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