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172 lines
8.2 KiB
Markdown
172 lines
8.2 KiB
Markdown
# Equation Forest Index v0.1 — Canonical Layer
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## 12 Foundation Kernels (Exact Solver Basis Vectors)
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| ID | Equation | Domain |
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|----|----------|--------|
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| F01 | Shannon_Entropy_Calculation | Entropy/Compression |
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| F02 | Information_Content_Measurement | Entropy/Compression |
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| F03 | Hierarchical_Entropy_Decomposition | Entropy/Compression |
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| F04 | Thermodynamic_Efficiency_Limit | Thermodynamic |
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| F05 | Computation_Energy_Bound | Thermodynamic |
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| F06 | Energy_Balance_Threshold | Thermodynamic |
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| F07 | Maxwell_Demon_Recovery | Thermodynamic |
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| F08 | Riemannian_Distance_Calculation | Geometry |
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| F09 | Geodesic_Connection_Coefficients | Geometry |
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| F10 | Single_Step_Geodesic_Integration | Geometry |
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| F11 | Aggregate_Load_Combination | Cognitive/Routing |
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| F12 | Intrinsic_to_Total_Ratio | Cognitive/Routing |
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## 5 Core Streets (Graph Collapse)
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1. **Entropy/Compression** (F01-F03) — Shannon entropy → hierarchical decomposition
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2. **Thermodynamic Admissibility** (F04-F07) — Carnot/Landauer → energy balance
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3. **Geometric Motion** (F08-F10) — Metric → connection → geodesics
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4. **Cognitive/Routing Load** (F11-F12) — Aggregate load → routing efficiency
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5. **DIAT/AVMR/S3C Bridge** — Shell → vector → witness → surface
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## 8 Bridge Nodes
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| Bridge | Connection |
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|--------|------------|
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| B1 | Entropy ↔ Load |
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| B2 | Entropy ↔ Landauer |
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| B3 | Energy ↔ Routing |
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| B4 | Geometry ↔ Routing |
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| B5 | DIAT ↔ Geometry |
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| B6 | AVMR ↔ Entropy |
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| B7 | S3C ↔ Codec |
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| B8 | PIST ↔ Surface |
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## 18-Bit Semantic Micro-ISA (Hardware Bridge)
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**Genome18 Structure:**
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- muBin: mutation/drift (routing load)
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- rhoBin: verification pressure (routing efficiency)
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- cBin: connectance (geometry/route neighborhood)
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- mBin: compression residue (entropy)
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- neBin: effective sample (entropy)
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- sigmaBin: fitness proxy (entropy)
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6 bins × 3 bits = 18 bits (262,144 states)
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**Address Calculation:**
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```
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addr = muBin * 32768 + rhoBin * 4096 + cBin * 512 + mBin * 64 + neBin * 8 + sigmaBin
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```
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**Kernel to Bin Mapping:**
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- F01-F03 → mBin, neBin, sigmaBin
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- F04-F07 → cost/failure mask
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- F08-F10 → cBin
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- F11-F12 → muBin, rhoBin
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- DIAT/AVMR/S3C/PIST → transition surface
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## Pipeline Architecture
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```
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raw equation
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→ F01-F12 kernel signature
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→ street / bridge assignment
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→ six 3-bit Genome18 bins
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→ 18-bit ISA/LUT address
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→ FPGA route expansion
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→ PIST/witness audit
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→ Lean/proof/executable check
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```
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## Best Street Through System
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Shannon entropy → hierarchical entropy decomposition → DIAT shell reduction → AVMR vector roll-up → S3C codec → cognitive load minimization → Riemannian/geodesic routing → Landauer/Carnot admissibility → PIST witness surface
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## Key Insight
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Shell structure = coordinate system (not predictor). 18-bit ISA = routing state class (not full math object). Value = structural organization for measuring constraint, compression, geometry, routing together (not magical prediction).
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## Graph Compression
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Raw: hundreds of equations
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→ After kernel signature: ~30-45 supernodes
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→ After Genome18 encoding: 262,144 LUT addresses
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→ Exact TSP becomes plausible
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## Files
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- `data/equations_forest.jsonl` — Full equation forest with signatures
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- `data/equations_forest_genome18.jsonl` — Genome18 encoded equations
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- `0-Core-Formalism/lean/Semantics/Semantics/Genome18.lean` — Lean formalization with theorems
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- `scripts/equation_forest_genome18_encoder.py` — Kernel to bin mapping
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- `MATH_MODEL_MAP.tsv` — Equation registry (source of truth, obeys this index)
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- `AGENTS.md` — Full specification (sections 9.1-9.12)
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## Minimap Visualization (Complex Roots Approach)
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**Inspiration:** Parametric complex roots visualization (@lbarqueira.bsky.social, inspired by @sconradi.bsky.social)
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**Concept:** Adapt complex roots visualization technique to create a 3D minimap of the Genome18 address space
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**Mapping:**
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- **Parameter domain:** Genome18 18-bit address space (262,144 states) instead of unit circle
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- **Roots being tracked:** Equation signatures as they traverse parameter space
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- **Color dimension:** Kernel signatures (F01-F12) instead of Im(t₂)
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- **Trajectory paths:** Street/bridge transitions through the space
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**Implementation approach:**
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- Use polynomial root finding to visualize how equation clusters shift as Genome18 bins vary
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- 6 bins × 3 bits = 18 parameters → traverse high-dimensional parameter space
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- Color by street assignment (Entropy/Compression, Thermodynamic, Geometric, Cognitive/Routing, DIAT/AVMR/S3C)
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- Show bridge transitions as trajectory lines between clusters
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**Benefits:**
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- Visual navigation system for 262,144-state Genome18 space
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- Identify equation family clustering patterns
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- Show parametric stability regions (bifurcation analysis)
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- Debug kernel-to-bin mapping by visualizing signature drift
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**Navigation/Positioning (Planet Beacon Concept):**
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- Dynamic positioning: show current Genome18 address as "you are here" beacon
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- Trajectory visualization: highlight paths to nearby states in the forest
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- Bridge transitions: animate movement across street/bridge connections
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- Relative positioning: understand your location within the full geometric space
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- Real-time feedback: see how kernel signature changes affect position
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- Exploration guidance: suggest optimal paths through equation space based on constraints
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**Visual Aesthetic (Liquid Metal Inspiration):**
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- Topographical feel: fluid, organic lines that swirl and cluster in dense areas
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- Depth/3D effect: some areas bulge forward (active states), others recede (inactive)
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- Warm metallic tones: champagne, rose gold, soft bronze mapping to street assignments
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- Dark shadows: coffee-colored shadows defining depth between equation families
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- Iridescent surface: high-sheen effect creating continuous motion despite static image
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- Pearlescent quality: natural seashell-like patterning for bridge transitions
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- Smooth, hypnotic flow: emphasizes continuous traversal through Genome18 space
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**Reference Implementation (Scale Space):**
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- Game: Scale Space by setz (itch.io, Steam coming)
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- Multi-scale navigation: quantum/microscopic/classical/cosmic scales
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- Physics controls: Equilibrium, Coherence, Viscosity, Mass for parameter space traversal
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- Mathematical shapes: vortices, Lissajous figures, knots
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- Emergent ecosystems: life competing for resources, dividing, playing, hunting
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- Zen-like experience: calming navigation through infinite parameter space
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- Tech stack: Unreal Engine Blueprints, WebGL, Three.js, Antigravity
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- Mapping to Equation Forest:
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- Physics controls → Kernel signature parameters (F01-F12)
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- Multi-scale → 5 street assignments (Entropy/Compression, Thermodynamic, Geometric, Cognitive/Routing, DIAT/AVMR/S3C)
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- Mathematical shapes → Equation signature clusters and bridge transitions
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- Emergent ecosystems → Equation families competing for representation
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- Parameter space traversal → Genome18 262,144-state navigation
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**Foundational Concepts (Scale Space Science):**
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- Scale-Space Theory: structures emerge at the right scale (applies to Genome18 address space)
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- Wavelet Transform: structure emerges from coherence and frequency (kernel signature coherence)
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- Renormalization Group: behaviors evolve consistently across scales (street assignment invariance)
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- Fractals & Scale-Invariance: recursive patterns across scales (equation family clustering)
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- Emergence & Complexity: complex structures from simple interactions (F01-F12 kernel interactions)
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- Cellular Automata: complexity from simple deterministic rules (Genome18 bin encoding)
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- Information & Entropy: measuring and guiding emergence (entropy/compression kernels F01-F03)
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- Thermodynamics: irreversible processes, phase transitions (thermodynamic kernels F04-F07)
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- Quantum theories: entanglement, Hilbert space (geometric kernels F08-F10)
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- Network Theory: graph topology, small-world networks (cognitive/routing kernels F11-F12)
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- Cymatics & Resonance: standing waves, Fourier transform (bridge transition resonance)
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- Swarm Intelligence: decentralized interactions producing global behavior (equation forest as CAS)
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- Twistor Theory: emergent dimensionality from relationships (Genome18 as emergent coordinate system)
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- Visualization: Unreal Engine Niagara particle systems (potential rendering backend)
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