Title: Exploring a Finite Multiway System: Exact Computability in the "S21" Discrete Quantum Gravity Model
Hello everyone,
I’ve been reviewing a recent theoretical framework called "S21 Theory," and it shares significant conceptual DNA with the Wolfram Physics Project—specifically regarding multiway systems and the emergence of continuous physics from discrete graphs [1, 2]. I thought it would be an interesting model to discuss here, particularly because of how it approaches the problem of infinite state spaces and exact computability.
Here is a breakdown of how the S21 model aligns with (and diverges from) the Wolfram approach:
1. The Foundation: A 6-Bit Postulate Instead of Arbitrary Rules While the Wolfram approach often searches empirically for generative rewriting rules across an infinite space of possible strings [2, 3], S21 derives its structure from a single discrete postulate: spacetime at the Planck scale admits exactly six binary degrees of freedom (bits) per cell [4]. * This 6-bit postulate creates a finite 64-state configuration space (the $Q_6$ hypercube) [5]. * Applying topological consistency and action minimization filters this down to exactly 21 stable configurations [6]. * 20 of these states form a connected visible-sector vacuum manifold ( $M_{20}$), while 1 isolated state becomes a Dark Matter candidate ( $\sigma$) [6].
2. Multiway Evolution and Exact Solvability S21 explicitly utilizes the multiway evolution paradigm [1]. Dynamics in the S21 vacuum occur as a multiway directed acyclic graph (DAG) where the system simultaneously explores all allowed paths on a 20-node physical transition graph ( $G_E$) [7]. * The "Wolfram Difference": The S21 author explicitly compares the two models, noting that because S21 is restricted to a finite 20-state manifold rather than an infinite state space, its multiway evolution is exactly solvable [2, 3]. * The discrete Feynman path integral (summing over all paths in the multiway graph) is evaluated exactly, matching matrix-inverted Green's functions to machine precision ( $10^{-14}$) [8]. This provides a convergent, explicit sum without the need for Monte Carlo approximations or dealing with divergent infinities [9].
3. Emergent Curvature (Ollivier-Ricci) Just as Wolfram models look for continuum limits of discrete hypergraphs, S21 proves that continuous relativistic geometry emerges from the discrete graph $G_E$. By computing the Ollivier-Ricci curvature using optimal transport (Wasserstein-1 distance) between the neighborhoods of adjacent vertices, the theory proves the graph has a uniform negative curvature ( $\kappa = -1/3$) [10, 11]. This establishes the vacuum as a constant-curvature homogeneous space satisfying the discrete Einstein equations [12].
4. Topological Origin of the Standard Model Instead of treating particle physics as an add-on, S21 claims the Standard Model is structurally inevitable from the graph topology: * Fermion Generations: The topological skeleton of the 20-state manifold has a first Betti number of $b_1 = 3$, which exactly matches the 3 generations of fermions [13]. * Particle Spectrum: The 43 "forbidden" states ( $F_{43}$) outside the vacuum manifold act as an encoding space for the particle spectrum. The boundary membrane between the forbidden sector and the vacuum yields exactly 39 observable states (1 Higgs + 12 gauge bosons + 16 quarks + 10 leptons), which perfectly divides into 13 particles across 3 generations [14, 15]. * Cosmology: The framework tracks the minimal CP-odd closed walk on the graph, finding a length of $l_{min} = 7$ [16]. This single integer invariant is used to derive both the baryon asymmetry ( $\sim 10^{-10}$) and the cosmological constant ( $\sim 10^{-119}$) [17, 18].
Discussion Prompt for the Forum: The S21 framework suggests that by restricting a multiway system to a highly constrained, finite topological manifold ( $M_{20}$), we can bypass the computational intractability of infinite state spaces and extract exact, quantitative cosmological parameters [2, 3].
Has anyone here experimented with similarly constrained, finite multiway systems? I’d be very interested in hearing the community's thoughts on using a strictly finite 6-bit partition to solve the path integral convergence problem in discrete quantum gravity.
