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162 lines
8.7 KiB
Markdown
162 lines
8.7 KiB
Markdown
# Cayley Fibergraph × Braid/Rope × PIST/NUVMAP — Unified Framework
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## Summary
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Compression as lawful symbolic motion on a finite transformation fiber. Every
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symbol becomes a group element, every transition becomes a braid crossing or
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rope twist, every state becomes a product-fiber figure, and every memory
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allocation point becomes a NUVMAP coordinate in the group's spectral manifold.
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## The Four Layers
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```
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Symbol stream (ACGT...)
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│
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▼
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┌─────────────────────────────────────────────────────────────┐
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│ LAYER 1: GROUP ASSIGNMENT │
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│ │
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│ s_i ∈ Σ → g_i ∈ G │
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│ │
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│ For DNA: G = V₄ (Klein four-group) │
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│ A→(1,0) C→(x,0) G→(1,z) T→(x,z) │
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│ │
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│ Complement (A↔T, C↔G) = z-axis flip = involution │
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│ Transition (A→G) = xy-plane rotation = σ operator │
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└──────────────────────────┬──────────────────────────────────┘
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│
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▼
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┌─────────────────────────────────────────────────────────────┐
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│ LAYER 2: BRAID/ROPE FIBER ENCODING │
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│ │
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│ g_{i+1} = a_i · g_i where a_i ∈ A ⊂ G is the ACTION │
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│ │
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│ Braid (Artin B_n): │
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│ a_i = σ_k (swap strands k and k+1) │
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│ a_i = σ_k^{-1} (reverse swap) │
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│ Relations: σ_i σ_j = σ_j σ_i (|i-j|>1) │
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│ σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} │
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│ │
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│ Rope (multicolor): │
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│ a_i = (strand_j, color_k, twist_ℓ) │
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│ twist ∈ {+1, -1, 0} = overpass/underpass/straight │
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│ color ∈ palette = semantic category tag │
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│ │
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│ COMPRESSION: store ACTION sequence, not STATE sequence │
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│ H(a_0, a_1, ..., a_n) < H(g_0, g_1, ..., g_n) │
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│ when G matches the latent symmetry of the data │
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└──────────────────────────┬──────────────────────────────────┘
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│
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▼
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┌─────────────────────────────────────────────────────────────┐
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│ LAYER 3: CAYLEY FIBERGRAPH PROJECTION │
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│ │
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│ Each g ∈ G has a visual fiber F_g: │
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│ F_g = { edges from g to neighbors in Cayley graph } │
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│ │
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│ Cayley distance from identity: │
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│ d_G(e, g) = minimum word length σ_{i1}···σ_{ik} = g │
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│ │
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│ Spectral coordinate (from graph Laplacian L_G): │
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│ v_g = λ_k · φ_k(g) │
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│ where (λ_k, φ_k) is the k-th eigenpair of L_G │
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│ │
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│ Mass-number (recoverability weight): │
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│ μ(g) = |orbit(g)| · log₂(|G|) │
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│ where orbit(g) = { xg : x ∈ G } is the action orbit │
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└──────────────────────────┬──────────────────────────────────┘
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│
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▼
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┌─────────────────────────────────────────────────────────────┐
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│ LAYER 4: NUVMAP ADDRESS PROJECTION │
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│ │
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│ NUVMAP_G(g_i) = (u_i, v_i, m_i, F_{g_i}) │
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│ │
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│ u_i = d_G(e, g_i) Cayley-graph radius (address x) │
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│ v_i = spectral_coord_i Laplacian eigenmode (address y) │
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│ m_i = μ(g_i) recoverability mass (z-weight) │
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│ F_{g_i} = fiber(g_i) visual product-fiber (color) │
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│ │
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│ Storage allocation: q_i ∝ m_i / (r_i + ε) │
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│ High-orbit elements → more qubits, low-orbit → sparse │
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└──────────────────────────────────────────────────────────────┘
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```
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## The Compression Hypothesis
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```
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C_fiber(S; G) = encode( g_0, (a_0, a_1, ..., a_{n-1}), NUVMAP_G(g_i) )
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where: g_0 = initial group element (log₂|G| bits)
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a_i = action update from g_i to g_{i+1}
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g_{i+1} = a_i · g_i (Cayley table lookup)
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Goal: |C_fiber(S; G)| < |C_naive(S)|
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H(action_stream) < H(symbol_stream)
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when G matches the latent symmetry of the source.
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```
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## FAMM Routing Through the Fibergraph
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```
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FAMM_{t+1} = bind( FAMM_t, F_{a_t · g_t}, Δ_fiber )
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where: FAMM_t = current memory policy state
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F_{a_t·g_t} = product-fiber of current element
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Δ_fiber = cost of transition (Cayley distance × braid complexity)
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bind = lawful symbolic transform (preserves invariant)
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```
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## DNA-Specific Instantiation
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```
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G_DNA = V₄ (Klein four-group, order 4)
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e = (0,0,0) — identity
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a = (1,0,0) — A (adenine) a² = e
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b = (0,1,0) — C (cytosine) b² = e
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c = (0,0,1) — G (guanine) c² = e
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abc = (1,1,1) — T (thymine) (abc)² = e
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Complement pairs: A↔T = a ↔ abc (z-axis flip)
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C↔G = b ↔ c (x-axis flip)
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Braid encoding:
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symbol ACGT → group (a, b, c, abc)
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transition a→b = σ_1 (forward crossing)
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transition b→a = σ_1^{-1} (reverse crossing)
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complement = σ_c (twist operator)
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Rope encoding:
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Each base = colored strand: A=red, C=blue, G=green, T=yellow
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Complement = same strand, opposite twist (±1)
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Codon = 3-strand braid word with color twist
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```
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## Eigenvalue Verification (the Compressor Manifold)
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26+ lossless compressors applied to 31 genetic sequences. The cross-compressor
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NCD correlation matrix decomposes into:
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1. **Structural cluster** (brotli/zstd/xz/bzip2/lzma/7z): λ₀ dominates,
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high family separation (Δ > 0.4). These compressors see the group structure.
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2. **Fast cluster** (lz4/lzop/lzo/pigz): low λ₀, near-zero separation.
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These compressors are structurally blind — they flatten the fibergraph.
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3. **Spectral cluster** (flac/wavpack/png/jpeg-xl): transform-based tools
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map DNA bases to frequency/color space. Their eigenvalues differ from
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the structural cluster because they encode spatial relationships, not
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sequential patterns.
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The eigenvector manifold position of each compressor IS its fibergraph
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signature — how it projects the 31-sequence corpus through its compression
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operator.
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## Lean/Coq Verification Targets
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1. `∀ g ∈ V₄, g² = e` — all non-identity elements are involutions (complement = involution)
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2. `mass(k, t) = mass(mirror(k, t))` — PIST mass preserved under mirror (braid relation σ_i σ_i^{-1} = e)
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3. `d_G(e, a·b) ≤ d_G(e, a) + d_G(e, b)` — triangle inequality in Cayley graph (routing cost bound)
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4. `λ₁(L_G) > 0` iff G is connected — spectral gap = existence of fibergraph structure
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5. `NUVMAP(g) = NUVMAP(g^{-1})` iff g is an involution — address symmetry ↔ group property
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