https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Wolfram Fundamental Physics Project sorted by new. https://community.wolfram.com/groups/-/m/t/3733334 ![Q102 Growth][1] Start with a *bare* hypergraph rewriting system in the spirit of the Wolfram Physics Project: featureless atoms, ternary edges, one local rewrite. Now add a single demand: that the system be **closed to efficient causation** in Robert Rosen's sense, so every producer in the system is itself produced by the system. That one demand is not free. It forces a chain of structure, and what grows out of it is the object in the animation above: a 6-vertex seed that grows, collapses to a 51-class gauge quotient, and doubles under charge conjugation to a 102-vertex object that carries the sign data of a real spectral triple of **KO-dimension 6**. Everything here is exact - Gaussian integers, no floating point, **no random seed** - so the headline count is reproducibly **102**. The full notebook is attached. ## What is Rosen closure? Robert Rosen was a theoretical biologist who asked what *minimally* separates an organism from a machine. His answer: a machine is built and maintained from the outside (its blueprint, its factory, its repairman are all external) whereas a living system is **closed to efficient causation**. "Efficient cause," in Aristotle's sense, is *the thing that brings something about*; closure means every process that produces or maintains a part of the system is itself produced by some part of the system. The system makes its own makers, and nothing in its organization is entailed from outside. Rosen formalized this as an **(M,R) system** (a metabolism together with repair-and-replication maps that the system must generate itself) and the consequence that matters here is that closure forces a **self-referential fixed point**: the system has to be able to produce its own production rules. That self-reference is the entire engine. A Wolfram model's rewrite rule is imposed from outside; demanding closure instead turns the rule into something the decorated hypergraph must reproduce *internally* and that single requirement (next section) is what forces three functional roles, ternary edges, and a continuous ℂ³ colour, with essentially no freedom left. This is *deliberately not* a stock `WolframModel`, and that distinction is the whole point. So, let me start there. ## Why it isn't a stock Wolfram model A `WolframModel` rewrites **bare** hypergraphs: atoms are featureless, and rules are pure connectivity patterns. The moment you impose Rosen closure, the bare atoms are *obstructed* out of every featureless option and **forced** to carry a continuous colour. The forcing chain (each link the unique survivor of an obstruction): - closure needs exactly three functional roles → edges are **ternary** (arity 3); - **no discrete labelling** can satisfy closure → every finite decoration is exhaustively obstructed (all 256 binary rules fail; ℤ₂, ℤ₃, …, ℤ₈ all fail); - the only continuous composition that closes is a **cross product on a 3-vector** (Hurwitz); - and the field must be **ℂ, not ℝ** → the self-referential edge needs an isotropic unit vector (`v·v = 0`), which ℝ³ has none of and ℂ³ does, e.g. `(1, i, 0)/√2`. So the colour isn't an attribute bolted onto a Wolfram model, it's **what closure does to one**. Each vertex carries a colour `ψ ∈ ℂ³`, and the composition rule is forced to be: compose[a_, b_] := Conjugate[Cross[a, b]] (* conj(a × b), exact, unnormalised *) (the conjugation is what keeps it SU(3)-equivariant). The gauge quotient identifies vertices whose colour vectors are linearly dependent over ℂ exactly, with no threshold: rayEquivQ[a_, b_] := AllTrue[Cross[a, b], # == 0 &] (* a ~ b ⇔ a × b = 0 *) Putting a colour on the atoms makes this a *decorated* (attributed) hypergraph rewriting system (close in spirit to **@Ray Aschheim**'s color spin networks, but here the colour, the dimension 3, the cross product, and the conjugation are all *forced*, not posited). ## Exactness matters: 98 vs 102 Because the quotient is by exact ray-equivalence over ℤ[i], the construction is fully deterministic and reproducible. This is not a cosmetic choice: a Float64 fidelity-threshold version of the same construction **merges four genuinely-distinct rays** and reports 98 instead of 102. Exact Gaussian-integer arithmetic gives 102 every time. (For the related question of detecting equivalence on hypergraphs *approximately*, cf. **@Regen Petu-Stiles**'s work on approximate isomorphism. The contrast between exact and thresholded equivalence is exactly where the 98-vs-102 discrepancy lives). Six canonical Gaussian-integer rays seed the system, on the complete ternary topology K₆³ (all 120 ordered triples of distinct vertices): icsBase = {{2+I, 1, 3-I}, {1, 2+I, 1-2 I}, {1-I, 3, 2+I}, {3, 1-I, 1+2 I}, {1+2 I, 2-I, 1}, {2, 1+I, 3}}; ## Growth to Q102 The multiway grows; the colour quotient closes almost immediately. Distinct colour-classes per generation: **6 → 21 → 51 → 51 → 51** — a fixed point at generation 2. ![Distinct Colour-classes vs Generation][2] Side by side: the raw multiway (left, coloured by generation) explodes to 456 vertices, while its exact gauge quotient (right, tier A/B/C) closes at 51. ![Frame 2][3] The charge-conjugation closure then doubles it. Growing the conjugate multiway from the conjugated initial colours and re-quotienting together gives **Q102 = Q51 ∪ C(Q51) = 102 vertices = 51 orig + 51 conj** (zero self-conjugate): ![Closure][4] ## KO-dimension 6 The involution `J` that swaps the two halves is a fixed-point-free permutation with `J² = +I`, and it **anticommutes** with the orig/conj grading `γ` (so `{J, γ} = 0`). Those are exactly the sign data of a real spectral triple of **KO-dimension 6**. Thi is the same KO-dimension the Standard Model's internal space carries in Connes' noncommutative geometry. The notebook checks it directly: ![KO][5] ## The multiway as a derivation graph Every non-seed vertex is produced by composing two parents, so each edge runs from an earlier generation to a later one. Drawn as concentric rings (one per generation), the multiway becomes a derivation graph: 6 seeds at the centre generating a ring of 15, then a ring of 30 and closing at 51. ![DAG Class][6] The full raw multiway (all 456 vertices over five generations) as a derivation mandala (vertices coloured by generation): ![Raw Multiway][7] ## All the gauge sectors on one hypergraph The colour sector gives SU(3). But the full gauge group `[SU(3) × SU(2) × U(1)_Y] × SU(3)_gen` is the **automorphism group of the decoration functor** — derived from closure on the possibilistic layer, not from the NCG bridge (the spectral triple handles anomaly cancellation, not the group). The other sectors ride on the *same* hypergraph as extra fibres; the colour quotient is unchanged: - **SU(2)_weak** — a doublet `w ∈ ℂ²` rotated by an SU(2) element built from the colour cross-product. The weak sector is literally *driven by* colour (it's the stabilizer of the colour output direction). This is the one float layer. - **U(1)_Y** — the residual phase of the weak doublet; not a separate layer. - **SU(3)_gen** — a generation amplitude `g ∈ ℤ[i]³` composing exactly like colour, so its own ray-quotient is a *second, independent* Q51: a decoupled family symmetry. - **ℤ₂ spinor** — a sign `ε` per vertex, the product of its three source signs (the chirality double-cover). ![Sectors][8] Three sectors as fibres on the identical colour-quotient Q51, plus generation as its own parallel quotient. (Unlike exceptional-group "theory of everything" embeddings [cf. **@J Gregory Moxness**'s H₄/E₈ work] here the gauge group is *derived by obstruction*, not embedded in a larger group.) ##A few things to keep straight: - This is an **independent research program** ("Closure Forces Structure"), adjacent to the official Wolfram Physics Project. The shared substrate is hypergraph rewriting; the extra ingredient is Rosen closure. - The notebook **reproducibly verifies the structural facts** (51, 102, the KO-dimension-6 signs, the four gauge-sector decorations). The *forcing arguments* (discrete obstruction, Hurwitz, isotropy, the gauge-group derivation) are theorems in the accompanying paper, not claims the notebook proves by itself. - One layer, the weak doublet, is **float** (an SU(2) rotation with `cos θ`, `sin θ` from `θ = Arg⟨ψ₁|ψ₂⟩`); everything else is exact Gaussian-integer arithmetic. - The object is studied by **observation** (slicing and characterizing the static multiway and its quotients) not by instantiating it as a running dynamical system. ## Run it yourself The attached notebook is self-contained and evaluates top to bottom (≈ a minute; the 456-vertex layout is the slow step). It prints a verification table you can check against the numbers above, then draws every figure in this post including an interactive generation slider: Manipulate[frameAt[G], {{G, 4, "generation"}, 0, 4, 1}] To export the animation: Export["q102_growth.gif", Append[growthFrames, closurePanel], "DisplayDurations" -> 1.2] Comments, criticism, and especially attempts to break it are welcome. &[CFS Notebook][9] --- ##References / related Community posts:* - *Original Reference: Aaron Green (2026). Closure Forces Structure: Rosen Closure and the Standard-Model Algebra. Researchers.One. [https://researchers.one/articles/26.02.00001v7][10]* - *@Ray Aschheim — [WSS21] Color spin networks — https://community.wolfram.com/groups/-/m/t/2311732* - *@Regen Petu-Stiles — [WSS22] Identifying homogeneity in hypergraphs via approximate isomorphism — https://community.wolfram.com/groups/-/m/t/2445699* - *@J Gregory Moxness — The Isomorphism of H₄ and E₈ — https://community.wolfram.com/groups/-/m/t/3073391* ##AI Usage Claude Opus 4.8 (Anthropic) assisted in the drafting of Wolfram code and generation of figures and text used in this post. [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=q102_growth.gif&userId=3733286 [2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_curve.png&userId=3733286 [3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_frame2.png&userId=3733286 [4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_closure.png&userId=3733286 [5]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_ko.png&userId=3733286 [6]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_dag_class.png&userId=3733286 [7]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_dag_raw.png&userId=3733286 [8]: https://community.wolfram.com//c/portal/getImageAttachment?filename=fig_sectors.png&userId=3733286 [9]: https://www.wolframcloud.com/obj/11f014a0-6090-45bb-994d-68402d1aa461 [10]: https://researchers.one/articles/26.02.00001v7 Aaron Green 2026-06-14T07:06:23Z https://community.wolfram.com/groups/-/m/t/3732396 ![Discrete differential geometry on meshes via the cotangent Laplacian][1] The whole notebook in one image. A triangle mesh has no curvature, no spectrum, no notion of distance — none of differential geometry — until you build it. The surprise is that a **single sparse matrix, the cotangent Laplacian**, manufactures all of it. Here is the same torus seen four ways through that one operator: its Gaussian **curvature**, its Laplace-Beltrami **vibration modes**, mean-curvature **flow**, and heat-method **geodesic distance**. &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Video-bf248396-73e0-42b0-994d-2814b8192166-ezgif.com-optimize.gif&userId=20103 [2]: https://www.wolframcloud.com/obj/200b007e-0739-4903-aea6-d66a604addac Marco Thiel 2026-06-12T12:46:06Z https://community.wolfram.com/groups/-/m/t/3691199 Hi everyone in the Wolfram Physics / Fundamental Physics Project community, I’ve just finished a 29-page cumulative working document that takes a strictly graph-first approach to electromagnetism, charge, chirality, and family replication. The core is built on a finite oriented graph with compact U(1) links — very much in the spirit of causal graphs and hypergraph models. Quick summary of what’s in the paper: Charge is literally an integer-valued flux-divergence defect on the graph. Exact regional Gauss law, topological quantization from link compactness, exact lattice continuity, Coulomb law from graph-Laplacian minimization, and full Maxwell dynamics emerging in the aligned deconfined phase — all proved cleanly. A common topological language (cellular gauge data + relative current classes + twisted Spinc index theory) unifies charge, chirality, and three-family replication. Explicit finite toy model: product graph $ G = \Lambda_4 \times I_s \times T^2_F $ with domain-wall/overlap fermions and internal flux index 3 gives minimal charge + chirality + three replicated families from one parent generation. Preferred strong ontology (the part I’m most excited about): charged particles are reinterpreted as self-bound local causal shells. A charged lepton is a trapped spinor core coupled to a bosonic shell-deformation sector on a closed 2D simplicial surface. The minimal regular shell is naturally tetrahedral; the leading deformation multiplet is a geometric triplet whose microscopic anisotropy + holonomy collapse to a single-vector operator that automatically produces the exact cosine spectrum used in Koide-type charged-lepton relations. Strong coefficient locking and a codimension-one dynamical balance surface emerge from identical-link tetrahedral geometry and radial-wall tension. The paper is deliberately layered: theorems vs. constructions vs. strong-but-incomplete branches vs. ruled-out ideas. It overlaps structurally with tetrahedral A₄ flavor models but supplies a concrete geometric engine inside the shell rather than abstract flavons. Full PDF (master_working_document_v8.pdf) is attached. Transparency note: I used Grok (xAI) heavily to help me refine the writing, tighten the LaTeX, organize the sections, and make the ideas as clear as possible. The core physics ideas, and ontology are my own — but the AI was instrumental for making this possible. I’d love honest feedback from this community, especially on: how the compact-graph U(1) core and causal-shell picture might connect to hypergraph / causal-graph dynamics, any overlaps or tensions with the Wolfram Physics framework, suggestions for next steps or things I should stress/test. Happy to discuss any part of it. Thanks in advance! Aiden Smith 2026-04-17T01:09:09Z https://community.wolfram.com/groups/-/m/t/3682254 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=8239hero.png&userId=20103 [2]: https://www.wolframcloud.com/obj/27b9cbf2-47f5-4888-81b8-aa014c78b0c6 Myrto Terpsiadou 2026-04-11T13:27:03Z 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/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/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/3636637 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/40f91cb1-021a-4e68-b55a-5249993c389c Donald Airey 2026-02-08T20:17:36Z 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/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/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/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/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/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/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/3515353 Hi all, My book introducing lay readers to the Physics Project is at [Amazon][1]. But community members who want a copy need only reach out to me. I'm in it for the dissemination of ideas, not the money. --- *Developing ideas first introduced in A New Kind of Science two decades ago, Stephen Wolfram and his collaborators in the Wolfram Physics Project have created a completely new paradigm of fundamental physics. Delving deeper than the Standard Model and Einsteinian relativity, Wolfram's new model not only accommodates the traditional models but actually incorporates and, finally, reconciles them.* *The deep conundrums of space, time, and energy turn out to be consequences of a much deeper, more profound reality many orders of magnitude smaller and less tangible. The Universe Within explores this inconceivably small realm, where space itself is infinitesimally tiny pieces of all-but nothingness which, continually transforming and evolving through time and on a vast scale, comprise our reality -- and many others.* *The Universe Within gives the ordinary non-scientist reader a window into what may well be the most radical, most profound, most fundamental model of physics ever proposed.* --- Patrick Sutton ![enter image description here][2] [1]: https://www.amazon.com/dp/B0FHFJLP3F [2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=81ZweIPOqL._SL1500_.jpg&userId=20103 Patrick Sutton 2025-07-19T05:14:09Z https://community.wolfram.com/groups/-/m/t/3497693 ![Units: How we got here and Methodology for Implementation in the Wolfram Physics][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=8694Hero.png&userId=20103 [2]: https://www.wolframcloud.com/obj/106cbf83-ccc0-4607-b14c-9308f182ff06 Matthew Fox 2025-07-10T05:58:55Z