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%  PIU-Psi V5.1  --  Master document
%  Universal Integrity Principle (PIU-Psi)
%  Manuel Alberto Celedon Mejia
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  pdftitle={Universal Integrity Principle (PIU-Psi) -- Version 5.1},
  pdfauthor={Manuel Alberto Celedon Mejia},
  pdfsubject={A foundations program: el Pleno as a single substrate},
  pdfkeywords={PIU-Psi, el Pleno, emergent gravity, fundamental constants,
               cosmology, dark energy, dark matter}}

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\begin{document}
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% ---------------------------------------------------------------------
%  COVER
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\begin{titlepage}
  \centering
  \vspace*{0.4cm}
  {\color{piuRule!55}\rule{0.5\textwidth}{1.2pt}}\par
  \vspace{0.6cm}
  {\Huge\bfseries Universal Integrity Principle\par}
  \vspace{0.35cm}
  {\Huge\bfseries\color{piuRule!85!black} PIU--$\Psi$\par}
  \vspace{0.6cm}
  {\color{piuRule!55}\rule{0.32\textwidth}{0.8pt}}\par
  \vspace{0.7cm}
  {\Large Version 5.1\par}
  \vspace{0.25cm}
  {\large 30 July 2026\par}
  \vspace{0.8cm}
  {\large Manuel Alberto Celed\'on Mej\'ia\par}
  \vspace{0.15cm}
  {\normalsize Mechanical Engineer\par}
  {\normalsize Medell\'in, Colombia\par}
  \vspace{0.7cm}
  \begin{tcolorbox}[ecuacionclave,title={One single thing}]
    PIU--$\Psi$ is a research program in theoretical physics ---in cosmology and
    quantum physics--- built on strict mathematics. It starts from a radical and
    precise hypothesis: that gravity, light, particles, space, time and the dark
    sectors are configurations of a single physical substrate ---\emph{el Pleno}---,
    described by one field $\Psi = S\cdot e^{i\theta}$ and six axioms. From two
    properties of the medium and its causal structure, the program recovers $G$ and
    $\hbar$ with no fitted coefficient, derives the Heisenberg
    uncertainty relation with its exact $\hbar/2$ floor, makes gravity emerge
    (Newton and the structure of Einstein's equations, with the geodesic and the
    Schwarzschild metric as a geometric reading of the medium) together with the
    structure of the Dirac equation, derives Maxwell and Yang-Mills from phase
    circulation, obtains the cosmological ratios to within $1\sigma$ of Planck, and
    reads Relativity as the rheology of a dilatant fluid. General Relativity and
    Quantum Mechanics cease to read as two theories awaiting reconciliation: they
    emerge as effective readings of the same medium, an integration already
    operative at the quantitative level. The program matches $\Lambda$CDM
    statistically with \emph{derived} parameters, not fitted ones, and adds
    falsifiable predictions via LISA and Fermi-LAT. Every step carries its algebra
    and its epistemic classification. The invitation is to verify it.
  \end{tcolorbox}
  \vspace{0.45cm}

  {\color{piuRule!40}\rule{\textwidth}{0.6pt}}\par
  \vspace{0.2cm}
  {\small\itshape https://www.piuniversal.com \,\textbullet\, Zenodo DOI 10.5281/zenodo.21724390\par}
\end{titlepage}

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%  OVERVIEW
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\section*{Overview}
\phantomsection
\addcontentsline{toc}{section}{Overview}

What if everything that exists ---gravity, light, particles, space, time, the dark
sectors that make up 95\,\% of the universe--- were the manifestation of one single
thing? PIU-$\Psi$ is a \textbf{research program in theoretical physics} ---in
cosmology and quantum physics--- that answers that question with strict mathematics.
It proposes that the entire universe is a single continuous physical substrate
---el Pleno--- and that every phenomenon physics describes separately is not an
independent entity but a \emph{configuration} of that single medium. The letter
$\Psi$ denotes the complete state of el Pleno, $\Psi = S\cdot e^{i\theta}$: the
modulus $S$ carries the structure, the phase $\theta$ carries the dynamics. From
that single equation, and from six structural axioms, the program makes known
physics emerge, each step with its explicit derivation and its epistemic
classification.

The scope is quantitative and verifiable. From two properties of the medium
---its density $\rho_P$ and its microscopic scale $\ell_P$--- and its causal
structure $c$, PIU-$\Psi$ reproduces the fundamental constants $G$ and $\hbar$ with no
fitted coefficient ---not by dimensional analysis, but because the statics of the modulus
and the quantisation of the rotor produce them from F1 with unit coefficient---; it derives the Heisenberg uncertainty
relation ---with its exact $\hbar/2$ floor---, and with it the Chandrasekhar limit
($\approx 1.456\,M_\odot$); it obtains the cosmological ratios
$\Omega_{MO}/\Omega_M = \sqrt{3}\,\pi$ and $\Omega_{EO} = f_c \approx 0.6869$ to
within $1\sigma$ of Planck, and the matter density $\Omega_m = 1-f_c \approx 0.3131$
---at $0.3\sigma$, without fitting. The cosmological background is derived from the
F1 Lagrangian as the elastic dynamics of the medium, with the Friedmann equation as
its linear limit, and its expansion history $H(z)$ fits the DESI BAO data as well as
the standard model with less freedom. The program reads the Hubble constant as a
\emph{positional} quantity ---the frequency of the Planck tick modulated by the age
of the universe in ticks---, which recasts the ``$10^{-123}$'' of the cosmological
constant problem as age in ticks rather than fine-tuning, and establishes that the
\emph{Hubble tension} is a method artifact that dissolves in model-independent
observables. Among its quantitative predictions is the time-dilation exponent of
type-Ia supernovae, $b=1$, which the program derives ---where the standard model
assumes it--- in agreement with the most precise measurement to $0.27\sigma$. It
Newton's three laws appear as the behaviour of a medium with a Lagrangian in the
low-energy limit, and what the programme adds is the origin of what Newton left primitive:
what mass is, where $G$ comes from, why inertia exists. The thermodynamics of the medium is
also native: the Bose-Einstein occupation and the coefficient $2\pi^2/45$ of the
Stefan-Boltzmann law are obtained by counting configurations of the eigenmodes of F1,
without assuming quantum statistics. It
derives gravity ---Newton in the weak field and Einstein's equations in the
relativistic regime--- as an emergent macroscopic property; it derives Maxwell
---and the non-abelian structure of Yang-Mills--- from the circulation of the phase
over the lattice; and it reads Special Relativity as the \emph{rheology} of a
dilatant fluid, where the Lorentz factor is the resistance of the medium to
deforming near its limiting speed. Further still: the incompatibility between
General Relativity and Quantum Mechanics dissolves, it is not forced, because both
are effective readings of the same substrate ---an integration already operative at
the quantitative level, whose combined confrontation the program declares as an open
thread.

PIU-$\Psi$ is not offered as a replacement for existing physics, but as an
explanatory layer beneath it. Newton, General Relativity and Quantum Mechanics work
in their validated regimes, and this program recovers them as correct effective
limits: what it proposes is \emph{why} they work. And it does so explicitly ---each
result carries its label of \texttt{\textbf{[DERIVED]}}, \texttt{[strong logical
deduction]}, \texttt{\textbf{[CALIBRATED]}} or \texttt{[hypothesis]}, and each open
thread is declared as such---, because a serious research program is measured by the
quality of its derivations and the candor about its own status: the boundary between
what is derived and what is open is declared with precision, and that precision is
part of the rigor, not a reservation about it.
That candor is accompanied by \emph{reproducibility}: all the constants and ratios
of the program are recomputed from their first-principles formulas by means of a
\textbf{verification notebook}, whose organisation into three layers ---what the medium
produces with no rod, the two numbers of the dictionary, and the reading in human units---
is set out in Annex \S{}V.9 and whose executable code (\texttt{verificar\_PIU.py}), together
with the results table it generates, accompanies the manuscript as supplementary material;
its execution reproduces the central quantities ---including $G$,
$\hbar$, $S_{\min}$, $S^*$, $f_c$, $E_0$, $\sqrt{3}\pi$ and the mass limits---
from their first-principles formulas, with no preloaded values. The complete mathematical development of each
derivation ---with all the intermediate steps--- is collected in a companion
document, the \textbf{Mathematical Compendium of PIU-$\Psi$}, which accompanies this
corpus.

The document is built in five Parts in strict order of ontological dependence:
el Pleno as a physical medium; its space of permitted configurations; the
phenomenological reading of the observed facts; the crunch-bounce cosmological cycle
and the formation of particles; and the quantitative confrontation with observation.
None is read in isolation: each Part operates on what the previous one established.

\clearpage
\begin{tcolorbox}[notabox,title={Status of the manuscript}]
The five Parts chain together in a logical sequence of dependence. What each one
establishes:

\textbf{Part I --- el Pleno as a medium.} The \textbf{genesis from A1 to F1} shows
how the whole apparatus hangs from the root axiom of finite, conserved energy
($E=Mc^2$ as a conversion factor, the Lorentz factor from the mass-shell constraint,
emergent covariance). From there the six axioms A1--A6, the F1 Lagrangian and the
reading of its parameters: \textbf{$G$ and $\hbar$ derived}, and the precision that
el Pleno is described by \textbf{two substantive properties and one causal
structure}. It includes the \textbf{emergence of Newtonian mechanics} as a
weak-field effective theory.

\textbf{Part II --- the space of configurations.} The \emph{Pleno}-ground state, the
diamond lattice, the mode spectrum and solitons as particles, with the closed forms
$S_{\min}^\text{estr} = 12(\pi^2-4)/\pi^4$ and $S_{\max,\text{geom}}^2 = (8/3)\sqrt{2/3}$.
The geometry of the lattice is \textbf{closed in its two pieces}: the tetrahedral
coordination $Z=4$ (D1-a, spherical 2-design) and the cubic stacking (D1-b, minimum
of the isotropic flexural functional, with fourth-order uniaxial anisotropy $4/81$
for wurtzite against $0$ for diamond).

\textbf{Part III --- phenomenological reading.} Gravity and \textbf{constants as
structural identities} (with the \textbf{geodesic} that closes the matter-geometry
pair and the \textbf{Schwarzschild metric} as a geometric reading of the static
configuration of the medium, with a sonic horizon and a regular core); gravitational
waves; the gauge sector (\textbf{Maxwell} from phase circulation with coefficient
$8/9$, non-abelian \textbf{Yang-Mills} as its extension, \textbf{confinement} as a
center obstruction $\mathbb{Z}_3$); the hadrons; the quantum reading
(\textbf{Heisenberg uncertainty} with an exact $\hbar/2$ floor, \textbf{Chandrasekhar}
with the complete coefficient $1.456\,M_\odot$, unitarity native to F1,
\textbf{integration of GR and QM} as the disappearance of the boundary); the
\textbf{cosmic microwave background and quantitative cosmology} (perfect blackbody,
nearly scale-invariant spectrum without inflation, $\Omega_m = 1-f_c$ at $0.3\sigma$
of Planck without fitting, the \textbf{cosmological background derived from F1} as
the elastic dynamics of the medium ---with Friedmann as its linear limit--- whose
$H(z)$ fits the DESI BAO data, and the \textbf{shared-factor theorem} that derives
redshift and time dilation from the same stretching of the medium, with the exponent
$b=1$ of type-Ia supernovae derived at $0.27\sigma$ of the datum); and el Pleno as a
Herschel-Bulkley \textbf{dilatant fluid}, where Relativity emerges as the rheology of
the medium. The hydrodynamic reading is completed with \textbf{incompressible
Navier-Stokes as the IR limit of F1} (the viscous term as the infrared imprint of the
flexural one), the \textbf{global regularity of the flow of el Pleno} (the A6 bound
plus divergent viscosity), the dissolution of the singularity under emergent time,
and the \textbf{critical point $S^*=\sqrt{S_{\min}S_{\max}}$} of minimal stiffness
between the two walls. The Hubble constant is read as a \textbf{positional quantity}
---$H_0=\sqrt{8\pi/3}\,(c/\ell_P)\sqrt{\Theta}$, with the ``$10^{-123}$'' of the
cosmological constant recast as the age of the universe in Planck ticks, and the
Hubble tension as a method artifact that dissolves in raw observables. Dark energy,
with exact $w=-1$, includes ---besides the homogeneous DE-Curtain--- the
\textbf{ponderomotive drag} of matter by the phase waves $\delta\theta$, derived from
the second order of the kinetic term of F1.

\textbf{Part IV --- the crunch-bounce cycle.} The dilatant rheology as the engine of
the cycle: the Gent wall and dilatancy regulate the bounce; the post-bounce
crystallization (latent heat $Z=12\to Z=4$) forms the three real particles (proton,
neutron, electron); the \textbf{phase freeze-out at $S^*$} (Kibble-Zurek) separates
dark energy (intra-domain coherent phase) from matter (boundary defects) and fixes
the positional status of $f_c$; the \textbf{viscous crystallization transient}
---the dynamics of the radial mode from $S^*$ to $S_{\min}$, integrated without a
free parameter and self-limited by the Planck bound--- is the event that
\emph{imprints} that partition in $\sim t_P$; the same potential gives the three
destinies of collapse (structure, black hole, Big Bounce), with primordial black
holes as incomplete crystallization; and the principle of the
interpretation-independent observable.

\textbf{Part V --- confrontation.} The catalogue of classified predictions (up to
P36, plus the photon series, the hadrons and $\Lambda_\text{PIU}$), the numerical
verification of $G$, $\hbar$ and the two cosmological ratios, the Bayesian selection
against the coincidence objection, and a transparent inventory of open threads that
distinguishes the \textbf{theory threads} (derivation gaps, absolute priority) from
the \textbf{confrontation threads} (submitting already-derived results to third-party
data, subordinate priority). It closes with two \textbf{dedicated appendices}: the
\textbf{Appendix \S{}V.7}, which derives the structure of the Dirac equation as the
effective theory of the phase modes over the diamond lattice ---by two convergent
routes, with the structure of the operator established as \epi{DERIVED}---; and the
\textbf{Appendix \S{}V.8}, on the ontological descent of the constants (Wien, Planck
and the reading of $G$ as the softness of the medium).

\smallskip
Each result carries its visible classification (\texttt{\textbf{[DERIVED]}},
\texttt{[strong logical deduction]}, \texttt{\textbf{[CALIBRATED]}}, \texttt{[hypothesis]}).
The numerical values follow the convention of \S\,I.1.3: computation on real values,
presentation to four figures and relative differences to two, with exact agreement by
construction for $G$ and $\hbar$ and residual $<0.1\%$ from rounding.
\end{tcolorbox}

\clearpage

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