https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Wolfram Fundamental Physics Project sorted by active. https://community.wolfram.com/groups/-/m/t/3690586 Hello Wolfram Community, I want to share a computational observation that I have not seen discussed in this precise form, and I would genuinely welcome input on whether there is an obvious derivation I am missing. ## The observation Consider the coefficient: Gcurvature[d_] := Gamma[1 + d]^2 / Gamma[1 + 2 d] For integer arguments, this equals `1/Binomial[2n, n]` — the reciprocal of the central binomial coefficients. The first few values: Gcurvature(1) = 1/2, Gcurvature(2) = 1/6, Gcurvature(3) = 1/20, Gcurvature(4) = 1/70 I have observed that the critical exponents of the three-dimensional Ising universality class appear to be expressible as exact rationals built from these four coefficients. ## The three main expressions eta3D = Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2) = 9/250 = 0.0360 nu3D = Gcurvature[3]^3 / (Gcurvature[1] * Gcurvature[2]^2 * Gcurvature[4]) = 63/100 = 0.6300 beta3D = Gcurvature[4]^2 / (Gcurvature[1]^2 * Gcurvature[3]^2) = 16/49 = 0.326531... The numerator of `nu3D` simplifies remarkably: `5040/8000 = 7!/20^3`. ## Comparison with the conformal bootstrap The state-of-the-art values from Chester et al. (2024, arXiv:2411.15300) are: eta = 0.0362976(5) — GOD prediction 9/250 — error 0.82% nu = 0.6299710(4) — GOD prediction 63/100 — error 0.005% beta = 0.3264187(6) — GOD prediction 16/49 — error 0.034% By standard scaling relations, the remaining exponents also become exact rationals: gamma = 30933/25000 (bootstrap: 1.237076, error 0.020%) alpha = 11/100 (bootstrap: 0.110087, error 0.079%) delta = 1241/259 (bootstrap: 4.789843, error 0.035%) ## The 2D case is exact For the 2D Ising model (Onsager 1944), the same framework gives exact results: eta_2D = Gcurvature[1]^2 = 1/4 (Onsager: 1/4) EXACT nu_2D = Gcurvature[0] = 1 (Onsager: 1) EXACT beta_2D = Gcurvature[1]^3 = 1/8 (Onsager: 1/8) EXACT Mean-field values (d >= 4) also match exactly: eta_MF = 0 EXACT nu_MF = Gcurvature[1] = 1/2 EXACT beta_MF = Gcurvature[1] = 1/2 EXACT ## Context This observation forms part of a broader framework I have been developing (GOD Theory, DOI: 10.5281/zenodo.19599917), in which these coefficients arise as curvature coefficients of a fractal algebra built on fractional derivatives. However, the numerical match above holds independently of that framework and can be verified by anyone with the attached notebook. ## What I am asking The 2D Ising match and the mean-field match are exact. The 3D match is empirical at the 0.005%-0.82% level. I do not have a rigorous derivation of the exponent patterns (why specifically `Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2)` for eta, for example). My question to the Community: does anyone see an obvious path to this via the renormalisation group, conformal field theory, or the structure of central binomial coefficients? Has any similar pattern been observed? The attached notebook contains all computations in roughly 30 lines of Wolfram Language. Please feel free to run it, criticize it, or extend it. Thanks in advance. Francisco Torrado Cano Independent Researcher Cáceres, Spain Francisco Torrado Cano 2026-04-16T11:00:14Z https://community.wolfram.com/groups/-/m/t/3674343 The rule "BA" -> "AB" is presented in Section 5.9 of the Physics Project book ("The Significance of Causal Invariance") as being causal invariant. The instances of its causal graph shown therein for 5 time steps and initial condition "BBBAA" is given as example, showing that all 5 instances are isomorphic to each other. I've run the example with the MultiwaySystem function (ResourceFunction["MultiwaySystem"][{"BA" -> "AB"}, "BBBAA", 5, "CausalGraphInstances"]) and the function output matched the book content. However: 1) Running the system for only 4 time steps, not all instances are isomorphic to each other, since in this case there 2 types of graphs (among the 5 instances) which are clearly not isomorphic to each other. 2) In the subsequent example in the section -- that changes the initial condition to "BBBBAAAA" -- the MultiwaySystem function couldn't generate the instance causal graph shown in the book, regardless of the number of time steps I tried (from 4 to 10). Actually, for all these time steps different graph types are generated which precludes the existence of isomorphism among them all. Pedro Paulo Balbi 2026-04-04T20:45:37Z https://community.wolfram.com/groups/-/m/t/2759491 Wolfram's Physics Project is a very interesting one and its recent evolution fascinates me. I think that the recent discovery of the ruliad is a good progress in the project. However, there is something I don't understand: According to Wolfram's writing about the ruliad (https://writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad/), there is only one ruliad which contains all possible formal systems. I understand why there should be only one ruliad as it would contain all possible computational rules by definition. However, I can imagine a formal system or abstract situation where somehow there could be multiple rulial spaces or "ruliads". So, at the end, wouldn't it be possible that there could be other ruliads (for computational systems, I'm not talking about other possible "ruliads" containing hypercomputation)? Nodu Agga 2023-01-04T15:08:10Z https://community.wolfram.com/groups/-/m/t/3663446 ![The Ruliad concept: some ideas and observations][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=TheRuliadconceptsomeideasandexamples.png&userId=20103 [2]: https://www.wolframcloud.com/obj/db1cd65f-d494-49cb-8414-bcaed8b4ffef Denis Ivanov 2026-03-17T04:57:25Z https://community.wolfram.com/groups/-/m/t/3636637 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/181086a3-2534-41da-92a6-762c33e102c4 Donald Airey 2026-02-08T20:17:36Z https://community.wolfram.com/groups/-/m/t/3642808 **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. Suhail Bachani 2026-02-21T00:32:44Z https://community.wolfram.com/groups/-/m/t/3631121 I am working on a model that requires a specific type of "Selection Bias" in the rule evolution, and I am looking for guidance on how to represent this using WolframModel or MultiwaySystem. The Theoretical Goal: I am trying to simulate a "stabilization" phase where the system transitions from a high-entropy state (random fluctuations) to a structured state (durable causal loops). I hypothesize that this requires a Parity-Breaking Bias (which I call "The Twist") that weights the path integral. The Mechanism I Want to Model: Instead of all branches in the Multiway System having equal weight, I want to penalize branches that are "topologically symmetric" and reward branches that exhibit a specific "Chiral Asymmetry" (Twist). >Hypothesis: This bias should force the Causal Graph to "lock" into durable subgraphs (particles/structures) rather than exploring the full infinite Ruliad. The Question: Is there a standard way in WolframModel to apply a "Selection Function" or "Path Weight" that prunes the Multiway Graph based on the topological properties of the hypergraph at that step? Context: This is part of a larger framework ("Universal Compression") linking causal graph dynamics to observer constraints. >System Architecture (Preprint): https://doi.org/10.5281/zenodo.18421925 >Specific Cosmological Derivation: https://doi.org/10.5281/zenodo.18421691 Any pointers on how to implement a "Selection Bias" function in the evolution step would be appreciated. Matt Prager 2026-01-30T11:29:43Z https://community.wolfram.com/groups/-/m/t/3589391 The Rule：{{x, y}, {y, z}} -> {{x, z}, {x, w}, {w, z}} This model investigates the emergence of causal geometry from a minimal graph-rewriting rule. Unlike standard branching trees, this rule facilitates state merging (interference), mimicking a Dynamic Programming optimization process within the causal graph. The evolution demonstrates Markovian properties where the spatial structure ('ripple') expands purely based on local connectivity, creating a discrete spacetime fabric that exhibits Causal Invariance. This serves as a computational candidate for interpreting the 'Many-Worlds' path integral as a deterministic graph optimization problem. Proposed Model Description (Short Explanation) Nodes represent discrete universe slices (microstates of spacetime). Each node encodes a complete instantaneous configuration of the universe. Directed edges represent causal relations between slices. An edge from node A to node B indicates that B is a possible successor state generated from A. The network is constructed through a recursive update process combining (1) a Markov-style probabilistic transition rule, and (2) a deterministic local causal rule. Together, these govern how new spacetime slices branch and evolve. A path from the root to any node corresponds to a possible history of the universe. Compressing such a path yields the emergent notions of time and macroscopic causality. The diagrams shown depict the evolving causal structure and the resulting spatial slice (wavefront) produced by these rules.![Causal Graph showing state merging and loop structures][1]![Final Spatial Slice exhibiting wavefront expansion][2]![20-times recursive version][3] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=yuz1.png&userId=3589356 [2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=yuz2.png&userId=3589356 [3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=222222.png&userId=3589356 Xin Wang 2025-12-11T13:21:31Z https://community.wolfram.com/groups/-/m/t/3608412 ![Holographic duality between causal and branchial graphs][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=adsss.png&userId=20103 [2]: https://www.wolframcloud.com/obj/5caba86f-013f-4f65-8d61-2d8d78f91365 Narmin Nasibova 2026-01-16T11:46:00Z https://community.wolfram.com/groups/-/m/t/3607969 ![Wolfram Physics Project Noncommutative Geometry Bridge][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=WPP-NCG-Bridge.png&userId=3607945 [2]: https://www.wolframcloud.com/obj/57ceecb3-2977-4fe1-9bc5-6e40e46dfd99 Jorge Plazas 2026-01-15T23:41:53Z https://community.wolfram.com/groups/-/m/t/3607514 ![Branchial graph stability][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2026-01-16004426.png&userId=3599863 [2]: https://www.wolframcloud.com/obj/9ee1946e-a652-4de6-a636-6efda46efac4 Rehab Almakrami 2026-01-15T21:49:27Z https://community.wolfram.com/groups/-/m/t/3599773 **This is a collaborative work by** [**Furkan Semih Dündar**][at0], [**Xerxes D. Arsiwalla**][at1], and [**Hatem Elshatlawy**][at2]. ![Figure adapted from (Dündar, Arsiwalla, Elshatlawy, 2025)][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=2712hero.png&userId=20103 [2]: https://www.wolframcloud.com/obj/6954e3c7-a10d-460a-9a9a-ade61ae80043 [at0]: https://community.wolfram.com/web/fsemihdundar [at1]: https://community.wolfram.com/web/xdarsiwalla [at2]: https://community.wolfram.com/web/hatemelshatlawy Furkan Semih Dündar 2026-01-01T12:58:27Z https://community.wolfram.com/groups/-/m/t/3593059 **The Core Idea: Physics as Discretization Error** In continuous calculus, Green's Theorem ($\int_R dA = \oint_{\partial R} ds$) is exact. However, when applied to a discrete lattice with resolution limits, this equality fails. There is a "residue" or "defect" left over because the discrete boundary cannot perfectly capture the bulk interactions. I have been working on a model suggesting **Physical Law is this residue**. Specifically, I observe the discrete Green's Theorem on a recursive lattice (a Pentatope characteristic network). When this lattice attempts to fold from Dimension 4 (Bulk) to Dimension 3, the "counting error" forces specific constants to emerge. ![The "Golden Stencil" generated by the attached notebook. Visualizing the recursive $D=4$ fold scaling by $1/\phi$. The color gradient (Blue $\to$ Cyan $\to$ White) represents the descent through topological strata, and the central density represents the geometric residue (Mass).][1] **The Results (Zero Import)** Remarkably, this geometric residue yields values that match fundamental constants to high precision, without manual tuning: - **Fine Structure Constant** ($\alpha$): Matches to within **$0.00002\%$** **(0.014$\sigma$)** (derived from the impedance mismatch of the fold). - **Nuclear Stability**: The geometric residues scale to predict the Binding Energies of the Periodic Table (e.g., He-4, O-16, U-235) purely from the entropy of the lattice edges. - **Cosmic Structure**: The ratio of Dark Matter to Baryonic Matter matches Planck 2018 data with **0.01%** precision. - **Dark Energy**: Matches the vacuum tension of the bulk with **0.002% deviation**. **Why I am posting here** I come from a CFD/Aerospace background (Ph.D.), where we constantly deal with discretization errors. This "rabbit hole" is the direct result of chasing the residual while developing a CFD method free of approximations for many years. I have been trying to "break" this model, but the geometric coherence holds up across scales (from Quantum to Cosmic). And, I believe this community and Wolfram's theory is remarkably closely aligned. Reproducibility I have attached the full Wolfram Notebook below. It performs the exact calculations from first principles (zero imports, self-contained derivation). The full theory is available on Zenodo https://doi.org/10.5281/zenodo.17957211. I would appreciate your thoughts: Does this specific "Green's Theorem Residue" map to any topological defects you see in hypergraph rewriting? Charles Cook, Ph.D. &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=lattice.png&userId=3593023 [2]: https://www.wolframcloud.com/obj/ec7f9154-bd59-463f-a07b-791912d006c4 Charles Cook 2025-12-17T16:25:42Z https://community.wolfram.com/groups/-/m/t/3593931 Previous work has proposed mapping external theoretical frameworks onto WPP hypergraph dynamics (e.g., Malicse 2025), but to my knowledge none have operationalized such mappings as measurable causal graph observables and run validation experiments. This paper reports the first such attempt: testing whether Energetic First Principles (E1P), a structural framework for process dynamics, appears in WPP hypergraph evolution. Method: E1P's four-phase cycle was operationalized as causal graph observables—branching events (CC), merging events (AC), cumulative open branches (CA), inverse activity (AA)—and tested across 15 hypergraph rewriting rules. Results: Phase ordering (CC → CA → AC): 9/9 CI rules ✓ CA accumulation → drainage: 9/9 CI rules ✓ Merge rate consistency across evaluation orders: 9/9 CI rules ✓ Discrete τ threshold at ~87%: No rules in 1-85% range Four-class taxonomy emerges: Generative CI, Conservative CI, FixedPoint, CV Implication: E1P phase structure appears to be discovered structure within CI dynamics, not imposed interpretation. This suggests external frameworks can be empirically validated in WPP—not just proposed. Full paper with methodology, data tables, and reproducibility details: https://doi.org/10.5281/zenodo.17979892 Interested in whether others have attempted similar operationalizations, or observed comparable phase patterns in CI rule evolution. Note: This is empirical validation, not derivation of physical constants. The claims are limited to observable phase patterns in causal graphs—not cosmological correspondence. All results reproducible in Mathematica 14.0+ with SetReplace.&[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/a93df282-9b3d-4893-b13c-71b80c7ee82d Catalin Leescu 2025-12-18T17:41:53Z https://community.wolfram.com/groups/-/m/t/3580238 I’ve been thinking about how the Wolfram model treats manifold-like behavior as an emergent, approximate property of the underlying hypergraph. In regions of extreme causal-edge density — for example near black hole interiors — the manifold approximation is expected to break down. My question is this: When the manifold structure collapses, does the distinction between different effective “universes” (different limiting foliations or rule-equivalence classes in the Ruliad) also collapse? In other words, if spatial distance and branchial distance both arise from constraints on causal reducibility, then in a region where those constraints fail (because the updating is too dense or too singular), do previously distant parts of branchial space become adjacent? The speculative idea is that a black hole interior might serve as a region where manifold structure, branchial separation, and even differences between emergent effective laws of physics lose their usual distinctions, because the underlying hypergraph is no longer well-approximated by any smooth foliation. Could this imply that a black hole is a nexus where different emergent universes in the Ruliad could become computationally adjacent in ways they normally aren’t, like an *even more* speculative variation on a wormhole? A bridge entangling parts of the Ruliad whose physics don't even look the same? I’m wondering whether anything in the current WPP formalism supports or contradicts this possibility, but this is well outside my area of study. If anyone deep into the Wolfram Physics Project has explored this rather sci-fi notion, I would be grateful to hear their musings! Brenden Martin 2025-11-22T20:03:09Z https://community.wolfram.com/groups/-/m/t/3497586 ![A Graph Theoretic Approach to Feynman Checkers][1] &[Wolfram Notebook][3] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=HeroImage.bmp&userId=3492774 [2]: https://www.wolframcloud.com/obj/f41c98b6-bfc7-4029-a616-666c2a8d4915 [3]: https://www.wolframcloud.com/obj/2bd729ea-edb4-4a60-aadd-512bdbbe2417 Gary Louw 2025-07-10T01:49:21Z https://community.wolfram.com/groups/-/m/t/3358036 ![Exploring ruliological engineering of well-defined geometries from hypergraph rewriting][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=projectImg.png&userId=2958656 [2]: https://www.wolframcloud.com/obj/df43d3c2-655c-44ae-91e6-c04c5ac15b89 Jacopo Uggeri 2025-01-16T18:01:51Z https://community.wolfram.com/groups/-/m/t/2959221 ![enter image description here][1]&[Wolfram Notebook][3] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=image.png&userId=2958976 [3]: https://www.wolframcloud.com/obj/0c130964-8090-46be-ab7d-83b7c6b4ccd8 Joel Choi 2023-07-12T20:33:45Z https://community.wolfram.com/groups/-/m/t/3534450 ![A Project to Find the Fundamental Theory of Physics by Stephen Wolfram][1] It's been five years since the original publication of *A Project to Find the Fundamental Theory of Physics* and the launch of the Wolfram Physics Project. Now the book is available in paperback for the first time! > **Order Now:** https://a.co/d/0uyBmEj The Wolfram Physics Project is a bold effort to find the fundamental theory of physics. It combines new ideas with the latest research in physics, mathematics and computation in the push to achieve this ultimate goal of science. Written with Stephen Wolfram's characteristic expository flair, this book provides a unique opportunity to learn about a historic initiative in science right as it is happening. *A Project to Find the Fundamental Theory of Physics* includes an accessible introduction to the project as well as core technical exposition and rich, never-before-seen visualizations. The paperback edition will be published on August 26, 2025. Preorders are available until that time. [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=4950Book-3-edited.jpg&userId=20103 Paige Vigliarolo 2025-08-22T17:35:59Z https://community.wolfram.com/groups/-/m/t/3357787 ![Entanglement on quantum optics kit towards creating physical correlates of multiway diagrams][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=WWS25Mark.png&userId=20103 [2]: https://www.wolframcloud.com/obj/345e80e4-210b-4724-b510-4f4060c1d312 Mark Merner 2025-01-16T15:50:43Z