\n The simulation above runs a hybrid cellular automaton on a Chinese-Remainder–structured hypergrid.\n Each node's birth, survival, and toppling thresholds are generated from its own phase coherence,\n so the rules evolve endogenously while the CRT topology drives non-local interaction.\n Use the tabs to switch between the geometric hypergrid, the phase-vector physics view,\n and the thermodynamic tile-domain board. Scroll for details.\n

\n```\n\nREPLACE with:\n```\n

\n The simulation runs a hybrid cellular automaton on a Chinese-Remainder–structured hypergrid\n whose nodes are anchored to a 3D Mandelbulb topological boundary. Each module's phase vector\n rotates its CRT residue point, and a polynomial Mandelbulb iteration measures how close that\n rotated point lies to the fractal skin. That boundary potential, together with gated phase\n covariance, endogenously generates birth, survival, and toppling rules. Switch tabs to see the\n boundary-colored hypergrid, the covariance-modulated phase-vector field, and the thermodynamic\n tile-domain board. Scroll for details.\n

\n```\n\nI should also maybe update the title in index? It says \"Endogenous Topological Life/Sand\". Could add Mandelbulb but not necessary. Maybe update h1? The README replaces content anyway. Keep title.\n\nNeed to ensure no runtime errors. Let's verify substrate.js with new methods. `_crtBasePoint` uses `this.crtAxes` which is set in constructor. Module init calls `_crtBasePoint(i)`; at that point `this.crtAxes` defined. Good.\n\n`boundaryPoint: { ...basePoint }` clones object.\n\n`_mandelbulbPotential` default parameters use `this.power`, `this.bulbIter`. In method signature, `power = this.power, maxIter = this.bulbIter`. That's valid.\n\n`_phaseCovariance` calls `this.legalityGate(m,n)` which uses `a.bulbPotential` etc. During first call in constructor? `_phaseCovariance` is only called in step after bulb metrics set. In step, before first call, bulbPotential initialized to 0.5 in module object. Good.\n\nIn `legalityGate`, boundaryFactor uses `a.bulbPotential||0`. Good.\n\nIn phase update loop, `m.bulbPotential` initial 0.5, so boundaryCoupling = (0.6+0.8*0.5)=1.0 * couplingStrength. Good. Later it varies.\n\nIn step, after phase commit, compute bulb metrics and covariance. Then state commit uses new rule which uses updated `m.bulbPotential`? Wait `_endogenousRule` is called earlier in first loop using previous frame's bulbPotential (since updated only after phase commit). Then state commit uses `m.nextState` computed with old rule but commits raw + topples. Then recomputes coherence. That means rule thresholds lag one frame behind boundary, fine. Could also recompute rule in final loop if desired, but not needed.\n\nPotential issue: `_gatedNeighborSum` called in first loop uses legalityGate with old bulbPotential; fine.\n\nPotential issue: In step, `topples` array is initialized; in first loop, when m.state >= threshold, topples[i] -= overflow and neighbors += share. However `topples[i]` initially 0, becomes negative. In final loop `raw = m.nextState + topples[i]`. If nextState set to m.state (because toppling branch doesn't set nextState? Actually in toppling branch `topples[i] -= overflow; ...` and nextState remains whatever initialized? The code sets `let nextState = m.state;` at start. In toppling branch, it doesn't assign `m.nextState = nextState`? It sets `m.nextState = nextState` after the if-else block, outside. Wait structure:\n\n```\n let nextState = m.state;\n\n // Sandpile toppling: overflow above threshold spreads to neighbors.\n if (m.state >= rule.toppleThreshold) {\n const overflow = m.state - rule.toppleThreshold + 1;\n const share = overflow / m.neighbors.length;\n topples[i] -= overflow;\n for (const nId of m.neighbors) {\n topples[nId] += share;\n }\n } else {\n // Game-of-Life layer\n ...\n m.nextState = nextState;\n }\n```\n\nWait `m.nextState = nextState;` is inside else? The current code as provided shows:\n```\n } else {\n // Game-of-Life layer\n if (m.state > 0) {\n if (gatedSum < rule.surviveMin || gatedSum > rule.surviveMax) {\n nextState = Math.max(0, m.state - 1);\n }\n } else {\n if (gatedSum >= rule.birthMin && gatedSum <= rule.birthMax) {\n nextState = 1;\n }\n }\n }\n\n m.nextState = nextState;\n```\nActually from provided code:\n```\n } else {\n // Game-of-Life layer\n if (m.state > 0) {\n if (gatedSum < rule.surviveMin || gatedSum > rule.surviveMax) {\n nextState = Math.max(0, m.state - 1);\n }\n } else {\n if (gatedSum >= rule.birthMin && gatedSum <= rule.birthMax) {\n nextState = 1;\n }\n }\n }\n\n m.nextState = nextState;\n```\nYes `m.nextState = nextState;` is outside else. Good. So toppling nodes nextState = current state, then topples subtract overflow, so final state = threshold-1 plus shares. Fine.\n\nNow ensure `_endogenousRule` toppleThreshold can be 1. If maxState=4, threshold can be 1 leading to constant toppling. With boundaryLean high, threshold may be 1. That might be too aggressive. But maxState 4. If threshold=1, any state>=1 topples. That could keep states low. Might be okay. But maybe set minimum 2 to preserve sandpile. User wanted life/sand. Maybe `Math.max(2,...)` is better. I changed to `Math.max(1,...)`. Should I revert to min 2? In the new formula, if boundary and coherence high, threshold=1 could collapse. Maybe better min 2. Let's use `Math.max(2, ...)` as original. But the user mentioned \"Mandelbulb physics on topological boundaries\" and might want strong effect. Keep min 1 could be chaotic. Let's set min 2 to keep some accumulation but lower threshold near boundary. I'll adjust replacement to `Math.max(2, ...)` instead of `Math.max(1, ...)`.\n\nActually original used `Math.max(2, Math.min(this.maxState, Math.floor(...)))`. I'll preserve min 2.\n\nNew replacement:\n```\n const toppleThreshold = Math.max(2, Math.min(this.maxState, Math.round(\n 2 + (1 - coh) * 2 + (1 - boundaryLean) * 2 + m.ruleBias * 2\n )));\n```\n\nNow what about the instruction to \"not ask questions\" and output changes. Also provide Summary at the end. The summary should be about actual changes, ~tweet length.\n\nAlso note asset generation not needed. No new assets.\n\nDouble-check vector-physics `_phaseColor`: It uses phase[0], phase[1]. phases can be large after many steps (not wrapped). `Math.atan2(Math.sin(phase[1]), Math.cos(phase[0]))` is weird mixing dims. Not changed. It's okay.\n\nPotential issue: In vector-physics.js edge strokeStyle uses `toFixed(3)`; fine.\n\nPotential issue: `m.boundaryPoint` assigned object in step; not used in renderers? We could use in render.js to adjust layout? Not necessary.\n\nPotential issue: README markdown in main.js renderer: It will render new README with headings/lists. Fine.\n\nNow produce final response with all edits. I need to format each file edit. Since many edits to same file, each in own code block labeled with filepath.\n\nI will output all SEARCH/REPLACE blocks. Then Summary.\n\nLet's write README new content. Make it comprehensive but not too huge. Include CRT topology, Mandelbulb boundary, polynomial functional, phase covariance, endogenous rules, tabs.\n\nREADME content:\n```\n# MHCGH-S – Endogenous Topological Life/Sand on a Mandelbulb Boundary\n\nA deterministic cellular automaton / sandpile hybrid whose nodes live on a Chinese-Remainder–structured hypergrid and are anchored to a 3D Mandelbulb topological boundary. Rules are not global: each module's birth, survival, and toppling thresholds are generated endogenously from its phase coherence, its Mandelbulb boundary potential, and its gated phase covariance with CRT neighbors.\n\n## 1. Files\n\n- `substrate.js` – `MHCGHSubstrate`: the simulation model.\n- `render.js` – `MHCGHRenderer`: geometric hypergrid view colored by state, coherence, and boundary potential.\n- `vector-physics.js` – `VectorPhysicsRenderer`: phase-vector field, coupling gates, and covariance-modulated edges.\n- `tile-domain.js` – `TileDomainRenderer`: thermodynamic tile-domain board with CRT skip links and Hessian curvature.\n- `main.js` – boots the substrate, wires the three tabs, fetches and renders `README.md`.\n- `index.html` – the page shell.\n\n## 2. Core Concepts\n\n- **Modules**: Nodes in a sparse non-local graph. Each carries a multi-dimensional phase vector, a discrete state, a private rule bias, and a CRT residue triple.\n- **CRT Topological Boundary**: The step sizes `[5, 7, 11]` are pairwise coprime. Each module's residues `(i mod 5, i mod 7, i mod 11)` are mapped to a point on the unit sphere — a fixed topological boundary anchor.\n- **Mandelbulb Potential**: The boundary point is rotated by the module's phase vector, then iterated through a power-8 Mandelbulb polynomial. The result is an escape-time `bulbEscape` and a boundary-distance `bulbPotential`. Nodes near the fractal skin couple more strongly and topple sooner.\n- **Phase Covariance**: A gated average cosine similarity between a module's phase vector and those of its neighbors. It measures the statistical glue of the rule complex.\n- **State**: An integer `0 … maxState`. `0` is empty/dead; higher values are active cells or sand.\n- **Phase Coherence (`pas_h`)**: Recomputed as a blend of self phase alignment and neighbor covariance, boosted by boundary proximity.\n- **Entropy**: Phase dispersion among CRT neighbors; high entropy desaturates color and can trigger decoherence.\n- **Decoherence**: A red halo appears when coherence drops below half the configured threshold.\n\n## 3. Topology: CRT Non-Local Hypergrid\n\nThe wiring is built from the coprime step sizes:\n\n```js\nconst steps = [5, 7, 11];\n```\n\nFor each node `i` and step `s`, neighbors are `(i ± s) mod N`; local `±1` links are also added. This produces short-range and long-range \"skip\" connections that the tabs render as straight or curved arcs.\n\n## 4. Endogenous Rules\n\nEach module evaluates:\n\n1. **Mandelbulb boundary map**: rotate `basePoint` by `phase`, run `power` iterations, read `bulbPotential`.\n2. **Phase covariance**: average gated cosine similarity with CRT neighbors.\n3. **Rule generation**: `toppleThreshold`, `birthMin/Max`, `surviveMin/Max` are functions of coherence, boundary potential, and `ruleBias`.\n4. **Update**:\n - Phases couple through gated sine differences, with coupling boosted near the boundary.\n - Sandpile toppling redistributes overflow to neighbors.\n - Game-of-Life layer applies birth/survival windows.\n - Coherence is refreshed from self-alignment, covariance, and boundary boost.\n\n## 5. Tabs\n\n- **Simulation**: Geometric embedding of the CRT hypergrid. Node color and size encode state, coherence, entropy, and Mandelbulb boundary proximity (violet shell).\n- **Vector Physics**: Phase-vector spokes per dimension, directed coupling arrows whose opacity reflects gate strength and phase covariance, and a violet shell showing boundary potential.\n- **Tile Domain**: A thermodynamic board where tile height combines state and boundary pressure, tile color maps solid/liquid/plasma phases, CRT skip links are curved arcs, and a small arc encodes local Hessian curvature.\n```\n\nNow index.html description as above.\n\nLet's also update the `

` in index? It says \"MHCGH-S – Endogenous Topological Life/Sand\". Could add \"Mandelbulb\". Maybe not required. The description covers it. Keep.\n\nNow final response.\n\nI should ensure all code blocks labeled with filename.\n\nAlso no extra narrative? The instructions say \"Afterwards, describe what you did.\" So I can put summary at end. The code edits come first.\n\nLet's produce.\n\nOne more thought: The user said \"non-mantissa systems ... could actually enhance the efficiency of the simulations by digging into polynomial functionals and their covariance\". I implemented polynomial Mandelbulb functional and covariance. Good.\n\nCheck for any `Math.random()` in tile-domain.js for flow particles; that's allowed outside Remotion. Fine.\n\nPotential issue: `_phaseCovariance` in step is called after phase commit but before state commit. It uses `this.modules[nId]` phases updated. Good. But `legalityGate` uses `a.bulbPotential` and `b.bulbPotential` which are old values? Actually in the same loop we set bulb metrics before covariance. So covariance uses updated bulbPotential. Good.\n\nPotential issue: In `step` first loop, `boundaryCoupling` uses `m.bulbPotential` from previous frame. Fine.\n\nPotential issue: `m.bulbPotential` initialized 0.5, so boundaryFactor in gate starts 0.775. Good.\n\nPotential issue: In `render.js`, boundary ring uses `m.bulbPotential`. Fine.\n\nPotential issue: `tile-domain.js` `_hessianCurvature` uses state only. Could include boundary potential gradient. But okay.\n\nPotential issue: README includes code block; main.js markdown renderer handles fences.\n\nPotential issue: The import map in index.html only has d3-interpolate, but vector-physics/render/tile-domain import it. Good.\n\nPotential issue: `_rotateByPhase` assumes phaseDimensions=3; if not, `phase[1]` etc default 0. Good.\n\nPotential issue: `crtAxes` from steps; if steps length <3, fallback [5,7,11]. Good.\n\nPotential issue: `_crtBasePoint` divides by `i % a / a`. If a=0? not possible.\n\nPotential issue: In module init, `crt: [i % this.crtAxes[0], ...]` good.\n\nPotential issue: performance: computing Mandelbulb per module per frame with `Math.pow(r,8)` and trig could be okay for 96 nodes. Good.\n\nPotential issue: `boundaryBoost` in coherence could push pas_h above 1; clamped. Good.\n\nPotential issue: `phaseCovariance` can be negative if phases anti-aligned; coherence mix could lower. Fine.\n\nPotential issue: `legalityGate` boundaryFactor uses max of a,b potentials; if both high, factor up to 1.0. Good.\n\nPotential issue: `_endogenousRule` birthMax uses `boundary` maybe up to 4, but maxState 4. That's fine for thresholds.\n\nNow output.\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nsubstrate.js\n```substrate.js\n```\n\nrender.js\n```render.js\n```\n\nrender.js\n```render.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\nvector-physics.js\n```vector-physics.js\n```\n\ntile-domain.js\n```tile-domain.js\n```\n\ntile-domain.js\n```tile-domain.js\n```\n\ntile-domain.js\n```tile-domain.js\n```\n\nindex.html\n```index.html\n```\n\nREADME.md\n```README.md\n```\n=======\n# MHCGH-S – Endogenous Topological Life/Sand on a Mandelbulb Boundary\nA deterministic cellular automaton / sandpile hybrid whose nodes live on a Chinese-Remainder–structured hypergrid and are anchored to a 3D Mandelbulb topological boundary. Rules are not global: each module's birth, survival, and toppling thresholds are generated endogenously from its phase coherence, its Mandelbulb boundary potential, and its gated phase covariance with CRT neighbors.\n\n## 1. Files\n- `substrate.js` – `MHCGHSubstrate`: the simulation model.\n- `render.js` – `MHCGHRenderer`: geometric hypergrid view colored by state, coherence, and boundary potential.\n- `vector-physics.js` – `VectorPhysicsRenderer`: phase-vector field, coupling gates, and covariance-modulated edges.\n- `tile-domain.js` – `TileDomainRenderer`: thermodynamic tile-domain board with CRT skip links and Hessian curvature.\n- `main.js` – boots the substrate, wires the three tabs, fetches and renders `README.md`.\n- `index.html` – the page shell.\n\n## 2. Core Concepts\n- Modules: Nodes in a sparse non-local graph. Each carries a multi-dimensional phase vector, a discrete state, a private rule bias, and a CRT residue triple.\n- CRT Topological Boundary: The step sizes `[5, 7, 11]` are pairwise coprime. Each module's residues `(i mod 5, i mod 7, i mod 11)` are mapped to a point on the unit sphere — a fixed topological boundary anchor.\n- Mandelbulb Potential: The boundary point is rotated by the module's phase vector, then iterated through a power-8 Mandelbulb polynomial. The result is an escape-time `bulbEscape` and a boundary-distance `bulbPotential`. Nodes near the fractal skin couple more strongly and topple sooner.\n- Phase Covariance: A gated average cosine similarity between a module's phase vector and those of its neighbors. It measures the statistical glue of the rule complex.\n- State: An integer `0 … maxState`. `0` is empty/dead; higher values are active cells or sand.\n- Phase Coherence (`pas_h`): Recomputed as a blend of self phase alignment and neighbor covariance, boosted by boundary proximity.\n- Entropy: Phase dispersion among CRT neighbors; high entropy desaturates color and can trigger decoherence.\n- Decoherence: A red halo appears when coherence drops below half the configured threshold.\n\n## 3. Topology: CRT Non-Local Hypergrid\nThe wiring is built from the coprime step sizes:\n\n```js\n```\n\nFor each node `i` and step `s`, neighbors are `(i ± s) mod N`; local `±1` links are also added. This produces short-range and long-range \"skip\" connections that the tabs render as straight or curved arcs.\n\n## 4. Endogenous Rules\nEach module evaluates:\n\n1. Mandelbulb boundary map: rotate `basePoint` by `phase`, run `power` iterations, read `bulbPotential`.\n2. Phase covariance: average gated cosine similarity with CRT neighbors.\n3. Rule generation: `toppleThreshold`, `birthMin/Max`, `surviveMin/Max` are functions of coherence, boundary potential, and `ruleBias`.\n4. Update:\n - Phases couple through gated sine differences, with coupling boosted near the boundary.\n - Sandpile toppling redistributes overflow to neighbors.\n - Game-of-Life layer applies birth/survival windows.\n - Coherence is refreshed from self-alignment, covariance, and boundary boost.\n\n## 5. Tabs\n- Simulation: Geometric embedding of the CRT hypergrid. Node color and size encode state, coherence, entropy, and Mandelbulb boundary proximity (violet shell).\n- Vector Physics: Phase-vector spokes per dimension, directed coupling arrows whose opacity reflects gate strength and phase covariance, and a violet shell showing boundary potential.\n- Tile Domain: A thermodynamic board where tile height combines state and boundary pressure, tile color maps solid/liquid/plasma phases, CRT skip links are curved arcs, and a small arc encodes local Hessian curvature.\n>>>>>>> REPLACE\n```\n\nSummary: I wired each CRT module to a real 3D Mandelbulb boundary map—its residues define a point on the unit sphere, its phase vector rotates that point, and a power-8 Mandelbulb iteration produces an escape-time/potential pair every frame. That boundary potential plus a new gated phase-covariance scalar now drive the endogenous rules, coupling strength, and all three visualizers, which now show violet boundary shells, covariance-tinted vectors, and boundary-pressured tiles.","page_description","cursor","eyJpZCI6IjRhZnFoMDMzcHc2aDFjZ3V5bmY4In0","token","included",[],"routes/browser/layout","actionData","errors"]

MHCGH-S – CRT Hypergrid Substrate(non-random probability substrate)