What question did this study set out to answer?

The aim is to show how general relativity and particle generation emerge from discrete topological constraints.

February 21, 2026Open Access

Fractal Entropic Geometrodynamics: Emergent Gravity and Three Particle Generations from Discrete Topological Constraints

Key Points

The aim is to show how general relativity and particle generation emerge from discrete topological constraints.
Developed an $O(N)$ simulation engine in Rust to model events in a 4D causal diamond.
Constructed Hasse diagrams to identify topological defects called Causal Prisms.
Established a discrete phase invariant that classifies prisms into three generations based on bulk momentum.
Conducted finite-size scaling analyses across various lattice sizes to observe boundary-sensitive behavior.
Detected a mass hierarchy for particle generations of 4.55 : 6.53 : 7.73.
Spectral dimensions identified varying from 1.95 in UV to crossing 4 at $\sigma ext{ approximately } 3$–4.
Validated that topological invariants and generation fractions are independent of particle number.
Found that the emergent fine structure constant is related to topological observables through a specific identity.

Abstract

We present a strictly discrete framework in which General Relativityand the three-generation structure of the Standard Model emerge fromtwo axioms and one principle. An O (N) simulation engine---implemented in Rust and freelyavailable---Poisson-sprinkles events into a 4D causal diamond, builds the Hasse diagram, and detects topological defects calledCausal Prisms (bipartite subgraphs). A discrete phase invariant based on the sign trichotomy of bulk momentumclassifies prisms into exactly three generations (g 3), producinga mass hierarchy Gen\, 1\,: \, Gen\, 2\,: \, Gen\, 3 = 4. 55: 6. 53: 7. 73and CPT symmetry---from pure combinatorics. The spectral dimension flows from (=1) = 1. 95 0. 002in the UV, crossing = 4 at 3--4. A finite-size scaling analysis across five lattice sizes (N = 10⁵ to 10⁷, M 8 realisations each) reveals thatall boundary-sensitive observables follow the 4D scaling lawO (N) = O_ + a\, N^-1/4 (R² = 0. 9996 for Qₓ₎₏₎), whiletopological invariants---mass ratios and generationfractions---are N-independent. The emergent fine structure constant = Qₓ₎₏₎/ (8) and self-energy ratio = 1/Qₓ₎₏₎ - 1are locked by the exact identity (1+) = 1/ (8) ;the bare topological coupling ₀^-1 = 165. 1 1. 0 setsthe UV-cutoff value for renormalization-group studies. Gravity is the Fisher information cost of macroscopic deformation;the Einstein equations are thermodynamic. Bounded Hasse degree (D 15) makes the computation O (N), providing a proof---not an assertion---that the physics is local.

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