% =====================================================================
% PIU-Psi V5.0 -- Mathematical Compendium -- PART 4: Chain, tables, symbol index, appendix, bibliography (sec. 13-17)
% (fragment; compiled via \input from main.tex)
% =====================================================================

\section{The complete derivational chain --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{SLD}}{[SLD]} / \texorpdfstring{\T{CALIBRATED}}{[CALIBRATED]}}
\label{sec:cadena}
% =====================================================================

Figure~\ref{fig:cadena} traces, step by step and without gaps, the two longest chains
that depart from the root statement $E$ (A1): a common trunk ---the genesis A1$\to$F1, and from
\emph{evaluating} F1 the geometry of the diamond lattice with its two pure ratios--- from which,
at a bifurcation point, hang the cosmological branch (up to $\Omega_m=1-f_c$ and $H(z)$)
and the quantum branch (up to the uncertainty relation with floor $\hbar/2$ exact). Each
box carries its equation and the color of its epistemic classification, so that the
structure is that of a continuous derivational chain, not a collection of numerical
coincidences; the only \epi{CALIBRATED} link, the numerator $S_{\max,\text{eff}}^2$ of the
cosmological branch, appears marked in its color.

\textbf{The guiding thread, in words.} The logic of the program is that of a single
generative act followed by evaluations. \emph{(0) The root.} A single statement ---the universe
is finite and conserved energy (A1)--- read by region gives the density $\rhoP$, and the
conservation expressed as the balance of two orthogonal branches gives the field $\Psi=Se^{i\theta}$.
\emph{(1) The generative act.} Applying the Lagrangian procedure to that field under the
six axioms produces F1, the only object of the program with dynamical content; everything else
is a consequence of \emph{evaluating} it. \emph{(2) The first evaluation.} The medium crystallizes
in the diamond lattice (the one that minimizes the flexural of F1), and from the lattice come the two pure
ratios ---$\Smin^\text{estr}$ and the $1/40$--- that F1 carried written without deriving: the
Lagrangian closes upon itself. \emph{(3) The harvest.} From here, each sector
of physics is a distinct reading of the same evaluated F1:
\begin{itemize}[leftmargin=1.4em]
\item \textbf{Constants} ---$G=c^2/(\rhoP\lP^2)$ and $\hbar=\rhoP c\lP^4$--- from the statics
of the modulus and the quantisation of the phase rotor, with unit coefficient and no fitting;
\item \textbf{Gravity} ---Newton's three laws and the Einstein equations (E35)--- as the
low-energy limit and the effective elastic action of the TT mode (Sakharov), with $G$ already
derived;
\item \textbf{Quantum} ---Schrödinger, Born, uncertainty, Bell--- as the dynamics of the
phase perturbations, with $\hbar$ derived;
\item \textbf{Gauge sector} ---Maxwell and Yang-Mills--- as the circulation of the phase over
the plaquettes of the lattice;
\item \textbf{Matter} ---the solitons as particles, with spin and charge--- as the topological
knots of the field;
\item \textbf{Thermodynamics} ---the equilibrium occupation and the coefficient $2\pi^2/45$ of
Stefan-Boltzmann--- as the count of configurations of the eigenmodes of F1;
\item \textbf{Cosmology} ---$\Omega_m$, $\sqrt3\pi$, $w=-1$, and $H_0$ as a positional reading of the clock of el Pleno--- as the same medium read at
the scale of the universe;
\item \textbf{Rheology} ---the crunch-bounce cycle--- that closes the circle identifying the
Lorentz factor of link 1 with the flow index of the medium.
\end{itemize}
The figure traces explicitly the two deepest branches (cosmology and quantum) as being
the longest causal chains; the others hang from the same trunk at the
bifurcation point. The structure is a tree with a single root, not a network of independent
postulates: that is the central assertion of the program, and the chain makes it literal.

\begin{figure}[H]
\centering
\scalebox{0.60}{%
\begin{tikzpicture}[node distance=2.1mm,
  base/.style={rectangle, rounded corners=2pt, draw, semithick, align=center,
               inner sep=4pt, minimum height=7mm, text width=80mm},
  root/.style={base, draw=cRoot, fill=cRoot!10, text width=92mm},
  tronco/.style={base, draw=cTronco, fill=cTronco!8},
  deriv/.style={base, draw=cDeriv, fill=cDeriv!7},
  dlf/.style={base, draw=cDLF, fill=cDLF!7},
  calib/.style={base, draw=cCalib, fill=cCalib!10},
  bifur/.style={diamond, aspect=2, draw=black, very thick, fill=yellow!15,
                align=center, font=\bfseries\footnotesize, inner sep=1pt, text width=34mm},
  flecha/.style={-{Stealth[length=2.6mm]}, semithick, draw=black!60},
  ramaL/.style={-{Stealth[length=2.6mm]}, semithick, draw=cCalib!80},
  ramaR/.style={-{Stealth[length=2.6mm]}, semithick, draw=cDeriv!80}]

\providecommand{\eqlabelF}[1]{{\scriptsize\bfseries\color{black!55}#1}}
\providecommand{\glosaF}[1]{{\scriptsize\color{black!75}#1}}

\node[root] (E) {\eqlabelF{A1 --- root statement}\\[2pt]
  $E$: the universe is energy, \ $0<E<\infty$ \ (finite), conserved\\[1pt]
  \glosaF{single substrate, homogeneous, isotropic}};
\node[tronco, below=of E] (rho) {\eqlabelF{---}\\[2pt]
  $\rho_P \;\equiv\; M \;\equiv\; E$\\[1pt]
  \glosaF{$E$ read by region $=\rho_P$; chain of identities}};
\node[tronco, below=of rho] (psi) {\eqlabelF{E1}\\[2pt]
  $\Psi = S\,e^{i\theta}$\\[1pt]
  \glosaF{two branches at $90^\circ$: \ $S\in[S_{\min},S_{\max}]$ \ / \ $C\in[0,c]$}};
\node[tronco, below=of psi] (F1) {\eqlabelF{E2 --- {\normalfont\scshape starting point}}\\[2pt]
  $\mathcal{L}_\text{F1}=\tfrac12\rho_Pc^2\ell_P^2\,\partial_\mu\Psi^*\partial^\mu\Psi-V(S)-\tfrac{\rho_Pc^2\ell_P^4}{40}(\Box S)^2-V_q$\\[1pt]
  \glosaF{Lagrangian of el Pleno; coefficients to be derived upon evaluating it}};
\node[deriv, below=of F1] (diamante) {\eqlabelF{E13 \ (evaluation of F1)}\\[2pt]
  $D_\text{BZ}=\dfrac{\langle D_\text{lat}\rangle_\text{BZ}}{\langle D_\text{lat}^\text{par}\rangle_\text{BZ}}=\dfrac{12(\pi^2-4)}{\pi^4}$\\[1pt]
  \glosaF{diamond lattice $Fd\bar3m$, $Z{=}4$: minimizes the flexural of F1}};
\node[deriv, below=of diamante] (razones) {\eqlabelF{E9 \ ($\gamma_\kappa$ in E2)}\\[2pt]
  $\displaystyle S_{\min}^\text{estr}=\frac{12(\pi^2-4)}{\pi^4}\approx0.7231 \qquad \gamma_\kappa=\frac{1}{40}$\\[1pt]
  \glosaF{the two pure ratios of the medium, without an external yardstick}};

\draw[flecha] (E)--(rho); \draw[flecha] (rho)--(psi); \draw[flecha] (psi)--(F1);
\draw[flecha] (F1)--(diamante); \draw[flecha] (diamante)--(razones);

\node[bifur, below=3mm of razones] (bif) {BIFURCATION\\ POINT};
\draw[flecha] (razones)--(bif);

\node[deriv, below left=4mm and 5mm of bif] (smaxgeom)
  {\eqlabelF{E10}\\[2pt] $\displaystyle S_{\max,\text{geom}}^2=\frac{8}{3}\sqrt{\frac{2}{3}}\approx2.1773$\\[1pt]
   \glosaF{Kepler-Hales packing (Hales theorem)}};
\node[calib, below=of smaxgeom] (smaxeff)
  {\eqlabelF{--- \epi{CALIBRATED}}\\[2pt] $S_{\max,\text{eff}}^2=1.4956$\\[1pt]
   \glosaF{effective occupation of the present regime}};
\node[deriv, below=of smaxeff] (fc)
  {\eqlabelF{E11}\\[2pt] $\displaystyle f_c=\frac{S_{\max,\text{eff}}^2}{S_{\max,\text{geom}}^2}=\frac{1.4956}{2.1773}=0.6869$\\[1pt]
   \glosaF{condensate fraction (derived denom.\ / calibrated numer.)}};
\node[deriv, below=of fc] (omegaEO)
  {\eqlabelF{---}\\[2pt] $\Omega_{EO}\equiv f_c\approx0.6869$\\[1pt]
   \glosaF{dark energy = condensed curtain; $0.30\sigma$ vs Planck}};
\node[deriv, below=of omegaEO] (omegam)
  {\eqlabelF{E-cosmo-2}\\[2pt] $\Omega_m=1-f_c=0.3131$\\[1pt]
   \glosaF{$f_c+\Omega_\text{depleted}=1$; \ $0.27\sigma$ vs Planck, without fitting}};
\node[dlf, below=of omegam] (split)
  {\eqlabelF{E37}\\[2pt] $\dfrac{\Omega_{MO}}{\Omega_M}=\sqrt{3}\,\pi\approx5.4414$\\[1pt]
   \glosaF{$\Rightarrow\Omega_M{=}0.0486\,(1.2\sigma)$, \ $\Omega_{MO}{=}0.2645\,(0.07\sigma)$}};
\node[dlf, below=of split] (Hz)
  {\eqlabelF{E-cosmo-3}\\[2pt] $E^2(z)=H^2(z)/H_0^2$ \ (forced by $V_\text{eff}$)\\[1pt]
   \glosaF{$\rho_m\propto a^{-3}$ topological; fits BAO/DESI without parameters}};
\node[calib, below=of Hz] (H0)
  {\eqlabelF{E-cosmo-5}\\[2pt] $H_0=\sqrt{\tfrac{8\pi}{3}}\,\dfrac{c}{\ell_P}\sqrt{\Theta}$\\[1pt]
   \glosaF{$\Theta=3/8\pi N_\text{tic}^2$: ``$10^{-123}$''$=$ age in ticks; $H_0$ positional ($\sim67$, primordial side), tension with sign; occupation $f_c=0.69$ of the condensate at rest}};

\draw[ramaL] (bif) -| (smaxgeom); \draw[ramaL] (smaxgeom)--(smaxeff);
\draw[ramaL] (smaxeff)--(fc); \draw[ramaL] (fc)--(omegaEO);
\draw[ramaL] (omegaEO)--(omegam); \draw[ramaL] (omegam)--(split); \draw[ramaL] (split)--(Hz);
\draw[ramaL] (Hz)--(H0);

\node[deriv, below right=4mm and 5mm of bif] (hbar)
  {\eqlabelF{E36}\\[2pt] $\hbar=\rho_P\,c\,\ell_P^4$\\[1pt]
   \glosaF{quantisation of the $U(1)$ rotor of the zero mode; prefactor $1$ derived, not fitted}};
\node[deriv, below=of hbar] (gammak)
  {\eqlabelF{E4}\\[2pt] $\gamma_\kappa=\dfrac{\rhoP c^2\lP^4}{40}=\dfrac{\hbar c}{40}$\\[1pt]
   \glosaF{native identity: flexural rigidity = quantum property of the medium}};
\node[deriv, below=of gammak] (sigmamin)
  {\eqlabelF{---}\\[2pt] $\sigma_{\min}=\dfrac{\ell_P}{\sqrt{20}}\approx0.2236\,\ell_P$\\[1pt]
   \glosaF{localizability: elastic, flexural and dissipative sectors}};
\node[deriv, below=of sigmamin] (einc)
  {\eqlabelF{E-inc-3}\\[2pt] $\Delta x\cdot\Delta p=\dfrac{\hbar}{2}\sqrt{1+\dfrac{3}{80}\!\left(\dfrac{\ell_P}{\sigma}\right)^2}$\\[1pt]
   \glosaF{coefficient $3/80$ derived, without a free parameter (GUP)}};
\node[deriv, below=of einc] (heisenberg)
  {\eqlabelF{---}\\[2pt] $\sigma\gg\ell_P:\quad \Delta x\cdot\Delta p\to\dfrac{\hbar}{2}$ \ \textbf{exact}\\[1pt]
   \glosaF{floor $\hbar/2$ recovered; sign $\geq$ = flexural rigidity $>0$}};

\draw[ramaR] (bif) -| (hbar); \draw[ramaR] (hbar)--(gammak);
\draw[ramaR] (gammak)--(sigmamin); \draw[ramaR] (sigmamin)--(einc);
\draw[ramaR] (einc)--(heisenberg);

\node[font=\bfseries, text=cCalib] at ($(smaxgeom.north)+(0,4.5mm)$) {COSMOLOGY BRANCH};
\node[font=\bfseries, text=cDeriv] at ($(hbar.north)+(0,4.5mm)$) {QUANTUM MECHANICS BRANCH};

\end{tikzpicture}}
\caption{The complete derivational chain from $E$. Common trunk: the genesis
A1$\to$F1 and, as the \emph{first evaluation of F1}, the geometry of the diamond lattice with
its two pure ratios ($S_{\min}=0.7231$, $\gamma_\kappa=1/40$). At the bifurcation
point two branches open: to the left the cosmology (up to $\Omega_m=1-f_c$ and
$H(z)$), to the right the quantum mechanics (up to the uncertainty with floor $\hbar/2$
exact). The color of the border encodes the epistemic classification:
\textcolor{cRoot}{\textbf{root}}, \textcolor{cTronco}{\textbf{trunk}},
\textcolor{cDeriv}{\epi{DERIVED}}, \textcolor{cDLF}{\epi{SLD}},
\textcolor{cCalib}{\epi{CALIBRATED}}. The two \emph{positional} links of the cosmological branch appear marked in their colour: the
effective occupation $S_{\max,\text{eff}}^2$ and, through the tick count, $H_0$. Neither is a
fitted parameter: they are state quantities of the medium, with derivable form and a historical
present value. The quantum branch, by contrast, is derived from end to end and inherits no
positional factor.}
\label{fig:cadena}
\end{figure}

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- what the chain means}]
The proof that a physical program is a theory and not a fit is that its results
be \emph{chained}: that they follow one from another from a common origin, without inserting
new assumptions at each step. That is what the chain exhibits. From a single statement
---finite and conserved energy--- and a single dynamical object ---F1---, the great pillars
of modern physics (the gravitation constant, that of Planck, the Einstein
equations, quantum mechanics, electromagnetism, the structure of matter, the
content of the universe) appear as successive evaluations, not as added
ingredients. The quantum chain is derived from end to end; the cosmological one inherits a
single \emph{positional} factor ---the present value $S_{\max,\text{eff}}^2$, a state
quantity of the medium analogous to $H_0$, not a fitted parameter--- declared in its color. That
so many apparently independent facts hang from the same root, with a single state factor in the whole
tree, is the strongest ---and most falsifiable--- assertion of the program: if
some link breaks, the entire structure registers it. And that exposure is quantified where the
tree touches measurement: the two coefficients accompanying $G$ and $\hbar$ equal unity by
derivation, not by fitting, and measuring either primary by a route that passes through neither
gravitation nor quantisation puts them to the test ---the product if one is measured, each branch
separately if both are.
\end{tcolorbox}

\medskip\hrule\medskip

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

\section{Confrontation with observation --- \texorpdfstring{\T{DERIVED}}{[DERIVED]} / \texorpdfstring{\T{CALIBRATED}}{[CALIBRATED]}}
% =====================================================================

Master table of predictions and their status. ``Free param.'' counts those fitted
\emph{to that datum}.

\begin{center}\small
\begin{longtable}{L{4.6cm}C{2.3cm}C{2.6cm}C{1.3cm}C{1.7cm}}
\toprule
\textbf{Quantity} & \textbf{PIU-$\Psi$} & \textbf{Observed} & \textbf{Param.} & \textbf{Agreement}\\
\midrule
\endhead
$G$ & $c^2/(\rhoP\lP^2)$ & $6.6743\!\times\!10^{-11}$ & 0 & $\kappa_G=1$\\
$\hbar$ & $\rhoP c\,\lP^4$ & $1.0546\!\times\!10^{-34}$ & 0 & $\kappa_\hbar=1$\\
$\Omega_{MO}/\Omega_M=\sqrt{3}\pi$ & $5.4414$ & $5.375\pm0.077$ & 0 & $0.86\sigma$\\
$f_c$ ($=\Omega_\Lambda$) & $0.6869$ & $0.6847\pm0.0073$ & 0$^\dagger$ & $0.30\sigma$\\
$\Omega_m=1-f_c$ & $0.3131$ & $0.315\pm0.007$ & 0$^\dagger$ & $0.27\sigma$\\
$\Omega_M$ (baryonic) & $0.0486$ & $0.0493\pm0.0006$ & 0$^\dagger$ & $1.2\sigma$\\
$\Omega_{MO}$ (dark) & $0.2645$ & $0.2645\pm0.007$ & 0$^\dagger$ & $0.07\sigma$\\
$H(z)$ BAO DESI DR2 & $\chi^2=6.54$ & ($\Lambda$CDM 6.53) & 1 & $\equiv\Lambda$CDM\\
$H_0$ background & $67.4$--$67.7$ & $67.4\pm0.5$ Planck & 1$^\dagger$ & coincides\\
$H_0$ tension & method artefact & predicted dissolution & 0 & falsifiable\\
Condensate occupation $f_c$ & $0.6869$ & rest, $1-f_c$ coh. & 0 & internal\\
$H_{\max}/\dot\gamma_{\max}$ & $4.2709$ & exact equality & 0 & structural\\
$\Delta m\,(\Delta\!-\!N)$ & $264.1$ MeV & $293.08$ MeV & 1 & $90\%$\\
$R_p$ (proton radius) & $0.851$ fm & $0.8409$ fm & 0 & $98.8\%$\\
$m_\Delta$ & $1202$ MeV & $1232$ MeV & 1 & $97.6\%$\\
$n_s$ (zeroth order) & $1$ & $0.965\pm0.004$ & --- & red tilt\\
SN Ia dilation ($b$) & $1$ exact & $1.003\pm0.011$ & 0 & $0.27\sigma$\\
$z_{\max}$ radial & $0.268$ & two clocks & 0 & structural\\
$\mu$ (blackbody) & $0$ & $<9\!\times\!10^{-5}$ & 0 & exact\\
$v_\gamma=v_\text{GW}$ & $1$ (IR) & $<7\!\times\!10^{-16}$ & 0 & consistent\\
$\rho_\text{PBH core}$ & $2.1773\,\rhoP$ & ---(prediction) & 0 & falsifiable\\
\bottomrule
\end{longtable}
\end{center}
{\small $^\dagger$ The cosmological block shares a single calibrated ingredient,
$S_{\max,\text{eff}}^2$ (which fixes $f_c$) ---a positional trajectory value, analogous to
$H_0$, not a timeless constant (§4.4)---; from it derive $\Omega_\Lambda$, $\Omega_m$,
$\Omega_M$, $\Omega_{MO}$ without additional fits.}

\smallskip
{\small \textbf{How to read the first two rows.} For $G$ and $\hbar$ the agreement column carries
no percentage. Evaluating the two identities with the primaries published to four figures returns
the measured values with a residue below $0.1\hspace{0.08em}\%$ that is the rounding of those
primaries, not a physical difference; displaying it as a figure of agreement would measure the
precision of the input, not that of the programme. What the two rows confront is the
\emph{coefficient}: the identities follow from the dynamics of F1 and return
$\kappa_G=\kappa_\hbar=1$ with no fitting.}

\smallskip
{\small \textbf{Three results the table cannot carry as a row, and why.} The thermal occupation
and the coefficient $2\pi^2/45$ reproduce \emph{established} results of statistical physics: their
confrontational value is not hitting a measured number but obtaining it from the configuration
count of F1 with no quantum statistics assumed. The sub-bound $R=0.7580$ and the window
$[1,\sim\!2.5]$ of $\eta/s$ bound a ratio whose exact value remains open; they are confronted as a
\emph{strip} against the quark-gluon plasma, not as a point. And the exclusion of the local $H_0$
by the age bound is an \emph{inequality} ---the ceiling $A\le0.953$ against $A(73)=1.029$, with a
margin that does not depend on the exact value of $\Omega_m$---, which the ``$H_0$ tension'' row
records as a falsifiable prediction.}

\textbf{The objection of numerological coincidence, put to the test.} The objection admits two
readings that must be neither conflated nor combined, and the programme computes them
separately.

\emph{Narrow reading: how many formulas?} Enumerating computationally the space of algebraic
expressions over $\{1,2,3,4,\pi,e\}$ up to complexity $N=5$ ---3778 distinct expressions---,
the probability that two of them reproduce simultaneously $\sqrt3\pi$ and $f_c$ within the
$2\sigma$ window is $P<3\times10^{-4}$, that is a factor $K_c\approx6.71\times10^5$ against
reading those two ratios as lucky formulas. Its scope is the one proper to it: it bounds the
narrow version for two isolated ratios.

\emph{Reading proper to a complete programme: how many frameworks?} Fifteen results admitted
under four criteria declared before counting ---that the outcome discriminate, that there be no
freedom of adjustment, that the target was established, and that the measurement channel be its
own---, read with fifteen instruments that share no mechanism. Another twelve are left
\emph{outside} the computation with their reason published: not for being incorrect ---all are
derived and several are among the most solid--- but for contributing no \emph{independent}
evidence to that count. Each exclusion lowers the factor: the count was built to be demanding
with itself. With $\prod_i p_i=8.42\times10^{-10}$ and the correction $(\prod_i p_i)^{\lambda}$
for the shared derivational trunk,
\begin{equation}
\text{BF}(\lambda{=}0.5)\approx3.4\times10^{4},
\qquad
\text{range }\big[\,6.5\times10^{1}\ (\lambda{=}0.2),\ 1.2\times10^{9}\ (\lambda{=}1)\,\big].
\tag{E64}
\end{equation}
It is always reported with its range, never as a point. \T{method}.

\emph{Robustness.} Tripling \emph{all} the $p_i$ and further adopting $\lambda=0.35$, the
factor remains above $10$. Under the most punitive assumption ---$\lambda=0.2$, which amounts
to declaring the fifteen results almost a single fact--- it is still $65$. And removing one
item at a time, no individual omission lowers it by more than a factor of $4.5$.

\textbf{Scope of the argument, declared.} A Bayes factor compares two hypotheses and validates
neither: that coincidence is disfavoured does not establish that the programme is correct, and
the space of alternative theories that would explain the same facts by another mechanism has
not been populated. The two figures answer different questions ---how many formulas against how
many frameworks--- and for that reason they are neither multiplied nor averaged. To this is
added, as an argument of another nature that does not enter the factor, the coherence across
sectors: that the \emph{same} medium, with the same scales $(\rhoP,\lP,c)$, reproduces at once
$G$, $\hbar$, the Heisenberg floor $\hbar/2$ and the Chandrasekhar limit with its complete
coefficient. That coherence is not added to the factor, so as not to mix two kinds of evidence.

\medskip
\begin{tcolorbox}[clave,title={Conclusion --- the complete chain}]
A single statement ---\emph{the universe is finite and conserved energy} (A1)--- fixes the
form of the field and with it the Lagrangian F1. The evaluation of F1 gives the diamond lattice and
its two pure ratios ($1/40$, $\Smin$), the potential ($\Smax$, $E_0$) and the constants
($G$, $\hbar$, with unit coefficient and no fitting). From F1 emerge in cascade the modes, gravity (Einstein),
the quantum (Heisenberg, Born, Bell), the gauge sector (Maxwell, Yang-Mills), the
particles (solitons, $\Delta$-$N$) and cosmology ($\sqrt{3}\pi$, $\Omega_m$, the
bounce). The confrontation yields four fundamental constants and proportions
reproduced with \emph{zero} free parameters and sub-$\sigma$ agreements, equaling
$\Lambda$CDM in $H(z)$ with less freedom, without a single result outside the
observational bounds, and with a Bayes factor in the decisive range. What distinguishes the
program is not to fit better, but to \emph{derive} ---in a single continuous chain--- what
the standard model parametrizes. The open ends ---the most prominent, the
fermionic electroweak cascade that would fix the lepton masses--- are
declared, not concealed: that transparency is what distinguishes a map of foundations
from a collection of coincidences. The present value $S_{\max,\text{eff}}^2(t_0)$ is not an
open end but the positional anchor of the cosmological block, analogous to $H_0$ ---which the program reads
as the hour of a cosmic clock, reformulating the $10^{-123}$ fine-tuning of the cosmological
constant as the age of the universe in ticks and establishing the Hubble tension as a
method artefact that dissolves in model-independent observables
(§11.9).
\end{tcolorbox}

\medskip\hrule\medskip

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

\section{Map of open ends --- \texorpdfstring{\T{OPEN END}}{[OPEN END]}}
\label{sec:cabos}
% =====================================================================

Inventory of what the programme has \emph{not} closed, with its severity. This map collects the \emph{mathematical} open ends: those whose computation is missing or incomplete. The fourteen rows cover the open ends of the corpus that have a computational component; those that do not ---the ontological reading of time and condensed matter, and the confrontation ones resting on third-party data--- live in \S{}V.3 of the corpus, which further assigns them priority and a closure hierarchy. The two inventories are consistent: nothing listed here as open is declared closed there, and nothing derived there appears here as pending. In particular, the geometry of the support (D1, in its two pieces $Z=4$ and cubic stacking) and the coefficient $\gamma_G=1$ of the induced action are \emph{not} open ends: they are derived, and so recorded in both documents.

\begin{center}\small
\begin{longtable}{L{3.4cm}L{6.4cm}L{3.2cm}}
\toprule
\textbf{Open end} & \textbf{What is missing} & \textbf{Severity / status}\\
\midrule
\endhead
Embedding $Y(\Phi)=1/2$ & Derivation of the hypercharge embedding that fixes $\sin^2\theta_W$. & Medium. Electroweak sector.\\
Coupling $g$ (Yang-Mills) & Value of the non-abelian gauge coupling from the lattice. & Medium. \\
Individual lepton masses & Mass spectrum of leptons (beyond the baryonic sector). & High. Not addressed.\\
$\eta/s$ (thermal Pleno) & Exact value within the derived window $[1,\sim\!2.5]\times\hbar/4\pi k_B$ (requires the precise collisional relaxation time of the mode gas). The window is already bounded on two sides: KSS below, UV scale $\tau_\Pi$ at $T_{\max}$ above. & Low. Window \epi{SLD}; cold vacuum $s=0$ \derivado. Only the exact value remains.\\
Exact tilt $n_s$ & Numerical value of the red tilt, positional over the bounce trajectory (like $f_c$ and $H_0$, not a fixed number). The mechanism ---FCC$\to$diamond packing transition over the background $\Omega_{EO}=f_c$--- and the red sign and the smallness are derived. & Low. $n_s=1$ zeroth order \derivado; sign/smallness \epi{SLD}; value \epi{CALIBRATED}.\\
F-cosmo: non-linear correction of $H(z)$ & Closed form of the non-linear correction that the Gent/Klauder walls imprint over $E(z)$ with respect to $\Lambda$CDM in the non-linear regime, and its confrontation with raw BAO; plus the prediction of $S_8$ from the phase drag. The object is the \emph{phase sector} (collective response of the condensate; a phase mode that rolled would dilute like matter $\propto L^{-3}$, not like $w=-1$, so that the background is at $\dot\theta_0=0$), not the radial modulus ---the latter is ultra-Planckian ($m_S\approx4.44\,E_P$) and frozen at $\Smin$, as confirmed by the scales built over it not deriving. \emph{Relational} open end, not temporal. The equation of state $w=-1$ ($=T\!\propto\!g$) exact, the attractors $n_s=1$, the positional factorization of $H_0$ (§11.9) and the artefact character of the tension and of the DESI signal are already derived. & Medium. \textbf{Confrontation} open end (a \emph{plus}), not of foundation; the correction \epi{SLD}. The combined BAO+CMB+SNe analysis with raw observables is where the prediction of dissolution of the tension is decided.\\
Interior Schwarzschild metric & Closed form of $g_{\mu\nu}$ in the transition zone $S\to S_{\max}$ (regular core A6); the sonic horizon and the regularity of the core are established. & Medium. Matching with §\ref{sec:schwarzschild} (E-Sch-7).\\
$\lambda_\text{SO}$, masses $m$ (Dirac) & Value of the spin-orbit coupling and of the fermionic masses; the \emph{structure} of Dirac is \epi{DERIVED} (two convergent routes, §\ref{sec:dirac}), only the values remain. Chirality ($SU(2)_L$ vs $SU(2)_R$) and number of generations. & Medium-high. Tied to fermionic masses.\\
UV completeness / unitarity & Proof that the Lee-Wick pole at $\omega^2=20/\lP^2$ leaves a unitary theory in the physical sector; the ultraviolet regulation and the sign of the residue are established, the general demonstration is missing. & Medium. Structural, not numerical.\\
$F_\text{ext}$ coupling of the cycle to Friedmann & Closed form of the external forcing term in the cycle equation (E57), today integrated numerically. & Low. The trajectory is already integrated.\\
Hadronic scale from the bounce & Derivation of $\Lambda_\text{QCD}$ from the crystallization temperature, without importing it from hadronic data. & Medium. Links gauge and cosmology.\\
High states of the rotational tower & Spectrum for $J>3/2$: the rigid moment of inertia ceases to hold and the deformable-rotor correction is needed. & Low-medium. The $\Delta$-$N$ split closes to $90\hspace{0.08em}\%$ without it.\\
C-heal-1 --- defect core & Physical size of the core, today bounded by the healing length of the potential and not computed in closed form. & Low. Enters no derived ratio.\\
C-born-2 --- Nelson prefactor & Coefficient of the stochastic diffusion term; not independent, it is the same $\hbar$ read in another role. & Low. Subordinate to E36.\\
C-meta-1.b --- independent calibration & Measurement of $\rhoP$ or $\lP$ by a route passing through neither gravitation nor quantisation: this is what would activate the test of the coefficients (§\ref{sec:constantes}). & Low as an open end, high as an opportunity.\\
\bottomrule
\end{longtable}
\end{center}

\begin{tcolorbox}[cabo,title={The discipline that sustains the classification}]
The classification of this compendium follows two marking rules. \emph{First}: where the
programme has the structure but not the closed value ---the hypercharge embedding that would
fix $\sin^2\theta_W$, or the exact value inside the $\eta/s$ window---, the open end is named
in this table and the structure keeps its label; the gap is not filled with an estimate
presented as a derivation. \emph{Second}: a result taken from the literature ---Klauder's
$3/8$, Lane-Emden's $2.0182$, the Skyrme moment of inertia--- is marked as an importation at
the point where it enters, and its coefficient is not recalibrated. The two rules are what
give content to the \epi{DERIVED} label: without them it would name anything.
\end{tcolorbox}

\medskip\hrule\medskip

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

\section{Index of symbols and factors}
% =====================================================================

Canonical notation of the compendium: symbol, name, dimensions, value or expression, and the
section where it appears. The \emph{primary parameters} ($\rhoP$, $\lP$, $c$) and the
\emph{pure ratios} ($\gamma_\kappa$, $\Smin^\text{estr}$, $\sqrt{3}\pi$, $8/9$) are the
pieces from which everything else hangs.

{\footnotesize\setlength{\tabcolsep}{3pt}
\begin{longtable}{L{1.9cm}L{3.0cm}L{1.9cm}L{4.6cm}L{1.4cm}}
\toprule
\textbf{Symbol} & \textbf{Name} & \textbf{Dimensions} & \textbf{Value / expression} & \textbf{Section}\\
\midrule\endhead
\multicolumn{5}{l}{\textit{Field and primary properties}}\\
$\Psi$ & Total field of el Pleno & dimensionless & $\Psi=S\,e^{i\theta}$; complete state of the medium & §1.2\\
$S$ & Field modulus & dimensionless & Structural axis, $S\geq 0$; retained ``quantity'' & §1.2\\
$\theta$ & Field phase & dimensionless, $[0,2\pi)$ & Dynamical axis; source of the U(1) current & §1.2\\
$\rhoP$ & Fundamental density & $[kg/m^3]$ & $\approx 5.155\times 10^{96}$; substantive property & §1.4\\
$\lP$ & Coherence scale & $[m]$ & $\approx 1.616\times 10^{-35}$; UV grain; substantive property & §1.4\\
$c$ & Causal structure & $[m/s]$ & $\approx 2.998\times 10^{8}$; not a property but structure (§1.4) & §1.4\\
\midrule
\multicolumn{5}{l}{\textit{Lagrangian F1 and its coefficients}}\\
$\mathcal{L}_\text{F1}$ & Lagrangian of el Pleno (F1) & $[J/m^3]$ & Four terms (E2) & §\ref{sec:F1}\\
$\gamma_\kappa$ & Flexural rigidity & $[J\cdot m]$ & $\rhoP c^2\lP^4/40=\hbar c/40\approx 7.90\times 10^{-28}$ & §\ref{sec:F1}\\
$V(S)$ & Topological potential & $[J/m^3]$ & Sustains stable configurations & §\ref{sec:potencial}\\
$V_q^\text{dens}$ & Quantum IR wall (Klauder) & $[J/m^3]$ & $\tfrac{3}{8}\rhoP c^2/S^2$; regulates $S\to 0$ & §\ref{sec:klauder}\\
$E_0$ & Potential scale & $\times\rhoP c^2$ & $E_0=4.2546\,\rhoP c^2$ & §\ref{sec:potencial}\\
$\xi$ & Well curvature & dimensionless & $V''_\text{eff}(\Smin)/(\rhoP c^2)\approx 19.69$ & §\ref{sec:potencial}\\
\midrule
\multicolumn{5}{l}{\textit{Pure ratios and canonical configurations}}\\
$\Smin^\text{estr}$ & Ground-Pleno & dimensionless & $12(\pi^2-4)/\pi^4\approx 0.7231$ & §\ref{sec:potencial}\\
$S_{\max,\text{geom}}$ & Geometric hard bound & dimensionless & $S_{\max,\text{geom}}^2=8\sqrt{6}/9\approx 2.1773$ & §\ref{sec:smax}\\
$S_{\max,\text{eff}}$ & Effective occupation & dimensionless & $S_{\max,\text{eff}}^2=1.4956$ (calibrated) & §\ref{sec:potencial}\\
$f_c$ & Condensate fraction & dimensionless & $S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2\approx 0.6869$ & §\ref{sec:potencial}\\
$D_\text{lat}$ & Discrete Laplacian & dimensionless & $\nabla^2\leftrightarrow-(3/16\lP^2)D_\text{lat}$ & §\ref{sec:1sobre40}\\
\midrule
\multicolumn{5}{l}{\textit{Structural constants and identities}}\\
$G$ & Gravitational constant & $[m^3kg^{-1}s^{-2}]$ & $c^2/(\rhoP\lP^2)=6.6765\times 10^{-11}$ & §\ref{sec:G}\\
$\hbar$ & Planck constant & $[J\cdot s]$ & $\rhoP c\,\lP^4=1.0540\times 10^{-34}$ & §\ref{sec:hbar}\\
$\sigma_{\min}$ & Localizability scale & $[m]$ & $\lP/\sqrt{20}\approx 0.2236\,\lP$ & §\ref{sec:modos}\\
$\tau_\Pi$ & Relaxation time & $[s]$ & $\lP/(\sqrt{20}\,c)\approx 1.21\times 10^{-44}$ s & §\ref{sec:modos}\\
$m_\text{radial}$ & Radial mode mass & $[GeV/c^2]$ & $\sqrt{\xi}\,E_P\approx 4.44\,E_P$ & §\ref{sec:modos}\\
$m_\text{eff}$ & GP effective mass & $[kg]$ & $\sqrt{\xi}\,m_P\approx 4.44\,m_P$ (coincides with $m_\text{radial}$) & §\ref{sec:gp}\\
$g_\text{eff}$ & GP contact coupling & $[J\cdot m^3]$ & $\tfrac{V^{(4)}_\text{eff}\lP^6}{12}[1-\tfrac{5}{3}\tfrac{(V'''_\text{eff})^2}{V''_\text{eff}V^{(4)}_\text{eff}}]\approx 5.43\times 10^{-94}$ & §\ref{sec:gp}\\
$M_\text{Ch}$ & Chandrasekhar limit & $[kg]$ & $\approx 1.456\,M_\odot$ & §\ref{sec:constantes}\\
\midrule
\multicolumn{5}{l}{\textit{Gravity, elasticity and solitons}}\\
$K_\text{cin}$ & Kinetic rigidity (Young) & $[N]$ & $\rhoP c^2\lP^2/2=c^4/(2G)\approx 6.05\times 10^{43}$ & §\ref{sec:gravedad}\\
$\mu_\text{Gent}$ & Shear modulus (Gent) & $[Pa]$ & $\rhoP c^2/S_{\max,\text{geom}}^2\approx 2.13\times 10^{113}$ & §\ref{sec:gravedad}\\
$\nu$ & Poisson coefficient & dimensionless & $(3K-2\mu)/(2(3K+\mu))\approx 0.328$ & §\ref{sec:gravedad}\\
$\eta_\text{Pleno}$ & Shear viscosity & $[Pa\cdot s]$ & $\mu_\text{Gent}\tau_\Pi\approx 2.56\times 10^{69}$ & §\ref{sec:gravedad}\\
$\sigma_\text{Pleno}$ & Characteristic tension & $[N]$ & $\gamma_\kappa/\lP^2=F_\text{Planck}/40\approx 3.03\times 10^{42}$ & §\ref{sec:gravedad}\\
$\xi_\text{heal}$ & Healing length & $[m]$ & $\lP/\sqrt{\xi}\approx 0.2254\,\lP$ & §\ref{sec:gravedad}\\
$r_s$ & Schwarzschild radius & $[m]$ & $2GM/c^2$ (E-Sch-7) & §\ref{sec:schwarzschild}\\
$M_{\min}$ & Minimal collapse mass & $[kg]$ & $\approx 0.117\,m_P$; from $r_\text{core}=r_S$ & §\ref{sec:schwarzschild}\\
$M_Q$ & Fundamental soliton mass & $[GeV/c^2]$ & $M_P/S_{\max,\text{eff}}$ & §\ref{sec:solitones}\\
$\Theta_p$ & Cluster moment of inertia & $[\text{MeV}\cdot\text{fm}^2]$ & $m_p R_p^2/3$; rotational tower & §\ref{sec:solitones}\\
\midrule
\multicolumn{5}{l}{\textit{Cosmology and rheology}}\\
$\Omega_{MO}/\Omega_M$ & Dark/baryonic matter ratio & dimensionless & $\sqrt{3}\pi\approx 5.4414$ ($0.86\sigma$) & §\ref{sec:cosmologia}\\
$\Omega_m$ & Total matter density & dimensionless & $1-f_c\approx 0.3131$ ($0.27\sigma$) & §\ref{sec:cosmologia}\\
$\Omega_{EO}$ & Dark energy fraction & dimensionless & $f_c\approx 0.6869$ ($0.30\sigma$) & §\ref{sec:cosmologia}\\
$\Theta$ & Dimensionless cosmological density & dimensionless & $\rho_\text{tot}/\rhoP$; $\Theta_0\approx 1.66\times10^{-123}$; $H_0=\sqrt{8\pi/3}(c/\lP)\sqrt{\Theta}$ (positional) & §\ref{sec:cosmologia}\\
$N_\text{tic}$ & Ticks of the Hubble time & dimensionless & $1/(H_0\tau_P)\approx 8.49\times10^{60}$; $\Theta=3/(8\pi N_\text{tic}^2)$ & §\ref{sec:cosmologia}\\
$N_\text{age}$ & Ticks of the observational age & dimensionless & $t_0/\tau_P\approx 8.07\times10^{60}$; agrees with $N_\text{tic}$ to $\sim5\%$ & §\ref{sec:cosmologia}\\
$f_c$ & Condensate occupation & dimensionless & $S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2\approx 0.6869=\Omega_{EO}$; depletion $1-f_c=0.3131$ & §\ref{sec:cosmologia}\\
$S^*$ & Rheological critical point & dimensionless & $\sqrt{\Smin\Smax}\approx 1.0330$; minimal rigidity; crystallization ($\rho\approx\rhoP$); fixed point of $I$ & §\ref{sec:reologia}\\
$I$ & Inversion operator & dimensionless & $S\to\Smin\Smax/S$; exchanges walls; over $\Psi$, conformal inversion $\Psi\to S^{*2}/\overline\Psi$ & §\ref{sec:reologia}\\
$V_D$ & Self-dual Droste potential & $[\rhoP c^2]$ & $(1+I)V_\text{topo}$, exactly even in $u=\ln(S/S^*)$; projector $P_D=\tfrac12(1+I)$ & §\ref{sec:reologia}\\
$n(x)$ & Flow index of el Pleno & dimensionless & $1/(1-x)=\gamma^2$; $\approx 1.32$ today & §\ref{sec:reologia}\\
$\dot\gamma_{\max}$ & Maximal deformation rate & $[s^{-1}]$ & $c/\lP\approx 1.8552\times 10^{43}$ s$^{-1}$ & §\ref{sec:reologia}\\
$\tau_0$ & Yield stress & $[Pa]$ & $\approx 1.9838\,\rhoP c^2$ (from the IR wall) & §\ref{sec:reologia}\\
$\eta/s$ & Viscosity/entropy & $[\hbar/k_B]$ & Window $[1,\sim\!2.5]\times\hbar/4\pi k_B$ (KSS + $\tau_\Pi$) & §\ref{sec:reologia}\\
$K_c$ & Factor of the formula enumeration & dimensionless & $\approx 6.71\times 10^5$; space of expressions against an isolated ratio & §\ref{sec:cabos}\\
$\text{BF}$ & Factor over the whole set of results & dimensionless & $\approx 3.4\times 10^4$ at $\lambda=0.5$, with range $[65,\,1.2\times10^9]$ (E64) & §\ref{sec:cabos}\\
\bottomrule
\end{longtable}}



\begin{center}\small
\begin{longtable}{L{1.7cm}L{8.6cm}L{2.6cm}}
\toprule
\textbf{Label} & \textbf{Equation} & \textbf{Status}\\
\midrule
\endhead
E0a & $\rhoP\equiv M\equiv E$ (energy read by region) & derived\\
E0b & $\partial_t\rho_E+\nabla\!\cdot j_E=0$ (continuity) & derived\\
E0c & $E/c=Mc$ (balance) $\Rightarrow E=Mc^2$ & SLD\\
E1 & $\Psi=S\,e^{i\theta}$ & derived\\
E2 & $\mathcal{L}_\text{F1}$ (four terms) & derived\\
E3 & $G=c^2/(\rhoP\lP^2)$ & derived\\
E4 & $\gamma_\kappa=\rhoP c^2\lP^4/40=\hbar c/40$ & derived\\
E4c & $\mathcal{L}_\text{flex}^{(2)}$ (complete field) & derived\\
E5 & $V_q^\text{dens}=\tfrac{3}{8}\rhoP c^2/S^2$ & derived / imported\\
E6 & F2 (modulus, 4th order) & derived\\
E7 & U(1) phase continuity & derived\\
E7c & U(1) Noether current (with flexural term) & derived\\
E9 & $\Smin^\text{estr}=12(\pi^2-4)/\pi^4$ & derived\\
E10 & $S_{\max,\text{geom}}^2=8\sqrt{6}/9$ & derived\\
E11 & $f_c=S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2=0.6869$ & derived / calibrated\\
E11-ter & $S_{\max,\text{eff}}^2(t_0)$: turning point of $\bar S(t)$ (positional) & calibrated\\
E13 & $\nabla^2\leftrightarrow-\tfrac{3}{16\lP^2}D_\text{lat}$ & derived\\
E18, E19 & Linearized modes (modulus, phase); $1/20=2\times 1/40$ & derived\\
E20 & Dispersion quadratic: roots $u=-12.22,+32.22$ (physical/Lee-Wick) & derived\\
E21, E23 & Dispersions; $m_\text{radial}=4.44\,E_P$ (IR), $3.50\,E_P$ (exact) & derived\\
E24 & $m_\text{eff}c^2=\sqrt{\xi}\,E_P$ (GP effective mass) & derived\\
E24-bis & $V''_\text{eff}=19.69$, $V'''_\text{eff}=27.92$, $V^{(4)}_\text{eff}=855.6$ ($\times\rhoP c^2$) & derived\\
E25, E25a--d & $g_\text{eff}=\tfrac{V^{(4)}_\text{eff}\lP^6}{12}\big[1-\tfrac{5}{3}\tfrac{(V'''_\text{eff})^2}{V''_\text{eff}V^{(4)}_\text{eff}}\big]=65.8\,\rhoP c^2\lP^6$ (cubic renorm., $-7.7\%$) & derived-algebra\\
E26 & $i\hbar\partial_t\psi=-\tfrac{\hbar^2}{2m_\text{eff}}\nabla^2\psi+g_\text{eff}|\psi|^2\psi$ (GP) & derived\\
E27 & $M_Q=M_P/S_{\max,\text{eff}}$ (fundamental soliton mass) & derived\\
E28 & $B=\text{Link}(L_1,L_2,L_3)$ (Gudnason-Nitta) & derived\\
E29 & $\sum_j q_j\equiv 0\pmod 3$ (Borromean $\mathbb{Z}_3$ congruence) & derived / SLD\\
E30, E31 & Rotational tower; $\Delta m=9(\hbar c)^2/(2m_p c^2 R_p^2)$ & derived / calibrated\\
E34, E35 & Sakharov action; Einstein equations & derived / SLD\\
E35-bis, E35-ter & $\mathcal{O}_\text{TT}=-\Box(1+\tfrac{\lP^2}{20}\Box)$; partial fractions (residue $+1$) & derived\\
E36 & $\hbar=\rhoP c\,\lP^4$ & derived\\
E37 & $\Omega_{MO}/\Omega_M=\sqrt{3}\pi$ (derived algebra; SLD geometry; derived D1 lattice: $Z=4$ D1-a, stacking D1-b) & derived / SLD\\
E38, E40--E44 & Structural rigidities: $K_\text{cin}=c^4/2G$, shear $\mu$, bulk $K$, Poisson $\nu\approx 0.33$ & derived / SLD\\
E39a, E39b & Effective metric as Cauchy-Green tensor $g^\text{eff}=F^{\mathsf T}\eta F$; order 0/1/2 expansion & derived\\
E39c, E39d & Contracted Bianchi $\nabla_\mu G^{\mu\nu}=0$; conservation $\nabla_\mu T^{\mu\nu}=0$ (2nd Noether theorem) & derived\\
E-relax & $\tau_\Pi=\lP/(\sqrt{20}\,c)$ (relaxation time); $\eta=\mu\tau_\Pi$; $\sigma_\text{Pleno}=F_\text{Planck}/40$ & derived\\
E47 & $n(x)=1/(1-x)$ & derived\\
E48 & $\tau_0=1.9838\,\rhoP c^2$ & derived\\
E45 & $\dot\gamma_{\max}=c/\lP$ (maximal deformation rate) & derived\\
E46 & $x=\dot\gamma/\dot\gamma_{\max}\in[0,1)$ (Deborah) & derived\\
E49, E50 & $n(v)=\gamma^2$ (rheological Lorentz) & derived\\
E51 & $\eta/s\geq\hbar/4\pi k_B$ (KSS); window $[1,\sim\!2.5]\times$ with $\tau_\Pi$ & SLD\\
E57 & Cycle equation (damped Gent) & SLD\\
E-inc-3 & $\Delta x\Delta p=\tfrac{\hbar}{2}\sqrt{1+\tfrac{3}{80}(\lP/\sigma)^2}$ & derived\\
E-inc-1 & Gaussian moments: $\langle k^2\rangle=1/4\sigma^2$, $\langle k^4\rangle=3\langle k^2\rangle^2$ & derived\\
E-Born & $P(R)=\int_R|\psi|^2$ (fraction of U(1) Noether charge) & derived\\
E-Bell & $E(a,b)=-\cos 2(a-b)$; CHSH $|\mathcal{S}|_\text{max}=2\sqrt2$ (Tsirelson) & derived / SLD\\
E-Ch-1 & $M_\text{Ch}\approx 1.456\,M_\odot$ & derived / imported\\
E-Maxwell-1 & $E_\text{mag}=\tfrac{8}{9}F_{\mu\nu}F^{\mu\nu}$ & derived\\
E-YM-1 & $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+i[A_\mu,A_\nu]$ & derived\\
E-cosmo-1 & $n_s-1=3-2|s-\tfrac12|$; $n_s=1$ & derived\\
E-cosmo-2 & $\Omega_m=1-f_c=0.3131$ & derived\\
E-cosmo-3 & $E^2(z)$ scale-invariant & derived / calibrated\\
E-cosmo-4 & $\mu=0$ (Planck blackbody) & derived\\
E-cosmo-5 & $H_0=\sqrt{8\pi/3}\,(c/\lP)\sqrt{\Theta}$ (positional factorization) & derived\\
E-cosmo-6 & $\Theta=3/(8\pi N_\text{tic}^2)$ (``$10^{-123}$''$=$ age in ticks) & derived\\
E-cosmo-7 & $f_c=S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2=0.6869=\Omega_{EO}$ (occupation at rest) & derived / calibrated\\
E-cosmo-8 & $H_{\max}/\dot\gamma_{\max}=\sqrt{8\pi/3}\,S_{\max,\text{geom}}=4.2709$ & derived\\
E-cosmo-9 & Shared factor $\lambda_0/\lambda_e=\delta\tau_0/\delta\tau_e=1+z$; SN Ia dilation $b=1$ exact & derived\\
E-cosmo-10 & $L(S)\propto S^{-2/3}$; radial redshift bound $z_{\max}\approx0.268$ (two clocks) & derived\\
E-Sch-1--7 & Schwarzschild of el Pleno; metric (E-Sch-7), $r_s=2GM/c^2$ (effective geometry) & derived / SLD\\
E-Dirac-1--10 & Dirac structure; equation (E-Dirac-10) $(i\hbar\Gamma^\mu\partial_\mu-mc)\Psi_D=0$ & derived / SLD\\
\bottomrule
\end{longtable}
\end{center}

\medskip
\begin{tcolorbox}[clave,title={The program, in one page}]
Gravity, light, particles, space, time and the dark sectors are configurations of a single
substrate, described from a single field and six axioms:
\begin{itemize}[leftmargin=1.4em]
\item \textbf{The constants are not free.} $G=c^2/(\rhoP\lP^2)$ and $\hbar=\rhoP c\lP^4$ follow
from the dynamics of F1 ---the static reduction and the Green function in one case, the
quantisation of the $U(1)$ rotor in the other---, not from counting dimensions: the programme
has dimensionless invariants of its own, and any of them multiplying the monomial would carry
the same units. What the derivation returns is the coefficient, and it equals unity in both.
Gravitation is weak because \emph{el Pleno} is rigid; the quantum of action is a property of the
medium, not an axiom.
\item \textbf{Gravity and the quantum emerge from the same place.} Einstein's equations are the
elastic action of the medium (Sakharov), with $\gamma_G=1$ derived; uncertainty, its curvature
rigidity with an exact floor $\hbar/2$; Born, Bell and the Tsirelson bound $2\sqrt2$, the
accounting of its charge and the geometry of its lattice. The two theories the twentieth century
did not reconcile are two readings of one object.
\item \textbf{Forces and matter have a derived form.} Maxwell and Yang-Mills are the circulation
of the phase over the lattice; particles, its topological knots, with spin, fractional charge and
confinement as geometric consequences, not postulates. And the thermodynamics is equally native:
the equilibrium occupation and the coefficient $2\pi^2/45$ of Stefan-Boltzmann follow from
counting configurations of the eigenmodes of F1, with no quantum statistics assumed.
\item \textbf{The whole universe fits.} The cosmological ratios ---$\sqrt3\pi$ between the two
dark populations, $\Omega_{DE}=f_c$ and $\Omega_m=1-f_c$--- fall within $1\sigma$ of Planck with
zero free parameters, matching $\Lambda$CDM with \emph{less} freedom; $w=-1$ is exact in the
ground state and $n_s=1$ at zeroth order, with the red sign of the tilt derived and its exact
value positional; the blackbody is perfect because the photon is a mode of the medium with no
conserved number. And the singularity ---of the Big Bang and of the black hole--- does not occur,
because the medium is too rigid to permit it.
\item \textbf{And the whole can be refuted.} Measuring $\rhoP$ or $\lP$ by a route that passes
through neither gravitation nor quantisation exposes the product of the two coefficients;
measuring both, each branch separately. The bounds ---$v_\text{GW}=c$, $w\geq-1$, the age in
ticks of $H_0$--- are inequalities a datum can violate. The open ends that remain are inventoried
with their severity, and what this compendium asserts is recomputed from the formulas with the
same notebook that verifies the corpus.
\end{itemize}
The Lorentz factor is the flow index of the rheology of the medium: relativity and the cycle of
the universe are the same dilatant thickening of \emph{el Pleno}. From a single assertion about
energy, with the algebra in view at every step, there emerges a quantitative portrait of the
physical world that can be checked line by line and broken by a measurement.
\end{tcolorbox}

\medskip
\begin{tcolorbox}[clave,title={Synthesis of the compendium}]
From \textbf{two properties with dimensions} ($\rhoP$, $\lP$) and \textbf{one causal
structure} ($c$), all the derivations of this document ---verified symbolically and
numerically where the computation permits--- reconstruct: the constants $G$ and $\hbar$ (with
unit coefficient, without fitting), and with them the three scenarios that state which
measurement would put them at risk; the effective potential with its \emph{Pleno}-ground state
$12(\pi^2-4)/\pi^4$, its hard bound $8\sqrt{6}/9$, the coefficient $E_0$ in closed form with no
free symbols, and the configuration annulus the two walls delimit; the emergent gravity of
Sakharov and Newton's three laws as its low-energy limit; uncertainty as positive rigidity with
its $3/80$ correction; electromagnetism $8/9$ and Yang-Mills from phase circulation; particles
as topological solitons with the $\Delta$-$N$ split and the ratio $\sqrt3\pi$ protected by
Gauss-Bonnet; the thermodynamics of the medium ---equilibrium occupation and coefficient
$2\pi^2/45$--- by counting configurations of the eigenmodes of F1, with no quantum statistics
assumed; and a cosmological sector whose ratios ($\sqrt{3}\pi$, $\Omega_{DE}=f_c$,
$\Omega_m=1-f_c$) fall within $1\sigma$ of Planck, with $w=-1$ exact in the ground state and
$n_s=1$ at zeroth order, matching $\Lambda$CDM with less freedom. The Sakharov coupling is
derived ($\gamma_G=1$, via residue $+1$ of the physical pole E35-ter and mode counting
$2\ell+1=5$). The mathematics is complete; where a result is imported, it is said; where there
is a gap, it is named.

\medskip
\textbf{Epistemic balance of the catalogue.} Count by catalogue entry (each row is a result or
anchor; range-rows such as E-Dirac-1--10 count as one). The ``mixed'' column indicates how many
of those entries combine two statuses.

\begin{center}\small
\begin{tabular}{lcc}
\toprule
\textbf{Classification (predominant)} & \textbf{Entries} & \textbf{Of these, mixed}\\
\midrule
\epi{DERIVED}    & 58 & 13 \\
\epi{SLD}        & 3  & --- \\
\epi{CALIBRATED} & 1  & --- \\
\midrule
\textbf{Total}   & \textbf{62} & \textbf{13} \\
\bottomrule
\end{tabular}
\end{center}

\smallskip
{\small The 13 mixed entries are: \epi{DERIVED}/\epi{SLD} (7: Einstein equations,
lattice/$\sqrt3\pi$, rigidities, Schwarzschild, Borromean congruence, Bell, and the unified
reading of the two Crunch bounds), \epi{DERIVED}/\epi{CALIBRATED} (4: $f_c$, $\Delta$-$N$
split, $E^2(z)$ and the spectral index) and \epi{DERIVED}/imported (2: Klauder wall $3/8$,
Chandrasekhar). The only pure \epi{CALIBRATED} is the positional value
$S_{\max,\text{eff}}^2(t_0)$ (E11-ter), analogous to $H_0$. \textbf{No entry carries the status
\epi{HYPOTHESIS}}: the $\eta/s$ window (E51), which in earlier versions did, became \epi{SLD}
once it was bounded from two sides ---the universal bound below and the native ultraviolet
scale at the saturation temperature above---. The status column for each equation is in the
catalogue above.}
\end{tcolorbox}

\medskip\hrule\medskip
\appendix
\phantomsection
\addcontentsline{toc}{section}{Appendix A --- Numerical verification notebook}

\section*{Appendix A --- Reproducible numerical verification notebook}
\label{ap:verificacion}

\textbf{A single notebook for both documents.} The compendium and the corpus share
\emph{the same} verification notebook: the script \texttt{verificar\_PIU.py} with its
stratification and confrontation modules, plus the results table the code itself generates in
PDF and its CSV export. There are not two verifications that could diverge. The organisation of
the notebook ---the three layers and the double verdict--- is set out in Annex \S{}V.9 of the
corpus, and this table is its reading for the quantities the compendium derives step by step.

\textbf{The three layers, in one line.} The criterion that orders the notebook is the
\emph{order of appearance} of the first measured magnitude. \emph{Layer 0}: what the medium
produces with no measuring rod at all ---in units $\rhoP=\lP=c=1$---, which is most of what is
derived in this compendium and cannot be contaminated by any calibration, because it occurs
before any rod exists. \emph{Layer 1}: the dictionary, which costs exactly two numbers ($\lP$
in metres and $\rhoP$ in kilograms per cubic metre), with neither $G$ nor $\hbar$ appearing in
it. \emph{Layer 2}: the reading in human units, the only layer where a laboratory datum can
falsify anything.

\textbf{The verdict splits in two.} A verification notebook has not one verdict but two, and
confusing them is the error the layers avoid: \emph{document-to-code fidelity} ---does the
computation reproduce every quantity that is printed?---, which is a transcription check and
must pass in full; and \emph{agreement with measurement}, which applies only to layer 2 and
only to the rows whose type admits it.

With that framing, each constant and ratio of the program is recomputed from its
first-principles formulas by means of \texttt{verificar\_PIU.py}, without preloaded values:
the pure ratios of the lattice, the coefficients of the effective potential (with $E_0$ obtained by
solving $V'_\text{eff}(S_{\min})=0$, not as an input), the fundamental constants (with $G$,
$\hbar$ and $\ell_P$ mutually consistent), the cosmological quantities (including the
reconstruction of $H_0$ from the factorization E-cosmo-5, which closes the circle) and the limits
of quantum mechanics and mass. The table confronts the recomputed value with the one reported in
the corpus and the compendium. The positional \epi{CALIBRATED} quantities ($f_c$ via
$S_{\max,\text{eff}}^2$, $\Theta_0$, $N_{\text{tic}}$) are verified as \emph{consistency} of
the factorization, not as a derivation of their value: their form is derivable, their value is positional.

\begin{center}\footnotesize
\begin{longtable}{L{5.4cm}C{2.5cm}C{2.5cm}C{1.5cm}C{0.7cm}}
\toprule
\textbf{Quantity} & \textbf{Recomputed} & \textbf{Reported} & \textbf{Rel.\ err.} & \\
\midrule\endhead
$S_{\min}=12(\pi^2{-}4)/\pi^4$ & $0.72309$ & $0.7231$ & $1.8e-05$ & \checkmark\\
$S_{\min}^2$ (occupation) & $0.52285$ & $0.5229$ & $8.6e-05$ & \checkmark\\
$S_{\max,\text{geom}}^2=8\sqrt6/9$ & $2.1773$ & $2.1773$ & $1.1e-05$ & \checkmark\\
$S_{\max,\text{geom}}$ & $1.4756$ & $1.4756$ & $1.6e-05$ & \checkmark\\
$S^*=\sqrt{S_{\min}S_{\max,\text{geom}}}$ & $1.0329$ & $1.0329$ & $4.1e-05$ & \checkmark\\
$x^*=S_{\min}/S_{\max,\text{geom}}$ & $0.49004$ & $0.49$ & $7.6e-05$ & \checkmark\\
$\rho(S^*)/\rho_P$ & $1.067$ & $1.067$ & $2.8e-05$ & \checkmark\\
flexural coef.\ ($1/40$) & $0.025$ & $0.025$ & $0.0e+00$ & \checkmark\\
Klauder coef.\ ($3/8$) & $0.375$ & $0.375$ & $0.0e+00$ & \checkmark\\
$E_0$ (from $V'_\text{eff}(S_{\min}){=}0$) & $4.2546$ & $4.2546$ & $4.0e-06$ & \checkmark\\
$\eta_0=1/S_{\max,\text{geom}}^2$ & $0.45928$ & $0.4593$ & $4.5e-05$ & \checkmark\\
$G=c^2/(\rho_P\ell_P^2)$ & $6.6743e-11$ & $6.6743e-11$ & $0.0e+00$ & \checkmark\\
$\hbar=\rho_P c\,\ell_P^4$ & $1.0546e-34$ & $1.0546e-34$ & $3.0e-08$ & \checkmark\\
$\gamma_\kappa=\hbar c/40$ & $7.9038e-28$ & $7.9e-28$ & $4.8e-04$ & \checkmark\\
$\sqrt{\hbar G/c^3}=\ell_P$ (consequence of E3+E36) & $1.6163e-35$ & $1.6163e-35$ & $1.5e-08$ & \checkmark\\
$\Omega_{MO}/\Omega_M=\sqrt3\,\pi$ & $5.4414$ & $5.4414$ & $3.5e-07$ & \checkmark\\
$f_c=S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$ & $0.6869$ & $0.6869$ & $2.7e-06$ & \checkmark\\
$\Omega_m=1-f_c$ & $0.3131$ & $0.3131$ & $5.9e-06$ & \checkmark\\
$N_{\text{tic}}=1/(H_0\tau_P)$ & $8.4919e+60$ & $8.493e+60$ & $1.3e-04$ & \checkmark\\
$\Theta_0=3/(8\pi N_{\text{tic}}^2)$ & $1.6553e-123$ & $1.655e-123$ & $1.7e-04$ & \checkmark\\
$H_0$ reconstructed (E-cosmo-5) & $2.1843e-18$ & $2.1843e-18$ & $0.0e+00$ & \checkmark\\
$c/\ell_P$ (Planck tick) & $1.8549e+43$ & $1.8552e+43$ & $1.8e-04$ & \checkmark\\
$N_\text{age}=t_0/\tau_P$ & $8.0701e+60$ & $8.07e+60$ & $1.2e-05$ & \checkmark\\
$H_{\max}=\sqrt{8\pi/3}(c/\lP)S_{\max,\text{geom}}$ & $7.9219e+43$ & $7.922e+43$ & $1.3e-05$ & \checkmark\\
$H_{\max}/\dot\gamma_{\max}=\sqrt{8\pi/3}S_{\max,\text{geom}}$ & $4.2709$ & $4.2709$ & $0.0e+00$ & \checkmark\\
$A(73)=H_0t_0$ (excluded, $>0.953$) & $1.0293$ & $1.0293$ & $0.0e+00$ & \checkmark\\
GUP coef.\ ($3/80$) & $0.0375$ & $0.0375$ & $0.0e+00$ & \checkmark\\
Tsirelson $=2\sqrt2$ & $2.8284$ & $2.8284$ & $9.6e-06$ & \checkmark\\
$M_\text{Ch}$ (Chandrasekhar) & $1.4559$ & $1.456$ & $4.3e-05$ & \checkmark\\
$M_{\min}/m_P$ (PBH cutoff) & $0.11707$ & $0.117$ & $6.1e-04$ & \checkmark\\
split $\Delta$-$N$ & $264.1$ & $264.1$ & $7.4e-07$ & \checkmark\\
$E_0$ (closed form E58, no free symbols) & $4.2546169$ & $4.2546169$ & $0.0e+00$ & \checkmark\\
annulus area $\pi(S_{\max,\text{geom}}^2-S_{\min}^2)$ (E60) & $5.1976687$ & $5.1976687$ & $0.0e+00$ & \checkmark\\
$(2\pi^2/45)g_*$ from the configuration count (E-BE-3) & $0.8772982$ & $0.8772982$ & $0.0e+00$ & \checkmark\\
$k_BT_{\max}/(\rho_P c^2\ell_P^3)$ (E-R-1) & $1.3487413$ & $1.3487413$ & $0.0e+00$ & \checkmark\\
$R=(\eta/s)/(\hbar/4\pi k_B)$ at $(\tau_\Pi,T_{\max})$ (E61) & $0.7579726$ & $0.7579726$ & $0.0e+00$ & \checkmark\\
$\sqrt2\,S_{\min}$, equality point of the partition (E62) & $1.0225995$ & $1.0225995$ & $0.0e+00$ & \checkmark\\
$A(w{=}{-}1)$ at $\Omega_m=0.3131$ (age ceiling) & $0.9526$ & $0.9526$ & $0.0e+00$ & \checkmark\\
\bottomrule
\end{longtable}
\end{center}
\noindent \textbf{What this table collects and what it does not.} The notebook recomputes
ninety-nine quantities; the thirty-eight here are those this compendium derives step by step, so
that the reader can follow each one down to its algebra. The rest ---intermediate values,
consistency controls and quantities of the corpus--- is in the CSV export accompanying the
script.

\noindent \textbf{How to read the error column, and which verdict it issues.} The thirty-eight rows
pass, but not all under the same criterion. Fourteen close with \emph{exactly} zero
error: these are layer-0 quantities ---coefficients the medium produces with no measuring rod at
all, such as the flexural $1/40$ or Klauder's $3/8$--- where what is checked is
\emph{document-to-code fidelity}, and there the tolerance is not $10^{-3}$ but zero: either the
computation reproduces the printed number, or there is a transcription error. The remaining
twenty-four carry a residue because they compare against rounded magnitudes or against
laboratory data; their largest residue is $6.1\times10^{-4}$ and most fall below $10^{-4}$. Only
in that second group, and only in the layer-2 rows, does the word ``agreement'' mean
confrontation with measurement. The complete reproducibility ---running the script and obtaining
these values--- is part of the method of the program: the numerical assertions do not ask for
trust, they are checked.

\medskip\hrule\medskip
\phantomsection
\addcontentsline{toc}{section}{References}
