https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Computational Systems sorted by active. https://community.wolfram.com/groups/-/m/t/3751127 ![Developing complex networks by modeling wealth exchange through agent based models][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2026-07-09at1.35.54%E2%80%AFPM.png&userId=3750720 [2]: https://www.wolframcloud.com/obj/8338bed2-7b18-4f8f-9ccb-23c7fec819c3 Ojasvi Singla 2026-07-09T19:23:08Z https://community.wolfram.com/groups/-/m/t/3749919 ![Encrypting, decrypting, and breaking cellular automata-based stream ciphers][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2026-07-08at3.49.01%E2%80%AFPM.png&userId=3733966 [2]: https://www.wolframcloud.com/obj/6888b579-5cb7-4e27-8ee4-6750999d353d Morgan Bergstrom 2026-07-09T16:00:26Z https://community.wolfram.com/groups/-/m/t/3749938 ![Searching for periodicity in vertical center columns of cellular automata][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2026-07-09003216.png&userId=3748468 [2]: https://www.wolframcloud.com/obj/d95e511e-6dd3-48e5-aaf9-7ed62777dc28 Aania Sayida Mir 2026-07-09T16:11:31Z https://community.wolfram.com/groups/-/m/t/3750050 ![Generalization of Turing machine to abstract graphs][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2026-07-09at12.23.40%E2%80%AFPM.png&userId=3657549 [2]: https://www.wolframcloud.com/obj/f1be97b5-0918-436c-b53c-e132542762db Aadi Chakraborty 2026-07-09T16:30:38Z https://community.wolfram.com/groups/-/m/t/3736626 Heya folks, I posted something last week for the first time here in relation to some CA oriented work I crossed into the Wolfram community to try and find some input. This work came from me doing fundamental work based on causal set theory, or onto-epistemic philosophy, or some some form of hybrid 'hopeium' based on conjecture and progressed with all hubris toward 'outcome'. I have no idea if this community is interested in the work I'm doing, but this the the article I made reference to: https://community.wolfram.com/groups/-/m/t/3729239 I have made a ton of progress leading up to that article, and more since. But, I'm not too sure how to introduce and discuss the work in this forum. I have attempted to build a demonstrator via [https://dev.opendata.ai][2] and I am attempting to progress releases there that are very much locked into the digital physics of CA. I'm keen to link in with anyone who is interested in helping me discover (sooner or later) that I'm totally nuts and misguided, or I'm simply totally full of Dunning-Kruger beans and need to learn a ton more before I hope so high :) Any takers? [2]: https://dev.opendata.ai Steven De Costa 2026-06-20T12:21:09Z https://community.wolfram.com/groups/-/m/t/3729239 # Informational Metabolics in a Dual-Phase Cellular Automaton: Verifiable Delay Functions, 4x21 Geometric Confinement, and the Algorithmic Exhaust of Observation ### Author’s Prologue: A Path to Cellular Automata Before diving into the mechanics of Linked Digital Dynamics (LDD), I want to briefly share the context of this work. My background did not begin in theoretical physics; it began in the social sciences. I was studying the emergent behaviors of complex networks and agents, which eventually required me to move under increasing confinement into the strict logic of number theory to test my intuition. My guiding thesis through this interdisciplinary journey was a simple epistemological equation: $1 = O + C + E$. For anything to be considered part of classical reality ($1$), it must be Observable ($O$), Communicable ($C$), and Effectual ($E$). For years, I operated on this intuitive conjecture, trying to map how ideal models resolve into logical, computational realities. As I refined this into Latin square dynamics, I belatedly realized that the mathematics of information, communication, and computation theory were always leading me to a single destination: Cellular Automata. Discovering that the Wolfram Physics Project and Mathematica had already established the exact computational environment needed to resolve these intuitive models was a revelation. As you read the following framework and experiment with the code, you will see how the LDD engine is simply the literal, discrete execution of this equation—where observation is geometric confinement, communication is phase-locking, and effect is the thermodynamic drag of computation. --- ### 1. Abstract The Wolfram Physics Project has provided a profound new lens for viewing spacetime as an emergent property of discrete computation. Yet, as a community, we often grapple with the combinatorial explosion inherent in un-pruned multiway systems—what might be thought of as a "Cosmological Halting Problem." This paper introduces **Linked Digital Dynamics (LDD)**, an exploratory cellular automaton that attempts to model a strict thermodynamic filter over these multiway branches. By treating the substate as a simple $6 \times 6$ dual-phase grid, we observe that even minimal localized interactions generate massive combinatorial phase spaces (e.g., $2^{72}$ conditions). To explore how a universe might compute this without freezing, LDD introduces a simulated physical cost: a **Verifiable Delay Function (VDF)**. Using a $4 \times 21$ geometric confinement matrix (the "Anvil Strike"), this toy model shears off incompatible geometries as **Algorithmic Exhaust** (entropy). Through an executable Wolfram Language Macro-Engine shared below, we can visually explore how this underlying "Informational Metabolic Hunt" naturally segregates a chaotic substate into stable, fast-ticking islands and dense, slow-ticking wells. LDD is presented here as a computational sandbox to invite community feedback on whether emergent gravity and relativistic time dilation might be fruitfully modeled as the macroscopic unevenness of computational friction. --- ### 2. Introduction: Exploring the Cosmological Halting Problem Like many in this community, I have been deeply inspired by the shift toward modeling physics as fundamental, discrete computation. However, while experimenting with localized multiway graphs, I repeatedly ran into a practical computational wall. If every point in space represents a discrete locus evaluating causal branches, the system must navigate a massive phase space of potential states. Consider a minimal collision of two $6 \times 6$ binary substates (an un-sponged False Vacuum). This simple interaction yields over 4.7 sextillion ($2^{72}$) possible combinatorial starting conditions. If the universe were purely probabilistic and smooth, searching through this phase space to find a stable interaction seems computationally prohibitive. Without a rigorous pruning mechanism, a discrete universe risks freezing under the weight of its own combinatorial explosion. **Testing a Thermodynamic Cost for Computation** Linked Digital Dynamics (LDD) is my attempt to build a computable model that addresses this by enforcing a strict thermodynamic cost for time. Inspired by cryptographic principles, I wanted to test what happens if the multiway graph is aggressively filtered by **Topological Drag**—a Verifiable Delay Function (VDF) that forces potential states to "prove" their structural integrity before solidifying. This led to the concept of **Informational Metabolics**. In this framework, a pro-causal trajectory does not calculate all possible futures; it "hunts" for adjacent structural support within a rigid $4 \times 21$ spatial confinement matrix. If the geometry perfectly phase-locks, it metabolizes that potential into a stable structure. If it fails, the unmatched data is sheared off into the substate as Algorithmic Exhaust. --- ### 3. Ontology of the Strata: Substate, Microstate, and Macrostate To circumvent the combinatorial explosion of the multiway graph, LDD proposes that reality does not compute in a single continuous layer. Information must survive severe geometric compression to ascend from pure potential to classical reality. * **I. The Substate (The False Vacuum):** The foundational layer, modeled computationally as a $6 \times 6$ branchial buffer. This is a region of pure, un-sponged potential—a completely frictionless state of non-constructive optimism where multiway branches freely generate. It possesses zero entropy. * **II. The Microstate (The Thermodynamic Forge):** Defined by the **$4 \times 21$ Geometric Confinement Matrix**. When two offset causal trajectories attempt to interact, their multiway potential is forced into an orthogonal cross-reading we call the **Anvil Strike**. It is a literal cryptographic pruning function testing if discrete geometries can support one another. * **III. The Macrostate (Linear Indifference):** The resulting classical reality, composed exclusively of localized motifs that survived the Anvil Strike. A Macrostate observer exists in a state of **Linear Indifference**—because its internal structure balances the thermodynamic drag, it feels no friction from its own existence, experiencing discrete processing purely as smooth spacetime. --- ### 4. Cryptographic Thermodynamics: The Cost of Observation In classical physics, entropy is a measure of disorder. In LDD, entropy is strictly redefined as **Algorithmic Exhaust**—the literal byproduct of failed computation. Because the substate is continuously attempting to expand, the universe requires an anchoring mechanism. The VDF acts as the fundamental clock speed of the universe, imposing **Topological Drag** modeled as a $+2, -1$ pro-causal ratchet. The $+2$ represents optimistic branchial generation, and the $-1$ represents the thermodynamic computational cost exacted by the universe. When a trajectory enters the $4 \times 21$ confinement cage, the VDF enforces **Mutually Assured Cooperation**. Optimistic multiway data only survives if it has adjacent structural support. Incompatible data snaps under the tension and is sheared off the event horizon back into the substate as Algorithmic Exhaust. --- ### 5. Informational Metabolics: The Darwinian Engine of Reality LDD requires a radical inversion of perspective: a localized multiway branch is an active, hungry computational agent. We term this survival mechanism the **Informational Metabolic Hunt**. The massive $2^{72}$ combinatorial phase space of the False Vacuum is the raw caloric potential of the universe. As a trajectory is pushed forward by the VDF, it must consume this substate potential to maintain phase-lock. A successful "hunt" occurs when two distinct trajectories possess the exact geometric seeds required to interlock. Conversely, trajectories with incompatible seeds cannot metabolize the local substate; they "starve" within the Anvil Strike and drop out of the macro-reality. Only the most geometrically optimized structures survive the friction of time. --- ### 6. Informational Relativity and Cosmogenesis Because LDD posits that time is locally computed labor rather than a universal background dimension, there is no absolute clock. By decoupling a local observer's VDF speed from the computational phase of an external node, we organically generate **Informational Relativity**. A stable macrostate motif requires an $8 \times 8$ harmonic membrane to resolve chiral pressure. The literal edge of this membrane acts as the particle's **Event Horizon**, which exhibits **Causal Elasticity**. An observer can tolerate a slight relational phase offset ($\Delta$) with an external node without breaking phase-lock. However, if the relativistic tension exceeds the harmonic capacity, the boundary violently shears into Algorithmic Exhaust. This perfectly mirrors Einsteinian mechanics: * **Spatial Expansion (Redshift):** If an external node's VDF computes slower ($\Delta > 0$), the computational channel stretches. * **Informational Collisions (Blueshift):** If the node computes faster ($\Delta < 0$), the computational distance collapses, aggressively forcing its multiway state onto the observer. --- ### 7. The LDD Engine: A Computable Laboratory To move from philosophy to falsifiable physics, I have developed a native Wolfram Language laboratory. **I. The Holographic Relativistic Micro-Engine** This module isolates a single $4 \times 21$ Anvil Strike. Using the dual-slider architecture, you can physically drag an external node through the entire spectrum of informational Doppler shift. You can also load highly ordered "Custom Seeds" (orthogonal arrays) to witness how perfect geometry acts as a "Still Life," surviving VDF loops without evaporating. ```mathematica ClearAll[SeedFrameA, SeedFrameB, InitializeFalseVacuum, VDFSubstateMix, AnvilStrike, LDDChronoverticalTick, LDDDashboard]; (* I. CUSTOM STARTING CONDITIONS *) SeedFrameA = {{1, 0, 1, 0, 1, 0}, {0, 1, 0, 1, 0, 1}, {1, 0, 1, 0, 1, 0}, {0, 1, 0, 1, 0, 1}, {1, 0, 1, 0, 1, 0}, {0, 1, 0, 1, 0, 1}}; SeedFrameB = {{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0}, {1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0}, {1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0}}; InitializeFalseVacuum[] := RandomInteger[1, {6, 6}]; VDFSubstateMix[grid_] := Mod[grid + ListConvolve[{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}, grid, 2], 2]; (* II. MICROSTATE CONFINEMENT & ALGORITHMIC EXHAUST *) AnvilStrike[gridA_, gridB_, ticksA_, ticksB_] := Module[ {trajA, trajB, phaseSpace, cage, stabilizedCage, preSum, postSum, entropy}, trajA = Join[Flatten[gridA], IntegerDigits[Mod[ticksA, 64], 2, 6]]; trajB = Join[Flatten[gridB], IntegerDigits[Mod[ticksB, 64], 2, 6]]; phaseSpace = Join[trajA, trajB]; cage = Partition[phaseSpace[[Mod[Range[84]*5, 84] + 1]], 21]; preSum = Total[Flatten[cage]]; stabilizedCage = cage * UnitStep[ListConvolve[{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}, cage, {2, 2}, 0] - 1]; postSum = Total[Flatten[stabilizedCage]]; entropy = preSum - postSum; <|"Microstate" -> stabilizedCage, "Entropy" -> entropy, "ExhaustBits" -> (cage - stabilizedCage)|> ]; (* III. CHRONOVERTICAL TICK *) LDDChronoverticalTick[gridA_, gridB_, localVDF_, deltaVDF_] := Module[ {ticksA, ticksB, mixedA, mixedB}, ticksA = localVDF; ticksB = Max[0, localVDF + deltaVDF]; mixedA = Nest[VDFSubstateMix, gridA, ticksA]; mixedB = Nest[VDFSubstateMix, gridB, ticksB]; AnvilStrike[mixedA, mixedB, ticksA, ticksB] ]; (* IV. INTERACTIVE DASHBOARD *) LDDDashboard[] := DynamicModule[ {gridA, gridB, result}, gridA = SeedFrameA; gridB = SeedFrameB; Manipulate[ result = LDDChronoverticalTick[gridA, gridB, localVDF, deltaVDF]; Column[{ Style["LDD: Holographic Relativistic CA Engine", Bold, 16, FontFamily -> "Helvetica"], Spacer[10], Row[{Style["Local VDF Speed: ", 12], Style[localVDF, Bold, 14], Spacer[20], Style["Relativity (\[Delta]): ", 12], Style[deltaVDF, Bold, 14], Spacer[30], Style["Algorithmic Exhaust (Entropy): ", 12, Red], Style[result["Entropy"], Bold, 16, Red]}], Spacer[15], Text@Grid[{{Column[{Style["Observer A Frame (Local)", 10], ArrayPlot[Nest[VDFSubstateMix, gridA, localVDF], ImageSize -> 130, Mesh -> True, ColorFunction -> "DeepSeaColors"]}], Column[{Style["Observer B Frame (External)", 10], ArrayPlot[Nest[VDFSubstateMix, gridB, Max[0, localVDF + deltaVDF]], ImageSize -> 130, Mesh -> True, ColorFunction -> "DeepSeaColors"]}]}}, Spacings -> {5, 1}], Spacer[20], Style["Phase-Locked Microstate (4x21 Confinement)", Bold, 12], ArrayPlot[result["Microstate"], ImageSize -> 550, Mesh -> True, ColorFunction -> "SunsetColors"], Spacer[10], Style["Non-Constructive Optimism (Sheared Entropy)", 10, Gray], ArrayPlot[result["ExhaustBits"], ImageSize -> 550, Mesh -> True, ColorFunction -> "Monochrome"] }, Alignment -> Center], {{localVDF, 42, "Observer A Baseline (Local VDF)"}, 1, 256, 1, Appearance -> "Labeled"}, {{deltaVDF, 0, "Informational Relativity (\[Delta])"}, -512, 512, 1, Appearance -> "Labeled"}, Row[{Button["Load Custom Seeds (A & B)", {gridA = SeedFrameA; gridB = SeedFrameB;}], Spacer[10], Button["Generate Random Vacuum", {gridA = InitializeFalseVacuum[]; gridB = InitializeFalseVacuum[];}]}], ControlPlacement -> Top, TrackedSymbols :> {localVDF, deltaVDF} ] ] LDDDashboard[] ``` **II. The Macroscopic Spacetime Lattice (Deep Time Simulation)** This module expands the engine into a macroscopic lattice of 144 interacting observers. Using Deep Time Controls, you can age the universe by hundreds of ticks instantly. You will observe the False Vacuum segregate into cold islands of stable matter (fast-ticking clocks / vacuum) and dense, hot entropic borders (slow-ticking clocks / time dilation). ```mathematica ClearAll[InitializeMacroVacuum, VDFSubstateMix, MacroAnvilStrike, MacroNetworkTick, LDDMacroDashboard]; InitializeMacroVacuum[size_] := Table[<|"Substate" -> RandomInteger[1, {6, 6}], "VDFClock" -> 0, "Entropy" -> 0.0|>, {size}, {size}]; VDFSubstateMix[grid_] := Mod[grid + ListConvolve[{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}, grid, 2], 2]; MacroAnvilStrike[nodeA_, nodeB_] := Module[ {trajA, trajB, phaseSpace, cage, stabilizedCage, preSum, postSum}, trajA = Join[Flatten[nodeA["Substate"]], IntegerDigits[Mod[nodeA["VDFClock"], 64], 2, 6]]; trajB = Join[Flatten[nodeB["Substate"]], IntegerDigits[Mod[nodeB["VDFClock"], 64], 2, 6]]; phaseSpace = Join[trajA, trajB]; cage = Partition[phaseSpace[[Mod[Range[84]*5, 84] + 1]], 21]; preSum = Total[Flatten[cage]]; stabilizedCage = cage * UnitStep[ListConvolve[{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}, cage, {2, 2}, 0] - 1]; postSum = Total[Flatten[stabilizedCage]]; preSum - postSum ]; MacroNetworkTick[network_] := Module[ {size, nextNetwork, n, s, e, w, localEntropy}, size = Length[network]; nextNetwork = network; Do[ n = network[[Mod[i - 1, size, 1], j]]; s = network[[Mod[i + 1, size, 1], j]]; e = network[[i, Mod[j + 1, size, 1]]]; w = network[[i, Mod[j - 1, size, 1]]]; localEntropy = Mean[{MacroAnvilStrike[network[[i,j]], n], MacroAnvilStrike[network[[i,j]], s], MacroAnvilStrike[network[[i,j]], e], MacroAnvilStrike[network[[i,j]], w]}]; If[localEntropy < 1.5, nextNetwork[[i,j, "VDFClock"]] += 1; nextNetwork[[i,j, "Substate"]] = VDFSubstateMix[network[[i,j, "Substate"]]];]; nextNetwork[[i,j, "Entropy"]] = localEntropy; , {i, size}, {j, size}]; nextNetwork ]; LDDMacroDashboard[] := DynamicModule[ {gridSize = 12, macroNetwork, totalAge = 0}, macroNetwork = InitializeMacroVacuum[gridSize]; Manipulate[ Column[{ Style["LDD: Macroscopic Spacetime Lattice", Bold, 16, FontFamily -> "Helvetica"], Spacer[5], Row[{Style["Age of Universe (Total Ticks): ", 14, White], Style[totalAge, Bold, 16, Cyan]}], Spacer[15], Row[{ Column[{Style["Algorithmic Exhaust (Gravitational Heatmap)", Bold, 12, White], Style["Dark = Perfect Phase Lock (Baryonic Matter/Vacuum)\nBright/Hot = High Entropy (Event Horizons/Chaos)", 10, Gray], MatrixPlot[Map[#["Entropy"] &, macroNetwork, {2}], ColorFunction -> "ThermometerColors", ImageSize -> 300, Frame -> False, Mesh -> True, MeshStyle -> Opacity[0.2]]}], Spacer[40], Column[{Style["Informational Relativity (Local VDF Ticks)", Bold, 12, White], Style["Bright = Fast Clocks (Low Drag)\nDark = Slow Clocks (Time Dilation/Gravity Wells)", 10, Gray], MatrixPlot[Map[#["VDFClock"] &, macroNetwork, {2}], ColorFunction -> "SunsetColors", ImageSize -> 300, Frame -> False, Mesh -> True, MeshStyle -> Opacity[0.2]]}] }] }, Alignment -> Center, Background -> Black], Row[{Style["Advance Time: ", Bold, 12], Button["+1 Tick", {macroNetwork = MacroNetworkTick[macroNetwork]; totalAge += 1;}], Button["+10 Ticks", {Do[macroNetwork = MacroNetworkTick[macroNetwork], {10}]; totalAge += 10;}], Button["+100 Ticks", {Do[macroNetwork = MacroNetworkTick[macroNetwork], {100}]; totalAge += 100;}]}], Spacer[10], Button["Inject Big Bang (Regenerate Network)", {macroNetwork = InitializeMacroVacuum[gridSize]; totalAge = 0;}], ControlPlacement -> Top ] ] LDDMacroDashboard[] ``` --- ### 8. Conclusion and Call to Action: Mining the Substate Linked Digital Dynamics demonstrates that a discrete, computable universe must possess a physical thermodynamic cost to prevent a combinatorial explosion. The Cosmological Halting Problem is solved by the Verifiable Delay Function—a pro-causal ratchet that violently shears off non-constructive optimism as Algorithmic Exhaust. In this model, gravity, time dilation, and event horizons are simply emergent, macroscopic artifacts of localized computational friction. I present this framework not as a completed map of physics, but as a foundational sandbox. The true work lies in exploring the $2^{72}$ phase space of the False Vacuum. I issue a direct challenge to the Wolfram Community: **Mine the Substate.** Using the provided Mathematica engine, search the combinatorial starting conditions for the exact binary seeds—the cryptographic "Latin squares" and orthogonal arrays—that yield perfectly stable, zero-entropy loops. By discovering which specific geometric matrices survive the thermodynamic forge of the $4 \times 21$ Anvil Strike, we can begin to isolate the literal discrete DNA of ordinary matter, dark matter, and the frictionless vacuum. Steven De Costa 2026-06-08T07:49:22Z https://community.wolfram.com/groups/-/m/t/3724930 ![ReloadCA: adaptive sparse and parallel cellular automata in Wolfram language][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=6486cover.png&userId=3207931 [2]: https://www.wolframcloud.com/obj/8b79a1ce-59b4-4d93-838c-629474b2b288 Austin Jiang 2026-06-01T04:57:33Z https://community.wolfram.com/groups/-/m/t/3720182 ![enter image description here][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=3470hero.png&userId=20103 [2]: https://www.wolframcloud.com/obj/7d116d6f-a55a-4d07-9873-c9201303db83 Anton Antonov 2026-05-21T22:16:16Z https://community.wolfram.com/groups/-/m/t/3720941 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/346760a8-fe52-47a7-b452-f945a4ca1986 Brian A. Mboya 2026-05-22T18:35:43Z https://community.wolfram.com/groups/-/m/t/3715029 A continuous-state cellular automaton on [0,1]³ where each cell carries an RGB vector. One axis — Blue — is subject to preferential decay through habituation, while a competing error-detection mechanism restores Blue when local chromatic discrepancies are large. The balance between these forces produces a sharp phase transition: below a critical decay rate, Blue persists and chromatic diversity is maintained. Above it, Blue collapses and the entire population converges to the Yellow vertex (1,1,0) — maximum capability on two axes, zero on the third. The post grounds the dynamics from the differentiation operator D(v) on {0,1}³ (Posts 1-2), demonstrates the emergent phase transition through bifurcation analysis, estimates the critical exponent, and visualizes the population trajectory through the binary cube as it falls from random initialization toward a specific vertex attractor. The terminal state histogram shows Red and Green distributed broadly across moderate-to-high values while Blue is compressed into a narrow spike near zero — the chromatic signature of a system that has lost its inhibitory axis. Five questions for the community: 1. Inevitability of collapse. In any three-axis system where one axis is subject to preferential degradation while the other two are not, is convergence to the two-axis vertex inevitable above some critical decay rate? Or can system topology — lattice geometry, boundary conditions, coupling structure — prevent the collapse entirely? 2. Universality class. The critical exponent β characterizes the universality class of the phase transition. Is the transition in the same universality class as known lattice models (Ising, percolation), or does the three-axis chromatic structure produce a novel class? 3. Negative feedback existence. The feedback loop (Blue decay reduces diversity → reduced diversity weakens error signals → weaker signals accelerate Blue decay) is self-reinforcing above the critical threshold. Is there a corresponding negative feedback loop below the threshold that actively stabilizes Blue, or does the subcritical regime simply lack positive feedback without possessing negative feedback? 4. Yellow maximization. The terminal attractor is the Yellow vertex (1,1,0), not an arbitrary point on the B=0 face. The surviving components are driven to their maximum values. Is this a general property of inhibitory decay in coupled systems — that the loss of the inhibitory axis maximizes the uninhibited axes — or is it specific to the coupling structure defined here? 5. Initial condition dependence. The system was initialized with uniform random RGB values. Does the critical threshold δ_c depend on initial conditions? Specifically, if the population starts near the Blue vertex (0,0,1), does δ_c increase, suggesting that initial proximity to the inhibitory axis provides transient protection against its decay? Notebook included. https://www.wolframcloud.com/obj/b1180013-2f9d-4af8-a808-8462ddceeb8d Dustin Sprenger 2026-05-12T11:33:32Z https://community.wolfram.com/groups/-/m/t/3701293 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/10063044-be48-4242-a0c3-602feb0b9f6e Marco Thiel 2026-04-22T23:43:13Z https://community.wolfram.com/groups/-/m/t/3695987 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/12016646-05c7-4e0f-aafc-a180d14e3a8e Ramón Eduardo Chan López 2026-04-19T00:41:46Z https://community.wolfram.com/groups/-/m/t/3696660 A chromatic cellular automaton on {0,1}³ generates a cruciform pattern from a single seed (established in Post 3). At every timestep, the null-space complement of the pattern — obtained by mapping each state s → 7−s — is computed and compared to the positive space using five independent complexity measures: Shannon entropy, joint entropy of adjacent pairs, spatial mutual information, 2×2 block entropy, and Kolmogorov complexity approximation via compression. Result: the first four measures show exact equality C(P,t) = C(N,t) at every step. The fifth (Kolmogorov complexity approximated via compression) shows near-equality with fractional difference under 3%, consistent with encoder-dependent variation in a computable upper bound rather than genuine structural asymmetry. The structural complexity of the complement matches the complexity of the pattern — not approximately, but identically — across all information-theoretic measures tested. The complement operator also reveals a vertex absent from the automaton's lifecycle. The lifecycle visits Red, Green, Blue, Yellow, and Magenta. Cyan (0,1,1) never appears. In the complement, Red maps to Cyan. Since Red is the most frequently renewed state (Dead → Idea), Cyan becomes the dominant active state in the null space. The vertex absent from the dynamics is the most prominent vertex of the shadow. Four questions for the community: 1. Continuous extension. The information symmetry C(P,t) = C(N,t) holds by construction for any bijective complement on a finite vertex set. Does it extend to continuous state spaces on [0,1]³? 2. Cyan dominance. The complement reveals Cyan (0,1,1) as the dominant active state of the null space. If Cyan were introduced into the lifecycle, would the complement become impoverished at a corresponding vertex? 3. Simultaneous threshold crossing. The null-space complement achieves identical structural complexity without its own dynamics. In any system where the positive space crosses a self-referential complexity threshold, does the complement cross simultaneously? 4. Universality challenge. Five complexity measures all show C(P,t) = C(N,t). Is there ANY computable complexity measure for which the equality fails? Notebook Attached. https://www.wolframcloud.com/obj/f17551bc-aec2-415b-9323-ad8865b5d46d Dustin Sprenger 2026-04-20T02:45:27Z https://community.wolfram.com/groups/-/m/t/3673620 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/ad350d33-f532-4a74-ba44-050ece07c9fb Ramón Eduardo Chan López 2026-04-03T07:56:17Z https://community.wolfram.com/groups/-/m/t/3673723 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/c4a1ef8d-8d48-4bf8-abe0-0eac4501058d Tigran Nersissian 2026-04-03T02:25:29Z https://community.wolfram.com/groups/-/m/t/3674654 ![Neural excitability, sparsity & the bridge to AI][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=5237hero.png&userId=20103 [2]: https://www.wolframcloud.com/obj/9dd8b8b7-ab62-4227-90c8-7b74b24ce525 Guhan Thiagarajan 2026-04-06T04:54:16Z https://community.wolfram.com/groups/-/m/t/3675246 A single Idea seed placed at the center of a 4-dimensional stochastic cellular automaton — two temporal layers coupled to a spatial grid — evolves through five semantic states: Idea, Concept, System, Flaw, and Dead. The update rule has three components: a cardinal-neighbor condition governing spatial propagation, stochastic decay with probability 0.3, and temporal coupling between layers via BitOr injection. No cross-shaped template is imposed. The cruciform pattern emerges from the interaction of cardinal propagation geometry with the lifecycle transition rules. This post implements the automaton defined in Mathematical Belief I (Sprenger, 2025), translating the original Python code into Mathematica with array-based state masking via Unitize. The simulation runs on a 31×31×2×2 grid for 150 steps and produces: (1) the emergent cross in all four temporal layers, (2) chromatic composition evolution over time, (3) Blue component tracking across the full simulation, (4) attractor analysis of late-time state fractions, (5) a decay-rate sensitivity sweep demonstrating that cross shape persists while chromatic balance varies, (6) breach analysis, and (7) the lifecycle path rendered on the {0,1}³ cube. The cross is the first of three emergent geometries. Post 7 extends the same lifecycle to 3D, producing cubic symmetry. Post 10 adds axial coupling, producing a tree. The operator is the same throughout. Only the dimensionality changes. Open Questions: 1. The cruciform symmetry emerges from the clean-tip propagation rule: exactly 1 cardinal System neighbor, 0 other active cardinals. Is this the minimal local rule producing stable cross-symmetry on a square lattice? 2. The four temporal layers develop distinct phase relationships through BitOr coupling. Does the system exhibit computational irreducibility? 3. The lifecycle visits 5 of 8 vertices of {0,1}³. What role would Cyan (0,1,1) play if introduced as a seventh lifecycle state? 4. Under what propagation rules does breach become inevitable? Is there a critical decay probability above which the cross destabilizes? Notebook Attached. https://www.wolframcloud.com/obj/58ccf07e-69a8-4d84-8864-0de8dccd2000 Dustin Sprenger 2026-04-07T04:31:38Z https://community.wolfram.com/groups/-/m/t/3671492 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/1f196033-714a-413f-90e4-7b22075ea1f4 Tigran Nersissian 2026-03-30T09:44:23Z https://community.wolfram.com/groups/-/m/t/3671124 ![Fireflies or nature's cellular automaton][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=4420FirefliesorNature%27sCellularAutomaton.gif&userId=20103 [2]: https://www.wolframcloud.com/obj/abfab849-1504-446c-90d3-4a5b862ab440 Kirill Vasin 2026-03-27T18:14:17Z https://community.wolfram.com/groups/-/m/t/3647733 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/b04b6551-fecf-465d-b02d-63d95abd751c Tigran Nersissian 2026-03-02T11:53:13Z