% =====================================================================
%  PIU-Psi V5.1 -- COMPLETE MATHEMATICAL COMPENDIUM (English)
%  Full derivations, unabridged; no plots.
%  Epistemic classification in the title, after a double dash.
%  Manuel Alberto Celedon Mejia
%  Compile: pdflatex (x3)
% =====================================================================
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\newcommand{\Smax}{S_{\max}}
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\newcommand{\lP}{\ell_{P}}
% -- Epistemic classification: system unified with the V5.1 corpus (\epi, UPPERCASE)
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% short prose labels (inherited; now refer to the \epi system)
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% label for titles: status in uppercase in brackets, compacted
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% --- macros for the derivational-chain diagram (moved from the body) ---
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\begin{document}
\sloppy
\pagenumbering{roman}

\begin{titlepage}
  \centering
  \vspace*{0.8cm}
  {\color{cTronco!55}\rule{0.5\textwidth}{1.2pt}}\par
  \vspace{0.6cm}
  {\LARGE\bfseries Universal Integrity Principle\par}
  \vspace{0.35cm}
  {\Huge\bfseries PIU--$\Psi$\par}
  \vspace{0.5cm}
  {\LARGE\bfseries Complete mathematical compendium\par}
  \vspace{0.35cm}
  {\large Full derivations, computations and comparisons with observation\par}
  \vspace{0.5cm}
  {\color{cTronco!55}\rule{0.32\textwidth}{0.8pt}}\par
  \vspace{0.6cm}
  {\large Version 5.1\par}
  \vspace{0.2cm}
  {\large 30 July 2026\par}
  \vspace{0.6cm}
  {\large Manuel Alberto Celedón Mejía\par}
  \vspace{0.15cm}
  {\normalsize Mechanical Engineer --- Medellín, Colombia\par}
  \vspace{1.0cm}

  \begin{tcolorbox}[clave,title={One single thing}]
  PIU-$\Psi$ is a \textbf{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 that make up 95\,\% of the universe are not
  independent entities, but \textbf{configurations of a single physical
  substrate} ---el Pleno---, described by a single field $\Psi=S\,e^{i\theta}$ and six
  axioms. From two properties of the medium ($\rhoP,\lP$) and its causal structure $c$,
  it recovers $G$ and $\hbar$ with no fitted coefficient; it derives the
  Heisenberg uncertainty with its exact floor $\hbar/2$; it makes gravity emerge (Newton and the
  Einstein structure) and the Dirac equation; it obtains Maxwell and Yang-Mills from
  phase circulation; it predicts the cosmological ratios to within $1\sigma$ of
  Planck; it reads the Hubble constant as the time of a cosmic clock ---reformulating
  the cosmological-constant problem, placing the background value on the
  primordial side ($\sim67$, with the local $\sim73$ structurally excluded) and establishing
  the Hubble tension as a method artefact that dissolves in model-independent
  observables---; and it reads Relativity
  as the rheology of a dilatant fluid. General
  Relativity and Quantum Mechanics cease to be two theories to reconcile: they emerge as
  effective readings of the same medium. It statistically matches $\Lambda$CDM with
  \emph{derived} parameters, not fitted, and adds falsifiable predictions via LISA and
  Fermi-LAT. Each step carries its algebra and its epistemic classification. The invitation
  is to verify it.
  \end{tcolorbox}

  \vspace{0.5cm}

  \vspace{0.4cm}
  {\color{cTronco!40}\rule{\textwidth}{0.6pt}}\par
  \vspace{0.2cm}
  {\small\itshape https://www.piuniversal.com \,\textbullet\, Zenodo DOI 10.5281/zenodo.21724390\par}

  \clearpage

  \begin{tcolorbox}[nota,title={What this document is, and how to read it}]
  This compendium gathers \textbf{all the mathematical part} of the theoretical-physics
  research program PIU-$\Psi$ (V5.1) with the \textbf{complete, unabridged}
  derivations: every closed form, every
  coefficient and every numerical value is obtained by showing the intermediate
  algebra. The order is not
  thematic but \textbf{derivational}: each section may use only what the preceding ones have
  already derived, which is why the sequence is also the proof that there is no circularity. It
  falls into three stretches. Sections §1--§5 are the \textbf{trunk}: they run through the seven
  links that lead from the root statement ---the universe is finite, conserved energy--- to F1
  completely fixed, with the lattice, the potential and the constants; their closing boxes carry
  the link number, and the last one reads ``end of the foundational phase'' because from there on
  no new ingredient enters. Sections §6--§12 are the \textbf{harvest}: each evaluates the same F1
  in a different regime ---modes, gravity, the quantum, the gauge sector, particles, cosmology,
  rheology--- and none adds axioms, so they may be read in any order once the trunk is closed.
  Sections §13--§17 are the \textbf{apparatus}: the complete chain at a glance, the confrontation
  with measurement, the map of open ends, the index of symbols and the verification notebook.
  Within each section, all the mathematics of a single matter falls under a single title. The key computations ---averages over the
  Brillouin zone, the coefficients $1/40$ and $8/9$, the minimization of the potential and the
  closed form of $E_0$, the area of the configuration annulus, the configuration count that
  delivers the thermal occupation and the coefficient $2\pi^2/45$, the Born rule and the
  Tsirelson bound, the Lane-Emden equation, the topological baryon number--- have been
  \textbf{verified symbolically and numerically}. Where the
  program imports an external result or declares an open end, it is signaled
  explicitly and no derivation is simulated.

  \smallskip
  These verifications are \textbf{reproducible}, and with \emph{a single} notebook for the whole
  package: the script \texttt{verificar\_PIU.py} and its modules ---supplementary material---
  recompute each constant and ratio from its first-principles formulas, without preloaded
  values, and are the same ones that verify what the corpus reports. There are not two
  verifications that could diverge. The notebook is organised in \textbf{three layers}
  according to the order of appearance of the first measured magnitude ---what the medium
  produces with no measuring rod at all, the dictionary that costs exactly two numbers, and the
  reading in human units--- and it issues \textbf{two verdicts that must not be conflated}:
  \emph{document-to-code fidelity}, which checks that the computation reproduces every printed
  quantity and must pass in full, and \emph{agreement with measurement}, which applies only to
  the layer where there is something to measure. Appendix~\ref{ap:verificacion} collects the
  reading of that notebook for the quantities this compendium derives step by step, and §V.9 of
  the corpus sets out its full organisation. This compendium
  accompanies the \textbf{main corpus of PIU-$\Psi$}, which presents the conceptual development and
  the physical reading of each result; here the complete mathematical apparatus is collected.

  \smallskip
  The \textbf{epistemic classification} of each block appears in its \textbf{title},
  in brackets: \epi{DERIVED} = complete proof; \epi{SLD} = strong logical
  deduction (structural argument without a conceptual gap); \epi{CALIBRATED} = value anchored
  to a datum with declared error; \epi{HYPOTHESIS} = motivated conjecture. Titles with mixed
  status carry the combination. The label \epi{CALIBRATED} never means a free
  parameter fitted to force the agreement: it covers either a \emph{ruler calibration}
  ---el Pleno has no units, and fixing the value of $\rhoP$ or $\lP$ is to introduce the
  ``human ruler'' of measurement, the only act that allows comparison with observation
  (§4.4)---, or a \emph{positional state quantity}, whose form is derivable and
  whose present value is a historical datum, like $H_0$ in any cosmology (§11.9).
  This discipline ---always declaring what is proven, what
  is deduced and what is anchored--- is part of the method. The boxes follow a colour code: the
  \textbf{maroon} ones carry the physical meaning of a result ---what the algebra just closed
  actually says---; the \textbf{green} ones close a section or a link of the chain; the
  \textbf{blue} ones bound the scope of a claim or forestall an excessive reading; and the
  \textbf{orange} ones mark what remains open.
  \end{tcolorbox}

\end{titlepage}

\tableofcontents
\clearpage
\pagenumbering{arabic}

% =====================================================================
% BODY OF THE COMPENDIUM -- divided into 4 parts
% =====================================================================
\input{parte1}
\input{parte2}
\input{parte3}
\input{parte4}

% =====================================================================
% BIBLIOGRAPHY
% =====================================================================

\begin{thebibliography}{99}

\bibitem{planck2018} Planck Collaboration (Aghanim N.\ et al.),
\textit{Planck 2018 results. VI. Cosmological parameters},
Astronomy \& Astrophysics \textbf{641}, A6 (2020).

\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{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
(2017).

\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''
Variables. I}, Physical Review \textbf{85}, 166 (1952).

\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{krusch} Krusch S.,
\textit{Homotopy of rational maps and the quantization of Skyrmions},
Annals of Physics \textbf{304}, 103 (2003).

\bibitem{gudnasonnitta} Gudnason S.\ B., Nitta M.,
\textit{Baryonic toroidal crystals and the Hopf charge}, Journal of High Energy
Physics \textbf{2020}, 4 (2020).

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

\bibitem{yukalov2007} Yukalov V.\ I.,
\textit{Bose-Einstein condensation and gauge symmetry breaking}, Laser Physics
Letters \textbf{4}, 632--647 (2007).

\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}

\end{document}
