Simulations — Cronos v3
The current numerical scheme: how the entropic friction that the treatise deduces becomes measurable cores and a falsifiable signature.
The simulations are not an illustration of the model: they are its operational translation into a language that observations can verify or falsify. The chain of collapses and the dynamics of Cronos become numerical evolution of fields and particles, and from there come concrete predictions about halos that can be compared with real catalogs.
The Cronos–KDK integrator
The treatise specifies a module that extends a Gadget-type N-body code, with the Kick–Drift–Kick leapfrog scheme, adding two ingredients that come directly from the Relative Dual Metric: a time step sensitive to the Cronos dilation and an extra force from the effective vacuum.
In the Newtonian limit, the comoving equation of motion incorporates a third term that does not appear in the standard scheme:
$$\ddot{\vec{r}}_i = -\frac{1}{a^3}\nabla\Phi_N - 2\frac{\dot{a}}{a}\dot{\vec{r}}_i - \tfrac{1}{2}\dot{\delta}\,\dot{\vec{r}}_i$$
The third term is the entropic friction of Cronos, the dynamical sibling of the gate friction of Chapter 11.Gravity additionally incorporates the sources of the dark sector by means of a modified Poisson equation, in which the densities of the two emergent channels enter as sources alongside matter.
The entropic time step
The heart of the scheme is the modification of the step in the Kick. In dense regions the entropic clock runs slower: the step is reduced and the dynamics are damped. It is the same lapse of Chapter 11 acting particle by particle.
$$\Delta t_i = \frac{\eta}{\sqrt{|\vec{a}_i|\,a}}\left[1 + \frac{1}{\alpha_{cr}}\left(\frac{\rho_i}{\rho_c}\right)^{3/2}\right]^{-1}, \qquad \eta = 0{,}025$$
With the local density obtained by octree or Voronoi tessellation. The sign is the one corrected in Observation 11.4, which replaces that of the old scheme.Why v3. The current scheme corrects the sign of the friction with respect to earlier versions and introduces the gate friction: it switches on only while the region collapses and switches off upon virialization. Any result obtained with the old scheme must be reproduced with this one before being considered valid.
The integration cycle
A conceptual detail worth keeping in mind: the evolution is parametrized in the ontological variable, with a step of $10^{-3}$. The integrator does not advance in a time postulated beforehand, but in tensional index. Time is an output of the calculation, not an input.
Implementation
The production variant Gadget-4-Cronos is specified in Appendix B.4, with flags for the time-step law, for the emergent channel as a passive particle-mesh grid and for the latent channel, and its configurations are in the repository; but it has not been run. What the repository runs today is Cronos v3 on a minimal particle-mesh grid, sufficient for collapse pairs with identical seeds, not for the production boxes.
In the treatise's design, the global synchronization maintains the usual power-of-two bitmask scheme, so that the parallelization is not broken, and the overhead of the adaptive step stays below 1%. These are design properties: they will be measured when the production variant is run.
| Scale | Box size | Resolution |
|---|---|---|
| Local | 25–50 h⁻¹ Mpc | 512³ |
| Meso | 100–200 h⁻¹ Mpc | up to 1024³ |
| Large-scale structure | 500–1000 h⁻¹ Mpc | ~1024³ |
From the earlier corpus to the current status
The comparisons are made with pairs of runs with identical seeds, one under ΛCDM and the other under the model, so that the differences cannot be attributed to the specific realization of the initial conditions.
| Test | Earlier corpus (historical record) | Current status |
|---|---|---|
| Isolated halo of $10^{11}\,M_\odot$ | Core of radius ≈ 2.3 kpc in 2 Gyr, as opposed to the ΛCDM cusp. | Core not produced. The preregistered Level A experiment gives an indeterminate outcome: the instrument does not converge in the interior of the halo. |
| Rotation curves (SPARC) | The mean squared error fell from 12% in ΛCDM to 4.5%. | SPARC not ingested. The cross-falsification with the Cronos amplitude frozen is preregistered and has not been run. |
| Substructure count | 45% fewer subhalos above $10^8\,M_\odot$. | Requires production with Gadget-4-Cronos, which has not been run. |
What the repository's experiments have in fact given is more modest and is published as is: on the minimal mesh, the gate friction closes exactly upon virialization, and the verdict on the core requires production resolution. A later test points against the calibrated amplitude: the vertical kinematics of the solar neighborhood excludes in the plane the Cronos amplitude fixed in Sculptor ($A \lesssim 0.06\,A_{\text{Sculptor}}$ with the declared slab).
The falsifiable signature
Here is what makes this more than a fit. The gate friction predicts a specific and testable behavior: the tension is active while the region collapses and subsides when the structure settles. In already virialized halos it must be off.
That is a prediction that can fail. If the effect were observed in virialized systems, or if it were not observed during the collapse, the mechanism would be called into question. Alongside it, the chapter on open fronts points to the bound on the Cronos amplitude, which must stay below $10^{-6}$.
Status. The figures of the earlier corpus are a historical record (FE-08-2026, E7). What is reproducible today are the experiments on the repository's minimal mesh, with their published outcomes, including the indeterminate ones. Running the production simulations with the corrected scheme is one of the seven open fronts: it is pending, not taken for granted.
The code, the dependencies and the reproducibility chain are in resources. The observational comparison, in MCMC versus ΛCDM. And the status of each result, in what is proven and what is not.