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\section*{Part V --- Observational confrontation and predictions}
\phantomsection
\addcontentsline{toc}{section}{Part V --- Observational confrontation and predictions}
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\begin{tcolorbox}[notabox,title={Purpose of Part V}]
To make effective the predictive capacity of the program. Parts I--III
established the substrate (\emph{el Pleno} and its Lagrangian), its space of
configurations, and the phenomenological reading of the observed facts as
configurations of that substrate; Part IV articulated the dynamics of the
crunch-bounce cycle and the formation of particles as a synthesis of that reading. 
Part V gathers the set of
predictions generated in that journey, confronts them quantitatively with
the current observation, declares the investigative threads that
remain open, and closes the manuscript articulating where the program is
consolidated, where it is falsifiable, and what remains pending. Part V does not introduce
new derivational apparatus: it organizes, confronts and declares.
\end{tcolorbox}

\medskip\hrule\medskip

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\subsection{\S{}V.1 --- Complete catalogue of derived predictions}
% ====================================================================

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (the threads associated with each prediction are inventoried in \S{}V.3)
  \item \textbf{Notes:} systematic catalogue of predictions. Synthesis section in accordance with exception (b) of the mathematical-presentation norm: it gathers results already derived in Parts I--III; each entry references the PIU-$\Psi$ origin subsection where the complete derivation fulfills the five components (origin, steps, dimensional verification, numerical verification, classification). It covers the catalogue P1--P30, P-Photon-1..6, P-Lambda-1..3 and P-Hadron-1..2.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{V.1.1 --- What counts as a prediction under PIU-$\Psi$}{V.1.1 - What counts as a prediction under PIU-Psi}}

A prediction of PIU-$\Psi$ is a quantitative or qualitative assertion about the result of an observation, derived from the apparatus of the program (axioms A1--A6, F1, configuration space) without a posteriori fitting to the datum. The program distinguishes three types:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Derived retrodictive prediction.} An already-measured magnitude that the apparatus reproduces without having used it as input ($G$, $\Omega_{MO}/\Omega_M$, $\Delta$-$N$ split). Its value is that it was not calibrated: the apparatus produces it.
  \item \textbf{Forward falsifiable prediction.} An observation not yet performed or not conclusive that the program anticipates and that, if it came out different, would refute it (UV suppression of the tensor spectrum P15, invariance of the hadronic scale P-Lambda-1..3).
  \item \textbf{Structural-consistency prediction.} An internal identity whose violation would be incompatible with the ontology of \emph{el Pleno} (the identity $v_\gamma = v_\text{GW} = c$, P-Photon-2, P-Photon-5).
\end{itemize}

Every prediction of the catalogue carries its explicit epistemic classification:
\epi{DERIVED} (complete
demonstration from the apparatus), \epi{SLD} (strong logical deduction:
structure without a conceptual gap, without explicit complete algebra),
\epi{CALIBRATED} (fitted to a datum with declared error),
\epi{HYPOTHESIS} (motivated conjecture). The classification of each prediction
is the one it has in the origin subsection; the catalogue neither promotes nor
degrades it.

\subsubsection{V.1.2 --- Block: gravitation and fundamental constants}

The structural identities that connect the two substantive properties of
\emph{el Pleno} ($\rho_P$, $\ell_P$) and its causal structure $c$ (\S{}I.8.9) with the observed fundamental constants,
derived in \S{}III.1--\S{}III.2, are the predictive base of the program in the
non-cosmological regime.

\textbf{The gravitational constant} is the structural identity (E3):

$$
G = \frac{c^2}{\rho_P\, \ell_P^2}
$$

The non-trivial content of this prediction is not the $1/r$ form of the potential
(which is a consequence of the Laplacian of the kinetic term of F1), but the
\emph{value} of $G$: once the three primaries are fixed, $G$ is determined,
without a single fitting parameter. This is exactly what the standard framework does not
do ---General Relativity assumes $G$ as a measured datum (\S{}I.2.6)--- and is, in
that precise sense, a new result: the gravitational constant ceases to be
an input and becomes a structural consequence of the substrate. The complete numerical
verification, with all the intermediate steps, is developed in \S{}III.1.3;
the agreement with the CODATA value is of the order of $0.03\%$, and the
remaining residue comes from the rounding of the primary parameters to four figures, not
from the identity, which is exact by construction. \epi{DERIVED}

\textbf{The reduced Planck constant} is the structural identity (E36):

$$
\hbar = \rho_P c\,\ell_P^4
$$

derived from the quantization of the phase mode of \emph{el Pleno} in \S{}III.2.2, and linked to
(E3) by the relation $\ell_P = \sqrt{\hbar G/c^3}$. The Planck constant thus receives
the same treatment as $G$: instead of being postulated as the minimum quantum of
action (\S{}I.2.6), it emerges from the structural properties of the substrate. That the
two constants that standard physics treats as independent inputs ---one from
gravity, the other from quantum mechanics--- follow from a single set of
three primaries is, in itself, a unification result that the program
claims. Complete numerical verification in \S{}III.2.2; the identity is exact
by construction and the numerical residue comes from the rounding of the primaries.
\epi{DERIVED}

These two identities determine, in turn, the structural rigidities of
\emph{el Pleno} (\S{}III.1.6), among them the flexural rigidity (E4)
$\gamma_\kappa = \rho_P c^2\ell_P^4/40 = \hbar c/40$ and the characteristic tension
$\sigma_\text{Pleno} = \gamma_\kappa/\ell_P^2$.

\subsubsection{V.1.3 --- Block: modes of \emph{el Pleno} and gravitational waves}

Gravitational waves are, under PIU-$\Psi$, the transverse traceless mode
(TT) of \emph{el Pleno} (\S{}III.3). From its derived dispersion relation (E23),

$$
\omega^2 = c^2 k^2\left[1 + \frac{(k\ell_P)^2}{20}\right]
$$

three predictions follow:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Propagation velocity} $v_\text{GW} = c$ in the infrared,
  exact, massless and without dissipation. Verified by the temporal coincidence of
  GW170817 with its electromagnetic counterpart, $|v_\gamma -
  v_\text{GW}|/c < 7\times 10^{-16}$. \epi{DERIVED}
  \item \textbf{Positive ultraviolet correction} of order $(k\ell_P)^2/20$,
  suppressed some $69$ orders of magnitude at LIGO/Virgo frequencies (which is why
  it is invisible there), but structurally present. \epi{DERIVED}
  \item \textbf{P15 --- ultraviolet suppression of the primordial tensor
  spectrum} around $k\sim\ell_P^{-1}$, generated by the sub-Planckian
  regulation pole $M_\text{LW} = \sqrt{20}/\ell_P$ (\S{}III.3.4).
  The flexural suppression operates at the Planck scale ($k\ell_P\gtrsim\sqrt{20}$), \emph{not}
  in the cosmic-microwave-background window (where $k\ell_P\sim 10^{-31}$); what
  the CMB can test is the scalar/tensor asymmetry and the red$\to$blue transition
  (\S{}III.8.4, \S{}III.8.8), not the flexural suppression directly. \textbf{[structural HYPOTHESIS]}
\end{itemize}

\subsubsection{V.1.4 --- Block: gauge sector: the photon}

The photon is the excitation of the gauge sector $U(1)_\text{em}$ induced over
\emph{el Pleno} after spontaneous symmetry breaking (\S{}III.4). Five predictions,
developed in \S{}III.4.3--\S{}III.4.6:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{P-Photon-1 --- structural zero mass.} $m_\gamma = 0$ exact, by
  the Goldstone-Higgs theorem plus the $U(1)$ conservation of \emph{el Pleno} (Axiom
  A5). Current bound $m_\gamma < 10^{-54}$ kg, broadly compatible. Falsifiable:
  any $m_\gamma > 0$ with statistical confidence would refute A5.
  \epi{DERIVED}
  \item \textbf{P-Photon-2 --- velocity equal to that of the gravitational
  waves.} $v_\gamma = v_\text{GW} = c$, both derived from the same fundamental $c$
  of \emph{el Pleno}. Verified by GW170817. \epi{DERIVED}
  \item \textbf{P-Photon-3 --- Planckian UV dispersion correction.}
  $\omega^2 = c^2 k^2[1 + (k\ell_P)^2/20 + \mathcal{O}((k\ell_P)^4)]$, with
  coefficient $+1/20$ derived from the flexural phase term of F1 (the same
  diamond-lattice regulator that fixes the TT mode, \S{}II.3.8 P9b, \S{}III.4.3).
  The coefficient is identical to that of the TT mode: both are modes of the phase sector.
  Testable in very-high-energy astrophysical propagation; suppressed tens of
  orders below the current bounds. \epi{DERIVED}
  \item \textbf{P-Photon-4 --- quantization of the electric charge.} All the
  observable charges are integer combinations of $\pm 1/3$, a consequence of the
  $\mathbb{Z}_3$ modular arithmetic of the tripartite cluster (\S{}III.4.6 and
  \S{}III.5.2). Falsifiable: an irrational or transcendental charge would refute the
  topological quantization. \epi{DERIVED}
  \item \textbf{P-Photon-5 --- identity of the fundamental velocity.} The $c$ of the
  photon is the same $c$ of the TT mode, of F1 and of the identity $G =
  c^2/(\rho_P\ell_P^2)$. The uniqueness of $c$ in all of physics is an ontological
  consequence of a single substrate. \epi{DERIVED}
  \item \textbf{P-Photon-6 --- the Maxwell equations from the phase
  circulation.} The free electromagnetic field emerges with its antisymmetric tensor
  $F_{\mu\nu}=\partial_\mu a_\nu-\partial_\nu a_\mu$ and its energy
  $(8/9)F_{\mu\nu}F^{\mu\nu}$ from the oriented circulation of the phase over the
  plaquettes of the diamond lattice; the homogeneous ones are the automatic Bianchi identity
  and the inhomogeneous ones in vacuum are the Euler-Lagrange equation of F1
  (\S{}III.4.4). The non-abelian Yang-Mills sector, with its term
  $i[A_\mu,A_\nu]$, emerges from the same mechanism with the matrix phase
  $\theta_-\in SU(2)$ (\S{}III.4.5), so that Maxwell is its abelian limit.
  Falsifiable: an electromagnetic structure that did not reduce to antisymmetric
  phase circulation would refute the reading. \epi{DERIVED} (free field);
  \epi{SLD} (with source).
\end{itemize}

The numerical value of the fine-structure constant $\alpha\approx 1/137.036$
is \emph{not} derived: it remains an open thread linked to the fermionic
electroweak cascade (\S{}V.3).

\subsubsection{\texorpdfstring{V.1.5 --- Block: hadronic: the baryon as a cluster}{V.1.5 - Block: hadronic: the baryon as a cluster}}

The baryon is a tripartite Borromean cluster of \emph{el Pleno} (\S{}III.5). With a
\emph{single} observational scale of the sector ---the hadronic operational scale
$\Lambda_\text{PIU}$, calibrated to the mass of the proton--- the apparatus reproduces the
rest of the baryonic spectrum. This deserves emphasis: Quantum Chromodynamics
also requires fixing a scale ($\Lambda_\text{QCD}$) by calibration, so
that PIU-$\Psi$ reaches here \textbf{structural parity with QCD using a single
observational input}, deriving from it radius, splits and rotational tower:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Radius of the proton.} $R_p^\text{pred} = 0.851$ fm vs observed
  $0.841$ fm; agreement at $98.9\%$ (deviation $1.1\%$). \S{}III.5.5.
  \textbf{[DERIVED, 1 input]}
  \item \textbf{Mass split $\Delta$-$N$.} From the rotational quantization of the
  cluster (E31), $\Delta m = 9(\hbar c)^2/(2 m_p c^2 d^2)$ with $d\approx R_p$, one
  obtains $\Delta m = 264$ MeV vs observed $293.7$ MeV; ratio $0.90$. \S{}III.5.4.
  \textbf{[computational SLD]}
  \item \textbf{P-Hadron-1 --- complete baryonic rotational tower.}
  $m(J) = m_N + [J(J+1) - 3/4]\,\hbar^2/(2\Theta_p)$ with moment of inertia
  $\Theta_p = m_p R_p^2/3$. Verification: $N(1680)$ at $97.8\%$, $\Delta(1950)$ at
  $84\%$ (consistent with typical rotational-deformation corrections).
  Falsifiable: a systematic deviation greater than the expectable corrections
  would be evidence against the compact cluster. \epi{SLD}
  \item \textbf{P-Hadron-2 --- separation of vortices equal to the radius.} The
  effective separation between the fractional vortices of the cluster is
  $d\approx R_p$ (verified at $5\%$). Falsifiable: an independent measurement of the
  radius of the cluster significantly distinct from $R_p$ would refute the compact
  cluster. \epi{SLD}
\end{itemize}

The stability of the proton is a topological property of the cluster (\S{}III.5.7),
not a numerical value: the linking invariant blocks the decay channel.

\subsubsection{\texorpdfstring{V.1.6 --- Block: invariant hadronic scale}{V.1.6 - Block: invariant hadronic scale}}

The operational scale $\Lambda_\text{PIU}$ is an operational constant
frozen in the primordial bounce, not a readjustable parameter in the current
relaxed regime (\S{}III.5.8). From its invariance three falsifiable predictions
follow:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{P-Lambda-1.} $\Lambda_\text{PIU}$ does not vary with the cosmological
  redshift $z$. A variation detected at $z>0$ (via hadronic observables
  of distant objects) would refute it. \epi{DERIVED}
  \item \textbf{P-Lambda-2.} $\Lambda_\text{PIU}$ does not vary with galactic
  or environmental location. Detected spatial variations would refute it.
  \epi{DERIVED}
  \item \textbf{P-Lambda-3.} $\Lambda_\text{PIU}$ does not admit readjustment with the cosmological
  epoch post-freeze-out. Any observational need to readjust it
  with the epoch would refute its operational constancy. \epi{DERIVED}
\end{itemize}

\subsubsection{\texorpdfstring{V.1.7 --- Block: structural cosmological}{V.1.7 - Block: structural cosmological}}

The cosmological predictions of the program, with zero cosmological fitting parameters, derived in \S{}III.6--\S{}III.8. First, the two structural background ratios:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{P16 --- dark / baryonic matter ratio.} The identity (E37)
  $\Omega_{MO}/\Omega_M = \sqrt{3}\,\pi \approx 5.4414$, derived from the discrete
  geometry of the substrate (diamond lattice $Z=4$ plus spinorial penalty $1/2$). vs
  Planck 2018 $\Omega_{dm}/\Omega_b = 5.375\pm 0.077$: deviation $1.23\%$,
  distance $0.86\sigma$. \epi{DERIVED}
  \item \textbf{P17 --- dark-energy fraction.} The identity (E33)
  $\Omega_{EO}\equiv f_c = S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2 \approx
  0.6869$. vs Planck 2018 $\Omega_\Lambda = 0.6847\pm 0.0073$: deviation
  $0.32\%$, distance $0.30\sigma$. \epi{DERIVED} (structure) /
  \epi{CALIBRATED} (numerator $S_{\max,\text{eff}}^2$)
\end{itemize}

And, on that base, the quantitative cosmic background derived in \S{}III.8:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{P27 --- matter density.} $\Omega_m = 1 - f_c \approx 0.3131$,
  a direct consequence of the conservation $f_c + \Omega_\text{depleted} = 1$. vs
  Planck 2018 $\Omega_m = 0.315\pm 0.007$: distance $0.3\sigma$, without a fitting
  parameter. \epi{DERIVED}
  \item \textbf{P28 --- expansion history.} The form of $H(z)$, scale-invariant
  and forced by the potential of the medium, fits the five BAO measurements of
  DESI DR2 with $\chi^2 = 6.54$ against $6.53$ of $\Lambda$CDM ---statistically
  indistinguishable, with $\delta_0$ as the only observable freedom. \epi{DERIVED}
  \item \textbf{P29 --- primordial spectral index.} $n_s = 1$ at zeroth order, from the
  native matter bounce ($w = 0$ derived, $\nu = 3/2$ via the master formula
  $n_s - 1 = 3 - 2\nu$), \epi{DERIVED}. The red tilt arises from the FCC$\to$diamond
  packing transition at the bounce ---both pure components give $s_z=-1$, so that the tilt
  requires the crossing, not the regime--- over the persistent background
  $\Omega_{EO}=f_c$ with $w=-1$: the red sign and the smallness are \epi{SLD};
  the exact value $0.965$ is positional, \epi{CALIBRATED} (\S{}III.8.4).
  \item \textbf{P30 --- cosmic microwave background.} Perfect blackbody with
  structural chemical potential $\mu = 0$ (the photon is a mode of the medium without conserved
  number); $|\mu|\lesssim 9\times 10^{-5}$ (FIRAS) is a consequence, not a fit.
  \epi{DERIVED}
\end{itemize}

Under flatness $\sum_i \Omega_i = 1$, P16 and P17 structurally determine the
four observable cosmological fractions ($\Omega_{EO}$, $\Omega_M$,
$\Omega_{MO}$ and their ratio), and P27 closes them with the total matter density. The
complete numerical computation of the background ratios, with intermediate steps and
comparison in units of $\sigma$, is consolidated in \S{}V.2.4.



\subsubsection{V.1.8 --- Block: implicit predictions of the apparatus}

Five quantitative consequences of the apparatus, articulated explicitly
(\S{}II and \S{}III), with their individual epistemic classification:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{P18 --- closed form of the \emph{Pleno}-ground state.} $S_{\min}^\text{estr} =
  12(\pi^2-4)/\pi^4 \approx 0.7231$ (E9), spectral average over the
  Brillouin zone of the lattice. Exact convergence to ten digits with the dynamical minimum
  of the tripartite effective potential by an independent route. Falsifiable: a
  deviation $>0.01\%$ between the two routes would break the convergence.
  \epi{DERIVED}
  \item \textbf{P19 --- structural minimum mass of a black hole.}
  $M_{\min} = m_P\sqrt{3/(32\pi S_{\max,\text{geom}}^2)} \approx 0.117\,m_P
  \approx 2.55\times 10^{-9}$ kg, from solving $r_\text{core}(M) = r_S(M)$.
  Coincides with the Hawking evaporation limit by a distinct ontological
  reason (interior geometric exhaustion). \epi{DERIVED}
  \item \textbf{P20 --- maximum Hubble bound.} $H_{\max}\approx 7.9\times
  10^{43}$ s$^{-1}$, minimum horizon $R_{H,\min}\approx 0.23\,\ell_P$. Falsifiable:
  a derivable inflationary scenario that required $H>H_{\max}$ would refute it.
  \epi{DERIVED}
  \item \textbf{P21 --- tripartite mode spectrum.} Mass of the radial mode
  $m_\text{radial}c^2\approx 4.44\,E_P$ (ultra-Planckian, which explains the
  non-detection of \emph{el Pleno} as a particle), with Lee-Wick dispersion in the
  UV regime. Signatures of the pole testable with LISA / ET / CE. \epi{DERIVED}
  \item \textbf{P22 --- the event horizon as a sonic surface.} Since the
  velocity of the modes of the medium is $c$ and the escape velocity at the horizon
  equals $c$, the horizon is the surface of Mach number unity of \emph{el Pleno}
  (exact Unruh acoustic gravity, not analogical). Structural identity
  $v_\text{escape} = v_\text{sound} = c$ ---non-trivial: in a generic fluid
  both velocities differ---; its observational violation would be incompatible with
  \emph{el Pleno} as a medium. \epi{SLD}
  \item \textbf{P23 --- ultraviolet correction to the uncertainty (derived
  GUP).} The uncertainty relation of \emph{el Pleno} takes the closed form
  $\Delta x\cdot\Delta p = (\hbar/2)\sqrt{1+(3/80)(\ell_P/\sigma)^2}$
  (\S{}III.2.6, E-inc-3): a generalized uncertainty principle with
  \emph{derived} coefficient ($3/80$, a combination of the flexural rigidity $1/20$,
  the gaussian quartic factor $3$ and the normalization $1/4$), not parametrized
  phenomenologically as in the usual GUP models. Falsifiable in principle:
  any measured excess over $\hbar/2$ must follow the law $(3/80)(\ell_P/\sigma)^2$
  with a fixed coefficient. At atomic scales the excess is $\sim 10^{-50}$,
  unobservable ---consistent with no experiment having detected a deviation
  from Heisenberg. \epi{DERIVED} in its form, \textbf{[hypothesis]} in its
  confrontation.
  \item \textbf{P24 --- Chandrasekhar limit as an internal balance of \emph{el Pleno}.}
  $M_\text{Ch} = (\sqrt{3\pi}/2)(-\xi^2\theta')_{n=3}(\hbar c/G)^{3/2}/(\mu_e m_u)^2
  \approx 1.456\,M_\odot$ (for $\mu_e=2$), derived with its complete coefficient
  (\S{}III.2.8): the maximum mass that the flexural rigidity of the medium sustains
  against its own gravitational response, using only scales that the program already
  contained ($m_P$ as the gravity-uncertainty equalization scale, the Planck-hadronic
  hierarchy $(m_P/\mu_e m_u)^2$). Agreement with the canonical value
  ($\approx 1.44\,M_\odot$). \epi{DERIVED}
\end{itemize}

The quasi-perfect character of the thermal \emph{Pleno} ($\eta/s$ close to the quantum bound
of Kovtun-Son-Starinets, \S{}III.9.6) is not included as a prediction of the catalogue:
the bound $\eta/s\geq\hbar/4\pi k_B$ is respected by every fluid, so that its
fulfillment is not a specific assertion of PIU-$\Psi$, and the proximity to the
bound is a motivated conjecture ---not a structural identity--- that remains a
\epi{HYPOTHESIS} characterization in \S{}III.9.6.

\textbf{Self-verification of the gravity of compact objects (Part IV).} The dynamical synthesis of Part IV establishes a strong qualitative confirmation that articulates several of these predictions under a single mechanism: the same Gent potential and the A6 hard bound produce, without fitted parameters, the three observed destinies of the gravitational collapse ---diffuse collapse that forms structure, concentrated collapse that forms stable black holes \emph{without singularity} (with the minimum mass P19), and global cosmological crunch that rebounds (Big Bounce, with the Hubble bound P20)---, distinguished only by the concentration of mass-energy and the confinement (\S{}IV.7). That PIU-$\Psi$ reproduces the gravitational phenomenology of compact objects ---including the existence of stable black holes, a basic observational fact--- with the same mechanism that resolves the singularity of the Big Bang is self-consistency with physical content, not fitting. The Hubble bound (P20) and the dilatant deformation bound ($\dot\gamma_{\max}=c/\ell_P$, \S{}III.9) turn out to be of the same order, which links the bounce cosmology and the rheology of the medium under a single physical bound (\S{}IV.5).

The fourteen inherited empirical predictions P1--P14 (masses, radii, PPN tests,
Regge trajectories, galactic cores, absolute maximum density
$\rho_{\max,\text{abs}} = 2.1773\,\rho_P$, among others) preserve their falsifiable
character and are reproduced without additional parameters by parametric
convergence.

\subsubsection{V.1.9 --- Master table of the catalogue}

The table gathers the complete catalogue with its epistemic classification and the
PIU-$\Psi$ subsection where the derivation fulfills the five components of the
mathematical-presentation norm. The scheme of identifiers combines three
families: the \emph{inherited empirical} ones are grouped in the initial row P1--P14
(masses, radii, PPN tests and other quantities recovered in \S{}III.5--\S{}III.7);
the \emph{new structural} ones are numbered P15 onward; and the \emph{thematic
blocks} ---photon, hadrons and $\Lambda_\text{PIU}$--- use a nominal prefix
(P-Photon, P-Hadron, P-Lambda) to keep together the predictions of a single
sector. The constants $G$, $\hbar$ and the velocity $v_\text{GW}$ lead by being
structural identities of the apparatus.

{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{1.7cm}L{6.0cm}L{2.9cm}L{2.6cm}}
\toprule
\textbf{ID} & \textbf{Statement / value} & \textbf{Classification} & \textbf{Origin} \\
\midrule\endhead
P1--P14 & Inherited empirical (masses, PPN, Regge, cores\ldots) & \epi{DERIVED/}\newline\epi{CALIB.} & \S{}III.5--\S{}III.7 \\
$G$ & $G = c^2/(\rho_P\ell_P^2)$; residue $<0.1\%$ vs CODATA & \epi{DERIVED} & \S{}III.1.3 \\
$\hbar$ & $\hbar = \rho_P c\,\ell_P^4$; residue $<0.1\%$ & \epi{DERIVED} & \S{}III.2.2 \\
$v_\text{GW}$ & $v_\text{GW} = c$ exact IR; GW170817 & \epi{DERIVED} & \S{}III.3.3 \\
P15 & UV suppression of primordial tensor spectrum, $k\sim\ell_P^{-1}$ & \textbf{[struct. HYPOTHESIS]} & \S{}III.3.6 \\
P-Photon-1 & $m_\gamma = 0$ exact; bound $<10^{-54}$ kg & \epi{DERIVED} & \S{}III.4.3 \\
P-Photon-2 & $v_\gamma = v_\text{GW} = c$; GW170817 & \epi{DERIVED} & \S{}III.4.3 \\
P-Photon-3 & Photon UV dispersion correction, coef.\ $+1/20$ (= TT mode) & \epi{DERIVED} & \S{}III.4.3 \\
P-Photon-4 & Electric charges $=$ integers of $\pm 1/3$ & \epi{DERIVED} & \S{}III.4.6 \\
P-Photon-5 & Uniqueness of $c$ in all of physics & \epi{DERIVED} & \S{}III.4.3 \\
P-Photon-6 & Maxwell from phase circulation, $(8/9)F_{\mu\nu}F^{\mu\nu}$; non-abelian Yang-Mills & \epi{DERIVED}/\newline\epi{SLD} & \S{}III.4.4 \\
$R_p$ & $0.851$ fm vs $0.841$ fm; $1$ input & \textbf{[DERIVED, 1 input]} & \S{}III.5.5 \\
$\Delta m$ & $264$ MeV vs $293.7$ MeV; ratio $0.90$ & \textbf{[comput. SLD]} & \S{}III.5.4 \\
P-Hadron-1 & Rotational tower $m(J)$; $N(1680)$ $97.5\%$ & \epi{SLD} & \S{}III.5.6 \\
P-Hadron-2 & Separation of vortices $d\approx R_p$ ($5\%$) & \epi{SLD} & \S{}III.5.5 \\
P-Lambda-1 & $\Lambda_\text{PIU}$ does not vary with $z$ & \epi{DERIVED} & \S{}III.5.8 \\
P-Lambda-2 & $\Lambda_\text{PIU}$ does not vary with location & \epi{DERIVED} & \S{}III.5.8 \\
P-Lambda-3 & $\Lambda_\text{PIU}$ frozen post-freeze-out & \epi{DERIVED} & \S{}III.5.8 \\
P16 & $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$; $0.86\sigma$ & \epi{DERIVED} & \S{}III.6.3 \\
P17 & $\Omega_{EO} = f_c \approx 0.6869$; $0.30\sigma$ & \epi{DERIVED}/\newline\epi{calib.} & \S{}III.7.2 \\
P18 & $S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ & \epi{DERIVED} & \S{}III.7.6 \\
P19 & $M_{\min}\approx 0.117\,m_P$ & \epi{DERIVED} & \S{}II.6 \\
P20 & $H_{\max}\approx 7.9\times 10^{43}$ s$^{-1}$ & \epi{DERIVED} & \S{}II.5 \\
P21 & $m_\text{radial}c^2\approx 4.44\,E_P$; Lee-Wick UV & \epi{DERIVED} & \S{}II.5 \\
P22 & Horizon $=$ sonic surface; $v_\text{esc}=v_\text{son}=c$ & \epi{SLD} & \S{}III.9.6 \\
P23 & Derived GUP $\Delta x\Delta p=(\hbar/2)\sqrt{1+(3/80)(\ell_P/\sigma)^2}$ & \epi{DERIVED}/\newline\epi{hyp.} & \S{}III.2.6 \\
P24 & $M_\text{Ch}\approx 1.456\,M_\odot$ (complete coef., internal) & \epi{DERIVED} & \S{}III.2.8 \\
P25 & Confinement $=$ $\mathbb{Z}_3$ center obstruction; singlet rule ($\sum$ winding $\equiv 0\bmod 3$) & \epi{DERIVED} & \S{}III.5.3 \\
P26 & Core density of primordial BH $=S_{\max,\text{geom}}^2\rho_P\approx 2.1773\,\rho_P$, universal and non-singular & \epi{SLD} & \S{}IV.7 \\
P27 & $\Omega_m = 1-f_c \approx 0.3131$; $0.3\sigma$ vs Planck, without fitting & \epi{DERIVED} & \S{}III.8.5 \\
P28 & $H(z)$ of scale-invariant form; fits BAO DESI DR2 ($\chi^2=6.54$ vs $6.53$) & \epi{DERIVED} & \S{}III.8.6 \\
P29 & $n_s = 1$ at zeroth order; red tilt by packing transition, positional value & \epi{DERIVED}/\newline\epi{SLD}/\newline\epi{calib.} & \S{}III.8.4 \\
P30 & Perfect blackbody CMB: $\mu=0$ structural (photon without conserved number) & \epi{DERIVED} & \S{}III.8.2 \\
P31 & Intrinsic TT attenuation $\alpha=(1/2\sqrt{20})(\ell_P/c^2)\omega^2$; absorption length $\sim 10^{31}$ Hubble radii at LISA (GW without absorption except at Planck scale) & \epi{DERIVED} & \S{}III.3.3 \\
P32 & Hubble tension as a method artefact: the ``primordial'' $H_0$ ($\sim67$) is the one $\Lambda$CDM needs for the CMB, not a datum; analyzing a medium with $w=-1$ exact and elastic $H(z)$ under the $\Lambda$CDM template biases the inference. Falsifiable prediction: the tension dissolves in model-independent observables (raw $D_H/r_d$, $\theta_*$) & \epi{SLD} & \S{}III.8.7 \\
P33 & Exact structural equality Hubble bound/dilatant bound: $H_{\max}/\dot\gamma_{\max}=\sqrt{8\pi/3}\,S_{\max,\text{geom}}=4.2709$ (the cosmological bound and the rheological bound are the same rigidity of the medium) & \epi{DERIVED} & \S{}III.9.2 \\
P34 & Time dilation of SN Ia: exponent $b=1$ exact (shared redshift/time factor by nullity of the TT mode), where $\Lambda$CDM assumes it; DES-SN5YR $b=1.003\pm0.011$ (0.27$\sigma$), tired light excluded $\sim90\sigma$ & \epi{DERIVED} & \S{}III.8.2-bis \\
P35 & DESI $w_0$--$w_a$ signal as an artefact of the $\Lambda$CDM parametrization of a medium with $w=-1$ exact; falsifiable prediction of dissolution in model-independent observables & \epi{SLD} & \S{}III.6.6 \\
P36 & Redshift bound of the radial sector: $z_{\max}\approx0.268$ ($(S^*/S_{\min})^{2/3}$), which forces the two-clock structure (frozen radial / phase governs the present) & \epi{DERIVED} & \S{}III.8.2-bis \\
\bottomrule
\end{longtable}}

\begin{tcolorbox}[notabox,title={Reading of the catalogue under triple validation}]
The catalogue \emph{interprets} (each prediction names what the phenomenon is as a
configuration of \emph{el Pleno}), \emph{explains} (each one is derived from the apparatus
A1--A6 + F1) and \emph{predicts} (each one is confronted with a measured datum or
announces a future falsifiable datum). No entry invokes a single one of the three
capacities as sufficient.
\end{tcolorbox}

\medskip\hrule\medskip

% ====================================================================
\subsection{\texorpdfstring{\S{}V.2 --- Confrontation with current observation}{Sec.V.2 - Confrontation with current observation}}
% ====================================================================

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own
  \item \textbf{Notes:} quantitative confrontation with observation. Master table of successes with numerical computation of visible steps for the central magnitudes. Bayesian model-selection argument ($K_c\approx 6.71\times 10^5$). Central thesis: statistical equivalence with $\Lambda$CDM (not superiority), with derived instead of fitted parameters.
\end{itemize}
\end{estadoseccion}

\subsubsection{V.2.1 --- The thesis of the confrontation}

In the cosmological sector, PIU-$\Psi$ reaches \textbf{statistical equivalence with
$\Lambda$CDM using the same observational degrees of freedom}: it reproduces the
central observables with comparable precision. But the equivalence in the fit
hides a decisive difference in the status of the parameters. Where $\Lambda$CDM
treats $\Omega_\Lambda$ and $\Omega_{dm}/\Omega_b$ as free parameters fitted
to the observation, PIU-$\Psi$ \emph{derives} them from the structure of the substrate. For
that reason the confrontation of this section is not played out in the $\chi^2$ ---where both
frameworks tie--- but in something deeper: magnitudes that in the standard model
are calibrated inputs, in PIU-$\Psi$ are computed outputs, and those outputs
fall within the observational error bars. To match $\Lambda$CDM in
precision with derived instead of fitted parameters is, precisely, the
type of advance that \S{}I.2 signaled as pending.

\medskip
The numerical verifications that follow ---and those of the rest of the program--- are
\emph{reproducible}: this work is accompanied by a verification notebook (script
\texttt{verificar\_PIU.py}, supplementary material) that recomputes each constant and ratio
from its first-principles formulas, without preloaded values. Its execution reproduces
the central quantities of the program ---$G$, $\hbar$, $\gamma_\kappa$, $S_{\min}$,
$S_{\max,\text{geom}}$, $S^*$, $E_0$, $\eta_0$, $f_c$, $\Omega_m$, $\sqrt{3}\pi$, $N_{\text{tic}}$,
$\Theta_0$, the Chandrasekhar limit, the mass cutoff $M_{\min}$, the Tsirelson bound and the
$\Delta$-$N$ split, among others. The notebook is organised into three layers according to the
order of appearance of the first measured magnitude, and its verdict splits in two ---fidelity
between document and code, and agreement with measurement---, so that no row can exhibit as
confirmation what merely checks transcription. The complete organisation, with the anchor
criterion and the resolution floor, is set out in Annex \S{}V.9, and the results table is
distributed with the executable code.
The positional \epi{CALIBRATED} quantities are verified as \emph{consistency} of the
factorization, not as derivation of their value.

\subsubsection{\texorpdfstring{V.2.2 --- Numerical verification of $G$}{V.2.2 - Numerical verification of G}}

\textbf{Structural identity:} $G = c^2/(\rho_P\ell_P^2)$ (E3). With the
canonical values of the primary parameters of \emph{el Pleno}:

\begin{enumerate}[leftmargin=1.6em]
  \item $c = 2.998\times 10^{8}$ m/s $\Rightarrow$ $c^2 = 8.9880\times 10^{16}$
  m$^2$/s$^2$.
  \item $\ell_P = 1.616\times 10^{-35}$ m $\Rightarrow$ $\ell_P^2 =
  2.6115\times 10^{-70}$ m$^2$.
  \item $\rho_P = 5.155\times 10^{96}$ kg/m$^3$ $\Rightarrow$ $\rho_P\,\ell_P^2 =
  5.155\times 10^{96}\cdot 2.6115\times 10^{-70} = 1.3462\times 10^{27}$ kg/m.
  \item $G = c^2/(\rho_P\ell_P^2) = 8.9880\times 10^{16}/1.3462\times 10^{27} =
  6.6765\times 10^{-11}$ m$^3$kg$^{-1}$s$^{-2}$.
\end{enumerate}

Dimensional verification:
$$
\frac{[c^2]}{[\rho_P][\ell_P^2]} = \frac{\text{m}^2\,\text{s}^{-2}}
{(\text{kg}\,\text{m}^{-3})(\text{m}^2)} =
\frac{\text{m}^2\,\text{s}^{-2}}{\text{kg}\,\text{m}^{-1}} =
\frac{\text{m}^3}{\text{kg}\,\text{s}^2} \quad\checkmark
$$

Comparison: $G_\text{CODATA} = 6.6743\times 10^{-11}$ m$^3$kg$^{-1}$s$^{-2}$;
the relative difference is $|6.6765-6.6743|/6.6743 = 0.03\%$. The meaning of this
number is direct: the intensity of gravity ---one of the parameters
not derived from fundamental physics--- is reproduced to four significant
figures from three properties of the substrate, without any fitting to the
datum. The residue is from rounding of the primaries, not from the identity.
\epi{DERIVED}

\subsubsection{\texorpdfstring{V.2.3 --- Numerical verification of $\hbar$}{V.2.3 - Numerical verification of hbar}}

\textbf{Structural identity:} $\hbar = \rho_P c\,\ell_P^4$ (E36).

\begin{enumerate}[leftmargin=1.6em]
  \item $\ell_P^4 = (1.616\times 10^{-35})^4 = 6.8197\times 10^{-140}$ m$^4$.
  \item $\rho_P c = 5.155\times 10^{96}\cdot 2.998\times 10^8 = 1.5455\times
  10^{105}$ kg\,m$^{-2}$\,s$^{-1}$.
  \item $\hbar = \rho_P c\,\ell_P^4 = 1.5455\times 10^{105}\cdot 6.8197\times
  10^{-140} = 1.0540\times 10^{-34}\ \text{J\,s}.$
\end{enumerate}

Dimensional verification:
$$
[\rho_P][c][\ell_P^4] = (\text{kg}\,\text{m}^{-3})(\text{m}\,\text{s}^{-1})
(\text{m}^4) = \text{kg}\,\text{m}^2\,\text{s}^{-1} = \text{J\,s}
\quad\checkmark
$$

Comparison: $\hbar_\text{CODATA} = 1.0546\times 10^{-34}$ J\,s. The identity
$\hbar = \rho_P c\,\ell_P^4$ is exact by construction: it reproduces the quantum of
action from the same three primaries that fix $G$. The numerical value
shown is sensitive to the rounding of $\ell_P$ (which enters to the fourth power) and to
the joint constraint $\ell_P = \sqrt{\hbar G/c^3}$, which couples $(\rho_P,\ell_P)$;
with the primaries at full precision the identity closes to $<0.1\%$, like $G$. The
joint calibration of $(\rho_P,\ell_P)$ at full precision is a step of
numerical consolidation registered in \S{}I.7, not an uncertainty of the
derivation. \epi{DERIVED}


\subsubsection{\texorpdfstring{V.2.3-bis --- What is at stake: the falsification test of the coefficients}{V.2.3-bis - What is at stake: the falsification test of the coefficients}}

The objection these two verifications must withstand is concrete: if $(\rho_P,\ell_P)$ are fixed using $G$ and $\hbar$, reproducing them afterwards proves nothing. The content of the agreement depends on which route produced the primaries, and the algebra that follows separates them: there is a scenario in which reproducing $G$ and $\hbar$ says nothing, and another in which it risks one figure per branch.

The two preceding verifications reproduce the measured values, and the residue below $0.1\,\%$ is the rounding of the primaries published to four figures, not a discrepancy. But the risked content of the programme is not in that residue: it is in the \emph{coefficients} of the two identities.

The two identities are not obtained by looking for which combination of the primaries has the right dimensions. They are obtained \emph{from} F1: the static reduction of the modulus equation to Poisson form, its Green function and the comparison of the interaction energy with $-Gm_1m_2/r$ produce $\rho_P c^2\ell_P^2 = c^4/G$ (\S{}III.1.2--III.1.3); and the quantisation of the $U(1)$ rotor of the phase sector produces $\hbar=\rho_P c\,\ell_P^4$ (\S{}III.2.2). Dimensional analysis intervenes afterwards, as a safety net: it verifies that the result closes in units. It is not the source, and it would not have sufficed ---no accounting of dimensions produces a force law or identifies which of the possible combinations is the gravitational constant.

The point holds even granting the objector everything they might ask for. Suppose it granted in advance that the gravity of the medium depends on nothing beyond $\rho_P$, $\ell_P$ and $c$ ---which is already granting A1 and F1---: even then, dimensions do not select the identity. The monomial $c^2/(\rho_P\ell_P^2)$ is unique in powers of the three primaries, but the programme contains dimensionless invariants of its own ---$S_{\min}^\text{estr}=0.7231$, $8\sqrt6/9$, $\xi=19.69$, the flexural $1/40$--- and any of them multiplying that monomial equally has the dimensions of $G$. The dimensionally admissible combinations are therefore infinite, as \S{}I.7.5 already argues. What selects the prefactor is not the counting of units but the dynamics: the derivation returns unity, and not $\xi$, nor $S_{\min}^{2,\text{estr}}$, nor $1/40$. That is what the programme risks, and that is why the test is posed on the coefficients.

To set up the test, two auxiliary coefficients are introduced. They are not magnitudes of the programme and they do not appear in F1, which contains no free number in front of its terms: they are an instrument of contrast, a hypothetical number placed in front of each already derived identity in order to ask what measurement would reveal it. That they equal unity is not a starting hypothesis but the result of the derivations of \S{}III.1.3 and \S{}III.2.2, written so that it can be put to the test.

Writing the two identities with those general coefficients,
\[
G = \kappa_G\,\frac{c^2}{\rho_P\ell_P^2}\,,
\qquad
\hbar = \kappa_\hbar\,\rho_P c\,\ell_P^4\,,
\]
and solving the pair for the two primaries,
\begin{equation}
\ell_P = \sqrt{\frac{\hbar G}{\kappa_G\,\kappa_\hbar\,c^3}}\,,
\qquad
\rho_P = \frac{\kappa_G^2\,\kappa_\hbar\,c^5}{G^2\hbar}\,.
\tag{E63}
\end{equation}
From (E63) three confrontation scenarios can be read, in increasing order of demand.

\begin{itemize}[leftmargin=1.4em,itemsep=3pt]
\item \textbf{With both primaries solved from the pair: zero bits.} If $(\rho_P,\ell_P)$ are obtained by solving the two identities against the measured values of $G$ and $\hbar$, then \emph{any} choice of $(\kappa_G,\kappa_\hbar)$ reproduces them exactly: two equations with two unknowns. What changes with the coefficients is not the predicted constants but the primaries that are inferred. Measuring $G$ and $\hbar$ does not discriminate between values of $\kappa$, and the programme must not present that agreement as though it did.
\item \textbf{With one primary measured by an independent route: one bit.} If $\ell_P$ or $\rho_P$ is obtained from physics that passes through neither gravitation nor quantisation, the first of (E63) exposes the \emph{product} $\kappa_G\kappa_\hbar$. This is a real confrontation, and it would falsify any pair whose product is not one. It does not separate the two branches: the pair $(2,\tfrac12)$ has unit product and would pass the test.
\item \textbf{With both primaries measured: two bits, one per branch.} Knowing $\rho_P$ and $\ell_P$, the pair (E63) inverts and returns the coefficients separately,
\[
\kappa_G = \frac{\rho_P\,G\,\ell_P^2}{c^2}\,,
\qquad
\kappa_\hbar = \frac{\hbar}{\rho_P\,c\,\ell_P^4}\,,
\]
so that each branch is subjected to its own test. Closing the calibration open end (C-meta-1.b, \S{}V.3) therefore yields more than confirming the primaries: it turns one chain into two independent confrontations, and measuring both is worth more than twice measuring one.
\end{itemize}

\begin{tcolorbox}[derivado,title={The two branches are not symmetric}]
$G$ and $\hbar$ are not two chains of different origin: they are the two polar parts of the \emph{same} kinetic term of F1, $\tfrac12\rho_P c^2\ell_P^2$. The modulus branch leads to $G$ through the static limit, the Poisson equation, the Green function and the comparison with Newton (\S{}III.1.2--III.1.3); the phase branch leads to $\hbar$ through the zero mode, the $U(1)$ rotor and its quantisation (\S{}III.2.2). They diverge completely after the common node, and for that reason they can fail separately.

The exposure to risk is not equal either. The phase branch incorporates three ingredients that the modulus branch never touches: the value of the modulus in the \emph{Pleno}-ground state, the cell volume $\ell_P^3$ of the diamond lattice, and the quantum number $n=1$ of the rotor. Each is a point at which that branch can break without dragging the other with it. The programme asserts $\kappa_G=\kappa_\hbar=1$ in both, and the phase branch is the more exposed of the two.
\end{tcolorbox}

\subsubsection{\texorpdfstring{V.2.4 --- Numerical verification of the cosmological predictions}{V.2.4 - Numerical verification of the cosmological predictions}}

\textbf{P16 --- dark / baryonic matter ratio.}

\begin{enumerate}[leftmargin=1.6em]
  \item PIU-$\Psi$ prediction: $\Omega_{MO}/\Omega_M = \sqrt{3}\,\pi$.
  \item $\sqrt{3} = 1.7321$; $\sqrt{3}\,\pi = 1.7321\cdot 3.1416 = 5.4414$.
  \item Planck 2018 observation~\cite{planck2018}: $\Omega_{dm}/\Omega_b = 5.375\pm 0.077$.
  \item Relative difference: $|5.4414-5.375|/5.375 = 1.23\%$.
  \item Distance in $\sigma$: $|5.4414-5.375|/0.077 = 0.0664/0.077 = 0.86\sigma$.
\end{enumerate}

Zero free parameters, zero observational input. \epi{DERIVED}

\textbf{P17 --- dark-energy fraction.}

\begin{enumerate}[leftmargin=1.6em]
  \item PIU-$\Psi$ prediction: $\Omega_{EO} = f_c =
  S_{\max,\text{eff}}^2/S_{\max,\text{geom}}^2$.
  \item $S_{\max,\text{eff}}^2 = 1.4956$ (calibrated to the current regime);
  $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3} = 2.1773$ (geometric hard bound).
  \item $f_c = 1.4956/2.1773 = 0.6869$.
  \item Planck 2018 observation: $\Omega_\Lambda = 0.6847\pm 0.0073$.
  \item Relative difference: $|0.6869-0.6847|/0.6847 = 0.32\%$.
  \item Distance in $\sigma$: $|0.6869-0.6847|/0.0073 = 0.0022/0.0073 =
  0.30\sigma$.
\end{enumerate}

The structure of the ratio is derived; the numerator $S_{\max,\text{eff}}^2$ is
calibrated to the current cosmological regime with declared error.
\epi{DERIVED} (structure) / \epi{CALIBRATED} (numerator)

\textbf{P27 --- total matter density.}

\begin{enumerate}[leftmargin=1.6em]
  \item PIU-$\Psi$ prediction: $\Omega_m = 1 - f_c$ (conservation: condensate plus
  depleted fraction sum to one).
  \item $\Omega_m = 1 - 0.6869 = 0.3131$.
  \item Planck 2018 observation: $\Omega_m = 0.315\pm 0.007$.
  \item Relative difference: $|0.3131-0.315|/0.315 = 0.60\%$.
  \item Distance in $\sigma$: $|0.3131-0.315|/0.007 = 0.0019/0.007 = 0.27\sigma$.
\end{enumerate}

Without a cosmological fitting parameter: $\Omega_m$ follows from $f_c$ by conservation,
it is not fitted to the observed value. \epi{DERIVED}

\textbf{P28 --- expansion history $H(z)$.} The scale-invariant form of
$H(z)$, forced by the potential of the medium (\S{}III.8.6), is confronted with the
five model-independent BAO measurements of DESI DR2 ($z = 0.51, 0.71, 0.93, 1.32,
2.33$): the fit of the program gives $\chi^2 = 6.54$ against $6.53$ of
$\Lambda$CDM ($\chi^2/\text{dof} = 1.63$ in both), statistically
indistinguishable, with $\delta_0$ as the only observable freedom. The program is not
falsified by the most precise cosmological datum available. \epi{DERIVED}

\subsubsection{\texorpdfstring{V.2.5 --- Master table of observational successes}{V.2.5 - Master table of observational successes}}

{\footnotesize\setlength{\tabcolsep}{3pt}
\begin{longtable}{L{2.9cm}L{2.7cm}L{2.9cm}L{1.6cm}L{2.3cm}}
\toprule
\textbf{Magnitude} & \textbf{PIU-$\Psi$ prediction} & \textbf{Observation} & \textbf{Agreement} & \textbf{Classif.} \\
\midrule\endhead
Constant $G$ & $c^2/(\rho_P\ell_P^2)$ & $6.6743\times 10^{-11}$ (CODATA 2018) & $\kappa_G=1$ & \epi{DERIVED} \\
Constant $\hbar$ & $\rho_P c\,\ell_P^4$ & $1.0546\times 10^{-34}$ J\,s (SI) & $\kappa_\hbar=1$ & \epi{DERIVED} \\
$v_\text{GW}$ & $c$ exact IR & GW170817, $<7\times 10^{-16}$ & exact & \epi{DERIVED} \\
$m_\gamma$ & $0$ exact & $<10^{-54}$ kg & compatible & \epi{DERIVED} \\
Proton radius $R_p$ & $0.851$ fm & $0.841$ fm & $1.1\%$ & [deriv., 1 in] \\
Split $\Delta$-$N$ & $264$ MeV & $293.7$ MeV & ratio $0.90$ & \epi{SLD} \\
$N(1680)$ & $1722$ MeV & $1680$ MeV & ratio $1.025$ & \epi{SLD} \\
$\Omega_{MO}/\Omega_M$ & $\sqrt{3}\pi = 5.4414$ & $5.375\pm 0.077$ & $0.86\sigma$ & \epi{DERIVED} \\
$\Omega_{EO}$ & $f_c = 0.6869$ & $0.6847\pm 0.0073$ & $0.30\sigma$ & \epi{DERIVED} \\
$\Omega_m$ & $1-f_c = 0.3131$ & $0.315\pm 0.007$ & $0.27\sigma$ & \epi{DERIVED} \\
$H(z)$ vs BAO DESI DR2 & form forced by $V_\text{eff}$ & $\chi^2=6.54$ (5 pts) & vs $6.53$ $\Lambda$CDM & \epi{DERIVED} \\
$H_0$ background & $67.4$--$67.7$ (primordial side) & $67.4\pm0.5$ Planck & coincides & \epi{CALIB.} pos. \\
$w$ (late sector) & $-1$ exact ($T_{\mu\nu}\propto g_{\mu\nu}$) & compatible & structural & \epi{DERIVED} \\
SN Ia time dilation & $b=1$ (shared factor) & $1.003\pm0.011$ DES-SN5YR & $0.27\sigma$ & \epi{DERIVED} \\
Hubble tension & method artefact & dissolution in raw obs. & falsifiable & \epi{SLD} \\
Hubble/dilatant bound & $H_{\max}/\dot\gamma_{\max}$ & $4.2709$ exact & equality & \epi{DERIVED} \\
$n_s$ primordial & $1$ (order 0); red tilt & Planck compatible & structural & \epi{DERIV.}/\epi{SLD} \\
$S_{\min}^\text{estr}$ & $12(\pi^2-4)/\pi^4$ & 2 routes $<10^{-10}$ & exact & \epi{DERIVED} \\
Color confinement & $\mathbb{Z}_3$ singlet rule & $qqq$, $q\bar q$ free; $q$, $qq$ confined & qualitative exact & \epi{DERIVED} \\
\bottomrule
\end{longtable}}

\subsubsection{\texorpdfstring{V.2.6 --- The objection of numerological coincidence, put to the test}{V.2.6 - The objection of numerological coincidence, put to the test}}

A programme that obtains closed numbers from a Lagrangian must answer why it is not the next entry on a familiar list. The Titius-Bode law fitted the orbital radii of the solar system with a simple geometric progression and meant nothing. Eddington's large numbers related constants with remarkable precision, and likewise. The objection, in its strong form, is not that the programme's arithmetic is wrong: it is that a sufficiently large set of simple algebraic expressions ends up grazing any observed value, so that hitting the target would prove not structure but abundance of attempts. The programme posed that objection to itself as a live possibility and submitted it to computation.

\textbf{Two tests for two different questions.} The objection admits two readings that must be neither conflated nor combined. One is narrow: \emph{how many formulas} of simple algebra would land on a given cosmological ratio? That question is answered by enumerating the space of expressions, and is treated in \S{}III.6.5 for $\sqrt3\pi$ and $f_c$. The other is the one appropriate to a complete programme: \emph{how many frameworks} would reproduce the entire set of facts the programme reproduces? That is the one computed here. They are different questions, with results that are neither multiplied nor averaged: each bounds one thing.

\textbf{The argument, before the number.} One equation, three parameters, zero adjustment knobs, and results in fifteen domains whose instruments do not touch one another: torsion balances and orbital dynamics, spectroscopy and interferometry, quantum metrology, Bell tests in quantum optics, heights of the acoustic peaks of the background, the sound-horizon scale as a standard ruler, stellar astrophysics, neutrality of matter, radiometry, precision electrodynamics, precession and deflection and gravitational waves, fine structure, statistics of quantum measurement, collider physics, degenerate matter. No established theory produces that set from three parameters. This is the argument; what follows measures it.

\textbf{What is admitted and what is not, before counting successes.} A result enters the computation if it satisfies four criteria: that its outcome discriminate between the hypothesis of structure and that of coincidence; that the programme have no freedom to adjust it; that its target was established beforehand; and that it be read through a measurement channel of its own, not shared with another item already counted. Applied strictly, they admit fifteen results and leave twelve others outside the computation.

That a result falls outside this count \textbf{does not mean that it is incorrect, nor that the programme fails to obtain it, nor that its epistemic status is lowered}. All those excluded remain results of the programme, with their derivation and their classification intact in the sections where they are established; several are among the most solid it has. They fall outside because this computation measures one very concrete thing ---how much \emph{independent} evidence they contribute against the hypothesis of coincidence--- and a result can be derived, correct and valuable without contributing new information to that particular count: because it is already counted inside another, because its target does not yet exist, because any framework with an action produces it, or because it is not a central value against which to fail. Excluding them makes the calculation \textbf{more demanding, not weaker}: each exclusion lowers the resulting factor. A count that admitted them would give a larger and less defensible number.

The twelve, with their reason: $f_c$, because it contains a positional quantity anchored to data; $\Omega_m$, $\Omega_{MO}$ and $\Omega_M$, because they are obtained from $f_c$ and $\sqrt3\pi$, already counted, and share their channel; the identity $\ell_P^2=\hbar G/c^3$, because it is an algebraic consequence of two items already present; the Newtonian potential $1/r$, because it is the intermediate step from which the identity of $G$ follows; the flexural coefficient $1/40$, the GUP correction $3/80$ and the Lee-Wick pole, because they have no established experimental target ---they are pending predictions, and their place is the falsifiability record, not the record of successes---; the $\Delta$-N split, which the programme derives and reproduces to ninety per cent with the pending rotor correction declared as a thread (\S{}V.3), because counting a partial agreement as a binary success would turn the analysis into a fit; the chemical potential $\mu=0$ of the black body, because it is an upper bound and not a measurement with a central value; and conservation of momentum, $\mathbf F=m\mathbf a$ and the exponent $b=1$ of the supernova time dilation ---all three derived in the programme and in agreement with the data, the last at $0.27\sigma$---, because they are also produced by any framework with an action or with metric stretching, so that their probability under the hypothesis of coincidence is practically one and their contribution to \emph{this} count is null. Publishing this list is part of the result: it measures the discipline with which the successes were counted, and makes plain that the number was not obtained by piling up everything the programme derives.

\textbf{The two classes of evidence.} The fifteen admitted are not homogeneous. Eight are \textbf{predictions}: no established theory produces them, the class of frameworks that would succeed is narrow and their probability under coincidence is low ---$G$ $(0.30)$, $\hbar$ $(0.20)$, the floor $\hbar/2$ $(0.15)$, the Tsirelson bound $(0.10)$, $\sqrt3\pi$ $(0.10)$, the Chandrasekhar limit $(0.20)$, the neutrality of matter $(0.05)$ and the Stefan-Boltzmann coefficient by counting $(0.15)$. Seven are \textbf{reconstructions}: the target was already nailed down by established physics and the class of formalisms that reach it is broad, so that their probability is high ---Maxwell $(0.60)$, the Einstein equations $(0.35)$, the Dirac structure $(0.40)$, the Born rule $(0.50)$, Yang-Mills $(0.55)$, Fermi-Dirac statistics $(0.45)$ and the expansion history against BAO $(0.60)$. The reconstructions count, and for a precise reason: a framework can fail at each one of them, and it did not fail. That their target was known does not make them free; it makes them \emph{cheap to reproduce for many formalisms}, and that is exactly what a high probability records, not an exclusion.

\textbf{The computation and its correction.} The product of the fifteen probabilities is $\prod p_i = 8.4\times 10^{-10}$. Multiplying them without more would assume total independence, which the programme cannot invoke: all its results hang from the same trunk. The correction applied is $P(\text{all}\,|\,\neg H) = (\prod p_i)^{\lambda}$, with $\lambda\in(0,1]$ measuring how much independent evidence remains after discounting derivational overlap. That correction does not penalise the economy of the framework: it discounts the possibility of counting twice one and the same link appearing in several results. The economy is charged elsewhere ---in that each result comes out forced, with no coefficient to adjust, which is precisely what makes each $p_i$ small---: a framework with a different structure per domain could multiply without correction, but each of its successes would be far more probable, because its structure was designed for that domain.

\begin{center}\small
\begin{tabular}{L{2.2cm}L{4.4cm}C{2.1cm}C{2.1cm}C{2.1cm}}
\toprule
$\lambda$ & Reading & Predictions only & Reconstr. only & Whole set \\
\midrule
$1.00$ & total independence & $7.4\times10^{6}$ & $1.6\times10^{2}$ & $1.2\times10^{9}$ \\
$0.60$ & moderate dependence & $1.3\times10^{4}$ & $2.1\times10^{1}$ & $2.8\times10^{5}$ \\
$0.50$ & strong dependence & $2.7\times10^{3}$ & $1.3\times10^{1}$ & $3.4\times10^{4}$ \\
$0.35$ & very strong dependence & $2.5\times10^{2}$ & $5.9$ & $1.5\times10^{3}$ \\
$0.20$ & almost a single fact & $2.4\times10^{1}$ & $2.8$ & $6.5\times10^{1}$ \\
\bottomrule
\end{tabular}
\end{center}

The full range is reported rather than a single value. The programme adopts $\lambda=0.5$ ---strong dependence--- as the reference reading, whence
\begin{equation}
\text{BF} \;\approx\; 3.4\times 10^{4}\,,
\tag{E64}
\end{equation}
with the range $[6.5\times10^{1},\,1.2\times10^{9}]$ according to the dependence assumption. \epi{method}.

\textbf{Robustness.} The result does not rest on the choice of probabilities nor on any particular success. Tripling \emph{all} the $p_i$ ---that is, assuming each was estimated three times too favourably--- and further adopting $\lambda=0.35$, the factor remains above $10$. Under the most punitive assumption in the table, $\lambda=0.2$, which amounts to declaring the fifteen results almost a single fact, the factor is still $65$ against coincidence. And removing one item at a time, no individual omission lowers the factor by more than a factor of $4.5$: there is no single result on which the argument hangs.

\textbf{What the number moves.} A Bayes factor does not fix a probability: it measures how far the evidence displaces the starting position of whoever evaluates it. With $\lambda=0.5$, a neutral position goes to $0.99997$; a sceptical one of $10^{-2}$, to $0.9971$; a very sceptical one of $10^{-3}$, to $0.9718$. And a hostile position assigning one chance in a million at the outset is displaced by more than four orders of magnitude. The sceptic and the sympathiser update in the same direction and by the same amount; that is what makes the calculation usable.

\textbf{The limits of the calculation.} Four, and they are declared with the same precision as the result. First: the $p_i$ are informed judgements with their justification written out, not measured frequencies; the robustness analysis bounds how much it would matter had they been systematically too favourable. Second: those of the reconstruction class require estimating the breadth of the family of formalisms that reproduces each structure, and that breadth is not measurable ---it is estimated from familiarity with the literature---; it is the weakest link of the computation, and that is why those probabilities are set high. Third: the verbal scales of Jeffreys and of Kass-Raftery were calibrated to compare models by way of marginal likelihoods, and here there is no marginal likelihood; they are cited as an order-of-magnitude reference and the argument does not rest on their labels. Fourth: 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.

\textbf{What is established.} Under any reasonable choice of assumptions, hostile ones included, the reading of the programme's successes as numerological coincidence is disfavoured. The burden that transfers is concrete: whoever maintains it must account for fifteen results read with fifteen instruments that share nothing, obtained from one equation with three parameters and no fitted coefficient. For the programme, the calculation closes no question of physics; it establishes that the route is worth pursuing, which is exactly what was asked of it.

\subsubsection{\texorpdfstring{V.2.7 --- Equivalence with $\Lambda$CDM and distinctive advantage}{V.2.7 - Equivalence with LambdaCDM and distinctive advantage}}

PIU-$\Psi$ reproduces the central cosmological observables with the same
statistical precision as $\Lambda$CDM, using the same degrees of freedom.
On that base of parity, the program contributes three features that $\Lambda$CDM does not
possess and that constitute its own contribution:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Derived, not fitted, parameters.} $\Omega_{EO}$ and
  $\Omega_{MO}/\Omega_M$ are derived from the geometry of the substrate; in $\Lambda$CDM
  they are fitted to the datum. To obtain these two cosmological fractions at $0.30\sigma$ and
  $0.86\sigma$ with zero free parameters is a result that, as far as
  the program reaches, no previous framework had produced.
  \item \textbf{Unique and falsifiable predictions.} The suppression of the
  primordial tensor spectrum and the scalar/tensor asymmetry with red$\to$blue transition
  (P15, testable by CMB-S4/LiteBIRD via the asymmetry, not the flexural suppression
  ---which operates at the Planck scale, \S{}III.8.8), the baryonic rotational
  tower (P-Hadron-1), the separation of vortices $d\approx R_p$ (P-Hadron-2),
  and the signatures of the Lee-Wick pole in GW data (P21, testable by LISA/ET/CE).
  $\Lambda$CDM formulates none of them; they are the proper terrain of PIU-$\Psi$ and
  the points where the program exposes itself, deliberately, to refutation.
  \item \textbf{Ontological reduction.} A single substrate read in distinct
  configurations, instead of an inventory of independent components
  (matter, dark matter, dark energy) each with its own postulated equation of
  state. The ontological economy is not an aesthetic argument: it is what
  allows the fractions to be derived instead of postulated.
\end{itemize}

\begin{tcolorbox}[ecuacionclave,title={Key advantage reaffirmed}]
PIU-$\Psi$ matches $\Lambda$CDM statistically with the same degrees of
freedom, but its parameters have physical derivation from first
principles, and it generates unique falsifiable predictions (LISA, Fermi-LAT, CMB-S4)
that $\Lambda$CDM cannot make.
\end{tcolorbox}

\medskip\hrule\medskip

% ====================================================================
\subsection{\S{}V.3 --- Open threads and investigative program}
% ====================================================================

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} this section \emph{is} the master inventory of threads of the program
  \item \textbf{Notes:} master inventory of threads of the program, classified by category (conceptual, computational, native), severity and priority. Native threads: D1 (geometric uniqueness, \S{}II.3) and F-cosmo-PIU-$\Psi$ (\S{}V.4). The threads are named by their physical content.
\end{itemize}
\end{estadoseccion}

\subsubsection{V.3.1 --- The status of an open thread}

An open thread is a question that the program recognizes as legitimate and has not yet closed. To declare them is discipline, not weakness: a foundational program under construction without open threads would be suspicious. Each one is classified by severity (how much it compromises coherence if not closed) and priority (suggested order of attack), and they are grouped into four categories.


\textbf{A prior and more fundamental distinction: theory threads and confrontation threads.} Every thread belongs to one of two classes with distinct priorities. A \textbf{theory thread} is a gap in the chain that goes from the axioms to a result: an incomplete derivation, an unclosed coefficient, a step that rests on a structural argument instead of an algebraic proof. A \textbf{confrontation thread} is of another class: the result is already derived with rigor, and what remains is to submit it to third-party data ---catalogues that evolve, measurements of other groups, foreign parametrizations.


The priority is unequivocal: \textbf{first the theory is closed with rigor; then that theory is used to analyze the observations}. The confrontation is an \emph{application} of the theory, not a precondition of its closure, and it depends on moving data and foreign infrastructure, not on the internal rigor; for that reason a theory thread weighs more than a confrontation thread of equal nominal severity. This is coherent with the discipline of the program: when a derivation reproduces by its own route an equation or a number already validated, what is audited is the correctness of the mathematics of the path, not the truth of the result ---which science already established---; to arrive at it by another path does not question it, but offers another way of seeing the same fact. The cosmological sector illustrates it: the background is closed as theory (\S{}III.8), and what remains ---the combined analysis with third-party catalogues--- is a confrontation thread, subordinate to the theoretical closure.

\subsubsection{V.3.2 --- Conceptual threads}

Structural questions whose answer requires new conceptual apparatus, not
only computation.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Fermionic electroweak cascade.} The program derives the induced
  gauge sector and the charge quantization, but not the left/right chirality, the hypercharge, the mass of the electron (the only lepton that is a
  persistent constituent), nor the cancellation of anomalies. According to the
  persistent-constituent principle (\S{}III.5.1), the obligation is restricted
  to what enters the stable assembly of matter; the masses of the
  transient fragments (muon, tau, heavy quarks) and the why of the three
  generations are threads of structural interest, not of obligation. It is the largest
  conceptual thread of the program. High severity, high priority. (The
  fermion/boson dichotomy is closed via the linking statistics of the
  cluster.)
  \item \textbf{Numerical value of the fine-structure constant.}
  $\alpha\approx 1/137.036$ depends on the hypercharge and on the fermionic masses;
  it is not an independent thread, but a consequence that closes automatically
  upon closing the fermionic electroweak cascade. Medium severity, priority
  subordinate to that of the thread on which it depends.
  \item \textbf{Recovery of the Standard Model as an effective limit.} The
  legitimate and verifiable demand is one of \emph{compatibility}: no prediction
  of PIU-$\Psi$ should contradict the established phenomenology of the Standard Model
  in its regime of validity. This is fulfilled where the program pronounces (charges
  $\pm 1/3,\pm 2/3$ by $\mathbb{Z}_3$ arithmetic, fermion/boson dichotomy by
  linking statistics, massless photon). The \emph{complete re-derivation} of the
  gauge group $SU(3)\times SU(2)\times U(1)$ and of all the content of matter as
  effective structure of \emph{el Pleno} is the long-term horizon of the gauge sector
  ---not a precondition of the current status---, whose first concrete step is the
  fermionic electroweak cascade (above) and whose color sector already has advance
  via topological confinement (\S{}III.5). Program horizon, not a closable
  thread.
  \item \textbf{Ultraviolet completeness and unitarity.} The sub-Planckian
  flexural regulation provides a Lee-Wick pole whose unitarity is
  established as a native consequence of F1 (\S{}III.3.4): the Ostrogradsky phase
  space is bounded by A2 (configuration) and the hard deformation bound
  $\dot\gamma\leq c/\ell_P$ (momentum), and the regulation root has $\omega^2<0$
  ---overdamped, it does not oscillate, it does not resonate---, so that there is no secular
  escape. What remains open is the rigorous formalization to all orders of that
  unitarity; the thread is reduced from imported prescription to a technical gap of
  formalization. Low-medium severity, medium priority.
  \item \textbf{Dynamical coupling of the cycle to the cosmological contraction.} 
  Part IV formalizes the crunch-bounce cycle with the equation E57 integrated
  numerically and the dilatant viscosity assembled without free parameters. The
  crystallization plateau now has a foundation: the thermodynamic
  bottleneck of the latent heat of the packing transition
  $Z=12\to Z=4$ ---of order $\mathcal{O}(1)\,\rho_P c^2$ per site by the sole
  difference of coordination--- is what sustains its quasi-constant conditions
  (\S{}IV.5). What remains open is not the mechanism of the cycle ---already
  derived--- but the exact form of the external force $F_\text{ext}$ that couples
  the Friedmann contraction to E57 from first principles, the numerical \emph{value}
  of the latent heat (which requires the computation with the Gent potential of the
  program, not a trial potential), the absolute temporal duration of the plateau
  (\S{}IV.8), and the gravitational-dragging dynamics that links a primordial black
  hole with its co-natal host galaxy (\S{}IV.7), which must be derived from
  the emergent gravity (\S{}III.1), not postulated. Low severity, medium
  priority.
  \item \textbf{Ontological reading of time, mass and condensed matter.} Several
  lines of investigation internal to el Pleno ---time and mass as rhythms of
  self-adjustment of the medium (the coarse-grained entropy production rate of the
  relaxation to Born), and conduction, magnetism and the conducting/insulating character
  as pore geometry of the saturated lattice--- produce ontological readings
  internally consistent with what is observed, but remain interpretation, not
  derivation, and introduce no new observational prediction. The objects that would
  make them quantitative are identified: that the gravitational dilation as a
  ratio of rates reproduce $\sqrt{1+2\Phi/c^2}$ and coincide with the emergent gravity
  (\S{}III.1); that the tail of the self-adjustment distribution evaluated at the core of a
  soliton reproduce the observed Compton frequencies (tied to C-heal-1); and the
  stress in the throat between pores and the fraction of open pores against the percolation
  threshold, which would decide conductor versus insulator from F1. Low severity,
  subordinate priority; structural interest, not confrontational.
\end{itemize}

\subsubsection{V.3.3 --- Computational threads}

Questions whose conceptual apparatus is available and that require explicit
computation to close.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Cosmological prediction of the hadronic scale.} To derive
  $\Lambda_\text{PIU}$ from the cosmology of the primordial bounce (energy density
  of the \emph{Pleno}-ground state at the freeze-out temperature, dynamics of
  dimensional transmutation of the confining sector). If it closes, the hadronic
  sector would go from $1$ observational input to $0$. Medium severity,
  medium-high plausibility.
  \item \textbf{High states of the baryonic rotational tower.} The growing
  deviations for $J\geq 7/2$ (ratios $m_\text{PIU}/m_\text{obs}$ of $1.23$ and $1.48$
  for $J=7/2,9/2$) do not close with a centrifugal-distortion term of a single
  parameter without compromising the calibration of the $\Delta$-N split; they require the
  deformation corrections of the skyrmion profile, so that the thread is
  linked to the closure of coefficients of the solitonic profile (\S{}III.9.5).
  Low-medium severity.
  \item \textbf{Coefficient of the Yang-Mills Lagrangian and coupling of
  Maxwell with source.} The Maxwell equations of the free field are derived
  with their lattice coefficient $8/9$ derived from the sum of plaquettes (\S{}III.4.4), and the complete
  Yang-Mills structure emerges from the non-abelian phase circulation
  (\S{}III.4.5); what remains a computation is the coefficient of the
  Yang-Mills Lagrangian (non-abelian analogue of the $8/9$), the explicit
  coefficient of the circulation-defect coupling that gives Maxwell with source
  $\partial_\mu F^{\mu\nu}=J^\nu$, and the full classification of the lattice constant
  $8/9$ by geometric families. The coupling constant $g$ is linked to the
  embedding of hypercharge (conceptual thread of the gauge sector). Low-medium
  severity, medium plausibility.
  \item \textbf{C-heal-1 --- Physical size of the defect core.} The healing
  length $\ell_\text{heal}=\ell_P/\sqrt{\xi}\approx 0.2254\,\ell_P$ (\S{}II.5.3) is
  sub-lattice, which leaves open whether the physical size of the vortex core is fixed by
  the healing length of the continuum or the curvature regulation of A4 at scale
  $\ell_P$. It requires solving the complete radial-mode equation (E18) with the flexural
  term $(\ell_P^2/20)\Box^2\delta S$, not at tree-level. Low severity.
  \item \textbf{C-born-2 --- Nelson prefactor.} The diffusion coefficient that
  closes the relaxation to Born is derived in structure and scale, $D=\hbar/(2m)$ via
  the Einstein relation without circularity (\S{}III.2.5, E-born-4); the exact $\mathcal{O}(1)$
  prefactor remains, which fixes the mass of the diffusing quantum and is deferred to the
  thread of fermionic masses, not to quantum mechanics. Low severity.
  \item \textbf{Value of $\lambda_\text{SO}$ and the fermionic masses.} The structure of the
  Dirac equation is \epi{DERIVED} by the covariant route (\S{}V.7.2), independently of the
  value of the spin-orbit coupling; and $\lambda_\text{SO}\neq 0$ is forced by the $SU(2)$
  character of the phase mode. What remains is the computation of the \emph{values}
  ---$\lambda_\text{SO}$ and the masses $m$--- from F1, tied to the electroweak cascade
  thread (\S{}III.4.7). Medium severity, medium-high priority.
\end{itemize}

\begin{tcolorbox}[notabox,title={On the vacuum energy and the radial mode}]
The vacuum energy of the programme is $V_\text{eff}(S_{\min}^\text{estr})$, the structural
energy of the medium in dynamical equilibrium (\S{}II.2.1), and its cosmological reading is
the dark-energy sector at $0.30\sigma$ with no parameters. It is \emph{not} a sum
$\sum\tfrac12\hbar\omega$ over vacuum modes: that construction presupposes the trivial vacuum
that A1 denies, and it is the one that produces the $120$-order discrepancy in the
cosmological constant (\S{}I.2.2). The cost of exciting one quantum of the radial mode is
given by its mass, $\sqrt{\xi}\,E_P=4.44\,E_P$ (\S{}III.1.9); an anharmonic Planckian
correction to that number alters no observable and constitutes no gap in the derivational
chain.
\end{tcolorbox}

\begin{tcolorbox}[notabox,title={Note on the unification of the infrared threads}]
Two calibrated threads of the program ---the cosmological numerator
$S_{\max,\text{eff}}^2(t_0)$ (\S{}II.4.2, \S{}III.7.2) and the hadronic scale
$\Lambda_\text{PIU}$ (\S{}II.6.3)--- share an identical epistemic status:
\epi{CALIBRATED}, a real physical property with a value anchored to the
current infrared regime of the field $\Psi$. They are \emph{not} linkable to each other by algebra
---they belong to distinct regimes of the same field (cosmological versus
hadronic), and any formula that connected them directly would be
\emph{number-matching}, prohibited by the validation protocol---, but both
hang from dynamics of the same field not yet integrated. Their closures are
\textbf{separate but of the same class}:
\begin{itemize}[leftmargin=1.4em]
  \item $S_{\max,\text{eff}}^2(t_0)$ is closed (in its trajectory, not in its
  present value; see \S{}III.7.2) via F-cosmo, by integration of the \emph{effective
  Friedmann equation of the relaxed regime} (\S{}III.6.6, Step 1),
  \[
  \ddot{\bar S} + 3H\dot{\bar S} + \frac{1}{\rho_P\ell_P^2}V_\text{eff}'(\bar S) = 0 ,
  \]
  whose friction is of Hubble ($3H\dot{\bar S}$) and \emph{without} external forcing.
  \item $\Lambda_\text{PIU}$ is closed via the density of the \emph{Pleno}-ground state at
  the \emph{freeze-out} of the primordial bounce (previous computational thread).
\end{itemize}
\textbf{Important technical precision.} The closure of $S_{\max,\text{eff}}^2$
does \emph{not} pass through the equation of the primordial cycle (\S{}IV.5), which is a
distinct equation from the background ---with dilatant friction $\eta(\dot\gamma)$ and forcing
$F_\text{ext}$, valid in the saturated regime $S\to S_{\max}$---, but through the
effective Friedmann one of the \emph{relaxed} regime written above. They are two equations
of the background with distinct terms; confusing them overestimates the difficulty of the thread.
$S_{\max,\text{eff}}^2(t_0)$ is the positional reading of $\bar S$ over the relaxation curve ---where the universe was left, fixed by the initial condition of the bounce--- a datum of the same class as $H_0$ (\S{}III.7.2). The reorganization does not alter any closure: it specifies that the two infrared
threads are projections of the same mother thread (the \emph{law of the relaxation curve} $\dot{\bar S}=f(\bar S)$ that fixes
the IR conditions of the current regime), which produces a cleaner and more
attackable map of threads, without linking them by algebra.
\end{tcolorbox}

\subsubsection{\texorpdfstring{V.3.4 --- Native threads of PIU-$\Psi$}{V.3.4 - Native threads of PIU-Psi}}

Threads that the architecture of the manuscript itself introduces or recognizes
explicitly.

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{D1 --- Geometry of the support: derived.} The geometry of the diamond
  lattice is not an open thread: it is a derived result in both of its pieces, and is
  included here only to record its status. The tetrahedral coordination $Z=4$ is a
  necessity of A1 (exact second-order isotropy) together with Maxwell isostaticity, with
  uniqueness demonstrated algebraically ---four unit vectors summing to zero and isotropic
  force $\cos\theta=-1/3$, the regular tetrahedron, a spherical 2-design unique up to
  rotation (D1-a, \S{}II.3.7, \epi{DERIVED}). The cubic stacking ($Fd\bar3m$) as against
  wurtzite and the remaining tetrahedral polytypes is the minimum of the isotropic flexural
  functional of F1: the fourth-order uniaxial anisotropy of the modulation is $0$ for the
  cubic case and $4/81$ (exact, parameter-free, anchored to the same invariant $1/40$ that
  fixes $\gamma_\kappa$) for the hexagonal one, so that the continuous medium relaxes to the
  only modulation that does not concentrate curvature (D1-b, \S{}II.3.7-bis, \epi{DERIVED}).
  The diamond lattice is the configuration of saturated \emph{el Pleno} with no residue of
  choice.
  \item \textbf{Non-linear correction of $H(z)$ and structure growth.} The
  background cosmological sector is closed (\S{}III.8): the form of $H(z)$, the
  matter density $\Omega_m = 1 - f_c$, the character of the cosmic microwave
  background, the nearly scale-invariant spectrum and the exact equation of state
  $w=-1$ of the late sector are derived, and the expansion history fits the
  model-independent BAO data as well as the standard model with less freedom. What
  remains open is the \emph{non-linear correction} of $H(z)$ at accessible $z$ ---and
  whether it is measurable with raw BAO $D_H/r_d$, in which case the programme would
  predict a residue with a definite sign---. The dynamical object that produces it is the
  \textbf{phase sector}, not the oscillation of the radial modulus: the background modulus
  is frozen at $S_{\min}^\text{estr}$ in the late epoch (\S{}III.6.7). Also open is the
  quantitative prediction of the fluctuation amplitude $S_8$ from that drag
  (\S{}III.7.5-bis): the local sign of the force is derived and its localisation at
  $z\to 0$ is \epi{SLD}, and the coupling is of pure momentum transfer, a class for which
  the net effect on the growth rate is one of \emph{suppression} ---the direction of the
  observed amplitude---; the net effect is not read off the isolated local force, and its
  magnitude requires the full coupled dynamics. The DESI $w_0$--$w_a$ signal and the Hubble
  tension are predicted as artefacts that dissolve in model-independent observables
  (\S{}III.8.7), not as quantities to be reproduced. It is a \textbf{confrontation
  thread}, not a theory one: the background result is already derived, and what remains is
  the computation of the non-linear correction and its contrast with third-party
  catalogues. For that reason its priority is subordinate to the closure of the theory
  threads, according to the hierarchy of \S{}V.3.1. It is developed in \S{}V.4. Low
  severity, subordinate priority.
\end{itemize}

\begin{tcolorbox}[derivado,title={The two roles of the local damping in the cycle}]
The damping factor of the radial mode, $\xi_\text{loc}(S)/20$ with
$\xi_\text{loc}(S)=V_\text{eff}''(S)/(\rho_P c^2)$, is not fixed: it equals $0.9845$
---quasi-critical, underdamped--- in the \emph{Pleno}-ground state of today
($S=S_{\min}^\text{estr}=0.7231$) and rises sharply as the medium is compressed towards
$S_{\max}$, crossing into overdamping at $S\gtrsim 0.77$ and reaching $\sim 10^4$ near the
hard bound. That profile is what \textbf{guarantees that the bounce is clean}: on being
compressed, the medium ceases to be a spring and becomes a wall, so that the radial mode
does not resonate near $S_{\max}$ but brakes without multiple rebounds.

The \textbf{temporal asymmetry} slow-crunch/fast-bounce of Part IV has a different origin,
and the two must not be conflated. Since $\xi_\text{loc}$ is a function of $S$ alone, the
local damping is the same going up as coming down through a given value of the modulus: it
is symmetric under $t\to-t$ and cannot, by itself, produce an asymmetry of times. The
asymmetry comes from the \textbf{contrast of scales between the engines}: the crunch is
forced by cosmological contraction, at the Hubble scale, whereas the bounce is free elastic
release at the Planck scale ($\omega_S=\sqrt{\xi}/\tau_P$, relaxing to $S_{\min}$ in
$\sim 14\,\tau_P$ from $S=1.3$). The ratio of those two scales is the origin of the
asymmetry, in agreement with the reading of Part IV (Gent wall plus dilatancy, \S{}IV.5).
\epi{DERIVED} in both roles: the cleanliness of the bounce from the profile
$\xi_\text{loc}(S)$, and the asymmetry from the contrast of engines. What remains ---the form
of $F_\text{ext}$ from first principles--- is the thread of the coupling of the cycle to
Friedmann, already inventoried in \S{}V.3.2.
\end{tcolorbox}

\begin{tcolorbox}[derivado,title={The two masses of the Higgs mode of el Pleno}]
The complete resolution of the characteristic equation of the modulus sector at $k=0$,
$W^2+20W-20\xi=0$ with $W\equiv(\omega\ell_P/c)^2$ and $\xi=19.69$, has two roots:
$W_+=12.22$, the physical propagating branch, and $W_-=-32.22$, the Lee-Wick regulation pole
with $\omega^2<0$ (overdamped, non-oscillating). Hence the radial mode has \textbf{two masses
and not one}: the elastic mass $\sqrt{\xi}\,E_P=4.44\,E_P$, which governs static rigidity
---bulk modulus, Poisson ratio, healing length, Gross-Pitaevskii limit---, and the
propagating mass $\sqrt{W_+}\,E_P=3.50\,E_P$, the gap of the regulated physical branch, with
ratio $0.788$. Both ultra-Planckian, both \epi{DERIVED} in \S{}III.1.9; the distinction is
ontological and alters no observable.
\end{tcolorbox}

\medskip
\textbf{The Chandrasekhar coefficient, derived.} The \emph{exact numerical
factor} of the Chandrasekhar limit from the structure of \emph{el Pleno} ---mode 4
of \S{}III.2.7, the support-collapse unification--- is \textbf{derived} in
\S{}III.2.8: the complete coefficient is obtained
($M_\text{Ch}\approx 1.456\,M_\odot$) using only
scales that the program contains, without external importation. There remains as a
minor thread only the fine quantitative matching between the regime of stellar
degeneracy and the regime of bounce.

\subsubsection{V.3.5 --- Table of the master inventory}

{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{5.7cm}L{2.2cm}L{1.9cm}L{1.9cm}}
\toprule
\textbf{Thread} & \textbf{Category} & \textbf{Severity} & \textbf{Priority} \\
\midrule\endhead
Fermionic electroweak cascade & conceptual & high & high \\
Value of $\lambda_\text{SO}$ and fermionic masses (Dirac structure \epi{DERIVED}, \S{}V.7.2) & computational & medium & medium-high \\
Numerical value of $\alpha$ (subordinate to the cascade) & conceptual & medium & subordinate \\
UV completeness / unitarity & conceptual & medium & medium \\
Coupling $F_\text{ext}$ of the cycle to Friedmann & conceptual & low & medium \\
Hadronic scale from the bounce (C-Cosmo-$\Lambda$) & computational & medium & medium-high \\
High states of the rotational tower & computational & low-medium & low \\
Coefficient of the Yang-Mills Lagrangian and Maxwell with source (\S{}III.4.4--III.4.5) & computational & low-medium & medium \\
C-heal-1 --- physical size of the defect core (\S{}II.5.3) & computational & low & low \\
C-born-2 --- Nelson prefactor (not independent: it is the fermionic-mass gap seen from Born diffusion, \S{}III.2.5) & computational & low & subordinate \\
Ontological reading of time, mass and condensed matter (structural interest, no new prediction) & conceptual & low & subordinate \\
C-meta-1.b --- calibration of $\rho_P$ or $\ell_P$ from non-gravitational/quantum physics; route of closure traced (saturation density $\rho_{\max}\approx 2.18\,\rho_P$ of a compact object, \S{}V.7.2) & native & low & subordinate \\
Non-linear correction to $H(z)$ and $S_8$ from the phase drag & confrontation & low & subordinate \\
\bottomrule
\end{longtable}}

The complete recovery of the Standard Model does not figure as a thread: the
\emph{compatibility} with it as an effective limit is a verifiable demand already
satisfied where the program pronounces, and its \emph{complete re-derivation} is a
program horizon whose first concrete step is the fermionic electroweak
cascade, already listed.

In addition to these, the program maintains of the order of fifteen programs of
refinement of lesser severity (precision improvements, independent
verifications, alternative formalizations) that do not compromise coherence and
are attacked opportunistically. Among them, the non-perturbative verification of the
spectrum and of the solitonic configurations by lattice Monte Carlo with the
discretized F1 action: it is not a structural gap but a desirable independent
confirmation of already-derived results.

\medskip\hrule\medskip

% ====================================================================
\subsection{\texorpdfstring{\S{}V.4 --- Quantitative cosmology: the closed background and the combined thread}{Sec.V.4 - Quantitative cosmology: the closed background and the combined thread}}
% ====================================================================

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic and numerical (\S{}III.8)
  \item \textbf{Open threads:} combined analysis BAO+CMB+SNe (F-cosmo, confrontation class) and prediction of $S_8$, high priority
  \item \textbf{Notes:} articulates the status of the quantitative cosmological sector. The background sector is closed and is developed in \S{}III.8; this section synthesizes that closure from the point of view of the confrontation and delimits with precision what remains open, preserving the discipline of not publishing a closure without the apparatus that supports it.
\end{itemize}
\end{estadoseccion}

\subsubsection{\texorpdfstring{V.4.1 --- What is the cosmic history under PIU-$\Psi$}{V.4.1 - What is the cosmic history under PIU-Psi}}

Under PIU-$\Psi$ there is no space that expands with things inside. There is a single
substrate whose state evolves. What standard cosmology reads as
``expansion of the universe'' is, in the reading of \emph{el Pleno}, the dispersion of the
solitonic configurations (the matter) over an invariant structural background
(the \emph{Pleno}-ground state, which is read phenomenologically as dark energy). The matter
is the perturbation over that background: $\rho_\text{matter} = \rho_P[S^2 -
S_{\min}^{2,\text{estr}}]$ (E32-bis). The background is not an additional substance
draggable by gravity, but a permanent structural state of the substrate. The
redshift and the temporal dilation observed are manifestations of the
evolution of the state of the medium, not of the stretching of a stage; the metric
effective emergent reproduces the FLRW phenomenology with this ontology (\S{}III.8.7).

\subsubsection{V.4.2 --- The background sector, closed}

The background sector of the cosmology of \emph{el Pleno} is closed, and its complete
quantitative development is found in \S{}III.8. Its results are:

\begin{itemize}[leftmargin=1.4em]
  \item The two static cosmological observables, with derived status and zero
  free parameters: the ratio $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ (P16,
  \S{}III.6.3, $0.86\sigma$) and the dark-energy fraction $\Omega_{EO} = f_c
  \approx 0.6869$ (P17, \S{}III.7.2, $0.30\sigma$).
  \item The matter density $\Omega_m = 1 - f_c \approx 0.3131$, derived from the
  condensate fraction without fitting, at $0.3\sigma$ of Planck (\S{}III.8.5).
  \item The functional form of $H(z)$, forced by the derived effective potential
  with a single observable freedom (the initial condition $\delta_0 \leftrightarrow
  w_0$), scale-invariant by analytical demonstration, and that fits the data
  BAO of DESI DR2 as well as the standard model with less theory freedom
  ($\chi^2 = 6.54$ against $6.53$, \S{}III.8.6).
  \item The perfect-blackbody character of the cosmic microwave background
  ($\mu = 0$ structural) and the nearly scale-invariant spectrum without inflation
  (native matter bounce, with $w = 0$ derived), with the native solution of the
  BKL problem by the dilatant viscosity (\S{}III.8.2--\S{}III.8.4, \S{}III.8.8).
  \item The asymptotic equation of state $w = -1$ in the relaxed regime, with
  $w > -1$ and $w_a < 0$ in the regime of active re-condensation, in sign
  consistency with DESI (\S{}III.7.5, \S{}III.8.7).
\end{itemize}

\subsubsection{V.4.3 --- What remains open: the combined analysis}

What the program does not close, and declares open with precision, is the \textbf{combined analysis BAO+CMB+SNe} (confrontation thread F-cosmo) and the quantitative prediction of $S_8$. The closed confrontation of the background is with the model-independent radial observable ($D_H/r_d$), the raw data layer; the DESI preference for a dark-energy dynamics ($w_0 > -1$, with apparent crossing of $w = -1$) comes from the combined analysis, which introduces infrastructure foreign to the program ($r_d$ inferred from the CMB, supernova sample, covariances, rigid CPL parametrization). To reproduce that preference with the initial condition $\delta_0$ of the program and to predict $S_8$ from the perturbation dynamics is the content of the thread.


What is pending is one of \emph{execution and confrontation}, not of foundation: all the conceptual pieces ---$V_\text{eff}$, induced action, \emph{Pleno}-ground state, re-condensation dynamics--- are derived. Since the program has $w \geq -1$ always (canonical kinetics), an eventual crossing $w < -1$ in the combined analysis must be read, before declaring it a tension, as a possible artifact of translating an emergent medium into the language of a fluid over Friedmann with CPL parametrization (\S{}III.8.9).

\subsubsection{V.4.4 --- The discipline of confrontation with the raw datum}

\begin{tcolorbox}[problemabierto,title={Explicit methodological discipline}]
The confrontation thread F-cosmo is deferred with criterion, and the omission is deliberate. The confrontation
of the program is made with the model-independent raw datum (the observable layer), never
with its model-dependent compression (the layer that assumes exact Friedmann, separation
of matter/dark-energy fluid and rigid CPL parametrization). To force the expansion
history of \emph{el Pleno} into a foreign parametrization, before having closed the apparatus of
perturbations that sustains it, would be to precipitate a result, not to demonstrate it. The rule
is firm and is a strength, not a limitation: if in the middle of a derivation the
need for a new mechanism appeared, the derivation stops, the mechanism is evaluated
separately under the filter of the axioms, and only afterward is it resumed. The background sector is
closed precisely because it respects this discipline; the combined analysis is deferred for the same
reason ---it depends on a moving datum (DESI DR2$\to$DR3, supernova sample, tension
BAO-CMB) and on infrastructure that is not of the program.
\end{tcolorbox}

\subsubsection{V.4.5 --- The falsifiability of the sector}

The cosmological sector is falsifiable in its closed part and in its open part. In the
closed part: the form of $H(z)$ is forced by the derived potential, so that
an expansion history incompatible with the model-independent BAO data would falsify
the program; and $\Omega_m = 1 - f_c$ is a prediction without fitting, so that a
measurement of the matter density incompatible with $0.3131$ by a wide margin
would strain the reading of the condensate. In the open part: the form of $w(z)$ derivable
from the equation of motion \textbf{must} coincide with the observation when the combined analysis
is executed; \textbf{if future measurements of DESI or Pantheon+ show a
$w(z)$ incompatible with the dynamics of the modulus of \emph{el Pleno} (which demands $w \geq -1$ and
convergence to $-1$ in the future), PIU-$\Psi$ would be falsified.} The cosmological sector
is, therefore, simultaneously one of the most solid of the program in its background and one of
the largest points of falsifiability in its fine dynamics. \textbf{[background derived; combined: open thread, high priority]}

\medskip\hrule\medskip

\subsection{\S{}V.5 --- Final synthesis and consolidated bibliography}
% ====================================================================

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own
  \item \textbf{Notes:} closure of the manuscript as a mirror of the Overview. Bibliography consolidated in two blocks: numbered academic references (thebibliography) and web references in a dedicated subsection. According to Rule 9 of the LaTeX protocol: without \texttt{\textbackslash section*\{References\}} before \texttt{thebibliography}; entry in the TOC via \texttt{addcontentsline}.
\end{itemize}
\end{estadoseccion}

\subsubsection{V.5.1 --- The program under triple validation}

The program set out to interpret, explain and predict simultaneously. At the
closure of the manuscript, the three capacities are articulated. The value of each
result is not measured in the abstract, but against the magnitude of the open problem
it attacks: the successes of PIU-$\Psi$ weigh exactly what the questions
that \S{}I.2 documented as unresolved weigh.

\textbf{Interpretation.} PIU-$\Psi$ offers a unique ontological reading: a single
continuous substrate of which gravity, the constants, the particles, the
electromagnetism, the cosmology and the dark sectors are configurations,
modes or states. There are no separate ontological categories; there is one \emph{el Pleno} and
distinct questions asked of it. This responds directly to the common root
that \S{}I.2.8 identified in the six problems: the difficulty of modern
physics in saying \emph{what} that which it studies is, beyond \emph{how it
behaves}. Where the standard framework postulates the vacuum as an unclarified entity
(problem 6), dark categories as operational labels for the $95\%$ of the
cosmic content (problem 2), and a space-time whose nature GR and QM
describe in incompatible ways (problem 4), PIU-$\Psi$ offers a single
ontological referent for the three.

\textbf{Explanation.} Each phenomenon is derived from a more basic apparatus: the value
of $G$ and $\hbar$ from the structural parameters; the Heisenberg uncertainty relation
from the flexural rigidity of the medium; the Chandrasekhar limit as an
internal balance between that rigidity and the gravitational response of \emph{el Pleno} itself;
Newtonian mechanics ---universal gravitation and the second law--- as the limit of
weak field and low velocity of the medium; the Einstein equations from
the induced action; the Maxwell equations of electromagnetism ---and the
non-abelian Yang-Mills structure--- from the circulation of the phase over the lattice;
the charge quantization from the modular arithmetic of the
cluster; the confinement from the $\mathbb{Z}_3$ center obstruction; the baryonic split from the rotational quantization; the cosmological
fractions from the geometry of the substrate. The explanatory chain is reconstructible
from the axioms A1--A6 and the Lagrangian F1. Against problem 5 ---the Standard
Model contains of the order of twenty non-derived constants, and GR and QM assume
$G$ and $\hbar$ as free parameters--- PIU-$\Psi$ exhibits $G$ and $\hbar$ as
structural identities of three primary inputs ---of which only two,
$\rho_P$ and $\ell_P$, are substantive properties, since $c$ is the causal structure
of the medium and not a free property (\S{}I.8.9)---, reducing the number of
unexplained inputs instead of increasing it.

\textbf{Prediction.} The catalogue of \S{}V.1 produces falsifiable quantitative
predictions, confronted in \S{}V.2: $G$ and $\hbar$ (unit coefficient, no fitting), $v_\text{GW}=c$
(GW170817), $R_p$ ($1.1\%$), $\Delta$-$N$ split (ratio $0.90$),
$\Omega_{MO}/\Omega_M$ ($0.86\sigma$), $\Omega_{EO}$ ($0.30\sigma$), $\Omega_m = 1-f_c$ ($0.27\sigma$) and the form of $H(z)$ that fits BAO DESI ($\chi^2=6.54$ vs $6.53$), the
Chandrasekhar limit P24 ($1.456\,M_\odot$, complete internal coefficient), plus the
forward falsifiable predictions P15, P-Hadron-1, P-Hadron-2, P-Lambda-1..3,
P21, the ultraviolet correction to the uncertainty P23 (derived GUP with
fixed coefficient $3/80$, not adjustable), and the structural-consistency one P22 (the horizon as a sonic surface
of \emph{el Pleno}, $v_\text{escape}=v_\text{sound}=c$). These predictions
are the response of the program to the Popperian demand of \S{}I.1.2: not
retrospective concordance, but declared empirical risk, with several points
(tensor UV suppression, Lee-Wick pole, cosmological $w(z)$) testable by
instruments in operation or under construction.

\textbf{A unifying reading.} The characterization of \emph{el Pleno} as a
Herschel-Bulkley dilatant fluid (\S{}III.9) closes circles of confirmation
between distinct domains: the Lorentz factor is
read as the dilatant resistance of the medium to deforming near its limiting
velocity, the light barrier as the jamming point ($n\to\infty$), the
event horizon as a sonic surface, and the hard density bound (A6)
as the rheological divergence of the medium. Relativity, the physics of black
holes and the structural bound are three manifestations of the same
rheology ---without introducing any new parameter and consistent with everything
derived.

\subsubsection{V.5.2 --- The consolidated, the falsifiable and the open}
\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Consolidated.} The substrate and its Lagrangian F1, with the genesis of the medium from the root axiom of finite and conserved energy; the structural identities of $G$ and $\hbar$ (derived from the properties of the medium, not postulated, with unit coefficient and no fitting); the emergence of gravity ---Newton in the weak field and the Einstein-Hilbert structure by induced action--- and of quantum mechanics as effective limits of the same medium, with the dissolution of the boundary between the two; the gauge sector: the photon and charge quantization, the Maxwell equations from phase circulation (derived coefficient $8/9$) and the non-abelian Yang-Mills structure, with confinement as $\mathbb{Z}_3$ center obstruction; the Heisenberg uncertainty relation as a structural property (exact floor $\hbar/2$) and the Chandrasekhar limit with its complete coefficient ($1.456\,M_\odot$); the thermodynamics of the medium obtained by counting configurations of the eigenmodes of F1 ---the equilibrium occupation and the coefficient $2\pi^2/45$ of the radiation entropy derived, not imported, and the dissipation bound read as the no-vacuum axiom in the language of transport---; the configuration space of $\Psi$ per cell as an annulus with the centre forbidden, and the coefficient of the constitutive potential in closed form in $\pi$ and $\sqrt6$; the baryon as a solitonic cluster with its radius and its rotational tower; the reading of Special Relativity as the temporal-conservative face of the rheology of a dilatant fluid, with Stokes' law as its spatial-dissipative face derived from F1 and the shear viscosity $\eta_\text{TT}$ of the gravitational-wave sector (via convergent Maxwell-Kubo and Green-Kubo); the crunch-bounce cycle and the formation of the three real particles, with the primordial black holes as incomplete crystallization of universal core density ($2.1773\,\rho_P$); and the cosmological sector: the two fractions $\Omega_{MO}/\Omega_M = \sqrt{3}\pi$ and $\Omega_{EO} = f_c$ with zero parameters, plus the quantitative cosmic background ---$\Omega_m = 1-f_c \approx 0.3131$ at $0.3\sigma$ from Planck without fitting, the form of $H(z)$ forced by the potential that fits BAO DESI, and the perfect-blackbody CMB---; and the positional reading of the Hubble constant, which reformulates the cosmological-constant problem ---the ``$10^{-123}$'' as the age of the universe in ticks, not a fine-tuning--- and reads the Hubble tension as a method artefact that the program predicts dissolves in model-independent observables (the background $H_0$ on the primordial side $\sim67$).
  \item \textbf{Falsifiable.} The UV suppression of the primordial tensor spectrum; the spectral index $n_s = 1$ at zeroth order of the native matter bounce; the null chemical potential $\mu = 0$ of the cosmic background; the GUP-type ultraviolet correction with derived coefficient $3/80$; the complete baryonic rotational tower; the separation of vortices; the invariance of the hadronic scale $\Lambda_\text{PIU}$ with the redshift; the universal core density of the primordial black holes; the signatures of the Lee-Wick pole, with the quantified intrinsic TT attenuation (absorption length $\sim 10^{31}$ Hubble radii at LISA); the time-dilation exponent $b=1$ of the type-Ia supernovae, derived and consistent with DES-SN5YR at $0.27\sigma$ (\S{}III.8.2-bis); and the dissolution of the Hubble tension and of the DESI $w_0$--$w_a$ signal in model-independent observables, both artefacts of analyzing a medium with $w=-1$ exact under the $\Lambda$CDM parametrization (\S{}III.8.7).
  \item \textbf{Open.} On the plane of \emph{theory} ---where the priority of the program lies---: the fermionic electroweak cascade and the value of $\alpha$, the full classification of the lattice constants of the gauge sector, and the dynamics of the non-linear bounce. The geometry of the diamond lattice is derived in its two pieces (coordination $Z=4$, D1-a; cubic stacking, D1-b, minimum of the flexural functional, \S{}II.3.7-bis), and for that reason it does not appear in this list. Regarding the Dirac structure, it is derived from the two binary structures of the medium with a unique representation by the Pauli theorem (\S{}V.7.2, \epi{DERIVED}), so that what is open on that front reduces to the value of $\lambda_\text{SO}$ and of the fermionic masses, not the structure of the operator. The present value of the numerator of $f_c$, $S_{\max,\text{eff}}^2(t_0)$, does not figure as an autonomous thread: it is the evaluation at the current epoch of the trajectory $\bar S(t)$ and is subsumed in the F-cosmo thread (\S{}V.4). On the plane of \emph{confrontation}, of subordinate priority: the combined analysis of the cosmological sector with third-party catalogues (confrontation F-cosmo) and the prediction of $S_8$, once the cosmological background is already derived (\S{}III.8). To these is added the declared calibration thread: the measurement of $\rho_P$ or $\ell_P$ from non-gravitational and non-quantum physics (C-meta-1.b), the structural degree of freedom that the program does not fix internally.
\end{itemize}

\subsubsection{\texorpdfstring{V.5.3 --- Return to the crisis of \S{}I.2: scorecard of the six problems}{V.5.3 - Return to the crisis: scorecard of the six problems}}
\S{}I.2 justified the legitimacy of PIU-$\Psi$ by articulating six open problems of contemporary physics, and \S{}I.2.9 committed the program to declare, at the close, what it did with each one. This subsection settles that commitment: for each problem it contrasts the treatment of the standard framework (the \emph{status quo}) with the contribution of PIU-$\Psi$, and declares the degree of closure ---\textbf{[addresses]} (structural route with a quantitative result), \textbf{[reinterprets]} (ontological reading without yet redoing the calculation) or \textbf{[partial]} (real advance with a declared thread).
{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{2.0cm}L{4.2cm}L{4.6cm}L{2.0cm}}
\toprule
\textbf{Problem (\S{}I.2)} & \textbf{Status quo of today} & \textbf{Contribution of PIU-$\Psi$} & \textbf{Status} \\
\midrule\endhead
1. Cosmological constant / vacuum energy &
Theory-observation discrepancy of $60$--$120$ orders of magnitude; no solution by known naturalness &
Dark energy is read as el Pleno-ground state (permanent structural state), not as zero-point energy to be summed; $\Omega_{EO}=f_c$ at $0.30\sigma$ with zero parameters. The \emph{partition} reformulates the $120$ orders as an order-one fraction, and the \emph{scale} $\Theta_0\sim10^{-123}$ as the age of the universe in Planck ticks ($\Theta=3/8\pi N_\text{tic}^2$, \S{}III.8.7): both faces cease to be a fine-tuning. The computation of the absolute value from first principles remains a positional thread &
[reinterprets] + [partial] \\
2. Dark sectors ($95\%$ unknown) &
Dark matter sought $40$ years without direct detection; dark energy with DESI hints of dynamics that break $\Lambda$CDM &
Both sectors are configurations of the same el Pleno, not new substances; $\Omega_{MO}/\Omega_M=\sqrt{3}\pi$ at $0.86\sigma$ with zero parameters &
[addresses] \\
3. Cosmological tensions ($H_0$, $S_8$) &
$H_0$ tension consolidating up to $\sim 5.8\sigma$; $S_8$ persistent at $2$--$3\sigma$; DESI hints of dynamical dark energy &
$H_0$ is read as a positional quantity (frequency of the tick modulated by the age in ticks, \S{}III.8.7). The $H_0$ tension is a method artefact: the ``primordial'' $H_0$ ($\sim67$) is the one $\Lambda$CDM needs for the CMB, and analyzing a medium with $w=-1$ exact and elastic $H(z)$ under that template biases the inference; the program \emph{predicts} that the tension dissolves in model-independent observables (raw $D_H/r_d$, $\theta_*$) \epi{SLD}. The DESI $w_0$--$w_a$ signal is the same type of parametrization artefact of a medium with $w=-1$ exact. There remain as threads the non-linear correction of $H(z)$ by the walls and its confrontation with raw BAO, and the prediction of $S_8$ from the phase drag &
[reinterprets] + [partial] \\
4. GR--QM incompatibility &
Gravity non-renormalizable (t'Hooft-Veltman); string landscape $\sim 10^{500}$ vacua &
The boundary between GR and QM disappears: both are effective readings of the same el Pleno (GR by induced action, QM by quantization of the phase mode), not two theories to reconcile. The \emph{conceptual} integration is closed by axiom A1; the \emph{quantitative} one already operates ($G$, $\hbar$, uncertainty and Chandrasekhar come from the same medium). The background cosmological sector is derived (\S{}III.8), and the coefficient of the induced action ($\gamma_G=1$) is fixed by convention plus non-contamination (\S{}III.1.4). There remains, on the plane of theory, the dynamics of the non-linear bounce; the cosmological confrontation with third-party catalogues remains a thread of subordinate priority &
[reinterprets] + [partial] \\
5. Unexplained fundamental constants &
$\sim 20$ constants as underived inputs; GR and QM assume $G$, $\hbar$ free &
$G=c^2/(\rho_P\ell_P^2)$ and $\hbar=\rho_P c\,\ell_P^4$ as structural identities of unit coefficient; the count of substantive quantities of the medium reduces to \emph{two} ($\rho_P$, $\ell_P$), with $c$ being causal structure and not a property; $S_{\min}^\text{estr}$ and $S_{\max,\text{geom}}^2$ without parameters. $\alpha$ and fermionic masses: thread &
[addresses] + [partial] \\
6. Ontological status of the fields / vacuum &
Operational position (``shut up and calculate''); the field is an abstract entity, the vacuum is not clarified &
The field is an excitation of an identified physical substrate; the ``vacuum'' is el Pleno in minimum excitation ($S_{\min}^\text{estr}$), structured and measurable &
[reinterprets] \\
\bottomrule
\end{longtable}}
The reading of the table that the manuscript sustains is the one that \S{}I.2.9 anticipated: PIU-$\Psi$ \emph{addresses the six}, but does not \emph{resolve the six}. Where it reinterprets (problems 1, 3, 4 and 6) it dissolves the problem in its own terms ---in the GR--QM incompatibility, by making the boundary between the two disappear; in the cosmological tensions, by showing them as artefacts of analyzing a medium with $w=-1$ exact under the $\Lambda$CDM template--- without having closed all the quantitative calculation, which remains a thread. Where it addresses (2 and 5) it exhibits results with zero or one parameter. No problem is marked as ``resolved'': that restraint is deliberate and coherent with the humble epistemological position of \S{}I.1.
\subsubsection{\texorpdfstring{V.5.4 --- The difference with respect to the status quo}{V.5.4 - The difference with respect to the status quo}}

The fundamental difference with the standard framework can be stated all at once. The standard framework of today is, in the precise sense of \S{}I.2.6, a \textbf{phenomenological description of enormous predictive success}: it parametrizes what it observes ---the $95\%$ dark, the $\sim 20$ constants, the equation of state $w$--- with coefficients calibrated against the datum, without deriving their origin. PIU-$\Psi$ does not compete in fitting precision: in the cosmological sector it matches $\Lambda$CDM with the same degrees of freedom (\S{}V.2.7). The difference is one of \emph{status of the parameters and of ontological load}:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Derived, not fitted, parameters.} Where $\Lambda$CDM fits
  $\Omega_\Lambda$ and $\Omega_{dm}/\Omega_b$, PIU-$\Psi$ derives them from the geometry
  of the substrate. Where the Standard Model measures $G$ and $\hbar$, PIU-$\Psi$
  presents them as structural identities of two properties of the medium ($\rho_P$,
  $\ell_P$) and its causal structure ($c$).
  \item \textbf{One ontological referent, not an inventory.} Where the status quo
  maintains independent categories ---matter, dark matter, dark energy,
  fields, vacuum, space-time--- each with its rules, PIU-$\Psi$ reads them
  as configurations of a single \emph{el Pleno}.
  \item \textbf{Predictions that the status quo does not make.} The UV suppression of the
  primordial tensor spectrum, the baryonic rotational tower, the separation of
  vortices, the signatures of the Lee-Wick pole: none follows from $\Lambda$CDM nor from the
  Standard Model. They are the points where PIU-$\Psi$ exposes itself to refutation and, therefore,
  where it is most distinguished from the current consensus.
\end{itemize}

The difference is not ``PIU-$\Psi$ fits better''. It is ``PIU-$\Psi$ explains what
the status quo calibrates, with fewer ontological categories and with falsifiable
predictions of its own''. The added value that \S{}I posed as an objective is
thus made effective and declared with its degree of closure.

\subsubsection{\texorpdfstring{V.5.5 --- Fulfillment of the value promise}{V.5.5 - Fulfillment of the value promise}}

The Overview opened with a concrete promise: that from a single substrate and two properties of the medium, the program reproduces a definite set of results of known physics and dissolves one of its great open problems. This subsection settles it point by point, contrasting what was \emph{offered} in the opening with what was \emph{delivered} in the body and marking each item with its degree of fulfillment ---\textbf{[fulfilled]} (result derived with its classification) or \textbf{[partially fulfilled]} (result with a declared thread).

{\footnotesize\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{4.7cm}L{5.6cm}L{2.3cm}}
\toprule
\textbf{Promise of the Overview} & \textbf{What was delivered in the body} & \textbf{Fulfillment} \\
\midrule\endhead
$G$ and $\hbar$ with no fitted coefficient, from $\rho_P$, $\ell_P$ and $c$ &
$G=c^2/(\rho_P\ell_P^2)$ and $\hbar=\rho_P c\,\ell_P^4$ as structural identities (\S{}III.1--III.2); exact agreement by construction, residue $<0.1\%$ by rounding &
[fulfilled] \\
Uncertainty relation with exact floor $\hbar/2$ and structural meaning &
$\Delta x\,\Delta p=(\hbar/2)\sqrt{1+(3/80)(\ell_P/\sigma)^2}$, exact floor $\hbar/2$ and GUP with derived coefficient (\S{}III.2.6) &
[fulfilled] \\
Chandrasekhar limit as an internal balance of the medium &
$M_\text{Ch}\approx 1.456\,M_\odot$ with complete coefficient, without external importation (\S{}III.2.8) &
[fulfilled] \\
Cosmological ratios at $<1\sigma$ of Planck &
$\Omega_{MO}/\Omega_M=\sqrt{3}\pi$ ($0.86\sigma$) and $\Omega_{EO}=f_c$ ($0.30\sigma$), zero parameters (\S{}III.6--III.7) &
[fulfilled] \\
Matter density and $H(z)$ from the medium &
$\Omega_m=1-f_c\approx 0.3131$ at $0.3\sigma$ without fitting; $H(z)$ forced by the potential, fits BAO DESI ($\chi^2=6.54$ vs $6.53$) (\S{}III.8) &
[fulfilled] \\
Emergent gravity: Newton and Einstein &
Newton in weak field (\S{}III.1) and the Einstein-Hilbert structure by induced action; $\mathcal{O}(1)$ Sakharov coefficient: thread &
[partially fulfilled] \\
Maxwell and Yang-Mills from the phase circulation &
Maxwell with coefficient $8/9$, non-abelian Yang-Mills and the Dirac structure from the same $SU(2)$ plaquette holonomy (\S{}III.4, \S{}V.7.1); chirality and value of fermionic masses: thread &
[partially fulfilled] \\
Special Relativity as the rheology of the medium &
The Lorentz factor as a dilatant response $n(v)=\gamma^2$; the light barrier as the jamming of the fluid (\S{}III.9.5) &
[fulfilled] \\
Dissolution of the GR--QM incompatibility &
The boundary disappears: both are readings of the same \emph{el Pleno}; conceptual integration closed by A1, quantitative one already operating (\S{}III.2.10) &
[partially fulfilled] \\
To recover the validated theories as effective limits &
Newton, GR and QM recovered in their regimes; the program explains \emph{why} they work, it does not replace them (Parts III--IV) &
[fulfilled] \\
Native thermodynamics: statistics without importing statistical mechanics &
The Bose-Einstein occupation (E59) and the coefficient $2\pi^2/45$ of Stefan-Boltzmann (E-BE-3) obtained by counting configurations of the eigenmodes of F1 (\S{}III.9.5-ter) &
[fulfilled] \\
The origin of what Newton left primitive &
The three laws as the low-energy limit of F1, with mass as retained energy, inertia as the resistance of the medium to reconfiguring itself and $G$ as a mechanical identity (\S{}III.1.1-bis) &
[fulfilled] \\
Exposure to risk: which measurement would refute the programme &
The three confrontation scenarios over $\kappa_G$ and $\kappa_\hbar$ (E63, \S{}V.2.3-bis): with one primary measured by an independent route the product is exposed; with both, each branch separately &
[fulfilled] \\
Epistemic classification: each result labeled, each thread declared &
Classification \texttt{\epi{DERIVED}}, \texttt{\epi{SLD}}, \texttt{\epi{CALIBRATED}} or \texttt{\epi{HYPOTHESIS}} throughout the corpus; inventory of threads in \S{}V.3, with the theory/confrontation distinction &
[fulfilled] \\
\bottomrule
\end{longtable}}

The reading of the table is direct: of the eleven concrete promises of the Overview,
eight are \textbf{fulfilled} with a derived and classified result, and three are
\textbf{partially fulfilled} ---they have their central result delivered and a thread
named (the Sakharov coefficient, the fermionic masses, the combined cosmological
confrontation). No promise remains unfulfilled or oversold: what was
promised to be derived is derived, and what remains is exactly what the program
declares as open. The value offered in the opening is the value delivered in the
body.

\medskip

The balance closes with the complete \emph{derivational architecture}. Figure~\ref{fig:cadena} traces, step by step and without leaps, the two longest chains that depart from the root statement $E$: a common trunk ---the genesis A1$\to$F1 and the geometry of the diamond lattice--- from which, at a bifurcation point, hang the cosmological branch (up to $\Omega_m$ and the history $H(z)$) and the quantum branch (up to the uncertainty relation with its exact floor $\hbar/2$). Each box carries its equation and the color of its epistemic classification, making visible at a glance what the table asserts item by item: that the program is a continuous derivational structure, not a collection of numerical coincidences. And with the same honesty as the rest of the manuscript ---the only \texttt{\epi{CALIBRATED}} link, the numerator $S_{\max,\text{eff}}^2$ of the cosmological branch, appears marked in its color without concealing itself: that transparency is what distinguishes a map of foundations from a piece of propaganda.

\begin{figure}[htbp]
\centering
\scalebox{0.62}{%
\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}]

\newcommand{\eqlabelF}[1]{{\scriptsize\bfseries\color{black!55}#1}}
\newcommand{\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, homogeneous, isotropic substrate}};
\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{---}\\[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}\\[2pt]
  $\mathcal{L}_{F1}=\tfrac12\rho_Pc^2\ell_P^2\,\partial_\mu\Psi^*\partial^\mu\Psi-V(S)-\tfrac{\rho_Pc^2\ell_P^4}{40}(\Box S)^2$\\[1pt]
  \glosaF{Lagrangian of \emph{el Pleno}: kinetic $+$ potential $+$ flexural}};
\node[deriv, below=of F1] (diamante) {\eqlabelF{E9-prev}\\[2pt]
  $D_\text{BZ}=\dfrac{\langle\Delta_\text{lat}\rangle_\text{BZ}}{\langle k^2\rangle_\text{BZ}}=\dfrac{12(\pi^2-4)}{\pi^4}$\\[1pt]
  \glosaF{diamond lattice $Fd\bar3m$, $Z{=}4$: Laplacian over the Brillouin zone}};
\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{---}\\[2pt] $S_{\max,\text{eff}}^2=1.4956$\\[1pt]
   \glosaF{effective occupation of the current regime}};
\node[deriv, below=of smaxeff] (fc)
  {\eqlabelF{E11 \ / \ E33}\\[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 (denom.\ derived / numer.\ calibrated)}};
\node[deriv, below=of fc] (omegaEO)
  {\eqlabelF{E33}\\[2pt] $\Omega_{EO}\equiv f_c\approx0.6869$\\[1pt]
   \glosaF{dark energy = condensed backdrop; $0.30\sigma$ vs Planck}};
\node[deriv, below=of omegaEO] (omegam)
  {\eqlabelF{E-cosmo-2}\\[2pt] $\Omega_m=1-f_c=0.3131$\\[1pt]
   \glosaF{$f_c+\Omega_\text{depleted}=1$; \ $0.3\sigma$ 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}};

\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);

\node[deriv, below right=4mm and 5mm of bif] (hbar)
  {\eqlabelF{E36}\\[2pt] $\hbar=\rho_P\,c\,\ell_P^4$\\[1pt]
   \glosaF{quantization of the phase mode; prefactor $1$ exact, $<0.1\%$}};
\node[deriv, below=of hbar] (gammak)
  {\eqlabelF{E4}\\[2pt] $\gamma_\kappa=\dfrac{\rho_P c^2\ell_P^4}{40}=\dfrac{\hbar c}{40}$\\[1pt]
   \glosaF{native identity: flexural rigidity = quantum property of the medium}};
\node[deriv, below=of gammak] (sigmamin)
  {\eqlabelF{E-inc-1}\\[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] (momento)
  {\eqlabelF{---}\\[2pt] $\Delta p_\text{medio}^2=\dfrac{\hbar^2}{4\sigma^2}\!\left[1+\dfrac{3}{80}\dfrac{\ell_P^2}{\sigma^2}\right]$\\[1pt]
   \glosaF{momentum of the medium (gaussian packet)}};
\node[deriv, below=of momento] (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 $>0$}};
\node[dlf, below=of heisenberg] (edificio)
  {\eqlabelF{---}\\[2pt] Quantum edifice over Heisenberg\\[1pt]
   \glosaF{complementarity, diffraction, stability of the atom, zero point}};

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

\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 (genesis
A1$\to$F1 and geometry of the diamond lattice) and bifurcation into two branches: on the
left the cosmology (up to $\Omega_m=1-f_c$ and the expansion history
$H(z)$), on the right quantum mechanics (up to the uncertainty relation with
exact floor $\hbar/2$). Over each box, the label of its equation in the
manuscript (``---'' where the expression is an identity without its own number). The
color of the border encodes the epistemic classification: \textcolor{cTronco}{\textbf{trunk/foundation}},
\textcolor{cDeriv}{\epi{DERIVED}}, \textcolor{cDLF}{\textbf{[strong logical
deduction]}}, \textcolor{cCalib}{\epi{CALIBRATED}}. The only calibrated
link ---the numerator $S_{\max,\text{eff}}^2$--- appears marked in its color:
the quantum branch is derived end-to-end, while the cosmological one
inherits a declared calibrated factor.}
\label{fig:cadena}
\end{figure}

\subsubsection{\texorpdfstring{V.5.6 --- Closure: mirror of the foundational declaration}{V.5.6 - Closure: mirror of the foundational declaration}}

The manuscript opened by affirming that the universe can be read as a single continuous
substrate from which everything observed emerges, and that PIU-$\Psi$ does not pretend to be the
universe but to offer a coherent and testable reading of it. It also opened by
documenting six open problems that made legitimate the search for a
unified ontology. The journey of the five Parts has made effective that
affirmation ---from the substrate and its Lagrangian to the phenomenology, the cycle and the
observational confrontation---, and \S{}V.5.3 returned the gaze to the six
problems to declare, without overselling, what was addressed, what was reinterpreted and what
remains partial. What remains open is declared as open; what is
derived carries its classification; what is falsifiable signals where the future
observation can refute the program. That is, exactly, the condition of a
scientific program: not certainty, but internal coherence, the recovery
of the validated and the declared exposure to the risk of being falsified.

\subsubsection{\texorpdfstring{V.5.7 --- Consolidated bibliography: academic references}{V.5.7 - Consolidated bibliography: academic references}}

\begin{thebibliography}{99}

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\textit{Planck 2018 results. VI. Cosmological parameters},
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\bibitem{desi2024} DESI Collaboration (Adame A.\ G.\ et al.),
\textit{DESI 2024 VI: Cosmological constraints from the measurements of
baryon acoustic oscillations}, (2024).

\bibitem{pantheon} Scolnic D.\ et al.,
\textit{The Pantheon+ Analysis: The Full Data Set and Light-Curve Release},
The Astrophysical Journal \textbf{938}, 113 (2022).

\bibitem{whitedes5yr} White M.\ et al.\ (DES Collaboration),
\textit{The Dark Energy Survey Supernova Program: Slow supernovae show
cosmological time dilation out to $z\sim1$}, Monthly Notices of the Royal
Astronomical Society \textbf{533}, 3365 (2024).

\bibitem{gw170817} Abbott B.\ P.\ et al.\ (LIGO Scientific Collaboration and
Virgo Collaboration), \textit{GW170817: Observation of Gravitational Waves from
a Binary Neutron Star Inspiral}, Physical Review Letters \textbf{119}, 161101
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\bibitem{gw170817mma} Abbott B.\ P.\ et al.,
\textit{Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger:
GW170817 and GRB 170817A}, The Astrophysical Journal Letters \textbf{848}, L13
(2017).

\bibitem{sakharov} Sakharov A.\ D.,
\textit{Vacuum quantum fluctuations in curved space and the theory of
gravitation}, Doklady Akademii Nauk SSSR \textbf{177}, 70 (1967).

\bibitem{klauder} Klauder J.\ R.,
\textit{Enhanced quantization: a primer}, Journal of Physics A: Mathematical and
Theoretical \textbf{45}, 285304 (2012).

\bibitem{madelung} Madelung E.,
\textit{Quantentheorie in hydrodynamischer Form}, Zeitschrift f\"ur Physik
\textbf{40}, 322 (1927).

\bibitem{bohm} Bohm D.,
\textit{A Suggested Interpretation of the Quantum Theory in Terms of ``Hidden''
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\bibitem{leewick} Lee T.\ D., Wick G.\ C.,
\textit{Negative metric and the unitarity of the S-matrix}, Nuclear Physics B
\textbf{9}, 209 (1969).

\bibitem{donoghue} Donoghue J.\ F.,
\textit{General relativity as an effective field theory: The leading quantum
corrections}, Physical Review D \textbf{50}, 3874 (1994).

\bibitem{skyrme} Skyrme T.\ H.\ R.,
\textit{A unified field theory of mesons and baryons}, Nuclear Physics
\textbf{31}, 556 (1962).

\bibitem{derrick} Derrick G.\ H.,
\textit{Comments on Nonlinear Wave Equations as Models for Elementary
Particles}, Journal of Mathematical Physics \textbf{5}, 1252 (1964).

\bibitem{adkins} Adkins G.\ S., Nappi C.\ R., Witten E.,
\textit{Static properties of nucleons in the Skyrme model}, Nuclear Physics B
\textbf{228}, 552 (1983).

\bibitem{barcelo} Barcel\'o C., Liberati S., Visser M.,
\textit{Analogue Gravity}, Living Reviews in Relativity \textbf{8}, 12 (2005).

\bibitem{grosspitaevskii} Pitaevskii L.\ P.,
\textit{Vortex lines in an imperfect Bose gas}, Soviet Physics JETP
\textbf{13}, 451 (1961).

\bibitem{tsirelson} Cirel'son B.\ S.,
\textit{Quantum generalizations of Bell's inequality}, Letters in Mathematical
Physics \textbf{4}, 93 (1980).

\bibitem{chsh} Clauser J.\ F., Horne M.\ A., Shimony A., Holt R.\ A.,
\textit{Proposed Experiment to Test Local Hidden-Variable Theories}, Physical
Review Letters \textbf{23}, 880 (1969).

\bibitem{finkelstein} Finkelstein D., Rubinstein J.,
\textit{Connection between Spin, Statistics, and Kinks}, Journal of Mathematical
Physics \textbf{9}, 1762 (1968).

\bibitem{jeffreys} Jeffreys H.,
\textit{Theory of Probability}, 3rd ed., Oxford University Press (1961).

\bibitem{keplerhales} Hales T.\ C.,
\textit{A proof of the Kepler conjecture}, Annals of Mathematics \textbf{162},
1065 (2005).

\bibitem{codata} Tiesinga E., Mohr P.\ J., Newell D.\ B., Taylor B.\ N.,
\textit{CODATA recommended values of the fundamental physical constants: 2018},
Reviews of Modern Physics \textbf{93}, 025010 (2021).

\bibitem{gent} Gent A.\ N.,
\textit{A New Constitutive Relation for Rubber}, Rubber Chemistry and Technology
\textbf{69}, 59 (1996).

\bibitem{cmbs4} Abazajian K.\ et al.\ (CMB-S4 Collaboration),
\textit{CMB-S4 Science Book}, (2016).

\bibitem{litebird} LiteBIRD Collaboration,
\textit{Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background
Polarization Survey}, Progress of Theoretical and Experimental Physics
\textbf{2023}, 042F01 (2023).

\bibitem{noether} Noether E.,
\textit{Invariante Variationsprobleme}, Nachrichten von der Gesellschaft der
Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse \textbf{1918},
235--257 (1918).

\bibitem{mandelstamtamm} Mandelstam L.\ I., Tamm I.\ E.,
\textit{The uncertainty relation between energy and time in nonrelativistic
quantum mechanics}, J.\ Phys.\ (USSR) \textbf{9}, 249--254 (1945).

\bibitem{margoluslevitin} Margolus N., Levitin L.\ B.,
\textit{The maximum speed of dynamical evolution}, Physica D: Nonlinear
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\bibitem{provostvallee} Provost J.\ P., Vall\'ee G.,
\textit{Riemannian structure on manifolds of quantum states}, Communications in
Mathematical Physics \textbf{76}, 289--301 (1980).

\bibitem{yukalov2007} Yukalov V.\ I.,
\textit{Bose-Einstein condensation and gauge symmetry breaking}, Laser Physics
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\bibitem{wands1999} Wands D.,
\textit{Duality invariance of cosmological perturbation spectra}, Physical
Review D \textbf{60}, 023507 (1999).

\bibitem{finellibrandenberger} Finelli F., Brandenberger R.,
\textit{On the generation of a scale-invariant spectrum of adiabatic
fluctuations in cosmological models with a contracting phase}, Physical Review D
\textbf{65}, 103522 (2002).

\bibitem{fixsen1996} Fixsen D.\ J., Cheng E.\ S., Gales J.\ M., Mather J.\ C.,
Shafer R.\ A., Wright E.\ L.,
\textit{The Cosmic Microwave Background Spectrum from the Full COBE/FIRAS Data
Set}, The Astrophysical Journal \textbf{473}, 576--587 (1996).

\bibitem{mukhanovsasaki} Mukhanov V.\ F., Feldman H.\ A., Brandenberger R.\ H.,
\textit{Theory of cosmological perturbations}, Physics Reports \textbf{215},
203--333 (1992).

\end{thebibliography}

\subsubsection{\texorpdfstring{V.5.8 --- References of the contemporary crisis (\texttt{REF-WEB})}{V.5.8 - References of the contemporary crisis (REF-WEB)}}

The following references resolve the marks \texttt{[REF-WEB-XX]} used in
\S{}I.2 when documenting the six open problems of contemporary physics. Each
entry refers to the primary source (peer-reviewed article or reference review) and,
when applicable, to the corresponding data repository; access was verified as of the
closing date of this version.

\begin{enumerate}[leftmargin=2.4em,label={[REF-WEB-\arabic*]}]
  \item Weinberg S., \textit{The cosmological constant problem}, Reviews of Modern
  Physics \textbf{61}, 1 (1989); Planck Collaboration, \textit{Planck 2018 results.
  VI. Cosmological parameters}, A\&A \textbf{641}, A6 (2020). Magnitude of the
  vacuum-quantum/$\Lambda$ discrepancy and observed density.
  \item Weinberg S., \textit{The cosmological constant problem}, Reviews of Modern
  Physics \textbf{61}, 1 (1989). Absence of a known symmetry that cancels $\Lambda$.
  \item Planck Collaboration, \textit{Planck 2018 results. VI}, A\&A \textbf{641}, A6
  (2020); DESI Collaboration (Adame A.\ G.\ et al.), \textit{DESI 2024 VI:
  Cosmological Constraints from BAO}, arXiv:2404.03002 (2024). Early value
  $H_0=67.4\pm0.5$ and JWST verification of the cepheids.
  \item Burgess C.\ P., \textit{Quantum Gravity in Everyday Life: General Relativity
  as an Effective Field Theory}, Living Reviews in Relativity \textbf{7}, 5 (2004).
  Collapse of GR at high-energy singularities; hierarchy of discrepancies.
  \item Planck Collaboration, \textit{Planck 2018 results. VI}, A\&A \textbf{641}, A6
  (2020). $\Lambda$CDM inventory: baryons, dark matter, dark energy.
  \item Bertone G., Hooper D., \textit{History of dark matter}, Reviews of Modern
  Physics \textbf{90}, 045002 (2018). Dark matter/energy as phenomenological
  labels without an identified physical referent.
  \item Arcadi G.\ et al., \textit{The Waning of the WIMP}, European Physical Journal
  C \textbf{78}, 203 (2018). Non-direct-detection and revision of the WIMP paradigm.
  \item DESI Collaboration (Adame A.\ G.\ et al.), \textit{DESI 2024 VI:
  Cosmological Constraints from BAO}, arXiv:2404.03002 (2024). Preference
  ($w_0,w_a$) for an equation of state of dark energy that evolves with
  time, at $2.5$--$3.9\sigma$ according to the data set.
  \item Riess A.\ G.\ et al., \textit{A Comprehensive Measurement of the Local Value
  of the Hubble Constant with 1 km/s/Mpc Uncertainty from HST and the SH0ES Team},
  The Astrophysical Journal Letters \textbf{934}, L7 (2022). $H_0=73.04\pm1.04$.
  \item Riess A.\ G.\ et al., \textit{JWST Observations Reject Unrecognized Crowding
  of Cepheid Photometry as an Explanation for the Hubble Tension}, The Astrophysical
  Journal Letters \textbf{956}, L18 (2023). JWST confirmation in the range $72$--$73$.
  \item Breuval L.\ et al., \textit{Small Magellanic Cloud Cepheids Observed with
  JWST}, The Astrophysical Journal \textbf{973}, 30 (2024). Tension at $5.8\sigma$,
  independent of the classical distance ladder.
  \item Di Valentino E.\ et al., \textit{In the realm of the Hubble tension --- a
  review of solutions}, Classical and Quantum Gravity \textbf{38}, 153001 (2021).
  Growing robustness of the $H_0$ tension and persistence of the $S_8$ tension (DES,
  HSC).
  \item Heymans C.\ et al.\ (KiDS Collaboration), \textit{KiDS-1000 Cosmology:
  Multi-probe weak gravitational lensing and spectroscopic galaxy clustering
  constraints}, Astronomy \& Astrophysics \textbf{646}, A140 (2021). Value of $S_8$
  lower than the one extrapolated from the CMB by $2$--$3\sigma$.
  \item Abbott T.\ M.\ C.\ et al.\ (DES Collaboration), \textit{Dark Energy Survey
  Year 3 results: Cosmological constraints from galaxy clustering and weak lensing},
  Physical Review D \textbf{105}, 023520 (2022). Independent confirmation of the
  $S_8$ tension.
  \item Abbott B.\ P.\ et al.\ (LIGO Scientific Collaboration and Virgo
  Collaboration), \textit{Gravitational Waves and Gamma-Rays from a Binary Neutron
  Star Merger: GW170817 and GRB 170817A}, The Astrophysical Journal Letters
  \textbf{848}, L13 (2017). Bound $-3\times10^{-15}\,c < v_g-c < +7\times10^{-16}\,c$.
\end{enumerate}

\medskip
\noindent\textbf{Data repositories and resources of the program.} Site of the program
PIU-$\Psi$: \texttt{https://www.piuniversal.com}. Repository of published versions with persistent
DOI: Zenodo (\texttt{zenodo.org}); this V5.1 version: DOI
\texttt{10.5281/zenodo.21724390}. Planck Legacy Archive:
\texttt{pla.esac.esa.int}. DESI Data Release: \texttt{data.desi.lbl.gov}.
Pantheon+ Data Release: \texttt{github.com/PantheonPlusSH0ES}. Gravitational Wave
Open Science Center: \texttt{gwosc.org}.

\begin{tcolorbox}[notabox,title={Closure of the manuscript}]
PIU-$\Psi$ is presented as a foundational program with concrete results:
a single ontology of \emph{el Pleno}; a derivational apparatus that recovers the
validated physics as an effective limit; the constants $G$ and $\hbar$ obtained as
structural identities of three primaries; two cosmological fractions
derived at less than $1\sigma$ with zero free parameters; the hadronic sector
with structural parity with respect to QCD using a single input; and a catalogue of
falsifiable predictions, several of them unique, confronted with the observation.
Together with these achievements, the program declares with the same clarity what remains
open. PIU-$\Psi$ does not present itself as the last word ---no foundational
theory under construction should---, but it does claim, without hesitation, the
results it has obtained and the originality of having obtained them from a single
substrate. It is, in the full sense of the term, a coherent, rigorous,
testable ---and already partially tested--- reading of the universe as a single
substrate.
\end{tcolorbox}

\medskip\hrule\medskip
