally ($\tau\propto\dot\gamma^{\,n}$ with $n>1$). The \emph{yield stress}, for its part, is a stress threshold below which the material does not flow at all ---like toothpaste or mayonnaise, which keep their shape until pushed with sufficient force. A fluid that combines both features ---yield threshold plus shear-thickening--- is a \textbf{Herschel-Bulkley fluid}, the most general rheological model for a medium whose response does not depend on time. The thesis of this section is that \emph{el Pleno} realizes exactly this behavior, with its three elements (threshold, power law, flow index) derived from the axioms, and that its extreme form ---resistance that \emph{diverges} at a maximum deformation rate--- is the rheological reading of the A6 hard bound.

The constitutive stress--rate relation of a time-independent fluid is modeled with the Ostwald-de Waele power law, $\tau = K\,\dot\gamma^{\,n}$, where the \textbf{flow index} $n$ classifies the medium: $n=1$ Newtonian, $n<1$ pseudoplastic (shear-thinning), $n>1$ \textbf{dilatant} (shear-thickening). The flow index is defined operationally as the local logarithmic slope of the flow curve, $n \equiv \mathrm{d}\ln\tau/\mathrm{d}\ln\dot\gamma$.

\textbf{Shear stress from the Gent potential.} \emph{El Pleno} admits a Gent hyperelastic characterization (\S{}III.1.7), with saturation energy


\begin{equation}
V_\text{topo}(S) = -\frac{\mu_\text{Pleno} J_m}{2}\log\!\left(1 - \frac{S^2}{S_{\max,\text{geom}}^2}\right), \qquad J_m = S_{\max,\text{geom}}^2\,.
\end{equation}

Writing the relative packing $u \equiv S^2/S_{\max,\text{geom}}^2$, the stress associated with the deformation is $\tau \propto \mathrm{d}V_\text{topo}/\mathrm{d}u$:


\begin{equation}
\tau \;\propto\; \frac{\mathrm{d}V_\text{topo}}{\mathrm{d}u} = \frac{\mu_\text{Pleno} J_m}{2}\,\frac{1}{1-u}\,,
\end{equation}

and since the flow starts from the relaxed state ($\tau\to 0$ when $u\to 0$), the reduced stress that measures the departure from rest is $\tau_\text{red}(u) = u/(1-u)$. Identifying the dynamical packing with the relative deformation rate, $u = x = \dot\gamma/\dot\gamma_{\max}$ (E46) ---both measure how close the medium is to its bound---, the flow curve of \emph{el Pleno} is $\tau_\text{red}(x) = x/(1-x)$.

\textbf{Flow index.} Since $x = \dot\gamma/\dot\gamma_{\max}$ implies $\mathrm{d}\ln x = \mathrm{d}\ln\dot\gamma$, the flow index is $n = \mathrm{d}\ln\tau_\text{red}/\mathrm{d}\ln x = x\,\mathrm{d}\ln\tau_\text{red}/\mathrm{d}x$. With $\ln\tau_\text{red} = \ln x - \ln(1-x)$,


\begin{equation}
n(x) = x\left(\frac{1}{x} + \frac{1}{1-x}\right) = 1 + \frac{x}{1-x} = \boxed{\;\frac{1}{1-x}\;}\,, \qquad x = \frac{\dot\gamma}{\dot\gamma_{\max}}\,. \tag{E47}
\end{equation}

The consequences are immediate and all with physical meaning:

\begin{itemize}[leftmargin=1.4em]
  \item $x\to 0$ (low rate): $n\to 1$ \emph{exactly}. \textbf{\emph{El Pleno} is Newtonian in the low-deformation-rate limit} ---which is why the non-Newtonian character of the vacuum is undetectable in ordinary conditions. This limit is model-independent: any smooth relation $x(\dot\gamma)$ that vanishes gives $n\to 1$.
  \item $0 < x < 1$: $n > 1$ always. \textbf{\emph{El Pleno} is dilatant} over the whole forced range, the more so the closer to the bound.
  \item $x\to 1$ ($\dot\gamma\to c/\ell_P$): $n\to\infty$. \textbf{Divergent dilatancy}: the resistance grows without limit as the maximum rate is approached. This is the rheological reading of the A6 hard bound and the mechanism of the absence of singularities: nothing can deform \emph{el Pleno} faster than $c/\ell_P$ because its resistance diverges.
  \item In the current universe, the \emph{Pleno}-ground state occupies $x = S_{\min}^{2,\text{estr}}/S_{\max,\text{geom}}^2 \approx 0.2401$, giving $n\approx 1.3160$: \textbf{slightly dilatant, almost Newtonian today}. The strong dilatancy is only activated upon forcing the medium toward $S_{\max}$ (primordial rebound, black-hole cores).
\end{itemize}

\textbf{Epistemic classification.} The Newtonian limit $n\to 1$ at $x\to 0$ is \epi{DERIVED} (model-independent). The dilatant character $n>1$ under forcing and the divergence at the bound are \epi{SLD} (they follow from the Gent form plus the UV bound). The closed form $n=1/(1-x)$ and the numerical values are \epi{SLD} (they depend on the constitutive packing model $x = S^2/S_{\max}^2$).


\subsubsection{\texorpdfstring{III.9.4 --- The yield stress and the Herschel-Bulkley model}{III.9.4 - The yield stress and the Herschel-Bulkley model}}

A shear-thickening fluid with a flow threshold is described by the Herschel-Bulkley model, $\tau = \tau_0 + K\,\dot\gamma^{\,n}$, where $\tau_0$ is the \textbf{yield stress}: the minimum stress to initiate the flow. \emph{El Pleno} possesses such a threshold, and it comes from the Klauder IR quantum wall (\S{}I.7.8): to deform the modulus of the medium one must overcome the restoring response of $V_q$. The restoring force per unit deformation is $-\mathrm{d}V_q/\mathrm{d}S$, with $V_q^\text{dens} = 3\hbar^2/(8\rho_P\ell_P^8 S^2)$ (E5). Using the native identity $\hbar = \rho_P c\,\ell_P^4$ (E36):


\begin{equation}
-\frac{\mathrm{d}V_q^\text{dens}}{\mathrm{d}S} = \frac{3\hbar^2}{4\rho_P\ell_P^8 S^3} = \frac{3}{4}\,\frac{\rho_P c^2}{S^3}\,,
\end{equation}

which evaluated at the \emph{Pleno}-ground state $S = S_{\min}^\text{estr}$ gives the structural yield stress:


\begin{equation}
\boxed{\;\tau_0 = \frac{3}{4\,(S_{\min}^\text{estr})^3}\,\rho_P c^2 \approx 1.9838\,\rho_P c^2\;} \tag{E48}
\end{equation}

\emph{El Pleno} is thus characterized as a \textbf{Herschel-Bulkley fluid}: yield stress $\tau_0 \approx 1.9838\,\rho_P c^2$ from the IR wall, plus a dilatant power law with index $n(x) = 1/(1-x)$ bounded by the UV wall. It is the most general rheological model for a time-independent fluid, and \emph{el Pleno} realizes its three elements from first principles.

\textbf{The consistency index $K$.} The third element of the model, the consistency index $K$ of the power term $K\dot\gamma^{\,n}$, is fixed without a free parameter by the same Gent structure. The excess stress over rest is $\tau_\text{exc}(x) = \tfrac{\mu_\text{Gent}J_m}{2}\,\dfrac{x}{1-x}$, and with $\mu_\text{Gent} = \rho_P c^2/S_{\max,\text{geom}}^2$ (\S{}III.1.6) and $J_m = S_{\max,\text{geom}}^2$ (\S{}III.9.3) the dimensional prefactor is $\tfrac{\mu_\text{Gent}J_m}{2} = \tfrac{1}{2}\rho_P c^2$. Writing the rate in units of its bound $\dot\gamma_{\max}$, the Ostwald change of variable $\tau_\text{exc} = K(x)\,x^{\,n(x)}$ gives the closed form

\begin{equation}
K(x) = \frac{1}{2}\,\rho_P c^2\;\frac{x^{\,x/(x-1)}}{1-x}\,, \qquad K_0 \equiv \lim_{x\to 0}K(x) = \frac{1}{2}\,\rho_P c^2\,. \tag{E48b}
\end{equation}

The consistency index is not, then, a free coefficient: in the Newtonian limit ($x\to 0$) it collapses exactly to half the stress scale of \emph{el Pleno}, $K_0 = \tfrac12\rho_P c^2$, and it diverges at the hard bound ($x\to 1$) in parallel with $n(x)$ ---intrinsic behavior of a variable-index fluid. At today's operating point ($x = 0.2401$), $K \approx 1.033\,\rho_P c^2$. \epi{DERIVED} the yield stress (via $V_q$ with the coefficient $3/8$ already derived) and the consistency index (closed form E48b, anchored in $K_0 = \tfrac12\rho_P c^2$); \epi{SLD} the complete Herschel-Bulkley structure.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{curva_reologica.png}
\caption{\emph{El Pleno} as a Herschel-Bulkley dilatant fluid. (A) The flow index $n(x)=1/(1-x)$ is greater than 1 over the whole forced range (dilatant: shear-thickening), tends to the Newtonian limit $n=1$ at low rate and diverges at the hard bound ($x\to 1$, Gent wall / A6). The current universe is at $x=0.240$, $n=1.32$: almost Newtonian, which explains why the non-Newtonian character of the vacuum is undetectable in ordinary conditions. (B) The flow curve $\tau=\tau_0+K\dot\gamma^{\,n}$ combines the yield threshold $\tau_0=1.9838\,\rho_P c^2$ (from the Klauder IR wall) with the dilatant thickening that diverges at the bound. The three elements of the model are derived from first principles.}
\label{fig:reologia}
\end{figure}


\subsubsection{\texorpdfstring{III.9.5 --- Rheological reading of relativity: the Lorentz factor as a dilatant response}{III.9.5 - Rheological reading of relativity}}

Lorentz covariance is already established in the program as an emergent property (axiom A3, Lorentz Emergence Theorem, \S{}III.3). The dilatant response of \emph{el Pleno} is an \textbf{equivalent reading} of that covariance ---in the same sense in which Bohm-Madelung mechanics is an equivalent reading of the identity of $\hbar$ (\S{}I.8.5)---, not a second independent derivation nor a replacement.

Consider a deformation that travels at velocity $v$ through \emph{el Pleno}, $S = S(x - vt)$. The kinetic invariant of F1 reduces to


\begin{equation}
(\partial_\mu S)(\partial^\mu S) = \frac{1}{c^2}\dot S^2 - (\nabla S)^2 = -\left(1 - \frac{v^2}{c^2}\right)(\partial_\xi S)^2\,, \tag{E49}
\end{equation}

so that the relativistic factor $(1 - v^2/c^2)$ \emph{emerges from the Lagrangian F1 itself}, without being postulated. The energy density ($T^{00}$ component of the tensor of F1) of this moving deformation is


\begin{equation}
T^{00}(v) = \frac{1}{2}\rho_P c^2\ell_P^2\left[\frac{1}{c^2}\dot S^2 + (\nabla S)^2\right] = \frac{1}{2}\rho_P c^2\ell_P^2\,(\partial_\xi S)^2\left[\frac{v^2}{c^2} + 1\right].
\end{equation}

To sustain the deformation at velocity $v$ the profile contracts according to the $c$-isotropy of axiom A3 ---proper width $L_0\to L = L_0\sqrt{1-v^2/c^2}$---, which amplifies the spatial gradient: $(\nabla S)^2 \to (\nabla S)^2/(1-v^2/c^2)$. The energy density becomes


\begin{equation}
T^{00}(v) = \frac{1}{2}\rho_P c^2\ell_P^2\,(\partial_\xi S)^2\left[\frac{v^2}{c^2} + \frac{1}{1-v^2/c^2}\right] \;\xrightarrow{\,v\to c\,}\; \gamma^2\cdot T^{00}(0)\,, \tag{E50}
\end{equation}

where the term $1/(1-v^2/c^2) = \gamma^2$ dominates near $c$, with $\gamma = (1-v^2/c^2)^{-1/2}$ the Lorentz factor. Equating this resistance with the dilatant apparent viscosity $\eta(x)\sim 1/(1-x)$ (E47) one obtains the identification, \emph{not postulated}, of the packing parameter with the velocity:


\begin{equation}
x = \frac{v^2}{c^2}\,, \qquad\Longrightarrow\qquad n(v) = \gamma^2\,.
\end{equation}

The flow index as a function of velocity \emph{is} the square of the Lorentz factor. The relativistic inertia ---the growing resistance to accelerate toward $c$--- is read as the dilatant hardening of \emph{el Pleno} upon being forced toward its limiting deformation rate.

\textbf{Recovery of the relativistic momentum and energy.} The momentum density is the $T^{0i}$ component of the tensor of F1, $T^{0i} = \rho_P c^2\ell_P^2\,(1/c^2)\dot S\,\partial_i S$; for the contracted profile, $T^{0i} = -(v/c^2)\gamma^2\,\rho_P c^2\ell_P^2(\partial_\xi S)^2$. Integrating over the volume, which contracts as $\mathrm{d}V = \mathrm{d}V_0/\gamma$, and defining the rest mass of the soliton by its proper energy $mc^2 \equiv \int T^{00}(0)\,\mathrm{d}V_0$:


\begin{equation}
p = \int T^{0i}\,\mathrm{d}V = \gamma^2 v\cdot\frac{1}{\gamma}\cdot m = \gamma m v\,, \qquad
E = \int T^{00}\,\mathrm{d}V = \gamma^2\cdot\frac{1}{\gamma}\cdot mc^2 = \gamma m c^2\,,
\end{equation}

which satisfy the invariant $E^2 - (pc)^2 = (mc^2)^2$ by construction. The rest case $v=0$ returns $E = mc^2$ (\S{}I.3.2). The \textbf{light barrier} is thus the \textbf{jamming barrier} of \emph{el Pleno}: $v\to c \Leftrightarrow x\to 1 \Leftrightarrow n\to\infty$.

\textbf{Epistemic classification.} The factor $(1-v^2/c^2)$ from F1 (E49) and the divergence of the resistance as $\gamma^2$ (E50) are \epi{DERIVED} (exact algebra plus A3). The identification $x = v^2/c^2$ is \epi{DERIVED} (consistency between the two readings of the same medium). The recovery of $p=\gamma mv$ and $E=\gamma mc^2$ is \epi{SLD} (schematic integration, pending the coefficients of the exact solitonic profile). The interpretive thesis ``relativity is the dilatant rheology of \emph{el Pleno}'' is \epi{SLD}: an equivalent reading of the already-derived covariance (A3/LET), not a second derivation.


\subsubsection{\texorpdfstring{III.9.5-bis --- The spatial-dissipative face: Stokes' law and the three-wall hierarchy}{III.9.5-bis - Stokes law and the three-wall hierarchy}}

\textbf{The three-wall hierarchy.} The Lorentz factor (\S{}III.9.5) is the \emph{temporal-conservative} face of the dilatant rheology; the medium opposes change by three mechanisms, one per axiomatic bound, which share the same flow index $n(x)=1/(1-x)$ evaluated at three distinct fractions:
\begin{center}
{\small
\begin{tabular}{lllll}
\toprule
\textbf{Axis} & \textbf{Bound} & \textbf{Ceiling} & \textbf{Mechanism} & \textbf{Manifestation} \\
\midrule
Temporal & A6 & $c$ & conservative dilatant & inertia / relativity ($\gamma^2 = n(v^2/c^2)$) \\
Spatial & A6 & $1/\sigma_\text{min}=\sqrt{20}/\ell_P$ & dissipative dilatant & ultraviolet Stokes \\
Amplitude & A2 & $S_{\min}$ & zero-point elastic & vacuum rigidity, threshold $\tau_0$ \\
\bottomrule
\end{tabular}}
\end{center}
The first two diverge by \emph{saturation} ($n\to\infty$ at the hard bound A6); the third by \emph{zero point} ($V_q\to\infty$ at $S\to 0$, A2), with a Bingham-type yield threshold $\tau_0 = (3/4)/(S_{\min}^\text{estr})^3\,\rho_P c^2 \approx 1.984\,\rho_P c^2$ (\S{}III.9.4). The three are the finiteness of energy that A1 imposes, seen on three axes ---temporal, spatial and amplitude. \epi{DERIVED} each piece; \epi{SLD} the unified organization.

\textbf{Why Stokes is not the law of the motion of bodies.} The classical Stokes law $F=6\pi\eta r v$ describes a rigid external object dragging through a fluid. Naively transferred to el Pleno (with $\eta_\text{Pleno}\approx 2.56\times10^{69}$ Pa$\cdot$s) it would predict an absurd drag on any moving body. The false premise is that the motion of a body is the translation of its material through the medium: in el Pleno the body \emph{is} the medium in a certain pattern. Moving a pattern of $S$ costs three distinct costs: non-relativistic inertia (kinetic of F1, $\tfrac12 Mv^2$, conservative), dilatant stiffening (the rheology of \S{}III.9.5, $\gamma Mc^2$, which diverges at $c$ and \emph{is} special relativity), and viscous dissipation (flexural, appreciable only if the profile varies at scale $\sim\ell_P$). The resistance to accelerate a macroscopic body is the dilatant stiffening, not the viscous drag: for a smooth profile ($L/\ell_P\sim10^{34}$) the dissipation is $\sim10^{-70}$ at any velocity. Ordinary motion costs kinetic energy, not drag.

\textbf{Stokes emerges from F1 in six links.} The Stokes regime ---linear overdamped viscous flow--- emerges from F1 as the behavior of el Pleno-ground state against the slow motion of a defect with an abrupt gradient. The starting point is the equation of motion of the modulus (E6), exact and conservative:
\begin{equation}
\underbrace{\rho_P c^2\ell_P^2\,\Box S}_{\text{(A) kinetic}} - \underbrace{\rho_P c^2\ell_P^2\,S(\partial_\mu\theta)(\partial^\mu\theta)}_{\text{(B) gauge}} - \underbrace{\frac{\rho_P c^2\ell_P^4}{20}\Box(\Box S)}_{\text{(C) flexural}} + \underbrace{V'(S) + V_q'(S)}_{\text{(D)+(E) potential+IR}} = 0. \tag{E-St-1}
\end{equation}
Since E-St-1 comes from a Lagrangian, it contains no first-order dissipation in $\partial_t$; Stokes is obtained by controlled reduction, link by link.

\emph{Link 1 --- the dissipative term is not postulated: it emerges from the flexural.} The term (C) introduces the Lee-Wick pole at the scale $\sqrt{20}/\ell_P$ (\S{}III.3.4). On integrating that fast mode over time scales $\gg\tau_\Pi$, its residual effect on the slow mode is a Maxwell-type memory constitutive relation,
\begin{equation}
\left(1 + \tau_\Pi\,\partial_t\right)\sigma = \mu_\text{Gent}\,\varepsilon, \tag{E-St-2}
\end{equation}
whose first-order expansion, $\sigma \approx \mu_\text{Gent}\,\varepsilon - \mu_\text{Gent}\,\tau_\Pi\,\partial_t\varepsilon$, contains the viscous correction $\eta\,\partial_t\varepsilon$ with
\begin{equation}
\eta = \mu_\text{Gent}\,\tau_\Pi, \tag{E-St-3}
\end{equation}
the viscosity as finite memory of the medium. It is not introduced by hand: it is the footprint of the Lee-Wick pole of the flexural.

\emph{Link 2 --- the spring, from (D)+(E).} At the minimum, $V'(S_{\min})+V_q'(S_{\min})=0$. Linearizing $S = S_{\min}+\delta S$ with $\delta S\ll S_{\min}$,
\begin{equation}
V'(S)+V_q'(S) \approx \big[V''(S_{\min})+V_q''(S_{\min})\big]\,\delta S \equiv \kappa\,\delta S, \qquad \kappa = \xi\,\rho_P c^2, \tag{E-St-4}
\end{equation}
with $\xi = V''_\text{eff}(S_{\min})/(\rho_P c^2) = 19.69$. This provides the effective restoring pressure $\nabla P_\text{eff} = \kappa\,\nabla\delta S$.

\emph{Link 3 --- the direct flexural decouples.} The ratio between the direct term (C) and the kinetic one (A) is $(\text{C})/(\text{A}) = (1/20)(\ell_P/L)^2$, which for a profile of scale $L\gg\ell_P$ is $\sim 10^{-70}$: the direct flexural is negligible as a term; what survives of it is its \emph{integrated} form, the viscosity (E-St-3).

\emph{Link 4 --- the kinetic splits.} With $\Box = (1/c^2)\partial_{tt} - \nabla^2$, the term (A) separates an inertial part $(1/c^2)\partial_{tt}$ and an elastic one $-\nabla^2$. In the overdamped regime ($\text{Re} = \rho_\text{exc} U L/\eta \ll 1$), the inertial part $\partial_{tt}$ is negligible against the viscous one; the Laplacian $\nabla^2$ survives.

\emph{Link 5 --- the phase motor.} The term (B), the phase flow $(\partial\theta)^2$, acts as a velocity field. The Goldstone mode of the phase ($\Box\theta = 0$) is incompressible, whence $\text{div}(v) = 0$.

Assembling the five links ---restoring pressure (E-St-4), viscosity (E-St-3), elastic Laplacian (link 4) and incompressible velocity field (link 5)--- the equation of motion of the slow mode is the Stokes equation of el Pleno:
\begin{equation}
\nabla P_\text{eff} = \eta\,\nabla^2 v, \qquad \text{div}(v)=0, \tag{E-Stokes}
\end{equation}
with each coefficient traced to F1: $\eta = \mu_\text{Gent}\,\tau_\Pi$ (flexural via Lee-Wick), $\kappa = \xi\,\rho_P c^2$ (potential) and $v$ (phase flow, U(1) sector). \epi{DERIVED} in its validity window.

\textbf{The Stokes regime is ultraviolet and narrow.} The relative dissipation of translating a profile of $S$ is $R = e_\text{dis}/e_\text{cin} \approx [1/(10 S_{\max}^2)]\,(\ell_P/L)^2/S^2$, \emph{independent of velocity}: it depends only on the sharpness of the gradient $\ell_P/L$, not on how fast or how large the body is. The boundary $R\sim1$ is at $L\approx 0.30\,\ell_P$, of the order of the Planck grain: a body with a Planckianly abrupt edge would dissipate, a smooth one would not. The Stokes window requires four simultaneous conditions, all from F1 ($U\ll c$, $\delta S\ll S_{\min}$, $\text{Re}\ll1$, $L\gtrsim\ell_P$): it is the intermediate strip between relativistic dilatancy (as $U\to c$) and the rigidity of the IR wall (as $S_{\min}$ is approached).

\textbf{The existence condition of the radial mode.} The radial mode of the modulus (homogeneous sector $k=0$) inherits from the flexural a fourth-order dynamics in time. Linearizing E6 in $S = S_{\min}+\delta S$ with $\delta S\propto e^{-i\omega t}$, the flexural operator $(\ell_P^2/20)\Box(\Box\,\cdot)$ contributes a term $\propto\omega^4$ and the potential one $\propto\xi$, giving the radial dispersion relation
\begin{equation}
\frac{\ell_P^2}{20}\left(\frac{\omega}{c}\right)^4 + \left(\frac{\omega}{c}\right)^2 - \frac{\xi}{\ell_P^2} = 0. \tag{E-rad-1}
\end{equation}
Multiplying by $20\,\ell_P^4$ and introducing the dimensionless variable $W\equiv(\omega\ell_P/c)^2$, the characteristic equation is
\begin{equation}
W^2 + 20W - 20\xi = 0, \tag{E-rad-2}
\end{equation}
whose physical root (the positive one) is $W_+ = -10 + \sqrt{100+20\xi}$, whence the propagating mass of the radial mode is
\begin{equation}
m_\text{prop} = \sqrt{W_+}\;E_P = \sqrt{-10 + \sqrt{100+20\xi}}\;E_P \approx 3.4959\,E_P, \tag{E-rad-3}
\end{equation}
against the elastic mass $m_S = \sqrt{\xi}\,E_P \approx 4.4373\,E_P$ (the bare curvature of the well). The condition $\xi < 20$ guarantees a real oscillating radial mode (underdamped side): with $\xi = 19.69 < 20$ it holds by a margin of $1.5\%$. \epi{DERIVED}. The sign $-20\xi$ in (E-rad-2) is the correct one: at a stable minimum $V''>0$ is restoring and the mode oscillates. A characteristic with sign $+20\xi$ would give discriminant $400+80\xi$ with root $W_+ = -10+\sqrt{100-20\xi}$ imaginary for $\xi>5$ ---an intrinsically complex false pole with a spurious factor--- which contradicts the stability of the minimum and is discarded: the dissipation of the medium does not come from a complex mode but from the genuine dissipative sector ($\tau_\Pi$ of the flexural).

\textbf{Convergence of the $\sqrt{20}$.} The number $1/\sqrt{20}\approx0.2236$ appears identical from two independent sectors: the dissipative one ($\tau_\Pi/(\ell_P/c) = \sqrt5/10 = 1/\sqrt{20}$) and the flexural one ($\sigma_\text{min}/\ell_P = 1/\sqrt{20}$, the minimum localizability). It is the angular word $20 = 40/2$ of the Laplacian of the diamond lattice declined in two sectors ---one geometry, multiple declensions. \epi{DERIVED}.

\textbf{Synthesis.} Stokes is not an independent law: it is the spatial-dissipative face of the same dilatant rheology whose temporal-conservative face is the Lorentz factor (\S{}III.9.5). Both hang from the hard bound A6; the Klauder IR wall (A2) adds the third face, elastic, which makes the cold vacuum a solid. \epi{DERIVED} the derivation of Stokes from F1; \epi{SLD} the unification of the three faces under the two bounds.

\textbf{The cold vacuum is not a fluid: it is an elastic solid.} In the relaxed regime, dark energy has equation of state $w=-1$ (\S{}III.7.5), so that $\rho + p = 0$ and the volumetric entropy density of el Pleno-ground state is \emph{null}: $s = (\rho+p)/T = 0$. The relaxed vacuum is a pure coherent condensate, of minimum entropy ---an elastic solid, not a viscous fluid---, which independently reinforces the DE-Curtain reading of \S{}III.7.5 from thermodynamics. \epi{DERIVED} (consequence of $w=-1$).

\subsubsection{\texorpdfstring{III.9.5-ter --- The thermal statistics of \emph{el Pleno} by configuration counting}{III.9.5-ter - The thermal statistics of el Pleno by configuration counting}}

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 (E17, \S{}II.5.2) has kinetic term $\tfrac12\rho_P c^2\ell_P^2 S_{\min}^{2,\text{estr}}(\partial\theta)^2$ and flexural term $-(\rho_P c^2\ell_P^4/40)S_{\min}^{2,\text{estr}}(\Box\delta\theta)^2$. Decomposing the perturbation into Fourier modes, $\delta\theta(\mathbf{x},t)=\sum_\mathbf{k} q_\mathbf{k}(t)\,e^{i\mathbf{k}\cdot\mathbf{x}}$, each $\mathbf{k}$ gives an 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 = \rho_P\ell_P^2 S_{\min}^{2,\text{estr}},
\qquad
K_k = \rho_P c^2\ell_P^2 S_{\min}^{2,\text{estr}}\left(k^2 + \frac{\ell_P^2 k^4}{20}\right),
\tag{E-BE-1}
\end{equation}
whence $\omega_k^2 = K_k/M_k = c^2k^2[1+(k\ell_P)^2/20]$, which is the dispersion relation already derived for the phase sector (E23, \S{}II.5.4), with its ultraviolet correction of coefficient $1/20$ and its luminal limit in the infrared. The phase field is therefore a set of harmonic oscillators, and the levels of each mode are $E_m = m\,\hbar\omega_k$ with $m=0,1,2,\dots$

\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,
\[
\Omega(q,N) = \frac{(q+N-1)!}{q!\,(N-1)!},
\]
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, \S{}V.8). Applying Stirling in the regime $q,N\gg1$ and writing the mean occupation $n\equiv q/N$,
\[
\ln\Omega \simeq (q+N)\ln(q+N)-q\ln q-N\ln N
= N\big[(1+n)\ln(1+n)-n\ln n\big],
\]
so that the entropy per oscillator is
\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 $E=q\,\hbar\omega$ and its entropy $S=N k_B\,\sigma(n)/k_B$, so that $\partial S/\partial E = (1/\hbar\omega)\,\partial\sigma/\partial n$. Differentiating (E-BE-2),
\[
\frac{\partial}{\partial n}\big[(1+n)\ln(1+n)-n\ln n\big] = \ln\frac{1+n}{n},
\]
and the equilibrium condition $\ln[(1+n)/n]=\hbar\omega/k_BT$ inverts to
\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{The 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 $x=\hbar ck/k_BT$,
\begin{equation}
\frac{s}{k_B} = \frac{g_*}{2\pi^2}\int_0^\infty k^2\,\sigma_k\,dk
= \frac{2\pi^2}{45}\,g_*\left(\frac{k_BT}{\hbar c}\right)^{3},
\tag{E-BE-3}
\end{equation}
which is the Stefan-Boltzmann law with its coefficient $2\pi^2/45$. The numerical check is 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; the result, $0.877298168985721$ for $g_*=2$, agrees with $(2\pi^2/45)g_*$ to all fifteen digits of double precision. \epi{DERIVED}, with no free parameter.

\begin{tcolorbox}[derivado,title={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 (\S{}III.2.2, E36); 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 (\S{}III.2.2): 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). With this, the thermodynamics of the mode gas belongs to the medium: its entropy is $\ln\Omega$ of the configurations of \emph{el Pleno}.
\end{tcolorbox}

\subsubsection{\texorpdfstring{III.9.6 --- Consequences: viscosity bound and horizon as a sonic surface}{III.9.6 - Consequences: viscosity bound and sonic horizon}}

\textbf{The thermal el Pleno and the viscosity bound.} In high-excitation regimes (primordial bounce, black-hole interior, early universe) the transverse modes of el Pleno (TT mode, massless, two polarizations, $g_*=2$; Result D2, \S{}III.3) form a radiation-type gas of quasiparticles, with entropy density $s = (2\pi^2/45)\,g_*\,k_B\,(k_B T/\hbar c)^3$ is obtained from the count of configurations of the modes of F1 (E-BE-3, \S{}III.9.5-ter). For any fluid, the ratio between shear viscosity and entropy density is bounded below by the universal quantum limit of Kovtun-Son-Starinets,
\begin{equation}
\frac{\eta}{s} \;\geq\; \frac{\hbar}{4\pi k_B}\,, \tag{E51}
\end{equation}
a bound saturated in strongly coupled theories with a gravitational dual (conformal) and approached in real plasmas like the quark-gluon one ($\sim 2$--$2.5$ times the bound). The relaxed cold vacuum, by contrast, has $w=-1$, hence $\rho+p=0$ and entropy density $s=(\rho+p)/T=0$: it is not a fluid but an elastic solid (see above). \epi{DERIVED}. The ratio $\eta/s$ of the thermal el Pleno ---the gas of TT modes in the high-excitation regime--- is bounded on two sides, one universal and one structural native, without needing the full collisional kinetics.

\textbf{The phase of the ratio.} The pairing of the viscosity of the saturated solid $\eta_\text{Pleno}=\mu_\text{Gent}\,\tau_\Pi$ (\S{}III.1.6) with the entropy of the TT radiation gas is inconsistent: $\eta_\text{Pleno}$ is the shear viscosity of the \emph{saturated elastic solid} (Gent shear times the relaxation time of the Lee-Wick pole), while $s$ is that of the TT \emph{radiation gas}, two distinct phases of the medium. The ratio so formed, evaluated at the saturation temperature, gives $\eta/s\approx 0.60\,\hbar/4\pi k_B$, below the universal bound E51 ---impossible for a physical fluid, which marks the pairing as incorrect. The physical ratio 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 in the relaxation-time approximation (RTA, Boltzmann), the shear viscosity and the entropy density are
\begin{equation}
\eta = \frac{1}{5}(\varepsilon + P)\,\tau_\text{col},
\qquad
s = \frac{\varepsilon + P}{T},
\end{equation}
with $\varepsilon=\rho_\text{rad}$ the energy density and $P=\rho_\text{rad}/3$ the pressure of the TT gas. 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{Maximum temperature by A6 saturation.} The maximum physical temperature of the TT gas is not free: it is the one at which its energy density reaches the maximum density of el Pleno (A6 saturation), the same ceiling $\rho_{\max}$ of the bounce. With the radiation density derived from the same count (\S{}III.9.5-ter, $g_*=2$),
\begin{equation}
\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\,\rho_P c^2,
\end{equation}
and the native identity $\hbar=\rho_P c\,\ell_P^4$ (E36), one solves
\begin{equation}
k_B T_{\max} = \Big(30\,S_{\max,\text{geom}}^2\,\rho_P c^2\,(\hbar c)^3/\pi^2 g_*\Big)^{1/4}
= \frac{\sqrt[4]{5}\;6^{7/8}}{3\sqrt\pi}\,\rho_P c^2\,\ell_P^3 .
\end{equation}

\textbf{Two-sided bound.} The lower bound is the universal E51, $R\equiv(\eta/s)/(\hbar/4\pi k_B)\geq 1$, valid for every fluid. The upper bound is structural and native: no process of el Pleno relaxes below the grain scale $\ell_P$, so that $\tau_\text{col}$ is bounded below by the ultraviolet regulation time $\tau_\Pi=\ell_P/\sqrt{20}\,c$ (\S{}II.5.4). Evaluating $\eta/s=\tfrac15\tau_\Pi T$ at $T_{\max}$:
\begin{equation}
R(\tau_\Pi, T_{\max}) = \frac{2\cdot 5^{3/4}\,6^{7/8}\sqrt\pi}{75} \approx 0.758 \;<\; 1.
\end{equation}
That the UV floor $\tau_\Pi$ gives a value below E51 means that the fastest conceivable relaxation time is incompatible with the universal bound: the bound E51 forces the real relaxation time to be slower than $\tau_\Pi$,
\begin{equation}
\tau_\text{col} \;\geq\; \frac{\tau_\Pi}{0.758} \approx 1.319\,\tau_\Pi .
\end{equation}
The upper end of the window is fixed by the natural thermal time $\tau_T=\hbar/k_B T_{\max}$, which satisfies $\tau_T/\tau_\Pi\approx 3.32$, corresponding to $\eta/s\approx 2.5\,$(E51). The ratio of the TT gas of el Pleno is bounded in a narrow window:
\begin{equation}
\boxed{\;1 \;\lesssim\; \frac{\eta/s}{\hbar/4\pi k_B} \;\lesssim\; 2.5\;}
\end{equation}
that is, $\tau_\text{col}$ of the order of $\tau_\Pi$ up to an $O(1)$ factor: exactly the strip of the quark-gluon plasma, the most perfect fluid known experimentally. \epi{SLD}: the lower bound is the universal KSS bound; the upper bound is the native ultraviolet relaxation scale $\tau_\Pi$ evaluated at the A6 saturation temperature; both combine without invoking the full collisional kinetics. The exact value within the window requires the precise collisional relaxation time of the mode gas, which remains a bounded thread.


\textbf{The sub-bound is a pure number: every constant self-annihilates.} The value $0.758$ depends on no measuring rod. Starting only from the three primary magnitudes, with $\hbar=\rho_P c\,\ell_P^4$ (E36), $\tau_\Pi=\ell_P/(\sqrt{20}\,c)$ and the saturation temperature
\[
k_B T_{\max} = \frac{5^{1/4}\,6^{7/8}}{3\sqrt\pi}\,\rho_P c^2\ell_P^3\,,
\]
the ratio $R=(4\pi/5)\,\tau_\Pi\,k_B T/\hbar$ evaluated at $T_{\max}$ leaves
\begin{equation}
R(\tau_\Pi,T_{\max}) = \frac{2\cdot 5^{3/4}\,6^{7/8}\sqrt\pi}{75}\,,
\tag{E61}
\end{equation}
with no $\rho_P$, no $\ell_P$, no $c$ and no $k_B$: the three primaries cancel by dimensional analysis and $k_B$ self-annihilates, because it enters twice ---once in writing temperature as energy per degree of freedom and once in the denominator of the bound--- and measures nothing of \emph{el Pleno} that is not already in the energy. Verified symbolically: zero free symbols. Its factors are words of the medium: the $5$ and the $\sqrt\pi$ of the angular family, the $\sqrt6$ of the tetrahedral bond.

\textbf{The Kovtun-Son-Starinets bound is the no-vacuum axiom read as transport.} \epi{SLD}. In a dilute gas nothing prevents $\eta/s$ from falling without limit: it suffices to dilute, because the quasiparticles can be isolated from one another and momentum transport shuts off. In \emph{el Pleno} that is impossible by construction. The medium is always full (A1) and dissipation is not collision of particles across an intervening vacuum but mode-to-mode transfer of the non-linear term of F1: no mode can be isolated because there is no region without medium. The existence of a positive floor for $\eta/s$ is then the impossibility of vacuum written in the language of transport. Where standard physics has a conjecture of holographic origin, the programme offers an axiomatic foundation. The scope of the claim is precise: what A1 forces is that a positive floor \emph{exists}, and the apparatus of the medium further fixes its value in the TT regime; the coefficient $1/(4\pi)$ of the universal bound is not derived from A1. And the thermal \emph{Pleno} does not saturate the bound ---it respects it with margin, within the window $[1,2.5]$---, so that the statement concerns the existence of the floor, not its saturation.

\textbf{The sign of the deviation between regimes is derived.} \epi{DERIVED}. Since dissipation is mode-to-mode transfer over the spectrum that F1 provides, a sector whose modes have a gap $\Delta$ cannot transfer momentum to states below $\Delta$: that channel is closed by energy conservation over the derived spectrum, relaxation is slower and the ratio rises. Hence an inequality with no parameters,
\[
\left.\frac{\eta}{s}\right|_{\text{with gap}} \;>\; \left.\frac{\eta}{s}\right|_{\text{without gap}}\,,
\]
which orders the regimes before any datum is consulted: the TT mode gas, massless and with open channels, must fall below a cold configuration populated by the gapped mode.

\textbf{The multi-domain signature.} Under A1 there is only one medium, so that the quark-gluon plasma, a strange metal and a unitary Fermi gas are the same \emph{Pleno} in three regimes. All three saturate the same order of magnitude of the bound, and none violates it, across twenty-one orders of magnitude in temperature:
\begin{center}\small
\begin{tabular}{L{3.9cm}L{2.6cm}L{6.4cm}}
\toprule
\textbf{System} & \textbf{Temperature} & \textbf{Transport observable} \\
\midrule
Quark-gluon plasma (RHIC/LHC) & $\sim10^{12}$ K & $\eta/s \sim 1$--$2.5$ in units of the bound \\
Strange metals (cuprates) & $\sim10^{2}$ K & Planckian dissipation, $\alpha = 1.2\pm0.4$ \\
Unitary Fermi gas ($^6$Li) & $10^{-9}$--$10^{-6}$ K & $(\eta/s)_{\min}\approx 5$ in units of the bound \\
\bottomrule
\end{tabular}
\end{center}
The specific coefficient of each regime is not universal, and the programme does not claim it: what is universal are the bound, the order of magnitude and the sign. The Fermi gas above the ceiling of the TT regime is the confirmation of the preceding inequality ---it populates the gapped mode---, not an anomaly. It is a signature that unit algebra cannot manufacture: it requires one and the same medium to order correctly three systems separated by twenty-one orders of magnitude in temperature and studied with instruments that share nothing.

\textbf{The event horizon is the sonic surface of el Pleno.} The propagation velocity of the modes of the medium is $c$ (TT mode, Result D2, \S{}III.3). At the horizon of a black hole the escape velocity equals $c$; therefore the horizon is exactly the surface where the flow of the medium reaches its speed of sound ---the acoustic-gravity analogue of Unruh~\cite{barcelo}, here exact and not analogical. The Mach number of el Pleno is $\mathrm{Ma} = v/c$, and the impossibility of free supersonic flow ($\mathrm{Ma}>1$) is the same bound as the light barrier and the dilatant jamming. The Hawking radiation is read as radiation from the sonic surface of the medium. \epi{SLD}: the identity $v_\text{sound} = v_\text{escape} = c$ is exact; the acoustic reading of the horizon is an interpretation consistent with the black-hole physics of the program (\S{}II.6, $M_{\min}$).

\subsubsection{III.9.7 --- Circles of confirmation and synthesis of the section}

The reading of \emph{el Pleno} as a Herschel-Bulkley dilatant fluid does not introduce new physics: it \emph{reorganizes} and \emph{confirms} results already derived, closing circles between domains that the manuscript treated separately.


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{With \S{}III.1.8 (continuous media):} it completes the constitutive card of the linear regime (Maxwell) with the non-linear rheology (dilatant Herschel-Bulkley). The medium is elastic at low rate and dilatant under forcing.
  \item \textbf{With A3/LET (\S{}III.3):} the Lorentz factor is the dilatant response; the light barrier is the jamming. An equivalent reading of the covariance, not a second derivation.
  \item \textbf{With the A6 hard bound:} the divergence $n\to\infty$ at $\dot\gamma\to c/\ell_P$ \emph{is} the mechanism of the maximum-density bound and of the absence of singularities.
  \item \textbf{With \S{}III.7.5 (EO-Backdrop):} $s=0$ in the relaxed vacuum reinforces from thermodynamics that dark energy is a structural property ($w=-1$), not a dynamical fluid.
  \item \textbf{With \S{}II.5 (black holes):} the horizon as a sonic surface connects with the structural minimum mass $M_{\min}$ and the regular core.
\end{itemize}

\emph{El Pleno} emerges as a quasi-perfect Herschel-Bulkley fluid ($\eta/s$ of the order of the KSS bound in its thermal regime), Newtonian at rest and dilatant under forcing, whose sonic barrier is simultaneously the relativistic limit, the event horizon and the hard density bound. Three domains ---relativity, black holes and structural bound--- turn out to be three manifestations of the same rheology.


\medskip\hrule\medskip

\subsubsection{\texorpdfstring{III.9.8 --- Incompressible Navier-Stokes as the hydrodynamic limit of F1}{III.9.8 - Incompressible Navier-Stokes as the hydrodynamic limit of F1}}

The decomposition $\Psi = S\,e^{i\theta}$ translates the dynamics of F1 into fluid variables, with identifications fixed by the conserved quantities:
\begin{equation}
\rho \equiv \rho_P\,S^2 \quad\text{(density)}, \qquad \mathbf v \equiv c^2\ell_P^2\,\nabla\theta \quad\text{(velocity)}.
\end{equation}
The coefficient $c^2\ell_P^2$ of the velocity is fixed by the Noether U(1) current (E7c, \S{}II.2): its spatial component $J^i = \rho_P c^2\ell_P^2 S^2\,\partial^i\theta$ equals the mass flow $\rho\,v^i = \rho_P S^2\,v^i$, whence $v^i = c^2\ell_P^2\,\partial^i\theta$. The flow is irrotational in the pure-phase sector ($\mathbf v = \nabla\phi$, $\phi = c^2\ell_P^2\theta$); the vorticity resides in the topological defects of $\theta$.

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

\textbf{Euler from the modulus equation.} The phase dynamics, in the IR limit, is a Hamilton-Jacobi equation for the velocity potential $\phi$:
\begin{equation}
\partial_t\phi + \tfrac12|\nabla\phi|^2 + \frac{\mu(\rho)}{\rho_P} + Q = 0, \qquad Q = -\frac{c^2\ell_P^2}{2}\frac{\nabla^2\sqrt\rho}{\sqrt\rho},
\end{equation}
with $Q$ the 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\;}\qquad \epi{DERIVED}
\end{equation}
with barotropic pressure $p(\rho)$ derived from $V+V_q$ (Gibbs-Duhem relation) and the quantum Bohm correction.

\textbf{The origin of the viscous term.} The Madelung derivation is conservative; the viscous term is not added by hand: its origin is in 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
\begin{equation}
\partial^i(\nabla^2\theta) = \nabla^2(\partial^i\theta) = \frac{1}{c^2\ell_P^2}\nabla^2 v^i.
\end{equation}
The current of el Pleno has the form of a diffusive current, $\mathbf J = \rho\mathbf v + (\text{coef})\nabla^2\mathbf v$; a current $\propto\nabla^2\mathbf v$ in the momentum transport is a viscous term. Closing the coefficient with $\tau_\Pi$ (Maxwell-Kubo), the viscosity is the already derived $\eta_\text{Pleno} = \mu_\text{Gent}\,\tau_\Pi$ (\S{}III.1.6). The flexural term $(\Box S)^2$ of F1 ---the ultraviolet regulator--- is the physical source of the infrared viscosity: UV regulation and IR dissipation are the same term at distinct scales.

\textbf{The dilute limit.} Three reductions controlled by $x = S^2/S_{\max,\text{geom}}^2$ (E46): (a) $Q\to 0$ at scales $L\gg\ell_P$ (the grain is invisible to the macroscopic flow); (b) $n(x)\to 1$ as $x\to 0$, the Newtonian regime (E47, \epi{DERIVED}), which provides $\eta\nabla^2\mathbf v$; (c) Mach number $\to 0$: $\rho\approx$ const, hence $\nabla\!\cdot\!\mathbf v=0$. The result is
\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
\begin{equation}
\nu = \frac{\eta_\text{Pleno}}{\rho_P S^2} = \frac{\sqrt5\,c\,\ell_P}{10\,S^2\,S_{\max,\text{geom}}^2},
\end{equation}
consistent with the TT kinematic viscosity of the corpus $\nu_\text{TT} = (1/\sqrt{20})\,c\,\ell_P$ (\S{}III.1.6, E-eta-TT) in the pure-shear sector. \epi{SLD}: continuity, Euler, Bohm and the Newtonian limit are \epi{DERIVED}; the identification of the viscous term with the flexural (verified symbolically) is \epi{SLD}; by Rule 19, the ensemble is \epi{SLD}. Navier-Stokes appears as an effective limit of F1, not as an origin.


\subsubsection{\texorpdfstring{III.9.9 --- Global regularity of the flow of el Pleno}{III.9.9 - Global regularity of the flow of el Pleno}}

\begin{tcolorbox}[notabox]
\textbf{Scope.} This result is not the proof of the Clay millennium problem, which concerns pure Navier-Stokes (constant viscosity, no kinematic bound). The flow of el Pleno is not pure Navier-Stokes: it possesses the A6 bound and divergent viscosity. The regularity of the flow \emph{of el Pleno} is established; the regularization comes from el Pleno departing from Navier-Stokes exactly where the latter could blow up.
\end{tcolorbox}

\textbf{The kinematic bound.} The maximum deformation rate is finite (E45): $\dot\gamma\le\dot\gamma_{\max}=c/\ell_P$, with $x=\dot\gamma/\dot\gamma_{\max}\in[0,1)$ always. This bounds the deformation part of the velocity gradient: $\dot\gamma=\sqrt{2S_{ij}S_{ij}}\le c/\ell_P$, with $S_{ij}=\tfrac12(\partial_i v_j+\partial_j v_i)$.

\textbf{The Beale-Kato-Majda criterion and the role of deformation.} The BKM criterion establishes that 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 $\boldsymbol\omega$ (a rigid rotation has $S_{ij}=0$, $\boldsymbol\omega\neq 0$). The resolution is physical: the only route to blow-up is vortex stretching, whose amplification rate is
\begin{equation}
\tfrac12\frac{D|\boldsymbol\omega|^2}{Dt}\Big|_\text{stretch} = \omega_i\,S_{ij}\,\omega_j .
\end{equation}
Only the deformation $S_{ij}$ stretches the vortices. The A6 bound on $S_{ij}$ therefore 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 $\|S\|_\text{op}\le\dot\gamma_{\max}/\sqrt2$:
\begin{equation}
\frac{d|\boldsymbol\omega|}{dt}\le\frac{\dot\gamma_{\max}}{\sqrt2}|\boldsymbol\omega|
\quad\Longrightarrow\quad
|\boldsymbol\omega(t)|\le|\boldsymbol\omega(0)|\,e^{\dot\gamma_{\max}t/\sqrt2}.
\end{equation}
The BKM integral converges for every finite $T$:
\begin{equation}
\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.
\end{equation}
There is no finite-time singularity: the flow is globally regular.

\textbf{Dynamical reinforcement.} The effective Herschel-Bulkley viscosity diverges on approaching the bound: $\eta_\text{eff}\propto d(x/(1-x))/dx = 1/(1-x)^2\to\infty$. This installs a negative feedback: the more the flow concentrates the gradient (approaching $\dot\gamma_{\max}$), the greater the dissipation that brakes it, so that the real growth is sub-exponential. \epi{SLD}: the pieces ($\dot\gamma_{\max}$, $n(x)$, BKM criterion) are derived or standard; the assembly into global regularity is a strong logical deduction.


\subsubsection{\texorpdfstring{III.9.10 --- The singularity under emergent time}{III.9.10 - The singularity under emergent time}}

\begin{tcolorbox}[notabox]
\textbf{Scope.} It is not a proof of the Clay problem in its terms. It shows that, under the PIU definition of time (time as accumulated change), the notion of a finite-time singularity dissolves. It is a reinterpretation of the terms of the problem.
\end{tcolorbox}

\textbf{Time is accumulated change.} PIU derives time, it does not postulate it: the internal change $C\in[0,c]$ has ceiling $c$, and time is read as the amount of accumulated change, measured against the maximum possible change whose scale fixes $c$ (A1, \S{}I; formal definition in \S{}I.8.9-bis). Physical time is not a background parameter: it is made of the change that occurs in the flow.

\textbf{Physical time is the Beale-Kato-Majda integral.} The local change of the fluid state is measured by $\|\nabla\mathbf v\|$. The accumulated physical time is $\tau_\text{phys}\propto\int\|\nabla\mathbf v\|\,dt$, which is identically the integrand of the BKM criterion. Hence, exactly:
\begin{equation}
\boxed{\;\text{blow-up (BKM)}\;\Longleftrightarrow\;\int\|\nabla\mathbf v\|\,dt=\infty\;\Longleftrightarrow\;\tau_\text{phys}=\infty\;}
\end{equation}
A blow-up is, by its own mathematical definition, an event at infinite physical time.

\textbf{Verification.} For the self-similar blow-up $\|\nabla\mathbf v\|=A/(T^*-t)$:
\begin{equation}
\tau_\text{phys}(T)\propto\int_0^T\frac{A\,dt}{T^*-t}=A\ln\frac{T^*}{T^*-T}\xrightarrow{T\to T^*}\infty .
\end{equation}
The gradient diverges at finite coordinate time $T^*$, but the physical time diverges logarithmically: the singularity is never reached in finite physical time.

\textbf{Synthesis.} A blow-up would require, simultaneously, (i) infinite local change ---forbidden by A1, $C\le c$; the A6 bound is A1 plus the grain, $\dot\gamma_{\max}=c/\ell_P$--- and (ii) in finite physical time ---impossible, since $\tau_\text{phys}$ diverges in the singular regime. Both fail for the same reason: change is finite and time is made of change. Infinite change does not accumulate in finite accumulated change. \epi{SLD}, internal to PIU (A1 + emergent definition of time).


\subsubsection{\texorpdfstring{III.9.11 --- Bilateral rheological structure and the critical point $S^*$}{III.9.11 - Bilateral rheological structure and the critical point S*}}

\textbf{The two walls and the two indices.} The modulus lives in $[S_{\min}, S_{\max}]$ between two walls: the Klauder IR wall ($S\to S_{\min}$, from A2) and the Gent hard wall ($S\to S_{\max}$, from A6). The flow index (E47) measures the proximity to the Gent wall:
\begin{equation}
n(x) = \frac{1}{1-x} = \frac{S_{\max}^2}{S_{\max}^2-S^2}, \qquad x = \frac{S^2}{S_{\max}^2}.
\end{equation}
Its mirror, derived by the same procedure (logarithmic slope of the restoring stress) toward the IR wall, is
\begin{equation}
m(\bar y) = \frac{1}{1-\bar y} = \frac{S^2}{S^2-S_{\min}^2}, \qquad \bar y = \frac{S_{\min}^2}{S^2}.
\end{equation}
The term-by-term of $n = 1 + x/(1-x)$: the $1$ is the Newtonian floor (\epi{DERIVED}, survives as $x\to 0$); the $x/(1-x)$ is the dilatant excess, whose denominator $(1-x)$ is the geometric clearance up to the bound ---the same factor that governs the Gent wall, the divergent viscosity and the regularization of the flow (\S{}III.9.9).

\textbf{The critical point $S^*$.} The bilateral index $N(S) = n + m = \tfrac{S_{\max}^2}{S_{\max}^2-S^2} + \tfrac{S^2}{S^2-S_{\min}^2}$ diverges at both walls and has a minimum (minimum rigidity, maximum fluidity) at
\begin{equation}
\boxed{\;S^* = \sqrt{S_{\min}\,S_{\max}}\approx 1.0330\;}\qquad \epi{DERIVED}
\end{equation}
the exact geometric mean of the walls (with $S_{\min}=0.7231$ and $S_{\max}=S_{\max,\text{geom}}=\sqrt{8\sqrt6/9}\approx 1.4756$). At $S^*$ the packing fractions are identical: $x^* = \bar y^* = S_{\min}/S_{\max}\approx 0.4900$ (exact, not an approximation of $1/2$).

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

\textbf{$S^*$ as the crystallization point.} $S^*$ is not a force equilibrium (the minimum of the potential is $S_{\min}$), but the point of minimum rigidity, where the elastic barrier to reorganize the lattice is minimal. A packing transition (compact $\leftrightarrow$ diamond) requires reorganizing the lattice, which costs elastic energy; that barrier is minimal at $S^*$, so that the lattice transition occurs at $S^*$. \epi{SLD}. The transition happens at density
\begin{equation}
\rho(S^*) = x^*\,\rho_{\max} = \frac{S_{\min}}{S_{\max}}\,S_{\max,\text{geom}}^2\,\rho_P \approx 1.067\,\rho_P \approx \rho_P,
\end{equation}
that is, the recrystallization of space occurs on crossing the Planck density.

\begin{tcolorbox}[ecuacionclave,title={Interpretation --- what el Pleno is as a fluid}]
The universe flows like a genuine physical 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 finite rigidity and a maximum deformation rate, the flow of el Pleno never develops the infinities that torment ideal hydrodynamics: a whirl can only sharpen up to the limit that the grain permits, and the viscosity grows without ceiling exactly when it comes closest, braking it. That same medium changes state with temperature, like water: cold it is a perfect solid (no friction, no entropy: the vacuum at rest); extremely hot ---in the bounce, inside a black hole, in the newborn universe--- it is an almost perfect liquid, in the strip of the quark-gluon plasma, the most perfect fluid measured in the laboratory. And its point of maximum fluidity, $S^*$, halfway between the two walls, is where space recrystallizes on compressing to the Planck density. The horizon of a black hole is, in this same reading, the surface where the fluid reaches the speed of sound: nothing escapes not by an imposed barrier, but because the flow becomes supersonic inward.
\end{tcolorbox}


\medskip\hrule\medskip

\subsubsection{\texorpdfstring{III.9.11-bis --- The inversion self-duality: the generator operator and the conformal inversion of the field}{III.9.11-bis - The inversion self-duality: generator operator and conformal inversion of the field}}

The inversion symmetry $I:S\to S_{\min}S_{\max}/S$ of \S{}III.9.11 ---which interchanges the two walls and fixes $S^*$--- admits two deeper readings. The first shows that the operator $I$ not only \emph{detects} the self-duality but \emph{constructs} it from a single wall. The second shows that, over the complete field $\Psi=S\,e^{i\theta}$, the inversion of the modulus is exactly the conformal inversion of the complex plane with respect to the circle $|\Psi|=S^*$. Both are a direct consequence of F1, with no new parameter.

\paragraph*{The inversion 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 $n(u)=m(-u)$ for the indices of the two walls (\S{}III.9.11). Applied to the single geometric wall of F1, the Gent UV wall
\begin{equation}
V_\text{topo}(S) = -\frac{\rho_P c^2}{2}\,\ln\!\left(1-\frac{S^2}{S_{\max,\text{geom}}^2}\right)
\qquad(\text{E10, \S{}II.2}),
\end{equation}
the operator generates its mirror image toward the lower wall:
\begin{equation}
I\!\left(V_\text{topo}\right)(S) = V_\text{topo}\!\left(\frac{S_{\min}S_{\max}}{S}\right)
= -\frac{\rho_P c^2}{2}\,\ln\!\left(1-\frac{S_{\min}^2}{S^2}\right),
\end{equation}
which is an IR wall of Gent form, divergent as $S\to S_{\min}$. Hence, 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\!\left(V_\text{topo}\right)\;},
\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 Language of el Pleno).

\paragraph*{The measured breaking: Klauder is not Gent.} The \emph{physical} potential of F1 is not $V_D$: its lower wall is the Klauder quantum wall (E5), not the geometric IR wall that $I$ generates. For the physical potential to be exactly self-dual, the coefficient of the real IR wall (Klauder, $3/8=0.375$) would have to coincide with that of the generated IR wall (Gent-IR, whose divergence coefficient is $S_{\min}^2=0.5229$). They do not coincide:
\begin{equation}
\frac{S_{\min}^2}{3/8} = \frac{0.5229}{0.375} = 1.394,
\qquad
S_{\min}^2 - \frac{3}{8} = 0.148.
\end{equation}
The ratio $1.394$ does not correspond to any known invariant of the program ($\sqrt2$, $\pi^2/7$, $1/f_c$ were discarded as false friends). This is not a failure but physical information: the difference $0.148$ measures the degree to which the IR wall of el Pleno is \emph{quantum} (Klauder, from A2) and not \emph{geometric} (Gent, from A6). The self-duality would be exact in a universe whose IR wall were purely geometric; that it is Klauder is what breaks it, by a calculable amount. The broken Droste is thus a thermometer of the quantum/geometric duality of the medium: exact at its center $S^*$ and in all the geometric structure, broken in the energetic dynamics by the quantum character of the lower wall. The generalized force $Q(u)=S\,V'(S)$ of the complete physical potential $V=V_q+V_\text{topo}$ is not odd under $u\to-u$ (symbolic verification), confirming that $u\to-u$ is not a symmetry of the dynamical potential of F1. \epi{DERIVED}.

\paragraph*{The phase is not a spectator.} The kinetic term of F1 is $\tfrac12\rho_P c^2\ell_P^2\big[(\partial S)^2 + S^2(\partial\theta)^2\big]$. The factor $S^2$ that weights the phase means that moving the phase costs according to the modulus: rigid where $S$ is large (Gent wall), soft where $S$ is small (Klauder wall). Under the inversion $I$ that weight is not fixed, it inverts: $S^2\to S^{*4}/S^2$. The complete symmetry that leaves F1 invariant is not only $u\to-u$ in the modulus, but a combined transformation ---invert the modulus and simultaneously dilate the phase gradient by $e^{2u}$---, an inversion with phase dilation. Under the Droste the two walls interchange not only the confinement of the modulus but also the rigidity of the phase sector. \epi{DERIVED}.

\paragraph*{The inversion of the modulus is the conformal inversion of the field.} Over the complete field $\Psi=S\,e^{i\theta}$, the inversion of the modulus is exactly the conformal inversion of the complex plane $\Psi$ with respect to the circle of radius $S^*$:
\begin{equation}
\boxed{\;\Psi \;\longrightarrow\; \frac{S^{*2}}{\overline{\Psi}}\;}.
\end{equation}
The verification is direct: 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 ($S\to S^{*2}/S$, the inversion $I$ of \S{}III.9.11 written with $S^{*2}=S_{\min}S_{\max}$) and \emph{preserves} the phase $\theta$. It is the classical spherical-mirror operation ---circle inversion--- in the plane $\Psi$, the same one that in complex-function theory 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}.

\paragraph*{The invariant set is a circle, not a point.} In the reading over the isolated modulus, $S^*\approx 1.0330$ 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. This gives physical meaning to the reading: the plane $\Psi$ is genuinely circular ---radius equal to the modulus $S$, angle equal to the phase $\theta$---, and the inversion reflects each interior point ($S<S^*$, toward the Klauder wall) to an exterior point ($S>S^*$, toward the Gent wall) over the same radius, conserving the phase. The central hole $|\Psi|<S_{\min}$ is the region forbidden by the Klauder wall, where the phase would be undefined at $\Psi=0$ ---exactly what the strict positivity of A2 prevents from being reached (\S{}I.5.6), by the same geometric route that connects positivity with the coherence of U(1) (A5). The modulus-only reading of \S{}III.9.11 is, therefore, the shadow in the modulus of this operation over the complete field. \epi{DERIVED}.

\begin{tcolorbox}[ecuacionclave,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 Klauder (quantum), not the Gent-IR that $I$ would generate, and the difference ---ratio $1.394$, or $S_{\min}^2-3/8=0.148$--- measures how quantum and not geometric that wall is. The self-similarity of el Pleno is bounded by construction: the minimum scale $\ell_P$ of axiom A4 (flexural regulation) 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}

\medskip\hrule\medskip

\subsection{\S{}III.10 --- Synthesis: the phenomenological catalogue as a reading of \emph{el Pleno}}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (synthesis section).
  \item \textbf{Notes:} integrating table phenomenon$\leftrightarrow$configuration of \emph{el Pleno}; distinctive advantage against $\Lambda$CDM.
\end{itemize}
\end{estadoseccion}

\subsubsection{III.10.1 --- The complete phenomenological reading}

The catalogue of configurations of \emph{el Pleno} (constructed in Part II) is read as a repertoire of observed physical phenomena, each with its phenomenological identification of complete derivational rigor: gravity as an emergent macroscopic property, the fundamental constants as structural identities, gravitational waves as transverse tensor modes, electromagnetism as an induced gauge sector ---with the Maxwell equations of the free field derived from phase circulation and the non-abelian Yang-Mills structure from the same mechanism---, the particles as solitons ---with confinement as a $\mathbb{Z}_3$ center obstruction---, cosmology as the evolution of the medium, the dark sectors as configurations of the \emph{Pleno}-ground state and its excitations.


The result is a unified reading: all the observed physical phenomena are configurations of a single physical medium. There are no ontologically distinct entities ---particles, fields, forces, geometry, dark sectors--- but distinct readings of the same \emph{el Pleno}.


\subsubsection{\texorpdfstring{III.10.2 --- Integrating table: observed phenomenon $\leftrightarrow$ configuration of \emph{el Pleno}}{III.10.2 - Integrating table: observed phenomenon <-> configuration of el Pleno}}

The following table maps each observed physical phenomenon to its configuration in the catalogue of \S{}II.8, the section of Part III where its reading is developed, and the epistemic status of the main result:


{\scriptsize\setlength{\tabcolsep}{3pt}
\begin{longtable}{L{2.4cm}L{2.0cm}L{1.2cm}L{4.2cm}L{2.1cm}}
\toprule
\textbf{Observed phenomenon} & \textbf{Configuration of \emph{el Pleno}} & \textbf{Reading} & \textbf{Main result} & \textbf{Status} \\
\midrule\endhead
Gravity / value of $G$ & Kinetic coefficient of F1 + TT mode & \S{}III.1 & $G = c^2/(\rho_P\ell_P^2)$, with $\kappa_G=1$ & \epi{DERIVED} \\
Einstein equations & Induced action over TT mode & \S{}III.1.4 & Structure, sign and scale derived; $\gamma_G = 1$ exact pending the Seeley-DeWitt factor & \epi{SLD} \\
Rigidities of \emph{el Pleno} & Elastic coefficients of F1 & \S{}III.1.6 & $K_\text{cin} = c^4/2G$, $\sigma_\text{Pleno} = F_P/40$ & \epi{DERIVED}/\newline\epi{SLD} \\
Saturation of the medium & Gent potential (bound $S_{\max}$) & \S{}III.1.7 & Hyperelastic medium, effective $T_P$ & \epi{SLD} \\
Planck constant $\hbar$ & Quantization of the phase mode & \S{}III.2.2 & $\hbar = \rho_P c\ell_P^4$, with $\kappa_\hbar=1$ & \epi{DERIVED} \\
Flexural rigidity $\gamma_\kappa$ & Native identity & \S{}III.2.3 & $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40 \approx 7.90\times 10^{-28}$ J$\cdot$m & \epi{DERIVED} \\
Gravitational waves & Transverse tensor mode TT & \S{}III.3 & $\Box h_+ = 0$, $v=c$ exact IR, without dissipation & \epi{DERIVED} \\
GW velocity = $c$ & TT mode of \emph{el Pleno} & \S{}III.3.3 & Satisfies GW170817 (\textasciitilde{}70 orders of margin) & \epi{DERIVED} \\
UV unitarity & Benign regulation pole & \S{}III.3.4 & $M_\text{LW} = \sqrt{20}/\ell_P$, without physical ghosts & \epi{SLD} \\
Photon / electromagnetism & Phase circulation over A/B & \S{}III.4 & Geometric $SU(2)\times U(1)$; massless photon; Maxwell ($8/9$) and non-abelian Yang-Mills & \epi{DERIVED} \\
Charge quantization & Topology of the phase $\theta$ & \S{}III.4.6 & $Q = n\hbar$, $n\in\mathbb{Z}$ & \epi{DERIVED} \\
Electroweak hierarchy & Dimensional transmutation & \S{}III.4.7 & Pole of the renormalization flow $\mu^*\approx 5.4\times 10^{17}$ GeV & \textbf{[cond. SLD]} \\
Quark charges $\pm 2/3, \pm 1/3$ & Borromean cluster & \S{}III.5.2 & $\mathbb{Z}_3$ modular arithmetic & \epi{SLD} \\
Baryon (proton / neutron) & Tripartite Borromean cluster & \S{}III.5.2 & $2+1$ structure, derived charges & \epi{DERIVED} \\
Split $\Delta$-$N$ & Rotational tower & \S{}III.5.4 & $\Delta m = 9(\hbar c)^2/(2m_p R_p^2)\approx 264$ MeV (90\%) & \epi{DERIVED} \\
Proton radius $R_p$ & Scale of the Borromean cluster & \S{}III.5.5 & $d\approx R_p$ consistent to 5\% & \epi{SLD} \\
Hadronic spectrum & Baryonic rotational tower & \S{}III.5.6 & $N, \Delta, N^*, \Delta^*$ reproduced & \epi{DERIVED} \\
Proton stability & Borromean entanglement & \S{}III.5.7 & Non-null Massey invariant & [topol. obs.] \\
Mass of the fundamental soliton & Soliton of \emph{el Pleno} & \S{}III.5.8 & $M_Q = M_P/S_{\max,\text{eff}}\approx 9.98\times 10^{18}$ GeV & \epi{DERIVED} \\
Quark confinement & Fractional-winding vortices & \S{}III.5.3 & $\mathbb{Z}_3$ center obstruction; singlet rule (only integer clusters free) & \epi{DERIVED} \\
Cosmic history & Cosmological regimes & \S{}III.6 & Cascade M$\to$MO$\to$EO, bounce without singularity & \epi{DERIVED} \\
Dark / baryonic matter ratio & Discrete geometry of the substrate & \S{}III.6.3 & $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$, 0.86$\sigma$ & \epi{DERIVED} \\
Inevitability of $\sqrt{3}\pi$ & Forced diamond lattice Z=4 & \S{}III.6.4 & 4 convergent arguments; enumeration of formulas, \S{}III.6.5 & \textbf{[DERIVED under substrate hypothesis]} \\
Dark energy & \emph{Pleno}-ground state (EO-Backdrop) & \S{}III.7.2 & $\Omega_{EO} = f_c\approx 0.6869$, 0.30$\sigma$ & \textbf{[SLD, calibrated numerator]} \\
EO equation of state & \emph{Pleno}-ground state, $T_{\mu\nu}\propto g_{\mu\nu}$ & \S{}III.7.5 & $w=-1$ derived, without rest frame & \epi{DERIVED} \\
DESI signal ($w_0$--$w_a$) & Artefact of $\Lambda$CDM parametrization & \S{}III.6.6 & predicted dissolution in raw observables & \textbf{[SLD]} \\
SN Ia time dilation & Shared factor (null TT mode) & \S{}III.8.2-bis & $b=1$ derived; DES-SN5YR $1.003\pm0.011$, 0.27$\sigma$ & \epi{DERIVED} \\
Background cosmology ($H(z)$, $\Omega_m$, CMB) & Elastic dynamics of the stretch & \S{}III.8 & $\Omega_m=1-f_c$ at 0.3$\sigma$; $H(z)$ fits BAO DESI & \textbf{[DERIVED / SLD]} \\
Non-linear correction of $H(z)$; $S_8$ & response of the phase sector to the potential with walls; phase drag & \S{}III.8.9 & thread: computation pending & [open thread] \\
Value of $S_{\min}$ & Spectrum of the diamond lattice & \S{}III.7.6 & $S_{\min} = 12(\pi^2-4)/\pi^4 = 0.7231$ & \epi{DERIVED} \\
Dark matter & Excited configurations without gauge & \S{}III.7.3 & $\rho_\text{DM} = \rho_P[S^2 - S^2_{\min}]$ & \epi{DERIVED} \\
\bottomrule
\end{longtable}}

The status column reflects the epistemic status of each result. \epi{DERIVED} results: $G$, $\hbar$, $\gamma_\kappa$, gravitational waves, photon, charge quantization, baryon, $\Delta$-N split, hadronic spectrum, soliton mass, cosmological cascade, ratio $\sqrt{3}\pi$, equation of state $w=-1$, value of $S_{\min}$, the matter density $\Omega_m = 1 - f_c \approx 0.3131$ (0.3$\sigma$ Planck, without fitting) and the scale invariance of the form of $H(z)$ (\S{}III.8). \epi{SLD} results: quark charges, proton radius, tension of \emph{el Pleno}, electroweak hierarchy, confinement, the perfect blackbody of the CMB and the nearly scale-invariant spectrum without inflation (\S{}III.8). The dark energy $\Omega_{EO} = f_c$ has a derived structure but one calibrated factor (the numerator $S_{\max,\text{eff}}^2$), which is declared explicitly (\S{}III.7.2). The inevitability of the diamond geometry is \textbf{[DERIVED under the diamond-substrate hypothesis]} by four convergent arguments. Declared open threads: the cosmological background sector is closed in \S{}III.8, and what remains open is the combined analysis BAO+CMB+SNe (confrontation thread F-cosmo, \S{}III.8.9) and the quantitative prediction of $S_8$ ---of lesser scope and better delimited than the original F-cosmo thread---; the threads of the gauge sector (chirality, fermionic masses, generations, anomalies) are inherited by Part V.


\subsubsection{\texorpdfstring{III.10.3 --- The distinctive advantage of PIU-$\Psi$ against $\Lambda$CDM}{III.10.3 - The distinctive advantage of PIU-Psi against LambdaCDM}}

The phenomenological reading of Part III makes explicit the distinctive advantage of the program: \textbf{PIU-$\Psi$ statistically matches the $\Lambda$CDM model using the same degrees of freedom, but its parameters have physical derivations from first principles ---they are not fits.} Where $\Lambda$CDM treats $\Omega_{EO}$ and $\Omega_{MO}/\Omega_M$ as free parameters fitted independently to observation, PIU-$\Psi$ derives them structurally:


\begin{itemize}[leftmargin=1.4em]
  \item $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ from the geometry of the substrate (0.86$\sigma$), entirely derived with zero free parameters.
  \item $\Omega_{EO} = f_c\approx 0.6869$ from the active condensate fraction (0.30$\sigma$), with derived structure and calibrated numerator.
\end{itemize}

In addition, PIU-$\Psi$ generates unique and falsifiable predictions that $\Lambda$CDM cannot make: the scalar/tensor asymmetry with structural suppression of the ratio $r$ and the red$\to$blue tilt transition of the primordial spectrum (testable by CMB-S4/LiteBIRD and Lyman-$\alpha$; \S{}III.8.4, \S{}III.8.8), the complete baryonic rotational tower (P-Hadron-1), the separation of the vortices of the cluster equal to the baryon radius (P-Hadron-2), and the correction $w(z)\neq -1$ comparable with DESI.


\subsubsection{III.10.4 --- Transition to Part IV}

The phenomenological reading of Part III culminated with \emph{el Pleno} as a Herschel-Bulkley dilatant fluid (\S{}III.9). \textbf{Part IV} takes that non-linear rheology and articulates it as the \textbf{dynamical engine} of the crunch-bounce cosmological cycle and of particle formation: how the Gent wall and the dilatancy regulate the rebound, how the post-bounce crystallization forms the three real particles (proton, neutron, electron) with their momentum and conditions, how the same potential gives the three destinies of the gravitational collapse (structure, black hole, Big Bounce), and the principle of the interpretation-independent observable. It is a dynamical synthesis that brings together, under a single mechanism, results already derived in Parts I--III, without new free parameters. \textbf{Part V} then organizes the set of predictions into a systematic catalogue of observational confrontation and articulates where the program is consolidated, where it is falsifiable, and what remains open.


\medskip\hrule\medskip


