High Energy Particle Physics

   

Emergent Spacetime and Protomatter from Ollivier-Ricci Flow with Discrete Cartan Torsion

Authors: Fabrizio Vassallo

We propose a discrete geometric model in which classical spacetime and matter emerge from a discrete random network equipped with two combinatorial structures: Ollivier-Ricci curvature (ORC), a metric-free notion of curvature defined via optimal transport on the network, and discrete Cartan torsion, a 2-cochain on a weighted simplicial complex measuring holonomy defects of parallel transport around elementary triangles. Both structures are intrinsically combinatorial and require no background geometry; classical geometry is an output, not an input, of the model. Building on Trugenberger's ORC-based network model, which exhibits a phase transition between a random hyperbolic phase and a geometric phase, we augment the Ricci flow with a nonlinear torsion coupling and demonstrate, in an explicit four-node toy model, that the resulting dynamical system possesses two distinct basins of attraction whose separation is topologically robust and independent of the specific toy model chosen. Regions of the network with vanishing torsion condense into a one-dimensional geometric phase (embryonic spacetime), while regions with non-vanishing torsion condense into a localized, topologically non-trivial configuration identified as a torsion defect carrying all quantum numbers equal to zero. A key structural result is that the torsion-bearing fixed point P T is a saddle point of the linearized discrete flow, with one expanding and one contracting direction in the (w, T) plane. The Jacobian entry J T w = 8λ/9 > 0 is derived exactly from the Wasserstein optimal transport, yielding real eigenvalues µ = 1 ± 8λη sech 2 (T *)/9 and a determinant det(J) = 1 − (8λη/9) sech 2 (T *) < 1. We conjecture-as a structural analogy motivated by the geometry of Einstein-Cartan theory, but not derivable from the linearized dynamics-that this fixed point is geometrically associated with the parametrization of the Cartan helix. This conjecture motivates, but does not rigorously imply, the hierarchical level model introduced in Section 7, in which the rotation angle and expansion factor of the helix are tentatively associated with spin halving and mass scaling between levels. We further propose a hierarchical particle spectrum-the level model-in which each level is characterized by doubled spacetime dimension, halved spin, and mass scaling by a factor 4α, where α is the 1 electromagnetic fine-structure constant and δ F ≈ 4.6692 is the Feigenbaum period-doubling constant. The conservative flow has a saddle-like structure at P T and cannot produce a Feigenbaum period-doubling cascade. We introduce a physically motivated dissipative extension of the torsion equation-a −ξ sin(T) restoring term-and show that the resulting effective torsion dynamics reduces, in the strongly dissipative limit, to the sine mapT → ξ sin(T) on [0, π], which belongs to the Feigenbaum universality class. The Feigenbaum constant δ F ≈ 4.6692 therefore emerges dynamically from the flow, and the empirical relation α ≈ 1/(2πδ 2 F) is found to be numerically consistent with this scaling, with the factor 2/π from the sine map amplitude. The physical origin of the dissipation is the geometric coarse-graining at each RG step: information is globally conserved, but the geometric distinguishabil-ity of sub-Planckian torsion configurations is reduced, creating equivalence classes in the sense of 't Hooft's dissipative quantum gravity program [82]. Our results and 't Hooft's program converge toward the same conceptual conclusion by independent routes: quantum-mechanical behavior can emerge from an effectively dissipative underlying structure, with information globally conserved. The continuum limit of the full construction is argued to reproduce Einstein-Cartan gravity in four dimensions. Falsifiable numerical predictions are formulated for simulation on synthetic networks.

Comments: 38 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Download: PDF

Submission history

[v1] 2026-05-09 22:27:05

Unique-IP document downloads: 5 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus