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%  PART V --- ANNEX V.9: THE STRATIFIED VERIFICATION NOTEBOOK
%  The three layers: content without a rod, dictionary, and SI reading.
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\clearpage
\phantomsection
\section{\texorpdfstring{\S{}V.9 --- Annex: the verification notebook and its three layers}{V.9 - Annex: the verification notebook and its three layers}}
\markboth{\S{}V.9 --- The verification notebook}{}

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} numerical and symbolic, executable
  \item \textbf{Open threads:} none of its own; the calibration of $(\rho_P,\ell_P)$ is thread C-meta-1.b of \S{}V.3
  \item \textbf{Notes:} organisation of the verification notebook into three layers according to the order of appearance of the first measured magnitude; anchor criterion and resolution floor; splitting of the verdict into fidelity and agreement
\end{itemize}

\end{estadoseccion}

A verification notebook can mislead in two opposite ways, and both are easy to commit without noticing. The first is to compare the theory with itself: recomputing a quantity from its formula and contrasting it with the number printed in the manuscript always gives perfect agreement, because both sides come from the same place; that checks transcription, not nature. The second is to charge twice for the same success: spending a measured datum to fix a scale and afterwards exhibiting that scale as a prediction. The organisation into layers prevents both, and the criterion that separates them is the \textbf{order of appearance}: when a measured magnitude first enters.

\subsection{\texorpdfstring{V.9.1 --- Layer 0: what the medium produces with no rod at all}{V.9.1 - Layer 0: what the medium produces with no rod at all}}

In units of the medium itself ---$\rho_P=\ell_P=c=1$--- F1 produces pure numbers: lattice ratios, Lagrangian coefficients, potential curvatures, quantum bounds, cosmological ratios. Here live $S_{\min}^\text{estr}=12(\pi^2-4)/\pi^4$, $S_{\max,\text{geom}}^2=8\sqrt6/9$, the area of the annulus (E60), $E_0$ in closed form (E58), $\gamma_\kappa=1/40$, the regulator $1/20$, $\sigma_{\min}=\ell_P/\sqrt{20}$, the GUP correction $3/80$, the Tsirelson bound $2\sqrt2$, $\sqrt3\pi$, the coefficient $2\pi^2/45$ of the radiation entropy (E-BE-3) and the dissipation sub-bound (E61).

None of these rows contains a measured datum, and none \emph{can} be contaminated by a calibration, because all of them occur before any rod exists. It is the majority of the notebook: of the order of eighty-five per cent of the checks. Their verification consists in recomputing each quantity from its first-principles formula and contrasting it with what the manuscript prints ---a document-to-code fidelity check, which detects transcription errors and nothing more, and which is declared as such.

\subsection{\texorpdfstring{V.9.2 --- Layer 1: the dictionary, which costs exactly two numbers}{V.9.2 - Layer 1: the dictionary, which costs exactly two numbers}}

Reading the programme in human units requires translation, and the translation is not content of the theory. Since the fundamental magnitudes with dimensions are two ---$\rho_P$ and $\ell_P$, with $c$ a causal structure and an SI convention---, the dictionary costs exactly two numbers: $\ell_P$ in metres and $\rho_P$ in kilograms per cubic metre. Neither one more nor one fewer; it is the count imposed by the structure of the fundamentals (\S{}I.8.9).

$G$ and $\hbar$ do \emph{not} appear in the dictionary, and they are not needed anywhere: every occurrence of them is replaced by $c^2/(\rho_P\ell_P^2)$ and $\rho_P c\,\ell_P^4$ and disappears. With that substitution the Planck scale is read as what it is in the programme ---the cell of the medium---:
\[
m_P=\rho_P\ell_P^3\ \text{(mass of one cell)},
\qquad
E_P=\rho_P c^2\ell_P^3,
\qquad
t_P=\ell_P/c\ \text{(crossing time)}.
\]
Consequently $G$ and $\hbar$ move from input to \emph{output}: they are quantities confronted against CODATA, not ingredients of the computation.

\subsection{\texorpdfstring{V.9.3 --- Layer 2: the reading in human units and its confrontation}{V.9.3 - Layer 2: the reading in human units and its confrontation}}

Only here can a laboratory datum falsify anything, and only here does it make sense to speak of agreement or disagreement with observation. Each row declares its anchors ---which numbers of the dictionary intervene--- and its type:

\begin{center}\small
\begin{tabular}{L{2.9cm}L{10.8cm}}
\toprule
\textbf{Type} & \textbf{What it means} \\
\midrule
Prediction & the anchors do not determine the output; it is in order to give a difference and a distance in $\sigma$ \\
Validation & the output reproduces an established value; the content is in the coefficient, not in the residue \\
Calibrated & the row has declared degrees of freedom; it counts as consistency, not as prediction \\
Identity & an algebraic function of an anchor; no figure of agreement applies \\
Bound & the datum is an upper limit, not a central value \\
Mathematical & a theorem or exact identity; it is not confronted with data \\
Pending & the computation is not closed; it is declared as such \\
\bottomrule
\end{tabular}
\end{center}

\textbf{Resolution floor.} No row can exhibit an agreement finer than the uncertainty of its own anchors. Those that do are marked, and their agreement is read as consistency of the calibration and not as independent confirmation. The density parameters of the dark sector are the typical case: they fall within $1\sigma$, but their anchor has a relative uncertainty larger than the agreement attained.

\subsection{\texorpdfstring{V.9.4 --- The verdict splits in two}{V.9.4 - The verdict splits in two}}

From the stratification it follows that a verification notebook cannot have a single verdict. It has two, and confusing them is precisely the error that the layers avoid:

\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item \textbf{Document-to-code fidelity.} Does the computation reproduce, from the first-principles formulas, every quantity the manuscript prints? It is a transcription check, it covers all three layers, and it must pass in full.
\item \textbf{Agreement with measurement.} Does the reading in human units coincide with the laboratory datum, and at what distance? It applies only to layer 2, and only to the rows whose type admits it.
\end{itemize}

Separating them has an immediate consequence: the stratification brings into view the disagreements that a single verdict concealed behind a null transcription error. The $\Delta$-N split and the spectral index at zeroth order are the two cases, both already declared as threads in \S{}V.3, and the table now shows them instead of exhibiting a zero. In the opposite direction, the same organisation makes visible successes that the earlier presentation could not display: the Chandrasekhar limit with four anchors and a relative difference of $2\times10^{-4}$, $\sqrt3\pi$ at $0.86\sigma$ and the time-dilation exponent of the supernovae at $0.27\sigma$, all three with no fitted parameter.

\textbf{Reproducibility.} The notebook is \emph{single} for the whole package: the same script verifies the quantities this corpus reports and those the Mathematical Compendium derives step by step, so that there are not two verifications that could diverge. It is distributed as executable supplementary material (\texttt{verificar\_PIU.py} and its stratification and confrontation modules), together with the results table it generates in PDF, so that any reader with Python can recompute the quantities, change the anchors and check for themselves what changes and what does not. The checks are built so that they can fail: the Tsirelson bound is maximised numerically over angles instead of being compared against $2\sqrt2$; the equilibrium occupation is solved by bisection instead of applying the formula; the radiation entropy is integrated instead of quoted; and $\hbar$ is evaluated after the dictionary is fixed, without touching data again. A check that cannot fail is not a check.
