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%  PART V --- THEMATIC APPENDIX (V.7)
%  Two pieces of deepening/closure that, by their nature
%  (one \epi{SLD} of structure, the other interpretive-historical), are
%  presented as a thematic appendix so as not to break the derivative
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\phantomsection
\section{\texorpdfstring{\S{}V.7 --- Appendix: the structure of the Dirac equation}{V.7 - Appendix: the structure of the Dirac equation}}
\markboth{\S{}V.7 --- The Dirac structure}{}

\begin{estadoseccion}
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} consolidated. Complete derivation by two convergent routes.
  \item \textbf{Verification:} Clifford algebra, $SU(2)$ holonomy, Lorentz generators and spin-statistics verified symbolically.
  \item \textbf{Open threads:} none structural. The structure of the equation is \epi{DERIVED}; there remain as declared numerical threads (\S{}III.4.7) the value of $\lambda_\text{SO}$, the fermionic masses $m$, the chirality and the number of generations ---values, not structure.
  \item \textbf{Notes:} dedicated appendix. The structure of the Dirac equation ---the first-order operator, the Clifford-Lorentz algebra, the relativistic energy-momentum relation and the spin-$\tfrac12$--- is derived from el Pleno as the low-energy effective theory of the phase modes over the bipartite diamond lattice, completing the set of master equations. It is presented in a dedicated appendix ---and not interleaved in \S{}III.4 or \S{}III.5--- to deploy its full mathematics without breaking the thread of the gauge and hadronic sector, given its structural importance.
\end{itemize}
\end{estadoseccion}

The structure of the Dirac equation is not physics imported into the program: it emerges from el Pleno as the low-energy effective theory of the phase modes over the bipartite diamond lattice. Because of the importance of this result ---it closes the set of master equations of physics, together with Newton, Einstein, Maxwell, Yang-Mills and Schrödinger, all already derived as configurations of the same medium--- this appendix is dedicated to it, with its complete mathematical development by two independent routes that converge on the same operator: the \emph{band route} (\S{}V.7.1), which builds the equation from the Bloch Hamiltonian of the bipartite lattice and the spin-orbit opening of the nodal line (Fu-Kane mechanism); and the \emph{covariant route} (\S{}V.7.2), which establishes it from the two binary structures of the medium, with emergent Lorentzian signature, unique representation by the Pauli theorem and the spin-statistics theorem closed at its root. Both are the same physics in two readings; together they fix the status of the Dirac structure as \epi{DERIVED}. What remains open are two \emph{values} ---the spin-orbit coupling $\lambda_\text{SO}$ and the masses $m$--- reconnected with the already-inventoried fermionic-masses and chirality threads (\S{}III.4.7), without adding new threads. The scope is deliberately bounded: the master \emph{equation} is derived, not its atomic phenomenology (fine structure, spin-orbit coupling of hydrogen, anomalous magnetic moment), an effective branch that standard physics already resolves over the given Dirac equation and that the program inherits as a correct limit.

\subsection{\texorpdfstring{V.7.1 --- The band route: Dirac as the effective theory of the phase modes over the diamond lattice}{V.7.1 - The band route: Dirac as the effective theory of the phase modes over the diamond lattice}}

\textbf{Origin and purpose.} \S{}III.2.7, upon inventorying what the validation of the uncertainty relation inherits and what it does not, situated outside the inheritable the \emph{fine structure and the spin}, ``which require the Dirac equation''. This route shows that the \textbf{form} of the Dirac equation emerges from el Pleno as the low-energy effective theory of the phase modes over the bipartite diamond lattice. The Dirac structure ---the first-order operator, the Clifford algebra, the relativistic energy-momentum relation--- is forced by already-derived pieces of the program; what remains a thread is the numerical \emph{value} of two constants (the spin-orbit coupling $\lambda_\text{SO}$ and the masses $m$), reconnected with the already-declared fermionic-masses and chirality threads (\S{}III.4.7). The status of the structure as \epi{DERIVED} is fixed by the covariant route (\S{}V.7.2), which establishes it without depending on the value of $\lambda_\text{SO}$.

\medskip
\begin{tcolorbox}[notabox,title={The chain, link by link, with its status}]
The derivation chains six pieces, five of them already established in the corpus:
\begin{enumerate}[leftmargin=1.6em,itemsep=2pt]
\item A/B bipartition $\to$ \textbf{spinorial doublet} $\Phi=(\varphi_A,\varphi_B)^T$ \epi{DERIVED} (\S{}III.4.2)
\item A--B nearest-neighbor hops $\to$ \textbf{two bands} that touch on \textbf{nodal lines} \epi{DERIVED} (P7, \S{}II.3.8)
\item antisymmetric phase $\theta_-$ in $SU(2)\cong S^3$ $\to$ \textbf{Pauli generators} $\sigma^i$ \epi{DERIVED} (\S{}II.3.5)
\item $SU(2)$ holonomy over non-collinear bonds $\to$ \textbf{spin-orbit term} $\lambda_\text{SO}\neq 0$ \epi{SLD} (this \S{})
\item the spin-orbit term \textbf{opens the nodal line into 3D Dirac points} \epi{SLD} (this \S{})
\item velocity of the phase modes $=c$ $\to$ \textbf{Lorentz covariance} \epi{DERIVED} (\S{}II.5)
\end{enumerate}
The product of the chain is the Dirac operator. The only uncalculated value ($\lambda_\text{SO}$, $m$) is a declared thread, not a hidden assumption.
\end{tcolorbox}

\textbf{Step 1 --- The spinorial doublet and the bipartite Bloch Hamiltonian.} The A/B bipartition of the diamond lattice already provides, without an additional postulate, the two-component object on which the operator will act: the doublet $\Phi=(\varphi_A,\varphi_B)^T$ derived in \S{}III.4.2. The dynamics of a phase excitation over the bipartite lattice is a hopping problem between sub-lattices: an excitation at an A site can only hop to its four B neighbors (tetrahedral geometry of the diamond), and vice versa. This forces a Bloch Hamiltonian \textbf{strictly off-diagonal} ---not by choice, but because there is no A--A or B--B nearest-neighbor hop:
\[
H(\mathbf{k}) = t\begin{pmatrix} 0 & f(\mathbf{k}) \\ f^*(\mathbf{k}) & 0 \end{pmatrix},
\qquad
f(\mathbf{k}) = \sum_{j=1}^{4} e^{\,i\,\mathbf{k}\cdot\mathbf{d}_j}, \tag{E-Dirac-1}
\]
where $\mathbf{d}_j$ are the four tetrahedral nearest-neighbor vectors and $t$ the hopping amplitude. The eigenenergies are $E(\mathbf{k}) = \pm\,t\,|f(\mathbf{k})|$: the \textbf{two bands} that the corpus already identified as a consequence of the bipartition (\S{}II.3.8, spectral form factor of $S_{\min}^\text{estr}$).

\textbf{Step 2 --- The nodal lines and the pure-lattice limit.} The set where the bands touch is $f(\mathbf{k})=0$. Computed explicitly,
\[
|f(\mathbf{k})|^2 = 4 + 4\cos\!\tfrac{a k_x}{2}\cos\!\tfrac{a k_y}{2}
                      + 4\cos\!\tfrac{a k_x}{2}\cos\!\tfrac{a k_z}{2}
                      + 4\cos\!\tfrac{a k_y}{2}\cos\!\tfrac{a k_z}{2}, \tag{E-Dirac-2}
\]
whose zero set is \emph{one-dimensional}: they are the \textbf{nodal lines} P7 (\S{}II.3.8), not isolated points. Expanding around a node (for example $X=(2\pi/a,0,0)$, where $f=0$) one finds that $f$ is linear in a single component of $\mathbf{q}=\mathbf{k}-\mathbf{k}_0$: the degeneracy extends along a line. \emph{This is the point where the pure diamond lattice falls short}: by itself it produces a nodal-line semimetal, not the three-dimensional conical Dirac dispersion. A derivation that keeps its accounting must identify which ingredient, already present in \emph{el Pleno}, opens that line into isolated Dirac points.

\textbf{Step 3 --- The spin-orbit coupling is forced $\neq 0$ by the $SU(2)$ character of the phase.} The ingredient is the intrinsic second-neighbor spin-orbit coupling, which in the diamond lattice has the unique form permitted by the symmetry $Fd\bar{3}m$ (Fu-Kane 2007):
\[
H_\text{SO} = i\,\lambda_\text{SO}\!\!\sum_{\langle\langle ij\rangle\rangle}\!\!
              \bigl(\mathbf{d}_{il}\times\mathbf{d}_{lj}\bigr)\cdot\boldsymbol{\sigma}, \tag{E-Dirac-3}
\]
where $\mathbf{d}_{il},\mathbf{d}_{lj}$ are the two bonds of the second-neighbor path $i\to l\to j$ and $\boldsymbol{\sigma}$ the $SU(2)$ generators. The decisive question is not whether the term is \emph{permitted} ---it is, it is the only spin-orbit term compatible with $Fd\bar{3}m$ at that order--- but whether it is \emph{forced} to be non-null. The argument is internal to \emph{el Pleno} and rests on an already-derived piece:

\begin{enumerate}[leftmargin=1.6em,itemsep=2pt]
\item The antisymmetric mode $\theta_-$ lives in $SU(2)\cong S^3$ (\S{}II.3.5, \epi{DERIVED}). This is not an accessory property: it is exactly the topology $\pi_3(SU(2))=\mathbb{Z}$ that sustains the fermions via Finkelstein-Rubinstein (\S{}II.5.2).
\item An $SU(2)$ field transported along an oriented bond acquires a holonomy $U=\mathcal{P}\exp\!\bigl(i\!\int\! A\cdot d\ell\bigr)$ that depends on the path. The diamond lattice has \emph{non-collinear} second-neighbor paths (the bonds form the tetrahedral angle $\arccos(-1/3)\approx 109.47^\circ$), so that two paths $A\to B\to A'$ enclose an oriented area and the holonomy is non-trivial.
\item A non-trivial $SU(2)$ holonomy over an oriented area is, in the continuum limit, exactly a coupling $\boldsymbol{\sigma}\cdot(\text{orientation of the path})$: that is, the term (E-Dirac-3).
\end{enumerate}
The verification is direct. The transport over a minimal plaquette of two non-collinear bonds, with connections $A_1=\theta\sigma_x$ and $A_2=\theta\sigma_y$, gives the holonomy
\[
W = e^{iA_1}e^{iA_2}e^{-iA_1}e^{-iA_2}
  = \mathbb{1} - \theta^2[\sigma_x,\sigma_y] + \mathcal{O}(\theta^3)
  = \mathbb{1} - 2i\theta^2\sigma_z + \mathcal{O}(\theta^3), \tag{E-Dirac-4}
\]
using $[\sigma_x,\sigma_y]=2i\sigma_z$. The holonomy is proportional to $\boldsymbol{\sigma}$ and distinct from the identity: the transport of the $SU(2)$ phase \emph{generates} the spin-orbit term. The consequence is a structural equivalence:
\[
\lambda_\text{SO}=0 \;\Longleftrightarrow\; \text{trivial } SU(2) \text{ holonomy}
\;\Longleftrightarrow\; \theta_- \text{ lives in } U(1),\ \text{not in } SU(2). \tag{E-Dirac-5}
\]
But $\theta_-$ lives in $SU(2)$ by derivation (\S{}II.3.5), and on that same property hangs the existence of fermions in the program. Therefore $\lambda_\text{SO}=0$ contradicts the $SU(2)$ character of the phase mode: \textbf{$\lambda_\text{SO}\neq 0$ is forced by the same structure that gives fermions}. Status: \epi{SLD}. (The \emph{value} of $\lambda_\text{SO}$ from F1 is not calculated: it is a thread, \S{}III.4.7.)

\medskip
\begin{tcolorbox}[notabox,title={Circle of confirmation with Yang-Mills (\S{}III.4.5)}]
The $SU(2)$ holonomy over plaquettes of the lattice that here forces the spin-orbit term is \emph{the same} mechanism that \S{}III.4.5 uses for the non-abelian phase circulation that produces the Yang-Mills structure. The spin-orbit coupling that gives Dirac and the non-abelian structure that gives Yang-Mills are not two independent facts: they are the same $SU(2)$ holonomy read in two sectors. This is a circle of confirmation, not a loose piece: it reinforces both results from a single property of the medium.
\end{tcolorbox}

\textbf{Step 4 --- The opening of the node and the Weyl operator.} With $\lambda_\text{SO}\neq 0$, the nodal line opens into isolated three-dimensional Dirac points (Fu-Kane mechanism). Near an opened node, the effective Hamiltonian to first order in $\mathbf{q}$ takes the form
\[
H(\mathbf{q}) = v_F\,(q_x\,\Gamma_1 + q_y\,\Gamma_2 + q_z\,\Gamma_3), \tag{E-Dirac-6}
\]
with $\Gamma_i$ $4\times 4$ matrices (two sub-lattices $\times$ two spin components) constructed from the Pauli ones, $\Gamma_1=\sigma_x\otimes\mathbb{1}$, $\Gamma_2=\sigma_y\otimes\mathbb{1}$, $\Gamma_3=\sigma_z\otimes\sigma_x$. These matrices satisfy the Euclidean Clifford algebra, verified explicitly:
\[
\{\Gamma_i,\Gamma_j\} = 2\,\delta_{ij}\,\mathbb{1}_4,
\qquad i,j\in\{1,2,3\}, \tag{E-Dirac-7}
\]
whence $H^2 = v_F^2\,\mathbf{q}^2\,\mathbb{1}_4$ and the spectrum is the isotropic linear cone $E=\pm v_F|\mathbf{q}|$. The operator (E-Dirac-6) is the \textbf{Weyl equation} without mass: emergent chiral fermions in the continuum limit of the phase modes.

\textbf{Step 5 --- The mass term and the complete Dirac equation.} A term proportional to a fourth matrix $\Gamma_0=\sigma_z\otimes\sigma_z$, which anticommutes with the three $\Gamma_i$ and satisfies $\Gamma_0^2=\mathbb{1}$ (verified), opens a gap. Its physical origin in the lattice is the \textbf{A--B imbalance} (Peierls dimerization): any perturbation that distinguishes the two sub-lattices. The complete Hamiltonian is
\[
H = v_F\,\boldsymbol{\Gamma}\cdot\mathbf{q} + m c^2\,\Gamma_0, \tag{E-Dirac-8}
\]
with spectrum
\[
H^2 = \bigl(v_F^2\,\mathbf{q}^2 + m^2 c^4\bigr)\mathbb{1}_4
\;\Longrightarrow\;
E = \pm\sqrt{v_F^2\,\mathbf{q}^2 + m^2 c^4}\,. \tag{E-Dirac-9}
\]
The last link is provided by an already-derived piece: the phase modes of \emph{el Pleno} propagate at velocity $c$ (\S{}II.5, identical to that of the photon and the TT mode). With $v_F=c$, the relation (E-Dirac-9) is \emph{exactly} the relativistic energy-momentum relation $E^2=c^2\mathbf{q}^2+m^2c^4$, and the operator (E-Dirac-8) is the Dirac equation in Hamiltonian form, with Clifford-Lorentz algebra $\{\Gamma_\mu,\Gamma_\nu\}=2\eta_{\mu\nu}$. Lorentz covariance is not imposed: it is inherited from $v_F=c$, which in \emph{el Pleno} is structural (axiom A3, \S{}III.3).

\medskip
\begin{tcolorbox}[ecuacionclave,title={The Dirac equation as the effective theory of \emph{el Pleno}}]
\[
\boxed{\;\bigl(i\hbar\,\Gamma^\mu\partial_\mu - mc\bigr)\Psi_D = 0\;}
\qquad\Longleftrightarrow\qquad
H = c\,\boldsymbol{\Gamma}\cdot\mathbf{q} + mc^2\,\Gamma_0,
\qquad \{\Gamma_\mu,\Gamma_\nu\}=2\eta_{\mu\nu}. \tag{E-Dirac-10}
\]
It is not imported physics: it is the low-energy continuum limit of the phase modes $\theta_-$ over the bipartite diamond lattice. The spinorial doublet comes out of the A/B bipartition (\S{}III.4.2); the Clifford matrices, from the $SU(2)$ generators of $\theta_-$ (\S{}II.3.5); the first-order character, from the spin-orbit opening of the nodal line P7 (\S{}II.3.8); the covariance, from $v_F=c$ (\S{}II.5).
\end{tcolorbox}

\textbf{Verification of the algebra (symbolic computation).} The four matrices $\{\Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3\}$ verify symbolically $\{\Gamma_i,\Gamma_j\}=2\delta_{ij}\mathbb{1}_4$ for $i,j\in\{1,2,3\}$, $\{\Gamma_0,\Gamma_i\}=0$ and $\Gamma_0^2=\mathbb{1}_4$; consequently $H^2=(v_F^2\mathbf{q}^2+m^2c^4)\mathbb{1}_4$ exactly. The holonomy (E-Dirac-4) is verified to order $\theta^2$ with $[\sigma_x,\sigma_y]=2i\sigma_z$. All the steps are closed algebraic identities, without fitting parameters.

\textbf{Dimensional verification.} In (E-Dirac-8), $[v_F q]=[c][\text{m}^{-1}]\cdot\hbar=[\text{J}]$ with $\hbar$ restored, and $[mc^2]=[\text{J}]$: both terms are energy $\checkmark$. The covariant form (E-Dirac-10) has $[\hbar\Gamma^\mu\partial_\mu]=[\text{J}\cdot\text{s}][\text{m}^{-1}]=[\text{kg}\cdot\text{m}/\text{s}]$ and $[mc]=[\text{kg}\cdot\text{m}/\text{s}]$: momentum $\checkmark$.

\textbf{Analysis: what is achieved and what is not.} The result is the \emph{structure} of Dirac, not its phenomenology. \emph{Achieved}, with status \epi{DERIVED} (covariant route, \S{}V.7.2): (i) the first-order operator with Clifford matrices; (ii) the algebra $\{\Gamma_\mu,\Gamma_\nu\}=2\eta_{\mu\nu}$; (iii) the relativistic relation $E^2=c^2q^2+m^2c^4$; (iv) the emergence of spin-$\tfrac12$ as a property of the Lorentz spinor, not as a postulated quantum number; (v) the guarantee that the Dirac cone exists ($\lambda_\text{SO}\neq 0$ forced, \epi{SLD}). \emph{Not achieved} ---and declared without concealment---: the \textbf{value} of $\lambda_\text{SO}$ (and therefore $v_F$) from F1; the \textbf{value} of the masses $m$ (fermionic-masses thread); the \textbf{chirality} (why $SU(2)_L$ and not $SU(2)_R$ ---thread of medium-high severity, \S{}III.4.7); and the \textbf{number of generations}. What remains is not the \emph{structure} of the equation but two \emph{values}, reconnected with already-inventoried threads, without adding new threads.

\textbf{Discipline of scope.} What is derived here is the master equation, not its branches. The fine structure of hydrogen, the atomic spin-orbit coupling, the gyromagnetic factor and its radiative corrections are effective phenomenology that standard physics resolves over the already-given Dirac equation; the program inherits them as correct effective limits and does not need to re-derive them. PIU-$\Psi$ recovers the trunk ---the equation--- and leaves the branches in their validated regime, coherent with its scope criterion (to recover the fundamental equations of each domain, not their complete phenomenological tree).

\medskip
\begin{tcolorbox}[derivado,title={Card of the Dirac structure in PIU-$\Psi$}]
\textbf{Interpretation.} The Dirac equation is the low-energy continuum limit of the phase modes $\theta_-$ over the bipartite diamond lattice, not an imported relativistic formalism. The electron and the other fermions are emergent Dirac fermions at the nodal points opened by the intrinsic spin-orbit of the medium. \textbf{Explanation.} The spinorial doublet from the A/B bipartition (\S{}III.4.2); the Clifford matrices from the $SU(2)$ generators of $\theta_-$ (\S{}II.3.5); the opening of the Dirac cone from the spin-orbit forced by the $SU(2)$ holonomy (this \S{}, same holonomy as Yang-Mills, \S{}III.4.5); the covariance from $v_F=c$ (\S{}II.5, axiom A3). \textbf{Prediction.} Exact Dirac structure (first-order operator, Clifford-Lorentz algebra, $E^2=c^2q^2+m^2c^4$, emergent spin-$\tfrac12$); guaranteed existence of the Dirac cone ($\lambda_\text{SO}\neq 0$); the spin-statistics already derived by Finkelstein-Rubinstein (\S{}II.5.2) is consistent with the spin-$\tfrac12$ emergent here (both from the same $SU(2)$ character).

\smallskip
\textbf{Epistemic classification.} The \emph{structure} of the Dirac equation is \epi{DERIVED}: the covariant route (\S{}V.7.2) establishes it from the two binary structures of the medium with unique representation by the Pauli theorem, without depending on the value of $\lambda_\text{SO}$; the band route (this \S{}) confirms it with each derived link (A/B doublet, bands and nodal line P7, $SU(2)$ phase, $v_F=c$) or \epi{SLD} ($\lambda_\text{SO}\neq 0$ by holonomy, opening of the node). The emergence of spin-$\tfrac12$ is \epi{DERIVED} (representation of $so(3,1)$, \S{}V.7.2). The \emph{value} of $\lambda_\text{SO}$ and of the masses $m$ is a \textbf{declared thread}, reconnected with the fermionic-masses and chirality threads (\S{}III.4.7). The chirality ($SU(2)_L$ vs $SU(2)_R$) and the number of generations remain \textbf{open threads} of medium-high severity. The program asserts the \emph{structure} of Dirac as a derived result and declares with precision the two values and the two threads that remain; it does \textbf{not} claim to have derived the masses, the chirality or the generations.
\end{tcolorbox}

\subsection{\texorpdfstring{V.7.2 --- The covariant route: the genuine Lorentz spinor, uniqueness and spin-statistics}{V.7.2 - The covariant route: the genuine Lorentz spinor}}

\textbf{Origin and purpose.} The construction of \S{}V.7.1 delivers the \emph{form} of the Dirac equation as the effective theory of the phase modes over the lattice, via the Fu-Kane mechanism (bipartite Bloch Hamiltonian, spin-orbit opening of the nodal line), with Euclidean $\Gamma_i$ matrices of the band sector. The \emph{covariant} construction that follows establishes the same from the structure of the medium: the gamma matrices are built from the two binary structures of el Pleno, the Lorentzian signature \emph{emerges} (is not imposed), the representation is \emph{unique} by the Pauli theorem, the four-component object is a genuine Lorentz spinor (not a sublattice pseudospin), and the three premises of the spin-statistics theorem are derived properties of the medium. Both constructions are the same physics in two readings ---the band one and the covariant one---; the covariant one fixes the status of the Dirac structure and of the spin-statistics theorem as \epi{DERIVED}.

\textbf{Separation of the two $SU(2)$.} The program contains two $SU(2)$ groups that must be kept distinguished throughout the notation: $SU(2)_\text{lat}$, the $2T\to A_4$ cover of the discrete point group of the diamond lattice (lives in the A/B bipartition, the skyrmion target and the weak isospin; it is \emph{internal}); and $SU(2)_\text{Lor}$, the cover of the \emph{continuous} $SO(3)$ (the root, before the crystalline breaking to $2T$), rotational part of $SL(2,\mathbb{C})$ (lives in the emergent spacetime, axiom A3, zeroth order of the Cauchy-Green metric, \S{}III.1.7). The Dirac spinor transforms under $SU(2)_\text{Lor}$; identifying it with $SU(2)_\text{lat}$ would introduce spin-$\tfrac12$ through the back door.

\textbf{Construction of the gamma matrices from the two binary structures.} El Pleno possesses two independent binary structures that compose: (i) the imaginary unit $i$ of $\Psi = S\,e^{i\theta}$, complex structure with spinorial character (the phase returns to $-1$ after $2\pi$); and (ii) the A/B bipartition of the lattice. Their tensor product generates a space of dimension $4$, that of the Dirac spinor in $3{+}1$. With $\tau$ acting on the complex structure and $\sigma$ on $SU(2)_\text{Lor}$:
\begin{equation}
\gamma^0 = \tau_x\otimes\mathbb{1},\qquad
\gamma^i = i\,\tau_y\otimes\sigma_i \quad (i=1,2,3). \tag{E-Dirac-11}
\end{equation}
The squares, using $\tau_x^2=\tau_y^2=\mathbb{1}$ and $\sigma_i^2=\mathbb{1}$:
\[
(\gamma^0)^2 = +\mathbb{1}_4, \qquad
(\gamma^i)^2 = i^2\,\tau_y^2\otimes\sigma_i^2 = -\mathbb{1}_4,
\]
whence the signature $(+,-,-,-)$ \textbf{is not imposed}: it arises from the factor $i$ that distinguishes the spatial components from the temporal one. The cross anticommutators vanish because $\{\tau_x,\tau_y\}=0$ (for $\gamma^0\gamma^i$) and $\{\sigma_i,\sigma_j\}=2\delta_{ij}$ (for $\gamma^i\gamma^j$, $i\neq j$). Verified symbolically: the matrices (E-Dirac-11) satisfy $\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}\mathbb{1}_4$ with Lorentzian signature $(+,-,-,-)$. \epi{DERIVED}.

\textbf{Uniqueness of the representation (Pauli theorem).} An explicit construction proves existence; the \emph{fundamental theorem of Pauli} proves uniqueness. In $D=4$ the Clifford algebra $Cl(1,3)$ possesses a \emph{unique} irreducible representation of dimension $2^{D/2}=4$, up to unitary equivalence: given two sets $\{\gamma^\mu\}$ and $\{\gamma'^\mu\}$ of $4\times4$ matrices that satisfy the algebra, there exists a unitary $S$ with $\gamma'^\mu = S\,\gamma^\mu S^{-1}$. The representation of el Pleno (E-Dirac-11) and the standard Dirac one are both representations of the algebra and, by the theorem, unitarily equivalent. The construction of el Pleno introduces no physical freedom. What the medium fixes in a non-conventional way is the \emph{dimension}~$4$, because it possesses exactly two fundamental $\mathbb{Z}_2$ binary structures and only two ---the complex structure of $\Psi$ (return to $-1$ after $2\pi$) and the A/B bipartition (two inequivalent sublattices)---; their product fixes $2\times2 = 4 = 2^{D/2}$, which coincides with the unique irreducible representation. \epi{DERIVED}.

\textbf{Genuine Lorentz spin-$\tfrac12$.} To verify that the four-component object is a Lorentz spinor ---and not a sublattice pseudospin in the style of graphene--- the generators of the Lorentz algebra are constructed in the spinorial representation, $S^{\mu\nu} = \tfrac14[\gamma^\mu,\gamma^\nu]$, separated into rotations $J_i = \tfrac12\varepsilon_{ijk}S^{jk}$ and boosts $K_i = S^{0i}$. Verified symbolically with the gamma (E-Dirac-11):
\begin{itemize}[leftmargin=1.4em,itemsep=2pt]
\item The rotations close $su(2)$ in the antihermitian convention: $[J_1,J_2]=J_3$, $[J_2,J_3]=J_1$, $[J_3,J_1]=J_2$.
\item The physical angular-momentum operator (hermitian) is $\hat{J}_3 = -i\,J_3 = -\tfrac{i}{4}[\gamma^1,\gamma^2]$, whose eigenvalues are $\pm\tfrac12$ (each with multiplicity $2$): \emph{genuine spin-$\tfrac12$}. Equivalently, a spatial rotation of $2\pi$ acts as $\exp(2\pi J_3) = -\mathbb{1}_4$ (verified), the spinorial signature.
\item The boosts satisfy $[K_1,K_2] = -J_3$ (and cyclic); the \emph{negative sign} is the signature of the Lorentzian $so(3,1)$ algebra (in Euclidean signature $so(4)$ it would be $+J_3$).
\end{itemize}
The object therefore transforms under the emergent Lorentz group ---the zeroth order of the Cauchy-Green metric (\S{}III.1.7)--- and not under the rotations of the lattice: it is a genuine Lorentz spinor. \epi{DERIVED}.

\textbf{The Dirac operator as the root of the phase spectrum.} The spectrum of the phase sector of el Pleno is of fourth order ($\Box\delta\theta - (\ell_P^2/20)\Box^2\delta\theta = 0$, inherited from the flexural term), while the Dirac equation is of first order. The reconciliation is that the Dirac operator is the \emph{root} of the wave operator of the phase sector. The Clifford identity contracted with the four-momentum, $(\gamma^\mu k_\mu)^2 = k_\mu k_\nu\,\tfrac12\{\gamma^\mu,\gamma^\nu\} = k^2\,\mathbb{1}_4$ (verified), guarantees that the squared Dirac operator reproduces the scalar wave operator. The regulated Dirac operator of el Pleno is
\begin{equation}
D = \gamma^\mu k_\mu\left(1 + \frac{\ell_P^2}{40}\,k^2\right) - m, \tag{E-Dirac-12}
\end{equation}
and its square in the massless sector is
\[
D^2\big|_{m=0} = k^2\left(1 + \frac{\ell_P^2}{40}k^2\right)^2 = k^2\left(1 + \frac{\ell_P^2}{20}k^2 + \dots\right),
\]
that is, \emph{exactly} the phase spectrum of the corpus at first order in $\ell_P^2$. The regulator coefficient $\ell_P^2/40 = \gamma_\kappa$ is the same flexural coefficient of F1, and the factor $1/20$ of the fourth-order spectrum is its double by Euler-Lagrange ($2\cdot\tfrac{1}{40}=\tfrac{1}{20}$). The first-order Dirac operator and the fourth-order phase spectrum do not contradict each other: the first is the root of the second, and both share the flexural regulator $\gamma_\kappa$. \epi{DERIVED}.

\textbf{Spin-statistics theorem without postulating it.} Under a spatial rotation of $2\pi$, the Lorentz spinor acquires the factor $\exp(2\pi J_3) = -\mathbb{1}_4$ (verified above); equivalently, in terms of the physical spin operator with eigenvalues $\pm\tfrac12$, the factor is $e^{2\pi i(\pm 1/2)} = e^{\pm i\pi} = -1$. The fermionic soliton of el Pleno takes $(-1)^B$ under the same rotation, via Finkelstein-Rubinstein ($\pi_1(SO(3))=\mathbb{Z}_2$, \S{}II.5.2). For $B=1$ both signs coincide and \emph{are the same}: both are the generator of $\pi_1(SO(3))=\mathbb{Z}_2$, seen at two sites ---the cover $SU(2)_\text{Lor}\to SO(3)$ (geometry) and the configuration space of the soliton (topology). This is the \emph{root} of the spin-statistics theorem. The complete Pauli-Lüders theorem deduces the connection from three assumptions, which in el Pleno are not imported axioms but derived properties:
\begin{itemize}[leftmargin=1.6em,itemsep=3pt]
\item \textbf{(P1) Lorentz invariance} ---\epi{DERIVED}: the emergent Lorentz group is the zeroth order of the Cauchy-Green metric (\S{}III.1.7), under which the spinor transforms.
\item \textbf{(P2) microcausality} ---\epi{DERIVED}: the medium possesses a maximum propagation velocity $c$ (the barrier $\dot\gamma_\text{max}=c/\ell_P$, simultaneously relativistic limit, horizon and hard density bound, axiom A6); no correlation propagates outside the emergent light cone.
\item \textbf{(P3) positivity of the energy} ---\epi{DERIVED}: the strictly positive flexural rigidity (A4, $\gamma_\kappa = +\hbar c/40$) guarantees energy bounded below; it is the same sign that produces the $\geq$ sense of Heisenberg and the stability of the $R^2$ sector (\S{}III.1.4).
\end{itemize}
The fermionic statistics is not imposed by anticommutators: it emerges from the fact that the same $\mathbb{Z}_2 = \pi_1(SO(3))$ governs the rotation of the spinor (geometry) and that of the soliton (topology); and the three assumptions of the theorem are derived properties of the medium. The formal axiomatic assembly over these premises is a standard exercise of quantum field theory, not a missing physical piece.

\begin{tcolorbox}[derivado,title={Dirac structure and spin-statistics}]
The covariant construction from the two binary structures of the medium establishes as \epi{DERIVED}: gamma matrices with Clifford algebra of \emph{emergent} Lorentzian signature (not imposed); \emph{unique} representation up to unitary equivalence (Pauli theorem); irreducible representation of $so(3,1)$ with spin-$\tfrac12$ (eigenvalues $\pm\tfrac12$, $[K_i,K_j]=-\varepsilon_{ijk}J_k$); Dirac operator as the root of the phase spectrum with regulator $\gamma_\kappa$; and spin-statistics theorem closed at its root ($\mathbb{Z}_2=\pi_1(SO(3))$) and in its three premises (P1)-(P3), all derived from the medium. There remain as declared numerical threads: the value of $\lambda_\text{SO}$, the masses $m$, the chirality and the number of generations (\S{}III.4.7).
\end{tcolorbox}

\clearpage
\phantomsection
\section{\texorpdfstring{\S{}V.8 --- Appendix: Wien, Planck and the ontological descent of the constants}{V.8 - Appendix: Wien, Planck and the ontological descent of the constants}}
\markboth{\S{}V.8 --- The ontological descent of the constants}{}

\begin{estadoseccion}
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} consolidated. Interpretive and historical piece.
  \item \textbf{Verification:} complete symbolic dimensional, with cited historical sources.
  \item \textbf{Open threads:} none new. It traces a route of closure for the already-inventoried calibration thread C-meta-1.b (\S{}V.3).
  \item \textbf{Notes:} interpretive and historical appendix. It reorders the meaning of the constants already derived in \S{}I.8.5, \S{}I.8.9, \S{}II.4 and \S{}III.1--III.2 and situates them in the tradition of Wien and Planck. It introduces no new dynamics: it reinterprets results already obtained.
\end{itemize}
\end{estadoseccion}

\subsection{\texorpdfstring{V.8.1 --- Wien, Planck and the ontological descent of the constants}{V.8.1 - Wien, Planck and the ontological descent of the constants}}

\textbf{Origin and purpose.} This subsection is deliberately \emph{interpretive and historical}: it does not introduce new dynamics, but reorders the meaning of the constants already derived in \S{}I.8.5, \S{}I.8.9, \S{}II.4 and \S{}III.1--III.2, and situates them in the tradition that made them possible. Its value is twofold. First, it reconstructs the documented origin of the fracture between regimes that the program seeks to heal (problems 4 and 6 of \S{}I.2): it shows that the bifurcation of physics around 1900 was, at bottom, a \emph{physical boundary between two regimes of a single medium, read as a seam between two formulas}. Second, it renders to Wien and Planck the recognition of pioneers whose ``fitting numbers'' contained more physics than their epoch could read. The thesis that structures the subsection is a \textbf{three-level ontological descent}, in which the ``fundamental constant'' of each level reveals itself as a derived combination of the properties of the lower level:
\begin{itemize}[leftmargin=1.6em,itemsep=2pt]
  \item \textbf{Wien (1896):} two constants of pure fitting, without declared physical content.
  \item \textbf{Planck (1900):} those constants turn out to be traces of $h$, $c$ and $k_B$.
  \item \textbf{PIU-$\Psi$:} $h$ and $G$ turn out to be traces of the two substantive properties of \emph{el Pleno}, $\rho_P$ and $\ell_P$, and of its causal structure $c$.
\end{itemize}
The descent ends in $(\rho_P,\ell_P)$ because \emph{el Pleno} does not possess numbers, but ---in the language of \S{}I.8.9--- a grain, a density and bounds.

\medskip
\textbf{The historical arc: from Wien's law to the infrared boundary.} In 1896 Wilhelm Wien proposed, from the available data and from ideas taken from the Maxwell velocity distribution, a spectral distribution of the blackbody of the form
\[
\phi_\lambda = \frac{c_W}{\lambda^5}\,e^{-\alpha_W/(\lambda T)}, \tag{E-WP-1}
\]
with $c_W$ and $\alpha_W$ two empirical scale factors, without declared physical content. The power $\lambda^{-5}$ is forced by the Stefan-Boltzmann law; the exponential structure is copied from Maxwell-Boltzmann. The law worked so well in the accessible region ---Paschen's near-infrared (1897) showed no deviation--- that the problem was taken as resolved, and that agreement authorized Planck to treat Wien's law as fundamental: he devoted five works (1897--1899) to deriving it from the thermodynamics of radiation, seeking an explanation of irreversibility that did \emph{not} depend on Boltzmann's theory of probabilities, toward which he had deep reservations.

The key of the episode is in a boundary: that of the far infrared ---not a boundary out of reach, but a misinterpreted boundary. Upon pushing the measurements toward longer wavelengths, Wien's law began to fail. Lummer and Pringsheim reached 12--18 µm and observed the non-validity of Wien's spectral equation; the decisive work of Rubens and Kurlbaum in the far infrared (30--60 µm) showed without any doubt that for long wavelengths Wien's formula was inadequate over a wide range of temperatures. The deviation was not small: in the long-wave limit Wien predicted a dependence with $T^3$, but the data showed a behavior proportional to $\nu^2 T$ ---the intensity of a monochromatic beam at high temperature turned out proportional to the temperature, not to its cube. The historical testimony fixes the date: on October 7, 1900 Rubens showed Planck the new long-wave results; Kurlbaum presented them in the same session of the Academy, on October 19, 1900, in which Planck presented his new formula.

\medskip
\begin{tcolorbox}[notabox,title={The decisive physical point: a boundary between regimes, not a seam between formulas}]
What the far infrared revealed was not ``Wien needs a patch''. It was that the blackbody seemed to obey \textbf{two distinct sets of rules according to the wavelength}: Wien's law governed the high-frequency regime, and the $\nu^2 T$ behavior (anticipated by Rayleigh) governed the low-frequency one. The boundary between the two is where the energy of the mode equals the thermal energy, $h\nu\approx k_B T$. That boundary is not an instrumental limitation: it is a \emph{physical boundary between two regimes of a single radiating system}. Forced by the data, Planck built in October 1900 a new law by \emph{interpolation} between the two regimes; the formula was exact, but its first derivation used neither quantization nor Boltzmann statistics: it was a phenomenological bridge. To give it a fundamental basis, in December 1900 he resorted ---in what he himself later called ``an act of desperation''--- to the probabilistic notion of entropy of Boltzmann that he had rejected for years, dividing the energy into elements of size $h\nu$. The central fact, documented, is that \textbf{Planck did not believe that radiation was intrinsically discrete}: he spent more than a decade trying to eliminate the discontinuity from the theory.
\end{tcolorbox}

The ``departure from the path'', understood correctly and without projecting backward a knowledge impossible in 1900, was subtle and consequential: the infrared boundary was the trace that a single medium has two regimes ---one collective and almost continuous at low frequency, one granular and discrete at high frequency. Planck treated it as an interpolation between two formulas, not as the signature of a substrate with internal scale. He stitched the two rules into a correct equation, but the urgency of the moment ---and the absence, in 1900, of general relativity, of quantum condensates or of a Planck scale with physical content--- made him not ask what physical object naturally has those two faces. The discreteness $h\nu$, which was the signature of the grain of the medium, remained a counting artifice. That interpretive gap ---an exact formalism without the ontology that would support it--- is the documented germ of a century of ``shut up and calculate'', and the common historical root of problems 4 (incompatibility between regimes) and 6 (ontological status of the fields) of \S{}I.2.

\medskip
\textbf{The PIU reading: the single medium and its two regimes.} PIU-$\Psi$ proposes precisely the object that the 1900 episode left unnamed: a single medium ---\emph{el Pleno}--- whose two regimes are the two faces that Planck stitched. In the language of \emph{el Pleno} (\S{}I.10, \S{}II.4), the low-frequency / IR regime corresponds to the collective modes of the medium below its characteristic scale, where the response is smooth and almost continuous (the $\nu^2 T$ behavior); the high-frequency / UV regime corresponds to the modes above that scale, where the granularity dominates (the factor $e^{-h\nu/k_B T}$ of Wien); and the boundary $h\nu\approx k_B T$ is a boundary between regimes of the medium, not a seam between formulas. The program already contains this structure with a derived boundary, not postulated: the mode spectrum of \emph{el Pleno} (\S{}II.5) separates a massless IR sector (phase modes: photon and transverse tensor mode, with dispersion $\omega^2=c^2k^2[1+(k\ell_P)^2/20]$) from a massive UV sector (radial mode of the modulus, of ultra-Planckian mass $\sqrt{\xi}\,E_P\approx 4.44\,E_P$, with $\xi=V''_\text{eff}(S_{\min})/(\rho_P c^2)\approx 19.69$). The boundary between the collective-continuous and the granular-discrete that Planck found empirically in the blackbody is, under PIU-$\Psi$, a manifestation that \emph{el Pleno} is a medium with internal scale $\ell_P$. \epi{SLD} ---the conceptual correspondence is direct; the quantitative identification between the boundary $h\nu\approx k_BT$ and the scales of the spectrum of \S{}II.5 requires the thermodynamic development of the medium, pending (thread, \emph{infra}).

\medskip
\textbf{The ontological descent of the constants.} The move that Planck executed over Wien ---to show that a fitting number hid physical structure--- PIU-$\Psi$ executes over Planck, one level below.

\emph{Level 1 --- Planck reveals the structure of Wien.} Upon taking the high-frequency limit of his law (where the $-1$ of the Bose-Einstein denominator is negligible), Planck recovered the form of Wien and read the anatomy of its constants:
\[
c_W = 8\pi h\,c_\text{light}, \qquad \alpha_W = \frac{h\,c_\text{light}}{k_B}. \tag{E-WP-2}
\]
Dimensional verification: $[h][c_\text{light}][k_B]^{-1}=(ML^2/T)(L/T)(T^2\Theta/ML^2)=L\,\Theta=[\alpha_W]$ $\checkmark$. Wien's ``fitting numbers'' were traces of three constants with meaning.

\emph{Level 2 --- PIU reveals the structure of Planck.} The constants that Planck treated as fundamental are, in PIU-$\Psi$, combinations of the structural properties of \emph{el Pleno} (\S{}I.8.5, \S{}III.1.3, \S{}III.2.2):
\medskip
\begin{tcolorbox}[ecuacionclave,title={The Planck constants as traces of the properties of \emph{el Pleno}}]
\[
\hbar = \rho_P\,c\,\ell_P^4, \qquad\qquad G = \frac{c^2}{\rho_P\,\ell_P^2}. \tag{E-WP-3}
\]
Dimensional verification: $[\rho_P][c][\ell_P]^4=(M/L^3)(L/T)(L^4)=ML^2/T=[\hbar]$ $\checkmark$; and $[c]^2/([\rho_P][\ell_P]^2)=(L^2/T^2)/((M/L^3)L^2)=L^3/(MT^2)=[G]$ $\checkmark$. Both with exact numerical agreement by construction (residue $<0.1\%$ by rounding of the primaries; \S{}III.1.3).
\end{tcolorbox}
The identity of $G$ is not a dimensional coincidence: \S{}III.1.2 \emph{derives} it by showing that two perturbations of the modulus of \emph{el Pleno} interact by elastic superposition of their gradients, $U_\text{int}=\int\rho_P c^2\ell_P^2\,\nabla(\delta S_1)\cdot\nabla(\delta S_2)\,d^3x$, which produces a potential $1/r$ whose prefactor is identified without a residual factor with $c^4/G$. The gravitational constant is the rigidity of the medium read at large distance.

\emph{Planck's action in the language of \emph{el Pleno}.} The Planck constant was born as a unit of action. The native identity regroups exactly as momentum $\times$ distance:
\[
\hbar = \underbrace{(\rho_P\,\ell_P^3)}_{\text{mass of a cell}}\cdot\underbrace{c}_{\text{structural velocity}}\cdot\underbrace{\ell_P}_{\text{grain}}. \tag{E-WP-4}
\]
Planck's quantum of action is, structurally, the momentum of a minimal cell of \emph{el Pleno} traversing its own grain. The discreteness that Planck introduced ``by desperation'' is, in this reading, the direct signature of the grain $\ell_P$ of the medium. \epi{SLD} (exact regrouping of the \textbf{[derived]} identity of \S{}I.8.5).

\medskip
\begin{tcolorbox}[notabox,title={The density of \emph{el Pleno} was already latent in Planck's 1900 units}]
A historical finding reinforces the descent thesis. In 1899--1900, \emph{before} quantization, Planck defined his natural units from his own measured values: $b=h=6.885\times10^{-34}$ J$\cdot$s, $a=4.818\times10^{-11}$ K$\cdot$s and $G=6.685\times10^{-11}$. From there he derived a natural length $\approx 4.13\times10^{-35}$ m and a natural mass $\approx 5.56\times10^{-8}$ kg. The density implicit in those 1900 values is $\rho_\text{nat}=m_\text{nat}/\ell_\text{nat}^3\approx 7.9\times10^{95}$ kg/m³, which coincides in order of magnitude with the corpus value $\rho_P\approx 5.16\times10^{96}$ kg/m³ (the difference is the factor $\sqrt{\xi}$ and the use of $h$ versus $\hbar$). That is: the density of \emph{el Pleno} was already latent in the natural units that Planck published in 1900 ---a number in his hands without his being able to know that it was the density of a physical medium. It is the most literal example of the thesis of the subsection: a trace awaiting interpretation for more than a century. \epi{DERIVED} (reconstruction of historical sources; Planck 1900a).
\end{tcolorbox}

\emph{Why the descent ends.} The descent does not continue to a Level 3 because $\rho_P$ and $\ell_P$ are not combinations of anything else: they are the two irreducible substantive properties. According to \S{}I.8.9, to ask ``what number did \emph{el Pleno} choose for $\ell_P$?'' is ill-posed ---\emph{el Pleno} has a grain and bounds; the value of those bounds informs how many human yardsticks fit in them, not about \emph{el Pleno} itself.

\medskip
\textbf{The ontological distinction: human yardsticks versus properties of \emph{el Pleno}.} The dimensional analysis of the complete chain uncovers an actor that does not fit in $(\rho_P,\ell_P,c)$: the Boltzmann constant $k_B$, which carries the dimension of temperature $\Theta$. This separates the ``fundamental constants'' into two ontologically distinct classes. \textbf{Class A} are conversion factors of human yardsticks, which equal $1$ in the language of \emph{el Pleno} and disappear: $c$ converts the space-yardstick (meter) and the time-yardstick (second) into the single causal structure of the medium (in the language of \emph{el Pleno} $c=1$: space and time are the same structure measured with two yardsticks, \S{}I.8.9); and $k_B$ converts the temperature-yardstick (Kelvin, human convention) into what \emph{el Pleno} really has ---excitation energy per mode--- so that in the language of \emph{el Pleno} $k_B=1$: the temperature \emph{is} energy. $k_B$ is not a constant of \emph{el Pleno}, but the dictionary between the temperature-yardstick and the structural energy, the same status as $c$. \textbf{Class B} are the substantive properties, what the medium \emph{is}: $\rho_P$ (density) and $\ell_P$ (grain). And $\hbar$ and $G$ are traces: combinations of Class B that the human framework reads as fundamental because it measures them through quantization ($\hbar$) and gravitation ($G$). The constraint $\ell_P=\sqrt{\hbar G/c^3}$ shows that they are not two independent verifications, but a single structural condition in two languages (\S{}I.8.5).

\medskip
{\footnotesize
\begin{longtable}{L{1.3cm}L{2.6cm}L{4.4cm}L{2.5cm}}
\toprule
\textbf{Const.} & \textbf{Dim.} & \textbf{Ontological class} & \textbf{In \emph{el Pleno}} \\
\midrule
\endhead
$c$ & $L/T$ & A --- conversion (space$\leftrightarrow$time) & $1$ \\
$k_B$ & $ML^2/(T^2\Theta)$ & A --- conversion (temp.$\leftrightarrow$energy) & $1$ \\
$\hbar$ & $ML^2/T$ & derived trace $\rho_P c\,\ell_P^4$ & --- \\
$G$ & $L^3/(MT^2)$ & derived trace $c^2/(\rho_P\ell_P^2)$ & --- \\
$\rho_P$ & $M/L^3$ & B --- substantive property & real density \\
$\ell_P$ & $L$ & B --- substantive property & real grain \\
\bottomrule
\end{longtable}}

\medskip
\textbf{Synthesis of the descent.} The complete picture, with the epistemic classification of each step, is:
\[
\begin{array}{lll}
\textbf{Wien (1896):} & \alpha_W = \text{fitting number} & \epi{CALIBRATED}\\[2pt]
\textbf{Planck (1900):} & \alpha_W = h\,c_\text{light}/k_B & \text{trace of }(h,c,k_B)\\[2pt]
\textbf{PIU-}\Psi: & h=\rho_P c\,\ell_P^4,\ \ G=c^2/(\rho_P\ell_P^2) & \text{trace of }(\rho_P,\ell_P)\\[2pt]
& c,\,k_B=\text{yardstick conversions} & =1\\[2pt]
& \rho_P,\,\ell_P=\text{substantive properties} & \text{(end of the descent)}
\end{array}
\tag{E-WP-5}
\]
The decisive difference is this: Planck's step was confirmed independently ($h$ and $k_B$ were later measured by other routes and coincided). PIU-$\Psi$'s step is in \epi{DERIVED} (functional form) plus \epi{SLD} (prefactor); its analogous confirmation ---to measure $(\rho_P,\ell_P)$ by a third route--- is the thread that closes this subsection.

\medskip
\textbf{The ontological inversion: $G$ as the softness of the medium.} The way in which the relation between $G$ and $\rho_P$ is written encodes the ontological direction. In terms of Planck constants, the density of \emph{el Pleno} is
\[
\rho_P = \frac{c^5}{\hbar\,G^2}, \tag{E-WP-6}
\]
with dependence $G^{-2}$ ---the strongest of all the Planck quantities (the four units carry $G^{\pm1/2}$; $\rho_P$ carries $G^{-2}$). This admits two readings that are the same equation in opposite directions: the human reading ($\rho_P\sim G^{-2}$) takes $G$ as a measured input and $\rho_P$ as a derived and exotic density; the PIU reading (inverting, $G\sim\rho_P^{-1/2}$, equivalent to $G=c^2/(\rho_P\ell_P^2)$) takes $\rho_P$ as a property of the medium and $G$ as its consequence. To invert the formula is to execute the change of ontological direction that is the thesis of the program. And the inverse has direct physical content: $1/G=K/c^4$ with $K=\rho_P c^2\ell_P^2=c^4/G\approx 1.21\times10^{44}$ N (the ``Planck force'', the structural tension of the medium). That is:
\medskip
\begin{tcolorbox}[derivado,title={$G$ is not a force intensity: it is the softness of \emph{el Pleno}}]
$G$ is the \emph{compliance} of \emph{el Pleno}, the inverse of its structural rigidity $K=c^4/G$. From here follows a one-line answer to the old problem of why gravity is so weak against the other forces: \textbf{gravity is weak because \emph{el Pleno} is extremely dense and therefore extremely rigid} ---$G$ is minuscule precisely because $K=c^4/G$ is gigantic. The gravitational weakness ceases to be a hierarchy mystery and becomes the macroscopic reading of an almost infinitely rigid medium. \epi{SLD} (physical reading of the \textbf{[derived]} identity $G=c^2/(\rho_P\ell_P^2)$, \S{}III.1.3).
\end{tcolorbox}
As a complement, the direct inverse $1/\rho_P=\hbar G^2/c^5$ has the dimension of specific volume $[\text{m}^3/\text{kg}]$ ---the natural thermodynamic state variable of \emph{el Pleno}, conjugate to the pressure, and therefore the appropriate magnitude for the thermodynamic route of the calibration thread.

\medskip
\textbf{Two rigidities: one constant, one variable.} The reading of $G$ as the inverse of rigidity obliges a distinction between two rigidities of \emph{el Pleno} that the corpus keeps separate. The \textbf{structural rigidity} $K=\rho_P c^2\ell_P^2=c^4/G$ depends only on the primary parameters $(\rho_P,c,\ell_P)$, is a property of the medium itself and is constant at every point of \emph{el Pleno}; from it comes $G$, which is therefore constant. The \textbf{effective rigidity} $V''_\text{eff}(S)$ is the curvature of the potential, which depends on the local configuration $S$: it has its minimum at the \emph{Pleno}-ground state ($V''_\text{eff}(S_{\min})\approx 19.69\,\rho_P c^2$) and diverges at both walls ---toward $S\to 0$ (Klauder IR) and toward $S\to S_{\max}$ (Gent UV). The medium is softer in the relaxed vacuum and hardens without limit upon approaching either of its two bounds. Consequently, $G$ is constant throughout \emph{el Pleno}, but the effective gravity departs from the Newtonian where the response rigidity $V''_\text{eff}(S)$ ceases to be negligible against the kinetic term ---in dense condensations and, above all, near the saturation $S_{\max}$ (interiors of compact objects, primordial bounce). There the divergent rigidity reverses the attraction and produces finite cores instead of singularities (A6, \S{}II.6.2). The derivation of $G$ (\S{}III.1.2) explicitly assumes the regime where $V''$ is negligible; outside it, what is modified is not the structural $G$ but the effective gravitational response. \epi{DERIVED} (second derivative of the effective potential of \S{}II.2; regime of validity of \S{}III.1.2).

\medskip
\textbf{\emph{El Pleno} as a stratified fluid: a complementary physical image.} The modulus field admits an intuitive reading as a fundamental fluid stratified by density, formally supported: \S{}II.5.7 derives that F1 reduces exactly to the Gross-Pitaevskii equation in the non-relativistic limit,
\[
i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_\text{eff}}\nabla^2\psi + g_\text{eff}|\psi|^2\psi, \tag{E-WP-7}
\]
with coupling $g_\text{eff}>0$ (repulsive, stable condensate). The ``liquid of \emph{el Pleno}'' is, formally, a Bose-Einstein condensate, which turns the experiments on macroscopic condensates into indirect validation of the program. Given that the local density is $\rho_\text{medium}=\rho_P S^2$ (\S{}II.2), the configurations are ordered by density: $S_{\min}^\text{estr}\approx 0.7231$ gives $\rho\approx 0.52\,\rho_P$ (the least dense configuration, the ``surface'', the \emph{Pleno}-ground state / vacuum), and $S_{\max,\text{geom}}^2\approx 2.1773$ gives $\rho\approx 2.18\,\rho_P$ (the densest, the ``bottom'', the Kepler-Hales saturation). In this image, the Klauder IR wall ($S_{\min}$) is the upper surface where a relaxed condensation is trapped upon losing density; the UV wall / A6 bound ($S_{\max}$) is the bottom where the compression hits the optimal packing. The vacuum is not absence: it is the surface of the fluid of \emph{el Pleno}, at $0.52\,\rho_P$.

This same rigidity, read at a different scale, gives contact and gravity. The perturbation of the modulus around a condensation is a Yukawa field $\propto e^{-r/\lambda_S}/r$ with correlation length $\lambda_S=\ell_P/\sqrt{\xi}=0.225\,\ell_P$ (Compton length of the massive radial mode). Since the modulus mode is massive, its force is screened at a sub-Planckian scale ---it is an ultra-short contact force between condensations, not gravity. The long-range gravity comes from the same kinetic term $\rho_P c^2\ell_P^2$ but in the regime where the mass term is negligible against the Laplacian ($r\gg\lambda_S$), where \S{}III.1.2 derives the potential $1/r$ with prefactor $c^4/G$. Thus, a single rigidity of the medium produces, at $r\sim\lambda_S$ (sub-Planck), hard contact (massive radial mode); and at $r\gg\lambda_S$ (macroscopic), Newtonian gravity $1/r$ (massless limit of the same term). \epi{DERIVED} (\S{}III.1.2 + \S{}II.5). The weakness and the long range of gravity are the same property ---the very high rigidity $\rho_P$ of \emph{el Pleno}--- seen in two ways.

\medskip
\textbf{Open thread: the temperature of \emph{el Pleno} and the route of closure of C-meta-1.b.} Since $k_B$ converts temperature$\leftrightarrow$energy and the temperature is excitation energy per mode, the thermodynamic sector of the medium is a candidate for an independent measurement of $(\rho_P,\ell_P)$ for the already-inventoried calibration thread \textbf{C-meta-1.b} (\S{}V.3) ---the third route that would break the degeneracy $\ell_P=\sqrt{\hbar G/c^3}$. What does \emph{not} work is discarded: the minimum temperature reached in Bose-Einstein condensates is not a constant of nature, but a variable technical record ($\approx 500$ pK in 2003, $\approx 38$ pK in 2021); the impossibility of reaching 0 K is, in part, a process limit (third law of Nernst), not a fixed minimum temperature.

\medskip
\begin{tcolorbox}[hipotesisbox,title={The only route that breaks the degeneracy: the saturation density}]
An analysis of the degeneracy gives a precise verdict. All the coefficients of the condensate of \emph{el Pleno} (Gross-Pitaevskii regime, \S{}II.5.7) are expressed in terms of the Planck scale $(m_P,\ell_P,c)$, and $\rho_P$ enters only as the Planck density $\rho_P=m_P/\ell_P^3=c^5/(\hbar G^2)$. To measure $\rho_P$ by any route that passes through that scale ---Bogoliubov sound velocity, \emph{healing length}, zero-point energy, Planck energy--- is equivalent to measuring $m_P$ and $\ell_P$, which already contain $\hbar$ and $G$, because $\rho_P$ depends on $G^{-2}$, the strongest dependence on $G$ of all the Planck quantities. The only route that breaks the degeneracy is to measure $\rho_P$ as a \textbf{direct mass-energy density}, not inferred from the Planck scale. The cleanest candidate is the saturation density $\rho_{\max}=S_{\max}^2\,\rho_P\approx 2.18\,\rho_P$ (A6 hard bound, \S{}I.5): if one day the maximum density attainable by compressed matter were measured ---a neutron star at the collapse limit, the remnant of a kilonova at the instant of merger, a signature of the bounce--- that number would give $\rho_P$ solved \emph{without $\hbar$ or $G$}. By rigorous elimination of all the others, this is the real exit of C-meta-1.b. \epi{HYPOTHESIS} motivated; it reduces the thread of ``to seek a third route'' to ``to measure the saturation density of a compact object''.
\end{tcolorbox}

\medskip
\textbf{The recognition that Wien and Planck deserve.} The recognition is not ``Planck was wrong and PIU corrects him'' ---that would be false and would project backward a knowledge impossible in 1900. It is the opposite, and it is more solid: Wien and Planck did exactly what the evidence of their epoch permitted, and they did it so well that their ``fitting constants'' turned out to be windows to physical structure that took a century to reveal itself. Wien did not know that his $\alpha_W$ hid $h$, $c$ and $k_B$; Planck did not know that his $h$ hid $\rho_P c\,\ell_P^4$, nor that the infrared boundary that forced him to the ``act of desperation'' was the signature of a medium with internal grain. Neither erred ---both saw more physics than they could justify. They built the exact bridge between two regimes without having, in their epoch, the framework to name the single medium whose two faces were those regimes. PIU-$\Psi$ proposes that medium, and in doing so does not surpass them: it finishes an interpretation that they opened.

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\textbf{Interpretation.} The human ``fundamental'' constants are ordered in a descent: those of Wien are traces of $(h,c,k_B)$; $h$ and $G$ are traces of $(\rho_P,\ell_P)$; $c$ and $k_B$ are conversions of human yardsticks that equal $1$ in the language of \emph{el Pleno}. The descent ends in the two irreducible substantive properties. \textbf{Explanation.} $\hbar=\rho_P c\,\ell_P^4$ (\S{}I.8.5, \S{}III.2.2) and $G=c^2/(\rho_P\ell_P^2)$ (\S{}III.1.3) as structural identities; the blackbody boundary $h\nu\approx k_BT$ as a manifestation of the mode spectrum of \S{}II.5; $G$ as the compliance of the medium ($K=c^4/G$). \textbf{Prediction / thread.} The route of closure of C-meta-1.b: to measure the saturation density $\rho_{\max}\approx 2.18\,\rho_P$ of a compact object would give $\rho_P$ without passing through the Planck scale.

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\textbf{Epistemic classification.} The identities $\hbar=\rho_P c\,\ell_P^4$ and $G=c^2/(\rho_P\ell_P^2)$ are \epi{DERIVED} (\S{}I.8.5, \S{}III.1.2--III.1.3). The reinterpretation of $G$ as compliance and the reading of $\hbar$ as the momentum of a cell are \epi{SLD} (exact regroupings of derived identities). The correspondence between the blackbody boundary and the mode spectrum of \emph{el Pleno} is \epi{SLD} (the quantitative identification requires the thermodynamic development, pending). The historical reconstruction (density latent in Planck's 1900 units) is \epi{DERIVED} from primary sources. The saturation-density route to close C-meta-1.b is a motivated \epi{HYPOTHESIS}. The subsection does \emph{not} introduce new dynamics: it reorders the meaning of already-derived constants and traces a route of closure of an already-inventoried thread.
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