Theoretical foundations of the MCMC
The structuring variable, the sequence of thresholds, the seals that fix the rules of the world and the instant at which time is born.
The model 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 and that of hosting, which inherits an irreducible imperfection from each cycle. That imperfection is at once the defect that prevents eternal inertia and the engine of everything that follows.
The primordial duality
The foundational object is not an empty spacetime but a dual unity with two inseparable aspects. The distinction between the two is internal, not spatial: they are not two things located in different places, but two capacities of one and the same reality, each present in its entirety at every point.
The determinate or condensable aspect: localization, density, structural closure. The tendency toward form.
The indeterminate aspect, or aspect of openness: extension, degrees of freedom, expansion. The tendency toward indefiniteness.
A primordial imperfection $\delta_0 > 0$ prevents the exact cancellation between the two. That mismatch is the only free input of the model, together with a handful of shape constants of order unity, and from it the driving tension is derived by means of a scaling law:
$$T_0 = \bar{c}\,\delta_0^{\,3}$$
The releasable tension of the cycle. It vanishes if and only if the imperfection vanishes: the perfect state is symmetric, inert and sterile.The internal setting of that duality is the Dual Plane, whose coordinates are the effective projections of both capacities. The tensional modulus measures how much reality has detached from the symmetric point; the conversion phase measures how much of what is condensed has become openness. And there is a diagonal where the two are equal, which is the axis of the system.
The structuring variable
The evolution is not indexed by time, which does not yet exist, but by a structuring variable $S$: a dimensionless, monotone non-decreasing index, defined as accumulated entropy production, which measures tensional reconfiguration. It is not physical time, nor is it supposed to be. Its monotonicity is not postulated: it is proven, by means of the Path Monotonicity Theorem, from the dissipative character of the flow.
The working gauge unit is $\Delta S = 10^{-3}$, and the treatise is explicit about its status: it is a choice of units, not a new physical constant. The quantum is real; its number is conventional. The later connection between $S$ and the observable redshift is established by the Law of Cronos at the background level, not the other way around.
Note on earlier versions. Earlier formulations of the model described $S$ as an entropic parameter bounded in the interval from 0 to 100, with a calibrated value of around 95 and a present epoch close to 90. The Tratado de Fundamentos does not contain that parametrization: it treats $S$ as a structural index, defined as accumulated entropy production and with no upper bound fixed in advance.
The sequence of thresholds
The discharge is not continuous: it proceeds by discrete thresholds, and at each one a rule of the world is irreversibly fixed. What is remarkable is that the four thresholds are not arbitrary stations but logarithmic echoes of a single transition repeated at a multiplicative scale. They reduce to three pieces of data: the Decade ratio, the gauge unit and a one-quantum rule.
$$S_k^{\text{col}} = \lambda^{\,k-2} - \Delta S \quad (k = 0,1,2), \qquad S_{\text{Florencia}} = \lambda^0 + \Delta S$$
With $\lambda = 10$ and $\Delta S = 10^{-3}$, the four thresholds are 0.009 · 0.099 · 0.999 · 1.001.| Threshold | What it opens or seals |
|---|---|
| $S_{0,009}$ | Dimensional collapse to $1D$: directional propagation appears. |
| $S_{0,099}$ | Dimensional collapse to $2D$. The sealing of the causal speed by the Delivery Bound is located in this vicinity. |
| $S_{0,999}$ | Dimensional collapse to $3D$. The seal of quadratic conversion, the mass-energy threshold, is located in this vicinity. |
| $S_{1,001}$ | Florencia threshold: signature change to $3{+}1D$. Relative time is born, and with it the Relative Dual Metric. This is where the electroweak scale, $v_3 = 246$ GeV, is sealed. |
There are two one-quantum shifts that are not technical details but ontological statements. The quantum of anticipation—the negative sign in the dimensional collapses—says that the structure that no longer fits in $d$ dimensions forces the next one a quantum before saturating the decade: the collapse is an anticipated act, not a late consequence. The quantum of confirmation—the positive sign at Florencia—says that time is not born at the perfect equilibrium between the two capacities, which is timeless by nature, but one quantum later. What is dated is the excess over the equilibrium.
The Chain of Algebras
Each dimensional collapse adds a generator without yet adding time. The working algebra of each stretch includes the spatial generators plus an entropic generator that marks the direction of structuring, all of positive norm: up to the last threshold, the whole chain is Euclidean.
$$C(2,0) \xrightarrow{\;S_{0,100}\;} C(3,0) \xrightarrow{\;S_{1,000}\;} C(4,0) \xrightarrow{\;S_{1,001}\;} C(3,1)$$
The last link is the signature change: a single generator changes norm and the light cone is born.On this chain the maximum speed of propagation is sealed by two independent routes: a transport bound and the gap of the Euclidean operator. It should be noted that the first of them, in its form as an identification of the speed of light with the transport speed, is classified in the treatise as conditional; the sealing by the gap route is independent of it.
Correction with respect to the earlier corpus. The treatise explicitly points out that the algebraic spine must include one more generator than the spatial dimension of the stretch. Earlier formulations labeled the stretches with one generator fewer, even though their own equations used the additional one for the direction of structuring. That entropic generator was the missing link.
The birth of time
At the final threshold no generator is added: one is rotated. Defining $\gamma^0 \equiv i\gamma_S$ yields the signature change to $(-,+,+,+)$, and the direction of structuring becomes the temporal direction. This is the point at which the decay rates become lived duration.
That rotation is not enough by itself. For it to produce a unitary history—a Hilbert space of positive norm, a Hamiltonian bounded from below and a unitary evolution—the prior Euclidean theory must satisfy reflection positivity with respect to the hyperplane of the threshold. The Precedence Lemma establishes that the prior sealing of the constants is a necessary condition for that reconstruction: time requires rules that are already fixed.
After the threshold
From there on, the same framework is projected onto the background cosmological evolution, structure formation and local gravitational environments. The conversion of mass into space generates two channels of ontologically distinct nature but phenomenologically coupled: one enters directly into the expansion and the other acts as potential mass in suspension, which does not appear as an observable vacuum density but subtly curves the metric.
In the ontological reading of the model, dark energy thus ceases to be an added fluid and becomes the late imprint of the pre-geometric landscape, and cosmic acceleration the macroscopic consequence of a conversion that is still taking place. Empirically, that reading is not established: the parameters of the channels are calibrated and, in the production fits, the amplitude of the transition $\varepsilon_\Lambda$ is compatible with zero (comparison).
The complete axiomatic architecture, with Adrián's Theorem, is in the eight axioms; the primordial state from which everything emerges, in The Root. The status of each result, in what is proven and what is not. The reading from philosophy, in The Living Grammar. And the full text in the Tratado de Fundamentos.