MCMC versus ΛCDM
Where they agree, where they diverge, and what concrete observation would decide between the two.
This page is a translation of the Spanish original (30 September 2026). Where they differ, the Spanish text and the Tratado de Fundamentos prevail.
An honest comparison with the standard model begins with an acknowledgement that alternative proposals usually omit: in the cosmological background, the MCMC remains nearly degenerate with ΛCDM. The acoustic peaks and the BAO phase are preserved at the sub-percent level. That degeneracy is not a defect—an extension that did not recover what is known would be ruled out from the outset—but it means that the background, on its own, does not distinguish between the two.
What is interesting lies where the degeneracy breaks, and in which of those windows the difference is large enough to be measured.
The structural difference
Both describe an expanding universe with baryonic content, radiation and a dark sector, and both reproduce the acoustic peaks of the cosmic background and the BAO scale. The MCMC contains ΛCDM as an exact limiting case.
In ΛCDM, the cosmological constant is a rigid parameter and dark matter a component added to the inventory. In the MCMC, both are consequences of one and the same process: the conversion of primordial mass into primordial space, which is still taking place.
The conversion generates two channels of an ontologically distinct nature. One enters the expansion directly; the other acts as potential mass in suspension, which does not appear as an observable vacuum density but subtly curves the metric. Dark energy ceases to be an added fluid and becomes the late-time imprint of the pre-geometric landscape.
$$H^2(z) = \frac{8\pi G_{\text{eff}}}{3}\Big[\rho_b(1+z)^3 + \rho_{\text{cdm}}(1+z)^3 + \rho_r(1+z)^4 + \rho_{id}(z) + \rho_{\text{lat}}(z)\Big] - \frac{k(1+z)^2}{a_0^2}$$
The tensional Friedmann equation. The last two terms in the bracket are the emergent channels that replace the rigid constant.From this follows a relative cosmological constant, which is not constant but transitions smoothly:
$$\Lambda_{\text{rel}}(z) = \Lambda_0\left[1 + \varepsilon_\Lambda \tanh\frac{z_{\text{trans}} - z}{\Delta z}\right]$$
With $\varepsilon_\Lambda = 0.012 \pm 0.003$ (calibrated, not derived), $z_{\text{trans}} = 8.9 \pm 0.4$ and $\Delta z \approx 1.5$. The effective equation of state goes from values close to zero at high redshift to $-1$ at low redshift. In the production fits, $\varepsilon_\Lambda$ is compatible with zero. C6 convention of FE-08-2026: $\varepsilon_\Lambda$ is not $\delta_0$, $\varepsilon_K$ or $\varepsilon_c$.The comparison in figures
The decisive tests have already been run in the code repository, and they are published with their outcome even when it is unfavorable. ΔAIC and ΔBIC are read as MCMC − ΛCDM: a positive value favors ΛCDM.
| Test | Data | Result |
|---|---|---|
| Production fit v1 (10 Aug 2026, corrected background) |
31 H(z) from chronometers + 6 BAO + 1371 SNe Ia (n = 1408) | ΔAIC = +4.00 · ΔBIC = +14.50. $\varepsilon_\Lambda$ compatible with zero; H₀ = 67.87 ± 0.68 km s⁻¹ Mpc⁻¹. |
| Production fit v2 (10 Aug 2026, corrected background) |
The v1 data + compressed CMB + 11 fσ₈ (n = 1422) | ΔAIC = +4.08 · ΔBIC = +14.60. $\varepsilon_\Lambda$ compatible with zero; H₀ = 68.19 ± 0.40 km s⁻¹ Mpc⁻¹; σ₈ = 0.799 ± 0.029. |
| DESI DR2, BAO (18 Aug 2026) |
13 DR2 components | Uniform verdict in favor of ΛCDM across eight configurations: ΔBIC between +4.47 and +4.90. $\varepsilon_\Lambda$ cannot be probed by BAO in this parametrization ($z_{\text{trans}} \approx 8.9$). |
The verdict is the same in all three tests: with current data, the model's additional parameters do not buy a fit that compensates for their cost, and the model recovers ΛCDM with no advantage. The figures that this page previously presented as corpus validations—a Hubble constant between 69 and 70, a slightly reduced σ₈ and a cored profile fitted to SPARC—come from the corpus and remain as a historical record (FE-08-2026, E5–E7). The full CMB spectrum and weak lensing do not yet enter these fits.
The recovery limit
An extension is only acceptable if it can give back the known physics, and the model does so explicitly. If the ontological tension switches off—when the transition parameter, the tensional field and the Cronos correction tend to zero and the gravitational constant reaches its fixed point—then the Relative Dual Metric collapses to the Friedmann–Robertson–Walker metric, the propagation speeds of gravitational waves and of scalar sound tend to unity, the latent channel is absorbed into a constant and the tensional Friedmann equation reduces exactly to the equation with constant $\Lambda$. The growth prediction then coincides with that of ΛCDM.
For primordial nucleosynthesis, consistency is not taken for granted: it has been tested. The test could have excluded the predicted value and does not exclude it; it is a consistency check passed, with no detection and no sign preference (FE-08-2026, E13), and the repetition in the repository, with its own preregistered band, gives the same outcome, provisional until the provenance of the data is verified. For recombination, the treatise argues for consistency, because at those redshifts the temperature comfortably exceeds the scales of the active thresholds.
What would decide between the two
This is the question that matters, and it is best posed the way a skeptical reviewer would pose it: what combination of current data would falsify the model against ΛCDM after penalizing for the number of parameters?
The program's answer was a Bayesian fit with the AIC and BIC information criteria, because a model with more freedom that fits equally well gains nothing: the comparison has to penalize that freedom. That fit has already been run on cosmic chronometers, BAO, supernovae and, in its v2 version, compressed CMB and growth of structure, and its verdict is the one in the table above: ΛCDM is favored. The full CMB spectrum and weak lensing have yet to be incorporated.
Beyond the background, the treatise points to three concrete empirical fronts where the model puts itself on the line: the comparison between the cosmological gravitational constant and the laboratory measurement, with a predicted deviation of around $-1.8\,\%$ that must be confronted with the bounds from nucleosynthesis and the cosmic background; the collapse gate in virialized halos, which must be active during collapse and switched off afterwards; and a bound on the Cronos amplitude below $10^{-6}$.
On the status of these figures. The tests in the table are negative results, published as such: with current data, the model does not improve on ΛCDM. The treatise described the behavior in the Hubble constant and in S₈ as directions consistent with the relief of those tensions and referred to a pending global fit; that fit has been carried out and does not confirm the relief. The corpus figures must not be cited as the current state.
It is worth adding the objection that any reviewer will raise, and which has not been resolved: as long as the background remains nearly degenerate with ΛCDM, it has to be demonstrated that the emergent dark sector is observationally distinguishable from an ordinary fluid or from a modified-gravity parametrization. Whether the model says something new or rewrites what is already known in another language depends on that answer.
The detailed status of each result is in what is proven and what is not, the numerical scheme in simulations, and the architecture of the model in the eight axioms.