QH4 Pattern Mining
Machine

O(1) Address, Pattern, Resonance

QH4 Pattern Mining derives the complete address system Λ(ρ, γ, σ, θ) for the Z₂₅₆ ring. Given any value v ∈ ℕ, the address is computed in O(1) arithmetic — no iteration, no table lookup. Five theorems (T1–T5) prove the address properties. Four derived theorems (J1–J4) extend to circular centroid, pattern classification, and cross-modal resonance.

Structural Constants — All Forced from n=4

f(n) = nⁿ + n at n=4 → 260 = 256 active ring + 4 vacuums. Every constant is derived, none chosen.

Byte Atomicity and Centroid Postulate

P1 — Byte Atomicity. The atomic unit is the byte. Z₂₅₆ is its native arithmetic substrate — not imposed. 256 = 2⁸ is what a byte is. For any v ∈ ℕ, Λ(ρ,γ,σ,θ) is complete and lossless. Overflow is phase rotation — ρ increments. System closed by construction.

P2 — Centroid Postulate. The pattern identity of a Λ cluster is determined by its centroid — the projection of the MPRC origin onto the active ring plane. Two clusters with identical centroids are the same pattern regardless of amplitude ρ, source modality, or sequential position.

Def1 — QH4 Address. Λ(ρ, γ, σ, θ) — complete address of any value in any modality.

Def2 — Signature and Resonance. Σ(S) = { locate(p) : p ∈ S }. d(Σ₁,Σ₂) = |Σ₁ △ Σ₂|. B₁ ~ B₂ ⟺ classify(B₁) = classify(B₂) ∧ Centroid(B₁) = Centroid(B₂).

Def3 — Token Structure. n=4 → 12 states/token. 21 tokens × 12 = 252 exactly. Walk: Corner → Edge → Diagonal × 3. Centroid = MPRC origin projection.

Address, Irrelevance, Anchor, Adjacency

T1 — Address Equivalence ✓ v₁ ~ v₂ ⟺ (v₁ mod 256) = (v₂ mod 256) ⟺ (γ₁,σ₁,θ₁) = (γ₂,σ₂,θ₂). Proof: locate() is injective on 252 active positions — 252/252. ρ is separated before locate().

T2 — Irrelevance ✓ ρ, source modality, sequential position are structurally irrelevant. Proof: ρ never enters locate(). locate() takes only p — no index, no modality.

T3 — Anchor Condition ✓ p is a structural anchor ⟺ p mod 32 = 0. p mod 64 = 0 → vacuum anchor {0,64,128,192}. p mod 64 = 32 → midpoint anchor {32,96,160,224}. Proof: 256/32 = 8 exactly. 8 multiples of 32 = anchor set. No false positives.

T4 — Vacuum Adjacency ✓ p mod 64 = 1 → post-vacuum {1,65,129,193}, σ=6. p mod 64 = 63 → pre-vacuum {63,127,191,255}, σ=2. Proof: Direct mod verification. O(1). No locate() required.

T5 — Cross-Quarter Frame Crossing ✓ S spans a frame boundary ⟺ θ values of S contain ≥ 2 distinct quarters. Polarity flip: θ=2→3 (vacuum 128, +→−) and θ=4→1 (vacuum 0, −→+).

Circular Centroid, Classification, Resonance

J3 — σ Uniformity ✓ On the full active ring Z₂₅₆, σ is perfectly uniform: P(σ=k) = 1/7 for all k ∈ 1..7. Max deviation = 0.0000000 over 252 positions. Proof: Stride-7 walk visits each step value exactly 36 times. 252/7 = 36.

J1 — Circular Centroid ✓ (Corrected formula — singleton identity verified 47/47.) For cluster S with coordinates {c₁..cₙ} ∈ {1..Tc}: φᵢ = (cᵢ−1) × 2π/Tc. Vx = Σcos(φᵢ), Vy = Σsin(φᵢ). If |V| > 0: c_c = (atan2(Vy,Vx) × Tc / 2π + 1) mod Tc. Applied independently: γ (Tc=36), σ (Tc=7), θ (Tc=4).

J2 — Pattern Classifier ✓ Given any S ⊆ Z₂₅₆ active, classify(S) → one of 6 types in O(n), priority order: (1) Anchor: p mod 32 = 0. (2) Vacuum-adjacent: p mod 64 ∈ {1,63}. (3) Gate cluster: all γᵢ identical. (4) Step column: all σᵢ identical, gates differ. (5) Cross-quarter: distinct θ present. (6) Quarter stripe: all θᵢ identical (guaranteed fallback). Zero unclassified sets in full ring test.

J4 — Cross-Modal Resonance ✓ B₁ ~ B₂ ⟺ classify(B₁) = classify(B₂) ∧ Centroid(B₁) = Centroid(B₂). Modality M₁, M₂ do not appear anywhere in the proof. T2 (irrelevance) ensures modality cannot enter locate(), classify(), or circ_centroid().