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\section*{Part I --- el Pleno as a physical medium}
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\textbf{Purpose of Part I.} To establish the interpretive apparatus of the program: what \emph{el Pleno} is, what dynamical laws it obeys and what structural properties characterize it, as the single physical medium of which the subsequent phenomenology is read as a configuration.

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\medskip\hrule\medskip

\subsection{\S{}I.1 --- Epistemological position and foundational statement}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (non-derivational section)
  \item \textbf{Notes:} opening section of the manuscript. Contains the foundational statement and the humble epistemological position, with the ontological reading of \emph{el Pleno} as a single substrate.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.1.1 --- The foundational statement}

The universe and its dynamics exist independently of any theory that describes them. Physics is the effort to understand that real universe through observation and mathematics, and PIU-$\Psi$ is a proposal within that effort: it does not claim to be the universe, but to offer a coherent reading of it, testable against observation.


The usual image of the cosmos presents it as a fundamentally empty stage where things happen ---particles that travel, fields that propagate, forces at a distance, geometry that curves---, presupposing a primary separation between the locus of phenomena and the phenomena themselves. PIU-$\Psi$ rejects that separation: it describes the entire universe as a single continuous physical substrate, with no vacuum, no gaps and no external agents. That substrate ---\emph{el Pleno}--- is the only thing there is, and everything else ---what is seen and what is not, what moves and what remains--- are configurations, modes or internal excitations of it. A particle is \emph{el Pleno} organized in a certain way; a gravitational wave, \emph{el Pleno} propagating a tensor excitation; a photon, a configuration of its internal dynamical sector. Matter and energy are not distinct ontological categories but names of configurations of \emph{el Pleno} in different states, interchangeable by special relativity: what is measured as mass or as energy is, in both cases, \emph{el Pleno}. Dark matter and dark energy are not hidden substances but structural states of the same \emph{el Pleno} that occupy the whole volume without manifesting as a visible object, and the space between two bodies is not empty: it is \emph{el Pleno} at minimal excitation.


\emph{El Pleno} is not an undifferentiated substrate, but a physical system with internal order: it obeys precise dynamical laws, evolves from cosmological initial conditions and admits characterization by measurable magnitudes. Its properties are described in the language of continuous media ---fundamental density, microscopic scale, propagation speed of its modes, rigidity against perturbations, infrared support wall, hard ultraviolet bound---, which are not analogies but the structural properties of the single substrate of the universe. From them falsifiable quantitative predictions are derived, and it is by those predictions, tested against observation, that PIU-$\Psi$ stands or is refuted.


The state of \emph{el Pleno} at each point is denoted $\Psi$, and the whole program consists in deriving from its structural properties and from the dynamics of $\Psi$ the observable consequences that physics records ---gravity, fundamental constants, particles, cosmology, dark matter and energy---, none of them an independent entity: all of them configurations of the same $\Psi$.


\subsubsection{\texorpdfstring{I.1.2 --- Triple validation: how PIU-$\Psi$ will be evaluated}{I.1.2 - Triple validation: how PIU-Psi will be evaluated}}

PIU-$\Psi$ is not validated by assertion of its principles; it is validated by its performance against three simultaneous questions:


\textbf{To interpret.} Does PIU-$\Psi$ offer a coherent ontological reading ---consistent, not arbitrary--- of what particles, fields, gravity and the dark sectors are? Interpretation is not philosophical decoration on top of the mathematics: it is what gives physical meaning to the equations. A theory without it can predict correct numbers without one understanding what they represent; PIU-$\Psi$ aspires to comprehension, not only computation.


\textbf{To explain.} Does PIU-$\Psi$ derive the observed properties from more basic principles, in reconstructible chains? This distinguishes a fundamental theory from a phenomenological fit: describing by parametrizing with calibrated coefficients is descriptive modelling; to explain is to show \emph{why} things are as they are, not only how. PIU-$\Psi$ aspires to explain ---deriving, for example, the fundamental constants from properties of \emph{el Pleno} rather than treating them as inputs.


\textbf{To predict.} Does PIU-$\Psi$ anticipate future observations or configurations not yet measured? This is the Popperian dimension: a theory must be falsifiable, must risk predictions that could turn out wrong. Predictive capacity is what subjects the program to empirical control and gives it scientific, not speculative, status.


The three capacities are distinct and necessary, and none suffices on its own: a theory can interpret well and predict badly, predict without interpreting (the ``shut up and calculate'' position of \S{}I.2), or explain retrospectively without predicting. PIU-$\Psi$ aspires to all three at once, explicit about the degree of closure of each. Every result carries its visible classification in the body of the text, not in footnotes: \texttt{\epi{DERIVED}} (complete proof), \texttt{\epi{SLD}} (structural argument without explicit algebra), \texttt{\epi{CALIBRATED}} (value anchored to a datum with declared error), \texttt{\epi{HYPOTHESIS}} (motivated conjecture, not proven). The \epi{CALIBRATED} label covers two situations that should not be conflated, neither of which is a fitted free parameter: a \emph{ruler calibration} (dimensional anchoring), where the value of a substantive property of el Pleno ($\rho_P$ or $\ell_P$) is fixed by introducing the human unit of measurement ---the unavoidable act of injecting the ``human ruler'' in order to compare with observation, since el Pleno operates in pure ratios and has no units of its own (\S{}I.8.9)---; and a \emph{positional state quantity}, where the \emph{form} of the magnitude is derivable but its \emph{present} value encodes the position of the medium along its trajectory $\bar S(t)$ ---like $H_0$ in any cosmology--- and is therefore not a timeless number to be derived but a positional observational datum (\S{}III.8.7). In no case does \epi{CALIBRATED} mean a free parameter tuned to force agreement.


\subsubsection{I.1.3 --- Numerical and presentation convention}

To guarantee the consistency of all values and comparisons throughout the
manuscript, the program adopts a uniform convention of computation and
presentation:

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Computation over real values.} Every derived quantity is
  computed from the primary parameters $(\rho_P, c, \ell_P)$ at their full
  precision. Comparisons with observational values ---including every
  percentage relative difference--- are computed over the \emph{real computed
  value}, never over its rounded presentation form. This avoids intermediate
  rounding bias: rounding first and comparing afterwards introduces a spurious
  error in the agreement figure.
  \item \textbf{Presentation of values to four decimal places.} Every numerical
  value is presented with four decimal places (in the mantissa, for quantities
  in scientific notation), rounding the last digit.
  \item \textbf{Presentation of percentages to two decimal places.} Every
  percentage relative difference is computed over the real value and presented
  rounded to two decimal places.
  \item \textbf{Intermediate steps.} When a step-by-step computation is shown,
  the intermediate factors are displayed to four figures as a reading aid, but
  the final reported result always comes from the full-precision computation, so
  that the presented value and its agreement percentage are mutually consistent.
  \item \textbf{Residue and origin.} When an identity is exact by construction
  (for example $G$ and $\hbar$ under $\ell_P = \ell_\text{Planck}$), the
  numerical residue reported reflects only the rounding of the primary
  parameters to four significant figures, and this is declared explicitly in
  each case.
\end{itemize}

This convention holds without exception in Parts I--IV and in the appendices.


\subsubsection{I.1.4 --- Why explore an alternative}

A basic question is in order: why explore an alternative theory, if General Relativity and Quantum Mechanics are enormously successful? The answer is not rhetorical but factual. Modern physics, for all its success, faces deep open problems that have resisted decades of effort and remain without an accepted solution ---recognized by the community itself, not inventions of this program. It is they that make the search for alternative frameworks scientifically legitimate.


\medskip\hrule\medskip

\subsection{\S{}I.2 --- Why it is legitimate to formulate an alternative: open conflicts of contemporary physics}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (non-derivational section)
  \item \textbf{Notes:} cognitively prepares the reader for \S{}I.3 (A priori motivation for \emph{el Pleno}). Every reference to external literature on the contemporary crisis is marked with \texttt{[REF-WEB-XX]} and resolved in \S{}V.5.8; the academic sources of the derivational apparatus are cited in place and consolidated in \S{}V.5.7.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.2.1 --- Opening: success and recognized problems of modern physics}

Modern physics is one of the most extraordinary intellectual achievements of humanity. Its founding is the work of Isaac Newton, whose classical mechanics ---the three laws of motion and the law of universal gravitation--- established for the first time that the heavens and the earth obey the same mathematical laws, and governed the description of motion without exception for more than two centuries; it remains today the language of engineering and the base on which everything else was built. Einstein's General Relativity describes gravitational behaviour at scales ranging from the solar system to the observable universe, with precision verified experimentally in numerous contexts: the bending of light by the Sun, the precession of Mercury's perihelion, the gravitational time dilation confirmed by GPS, the direct detection of gravitational waves by LIGO and Virgo since 2015, the image of the M87\textbackslash{}* black hole by the Event Horizon Telescope in 2019. Quantum Mechanics and its extension to Quantum Field Theory describe the behaviour of the subatomic world with unparalleled precision: the anomalous magnetic moment of the electron is predicted and measured to an agreement of twelve significant figures. The Standard Model of particle physics, completed with the detection of the Higgs boson in 2012 at the Large Hadron Collider, organizes the catalogue of known elementary particles and their electromagnetic, weak and strong interactions into a coherent structure. And the standard cosmological model $\Lambda$CDM, calibrated against data from the cosmic microwave background and large-scale structure, describes with remarkable precision the evolutionary history of the universe from the second after the Big Bang to today.


Any proposal that aspires to contribute to the understanding of the cosmos must start from an honest recognition of these successes. PIU-$\Psi$ is no exception. This section is not a destructive critique of established physics nor a claim that current theories are incorrect in the regimes where they are validated; it is \textbf{recognition of the real state of the field}: a modern physics that is enormously successful and that, at the same time, faces six deep open problems, all recognized by the community itself, none resolved. They are not inventions of this program; they are field consensus, and their persistence justifies that searching for alternative frameworks is scientifically healthy.


\subsubsection{I.2.2 --- Problem 1: The cosmological constant and the vacuum energy}

The first open problem, and possibly the most severe of modern physics, is the discrepancy between the vacuum energy predicted by Quantum Field Theory and the observational value of the cosmological constant.


In Quantum Field Theory, the ground state ---the ``quantum vacuum''--- is not empty: it contains zero-point fluctuations in every mode. Its energy, integrated up to a natural cut-off scale (the Planck scale, or more conservatively the supersymmetry-breaking scale), produces a theoretical vacuum energy density. Cosmological observations ---acceleration of the universe (Ia supernovae, 1998), confirmed by the cosmic microwave background and the baryon acoustic oscillations--- indicate instead a density associated with the cosmological constant $\Lambda$ of order $\rho_{\Lambda}^{\text{obs}} \approx 5.96 \times 10^{-27}$ kg/m$^{3}$ \texttt{[REF-WEB-01]}.


The theoretical prediction with a cut-off at the Planck scale exceeds the observed value by some \textbf{120 orders of magnitude}; in the most conservative versions the discrepancy drops to \textbf{60 orders} \texttt{[REF-WEB-01]}, and expressed on an energy scale to \textbf{30 orders} \texttt{[REF-WEB-04]} ---a magnitude without parallel in any other discipline. Hence it is described as \textit{``the worst theoretical prediction in the history of physics''} \texttt{[REF-WEB-01]}: not hyperbole, but a technical observation that no other theory in use disagrees with experiment by 60 to 120 orders of magnitude.


The proposed solutions ---supersymmetry, anthropic principles, fine-tuning, modified gravity, string landscapes--- have been explored for five decades without resolving the question: \textit{``no known symmetry requires the cosmological constant to vanish''} \texttt{[REF-WEB-02]}, and the problem remains as the principal obstacle to a complete theory of quantum gravity and cosmology. It is not a mere numerical disagreement, but a structural symptom: the very notion of ``vacuum'' in Quantum Field Theory, coupled to Einsteinian gravity, is not ontologically coherent. Something, in the current image of the universe, is fundamentally misunderstood.


\subsubsection{I.2.3 --- Problem 2: The dark sectors and the 95\% of the unknown universe}

Cosmological observations indicate that the energy content of the universe in its present state is distributed, under the $\Lambda$CDM model, among three components with approximate proportions \texttt{[REF-WEB-05]}:


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Ordinary (baryonic) matter:} approximately 5\%. It is all the matter described by the Standard Model of particle physics --- atoms, nuclei, electrons, neutrinos. Everything seen directly with telescopes, everything measured in the laboratory, all the matter of which human beings, planets and stars are made.
  \item \textbf{Dark matter:} approximately 27\%. A component whose existence is inferred from gravitational effects on multiple scales: galaxy rotation curves, dynamics of galaxy clusters, weak gravitational lensing, large-scale structure formation, cosmic microwave background anisotropies. It does not emit, absorb or reflect detectable light. Its physical origin remains unknown despite decades of direct experimental searches (XENON, LUX, CDMS, LZ) and indirect ones (search for annihilation in cosmic rays), all with negative results to date.
  \item \textbf{Dark energy:} approximately 68\%. A component responsible for the observed acceleration of cosmic expansion. Under $\Lambda$CDM it is identified with a cosmological constant of small positive value. Its physical origin is the problem described in \S{}I.2.2.
\end{itemize}

The situation, in precise terms: \textbf{95\% of the energy content of the universe is described by categories whose physical origin is unknown}. The $\Lambda$CDM model parametrizes both components ---dark matter as massive non-interacting particles, dark energy as a constant $\Lambda$--- calibrating them against observation without explaining what they are. Modern physics, for all its success, \textbf{directly describes only 5\% of the universe}; the remaining 95\% are operational labels that recognize a phenomenological need without identifying its physical referent \texttt{[REF-WEB-06]}.


Both categories face, moreover, growing internal problems. The WIMPs that dominated three decades of searching have produced no direct detection, to the point that the community speaks of the \textit{``twilight of the WIMP''} \texttt{[REF-WEB-07]}; the alternative candidates (axions, light or interacting dark matter) have not yielded positive evidence either. Dark energy, for its part, shows in DESI 2024-2025 preliminary evidence of \textbf{temporal dynamics} that $\Lambda$CDM with fixed $\Lambda$ does not predict: the DR2 data combined with CMB and supernovae indicate a probability above 95\% that its equation of state evolves with time \texttt{[REF-WEB-08]}. If confirmed, the cosmological constant is not constant, and the question of what entity produces cosmic acceleration reopens.


In synthesis: \textbf{a component sought for 40 years without direct detection, and another with growing indications of a dynamics that breaks the model under which it was postulated}.


\subsubsection{\texorpdfstring{I.2.4 --- Problem 3: Growing cosmological tensions (H$_{0}$, S$_{8}$)}{I.2.4 - Problem 3: Growing cosmological tensions (H0, S8)}}

Beyond the unknown components, $\Lambda$CDM faces growing internal discrepancies between measurements of the early universe and those of the late one. The two largest are the \textbf{Hubble tension (H$_{0}$)} and the \textbf{S$_{8}$ tension}.


\textbf{Hubble tension.} The current expansion rate $H_0$ can be measured in two ways that, under $\Lambda$CDM, should coincide. From the \textit{early universe}, Planck (CMB anisotropies extrapolated) obtains $H_0 = 67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ and DESI (BAO + CMB) $68.53 \pm 0.80$ \texttt{[REF-WEB-03]}. From the \textit{late universe}, SH0ES (Cepheid distance ladder + Ia supernovae) obtains $73.04 \pm 1.04$ \texttt{[REF-WEB-09]}, confirmed by JWST in the range $72$--$73$ ($72.6 \pm 2.0$ with JWST calibrators) \texttt{[REF-WEB-10]}.


The discrepancy is about $5\sigma$ ---less than 1 in 3.5 million probability of being a fluctuation--- and recent analyses (2025-2026) independent of the distance ladder raise it to $5.8\sigma$ \texttt{[REF-WEB-11]}. The hypothesis of a systematic error in the Cepheids was tested with JWST and refuted \texttt{[REF-WEB-03]}. Far from dissolving, the tension \textit{``has become more robust and harder to dismiss''} \texttt{[REF-WEB-12]}.


\textbf{S$_{8}$ tension.} The parameter $S_8 \equiv \sigma_8 \sqrt{\Omega_m / 0.3}$ measures the amplitude of matter fluctuations at $8 h^{-1}$ Mpc. The CMB extrapolated under $\Lambda$CDM predicts one value; direct measurements of the late universe (weak lensing from KiDS, DES, HSC; redshift-space distortions; cluster abundance) find a smaller one by $2$--$3\sigma$ \texttt{[REF-WEB-13, REF-WEB-14]}: the observed structure is less pronounced than that extrapolated from the CMB.


Like the H$_{0}$ tension, the S$_{8}$ tension has persisted across multiple independent experiments over several years. The 2025 report \textit{``Cosmological tensions in the era of precision cosmology''} concludes that \textit{``the S$_{8}$ tension, far from fading, remains statistically significant when the weak-lensing data from DES and HSC are combined''} \texttt{[REF-WEB-12]}.


\textbf{Synthesis of problem 3.} The $\Lambda$CDM model, calibrated against observations of the early universe, fails to predict simultaneously the current expansion rate and the amplitude of structure in the late universe. The two tensions could reflect unidentified systematic errors, or could indicate physics beyond the standard model. The current state of the field does not allow resolving between the two possibilities; but the persistence and growing robustness of the tensions, with each new data set, justifies exploring alternative frameworks that offer possible routes to resolution.


\subsubsection{I.2.5 --- Problem 4: Structural incompatibility between General Relativity and Quantum Mechanics}

The two fundamental theories of modern physics --- General Relativity for gravity and large-scale geometry, Quantum Mechanics (and its extension to Quantum Field Theory) for the subatomic world --- are individually successful but \textbf{fundamentally incompatible} in their current formulation.


The incompatibility is structural. General Relativity describes gravity as the dynamical geometry of space-time: masses and energies curve space-time, and space-time directs the motion of masses and energies. Space-time is a continuous, smooth, classical (non-quantum) entity. Quantum Mechanics, in contrast, operates on a fixed background space-time and describes the material fields as quantized entities, with superposition, indeterminacy and discreteness.


On quantizing gravity by the standard methods of Quantum Field Theory ---treating the gravitational field as one more, with a graviton analogous to the photon--- the procedure fails: the resulting perturbative theories are \textbf{non-renormalizable}, with divergences that are not absorbed into a finite number of parameters. The difficulty, established by 't Hooft and Veltman in the 1970s, has not been overcome in five decades within canonical quantization. The alternative programs ---string theory, loop quantum gravity, spin foams, asymptotic safety--- have produced elegant constructions and partial results, but none reaches the status of a complete, verified and accepted theory; string theory, in particular, faces the ``landscape problem'': of order $10^{500}$ possible vacua with no internal criterion to select ours.


The implications are severe: where the curvature is comparable to the quantum scales ---black hole interiors, primordial instants, the Planck scale--- \textbf{there exists no accepted theory that describes gravity and quantum effects at once}. General Relativity predicts singularities that Quantum Mechanics indicates should not exist, and the unified theory that resolves the tension remains open. As the literature summarizes, \textit{``at high energies General Relativity predicts its own collapse into singularities''} \texttt{[REF-WEB-04]}, while at low energies dark energy drives an accelerated expansion: problems at both ends of the scale, with no framework connecting them. This incompatibility has resisted fifty years of dedicated effort.


\subsubsection{I.2.6 --- Problem 5: The fundamental constants are not explained}

Modern physics, in its standard formulation, contains a set of \textbf{fundamental constants} whose numerical values are not derived from more basic principles. They are observational inputs: they are measured, included in the theories, and used to make predictions. But they are not explained.


The most fundamental constants include:


\begin{itemize}[leftmargin=1.4em]
  \item Newton's gravitational constant, $G \approx 6.674 \times 10^{-11}$ m$^{3}$ kg$^{-1}$ s$^{-2}$. It measures the strength of the gravitational interaction. General Relativity assumes it as a free parameter.
  \item The reduced Planck constant, $\hbar \approx 1.055 \times 10^{-34}$ J$\cdot$s. It quantifies the minimal scale of action in quantum mechanics. Quantum Mechanics assumes it as a free parameter.
  \item The speed of light in vacuum, $c \approx 3 \times 10^8$ m/s. A universal limiting speed. Special Relativity assumes it as a structural property of space-time, without explaining why it has that specific value.
  \item The fine-structure constant, $\alpha \approx 1/137.036$. Dimensionless. It characterizes the strength of the electromagnetic interaction. The Standard Model assumes it as a free parameter.
  \item The masses of the fundamental fermions (electron, quarks, neutrinos), the mixing angles, the coupling constants of the fundamental forces. The Standard Model contains approximately \textbf{19 free parameters} whose values are determined experimentally, not derived.
\end{itemize}

This state is recognized as a deep problem: no accepted theory explains the constants, it only measures and uses them. \textit{``Why does $G$ have this value and not another?''} or \textit{``why is the electron mass what it is?''} have no answer in the standard framework. The existing proposals ---grand unification, extra dimensions, anthropic principles, multiverse--- have reached neither consensus nor verification. A theory that assumes as non-derived inputs the very quantities it intends to describe is, strictly speaking, a \textbf{phenomenological theory of fitting}: it relates observables to observables without explaining the origin of the initial ones. The aspiration of fundamental physics, since Newton, has been to reduce the number of unexplained constants; with about twenty still underived, that aspiration has not been fulfilled, and any framework that derives even a few from structural principles addresses a recognized problem.


\textbf{How PIU-$\Psi$ addresses it.} The program characterizes \emph{el Pleno} with a minimal set of structural parameters ---two substantive properties ($\rho_P$, $\ell_P$) and one causal structure ($c$)--- and derives from them the other constants as consequences, instead of taking them as inputs. The structural identities for $G$ and $\hbar$ (which reproduce the measured values with unit coefficient, not fitted) and the dimensionless values of the medium (with no free parameters) are established in place: $G$ in \S{}I.7.5, $\hbar$ in \S{}I.8.5, the geometric constants in \S{}I.8.8. The natural objection ---whether those agreements could be a chance numerological coincidence--- is one the programme poses to itself and submits to computation: fifteen results read with fifteen instruments that share no measurement mechanism give a Bayes factor against the coincidence hypothesis of $3.4\times 10^4$ under the assumption of strong dependence between results, and above $65$ even under the most punitive assumption. The full development, with the admission criteria and with the twelve results the programme itself leaves outside the computation ---left outside for contributing no independent evidence to that count, not for being incorrect---, is in \S{}V.2.6.


\subsubsection{I.2.7 --- Problem 6: The ontological status of quantum fields and the vacuum problem}

The sixth problem is of a different nature from the previous five. It is neither a numerical discrepancy nor an observational tension: it is an \textbf{open ontological question} about the very nature of the entities that Quantum Field Theory manipulates.


Quantum Field Theory, in its standard operational formulation, treats quantum fields as mathematical entities that produce correct experimental predictions. That operational formulation has been extraordinarily successful: the anomalous magnetic moment of the electron, the details of the hydrogen spectrum, the properties of the Higgs boson, the cross-sections of high-energy processes --- all are predicted with outstanding precision.


But the ontological question remains open: \textbf{on what do quantum fields act? What are they made of?}


In the standard formulation, the quantum field is postulated as a fundamental entity. The electromagnetic field, for example, is not a vibration of something: it is directly an abstract quantum entity whose excitation is called a ``photon''. This position is known as the ``operational interpretation'' of Quantum Field Theory, and is summarized by the phrase attributed to several physicists: \textit{``shut up and calculate''} --- do not ask what the fields are, just use them to predict experimental results.


This operational position is legitimate as scientific pragmatics, but it leaves deep physical questions unanswered. The fluctuations of the quantum vacuum --- which are experimentally real, manifest in the Casimir effect and the Lamb shift --- fluctuations of what? If the quantum vacuum is structured, what is it that structures it? The very existence of universal constants such as the speed of light, which has the same value for all observers everywhere in the universe, suggests the presence of a common substrate; but the standard formulation does not identify that substrate.


This ontological gap connects with problem 1 (cosmological constant): the difficulty of correctly computing the vacuum energy comes, at least in part, from not having clarity about what the vacuum is. It connects with problem 4 (GR-QM incompatibility): the difficulty of quantizing gravity comes in part from not having clear what is being quantized --- whether space-time, the gravitational field, or something more fundamental underlying both.


The field literature recognizes this gap. Programs such as \textbf{emergent gravity} (Verlinde, Padmanabhan, others), the theories of \textbf{emergent space-time} (Jacobson, Carlip), the models of \textbf{structured vacuum} (Penrose, Volovik with superfluid analogues, others), all address, from different angles, the question of what underlies the entities that the standard theories manipulate.


None of these programs has reached consensus or decisive experimental verification. But their existence and persistence, over more than two decades of intense research activity, indicates that the question about the ontological status of quantum fields \textbf{is recognized as a legitimate open question of the field}, not mere philosophical speculation. Exploring frameworks based on a physical substrate underlying the quantum fields is, then, within the accepted research program of contemporary theoretical physics.


\subsubsection{I.2.8 --- Synthesis: what these problems have in common}

The six problems articulated in the previous sections are of diverse natures: catastrophic numerical discrepancy (problem 1), unknown components of the cosmic content (problem 2), growing internal tensions with recent data (problem 3), structural incompatibility between fundamental theories (problem 4), unexplained fundamental constants (problem 5), ontological gap about the nature of quantum fields (problem 6).


However, a careful reading reveals a common pattern. The six problems share, in different forms, \textbf{an ontological root}: the difficulty of modern physics in articulating \textbf{what} the thing it studies \textbf{is}, beyond \textbf{how it behaves}.


\begin{itemize}[leftmargin=1.4em]
  \item The cosmological constant is problematic because it is not understood what the vacuum is from which its energy is computed.
  \item The dark sectors are problematic because it is not known what they are, only their presence is inferred from gravitational effects.
  \item The cosmological tensions may indicate physics beyond $\Lambda$CDM, but that ``beyond'' does not yet have a defined ontological form.
  \item The GR-QM incompatibility is hard to resolve in part because there is no agreement on what the space-time is that both theories describe so differently.
  \item The fundamental constants are not explained because they have not been derived from a more basic ontology that contains them.
  \item The ontological status of quantum fields is directly the question of what entities underlie the successful mathematics.
\end{itemize}

This common reading suggests that the six problems may not be independent. They may be \textbf{symptoms of a single structural lack}: the absence, in modern physics, of a unified physical ontology that articulates what the universe is, of which the different successful theories are regional and effective readings.


This observation does not constitute a proof. It is not claimed here that ontological unification resolves all the problems. It is claimed that \textbf{one of the scientifically legitimate directions of search, in the face of this set of open problems, is the exploration of frameworks that postulate such a unified ontology and derive the standard theories as effective limits}.


\subsubsection{\texorpdfstring{I.2.9 --- Honest positioning of PIU-$\Psi$}{I.2.9 - Honest positioning of PIU-Psi}}

PIU-$\Psi$ is one of those explorations. Not the only one, nor necessarily the best; one possible one.


The PIU-$\Psi$ program inscribes itself within the broader effort of contemporary theoretical physics to confront the problems articulated in the previous sections. Its specific contribution consists in proposing a single physical substrate --- \emph{el Pleno}, described formally by the complex scalar field $\Psi$ --- from which gravity, quantum mechanics, fundamental constants, particles, electromagnetism, cosmology and dark sectors are derived as distinct readings of the same underlying object.


PIU-$\Psi$ \textbf{is not offered as a replacement for existing physics}. A criterion operates here that holds for \emph{any} foundations framework, not only this one: a theory that aspires to be more basic than Newtonian mechanics, General Relativity or Quantum Mechanics is obliged to recover them as effective limits in the regimes where they are already experimentally validated ---otherwise it would contradict the evidence and be immediately refuted. Recovering known physics is therefore not a concession nor a redundancy: it is a requirement of admissibility. What PIU-$\Psi$ adds on top of that requirement is to articulate \textbf{an underlying ontology} from which those theories emerge as regional effective descriptions, and to explain \emph{why} each one works in its domain.


PIU-$\Psi$ \textbf{does not claim to resolve all the problems listed in this chapter}. Its research program declares explicitly which problems it addresses (all six, to varying degrees), which consolidated results it offers (structural cosmological predictions with zero free parameters, structural parity with QCD in the hadronic sector with one observational input, structural identities for G and $\hbar$, among others), and which problems remain open as declared research threads (UV completeness, full quantitative confrontation with DESI and Pantheon+, geometric uniqueness of the solution space, among others).


PIU-$\Psi$ \textbf{submits to the triple validation articulated in \S{}I.1}: its value depends on its combined capacity to interpret, explain and predict, and on its degree of closure in each. Before constructing the formal apparatus, it remains to be shown that postulating \emph{el Pleno} as a unifying substrate is not arbitrary: there are a priori physical arguments that make it reasonable ---not as an absolute claim, but as a hypothesis that the rest of the manuscript develops and confronts with observation.


\medskip\hrule\medskip

\subsection{\S{}I.3 --- A priori motivation for postulating el Pleno}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (a priori argumentation section, non-derivational)
  \item \textbf{Notes:} section that articulates the specific physical arguments that make it reasonable to postulate a continuous physical substrate as the candidate starting point of the program. Positive construction, complementary to \S{}I.2 (which articulated why to search for alternatives). Prepares \S{}I.5 (structural axioms).
\end{itemize}
\end{estadoseccion}

\subsubsection{I.3.1 --- From the legitimacy of searching to the reasonableness of postulating}

The previous section justified \emph{searching} for alternatives, without pronouncing on what kind of alternative to explore. PIU-$\Psi$ postulates a single continuous physical substrate ---\emph{el Pleno}--- from which all the observed configurations emerge. The question of this section is: given that searching is legitimate, \textbf{are there specific physical reasons that make it reasonable to postulate precisely a continuous physical substrate?}


The answer is affirmative, sustained by six convergent physical arguments. None, in isolation, demonstrates the existence of \emph{el Pleno}; but their cumulative convergence establishes that the single-substrate hypothesis is not arbitrary speculation but a natural response to known physical observations. They are articulated from the most empirically established to the most conceptual.

\textbf{On the direction of the reasoning.} These arguments start from already-validated equations and facts ---$E = mc^2$, the constancy of $c$, gravitational waves--- and climb back toward the postulate that would make them natural: the reasoning here is inductive, from the known to the substrate that would explain it. Once the postulate is adopted, the direction reverses and the same set of facts is recovered as a consequence. That a physical fact appears first as evidence for postulating \emph{el Pleno} and afterwards as a consequence of having postulated it is not circularity: it is the mark of a well-chosen postulate, whose virtue is measured by the naturalness with which it reproduces what motivated its adoption.


\subsubsection{I.3.2 --- Argument 1: Mass-energy equivalence implies a common physical denominator}

The first argument starts from one of the most established results of modern physics: mass-energy equivalence, expressed by the relation $E = mc^2$ obtained by Einstein in 1905 within Special Relativity. This relation is not speculative theory: it is experimentally confirmed in countless contexts, from nuclear reactions to particle accelerators, from energy generation in stars to everyday processes in medical physics.


The equivalence has an ontological implication that is rarely articulated explicitly. If mass and energy can convert into one another completely and reversibly, then \textbf{they are not distinct ontological categories}. They are not two different substances that coexist: they are two distinct readings of the same underlying physical quantity. Mass is a specific form of manifestation; energy is another form of manifestation; but both are aspects of one and the same common denominator.


Under the usual image of the cosmos --- vacuum + independent things ---, this equivalence is presented as an operational fact without ontological commitment: things with mass can convert into other things with energy and vice versa, and that is all. But the equivalence is deeper than a mere fact of exchange: it indicates that \textbf{what is called ``matter'' and what is called ``energy'' are configurations of the same underlying physical object}.


The hypothesis of \emph{el Pleno} is a natural reading of this observation. If matter and energy are distinct readings of something common, that common something admits being thought of as an underlying physical substrate. PIU-$\Psi$ does not invent this ontological reading: it articulates it explicitly and turns it into the program's starting point. What is called ``matter'' is \emph{el Pleno} in a specific configuration; what is called ``energy'' is \emph{el Pleno} in another configuration; the equivalence $E = mc^2$ is the natural consequence of both being configurations of the same substrate.


This argument does not demonstrate the existence of \emph{el Pleno}. But it establishes that \textbf{postulating a common physical substrate to which mass and energy reduce is coherent with one of the most established results of modern physics}. The hypothesis does not contradict Special Relativity --- it reads it ontologically. At this inductive level it suffices to note that coherence; the deductive unfolding, in which $E = Mc^2$ is obtained as a consequence instead of taken as a datum, arrives with the genesis of \emph{el Pleno}.


\subsubsection{I.3.3 --- Argument 2: The universal constancy of c implies a common isotropic substrate}

The speed of light in vacuum, $c \approx 3 \times 10^8$ m/s, has a notable property: \textbf{it is the same for all observers everywhere in the universe}, independently of their state of motion. This is the basis of Special Relativity, derived by Einstein from the analysis of the Michelson-Morley and other experiments: the speed of light does not depend on the observer, does not depend on the direction, does not depend on the position.


Contemporary cosmological observations confirm this property at extraordinary scales. The electromagnetic radiation arriving from galaxies at distances of billions of light-years propagates at the same speed as light emitted in a terrestrial laboratory. The electromagnetic laws, in which the speed c is a fundamental parameter, operate identically in regions of the universe observed billions of years ago and in present laboratory conditions.


This constancy also has an ontological implication that the standard formulation does not articulate. \textbf{A universal speed can only be such if there exists a common isotropic substrate throughout the universe that sustains it}. If the universe were fundamentally empty with independent things, there would be no reason for the propagation speed of electromagnetic radiation to be the same everywhere --- independent things do not share structural properties by themselves. That c is universal indicates that there is \textbf{something physical common to all of space} whose structural properties determine that speed. Under the standard image, that something is called ``space-time'' and is assigned the status of an abstract geometric entity. Under PIU-$\Psi$, that something is called \emph{el Pleno} and is assigned the status of a continuous physical medium with defined structural properties --- c being one of those structural properties.


The difference between the two readings decides the ontological status of the substrate. Calling the underlying entity ``space-time'' and assigning it geometric properties does not answer the question: what is space-time? What is it made of? General Relativity operates on that entity without providing its ontology. PIU-$\Psi$ offers a candidate answer: the underlying entity is a physical medium (\emph{el Pleno}), whose structural properties produce c as the propagation speed of its modes. The universal constancy of c becomes thus, in the PIU-$\Psi$ reading, a natural consequence of the existence of a common physical substrate --- exactly analogous to the speed of sound in air, which is the same for all observers at rest relative to the air, because it is determined by the structural properties of the medium.


This second reason reinforces the first. If mass-energy equivalence suggests a common physical denominator, the universal constancy of c suggests that that common denominator is an isotropic substrate extended throughout the universe. Both observations converge in the same ontological direction.

The analogy with the speed of sound fulfils its motivating function well at this inductive level, but it has a limit. In \S{}I.8.9, $c$ is not, strictly, ``one more propagation speed'' determined by the rigidity of the medium in the manner of the speed of sound, but the \emph{causal structure} of \emph{el Pleno}: the dimensional name for the fact that space and time are directions of the same manifold. For that reason, unlike the speed of sound ---which is indeed a contingent property of air---, the value of $c$ is not a property that \emph{el Pleno} possesses, but an artefact of measuring length and time with separately calibrated rods. The analogy is right that a universal speed requires a common substrate; it falls short in suggesting that $c$ is of the same kind as the speed of sound. \S{}I.8.9 develops that refinement.


\subsubsection{I.3.4 --- Argument 3: Gravitational waves propagate with finite speed in a medium}

On September 14, 2015, the LIGO observatories detected for the first time the direct signal of gravitational waves, produced by the merger of two black holes at approximately 1.3 billion light-years from Earth. This detection, awarded the Nobel Prize in 2017, opened a new observational window onto the universe. In the following years, LIGO, Virgo and KAGRA detected dozens of additional events --- black hole mergers, neutron star mergers, and combinations of both.


One particular event, GW170817, detected on August 17, 2017, was extraordinarily informative. The merger of two neutron stars produced gravitational waves and electromagnetic radiation simultaneously (a short gamma-ray burst, GRB 170817A). The electromagnetic signal was detected 1.74 seconds after the gravitational signal, after travelling approximately 130 million light-years. This temporal coincidence made it possible to measure, with unparalleled precision, the propagation speed of gravitational waves compared with the speed of light.


The result: the speed of gravitational waves coincides with the speed of light to within $\sim 10^{-15}$. More specifically, the relative difference is bounded by $-3 \times 10^{-15} c < v_g - c < +7 \times 10^{-16} c$ \texttt{[REF-WEB-15]}. This is one of the most precise experimental verifications of any physical prediction, and it rules out entire families of modified-gravity theories that predicted detectable deviations.


The implication for PIU-$\Psi$ is direct: \textbf{gravitational waves are perturbations that propagate through something}. Propagation with finite speed is propagation through a medium ---if they were instantaneous they would be action at a distance, not propagation; if they were null they would not propagate. That they have finite speed equal to $c$ is a natural consequence of a medium whose structural properties fix that speed. Where the standard image calls them ``perturbations of space-time'' without committing ontologically, PIU-$\Psi$ reads them as \textbf{transverse tensor modes of \emph{el Pleno} propagating at the structural speed of the medium}, and the equality $v_g = c$ is not coincidence but identity: both are speeds of modes of the same medium. This argument is stronger than the previous ones because propagation with finite speed is direct observation, not inference: the observed gravitational waves exist and propagate, and the question of the medium that sustains them has an empirical basis.


\subsubsection{I.3.5 --- Argument 4: Nature effectively realizes macroscopic quantum media}

The previous arguments point to the conceptual coherence of postulating a substrate. This fourth one adds distinct empirical evidence: \textbf{nature effectively realizes, in the laboratory, macroscopic quantum substrates with properties analogous to those that PIU-$\Psi$ attributes to \emph{el Pleno}}. Four families confirm it:


\textbf{Bose-Einstein condensates (BEC).} Predicted by Bose and Einstein (1924-1925), observed in 1995 (Nobel 2001): millions of atoms at nanokelvin temperatures that behave as a single quantum entity with a common wavefunction $\Psi$, with measurable density, microscopic scale, propagating modes (Bogoliubov excitations) and rigidity. It is, literally, a macroscopic quantum medium with the formal structure of \emph{el Pleno}.


\textbf{Helium superfluids.} He-4 below 2.17 K, He-3 below 2.5 mK: quantum fluids of zero viscosity, dissipationless propagation and topological rigidity against vortices ---the subject of more than five Nobel Prizes.


\textbf{Cooper-pair condensates (superconductors).} Electrons coupled into pairs that condense into a macroscopic state described by BCS theory (1957, Nobel 1972); its Ginzburg-Landau order parameter is a complex scalar field formally analogous to the $\Psi$ of PIU-$\Psi$.


\textbf{Ferromagnetic condensates.} Below the Curie temperature, the spontaneous magnetization is a collective order parameter: another macroscopic system described by a continuous field with defined structural properties.


\textbf{It is not a theoretical invention that nature admits macroscopic quantum media with this structure: it is confirmed experimental physics}, with established field equations (Gross-Pitaevskii, Ginzburg-Landau). The hypothesis of \emph{el Pleno} does not postulate an entity of a new kind, but rather that \textbf{the entire universe is a macroscopic quantum medium} analogous to those terrestrial systems, scaled to cosmological dimensions, of which the laboratory condensates are local realizations. It has, then, direct experimental precedent.

\subsubsection{I.3.6 --- Argument 5: The universal fundamental constants suggest structural properties of a substrate}

The fundamental constants ---$G$, $\hbar$, $c$, $\alpha$, fermion masses, couplings--- share a notable property: \textbf{they are universal}, with the same value in every place, epoch and observer. This universality is one of the firmest observational facts of physics: the spectral lines of quasars emitted billions of years ago coincide with laboratory ones to parts per billion. The standard image assumes it without explaining it, but the universality itself demands an answer: why does the entire universe share the same values, without varying between regions or with time?


The natural answer under the hypothesis of \emph{el Pleno} is: \textbf{because they are structural properties of the same substrate that constitutes the entire universe}. If the cosmos is a single continuous medium, its properties are the same at every point because it is the same medium: the speed of its modes ($c$), the minimal scale of action ($\hbar$), the gravitational strength ($G$) and those of the gauge sector ($\alpha$) are distinct readings of one and the same \emph{el Pleno}. This does not demonstrate its existence, but it offers a \textbf{natural explanation} of the universality where the standard image leaves it as a brute fact: any framework that aspires to explain the constants, not only describe them, must derive them from an underlying physical object, and the hypothesis of \emph{el Pleno} articulates precisely that object.


\subsubsection{I.3.7 --- Argument 6: The quantum ``vacuum'' is not empty}

The sixth argument starts from an observation that Quantum Field Theory manipulates with operational success but leaves ontologically open: \textbf{the quantum vacuum is not empty, but has experimentally real fluctuations}. The \textbf{Casimir effect} (Casimir 1948, confirmed since the 1990s) shows two conducting plates attracting each other through the fluctuations of the vacuum electromagnetic field; the \textbf{Lamb shift} (1947) arises from the interaction of the atomic electron with those fluctuations; and the \textbf{anomalous magnetic moment of the electron}, of unparalleled precision, requires their contribution. Among the most precise results in physics, they establish that the vacuum fluctuations \textbf{exist physically}.


The question is inescapable: \textbf{fluctuations of what}. The standard formulation answers ``of the ground state of the field'', with no further ontological commitment ---pragmatically defensible, but if there are fluctuations, there is something that fluctuates. The hypothesis of \emph{el Pleno} gives the direct answer: what fluctuates is \emph{el Pleno} at its minimal excitation. The ``quantum vacuum'' is the \textbf{ground-state \emph{el Pleno}}; the Casimir effect, the pressure of the ground-state \emph{el Pleno} on the objects; the Lamb shift, the coupling of electrons with its excitations; the anomalous moment, the interaction of the electronic soliton with the surrounding ground-state \emph{el Pleno}. The reinterpretation does not change the numbers ---Quantum Electrodynamics keeps giving the same predictions---, but rather the \textbf{ontological reading} of what the entities that the theory manipulates are, answering what the standard formulation leaves open. And it connects with problem 1 of \S{}I.2: where the theoretical vacuum energy exceeds the observed one by 60-120 orders, the ground-state \emph{el Pleno} is a real physical state with defined structure, not a structureless infinity.


\subsubsection{I.3.8 --- Synthesis: convergence of the six arguments}

The six articulated arguments are not independent. They point, from distinct directions, toward the same candidate ontological position:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{4.40cm}L{4.40cm}L{4.40cm}}
\toprule
\textbf{Argument} & \textbf{Physical observation} & \textbf{Ontological implication} \\
\midrule\endhead
1. Mass-energy equivalence & $E = mc^2$ experimentally confirmed & Matter and energy are readings of the same underlying object \\
2. Universal constancy of $c$ & $c$ identical for all observers & There exists a common isotropic substrate throughout the universe \\
3. Gravitational waves with $v_g = c$ & GW170817: coincidence to $10^{-15}$ & Propagation with finite speed requires a medium \\
4. Macroscopic quantum media & BEC, superfluids, superconductors & Nature effectively realizes macroscopic quantum substrates \\
5. Universal constants & $G$, $\hbar$, $c$, $\alpha$ identical throughout the cosmos & They suggest structural properties of a single medium \\
6. Quantum vacuum fluctuations & Casimir effect, Lamb shift, $g-2$ & The ``vacuum'' has physical content --- fluctuations of something \\
\bottomrule
\end{longtable}}

The convergence is structural, not anecdotal: the six arguments point to the same hypothesis ---\textbf{the universe is coherently described as the manifestation of a single continuous physical substrate with defined structural properties}---, a natural reading of established observations, not arbitrary invention. But it must be recognized that \textbf{convergence of a priori arguments is not a proof}: they justify that postulating \emph{el Pleno} is reasonable, not that it is true. The truth of the hypothesis will be decided by the performance of the program under the triple validation of \S{}I.1 ---if it interprets, explains and predicts better than the alternatives, it stands; if not, it must be modified or abandoned. What this section establishes is that starting from \emph{el Pleno}, instead of abstract quantum fields or geometries without ontology, is justified by convergent observations: it is not a speculative leap, but a reasonable response to established physical facts.


\subsubsection{I.3.9 --- From the general framework to the formal apparatus}

Postulating \emph{el Pleno} is, up to here, a reasonable but unformalized hypothesis. The next step is to give it a mathematical apparatus: a single primordial statement about energy ---finite at each point and conserved--- from which everything else follows. From that root statement the field $\Psi$ and its dynamics emerge; the six structural axioms articulate it in operative form, each stated first in conceptual language and then in its precise mathematical form.


\medskip\hrule\medskip

\subsection{\texorpdfstring{\S{}I.4 --- The genesis of el Pleno: from A1 to F1}{Sec.I.4 - The genesis of el Pleno: from A1 to F1}}

\begin{estadoseccion}

\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} structural (foundational reorganization, no new numbers)
  \item \textbf{Open threads:} none of its own; the fragile link of A3 (Lorentz from A1) is declared and referred to \S{}III.9.5
  \item \textbf{Notes:} conceptual entry door of the program. Shows that the entire apparatus ---the two walls, the flexural term, $c$, time, covariance and phase, and the form of F1--- follows from the root axiom (finite, conserved energy). Global classification \texttt{\textbf{[SLD of foundation]}}. Zero new magnitudes. Prepares \S{}I.5 (axioms) and \S{}I.6 (Lagrangian formalism).
\end{itemize}
\end{estadoseccion}


The entire apparatus of the program ---the two walls of the potential, the flexural
term, the speed $c$, the origin of time, covariance, phase symmetry and the very
form of F1--- follows from a single primordial statement about energy: that it is
finite at each point and conserved. The construction proceeds in four stages.
Stage~0 fixes the root statement and a reading rule; Stage~1 ---the \emph{being}---
asks what energy \emph{is}; Stage~2 ---the two branches--- asks how a finite but
variable energy is described; Stage~3 ---the \emph{doing}--- asks how it evolves
from conservation. The closure gathers everything into F1.

\bigskip\hrule\medskip

\subsubsection*{Stage 0 --- The root statement and a reading rule}
\addcontentsline{toc}{subsubsection}{Stage 0 --- The root statement and a reading rule}

The whole tree starts from a single assertion about the universe:

\begin{tcolorbox}[ecuacionclave,title={A1 in genesis form --- the root statement}]
The universe is energy: a \textbf{single substrate} ---\emph{el Pleno}---,
\textbf{homogeneous} (the same laws at every point, with no privileged place) and
\textbf{isotropic} (with no privileged direction). That energy is \textbf{(i) finite
everywhere}: at each point, $0 < E < \infty$, bounded on both sides; and
\textbf{(ii) conserved}: it is neither created nor destroyed, only transformed and
redistributed.
\end{tcolorbox}

Here the central term of the program appears for the first time: \emph{el Pleno}, the
single physical substrate of which everything else will be a configuration. It is not
``space'', nor a field \emph{within} space, nor the quantum vacuum with its
fluctuations: it is the only physical constituent that exists, a continuous medium of
which space, time, matter and forces are \emph{modes of being}, not tenants. The
inversion relative to the usual image is total. Standard physics puts first a stage
(space-time) and places things on top of it (fields, particles); PIU-$\Psi$ puts the
medium first, and lets the stage and the things emerge as readings of its state. The
water analogy orients the idea: waves and whirlpools are not objects added to the
water, but forms the water adopts; in the same way, an electron or a gravitational
field are not things \emph{in} \emph{el Pleno}, but configurations \emph{of} \emph{el
Pleno}. Homogeneity and isotropy are not ornament: homogeneity is what will make the
ceiling of change be the \emph{same} at every point, and hence its universality.

Clause (ii) is the principle of conservation of energy, established by Mayer, Joule
and Helmholtz in the 1840s and raised to a general principle by Helmholtz (1847). Its
modern status comes from Noether's theorem (1918)~\cite{noether}: the conservation of
energy is the consequence of invariance under time translation. \textbf{The program
adopts this principle; it does not derive it.} Clause (i) ---the pointwise finiteness
of energy--- is the program's own postulation, and from it the complete architecture
of the medium follows.

\medskip

\textbf{The reading rule: for el Pleno there are no numbers, there are ceilings.} A1
fixes that energy is finite; from there \emph{edges} follow: a ceiling on energy per
region, a minimal grain of structure, a lower floor, a ceiling on the rapidity of
change. Two things that ordinary language confuses must be distinguished. That an edge
\textbf{exists} is a property of \emph{el Pleno}: it is ontological, independent of any
measurement. That an edge \textbf{has such-and-such a number} is not: a number appears
only when one measures, and to measure is to compare against a rod manufactured by us
---the metre, the second, the kilogram. Thus, when $\rho_P$, $\ell_P$ or $c$ appear,
their ``value'' will be how many human rods fit in the corresponding ceiling; in the
units of the medium itself ($c = 1$, $\ell_P = 1$) those numbers vanish, but the
ceilings remain. The manuscript already applies this criterion to $c$ (\S{}I.8.9); the
genesis extends it to the three structural edges. What \emph{is} derived from \emph{el
Pleno} with a value of its own are the \textbf{pure ratios} between structures
($\gamma_\kappa = 1/40$, $\sqrt{3}\pi$, $S_{\min}^\text{estr}$,
$\sin^2\theta_\text{tet} = 8/9$), which the diamond geometry produces without any rod.
This distinction ---\emph{pure ratio}, which is derived and has a value, versus
\emph{ceiling}, which exists without a value of its own--- is formalized in \S{}I.8.9
and governs the whole reading that follows. For the complete entry of any symbol
(dimensions, first appearance), the appendices offer the index of variables (\S{}V.6.1)
and the index of equations (\S{}V.6.2).

\bigskip\hrule\medskip

\subsubsection*{Stage 1 --- The being: what energy \emph{is}}
\addcontentsline{toc}{subsubsection}{Stage 1 --- The being: what energy is}

Energy is defined by pure identities, without
introducing dynamics yet. The order matters: finiteness is stated \emph{before}
density, because it is the condition that makes density well-defined. For that reason
the symbol of density is born only at the third step, when it is already justified.

\begin{enumerate}[leftmargin=1.6em,topsep=2pt,itemsep=2pt]
  \item \emph{The universe is energy.} Let $E$ be that energy. It is the only substance;
        a pure ontological identity.
  \item \emph{Energy is finite at each point.} Nowhere is it null or infinite: neither
        vacuum nor singularity. This finiteness is predicated \emph{point by point}
        ---not of the total energy, which depends on the volume (Part IV). To speak of
        ``energy at each point'' is already to speak of a \emph{quantity per region}, and
        that is what the next step names.
  \item \emph{Energy read per region is the density.} Its name in the program
        is $\rho_P$, one of the two \emph{substantive properties} of \emph{el Pleno} (the
        other is the scale $\ell_P$): a property with dimensions that the medium
        possesses (\S{}I.8.9). Only now, with density named, the finiteness of
        step 2 is written with a symbol:
        \[
          0 < \rho_P(x) < \infty .
        \]
  \item \emph{$E = \rho_P$.} Read per region, energy \emph{is} its density.
\end{enumerate}

That same quantity admits two other readings. Read as \textbf{inertia} ---the
resistance to being accelerated--- it is the mass $M$; and read as substance it is $E$. Under
the units of the medium itself ($c = 1$ and $\ell_P = 1$, so that the volume of a
cell is $V = \ell_P^3 = 1$ and $E = Mc^2$ reads $E = M$), the three readings collapse
into a single identity:
\[
  \rho_P \;\equiv\; M \;\equiv\; E .
\]
Here $\ell_P$ appears for the first time: it is the \textbf{coherence scale} of
\emph{el Pleno}, its minimal grain of structure, the second substantive property with
dimensions (\S{}I.8.9); its full role arrives in Stage~3, with the flexural term.
Density, mass and energy are not three things that are generated one from another:
they are a single object read from three angles ---quantity per region ($\rho_P$),
inertia ($M$), substance ($E$). There is no causal or temporal arrow; there is a
\textbf{chain of identities}. It is where a logical leap most easily slips in: never
``from $E$ arises $\rho_P$'', always ``$E$, read per region, \emph{is} $\rho_P$''. The
block of being fits in one sentence: \emph{the universe is energy, and its density is
finite everywhere}.

\bigskip\hrule\medskip

\subsubsection*{Stage 2 --- The two branches: from being to the field \texorpdfstring{$\Psi$}{Psi}}
\addcontentsline{toc}{subsubsection}{Stage 2 --- The two branches: from being to the field Psi}

Energy is finite but \emph{variable}. Describing \emph{how} it varies is not achieved
with a single reading: two are required, complementary and orthogonal. Energy has a
branch of \emph{how much there is} and a branch of \emph{how it changes}, and each
inherits from A1 its own wall of finiteness. This stage constructs them and, with them,
introduces all the structural terms of the field.

\medskip

\textbf{The branch of quantity: the modulus $S$ and its two walls.} ``How much energy
there is'' is captured by a real modulus $S$. Being a quantity of energy, $S$ inherits
the finiteness of A1 on both sides:
\[
  S \in [\,S_{\min},\, S_{\max}\,],
  \qquad
  S_{\min} > 0 \ \text{(floor, from } E>0\text{)},
  \qquad
  S_{\max} < \infty \ \text{(ceiling, from } E<\infty\text{)}.
\]
The floor $S_{\min} > 0$ says that \emph{el Pleno} cannot switch off at any point; the ceiling
$S_{\max}$, that it cannot saturate. They are the \textbf{two walls of the finiteness of the
quantity}, and they are realized dynamically as two terms of the potential (an IR
wall at $S\to S_{\min}$ and a UV wall at $S\to S_{\max}$, \S{}I.7). The ceiling follows
directly from $E < \infty$; the floor, from $E > 0$. The only thing the genesis needs here
is that reading: the floor $S_{\min} > 0$ is the \emph{lower face of the finiteness of
A1-i}. The manuscript lodges it in axiom A2 for its technical role ---it enables
differential calculus on the half-line $S \in (0, \infty)$---, and there (\S{}I.5.3) the
floor is examined with the detail it deserves, including the readings that reinforce it
by independent routes. Of A2, what neither finiteness nor conservation delivers is the
continuity and differentiability of the field: that is its genuinely independent
technical content.

\medskip

\textbf{The branch of change: $C$ and its ceiling $c$.} ``How'' the energy changes is
captured by a second magnitude, and here the program carefully makes a distinction that is
a habitual source of confusion:

\begin{tcolorbox}[derivado,title={$C$ versus $c$ --- change and its ceiling}]
\textbf{$C$ is the change:} how much a configuration of energy is changing in an
instant. It is internal to the energy, prior to any metric; it presupposes neither space nor
time.

\smallskip
\textbf{$c$ is the maximum capacity for change:} the upper edge that $C$ cannot
exceed. It is not \emph{the} change; it is its wall.
\[
  C \in [\,0,\, c\,].
\]
$C = 0$ is energy at rest (rest mass); $C = c$ is change at the ceiling (the \emph{massless
mode}, which always travels at $c$, like the photon or the gravitational wave). Matter
sits between the two extremes.
\end{tcolorbox}

The parallel with the branch of quantity is exact and is the heart of this stage: just
as the modulus is bounded above by $S_{\max}$, change is bounded by $c$.
\textbf{$c$ is to the branch of change what $S_{\max}$ is to the branch of quantity}:
the same kind of finiteness edge, read on the axis of ``how it changes''. This
reframes $c$ at its root: it is not ``the maximum speed'' ---a speed presupposes
space and time already given--- but the maximum rate of change of energy itself, a
property prior to the metric. That is the ontological advantage: it allows time to
\emph{emerge} from $c$ instead of being presupposed.

There is a second face of $c$, deeper, whose full development is in \S{}I.8.9. If
$c$ is the maximum rate of change, it is what converts length into time. In a medium
with the covariance recovered in Stage~3, space and time are not two domains that
$c$ connects from outside: they are the same manifold, and $c$ is the \textbf{dimensional
name} of that unification. For that reason the program classifies $c$ apart from
$\rho_P$ and $\ell_P$. Those two are \textbf{substantive properties} (the medium could
be more or less dense, of finer or coarser grain); $c$ is \textbf{structure}
---the signature of how \emph{el Pleno} organizes its dimensions, not a knob that the medium
chooses. The two faces are the same: $c$ is the ceiling of change (operational) and, by
being so, the causal structure of the medium (ontological, \S{}I.8.9). The formal
development ---supported on the Fubini-Study metric (Provost-Vallée, 1980~\cite{provostvallee})
and on the quantum speed limits (Mandelstam-Tamm, 1945~\cite{mandelstamtamm};
Margolus-Levitin, 1998~\cite{margoluslevitin}), with the universality coming out of the
homogeneity of A1--- is presented in \S{}I.8.9.

\medskip

\textbf{Time as accumulated change.} With $c$ available, time is read as
the amount of accumulated change, measured against the maximum change that $c$ fixes. Without a
ceiling on change (if $c \to \infty$) there would be no common rhythm and time would not be
defined. The causal direction is not circular: the change $C$ is internal to the energy
and requires no clock; from it its extreme $c$ is taken; and from $c$ time is obtained. The
time is the last thing to appear in this stage: it emerges, it is not postulated, in line with
the emergence thesis that the manuscript already maintains (\S{}I.5.2).

\medskip

\textbf{The two branches at $90^\circ$: the field $\Psi$.} We have two readings of the
same energy: the branch of quantity (modulus $S$, permanence) and that of change (a phase
that rotates, propagation). Energy can \emph{remain} (with $S$ fixed) \emph{while}
it changes (rotating its phase) without one reading interfering with the other;
that requires both branches to be orthogonal. The imaginary unit $i$ is the operator
of $90^\circ$ rotation that maintains that orthogonality. Hence the form of the field:
\[
  \Psi = S\,e^{i\theta},
\]
with $S$ the branch of permanence and $\theta$ that of change. The status of this step is
clear: the manuscript \emph{postulates} $\Psi = S\,e^{i\theta}$ and founds the positivity
of $S$ on A2 and the phase freedom on A5; the genesis offers the ontological \emph{why}
of that form ---energy with two branches at $90^\circ$--- so that the language of the program
is continuous from energy to F1, without an unexplained initial leap. It motivates the form
of $\Psi$; it does not derive it.

\medskip

\textbf{Table of Stage~2 terms.} The structural symbols of the medium,
each with its branch and its status, at a glance (complete entry in \S{}V.6.1).

{\small
\begin{longtable}{L{2.1cm}L{4.6cm}L{2.6cm}L{3.0cm}}
\toprule
\textbf{Symbol} & \textbf{What it is} & \textbf{Branch} & \textbf{Status} \\
\midrule\endhead
$\rho_P$ & energy density ($E$ per region) & identity ($\rho_P\equiv M\equiv E$) & substantive property; ceiling without a value of its own \\
$S$ & modulus: how much energy there is & quantity / permanence & bounded $S\in[S_{\min},S_{\max}]$ \\
$C$ & the change: how much it changes & change / propagation & bounded $C\in[0,c]$ \\
$c$ & the maximum capacity for change; causal structure & change (ceiling) & structure, not property (\S{}I.8.9) \\
$\ell_P$ & coherence scale: minimal grain of structure & (curvature, Stage~3) & substantive property; ceiling without a value of its own \\
\bottomrule
\end{longtable}}

\bigskip\hrule\medskip

\subsubsection*{Stage 3 --- The doing: dynamics from conservation}
\addcontentsline{toc}{subsubsection}{Stage 3 --- The doing: dynamics from conservation}

With the block of being closed and the field constructed, the dynamics enter. Its first
link is clause (ii) of A1: energy is conserved, it is only transformed. The
program demands saying \emph{into what} it transforms, without vague verbs.

\medskip

\textbf{The balance between branches.} If energy can only be confined
(permanence, mass) or propagating (change, radiation), there is no third fate: every
transformation is a transit between those two branches. The confined energy that
``disappears'' reappears as propagating, and vice versa, because (ii) forces it. With
$E_\text{conf}$ and $E_\text{prop}$ the energy in each branch, local conservation is
the balance
\[
  \Delta E_\text{conf} + \Delta E_\text{prop} = 0
  \;\;\equiv\;\;
  \partial_t \rho_E + \nabla \cdot j_E = 0 ,
\]
where $\rho_E$ is the local energy density (the $\rho_P$ of Stage~1 viewed varying
in time) and $j_E$ its current. What leaves one branch enters the other, with net
sum zero. This balance is more basic than $E = Mc^2$, which turns out to be
its case of total conversion: the conservation of energy is already a continuity
equation, before invoking the phase.

\medskip

\textbf{$E = Mc^2$ as the conversion factor of the balance.} The balance says how much
energy is conserved, not in what proportion one branch converts into the other. That
proportion is fixed by the change ceiling $c$. To go from confined to propagating is to release
the energy to the maximum change, and the conversion touches \emph{both} branches: a
power of $c$ appears on releasing the branch of quantity and another on its being received by
the branch of change. If the conversion is written exhibiting only one of those powers on each
side, the intermediate form $E/c = Mc$ remains ---one $c$ per branch, in view---;
together, the two powers give the factor $c \cdot c = c^2$:
\[
  E/c = Mc
  \quad\Longrightarrow\quad
  E = Mc^2 .
\]
The $c^2$ is not opaque: it is the same $c$ of Stage~2 ---the ceiling of change---
acting once for each branch that the transit connects. In this reading $E = Mc^2$
is not a law imported from outside, but the conversion factor of the balance.
The scope of the argument is limited, for here is one of the two fragile links:
``one $c$ per branch'' is an \emph{ontological count}, not an algebraic
derivation, and it inherits the dependence that $c$ be causal structure
(\S{}I.8.9). This foundational reading \emph{coexists} with the algebraic derivation
of the corpus ---$E^2 = (Mc^2)^2 + (pc)^2$ from the kinetic invariant $(\partial S)^2$
of F1, \S{}III.9.5, E49--E50---: they do not compete, they are two readings of the same fact at
different depth.

\medskip

\textbf{The five forms of the mass-energy relation, ranked.} The
central point of this stage is that the relation between mass and energy does not have \emph{one}
correct form and others approximate: it has a single identity read at five
depths. They do not compete; they are ordered. Each level conserves what the previous one
says and adds a layer of physical content. Read from the most fundamental to the most
complete:

\begin{enumerate}[leftmargin=2.0em,topsep=4pt,itemsep=5pt]
  \item $\rho_P \equiv M \equiv E$ \quad---\quad \textbf{the origin (identity).}
        Density, mass and energy are a single object read from three angles
        (Stage~1). There is no conversion or dynamics here: it is the assertion that the
        three words name the same thing. Everything else unfolds from here.
  \item $\Delta E_\text{conf} + \Delta E_\text{prop} = 0$ \quad---\quad \textbf{the most
        basic dynamics (conservation).} When energy transits between the confined and
        the propagating branch, what leaves one enters the other. This level asserts
        \emph{that} something is conserved, still without saying in what proportion one
        branch converts into the other: the conversion factor $c^2$ has not yet entered.
  \item $E/c = Mc$ \quad---\quad \textbf{one $c$ per branch (the count made
        visible).} Rewritten this way, the relation shows its anatomy: a power of
        $c$ on the confined side and another on the propagating one. It is the form where the ``ontological
        count'' of the conversion factor is in view, before collapsing
        into the usual notation.
  \item $E = Mc^2$ \quad---\quad \textbf{the recognizable face.} The two powers of
        $c$ of the previous level, together, give the familiar $c^2$. It is the standard
        form of the total conversion between rest and energy, now read as a consequence
        of the three levels that precede it, not as a starting point. Only here, with
        the conversion factor already constructed, the balance of level~2 in total
        conversion acquires its closed form: the confined energy $E_\text{conf} = Mc^2$
        is released entirely as propagating $E$, that is $E - Mc^2 = 0$.
  \item $E^2 = (Mc^2)^2 + (pc)^2$ \quad---\quad \textbf{the complete form, with
        motion.} When a fraction of the energy is confined and another
        propagating \emph{at once}, the total conversion generalizes to a split
        tied by the conserved energy. $E = Mc^2$ is its particular case of null
        momentum ($p = 0$). This is the level from which the Lorentz factor
        follows.
\end{enumerate}

The virtue of this hierarchy is that no level negates another: $E = Mc^2$ does not
``correct'' the conservation balance $\Delta E_\text{conf} + \Delta E_\text{prop} = 0$,
it specifies it by fixing the conversion proportion; $E^2 = (Mc^2)^2 + (pc)^2$ does not replace
$E = Mc^2$, it contains it as the rest case. That the same identity admits this
staircase ---from the pure \emph{being} of Stage~1 to the \emph{doing} with motion--- is
the mark that the mass-energy relation is not an external datum imported into the
theory, but a reading of the balance between the two branches of \emph{el Pleno}.

\medskip

\textbf{The balance with motion, the Lorentz factor and covariance.} When one
part of the energy is confined and another propagating at once ---matter in
motion, with $E$ conserved---, the split is the \textbf{mass-shell constraint}. It is the complete form of the ``$= 0$'' above:
\[
  E^2 = (Mc^2)^2 + (pc)^2
  \;\;\equiv\;\;
  E^2 - (Mc^2)^2 - (pc)^2 = 0 ,
\]
where $p$ is the \textbf{momentum}: the measure of the change that the propagating fraction
carries with it. The split between the confined fraction ($Mc^2$) and the propagating
($pc$) is not free; the total energy ties it. The algebra is direct. Define the
confined fraction as $1/\gamma \equiv Mc^2/E$, so that $E = \gamma Mc^2$.
Substituting into the constraint,
\[
  (\gamma Mc^2)^2 = (Mc^2)^2 + (pc)^2
  \;\Longrightarrow\;
  (pc)^2 = (Mc^2)^2(\gamma^2 - 1)
  \;\Longrightarrow\;
  p = \gamma M c\sqrt{1 - \gamma^{-2}} .
\]
Identifying the \textbf{velocity} $v$ by $v/c \equiv pc/E = pc/(\gamma Mc^2)$, one
obtains $v/c = \sqrt{1 - \gamma^{-2}}$, that is $\gamma^{-2} = 1 - v^2/c^2$, and
therefore
\[
  \boxed{\,\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\,,\qquad E = \gamma Mc^2,\qquad p = \gamma M v.}
\]
The Lorentz factor $\gamma$ is the split of the capacity for change between the two
branches, tied by the conservation of energy; it is not the deformation of a prior
space-time stage. The Lorentz space-time \emph{describes} this split a
posteriori; the constraint is what is primary. Thus covariance (today A3) emerges from the
balance, not from a postulated stage.

Here is the second fragile link. The quantitative derivation of $\gamma$ lives in
\S{}III.9.5: the factor $(1 - v^2/c^2)$ emerges from F1 from the kinetic invariant
$(\partial S)^2$ (E49, \texttt{\epi{DERIVED}}). That rheological reading and the
capacity-split of this stage are the same route in two vocabularies, as
\S{}III.9.5 declares. The contribution of the genesis is not a second derivation but
an axiomatic reframing: the structure of A3 ---finite capacity for change with a single ceiling
$c$, split under conservation--- is already contained in A1. The status is
\texttt{\epi{SLD}}: the reframing does not raise A3 to \texttt{\epi{DERIVED}} and leaves standing
the fragile point ---that ``finite capacity for change plus homogeneity'' does not
smuggle in the principle of relativity nor a prior space-time. The relocation
of A3 figures in \S{}I.5.4.

\bigskip\hrule\medskip

\subsubsection*{Closure --- F1 and the axiomatic economy}
\addcontentsline{toc}{subsubsection}{Closure --- F1 and the axiomatic economy}

\textbf{The axiomatic economy via Noether.} The pattern that organizes the symmetries
of \emph{el Pleno} is uniform: each structural symmetry is the obverse of a conservation
law (Noether, 1918), and each conservation law follows from A1.

{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{4.40cm}L{4.90cm}L{3.40cm}}
\toprule
\textbf{Symmetry of el Pleno} & \textbf{Conservation (inherited)} & \textbf{Clause of A1} \\
\midrule\endhead
Time translation & Energy (Mayer-Joule-Helmholtz) & A1-ii \\
Spatial translation and isotropy & Momentum and angular momentum (Noether) & A1 (single homogeneous substrate) \\
Boost with finite invariant speed $\to$ Lorentz (A3) & Causal structure & A1-i + A1-ii \\
Phase rotation $\to$ U(1) (A5) & Charge (Weyl; Noether) & A1-i ($E > 0 \Rightarrow$ defined phase) \\
\bottomrule
\end{longtable}}

The U(1) symmetry (today A5) is the cleanest case: charge conservation is a
fundamental law, and by Noether it is the obverse of phase invariance (Weyl). In the
genesis that invariance is already in $\Psi$ by A1: since $E > 0$ (A1-i) forces
$S > 0$, the phase $\theta$ is always well-defined, and rotating it does not alter $S$ (they are
at $90^\circ$ by the imaginary unit). That freedom \emph{is} the U(1). Lorentz
covariance (today A3) is the deepest and least closed case, because of the fragile link
already noted. The continuity and differentiability of the field, on the other hand, are
delivered neither by finiteness nor by conservation ---and F1 needs them, since it operates with derivatives of
$\Psi$. They remain, therefore, as the only genuinely independent technical axiom.

\medskip

\textbf{F1 verified term by term.} With all the pieces gathered, the
object that contains them all becomes legible: \textbf{F1, the Lagrangian of \emph{el Pleno}} ---the density
whose integral, made stationary, gives the dynamics of the medium (\S{}I.7). It is
\[
  \mathcal{L}_\text{cin} = \tfrac{1}{2}\rho_P c^2 \ell_P^2
     \big[(\partial_\mu S)(\partial^\mu S) + S^2(\partial_\mu\theta)(\partial^\mu\theta)\big],
  \quad
  \mathcal{L}_\text{pot} = -V(S),
  \quad
  \mathcal{L}_\text{flex} = -\tfrac{\rho_P c^2 \ell_P^4}{40}(\Box S)^2 .
\]
Each term is read from a clause of A1 and gathers pieces already constructed. The
Lagrangian formulation itself follows from (ii): conservation, symmetry and stationary
action are three faces of the same fact (Noether); the manuscript presents it in
\S{}I.6 as a choice, and the genesis shows it as obligatory.

\begin{itemize}[leftmargin=1.6em,topsep=2pt,itemsep=3pt]
  \item \textbf{The kinetic term} splits exactly into the two branches of
        Stage~2: $(\partial S)^2$ is the permanence (branch of quantity) and
        $S^2(\partial\theta)^2$ is the change (branch of propagation). They are weighted by the
        two substantive properties, $\rho_P$ and $\ell_P$, of Stage~1.
  \item \textbf{The potential $V(S)$} gathers the two walls of the finiteness of the
        quantity: the IR wall that sustains $S_{\min}$ (from $E > 0$, A1-i; what the
        manuscript presents as the positivity of A2) and the UV wall that sustains
        $S_{\max}$ (from $E < \infty$, A1-i, today tied to A6). Its concrete
        dynamical realization ---the Klauder and Gent potentials--- is developed in \S{}I.7.
  \item \textbf{The flexural term} is the same finiteness read over the curvature. The
        quantity $\Box S = \partial_\mu\partial^\mu S$ ---the d'Alembertian of the
        modulus--- measures the \emph{structural curvature} of the field: how much $S$ bends
        from one point to another. An infinite curvature would give infinite local
        energy, forbidden by A1-i; the term $(\Box S)^2$, weighted by $\ell_P^4$,
        penalizes it (today tied to A4). Here $\ell_P$ ---the coherence scale of
        Stage~1--- earns its role: it fixes at what scale the curvature begins to cost.
\end{itemize}

The cleanest foundational claim is this: the two walls of the potential, which the
manuscript derives from \emph{two} distinct axioms (A2 and A6), are in the genesis the
\emph{two bounds of the same} A1-i ---the floor of $E > 0$ and the ceiling of $E < \infty$. The
potential does not have two walls postulated separately; it has the two faces of a
single finiteness.

The claim ``from six axioms to two'' is real \emph{at the level of foundation}, but it does not
erase the six of the manuscript: they are hierarchically reordered into one root (A1), one
independent technical one (continuity and differentiability, today part of A2) and four that
come to be read as theorems or readings of A1 ---A2-positivity, A4 and A6 as the
three faces of finiteness; A3 and A5 as Noether symmetries. No content is lost:
it is hierarchized. The subsections \S{}I.5.2 to \S{}I.5.7 conserve the complete
presentation of each axiom; the backbone that connects them is that hierarchy. The genesis is, therefore, \texttt{\textbf{[SLD of foundation]}}: it contributes
origin, coherence and structure, not predictions or magnitudes. And everything inherited
---conservation of energy (Mayer, Joule, Helmholtz), Noether's theorem (1918),
phase invariance and charge (Weyl), Fubini-Study metric (Provost-Vallée), quantum
speed limits (Mandelstam-Tamm, Margolus-Levitin)--- carries its attribution: the
program adopts it, it does not derive it.

\medskip\hrule\medskip
\subsection{\S{}I.5 --- The six structural axioms of el Pleno (A1--A6)}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (non-derivational section)
  \item \textbf{Notes:} section that presents the six structural axioms of the program under the architectural convention of PIU-$\Psi$: conceptual language first, mathematical form afterwards. Each axiom with a justification that connects to \S{}I.3 where applicable. Prepares \S{}I.6 (Lagrangian formalism) and \S{}I.7 (F1 as the Lagrangian of \emph{el Pleno}).
\end{itemize}
\end{estadoseccion}

\subsubsection{I.5.1 --- Opening: why axioms, and how these were chosen}

Every physical theory is built on a minimal set of assumptions that are not proven within the framework but postulated: its \textbf{axioms}. General Relativity starts from equivalence and covariance; Quantum Mechanics from superposition, measurement and indeterminacy; Thermodynamics from its four principles. Choosing a good axiomatic set ---few, clear, consistent and sufficient to derive the phenomenology--- is one of the central creative acts in the construction of a theory.


PIU-$\Psi$ is built on six structural axioms, \textbf{A1 to A6}. They are not arbitrary: each one responds to a physical observation of \S{}I.2 and \S{}I.3, and together they are the minimal set for \emph{el Pleno} to admit the dynamics that the rest of the manuscript develops. Each axiom is articulated at four levels: \textbf{conceptual statement} (what it asserts, without formalism), \textbf{justification} (why it is reasonable, connecting with the observations), \textbf{mathematical form} (the precise expression) and \textbf{immediate implications} (what it derives or prepares).

\textbf{The hierarchy of the six axioms.} The six axioms receive equal detail ---each with its statement, its justification and its mathematical form---, but that flat treatment is a pedagogical choice, not a claim that they are at the same logical level. They are not: the genesis of \S{}I.4 makes them hang from a single primordial statement about energy, and only one ---the continuity and differentiability of the field--- remains as an independent technical axiom. Equality of treatment is not equality of rank.



\medskip

\subsubsection{I.5.2 --- Axiom A1: Existence and uniqueness of el Pleno}

\textbf{Conceptual statement.} There exists a single continuous physical substrate that fills the entire universe at every instant. This substrate is \emph{el Pleno}. There are not two substrates. There are no regions of the universe without \emph{el Pleno}. There are no agents external to \emph{el Pleno} that act on it. Everything that happens in the cosmos is internal behaviour of \emph{el Pleno}.


\textbf{Justification.} This axiom formally articulates the ontological position established in \S{}I.1 and sustained by the six convergent arguments of \S{}I.3. If mass and energy are readings of the same underlying object (Argument 1), if the universal constancy of c indicates a common isotropic substrate (Argument 2), if gravitational waves propagate with finite speed through a medium (Argument 3), and if the fundamental constants are universal by being structural properties of something common (Argument 5), then postulating a single substrate is the natural consequence. Axiom A1 turns this postulation into an explicit assumption of the program, not a rhetorical suggestion.


Uniqueness --- a single \emph{el Pleno}, not several --- is essential. If there were two or more coexisting substrates, the universally observed properties (the equality of c everywhere, the universality of the constants) would require coincidences between independent substrates that would be unjustifiable. Uniqueness is the most parsimonious ontological position compatible with the observations.


\textbf{Mathematical form.} \emph{El Pleno} is denoted $\mathcal{P}$. Its state at each point is described by a single complex field $\Psi(x^\mu)$ defined over the whole cosmological manifold $\mathcal{M}$, where the coordinates $x^\mu$ parametrize the points of \emph{el Pleno} both in its spatial extension and in its temporal evolution:


\[
\Psi: \mathcal{M} \to \mathbb{C}, \qquad \Psi(x^\mu) \in \mathbb{C} \quad \forall\, x^\mu \in \mathcal{M}
\]

\textbf{What space and time are under PIU-$\Psi$.} Here the ontological
inversion of the program reaches its most radical form. In standard physics space-time is the first stage:
a geometric entity that exists in itself and on which everything else
happens. Under PIU-$\Psi$ there is no such prior stage. Space and time are not
the container of \emph{el Pleno}: they are \emph{properties of \emph{el Pleno} itself}, two ways of
reading its structure. \textbf{Space} is the extension of the medium ---the fact
that \emph{el Pleno} has distinguishable parts, neighbourhood, distance: that a
configuration here is not the same as one there. It is not a vacuum that \emph{el Pleno} occupies,
but the internal organization of the medium itself; where there is no \emph{el Pleno} there is no
``empty space'', simply nothing. \textbf{Time} is the dimension along
which \emph{el Pleno} changes: it is read as the amount of accumulated change,
measured against the maximum possible change (whose scale $c$ fixes). Without a ceiling on
change there would be no common rhythm and time would not be defined ---hence
time is not a prior datum, but a consequence of the change of the medium
being bounded (\S{}I.4). An image orients the idea: in a liquid, ``distance''
and ``duration'' are not things that exist before the liquid; they are ways of
describing how it is organized and how it evolves. The same here, taken to the
ontological limit: what General Relativity calls ``Minkowski space-time''
is, under PIU-$\Psi$, the \textbf{emergent effective metric} that describes how the
modes of \emph{el Pleno} propagate in the present relaxed regime ---not the floor on
which \emph{el Pleno} rests, but a summary of its behaviour. This reading is what
dissolves, at the root, a part of the conflict between Relativity and Quantum
Mechanics: a good part of that conflict arises from quantizing space-time
treating it as fundamental; if space-time is already an effective description
of a medium whose grain ($\ell_P$) is the quantum substrate, the question changes
form (\S{}III.2.9). Notation convention: when this manuscript uses
``space-time'' without nuance, it must be understood as the coordinates $x^\mu$ of
\emph{el Pleno} or the emergent effective metric, never as an ontologically
primitive entity.

The uniqueness of the field $\Psi$ formalizes the uniqueness of \emph{el Pleno}: there are no $\Psi_1, \Psi_2, \ldots$ --- there is a single $\Psi$.


\textbf{Immediate implications.} Axiom A1 establishes the ontological domain of the program. Everything the manuscript will derive --- particles, waves, gravity, constants, cosmology --- will be constructed as properties, configurations, modes or excitations of the single $\Psi$ postulated here. No physical entity will appear in the manuscript that is not reducible to this single field: \emph{el Pleno} before its consequences.


\subsubsection{I.5.3 --- Axiom A2: Continuity, differentiability and strict positivity of the modulus}

\textbf{Conceptual statement.} \emph{El Pleno} is continuous and smooth. Its configurations admit description by differentiable fields; its dynamics admit description by differential equations. There are no structural discontinuities in \emph{el Pleno} --- no jumps, no holes, no excluded regions. Additionally, the modulus of the field is strictly positive: $S > 0$ at every point of \emph{el Pleno}, with no admission of the configuration $S = 0$. This strict positivity is a structural property of the medium: \emph{el Pleno} cannot ``switch off'' at any point. Continuity, differentiability and strict positivity are structural properties of the medium, not effective approximations.


\textbf{Justification.} This axiom enables the use of the mathematical tools of differential and integral calculus on \emph{el Pleno}. Without continuity and differentiability, basic concepts such as gradient, time derivative, divergence, Laplacian, action integral --- all central to physics --- could not be applied rigorously to the field $\Psi$. The physical justification is twofold: on the one hand, the experimentally confirmed macroscopic physical media (mentioned in \S{}I.3.5: BEC, superfluids, superconductors) are described by smooth continuous fields; on the other hand, the observed dynamics of the universe at all accessible scales is successfully described by differential equations, which would be inconsistent if the underlying substrate were fundamentally discontinuous.


The strict-positivity component ($S > 0$) admits three readings, which are ordered here as a hierarchy ---not as rivals--- because they are three faces of the same floor. The first is \textbf{ontological}: \emph{el Pleno} cannot switch off because energy is strictly positive at each point ($E > 0$, clause A1-i), read through the constitutive identification $S^2 \leftrightarrow$ relative number density (\S{}II.2). The second is \textbf{geometric and independent of energy}: if the modulus $S$ could reach the value zero, the field $\Psi = S \cdot e^{i\theta}$ would have points where the phase $\theta$ is mathematically undefined, which is incompatible with the global U(1) symmetry of axiom A5 (\S{}I.5.6); this route reinforces the first by a different path, without depending on it. The third is \textbf{dynamical}: the positivity manifests operationally as a \textbf{quantum IR structural wall} of \emph{el Pleno} at $S \to 0^+$, sustained by Klauder's quantum potential (\S{}I.7.8), which materializes what the two previous readings demand. The three point to the same floor. For that reason strict positivity is a requirement of internal coherence of the program, not an additional postulation: in the hierarchical reordering of foundation (\S{}I.4) it is read as the \emph{lower face of the finiteness of A1-i}, while continuity and differentiability ---which neither finiteness nor conservation deliver--- is what remains of A2 as genuinely independent technical content.


Continuity does not contradict ultraviolet regulation (axiom A4) nor the structural upper bound (axiom A6, \S{}I.5.7). A medium can be continuous and simultaneously have structural characteristic scales --- lower (via A4) and upper (via A6) --- where the effective descriptions change. This is what happens with the known material media: water is continuously describable by fluid mechanics in its typical regimes of validity, but it has structural transitions (crystallization into ice, evaporation) that delimit the regimes of applicability. \emph{El Pleno} has an analogous structure: continuity and differentiability in the operational range, with structural walls that delimit that range.


\textbf{Mathematical form.} The field $\Psi$ is of class $C^\infty$ over the cosmological manifold $\mathcal{M}$ almost everywhere (the exceptions, localized solitonic configurations, are specific configurations of $\Psi$, not failures of continuity of the medium):


\[
\Psi \in C^\infty(\mathcal{M}, \mathbb{C}) \quad \text{(almost everywhere)}
\]

with the additional condition of strict positivity of the modulus:


\[
S(x^\mu) > 0 \quad \forall\, x^\mu \in \mathcal{M}
\]

The differential operators $\partial_\mu$, $\Box \equiv \partial_\mu \partial^\mu$, gradients and Laplacians apply rigorously over $\Psi$. The condition $S > 0$ defines the modulus of the field over a configuration manifold of the type \textbf{open half-line} $(0, \infty)$, a property that has important technical consequences in the quantization of the modulus (Klauder's affine quantization instead of standard canonical quantization, developed in \S{}I.7.8).


\textbf{Immediate implications.} A2 enables the Lagrangian formulation of the program, which requires derivatives of the field. The Euler-Lagrange equations applied to the F1 Lagrangian (presented in \S{}I.7) produce the dynamical equations of \emph{el Pleno} as differential equations over $\Psi$. The strict-positivity component has as an operational consequence the presence of Klauder's quantum potential $V_q$ in F1, which diverges when $S \to 0$ and dynamically maintains the inaccessibility of the value zero. Without A2 (in its complete form), neither the Lagrangian formulation nor the internal coherence of the field $\Psi$ would be possible.


\subsubsection{I.5.4 --- Axiom A3: Covariant dynamics}

\textbf{Conceptual statement.} The dynamics of \emph{el Pleno} respects a covariant structure: the laws that govern its evolution are the same for every observer. No observer is privileged relative to others. The form of the field equations is invariant under the group of transformations that connects distinct observers. In the low-energy effective limit, this covariance produces the Lorentz invariance that Special Relativity postulates as a fundamental property of space-time.


\textbf{Justification.} This axiom formally articulates the observation of Argument 2 of \S{}I.3: the universal constancy of the speed of light, identical for all observers. If the dynamics of \emph{el Pleno} did not respect a covariant structure, observers in relative motion would see distinct physical laws, and the experimentally observed universality of the constants would be impossible. Lorentz invariance is not postulated in PIU-$\Psi$ as an independent axiom --- it emerges as an effective consequence of the structural covariance of \emph{el Pleno}, a result known as the Lorentz Emergence Theorem and developed in later parts of the manuscript.


The difference between the structural covariance of \emph{el Pleno} and the postulated Lorentz invariance of Special Relativity is significant. Special Relativity assumes that Minkowski space-time is the fundamental geometry on which physics happens. PIU-$\Psi$ assumes that \emph{el Pleno} is the fundamental physical object, and shows that its covariant dynamics produces a Minkowski-type effective metric as a description of the propagation of the modes of \emph{el Pleno} in the low-energy regime. Special Relativity is thus recovered as a correct effective limit, not contradicted.


\textbf{Mathematical form.} The action of \emph{el Pleno} is invariant under the group of transformations that preserves the covariant structure:


\[
S[\Psi] = \int \mathcal{L}(\Psi, \partial_\mu \Psi, \partial_\mu \partial_\nu \Psi)\, d^4x
\]

where the Lagrangian $\mathcal{L}$ is a scalar under such transformations. The explicit form of $\mathcal{L}$ is presented in \S{}I.7.


\textbf{Immediate implications.} A3 guarantees that the predictions of the program are independent of the observer's reference frame, a property necessary for any fundamental physical theory. It also guarantees, via Noether's theorem (which will be introduced in \S{}I.6), that every continuous symmetry of the action generates a conserved current --- a fundamental tool for connecting the dynamics of \emph{el Pleno} with the conservation laws observed in phenomenological physics.

\begin{tcolorbox}[notabox,title={Foundation note --- A3 as a consequence of A1}]
The genesis of \S{}I.4 goes one level below this statement. There it is argued
that the structure of A3 need not be postulated separately, because it is already contained
in A1: a finite capacity for change, with a single ceiling $c$ identical at every point
by homogeneity, split between the two branches under conservation of energy,
selects the Lorentz structure (homogeneity and isotropy with finite invariant speed
admit only the Galileo group with infinite invariant speed or that of
Lorentz with finite invariant speed; the finiteness of energy selects the
second). This relocation is \texttt{\epi{SLD}}, not an algebraic
derivation: it conserves an explicit fragile link ---that ``finite capacity for change
plus homogeneity'' does not smuggle in the principle of relativity nor a prior space-time---
whose quantitative closure (that the split give exactly $\gamma$) is obtained
from F1 in \S{}III.9.5 (E49--E50). For that reason A3 is kept in the list of axioms with
its full content: the genesis shows what it hangs from, it does not eliminate it.
\end{tcolorbox}


\subsubsection{I.5.5 --- Axiom A4: Ultraviolet regulation}

\textbf{Conceptual statement.} \emph{El Pleno} has a characteristic microscopic scale, $\ell_P$, that acts as a structural lower bound on the accessible wavelengths. \emph{El Pleno} does not admit excitations of arbitrarily short wavelength; any attempt to excite the medium with energies that would require wavelengths smaller than $\ell_P$ encounters a structural rigidity that regularizes the behaviour. This regulation is an intrinsic property of the medium, not a mathematical artefact.


\textbf{Justification.} This axiom resolves one of the most serious structural problems of standard Quantum Field Theory: the appearance of ultraviolet divergences in perturbative calculations. Under standard QFT, the contributions to observables (vacuum energy, self-energies, radiative vertices) involve integrals over all modes of the field down to zero wavelength, which produces divergent results that can only be handled by a posteriori renormalization procedures. This situation is widely recognized as unsatisfactory: the divergences are a symptom that the theory is not complete in the UV regime.


PIU-$\Psi$ resolves this problem ontologically, not by a mathematical trick. If \emph{el Pleno} is a real physical medium with a characteristic microscopic scale (analogous to the interatomic spacing in material media), then wavelengths smaller than that scale simply are not accessible physical configurations of the medium. The ``UV divergences'' of the standard theory are artefacts of treating the physical substrate as if it were mathematically continuous down to zero scale --- an assumption that axiom A4 explicitly rejects.


The physical justification of axiom A4 also comes from the analogy with the known material media. Every real physical medium --- water, air, crystal lattices --- has a microscopic scale below which the continuous-medium description ceases to be applicable. The absence of such a scale in \emph{el Pleno} would be contrary to all known physical precedent.


\textbf{Mathematical form.} The scale $\ell_P$ enters the Lagrangian of \emph{el Pleno} through a quartic flexural rigidity term in derivatives:


\[
\mathcal{L}_\text{flex} \propto -\rho_P c^2 \ell_P^4\, (\Box S)^2
\]

where $\Box S \equiv \partial_\mu \partial^\mu S$ is the d'Alembertian of the modulus of the field. This term penalizes field configurations with high structural curvature --- that is, configurations that require rapid variations on scales comparable to $\ell_P$. In the UV regime, $\mathcal{L}_\text{flex}$ regularizes the behaviour of the field, preventing excitations of wavelength smaller than $\ell_P$.


\textbf{Immediate implications.} A4 guarantees the ultraviolet consistency of the program without resorting to a posteriori renormalization. The integrals over modes of the field are naturally bounded in the UV regime by the presence of the flexural term. Additionally, A4 connects directly with the resolution of the cosmological singularities: under PIU-$\Psi$, the extreme-curvature configurations (classical singularities in GR) are regulated by the structural rigidity of \emph{el Pleno}, avoiding the infinities that General Relativity cannot handle internally. This result will be developed in Part III of the manuscript.


\subsubsection{I.5.6 --- Axiom A5: Conserved U(1) symmetry}

\textbf{Conceptual statement.} The phase $\theta$ of the complex field $\Psi = S \cdot e^{i\theta}$ is associated with a continuous symmetry of \emph{el Pleno}: the dynamical laws of the medium are invariant under global transformations of the phase. By Noether's theorem (presented in \S{}I.6), this continuous symmetry produces a conserved current. This current, in the low-energy effective limit, is identified with the electric current of standard electrodynamics.


\textbf{Justification.} This axiom articulates the observation that the phase of the complex field of \emph{el Pleno} has physical meaning --- it is not a mere mathematical convention. If it were a convention, the dynamics of \emph{el Pleno} would be invariant under arbitrary redefinitions of the phase and would not produce conserved currents. The existence of conserved charges in phenomenological physics (electric charge, baryon number, lepton number) suggests that \emph{el Pleno} admits continuous symmetries that produce these conservations via Noether.


PIU-$\Psi$ explicitly postulates the global U(1) symmetry as a fundamental axiom. This symmetry generates, through the Higgs mechanism applied to \emph{el Pleno} (developed in Part III of the manuscript), the observed electromagnetic sector, with the photon as the induced gauge mode and the quantization of electric charges as structural properties of the ground-state \emph{el Pleno}. This derivation is one of the key results of the program.


\textbf{Mathematical form.} The action of \emph{el Pleno} is invariant under global phase transformations:


\[
\Psi \to e^{i\alpha} \Psi, \qquad \alpha \in \mathbb{R} \quad \text{constant}
\]

Equivalently, $S \to S$ and $\theta \to \theta + \alpha$. The Noether current associated with this symmetry is:


\[
J^\mu \propto i(\Psi^* \partial^\mu \Psi - \Psi \partial^\mu \Psi^*)
\]

and it satisfies the conservation equation $\partial_\mu J^\mu = 0$ on the solutions of the dynamics of \emph{el Pleno}.


\textbf{Immediate implications.} A5 is the structural base on which all the electrodynamics derived by the program is built. Without A5, the photon could not emerge as a specific configuration of \emph{el Pleno}, and the quantization of charges would have no structural origin. A5 also prepares the presentation of the topological charges developed in Part III of the manuscript, where the electric charges observed as simple fractions emerge from modular $\mathbb{Z}_3$ arithmetic applied to tripartite Borromean configurations of \emph{el Pleno}.


\subsubsection{I.5.7 --- Axiom A6: Hard structural upper bound}

\textbf{Conceptual statement.} The modulus $S$ of \emph{el Pleno} has an inaccessible structural upper bound: there exists a value $S_{\max}^\text{geom}$ such that $S < S_{\max}^\text{geom}$ at every point of \emph{el Pleno} and in every physical configuration. The medium cannot ``saturate completely'' --- every configuration of \emph{el Pleno} preserves a strict separation from the maximum bound. This upper bound is a structural property of the medium, not an effective approximation, and it delimits from above the operational range of the modulus of the field.


\textbf{Justification.} This axiom articulates the observation that a continuous physical medium does not admit arbitrarily high densities: there exists a structural limit above which the continuous-medium description ceases to apply. In \emph{el Pleno}, this hard bound has a specific function: it structurally blocks the cosmological and black-hole singularities. General Relativity predicts singularities (scale factor $\to 0$ in the Big Bang, curvature $\to \infty$ in the interior of black holes) where the theory itself ceases to apply. Under PIU-$\Psi$, these singularities \textbf{do not exist}: the hard bound $S_{\max}^\text{geom}$ prevents configurations of infinite density, and the dynamics of \emph{el Pleno} produces a structural bounce (primordial bounce, finite cores in black holes) instead of singularities.


A6 also has a physical justification from the analogy with known material media. Every real physical medium has a saturation density: water cannot be compressed indefinitely without a phase change; crystals cannot be packed beyond the geometric packing limit (Kepler-Hales for sphere packing in 3D). \emph{El Pleno} shares this structural property with all known physical media.


The specific value of the bound is obtained as a \textbf{derived theorem}, not as an additional axiom: the geometry of saturated \emph{el Pleno} fits the Kepler-Hales family of maximum packing, and the Kepler-Hales theorem (Hales 2005, \emph{Annals of Mathematics}; formal verification Flyspeck 2014) produces the canonical structural value:


\[
S_{\max}^{\text{geom}\,2} = \frac{8}{3}\sqrt{\frac{2}{3}} \approx 2.1773
\]

A6 declares that a hard structural bound \textbf{exists}; the specific value is a consequence of the packing geometry.


\textbf{Mathematical form.} The hard bound of the modulus is expressed as a structural condition on the field:


\[
S(x^\mu) < S_{\max}^\text{geom} \quad \forall\, x^\mu \in \mathcal{M}, \quad S_{\max}^{\text{geom}\,2} = \frac{8}{3}\sqrt{\frac{2}{3}}
\]

Operationally, this bound manifests in the F1 Lagrangian (\S{}I.7) through a component of the potential $V(S)$ --- specifically the topological potential $V_\text{topo}(S)$ with logarithmic Gent form --- that diverges when $S \to S_{\max}^\text{geom}$. The divergence makes the configuration $S = S_{\max}^\text{geom}$ dynamically inaccessible and produces a \textbf{structural ultraviolet wall} (UV in the sense of maximum density of the medium, complementary to the IR wall of A2 in the sense of $S \to 0$). This UV wall of A6 ---on the maximum \emph{value} of the modulus--- must not be confused with the regulation of A4, which acts on the \emph{curvature} of the modulus (the rapid spatial variation); both are ``ultraviolet'' in distinct but complementary senses, one in amplitude and the other in frequency.


\textbf{Immediate implications.} A6 guarantees the absence of singularities in the dynamics of \emph{el Pleno}. The primordial cosmological bounce (a bounce instead of the singular Big Bang) and the finite cores of black holes (compact interiors instead of a central singularity) are direct consequences of A6. Additionally, A6 establishes the absolute maximum bound of structural density of \emph{el Pleno}: $\rho_{\max,\text{abs}} = S_{\max}^{\text{geom}\,2} \cdot \rho_P \approx 2.1773\,\rho_P$, a quantitative prediction of the program with observable consequences for the cosmology of the primordial bounce (developed in \S{}II.7 and \S{}V.4).


The UV wall of A6 is complementary to the IR wall of A2 (strict positivity at $S \to 0$): together they bound the modulus of \emph{el Pleno} at both extremes. The overview of this double-wall architecture ---the operational range $S \in (0, S_{\max}^\text{geom})$ as an open interval, and its role in the stability of the medium--- is gathered in the synthesis of \S{}I.5.8.


\subsubsection{I.5.8 --- Synthesis: the six axioms as a minimal coherent structure}

The six axioms presented form a minimal and coherent structure. Each one fulfils a specific structural function:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{4.40cm}L{4.40cm}L{4.40cm}}
\toprule
\textbf{Axiom} & \textbf{Structural function} & \textbf{What it enables} \\
\midrule\endhead
\textbf{A1} --- Existence and uniqueness & Establishes the ontological domain & All phenomenology as configurations of $\Psi$ \\
\textbf{A2} --- Continuity, differentiability and strict positivity & Enables differential calculus; sustains the IR wall at $S \to 0$ & Lagrangian formulation; internal coherence of the field $\Psi = S e^{i\theta}$ \\
\textbf{A3} --- Covariant dynamics & Guarantees invariance under observers & Recovery of Lorentz, Noether's theorem \\
\textbf{A4} --- Ultraviolet regulation & Establishes the microscopic scale $\ell_P$ & UV consistency without renormalization \\
\textbf{A5} --- Conserved U(1) symmetry & Enables conserved currents & Derived electrodynamics, charge quantization \\
\textbf{A6} --- Hard structural upper bound & Blocks singularities; sustains the UV wall at $S \to S_{\max}^\text{geom}$ & Cosmological bounce, finite cores in black holes \\
\bottomrule
\end{longtable}}

The structure is minimal: no axiom can be removed without sacrificing functionality. The structure is coherent: no axiom contradicts another. The structure is complete for the development of the program: the six axioms are sufficient to construct the F1 Lagrangian (\S{}I.7) and, from it, derive all the phenomenology that the rest of the manuscript will develop.


\textbf{Structure of the walls of el Pleno.} Two axioms establish the walls that bound the range of the modulus $S$: A2 sustains the quantum IR wall at $S \to 0^+$ (strict positivity) and A6 the hard UV bound at $S \to S_{\max}^\text{geom}$ (maximum saturation). \emph{El Pleno} operates in the open interval $S \in (0, S_{\max}^\text{geom})$, with both walls dynamically inaccessible. A4 adds a distinct mechanism, which is not a wall on the value of $S$ but on its \emph{curvature}: the microscopic scale $\ell_P$ regulates the rapidly varying configurations, penalizing high structural curvature. Range walls (A2, A6) and curvature regulation (A4) operate together, and it is that combination that makes \emph{el Pleno} a stable physical medium with no singularities in any regime.

\textbf{The minimal structure, read from its root.} In accordance with \S{}I.4, this structure of six admits a hierarchical reading that does not alter its content but reveals its economy. The positivity of A2, the regulation of A4 and the bound of A6 ---the three that delimit the range of $S$--- are the three faces of a single pointwise finiteness of energy ($0 < E < \infty$): the floor, the curvature cut and the ceiling. The covariance of A3 and the U(1) symmetry of A5 are symmetries whose conservation laws (causal structure and charge) follow from that same energy by Noether's theorem. The only thing that finiteness and conservation do not deliver ---the continuity and differentiability of the field, today lodged in A2--- remains as the only independent technical axiom. Thus the six axioms are not six loose postulates of equal rank, but a root statement about energy, a technical requirement of smoothness, and four structural consequences of the former. The minimality of the list becomes, under this reading, minimality of foundation.


What these axioms are \textbf{not}: they are not self-evident truths; they are not arbitrary postulates; they are not creative inventions without support. They are \textbf{minimal physical assumptions} about \emph{el Pleno} as a continuous physical medium, each justifiable from established physical observations, each necessary for the development of the program.


What these axioms are also not: they are not the totality of the program. The axioms establish the framework; the specific dynamics of \emph{el Pleno} is encoded in the F1 Lagrangian (\S{}I.7), constructed as the minimal mathematical expression compatible with the six axioms.


\medskip\hrule\medskip

\subsection{\S{}I.6 --- The Lagrangian formalism: what it is, why it is used, how it is constructed}

\begin{estadoseccion}


\begin{itemize}[leftmargin=1.4em]
  \item \textbf{Maturity:} reviewed
  \item \textbf{Verification:} symbolic
  \item \textbf{Open threads:} none of its own (pedagogical section)
  \item \textbf{Notes:} pedagogical section. Prepares the reception of F1 (\S{}I.7): what a Lagrangian is, why fundamental physics is formulated this way, how it is constructed and what is done with it.
\end{itemize}
\end{estadoseccion}

\subsubsection{I.6.1 --- What a Lagrangian is}

A \textbf{Lagrangian} is a mathematical object --- a function --- that encodes the complete dynamics of a physical system in a single expression. Given the Lagrangian, a standard mathematical procedure produces all the behaviour of the system: the equations that govern its temporal and spatial evolution, the conserved quantities, the modes of oscillation, the stable solutions. The Lagrangian is the mathematical condensation of the physical system.


Joseph-Louis Lagrange showed in the 18th century that Newton's mechanics, formulated in terms of forces and accelerations, could be reformulated entirely by a single function --- the Lagrangian $L$ --- that for a simple mechanical system takes the form $L = T - V$, with $T$ the kinetic energy and $V$ the potential. From that single function the equations of motion are systematically obtained. The reformulation was not a change of notation: it revealed a deep structure. The real physical trajectories are those that make a quantity called the \textbf{action} \textbf{stationary}, the integral of the Lagrangian over time:


\[
S = \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt
\]

This is the \textbf{principle of least action} (or of stationary action, Hamilton's principle): among all the possible trajectories between two points, the system follows the one for which the action is stationary. The dynamics of nature has a \textbf{variational} structure.


Since the 19th century the formalism was extended from particle mechanics to continuous fields (Maxwell), to Special and General Relativity (Einstein, Hilbert), to Quantum Mechanics (Dirac, Feynman) and to Quantum Field Theory. \textbf{All contemporary fundamental physics is formulated by Lagrangians}; it is not a stylistic preference but the standard mathematical language. For a field $\phi(x^\mu)$ the Lagrangian generalizes to a \textbf{Lagrangian density} $\mathcal{L}(\phi, \partial_\mu \phi, ...)$, and the action is obtained by integrating it over space-time:


\[
S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi, ...)\, d^4x
\]

where $d^4x = dt\, dx\, dy\, dz$. The F1 Lagrangian of PIU-$\Psi$ --- presented in \S{}I.7 --- is a specific Lagrangian density for the field $\Psi$ of \emph{el Pleno}.


\subsubsection{I.6.2 --- Why fundamental physics uses Lagrangians}

Three convergent reasons make the Lagrangian formulation the standard in fundamental physics, and they are the same ones for which PIU-$\Psi$ is formulated by a Lagrangian.


\textbf{First reason: symmetries are encoded explicitly.} In the Lagrangian formulation, the symmetries of a physical system appear as invariances of the Lagrangian under specific transformations. If the Lagrangian does not change when a certain transformation is applied, the system described by that Lagrangian respects that symmetry. This is fundamental because, in physics, symmetries are directly connected to the conservation laws: \textbf{Noether's theorem} (Emmy Noether, 1918) establishes that every continuous symmetry of the Lagrangian produces a conserved quantity in the dynamics.


The equivalences produced by Noether's theorem are central to all fundamental physics:


{\small\setlength{\tabcolsep}{4pt}
\begin{longtable}{L{6.60cm}L{6.60cm}}
\toprule
\textbf{Symmetry of the Lagrangian} & \textbf{Conserved quantity} \\
\midrule\endhead
Translation in time & Energy \\
Translation in space & Linear momentum \\
Spatial rotation & Angular momentum \\
U(1) phase rotation & Electric charge (or analogous quantum number) \\
SU(2) gauge transformation & Weak isospin charge \\
SU(3) gauge transformation & Colour of the quarks \\
\bottomrule
\end{longtable}}

In each case, a continuous symmetry of the Lagrangian produces a conserved current in the dynamics. This correspondence is the structural base on which all modern physics is built. \textbf{Axiom A5 of the PIU-$\Psi$ program --- conserved U(1) symmetry --- works precisely via this theorem}: the invariance of the F1 Lagrangian under global rotations of the phase of the field $\Psi$ produces, by Noether, a conserved current that in the effective limit is identified with the electric current.


\textbf{Second reason: the equations of motion are generated systematically.} Given a Lagrangian density $\mathcal{L}(\phi, \partial_\mu \phi)$, the principle of stationary action --- the condition that the action $S = \int \mathcal{L}\, d^4x$ be stationary under variations of the field $\phi$ --- produces the \textbf{Euler-Lagrange equations}:


\[
\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} = 0
\]

This is a differential equation for the field $\phi$ that completely describes its dynamics. For the Lagrangian of classical particle mechanics, the Euler-Lagrange equations reproduce Newton's equations. For the Lagrangian of Maxwell's electrodynamics, they reproduce Maxwell's equations. For the Lagrangian of General Relativity (Einstein-Hilbert action), they reproduce Einstein's equations. \textbf{For the F1 Lagrangian of the PIU-$\Psi$ program, the Euler-Lagrange equations will produce the dynamical equation of \emph{el Pleno}} --- the equation from which all the subsequent phenomenology is derived.


The systematicity is key: given a Lagrangian, the equations of motion are obtained by a mechanical procedure, without requiring additional creativity. This makes the Lagrangian the fundamental object: if the Lagrangian is known, the complete dynamics is known.


\textbf{Third reason: quantization is systematic.} Once the classical theory is formulated by a Lagrangian, there exist systematic procedures to construct its quantum version: canonical quantization (via the Hamiltonian derived from the Lagrangian) or path integral (Feynman, based on the action). Both require the Lagrangian as a starting point. For PIU-$\Psi$ this is essential: F1 admits quantization by the Klauder procedure (affine quantization), developed in later parts of the manuscript.


\subsubsection{I.6.3 --- How a Lagrangian is constructed}

Constructing a Lagrangian for a new physical system follows a constructive discipline of four steps:


\textbf{Step 1 --- Identify the fundamental fields of the system.} What physical entities does the system describe? A real scalar field? A complex scalar field? A vector field? A tensor field? This identification corresponds to the mathematical object on which the Lagrangian will be constructed. For PIU-$\Psi$, the fundamental field is the complex field $\Psi = S \cdot e^{i\theta}$ of \emph{el Pleno}, with real modulus $S$ and real phase $\theta$.


\textbf{Step 2 --- Identify the symmetries the system must respect.} What transformations leave the physics of the system invariant? Translations in space-time? Lorentz rotations? Internal transformations such as global U(1) or SU(2) gauge? The Lagrangian must be invariant under all those symmetries. For PIU-$\Psi$, the required symmetries come directly from the axioms: structural covariance (A3) and global U(1) symmetry (A5), plus the symmetries implicit in A2 (continuity, which admits translations and rotational invariance).


\textbf{Step 3 --- Construct the terms compatible with the symmetries.} Given the field and the symmetries, one enumerates all the terms that can appear in the Lagrangian respecting those symmetries. Typically this includes a \textbf{kinetic term} (which encodes the propagation of the field), a \textbf{potential term} (which encodes the self-interactions of the field), and possibly \textbf{higher-order terms} (which encode corrective effects or regularizations). Each term must be constructed as a suitable combination of the field and its derivatives, so that the resulting expression is invariant under all the identified symmetries.


\textbf{Step 4 --- Assign coefficients to each term.} Each term of the Lagrangian comes multiplied by a coefficient that encodes the intensity or scale associated with that term. These coefficients can be free parameters (calibrated against observation) or can be derived from more fundamental structural properties. For PIU-$\Psi$, the coefficients are constructed from three fundamental structural parameters of \emph{el Pleno} --- fundamental density $\rho_P$, microscopic scale $\ell_P$, structural speed $c$ --- which will be interpreted in \S{}I.8 as properties of the medium.


The resulting Lagrangian is the \textbf{minimal mathematical expression compatible with the structural axioms of the system}. ``Minimal'' means that no unnecessary terms are added: each term fulfils a specific structural function. This minimality is not aesthetic but methodological discipline --- an overloaded Lagrangian introduces free parameters without justification and weakens the predictive power. It is the procedure that \S{}I.7 applies to produce F1: field ($\Psi$), symmetries (covariance, U(1), UV regulation), compatible terms (kinetic, potential, flexural) and structural coefficients ($\rho_P$, $\ell_P$, $c$).


\subsubsection{I.6.4 --- What is done with a Lagrangian}

Once the Lagrangian is constructed, what follows is a program of systematic derivation:


\textbf{Step 1 --- Apply the Euler-Lagrange equations to obtain the field equation.} This is the dynamical equation that governs the evolution of the field through space-time. For PIU-$\Psi$, this step produces the dynamical equation of \emph{el Pleno}, which describes how $\Psi$ evolves at each point of space-time under the laws established by the six axioms.


\textbf{Step 2 --- Identify the stable solutions of the field.} The stable solutions are configurations of the field that persist in time without disintegrating --- they are the ``natural states'' of the system. For PIU-$\Psi$, the stable solutions will include the ground-state \emph{el Pleno} (state of minimal excitation, identified phenomenologically with the ``vacuum'') and localized solitonic configurations (identified phenomenologically with the particles).


\textbf{Step 3 --- Identify the propagation modes.} Small perturbations around a stable solution propagate through the system as waves. The mathematical structure of the Lagrangian determines the speed, the polarizations, and the dispersion relations of these waves. For PIU-$\Psi$, the modes will include the transverse tensor mode (identified phenomenologically with the gravitational waves) and the induced gauge mode (identified phenomenologically with the photon).


\textbf{Step 4 --- Identify the conserved quantities via Noether's theorem.} Each continuous symmetry of the Lagrangian produces a conserved current. These currents are identified phenomenologically with the observed conservation laws (energy, momentum, electric charge, etc.).


\textbf{Step 5 --- Derive specific observables for confrontation with experiment.} The stable solutions, the propagation modes, and the conserved quantities have specific observable consequences that can be compared with experimental data. For PIU-$\Psi$, these consequences include the value of the gravitational constant $G$, the value of the Planck constant $\hbar$, the spectrum of the elementary particles, the cosmological predictions $\Omega_{MO}/\Omega_M$ and $\Omega_{EO}$, and others.


This five-step program --- field equation, stable solutions, modes, conservations, observables --- is that of the manuscript from \S{}I.7 onward: F1 is the input, and Parts II, III and IV its development. F1 does not appear out of nowhere: it is the consequence of the standard procedure of Lagrangian construction applied to a physical medium with the axioms A1--A6 as structural assumptions.


\medskip\hrule\medskip

