The eight axioms and Adrián's Theorem

The deductive architecture of the Multiple-Collapse Cosmological Model: from a single inherited imperfection to basal Theorem 13.1a, proven within the axiomatic system; the complete sealing of the constants remains a conjecture (13.1b), according to the errata FE-08-2026.

Source: Modelo Cosmológico de Múltiples Colapsos — Tratado de Fundamentos (Multiple-Collapse Cosmological Model — Treatise of Foundations), full edition. Current scientific record: v35, published on 19 June 2026 · DOI 10.5281/zenodo.20765373, with the errata FE-08-2026 (4 August 2026), which prevails where it reclassifies a status · next version: v36 · permanent link to the latest version: 10.5281/zenodo.14167831. Author: Adrián Martínez Estellés · ORCID 0009-0009-4314-9642.

This page is a translation of the Spanish original (30 September 2026). Where they differ, the Spanish text and the Tratado de Fundamentos prevail.

The MCMC describes the universe as a cyclic process of tension discharge. At its origin there is no space, no time and no matter: there is a single reality with two inseparable capacities, that of condensing—Primordial Mass, $M_p$—and that of hosting—Primordial Space, $E_p$. That reality inherits from each cycle an irreducible imperfection, $\delta_0$, which is at once the defect that prevents eternal inertia and the engine of everything that follows.

What distinguishes the Tratado de Fundamentos from earlier formulations of the model is that this account ceases to be a narrative illustrated with formalism and becomes a closed deductive chain. Each link—“imperfection generates tension”, “tension discharges along the route of least resistance”, “time is born when a direction rotates”—ceases to be metaphor and becomes a theorem, a declared condition or a named conjecture. The whole construction starts from eight ontological axioms that are its only conceptual input: everything that follows is definition, theorem or conjecture explicitly declared as such.

The eight axioms

Axiom 1 Dual unity

Reality is a single unity with two inseparable capacities: to condense ($M_p$) and to host ($E_p$). Neither exists without the other, and their distinction is internal, not spatial.

Axiom 2 Inherited imperfection

Every cycle inherits an imperfection $\delta_0 > 0$. The case $\delta_0 = 0$ defines the perfect state: symmetric, inert and sterile.

Axiom 3 Primordial Tension

There exists a releasable energy of the cycle, $T_0 \geq 0$, with $T_0 = 0 \iff \delta_0 = 0$.

Axiom 4 The Path

The fundamental evolution is discharge of tension along the route of least resistance; it produces structure and admits no reversal.

Axiom 5 Collapses and seals

The discharge proceeds through discrete thresholds; at each seal a rule of the world is irreversibly fixed.

Axiom 6 Pre-geometric atemporality

Before the final threshold there is no time: only correlations ordered by the structuring $S$.

Axiom 7 Law of Cronos

Time is born as a causal projection of the flow of $S$; its local flow rate is the local rate of discharge.

Axiom 8 Victoria

The residue of each discharge founds the next cycle. The return is not guaranteed: it is won.

The deductive chain is linear. Axioms 1 to 3 fix the internal kinematics; axiom 4, the dynamics; axioms 5 and 6, the Euclidean stretch; axiom 7, the birth of time; and axiom 8, the cyclic closure. No later construction adds new ontological assumptions.

The deductive chain

On the pre-geometric stretch $S_0 \rightarrow S_{1.001}$—from the primordial state to the threshold at which time is born—five pieces are built. On them the five open problems are developed in full, and the mass spectrum closes the construction before the synthesis.

Ch. 2
The Dual Plane The internal arena of the duality, $\mathcal{D} \cong \mathbb{R}^2$ with coordinates $(\varphi_M, \varphi_E)$. It is legitimate even on a zero-dimensional support: the duality of internal dimension two is compatible with a point without dimensions, which dissolves the apparent paradox of the origin. It includes the Dimensional Orthogonality Lemma.
Ch. 3
The Basal potential It realizes the axiom of tension with a landscape whose only defect is $\delta_0$. In the exact limit $\delta_0 = 0$ the well is flat to sixth order: symmetric, inert, with nothing that begins. Hence the scaling law $T_0 = \bar{c}\,\delta_0^{\,3}$ and the false-vacuum structure.
Ch. 4
The Path flow The dissipative dynamics: not a force applied to the Field, but the descent along the route of least resistance. The Path Monotonicity Theorem proves the arrow of structuring instead of postulating it—the temporal arrow is induced afterwards by the Florencia Rotation (FE-08-2026, E11)—and the Sealing Proposition identifies the seals with fixed points of the flow.
Ch. 5
The Chain of Algebras Each collapse adds a Clifford generator without yet adding time. The speed of light is sealed by two independent routes: the Delivery Bound and the Euclidean gap.
Ch. 6
The Florencia Rotation At the final threshold no generator is added: one is rotated. Defining $\gamma^0 \equiv i\gamma_S$, one passes from $C(4,0)$ to $C(3,1)$ with signature $(-,+,+,+)$. A single rotation turns the direction of structuring into a temporal direction. Here Lorentzian time and the canonical Relative Dual Metric are born.
Ch. 7
Reflection positivity For the rotation to produce a unitary history, the Euclidean theory must satisfy $\langle \vartheta(\bar{F})F \rangle \geq 0$ on the Florencia slice. The Florencia Positivity Theorem and the Osterwalder–Schrader reconstruction give unitary time; the Precedence Lemma establishes that prior sealing is a necessary condition.
Ch. 8
The Law of the Decade The thresholds are not arbitrary stations but logarithmic echoes of a single transition repeated on a multiplicative scale, a signature of discrete scale invariance produced by complex critical exponents: $\lambda = e^{\pi/s_0}$, which for $\lambda = 10$ gives $s_0 = \pi/\ln 10 \approx 1.3644$. $\lambda = 10$ itself is calibrated, not derived: deciding whether it is physics or gauge convention is the second open front. It is completed by the Discriminant Mechanism.
Ch. 9
Gea, Newton and Atlas The gravitational sector. The spatial metric is the tensorial face of the Adrián Field. General relativity is not postulated: it is the seal where the foliation symmetry reaches its fixed point, $(\lambda_K, \xi) \rightarrow (1,1)$. The Atlas Identification recognizes the extra scalar mode of the foliation as the tensional field $\Phi_{\text{ten}}$ itself, so that there is no need to postulate a separate kinetic term for it.
Ch. 10
The Victoria cycle A hybrid system of flow plus restart, with return map $F_{\text{Vic}}: \delta_0 \mapsto \delta_0'$. It incorporates the Memory mode as an $O(2)$ pseudo-Goldstone, the activation floor and the Lydia exponent $\nu$, which decides whether the return grows or dies out.
Ch. 11
The Weak Law of Cronos The observable version: where mass condenses, conversion stagnates; where it stagnates, time thickens. Three falsifiable corrections and the Cronos v3 numerical scheme.
Ch. 12
The Path Functional The mass spectrum. Each mass is written as a weighted sum of the scales of the seals, $m_i = \sum_n c_{in}\,\vartheta_n v_n$: every particle remembers the tensional history of the collapses that created it.

Adrián's Theorem

Chapter 13 gathers the eight axioms into a single statement. The errata FE-08-2026 (E1/E2), which prevails over v35 until the publication of v36, corrects its declared scope without modifying any equation or proof: what v35 presented as a single theorem, with all the constants sealed, is divided into two pieces of distinct status.

Theorem 13.1a — Adrián, basal · proven

The core of the theorem: the scalar sector of the pre-geometric stretch $S_0 \rightarrow S_{1.001}$. It is an internal theorem: valid within the axiomatic system; not by itself an empirical confirmation.

Conjecture 13.1b — Adrián, complete · program

The sealing of all the constants and the extension of the integration to fermions, the gauge sector, gravity and the masses. It is not a closed theorem but a work program; its open pieces appear among the conditional results and the seven fronts on this page.

For reference, this is the single statement of v35, already reclassified. Its clause iii has the E11 correction applied—what the flow proves before Florencia is the arrow of structuring, not the arrow of time—and clause v notes the E3 reclassification.

Statement of v35 (Theorem 13.1) · reclassified by FE-08-2026

Under Axioms 1–8, the pre-geometric stretch $S_0 \rightarrow S_{1.001}$ defines a Lorentzian and unitary quantum field theory, in the sense that:

  1. The Adrián Field is well defined on every dimensional stretch, including $V_{0D}$.
  2. There exists a driving tension $T_0 = \bar{c}\,\delta_0^{\,3} > 0$ that vanishes only in the perfect state.
  3. The evolution is monotonic—$V \downarrow$, $S \uparrow$, $T \downarrow$—and the arrow of structuring is a theorem, not a postulate; the temporal arrow is induced afterwards by the Florencia Rotation.
  4. The constants $c$, $c^2$ and $m_{\min}$ are sealed as fixed points of the flow.
  5. The Euclidean theory is reflection-positive on the Florencia slice, so that the Florencia Rotation reconstructs a positive-norm Hilbert space with Hamiltonian $\hat{H} \geq 0$ and unitary evolution $e^{-it\hat{H}}$, prior sealing being a necessary condition for that reconstruction. Reflection positivity of the spinor sector (Wilson) becomes a conditional proposition (E3).
  6. The cosmic cycle is the fixed point of the Victoria return map, attractive under explicit conditions.

In its basal part, the proof chains the cited results without circularity: the dependence is a partial order without cycles, and each result uses only results earlier in the chain. Thus, the stretch that runs from the dual unity to the threshold at which time is born is written as a chained mathematical statement; the sealing of all the constants is the part that remains open (13.1b).

What is proven, what is conditional, what remains open

The treatise subjects each part to twenty-four internal consistency checks of the implementation—identities, signs, recovery limits and negative controls—all of them passed. They establish that the formalism does what it declares; they do not establish, by themselves, that the ansatz is physically true, that its parameters come from nature, or that the extrapolation to the continuum is valid (FE-08-2026, E8). The classification that follows is part of the model's own method: to distinguish precisely what holds from what does not yet.

Proven

Dimensional Orthogonality; structure of vacua and $T_0 = \bar{c}\,\delta_0^{\,3}$; Path Monotonicity; Euclidean ellipticity; Florencia Positivity and Osterwalder–Schrader reconstruction; Precedence; the Law of the Decade as an arithmetic reduction; the Discriminant Mechanism; Path Exclusion; the $O(2)$ Memory mode; entropic saturation; the Return of Victoria given hypotheses (A2)–(A4); the modified geodesic and its three corrections; and Adrián's basal Theorem 13.1a.

Conditional

Reflection positivity of the spinor sector (Wilson); stability of the Atlas mode with $\lambda_K \downarrow 1^+$; the sign of the Lydia exponent, $\nu > 0$; the Residues Conjecture; and the Delivery Bound as the identification $c \equiv v_{LR}$.

Reclassified by FE-08-2026

Adrián's Theorem in its complete form, which becomes Conjecture 13.1b (program); the count $N_{\text{gen}} = 3$, which becomes a structural identification (Prop. 12.4 without proof), with its derivation as a sub-front of front 7; Wilson reflection positivity, which becomes a conditional proposition; and the twenty-four verifications of Appendix H, which become internal consistency checks of the implementation.

Open

Seven declared research fronts, each reduced to a concrete calculation or an observational test. They are detailed below.

On the recovery limits. The match with known physics is checked in three limits: $\delta_0 \rightarrow 0$ gives back the eternal inertia of the perfect state; $(\lambda_K, \xi) \rightarrow (1,1)$ gives back general relativity through Newton's Seal; and $\varepsilon_c \rightarrow 0$ gives back the standard Friedmann–Robertson–Walker geodesic.

The seven open fronts

The open program is not a list of gaps but a map of bifurcations: each front admits several ways out, and the ontology of the MCMC predicts what its reading would be in each case.

Complete reflection positivity Detailed treatment of the positivity of the spinor sector without doublers, and extension to non-stationary $K(S)$ couplings far from the seals. The underlying question is whether time can be born on rules that are still changing. Status as of 30 Sep 2026. Toy model: specularity is sufficient, without proof that it is necessary. Wilson positivity remains open and is now a conditional proposition (FE-08-2026, E3).
Derivation of $\lambda$, the Number of the Path Integrate the flow functions of the Basal potential and read $\Delta \ln S_{\text{walk}}$ from the discriminant, to decide whether $\lambda = 10$ is physics or gauge convention. Status as of 30 Sep 2026. Two published negative results: the Fokker–Planck β (outcome A at the preregistered point) and the preregistered screening of 28 September (147 runs, no crossing of the discriminant) do not derive the Decade. $\lambda = 10$ remains calibrated.
The closure of Gea Closure of the constraint algebra with local $N(x,S)$, the strong-coupling regime of the Atlas mode, and derivation of $\Lambda_{\text{ten}}(\Phi_{\text{ten}})$ from the Basal potential. Status as of 30 Sep 2026. Newton's Seal is exact in the suite and the Atlas channel is derived from the action (internal check, E8); the preferred-frame PPN bounds set $\alpha_a \le 8\times10^{-7}$. Atlas remains conditional.
The Return of Victoria Compute the sign of the Lydia exponent $\nu$ and the Victoria Ceiling $W_{\max}$ from the microdynamics, and estimate the attractor $\delta_0^{*}$. On this depends whether the cosmos is fertile or whether there is a terminal cycle without offspring. Status as of 30 Sep 2026. The circle of $\delta_0$ reduces to a single number, the required Ceiling: $W_{\max} \approx 1.87\times10^{-4}$ with the fiducial shapes. The sign of $\nu$ remains conditional.
Cronos v3 in production Run the simulations with the gate friction, the exponent 2 and the force $+c^2\nabla\varepsilon_c$; verify the bound $\alpha_0^{-1} \lesssim 10^{-6}$. Status as of 30 Sep 2026. No production: the core has not been produced. Level A, the Cronos–Jeans criterion (rounds 1 and 2) and the halo and shell series return INDETERMINATE. The vertical kinematics of the solar neighborhood (Oort–$K_z$) excludes, in the plane, the amplitude calibrated in Sculptor ($A \lesssim 0.06\,A_{\text{Sculptor}}$ with the declared slab). SPARC and the Sculptor profile have not been ingested.
The Residues contrast Confront $G_{\text{cosmo}}/G_N - 1 \approx -1.8\%$ with the bounds from primordial nucleosynthesis and the CMB versus Cavendish-type measurements. Status as of 30 Sep 2026. Primordial nucleosynthesis, preregistered: consistency check passed at 0.32σ, with no sign significance (E13); the front remains refutable. Data transcribed from the publications, provisional on provenance grounds.
The Path Functional and the masses WKB calculation of the weights $c_{in}$ and independent derivation of $\beta_3$. Status as of 30 Sep 2026. The C¹ splice measures $\delta_H \approx 0.055$ without circularity (distinct from $\delta_0^{*}$, C6 convention); $\beta_3$ remains conditional and the mass spectrum, calibrated. $N_{\text{gen}} = 3$ becomes a sub-front (E4).

In sum, there are three concrete empirical fronts where the model puts itself on the line: the $G_{\text{cosmo}}/G_N$ contrast, the collapse gate in virialized halos, and the bound $\alpha_0^{-1} \lesssim 10^{-6}$ on the Cronos amplitude. As of 30 September 2026, none has yielded a favorable outcome: the Residues contrast is consistent without detection, the gate is implemented but without production, and the Cronos amplitude calibrated in Sculptor is excluded in the solar plane, conditional on the declared slab and on the weak form of the law. The detail, row by row, is in the computational status.

The reading of this same chain from the Philosophy of the Art of the Path—the correspondence between each mathematical structure and its lived meaning—is developed in the table of correspondences. The full text, with the complete proofs and the validation appendix, is in the Tratado de Fundamentos, and the complete chronology of deposits in publications.