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\section{\texorpdfstring{\S{}V.6 --- Indices: variables, equations and figures (Appendices)}{V.6 - Indices: variables, equations and figures (Appendices)}}
\markboth{\S{}V.6 --- Indices (Appendices)}{}
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\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed (indices consolidated for the complete manuscript, Parts I--V)
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none foreseen
  \item \textbf{Notes:} cross-reference section of the manuscript. It contains two subsections: \S{}V.6.1 Index of variables (canonical notation table with symbol, name, dimensions, description, first appearance) and \S{}V.6.2 Index of equations (sequentially numbered list with identifier, content, section of appearance, description). The important equations of the manuscript are numbered consecutively with labels (E1), (E2), (E3)... The section gathers the consolidated catalogue of the complete manuscript (Parts I--V).
\end{itemize}
\end{estadoseccion}

\subsection{\S{}V.6.1 --- Index of variables}

Canonical notation table of the manuscript. The variables listed here are those that appear recurrently in the body of the manuscript; local variables of restricted use to a single section are not included.


{\footnotesize\setlength{\tabcolsep}{3pt}
\begin{longtable}{L{2.2cm}L{2.7cm}L{2.2cm}L{4.0cm}L{1.7cm}}
\toprule
\textbf{Symbol} & \textbf{Name} & \textbf{Dimensions} & \textbf{Description} & \textbf{First appearance} \\
\midrule\endhead
$\Psi$ & Total field of el Pleno & dimensionless & Complete state of el Pleno at each of its points; $\Psi = S \cdot e^{i\theta}$ & \S{}I.1 \\
$E$ & Energy & $[J]$ & Single substance of the universe (A1); finite and conserved. Read by region it is $\rho_P$; by inertia, $M$ & \S{}I.4 \\
$M$ & Mass & $[kg]$ & Energy read as inertia; $E=Mc^2$. Confined fraction of energy & \S{}I.4 \\
$C$ & The change & dimensionless & How much a configuration of energy changes; magnitude prior to the metric; $C\in[0,c]$ & \S{}I.4 \\
$p$ & Momentum & $[kg\cdot m/s]$ & Momentum of the propagating fraction; $p=\gamma M v$; constraint $E^2=(Mc^2)^2+(pc)^2$ & \S{}I.4 \\
$v$ & Velocity & $[m/s]$ & Ratio between propagating momentum and confined energy per unit of $c$; $v/c=pc/E$ & \S{}I.4 \\
$\gamma$ & Lorentz factor & dimensionless & $\gamma=1/\sqrt{1-v^2/c^2}$; partition of the change capacity between branches (\S{}I.4); $n=\gamma^2$ flow index (\S{}III.9) & \S{}I.4 \\
$S$ & Modulus of the field & dimensionless & Structural part of the field, $S \geq 0$ & \S{}I.1 \\
$\theta$ & Phase of the field & dimensionless, $[0, 2\pi)$ & Induced gauge dynamical part, source of conserved charges & \S{}I.1 \\
$\rho_P$ & Fundamental density of el Pleno & $[kg/m^3]$ & Primary structural parameter; substantive property & \S{}I.7 \\
$\ell_P$ & Coherence scale of el Pleno & $[m]$ & Primary structural parameter, minimum grain, UV regulation; substantive property & \S{}I.7 \\
$c$ & Causal structure (propagation velocity) & $[m/s]$ & Primary input; causal structure (not a substantive property, \S{}I.8.9); maximum change capacity (\S{}I.4); velocity of all modes & \S{}I.7 \\
$\gamma_\kappa$ & Flexural rigidity of el Pleno & $[J \cdot m]$ & Coefficient of the flexural term in F1 & \S{}I.7 \\
$G$ & Gravitational constant & $[m^3 kg^{-1} s^{-2}]$ & Structural identity: $G = c^2/(\rho_P \ell_P^2)$ & \S{}I.7 \\
$\hbar$ & Reduced Planck constant & $[J \cdot s]$ & Derived structural identity & \S{}I.7 \\
$V(S)$ & Topological potential of el Pleno & $[J/m^3]$ & Potential component of F1 that sustains stable configurations & \S{}I.7 \\
$V_q^\text{dens}(S)$ & Klauder IR quantum wall & $[J/m^3]$ & Potential component of F1 that regulates the IR regime & \S{}I.7 \\
$S_{\min}^\text{estr}$ & Structural el Pleno-ground state & dimensionless & Minimum-excitation configuration, value $12(\pi^2-4)/\pi^4 \approx 0.7231$ & \S{}I.7 \\
$S_{\max,\text{geom}}$ & Upper critical configuration & dimensionless & $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}\approx 2.1773$ & \S{}I.7 \\
$S^*$ & Rheological critical point & dimensionless & Geometric mean of the walls, $S^*=\sqrt{S_{\min}S_{\max}}\approx 1.0330$; minimum rigidity of the modulus; crystallization point ($\rho\approx\rho_P$); fixed point of the inversion $I$ & \S{}III.9.11 \\
$I$ & Wall inversion operator & dimensionless & $I:S\to S_{\min}S_{\max}/S$; interchanges the two walls ($x\leftrightarrow\bar y$, Gent$\leftrightarrow$Klauder); $I(S^*)=S^*$; over $\Psi$ it is the conformal inversion $\Psi\to S^{*2}/\overline\Psi$ & \S{}III.9.11-bis \\
$V_D$ & Droste self-dual potential & $[\rho_P c^2]$ & $V_D=(1+I)V_\text{topo}$, exactly even in $u=\ln(S/S^*)$; symmetrization of the Gent wall under the projector $P_D=\tfrac12(1+I)$ & \S{}III.9.11-bis \\
$\mathcal{L}_{F1}$ & Lagrangian density of el Pleno & $[J/m^3]$ & F1 --- complete Lagrangian of el Pleno & \S{}I.7 \\
$\xi$ & Dimensionless curvature of the well & dimensionless & $\xi = V''_\text{eff}(S_{\min}^\text{estr})/(\rho_P c^2) \approx 19.69$; gap of the radial mode & \S{}I.8.8 \\
$\sigma_{\min}$ & Localizability scale of el Pleno & $[m]$ & $\sigma_{\min} = \ell_P/\sqrt{20}\approx 0.2236\,\ell_P$; hard grain of localization & \S{}III.2.6 \\
$\xi_\text{loc}(S)$ & Local curvature of the potential & dimensionless & $V''_\text{eff}(S)/(\rho_P c^2)$; dynamical profile of $\xi$; damping regime $=\xi_\text{loc}/20$ & \S{}V.3 \\
$M_\text{Ch}$ & Chandrasekhar limit & $[kg]$ & $\approx 1.456\,M_\odot$; internal flexural-rigidity--gravity balance of el Pleno & \S{}III.2.8 \\
$\mathcal{C}$ & Configuration annulus & dimensionless & Admissible region of $\Psi$ per cell: ring between the two walls with the centre $\Psi=0$ forbidden by A1 & \S{}II.4.7 \\
$\mathcal{A}$ & Area of the annulus & dimensionless & $\pi(S_{\max,\text{geom}}^2-(S_{\min}^\text{estr})^2)=5.197669$; configuration margin of the medium (E60) & \S{}II.4.7 \\
$\sigma(n)$ & Entropy per oscillator & $[k_B]$ & $(1+n)\ln(1+n)-n\ln n$; microcanonical count of the phase modes (E-BE-2) & \S{}III.9.5-ter \\
$\langle n\rangle$ & Equilibrium occupation & dimensionless & $1/(e^{\hbar\omega/k_BT}-1)$, derived from the count and not assumed (E59) & \S{}III.9.5-ter \\
$R$ & Dissipation ratio & dimensionless & $(\eta/s)/(\hbar/4\pi k_B)$; window $1\lesssim R\lesssim 2.5$ in the TT regime, sub-bound $0.757973$ (E61) & \S{}III.9.6 \\
$\kappa_G,\ \kappa_\hbar$ & Contrast coefficients & dimensionless & Auxiliary numbers placed in front of (E3) and (E36) to set up their falsification test; they do \emph{not} appear in F1, and the derivations return them equal to $1$ & \S{}V.2.3-bis \\
\bottomrule
\end{longtable}}

{\footnotesize\setlength{\tabcolsep}{3pt}
\begin{longtable}{L{2.4cm}L{2.9cm}L{1.9cm}L{4.1cm}L{1.5cm}}
\toprule
\textbf{$W_\text{bare}(S)$} & \textbf{Bare hyperelastic potential} & \textbf{$[J/m^3]$} & \textbf{Component of $V(S)$; $W_\text{bare} = (E_0/2)S^4/(S_{\max}^2-S^2)^2$} & \textbf{\S{}I.7} \\
\midrule\endhead
$E_0$ & Characteristic energy density & dimensionless ($\times\rho_P c^2$) & $E_0 = 4.2546\,\rho_P c^2$; scale of the effective potential & \S{}I.7 \\
$S_{\max,\text{geom}}$ & Geometric hard bound & dimensionless & $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}\approx 2.1773$ (Kepler-Hales packing) & \S{}I.8 \\
$S_{\max,\text{eff}}$ & Effective occupation of the current regime & dimensionless & $S_{\max,\text{eff}}^2 = 1.4956$ (calibrated to the current regime) & \S{}I.8 \\
$f_c$ & Fraction of active condensate & dimensionless & $f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2\approx 0.6869$ & \S{}I.8 \\
$D_\text{lat}$ & Discrete Laplacian of the lattice & dimensionless & Operator of the diamond lattice; $\nabla^2\leftrightarrow -(3/16\ell_P^2)D_\text{lat}$ & \S{}II.3 \\
$\tau_\Pi$ & Relaxation time of el Pleno & $[s]$ & $\tau_\Pi = \sqrt{5}\,\ell_P/(10\,c) = \ell_P/(\sqrt{20}\,c)\approx 1.21\times 10^{-44}$ s; inverse of the frequency of the Lee-Wick regulation mode (E-relax) & \S{}III.1.6 \\
$m_\text{eff}$, $m_\text{radial}$ & Mass of the radial mode & $[kg]$ or $[GeV/c^2]$ & $m_\text{radial}c^2 = \sqrt{\xi}\,E_P\approx 4.44\,E_P\approx 5.42\times 10^{19}$ GeV & \S{}II.5 \\
$g_\text{eff}$ & Gross-Pitaevskii quartic coupling & $[J\cdot m^3]$ & $g_\text{eff} = (V^{(4)}_\text{eff}\ell_P^6/12)[1-\tfrac53(V''')^2/(V''V'''')]\approx 5.43\times 10^{-94}$ J$\cdot$m$^3$ (repulsive; includes cubic renormalization, $-7.7\%$) & \S{}II.5 \\
$M_Q$ & Mass of the fundamental soliton & $[kg]$ or $[GeV/c^2]$ & $M_Q = M_P/S_{\max,\text{eff}}\approx 9.98\times 10^{18}$ GeV & \S{}II.6 \\
$\Lambda_\text{PIU}$ & Hadronic operational scale & $[MeV]$ & Single scale of the hadronic sector (analogous to $\Lambda_\text{QCD}$) & \S{}II.7 \\
$\Omega_{EO}$ & Dark-energy fraction & dimensionless & $\Omega_{EO}\equiv f_c\approx 0.6869$ (0.30$\sigma$ vs Planck) & \S{}III.7 \\
$\Omega_{MO}, \Omega_M$ & Dark / baryonic matter fractions & dimensionless & $\Omega_{MO}/\Omega_M = \sqrt{3}\pi\approx 5.4414$ (0.86$\sigma$) & \S{}III.6 \\
$\Omega_m$ & Total matter density & dimensionless & $\Omega_m = 1-f_c\approx 0.3131$ (0.3$\sigma$ vs Planck, derived without fitting) & \S{}III.8 \\
$E(z)$ & Normalized expansion history & dimensionless & $E(z) = H(z)/H_0$; scale-invariant form, forced by $V_\text{eff}$ & \S{}III.8 \\
$\Theta$ & Dimensionless cosmological density & dimensionless & $\Theta=\rho_\text{tot}/\rho_P$; state factor of $H_0=\sqrt{8\pi/3}\,(c/\ell_P)\sqrt{\Theta}$; $\Theta_0\approx 1.66\times10^{-123}$ (positional, not a constant) & \S{}III.8.7 \\
$N_\text{tic}$ & Ticks of the Hubble time & dimensionless & $N_\text{tic}=1/(H_0\tau_P)\approx 8.49\times10^{60}$; $\Theta=3/(8\pi N_\text{tic}^2)$: the ``$10^{-123}$'' is the age in ticks & \S{}III.8.7 \\
$\delta_0$ & Cosmological initial condition & dimensionless & Displacement of the field with respect to $S_{\min}$ today; single observable freedom, $\delta_0\leftrightarrow w_0$ & \S{}III.8 \\
$n_s$ & Scalar spectral index & dimensionless & $n_s - 1 = 3 - 2|s-\tfrac12|$; $n_s=1$ at zeroth order (matter bounce); calibrated red tilt & \S{}III.8 \\
$\nu, s$ & Exponents of the primordial spectrum & dimensionless & $z\propto(-\eta)^s$, $\nu=|s-\tfrac12|$; $s=-1$, $\nu=\tfrac32$ from $S^2\propto a^{-3}$ & \S{}III.8 \\
$w_\text{eff}$ & Effective deviation from $w=0$ & dimensionless & $|w_\text{eff}|\approx 0.006$ for the observed red tilt of the CMB & \S{}III.8 \\
$\mu$ & Chemical potential of the CMB & dimensionless & $\mu=0$ structural (photon without conserved number); $|\mu|\lesssim 9\times 10^{-5}$ (FIRAS) & \S{}III.8 \\
$K_\text{cin}$ & Kinetic rigidity (effective Young) & $[N]$ & $K_\text{cin} = \rho_P c^2\ell_P^2/2 = c^4/(2G)\approx 6.05\times 10^{43}$ N & \S{}III.1 \\
$\mu_\text{Pleno}$ & Shear modulus of el Pleno (Gent) & $[J/m^3]$ or $[Pa]$ & $\mu_\text{Pleno} = \rho_P c^2/S_{\max,\text{geom}}^2\approx 2.13\times 10^{113}$ Pa & \S{}III.1 \\
$\eta_\text{Pleno}$ & Shear viscosity of el Pleno & $[Pa\cdot s]$ & $\eta_\text{Pleno} = \mu_\text{Pleno}\tau_\Pi\approx 2.56\times 10^{69}$ Pa$\cdot$s (Maxwell-Kubo) & \S{}III.1 \\
$\eta_\text{TT}$, $\nu_\text{TT}$ & Shear viscosity of the TT mode & $[Pa\cdot s]$, $[m^2/s]$ & $\eta_\text{TT} = \rho_P c^2\tau_\Pi = (\sqrt5/10)\rho_P c\,\ell_P\approx 5.58\times 10^{69}$ Pa$\cdot$s; kinematic $\nu_\text{TT}=\eta_\text{TT}/\rho_P=(1/\sqrt{20})c\,\ell_P$; fixes the intrinsic TT attenuation (Maxwell-Kubo and Green-Kubo) & \S{}III.1, \S{}III.3 \\
$\mu$ (shear) & Propagation shear modulus & $[Pa]$ & $\mu = \rho_P c^2\approx 4.63\times 10^{113}$ Pa (fixes $v_T = c$) & \S{}III.1 \\
$K$ & Volumetric (bulk) modulus & $[Pa]$ & $K = \tfrac{1}{4}\xi\rho_P c^2 S_{\min}^{2,\text{estr}}\approx 1.19\times 10^{114}$ Pa & \S{}III.1 \\
$\lambda$ (Lamé) & First Lamé parameter & $[Pa]$ & $\lambda = K - \tfrac{2}{3}\mu\approx 8.84\times 10^{113}$ Pa & \S{}III.1 \\
$E$ (Young) & Young's modulus of el Pleno & $[Pa]$ & $E = 9K\mu/(3K+\mu)\approx 1.23\times 10^{114}$ Pa & \S{}III.1 \\
$\nu$ (Poisson) & Poisson's ratio of el Pleno & dimensionless & $\nu = (3K-2\mu)/(2(3K+\mu))\approx 0.328$ (stable medium) & \S{}III.1 \\
$\xi_\text{heal}$ & Healing length of the condensate & $[m]$ & $\xi_\text{heal} = \ell_P/\sqrt{\xi}\approx 0.225\,\ell_P$; coherence of the Higgs mode & \S{}III.1 \\
$\sigma_\text{Pleno}$ & Characteristic tension of el Pleno & $[N]$ & $\sigma_\text{Pleno} = \gamma_\kappa/\ell_P^2 = F_\text{Planck}/40\approx 3.026\times 10^{42}$ N & \S{}III.1 \\
$T_P$ & Effective temperature of el Pleno & $[K]$ & $T_P = \rho_P c^2\ell_P^3/k_B\approx 1.42\times 10^{32}$ K (Planck temperature) & \S{}III.1 \\
$M_\text{LW}$ & Scale of the UV regulation pole & $[m^{-1}]$ & $M_\text{LW} = \sqrt{20}/\ell_P\approx 5.46\times 10^{19}$ GeV; Lee-Wick pole, sub-Planckian in length & \S{}III.3 \\
$\tau_\Pi, t_\text{dec}$ & Intrinsic decoherence time & $[s]$ & $t_\text{dec}^\text{Planck} = (20)^{1/4}\tau_\Pi\approx 2.55\times 10^{-44}$ s & \S{}III.2 \\
$E(\mathbf{a},\mathbf{b})$ & Spin correlation function & dimensionless & $E = -\cos(2(a-b))$; Tsirelson bound $2\sqrt{2}$ (CHSH) & \S{}III.2 \\
$\Theta_p$ & Moment of inertia of the cluster & $[\text{MeV}\cdot\text{fm}^2]$ & $\Theta_p = m_p R_p^2/3$; governs the baryonic rotational tower $m(J)$ & \S{}III.5 \\
$M_{\min}$ & Structural minimum mass of a black hole & $[kg]$ & $M_{\min}\approx 0.117\,m_P$; from $r_\text{core}=r_S$ & \S{}II.6 \\
$H_{\max}$ & Maximum bound of the Hubble constant & $[s^{-1}]$ & $H_{\max}\approx 7.9\times 10^{43}$ s$^{-1}$; minimum horizon $0.23\,\ell_P$; exact equality $H_{\max}/\dot\gamma_{\max}=\sqrt{8\pi/3}\,S_{\max,\text{geom}}=4.27$ & \S{}II.5, \S{}III.9.2 \\
$K_c$ & Factor of the formula enumeration & dimensionless & $K_c\approx 6.71\times 10^5$; space of algebraic expressions against an isolated ratio (\S{}III.6.5). The factor over the whole set of results of the programme is a different quantity and lives in \S{}V.2.6 & \S{}III.6.5 \\
$w(z)$ & Equation of state of dark energy & dimensionless & $w=-1$ relaxed; $w\geq -1$ (canonical kinetics); non-monotonic integrated form forced by $V_\text{eff}$, shallow minimum toward $z\approx 0.5$--$0.7$, with $\delta_0$ as the single positional freedom; magnitude of the shift and combined analysis: F-cosmo confrontation thread & \S{}III.6--\S{}III.8, \S{}V.4 \\
$\dot\gamma_{\max}$ & Maximum deformation rate & $[s^{-1}]$ & $\dot\gamma_{\max} = c/\ell_P = 1/t_P\approx 1.8552\times 10^{43}$ s$^{-1}$ (Planck frequency) & \S{}III.9 \\
$x$, $\mathrm{De}$ & Relative rate / Deborah number & dimensionless & $x = \dot\gamma/\dot\gamma_{\max}$; packing degree; fluid-solid transition at $x\to 1$ & \S{}III.9 \\
$n$ & Flow index of el Pleno & dimensionless & $n(x) = 1/(1-x) = \gamma^2$; $n=1$ Newtonian, $n>1$ dilatant; $\approx 1.3160$ today & \S{}III.9 \\
$\tau_0$ & Yield stress & $[Pa]$ & $\tau_0 \approx 1.9838\,\rho_P c^2$; from the Klauder IR wall ($V_q$) & \S{}III.9 \\
$K_\text{HB}$ & Herschel-Bulkley consistency index & $[Pa]$ & $K(x) = \tfrac12\rho_P c^2\,x^{x/(x-1)}/(1-x)$; Newtonian anchor $K_0 = \tfrac12\rho_P c^2$ & \S{}III.9 \\
$\eta/s$ & Viscosity / entropy ratio & $[\hbar/k_B]$ & Derived window $1\lesssim(\eta/s)/(\hbar/4\pi k_B)\lesssim 2.5$ (KSS below, UV scale $\tau_\Pi$ at $T_{\max}$ above); almost-perfect fluid [SLD] & \S{}III.9.6 \\
$\dot\gamma$ & Volumetric deformation rate & $[s^{-1}]$ & $\dot\gamma = 2|\dot S|/S$ (from $\rho_\text{medium}=\rho_P S^2$); physical rate that governs the dilatancy in the cycle & \S{}IV.4 \\
$\eta(\dot\gamma)$ & Effective dilatant viscosity & $[Pa\cdot s]$ & $\eta = \mu_\text{Gent}\tau_\Pi/(1-\dot\gamma/\dot\gamma_{\max})$; assembled, without free parameter; regulates the bounce & \S{}IV.4 \\
$V_\text{eff}(S)$ & Effective potential of the cycle & $[\rho_P c^2]$ & Well at $S_{\min}$, Gent wall at $S_{\max}$, IR wall; (E52), from \S{}II.2 & \S{}IV.3 \\
$f_c$ & Fraction of the DE-Curtain & dimensionless & $f_c = 0.6869$; DE population gravitationally inert; $1-f_c=0.3131$ is the matter that cycles & \S{}IV.2 \\
\bottomrule
\end{longtable}}

\textit{[Additional variables are added to the table as the manuscript advances.]}

\subsection{\S{}V.6.2 --- Index of equations}

Sequentially numbered list of the important equations of the manuscript. Each equation carries a unique identifier of the form (EX) where X is a positive integer assigned in order of appearance. The equations referenceable from the body of the text use this identifier.


Format of the entries: two rows per equation. The first row contains \textbf{ID, equation, section} of appearance; the second row contains the \textbf{description} of the physical content of the equation.


\medskip\hrule\medskip

\textbf{(E1)} $\quad\Psi(x^\mu) = S(x^\mu) \cdot e^{i\theta(x^\mu)} \quad$ --- \S{}I.7.2


\textit{Canonical decomposition of the field of \emph{el Pleno} into modulus $S$ and phase $\theta$.}


\medskip\hrule\medskip

\textbf{(E2)} $\quad\mathcal{L}_{F1} = \dfrac{1}{2}\rho_P c^2 \ell_P^2 \, \partial_\mu \Psi^* \partial^\mu \Psi - V(S) - \dfrac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2 - V_q^\text{dens}(S) \quad$ --- \S{}I.7.4


\textit{Lagrangian F1: Lagrangian density of \emph{el Pleno} under PIU-$\Psi$. Minimal mathematical expression compatible with the six axioms A1--A6.}


\medskip\hrule\medskip

\textbf{(E3)} $\quad G = \dfrac{c^2}{\rho_P\, \ell_P^2} \quad$ --- \S{}I.7.5


\textit{Structural identity that derives the gravitational constant from the parameters of \emph{el Pleno}. Numerical verification with exact agreement by construction; residue $<0.1\%$ by rounding of the primary parameters.}


\medskip\hrule\medskip

\textbf{(E4)} $\quad\gamma_\kappa = \dfrac{\rho_P c^2 \ell_P^4}{40} = \dfrac{\hbar c}{40} \approx 7.90 \times 10^{-28}\ \text{J}\cdot\text{m} \quad$ --- \S{}I.7.7


\textit{Flexural rigidity of \emph{el Pleno}, structural analogue of the bending rigidity EI in the Euler-Bernoulli theory of elastic beams.}


\medskip\hrule\medskip

\textbf{(E5)} $\quad V_q^\text{dens}(S) = \dfrac{3\hbar^2}{8 \rho_P \ell_P^8 S^2} \quad$ --- \S{}I.7.8


\textit{Klauder quantum potential. IR quantum wall that prevents $S \to 0$, derived uniquely from the affine quantization of the field of \emph{el Pleno}.}


\medskip\hrule\medskip

\textbf{(E6)} $\quad\rho_P c^2 \ell_P^2 \left[ \Box S - S(\partial_\mu \theta)(\partial^\mu \theta) \right] - \dfrac{\rho_P c^2 \ell_P^4}{20} \Box(\Box S) + V'(S) + V_q'(S) = 0 \quad$ --- \S{}I.7.10


\textit{Equation F2 --- dynamical equation of the modulus of \emph{el Pleno} (primordial equation). Result of applying Euler-Lagrange to F1 with respect to $S$. Fourth-order differential equation.}


\medskip\hrule\medskip

\textbf{(E7)} $\quad\partial_\mu (S^2 \partial^\mu \theta) - \dfrac{\ell_P^2}{20}\,\Box(S^2\,\Box\theta) = 0 \quad$ --- \S{}I.7.10


\textit{Equation of motion of the phase sector with ultraviolet regulation. The conserved Noether current of the U(1) symmetry (axiom A5) is $J^\mu = \rho_P c^2\ell_P^2 S^2[\partial^\mu\theta + (\ell_P^2/20)\partial^\mu(\Box\theta)]$ (E7c), with $\partial_\mu J^\mu = 0$ exact over the dynamics. On the massless physical branch ($\Box\theta=0$) the flexural term of the current vanishes and $J^\mu$ reduces to the standard current, identifiable in the effective limit with the electric current of electromagnetism. The flexural term fixes the ultraviolet dispersion correction $(k\ell_P)^2/20$ common to the photon and the TT mode (\S{}II.3.8 P9b, \S{}III.3--\S{}III.4).}


\medskip\hrule\medskip

\textbf{(E-tiempo-1)} $\quad \tau = \dfrac{1}{c}\displaystyle\int_\gamma C(\lambda)\,d\lambda \quad$ --- \S{}I.8.9-bis
\textit{Time as the arc of accumulated change of el Pleno over its trajectory $\gamma$, normalized by the change ceiling $c$ (A1). Time is a derived magnitude, not a background parameter: it is constructed from the medium's own change, with the non-circular chain $C\to c\to\tau$ (change is prior to time and requires no clock). It coincides with the proper time of the geodesic (\S{}III.1.5, same integral over the effective metric) and becomes observable from within by the Page-Wootters correlation (\S{}III.2). \epi{SLD}; non-circularity \epi{DERIVED}.}


\medskip\hrule\medskip

\textbf{(E8)--(E33)} --- Part II (configuration space). Equations of the effective potential and the \emph{Pleno}-ground state (E8: $V_\text{eff} = W_\text{bare}+V_q+V_\text{topo}$; E9: $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$; E10: $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$), the discrete/continuum Laplacian relation (E13: $\nabla^2 \leftrightarrow -(3/16\ell_P^2)D_\text{lat}$), the mode spectrum and the dispersion relations (E17--E23, with the UV correction $(k\ell_P)^2/20$), the solitonic configurations and the rotational split (E30--E31), the matter-density formula (E32-bis: $\rho_\text{matter} = \rho_P[S^2 - S_{\min}^{2,\text{estr}}]$), and the active condensate fraction (E33: $\Omega_{EO} \equiv f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$). Defined and derived in \S{}\S{}II.2--II.6.


\medskip\hrule\medskip

\textbf{(E34)} $\quad \Gamma_\text{ind}[g] = -\dfrac{c^4}{16\pi G}\int d^4x\sqrt{-g}\,R + \mathcal{O}(R^2/E_P^2) \quad$ --- \S{}III.1.4


\textit{Effective Einstein-Hilbert action induced over the TT mode of \emph{el Pleno} (Sakharov mechanism).}


\medskip\hrule\medskip

\textbf{(E35)} $\quad R_{\mu\nu} - \tfrac{1}{2}R\,g_{\mu\nu} = \dfrac{8\pi G}{c^4}T_{\mu\nu}^\text{(Pleno)} \quad$ --- \S{}III.1.4


\textit{Einstein equations as the effective regime of \emph{el Pleno}, with fixed point $\gamma_G = 1$.}


\medskip\hrule\medskip

\textbf{(E39a)--(E39d)} --- \S{}III.1.7. Effective metric as the Cauchy-Green tensor of the hyperelastic medium, $g^\text{eff}_{\mu\nu} = F^a{}_\mu\,\eta_{ab}\,F^b{}_\nu$ with $F^a{}_\mu = \partial_\mu X^a$ (E39a); expansion in orders, with zeroth order $\eta_{\mu\nu}$, first order the TT mode of the corpus and second order $\eta_{ab}\partial_\mu u^a\partial_\nu u^b$ (E39b); contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ emergent from the second Noether theorem (E39c); covariant conservation $\nabla_\mu T^{\mu\nu}=0$ (E39d).


\medskip\hrule\medskip

\textbf{(E36)} $\quad \hbar = \rho_P c\,\ell_P^4 \quad$ --- \S{}III.2.2


\textit{Structural identity of the Planck constant by quantization of the phase mode. Linked to (E3) by $\ell_P = \sqrt{\hbar G/c^3}$.}


\medskip\hrule\medskip

\textbf{(E37)} $\quad \dfrac{\Omega_{MO}}{\Omega_M} = \sqrt{3}\,\pi \approx 5.4414 \quad$ --- \S{}III.6.3


\textit{Dark/baryonic matter ratio from the discrete geometry of the substrate. Agreement at 0.86$\sigma$, entirely derived.}


\medskip\hrule\medskip

\textbf{(E38)} $\quad K_\text{cin} = \dfrac{c^4}{2G}\approx 6.05\times 10^{43}\ \text{N}; \quad \mu_\text{Pleno} = \dfrac{\rho_P c^2}{S_{\max,\text{geom}}^2}\approx 2.13\times 10^{113}\ \text{Pa}; \quad \eta_\text{Pleno} = \mu_\text{Pleno}\tau_\Pi\approx 2.56\times 10^{69}\ \text{Pa}\cdot\text{s}; \quad \sigma_\text{Pleno} = \dfrac{F_\text{Planck}}{40}\approx 3.03\times 10^{42}\ \text{N} \quad$ --- \S{}III.1.6


\textit{Structural rigidities of \emph{el Pleno} as a mechanical medium: effective Young's modulus ($K_\text{cin}$), Gent shear modulus ($\mu_\text{Pleno}$), Maxwell-Kubo shear viscosity ($\eta_\text{Pleno}$) and characteristic tension ($\sigma_\text{Pleno}$). All evaluated numerically from the primary parameters $(\rho_P, c, \ell_P)$ without additional free parameters.}


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\textbf{(E-relax)} $\quad \tau_\Pi = \dfrac{1}{\omega_\text{LW}} = \dfrac{\ell_P}{\sqrt{20}\,c} = \dfrac{\sqrt{5}\,\ell_P}{10\,c}\approx 1.21\times 10^{-44}\ \text{s} \quad$ --- \S{}III.1.6

\textit{Relaxation time of \emph{el Pleno} derived as the inverse of the frequency of the flexural regulation mode (Lee-Wick pole $M_\text{LW}=\sqrt{20}/\ell_P$, \S{}III.3.4). Without a fitting parameter; consistent with the fluctuation-dissipation relation $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$ (E-TP). Fixes the Maxwell viscosity of the medium (E38) and the dilatant viscosity of the cycle (E54).}


\medskip\hrule\medskip

\textbf{(E-TP)} $\quad T_P = \dfrac{\rho_P c^2\ell_P^3}{k_B}\approx 1.42\times 10^{32}\ \text{K}; \quad M_\text{LW} = \dfrac{\sqrt{20}}{\ell_P}\approx 5.46\times 10^{19}\ \text{GeV} \quad$ --- \S{}III.1.7, \S{}III.3.4


\textit{Effective temperature of \emph{el Pleno} (Planck temperature, by fluctuation-dissipation, cross-verification $\tau_\Pi k_B T_P/\hbar = 1/\sqrt{20}$) and scale of the Lee-Wick ultraviolet regulation pole.}


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\textbf{(E40)} $\quad \sigma_{ij} = \lambda\,(\operatorname{tr}\varepsilon)\,\delta_{ij} + 2\mu\,\varepsilon_{ij} \quad$ --- \S{}III.1.8


\textit{Generalized Hooke's law (constitutive stress-strain relation) of \emph{el Pleno} as an isotropic medium. The strain is built over the TT mode (shear) and the relative change of the modulus (compression), both covariant objects.}


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\textbf{(E41)} $\quad \mu = \rho_P c^2\approx 4.63\times 10^{113}\ \text{Pa} \quad$ --- \S{}III.1.8


\textit{Propagation shear modulus of \emph{el Pleno}, fixed by the condition $v_T = \sqrt{\mu/\rho_P} = c$ (TT mode, Result D2). Distinct from the static Gent shear $\mu_\text{Gent} = \rho_P c^2/S_{\max,\text{geom}}^2$.}


\medskip\hrule\medskip

\textbf{(E42)} $\quad K = \tfrac{1}{4}\xi\rho_P c^2 S_{\min}^{2,\text{estr}}\approx 1.19\times 10^{114}\ \text{Pa} \quad$ --- \S{}III.1.8


\textit{Bulk (volumetric) modulus of \emph{el Pleno}, derived from the curvature of the potential $V''_\text{eff} = \xi\rho_P c^2$ and the volumetric deformation $\varepsilon_\text{vol} = \delta(S^2)/S_{\min}^{2,\text{estr}}$ fixed by the constitutive identification $S^2 \leftrightarrow n/n_0$.}


\medskip\hrule\medskip

\textbf{(E43)} $\quad \nu = \dfrac{3K - 2\mu}{2(3K+\mu)}\approx 0.328; \quad \lambda = K-\tfrac{2}{3}\mu\approx 8.84\times 10^{113}\ \text{Pa}; \quad E = \dfrac{9K\mu}{3K+\mu}\approx 1.23\times 10^{114}\ \text{Pa} \quad$ --- \S{}III.1.8


\textit{Poisson's ratio, first Lamé parameter and Young's modulus of \emph{el Pleno}, exact algebraic identities of isotropic elasticity. $\nu \approx 0.33$ (comparable to steel) locates \emph{el Pleno} in the range of thermodynamic stability, a verification of internal consistency.}


\medskip\hrule\medskip

\textbf{(E44)} $\quad K = \tfrac{1}{4}\xi\rho_P c^2 S_{\min}^{2,\text{estr}} = \tfrac{1}{4}\left(m_\text{radial}c^2/E_P\right)^2\rho_P c^2 S_{\min}^{2,\text{estr}} \quad$ --- \S{}III.1.9


\textit{The bulk modulus linked to the square of the mass of the Higgs (amplitude) mode of the condensate of \emph{el Pleno}. The phase mode (Goldstone, gapless) gives the shear $\mu$; the amplitude mode (Higgs, gap $\sqrt{\xi}E_P$) gives the bulk $K$. The Poisson's ratio $\nu$ measures the balance between the two modes.}


\medskip\hrule\medskip

\textbf{(E45)} $\quad \dot\gamma_{\max} = \dfrac{c}{\ell_P} = \dfrac{1}{t_P} \approx 1.8552\times 10^{43}\ \text{s}^{-1} \quad$ --- \S{}III.9.2


\textit{Maximum deformation rate of \emph{el Pleno} (Planck frequency), rheological reading of the UV cutoff of axiom A4. Defines the Deborah number $x = \dot\gamma/\dot\gamma_{\max}$.}


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\textbf{(E46)} $\quad x \equiv \dfrac{\dot\gamma}{\dot\gamma_{\max}} = \dot\gamma\,\dfrac{\ell_P}{c} \in [0,1) \quad$ --- \S{}III.9.2


\textit{Dimensionless relative deformation rate, identical to the Deborah number of \emph{el Pleno}. Measures the degree of packing; the fluid-solid transition coincides with the hard bound $x\to 1$.}


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\textbf{(E47)} $\quad n(x) = \dfrac{1}{1-x} \quad$ --- \S{}III.9.3


\textit{Flow index of \emph{el Pleno}. Newtonian ($n=1$) at $x\to 0$; dilatant ($n>1$) under forcing; divergent ($n\to\infty$) at the bound $\dot\gamma\to c/\ell_P$. In the current universe $x\approx 0.2401$, $n\approx 1.3160$.}


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\textbf{(E48)} $\quad \tau_0 = \dfrac{3}{4\,(S_{\min}^\text{estr})^3}\,\rho_P c^2 \approx 1.9838\,\rho_P c^2 \quad$ --- \S{}III.9.4


\textit{Yield stress of \emph{el Pleno}, derived from the Klauder IR quantum wall $V_q$ (E5). Completes the Herschel-Bulkley model: $\tau = \tau_0 + K\dot\gamma^{\,n}$.}


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\textbf{(E49)} $\quad (\partial_\mu S)(\partial^\mu S) = -\left(1 - \dfrac{v^2}{c^2}\right)(\partial_\xi S)^2 \quad$ --- \S{}III.9.5


\textit{The relativistic factor $(1-v^2/c^2)$ emerges from the kinetic invariant of F1 for a moving deformation $S(x-vt)$, without being postulated.}


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\textbf{(E50)} $\quad \dfrac{T^{00}(v)}{T^{00}(0)} \longrightarrow \dfrac{1}{1-v^2/c^2} = \gamma^2 \quad$ --- \S{}III.9.5


\textit{The resistance of \emph{el Pleno} to the moving deformation diverges as the square of the Lorentz factor. Identifies $x = v^2/c^2$ and gives $n(v) = \gamma^2$: the relativistic inertia as a dilatant response. Equivalent reading of the A3/LET covariance.}


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\textbf{(E51)} $\quad \dfrac{\eta}{s} \geq \dfrac{\hbar}{4\pi k_B} \quad$ --- \S{}III.9.6


\textit{Quantum bound of Kovtun-Son-Starinets for the viscosity/entropy ratio, valid for every fluid. For the thermal el Pleno (TT-mode gas) the ratio is bounded in a narrow window $1\lesssim(\eta/s)/(\hbar/4\pi k_B)\lesssim 2.5$ (KSS below; UV scale $\tau_\Pi$ at $T_{\max}$ above), the quark-gluon-plasma strip \epi{SLD}. The relaxed cold vacuum has $s=0$ (elastic solid, $w=-1$).}


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\textbf{(E-Dr-1)} $\quad I:\;S \to \dfrac{S_{\min}S_{\max}}{S},\qquad I(S^*)=S^*,\qquad n(u)=m(-u) \quad$ --- \S{}III.9.11-bis
\textit{Wall inversion operator in the logarithmic variable $u=\ln(S/S^*)$. It interchanges the packing fractions ($x\leftrightarrow\bar y$, Gent$\leftrightarrow$Klauder); its only fixed point is $S^*=\sqrt{S_{\min}S_{\max}}$. \epi{DERIVED}.}


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\textbf{(E-Dr-2)} $\quad V_D = (1+I)\,V_\text{topo} = V_\text{topo} + I(V_\text{topo}),\qquad P_D=\tfrac12(1+I),\ \ P_D^2=P_D \quad$ --- \S{}III.9.11-bis
\textit{Droste self-dual potential: canonical symmetrization of the Gent wall (E10) under the idempotent projector $P_D$. It generates the mirror IR wall $-\tfrac{\rho_P c^2}{2}\ln(1-S_{\min}^2/S^2)$ from a single geometric wall; exactly even in $u$. The difference with the physical IR wall (Klauder, E5) measures the quantum/geometric duality: $S_{\min}^2-3/8=0.148$. \epi{DERIVED}.}


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\textbf{(E-Dr-3)} $\quad \Psi \to \dfrac{S^{*2}}{\overline{\Psi}} = \dfrac{S^{*2}}{S}\,e^{i\theta},\qquad \{|\Psi|=S^*\}\ \text{invariant} \quad$ --- \S{}III.9.11-bis
\textit{The inversion of the modulus is, over the complete field $\Psi=S\,e^{i\theta}$, the conformal inversion of the complex plane with respect to the circle $|\Psi|=S^*$ (inverts the modulus, preserves the phase). The fixed set is not a point but the circle $|\Psi|=S^*$; the central hole $|\Psi|<S_{\min}$ is the region forbidden by A2 (phase undefined at $\Psi=0$). \epi{DERIVED}.}


\medskip\hrule\medskip

\textbf{Reappearances in Part III (without their own tag; the result keeps its original numbering).} Part III phenomenologically re-reads several results already derived in Parts I--II; these reappearances do not receive a new tag, but reference the original equation:


\begin{itemize}[leftmargin=1.4em]
  \item $G = c^2/(\rho_P\ell_P^2)$ --- derived by induced action in \S{}III.1.3, keeps the numbering \textbf{(E3)} (\S{}I.7.5).
  \item $\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$ --- rewritten as a native identity in \S{}III.2.3, keeps \textbf{(E4)} (\S{}I.7.7).
  \item $\omega^2 = c^2 k^2[1 + (k\ell_P)^2/20]$ --- dispersion relation of the TT mode in \S{}III.3.3, keeps \textbf{(E23)} (\S{}II.5.4).
  \item $\Delta m = 9(\hbar c)^2/(2 m_p c^2 d^2)$ --- $\Delta$-N split read phenomenologically in \S{}III.5.4, keeps \textbf{(E31)} (\S{}II.6.7).
  \item $\Omega_{EO} \equiv f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2 \approx 0.6869$ --- dark-energy fraction in \S{}III.7.2, keeps \textbf{(E33)} (\S{}II.7.6).
\end{itemize}

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\textbf{Part IV --- the dilatant crunch-bounce cycle and particle formation.} Part IV introduces six display equations with a new global tag, which articulate the dynamics of the cycle from quantities already derived in Parts I--III:

\medskip

\textbf{(E52)} $\quad V_\text{eff}(S) = \dfrac{E_0}{2}\dfrac{S^4}{(S_{\max}^2-S^2)^2} - \dfrac{\rho_P c^2}{2}\log\!\left(1-\dfrac{S^2}{S_{\max}^2}\right) + \dfrac{3\rho_P c^2}{8 S^2} \quad$ --- \S{}IV.3

\textit{Effective potential of the cycle: well at $S_{\min}$, hard Gent wall at $S_{\max}$, IR quantum wall. Operative rewriting of the tripartite potential of \S{}II.2 with $E_0=4.2546\,\rho_P c^2$ derived; $dV_\text{eff}/dS=0$ exact at $S_{\min}$ (verified, 10 digits).}

\medskip

\textbf{(E53)} $\quad -\dfrac{dW_\text{bare}}{dS} = -\dfrac{2\,E_0\,S^3\,S_{\max}^2}{(S_{\max}^2-S^2)^3} \quad$ --- \S{}IV.3

\textit{Restoring force of the Gent wall. The denominator $(S_{\max}^2-S^2)^3$ explodes near $S_{\max}$: between $S=1.30$ and $1.40$ the force grows $\times 14$. It is the signature of the limited extensibility (A6) that produces the structural rebound.}

\medskip

\textbf{(E54)} $\quad \eta(\dot\gamma) = \dfrac{\mu_\text{Gent}\,\tau_\Pi}{1-\dot\gamma/\dot\gamma_{\max}} \quad$ --- \S{}IV.4

\textit{Effective dilatant viscosity, assembled from $\mu_\text{Gent}=\rho_P c^2/S_{\max}^2$ and $\tau_\Pi=\sqrt{5}\,\ell_P/(10c)$ (both derived). Diverges at the deformation bound: regulates the bounce.}

\medskip

\textbf{(E55)} $\quad \tilde\eta(x) = \dfrac{\sqrt{5}/(10\,S_{\max}^2)}{1-x} \approx \dfrac{0.1027}{1-x} \quad$ --- \S{}IV.4

\textit{Dimensionless form of (E54). The coefficient $\sqrt{5}/(10 S_{\max}^2)$ reproduces the Maxwell viscosity of \S{}III.1.6 upon nondimensionalizing (verified).}

\medskip

\textbf{(E56)} $\quad \dot\gamma = \left|\dfrac{d\ln V}{dt}\right| = 2\,\dfrac{|\dot S|}{S}, \qquad \dot\gamma \leq \dfrac{c}{\ell_P} \quad$ --- \S{}IV.4

\textit{Physical deformation rate as volumetric deformation, from the identification $\rho_\text{medium}=\rho_P S^2$ (\S{}II.2.2). Key link: the dilatancy is governed by this rate, not by $\dot S$ directly.}

\medskip

\textbf{(E57)} $\quad \rho_P\ell_P^2\,\ddot S + \eta(\dot\gamma)\,\dot S + \dfrac{dV_\text{eff}}{dS} = F_\text{ext} \quad$ --- \S{}IV.5

\textit{Equation of the cycle: Gent oscillator with dilatant damping. $F_\text{ext}$ is the cosmological forcing (Friedmann contraction, $S^2\propto 1/a^3$). Its numerical integration confirms that the dilatancy keeps $\dot\gamma\leq c/\ell_P$ (subluminal bounce).}

\medskip
\textbf{(E58)} $\quad E_0 = \dfrac{(S_{\max}^2-S^2)^3}{2S^3S_{\max}^2}\left[\dfrac{3}{4S^3}-\dfrac{S}{S_{\max}^2-S^2}\right]\rho_P c^2\Big|_{S=S_{\min}^\text{estr}} = 4.254617\,\rho_P c^2 \quad$ --- \S{}II.4.4

\textit{Closed form of the coefficient of the constitutive potential. Solving (E8), linear in $E_0$; with the two bounds substituted it becomes a function of $\pi$ and $\sqrt6$ alone, with no free symbols.}

\medskip
\textbf{(E59)} $\quad \langle n\rangle = \dfrac{1}{e^{\hbar\omega/k_BT}-1} \quad$ --- \S{}III.9.5-ter

\textit{Equilibrium occupation of the phase modes, derived from the microcanonical count over the oscillators that F1 provides (E-BE-1, E-BE-2) and not assumed. From it follows the coefficient $2\pi^2/45$ of the radiation entropy (E-BE-3).}

\medskip
\textbf{(E60)} $\quad \mathcal{A} = \pi\left(S_{\max,\text{geom}}^2-(S_{\min}^\text{estr})^2\right) = 5.197669 \quad$ --- \S{}II.4.7

\textit{Area of the configuration annulus: admissible region of $\Psi$ per cell between the two walls, with the centre $\Psi=0$ forbidden by A1. A pure number.}

\medskip
\textbf{(E61)} $\quad R(\tau_\Pi,T_{\max}) = \dfrac{2\cdot 5^{3/4}\,6^{7/8}\sqrt\pi}{75} = 0.757973 \quad$ --- \S{}III.9.6

\textit{Ultraviolet sub-bound of the ratio $\eta/s$ in the TT regime. A pure number: the three primaries cancel by dimension and $k_B$ self-annihilates.}

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\textbf{(E62)} $\quad \Omega_\text{DE}(x) = \dfrac{1}{(1+x)^2}, \qquad x \equiv \dfrac{S_\text{tot}-S_{\min}^\text{estr}}{S_{\min}^\text{estr}} \quad$ --- \S{}IV.5-bis

\textit{Partition between the coherent floor and the excess, with $S_{\min}^\text{estr}$ cancelled. The equality $\Omega_\text{DE}=\tfrac12$ occurs at $x=\sqrt2-1$, that is $S_\text{tot}=\sqrt2\,S_{\min}^\text{estr}=1.022600$, exact and uncalibrated.}

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\textbf{(E63)} $\quad \ell_P = \sqrt{\dfrac{\hbar G}{\kappa_G\kappa_\hbar c^3}}, \qquad \rho_P = \dfrac{\kappa_G^2\kappa_\hbar c^5}{G^2\hbar} \quad$ --- \S{}V.2.3-bis

\textit{The pair solved for the primaries with general coefficients in the two identities. Basis of the falsification test: the three scenarios according to how many primaries are measured by an independent route.}

\medskip
\textbf{(E64)} $\quad \text{BF} = \left(\prod_i p_i\right)^{-\lambda} \approx 3.4\times 10^4 \ \ (\lambda=0.5) \quad$ --- \S{}V.2.6

\textit{Factor against the hypothesis of numerological coincidence over the set of admitted results. Reported with its full range in $\lambda$; label \epi{method}.}

\medskip
\textbf{Part III --- extension of the quantum reading (V4.4).} Two new closed forms are incorporated into the phenomenological reading:

\smallskip
\textbf{(E-inc-3)} $\quad \Delta x\cdot\Delta p = \dfrac{\hbar}{2}\sqrt{1 + \dfrac{3}{80}\left(\dfrac{\ell_P}{\sigma}\right)^2} \quad$ --- \S{}III.2.6
\textit{Uncertainty relation of \emph{el Pleno} (derived GUP). Recovers exact $\hbar/2$ for $\sigma\gg\ell_P$; the coefficient $3/80$ is a pure number from the flexural rigidity.}

\smallskip
\textbf{(E-Ch-1)} $\quad M_\text{Ch} = \dfrac{\sqrt{3\pi}}{2}\bigl(-\xi^2\theta'\bigr)_{n=3}\dfrac{(\hbar c/G)^{3/2}}{(\mu_e m_u)^2} \approx 1.456\,M_\odot \quad$ --- \S{}III.2.8
\textit{Chandrasekhar limit as an internal balance of \emph{el Pleno}; complete coefficient derived, without external importation.}

\smallskip
\textbf{(E-Maxwell-1)} $\quad E_\text{mag} = \dfrac{8}{9}\,F_{\mu\nu}F^{\mu\nu},\quad F_{\mu\nu} = \partial_\mu a_\nu - \partial_\nu a_\mu \quad$ --- \S{}III.4.4
\textit{Magnetic energy of \emph{el Pleno} from the oriented circulation of the phase over the plaquettes of the diamond lattice; antisymmetric tensor with lattice coefficient $8/9$ derived from the sum of plaquette circulations over the diamond lattice. The Maxwell equations of the free field are its Euler-Lagrange equations and the Bianchi identity.}

\smallskip
\textbf{(E-YM-1)} $\quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i\,[A_\mu, A_\nu] \quad$ --- \S{}III.4.5
\textit{Yang-Mills tensor of \emph{el Pleno} from the non-abelian plaquette holonomy ($\theta_-\in SU(2)$). The term $i[A,A]$ is born from the non-commutativity of the generators; the abelian limit recovers (E-Maxwell-1).}

\medskip
\textbf{Part III --- the quantitative cosmic background.} Four closed forms sustain the background cosmology of \emph{el Pleno}:

\smallskip
\textbf{(E-cosmo-1)} $\quad n_s - 1 = 3 - 2\nu = 3 - 2\left|s - \tfrac{1}{2}\right| \quad$ --- \S{}III.8.3
\textit{Master formula of the primordial spectral index. The exponent $s$ of the background ($z\propto(-\eta)^s$) fixes $n_s$; with $S^2\propto a^{-3}$ ($w=0$) one obtains $s=-1$, $\nu=3/2$ and $n_s=1$ at zeroth order (native matter bounce, without inflation).}

\smallskip
\textbf{(E-cosmo-2)} $\quad \Omega_m = 1 - f_c = 1 - 0.6869 = 0.3131 \quad$ --- \S{}III.8.5
\textit{Total matter density as the depleted fraction of the condensate, derived from the conservation $f_c+\Omega_\text{depleted}=1$. At $0.3\sigma$ of Planck without a fitting parameter. With $\Omega_{MO}/\Omega_M=\sqrt{3}\pi$ it decomposes into $\Omega_M=0.0486$, $\Omega_{MO}=0.2645$.}

\smallskip
\textbf{(E-cosmo-3)} $\quad E^2(z) = \dfrac{\Omega_m(1+z)^3 + \rho_S(z)/\rho_{c,0}}{\Omega_m + \rho_S(0)/\rho_{c,0}} \quad$ --- \S{}III.8.6
\textit{Normalized expansion history $E(z)=H(z)/H_0$. Its form is scale-invariant (every dimensional scale cancels after normalizing), forced by the derived potential with the initial condition $\delta_0$ as the only freedom. Fits BAO DESI DR2 with $\chi^2=6.54$ against $6.53$ of $\Lambda$CDM.}

\smallskip
\textbf{(E-cosmo-4)} $\quad n(\nu) = \dfrac{1}{e^{h\nu/k_B T}-1},\qquad \mu = 0 \quad$ --- \S{}III.8.2
\textit{Exact Planck distribution of the CMB. The chemical potential vanishes structurally because the photon is a mode of the medium without conserved number; the perfection of the blackbody ($|\mu|\lesssim 9\times 10^{-5}$, FIRAS) is a consequence, not a fit.}

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\textbf{(E-cosmo-5)} $\quad H_0 = \sqrt{\dfrac{8\pi}{3}}\,\dfrac{c}{\ell_P}\,\sqrt{\Theta(t)} \quad$ --- \S{}III.8.7
\textit{Factorization of the Hubble constant in the Language of el Pleno: geometric Friedmann ratio $\sqrt{8\pi/3}$ (\epi{DERIVED}), anchor $c/\ell_P=1/\tau_P$ (frequency of the Planck tick, $=\dot\gamma_{\max}$, E45) and state factor $\sqrt{\Theta}$ (\epi{CALIBRATED} positional). It is obtained by eliminating $G$ with the native identity $G=c^2/(\rho_P\ell_P^2)$ (E3); the anchor-density $\rho_P$ cancels.}

\smallskip
\textbf{(E-cosmo-6)} $\quad \Theta(t) = \dfrac{3}{8\pi\,N_\text{tic}^2(t)},\qquad N_\text{tic}=\dfrac{1}{H_0\tau_P} \quad$ --- \S{}III.8.7
\textit{The dimensionless cosmological density of today is the age of the universe in Planck ticks, squared. Exact identity that reformulates the ``$10^{-123}$'' of the cosmological constant as age in ticks ($N_\text{tic}\approx 8.49\times10^{60}$), not as a fine-tuning; verification $3/(8\pi N_\text{tic}^2)=1.66\times10^{-123}=\Theta_0$.}

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\textbf{Part III --- geodesic and Schwarzschild metric.} The two added gravity subsections complete the matter$\leftrightarrow$geometry pair and the static strong-field reading:

\smallskip
\textbf{(E-geo-1)--(E-geo-5)} --- \S{}III.1.5. Proper time over the effective metric, $c^2 d\tau^2=-g^\text{eff}_{\mu\nu}dx^\mu dx^\nu$ (E-geo-1); equation of the geodesic from $\delta\tau=0$, with Christoffel symbols built only from $g^\text{eff}_{\mu\nu}$ (E-geo-2); weak-field limit $d^2x^i/dt^2=-c^2\Gamma^i_{00}$ (E-geo-3); $\Gamma^i_{00}=\partial_i\Phi/c^2$ by symbolic computation (E-geo-4); recovery of Newton $\mathbf{a}=-\nabla\Phi$ (E-geo-5).
\textit{The geodesic as free motion that extremizes the proper time over the emergent effective metric; corollary of consistency and completeness that closes the matter-geometry pair together with (E35). Newtonian limit \epi{DERIVED}, coincident with \S{}III.1.2 by an independent route.}

\smallskip
\textbf{(E-Sch-1)--(E-Sch-7)} --- \S{}III.1.10. Effective potential $\Phi=-GM/r$ from the static configuration of the modulus (E-Sch-1); $g_{00}=-(1-2GM/c^2r)$ by consistency with Newton (E-Sch-2); $R_{\mu\nu}=0$ in the relaxed exterior (E-Sch-3); $AB=\text{const}$ from the Ricci and asymptotic matching (E-Sch-4); $A=1-r_s/r$, $B=1/A$ (E-Sch-5); $r_s=2GM/c^2$ by Newtonian matching (E-Sch-6); complete Schwarzschild metric (E-Sch-7).
\textit{The Schwarzschild metric as the geometric reading (\S{}III.1.7) of the static configuration of the modulus, forced in its form by the emergent Einstein equations (E35). Corollary \epi{DERIVED}/\epi{SLD} with a sonic horizon (\S{}III.9.6) and a regular core by A6 (\S{}IV.7, \S{}II.6.2); the two cures are its distinctive content. The exact coefficient of $r_s$ from first principles refers to settled in \S{}III.1.4.}

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\textbf{Thematic appendix \S{}V.7 --- Dirac structure and descent of the constants.}

\smallskip
\textbf{(E-Dirac-1)--(E-Dirac-12)} --- \S{}V.7. \emph{Band route} (\S{}V.7.1): bipartite off-diagonal Bloch Hamiltonian $H(\mathbf{k})$ (E-Dirac-1); $|f(\mathbf{k})|^2$ and nodal lines P7 (E-Dirac-2); Fu-Kane spin-orbit term (E-Dirac-3); $SU(2)$ holonomy $W=\mathbb{1}-2i\theta^2\sigma_z$ (E-Dirac-4); equivalence $\lambda_\text{SO}=0\Leftrightarrow$ trivial holonomy $\Leftrightarrow\theta_-\in U(1)$ (E-Dirac-5); Weyl operator $v_F\,\Gamma_i q_i$ (E-Dirac-6); Clifford algebra $\{\Gamma_i,\Gamma_j\}=2\delta_{ij}$ (E-Dirac-7); mass term $\Gamma_0$ (E-Dirac-8); relation $E^2=c^2q^2+m^2c^4$ (E-Dirac-9); covariant Dirac equation $(i\hbar\Gamma^\mu\partial_\mu-mc)\Psi_D=0$ (E-Dirac-10). \emph{Covariant route} (\S{}V.7.2): gamma matrices from the two binary structures with emergent Lorentzian signature (E-Dirac-11); regulated Dirac operator as the root of the phase spectrum (E-Dirac-12).
\textit{The Dirac structure as the low-energy effective theory of the phase modes $\theta_-$ over the bipartite diamond lattice; \emph{structure} \epi{DERIVED} by the covariant route (gamma matrices, Pauli uniqueness, Lorentz spin-$\tfrac12$, spin-statistics), with $\lambda_\text{SO}$ and the masses $m$ as a thread of \emph{values} (reconnected with \S{}III.4.7). Same $SU(2)$ holonomy that sustains Yang-Mills (\S{}III.4.5).}

\smallskip
\textbf{(E-WP-1)--(E-WP-7)} --- \S{}V.7.2. Wien's law (E-WP-1); anatomy of the Wien constants $c_W=8\pi h c$, $\alpha_W=hc/k_B$ (E-WP-2); structural identities $\hbar=\rho_P c\ell_P^4$, $G=c^2/(\rho_P\ell_P^2)$ (E-WP-3); $\hbar$ as the momentum of a cell (E-WP-4); table of the ontological descent (E-WP-5); $\rho_P=c^5/(\hbar G^2)$ (E-WP-6); reduction of F1 to Gross-Pitaevskii (E-WP-7).
\textit{The ontological descent of the constants: those of Wien are traces of $(h,c,k_B)$; $\hbar$ and $G$ are traces of $(\rho_P,\ell_P)$. Interpretive-historical subsection; reorders already-derived constants (\S{}I.8.5, \S{}III.1.3, \S{}III.2.2) and traces the route of closure of C-meta-1.b by saturation density.}

\medskip\hrule\medskip

\textbf{Part V --- observational confrontation.} Part V reuses the catalogue (E1)--(E62) derived in Parts I--IV and adds two equations of its own, both of method: (E63), the pair solved with general coefficients that underpins the falsification test of \S{}V.2.3-bis, and (E64), the factor against the coincidence hypothesis of \S{}V.2.6. For the rest it is a part of organization, confrontation and declaration. The numerical confrontation of \S{}V.2 reuses mainly \textbf{(E3)} ($G$), \textbf{(E36)} ($\hbar$), \textbf{(E37)} ($\Omega_{MO}/\Omega_M=\sqrt{3}\pi$) and \textbf{(E33)} ($\Omega_{EO}=f_c$). The catalogue of predictions \S{}V.1 references the origin subsections of each result.

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\textit{[Catalogue of equations consolidated for Parts I--V. Complete manuscript.]}


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\subsection{\S{}V.6.3 --- Index of figures}

The thirteen figures of the manuscript, with the reading of what each one establishes and the
page where it appears. Most share the format of labeled panels
A--D with the parameters derived without fitting indicated in the caption; the trajectory of
$f_c$ uses two panels, and the last is the diagram of the derivational architecture
of the program.

{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{3.0cm}L{9.2cm}L{1.2cm}}
\toprule
\textbf{Figure} & \textbf{What it shows and what it establishes} & \textbf{Pg.} \\
\midrule\endhead
Effective potential $V_\text{eff}(S)$ & Central object of the program. Complete potential on logarithmic scale (Klauder IR wall, well at $S_{\min}=0.7231$, Gent wall at $S_{\max}=1.4756$); decomposition into the three derived components ($W_\text{bare}$, $V_\text{topo}$, $V_q$); zoom of the well with the curvature $V''(S_{\min})=19.69\,\rho_P c^2$; packing $x$ and dilatant index $n(x)$. Fixes the \emph{Pleno}-ground state and the hard bound. & \pageref{fig:potencial} \\
Diamond lattice $Fd\bar{3}m$ & Structure that \emph{el Pleno} adopts in its saturated phase. Conventional cell with A/B bipartition; tetrahedral coordination $Z=4$ (Maxwell isostatic minimum); the spectral average of the lattice Laplacian produces $S_{\min}=12(\pi^2-4)/\pi^4$. & \pageref{fig:diamante} \\
Mode spectrum & Dispersion relation of \emph{el Pleno}: the radial mode is massive (gap $\sqrt{\xi}\,c/\ell_P$, elastic mass $4.44\,E_P$ and propagating $3.50\,E_P$, \S{}III.1.9), the phase and TT modes are massless and are the observables. Explains the non-detection of \emph{el Pleno} as a particle. & \pageref{fig:dispersion} \\
Cosmological chronology & The complete crunch-bounce cycle: compression from $S_{\min}$, braking by the Gent wall before $S_{\max}$, bounce without singularity, freeze-out and return to the relaxed regime. Asymmetric cycle; our universe is the phase after the last bounce. & \pageref{fig:cronologia} \\
Uncertainty of \emph{el Pleno} & The uncertainty relation as a property of \emph{el Pleno}: convergence of $\sqrt{20}$ in three sectors (scale $\ell_P/\sqrt{20}$), the GUP form $\Delta x\,\Delta p=(\hbar/2)\sqrt{1+(3/80)(\ell_P/\sigma)^2}$ with exact floor $\hbar/2$, the two scales instantaneous/relaxed and their ratio $\sqrt[4]{5}$, and the deficit $\xi<20$ as a condition of existence of the oscillating mode. & \pageref{fig:incertidumbre} \\
Mass limits of \emph{el Pleno} & The Chandrasekhar limit as an internal balance: anatomy of the coefficient ($m_P$, Planck-hadronic hierarchy, polytrope $n=3$) and the three mass limits of the medium ($M_{\min}$, $M_Q$, $M_\text{Ch}\approx 1.456\,M_\odot$), all balances of the same \emph{el Pleno}. & \pageref{fig:limites_masa} \\
Quantitative cosmic background & The cosmic background of \emph{el Pleno} against the data: $D_H/r_d$ vs BAO DESI DR2 (5 points, $\chi^2=6.54$ against $6.53$ of $\Lambda$CDM); the expansion history $E(z)$ with the form forced by the potential; the normalized residuals; and the equation of state $w(z)\geq -1$ with a shallow minimum toward $z\approx 0.5$. The closed background sector, with $\delta_0$ as the only freedom. & \pageref{fig:fondocosmico} \\
Trajectory of $f_c$ and status of $S_{\max,\text{eff}}^2$ & Why the numerator of $f_c$ is positional \epi{CALIBRATED}, not derivable as a number. (A) The effective well $V_\text{eff}$ (derived closed form) with the rest at $S_{\min}=0.7231$ \epi{DERIVED} and the upper turning point $S_{\max,\text{eff}}=1.2229$ fixed by the available oscillation energy today $E_\text{disp}(t_0)=10.003\,\rho_P c^2$ (positional, like $H_0$). (B) The cosmic clock $f_c(t)$ growing toward its hard bound $f_c<1$ by re-condensation (second law), with TODAY ($f_c=0.687$) marked as a position, not as a constant. Fixes the nature of the only calibrated factor of the cosmological sector. & \pageref{fig:fc_trayectoria} \\
Rheological curve & \emph{El Pleno} as a Herschel-Bulkley dilatant fluid: flow index $n(x)=1/(1-x)>1$ (shear-thickening), flow curve $\tau=\tau_0+K\dot\gamma^n$ with yield threshold from the IR wall. The current universe at $x=0.240$, almost Newtonian. & \pageref{fig:reologia} \\
Gent wall & The potential and the restoring force exploding toward $S_{\max}$; decomposition into the three terms (only Gent fires); root cause in the gap $(S_{\max}^2-S^2)^3$ of the denominator. The regulator of the bounce. & \pageref{fig:pared_gent} \\
Integration of the cycle & Numerical integration of the equation of the cycle (E57): with dilatant viscosity the modulus oscillates bounded by $x\leq 1$ and the bounce remains subluminal; without the dilatant mechanism the rate exceeds the bound by three orders (impossible). The dilatancy is the regulator of the rebound. & \pageref{fig:integracion} \\
Particle formation & The crystallization plateau of the crunch-bounce cycle: the trajectory $S(t)$ from the rest at $S_{\min}$, the compression ($S^2\propto 1/a^3$), the bounce regulated by dilatancy and the formation of the three real particles (proton, neutron, electron) with their origin, moment, conditions and mass, consistent with the hadronic sector of the program. & \pageref{fig:formacion} \\
Viscous crystallization transient & The dynamics of the radial mode that imprints the partition $f_c$ (\S{}IV.5-ter): integration of the viscous equation of motion with all derived coefficients. The modulus $S(t)$ descends from $S^*=1.033$ to $S_{\min}=0.7231$ and settles in a few $t_P$; the deformation rate $\dot\gamma(t)$ reaches $0.98$ without exceeding the Planck bound $\dot\gamma_{\max}=c/\ell_P=1$, which the dilatant viscosity self-limits. It shows the temporal mechanism ---transient $\sim t_P$, rest for the remainder of history--- by which the geometric partition is imprinted. & \pageref{fig:transitorio} \\
Derivational chain from $E$ & Complete derivational architecture of the program: common trunk (genesis A1$\to$F1 and geometry of the diamond lattice) and bifurcation into the cosmological branch (up to $\Omega_m=1-f_c$ and the expansion history $H(z)$) and the quantum branch (up to the uncertainty relation with exact floor $\hbar/2$). The color of the border encodes the epistemic classification of each link; it makes visible that the quantum branch is derived end-to-end, while the cosmological one inherits the only declared calibrated factor. & \pageref{fig:cadena} \\
\bottomrule
\end{longtable}}

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