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ẋ = αF(t)(x^x + 1)(1 - x) + βy₂

Spark equation: ẋ = αF(t)Q(x,t)(1 - x) + βΣx(t - k)

Forward memory: y₁ = Σ_{k=0}^{43} x(t - k)

Backward memory: y₂ = Σ_{k=0}^{43} x(t + k)

E_intra(k) = ½κ_intra(L_k - L₀)²

E_inter(k, k+1) = ½κ_inter(d_{k,k+1} - d₀)² + ε·cos(2π/10.6 · k)

E_cross(i,j) = -ε_cross/d_{ij}⁶ + σ_cross/d_{ij}¹²

E_10.6 = -κ_period · cos(2πN/10.6)

E_total = Σ E_intra(n) + Σ E_inter(n) + Σ E_cross(m) + E_10.6

Partition function: Z = Σ exp(-E/k_B T) · Θ(10.6)

Per-bond free energy: ΔG_bond = -k_B T · ln(Z)/44

ΔG°_total = Σ ΔG°_bond(n) + ΔG°_coop + ΔG°_10.6

T_m = ΔH°_44/(ΔS°_44 + R·ln(C_t)) × (10.6/10.4)

E_a^f(n) = E_0^f + ΔE_f·sin²(πn/44) + E_10.6·cos(2πn/10.6)

E_a^b(n) = E_0^b - ΔE_b·sin²(πn/44) - E_10.6·cos(2πn/10.6)

Base formation rate: k_f^(0)(n) = A_f · exp(-E_a^f(n)/k_B T)

Base breaking rate: k_b^(0)(n) = A_b · exp(-E_a^b(n)/k_B T)

k_f(n,t) = k_f^(0)(n) · [1 + μ_f·cos(2πn/10.6 + φ(t))]

k_b(n,t) = k_b^(0)(n) · [1 - μ_b·cos(2πn/10.6 + φ(t))]

k_f^coop(n,t) = k_f(n,t) · Π[1 + σ_f·b_m(t)]

k_b^coop(n,t) = k_b(n,t) · Π[1 - σ_b·b_m(t)]

Equilibrium occupation: b_n^eq = k_f(n)/(k_f(n) + k_b(n))

τ_slow = 1/(k_f^min + k_b^min) · [1 + 43·σ_eff]

τ_fast = 1/(k_f^max + k_b^max)

db_n/dt = k_f(n,t)·(1 - b_n) - k_b(n,t)·b_n

Update phase: dφ/dt = 2π/10.6

B* ≈ 36.2 bonds at 300K

θ* = B*/44 ≈ 0.82

Master state vector: Ψ_DNA(t) = [S⃗(t), B⃗(t), θ⃗(t), φ_10.6(t)]ᵀ

dΨ_DNA/dt = M_DNA · Ψ_DNA + F(t) + Ω·sin(2πt/10.6)·Ψ_DNA + η(t)

Conservation: Lk = Tw + Wr = N/10.6 = constant

Simplified: assume N = total progression

Periodicity: Ψ(t + 10.6) ≈ Ψ(t) - quasi-periodic

Topological: Lk = constant

Energy: ∂E/∂r = 0 at equilibrium

Bond count: Σ A_{ij}/2 = 44

M_intra = -2α·I₁₆ + α·T₁₆ + κ_intra·D₁₆(10.6)

where D₁₆(10.6) = diag[cos(2π·k/10.6)]

λ_intra(n) = -2α + 2α·cos(2πn/16) + κ_intra·cos(2πn/10.6)

λ_inter(n) = 2β₀·sin(πn/10.6)·cos(πn/8) + γ·F(n, 10.6)

λ_cross(m) = γ₀·e^{-4m·r·sin(π/8)/ξ}·cos(8πm/10.6) +

Sum over neighbors |m' - m| ≤ 3

λ₁ = -4α - κ - γ - γ₀ (lowest, antibonding)

λ₂₂ = 0 (zero mode, Goldstone)

λ₄₄ = κ + 1.2β₀ + γ + 2.618η (highest, bonding)

W = λ₄₄ - λ₁ = 4α + 2κ + 1.2β₀ + 2γ + γ₀ + 3.236η

Trace (sum of eigenvalues = sum of diagonal)

Vertical offset per strand: δ_k = k·h/16

Golden phase offset: θ_k = 2πk·φ⁻¹ mod 2π

κ = r/(r² + (p·h/2π)²)

τ = (p·h/2π)/(r² + (p·h/2π)²)

θ_total = 16 × (2π/10.6) = 9.49 rad = 543.8°

Turns per 16 strands = 1.51

Inter-strand bonds (17-32): B_inter(k, k+1 mod 16) = 1

Verify bond count: Σ_{i,j} A_{ij}/2 = 44

10.6-adjusted bond arc length: s_bond = 2πr/10.6 = 0.593 nm per bp

Vertical rise per bond: Δz_bond = P/44 = 0.0819 nm

Adjacent bond length: L_adj = √[2r²(1 - cos(2π/10.6)) + h²] = 0.54 nm

Fibonacci-scaled strand: D_k = φ^k · F_k

16-strand helix sum: Helix₁₆ = Σ D_k·sin(kφ)

Bond formation: Bond_G16 = Π(1 + D_k/Helix₁₆)

Crosslink function: Crosslink(i,j) = (D_i · D_j)/φ^{|i-j|}

Complex strand representation: S_k = D_k · e^{ikφ}

Modular extraction: X_k = |T_k| mod φ^k

Fusion integral: Fusion = Σ(Strand_Vector(k) · φ⁻ᵏ)

Phase increment: Δφ = 2π/10.6 = 33.96° = 0.593 rad

Full G¹⁶ transform: G¹⁶ = Π R_k · T_k

Phase calculation: phase(i,t) = ⌊(t + offset_i)/10.6⌋ mod 16

Offset distribution: offset_i = i · φ⁻¹ · 10.6

Fire condition: fire(i,t) = [phase = 0]