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“Accepted” entries belong to the mathematical catalog. “Rejected” means the extracted span is not a complete mathematical record; it is still preserved here and in its full source. Where an adjacent source passage supplies the complete expression, the recovered replacement is linked.

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

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

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

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

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

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

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

N_intra(k) = {k-1, k+1} mod 16

N_inter(k) = {k, k+16}

N_cross(m) = {m-1, m+1, m+4}

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

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

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

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

FastRelaxationTime: τ_fast = 1/(k_f^max + k_b^max)

SingleBondDerivative: db_n/dt = k_f·(1-b_n) - k_b·b_n

Intra-strand (n = 1..16)

Inter-strand (n = 17..32)

Cross-helix (n = 33..44)

SteadyStateOccupation returns θ* = B*/44

FYad: F_Yad(x) = x^x + 1 (Foundation)

FLashon: F_Lashon(x) = Σ_n a_n·sin(nπx) (Language - Fourier series)

FAthena: F_Athena(x) = argmax_y U(x,y) (Strategy - utility maximization)

Simplified: find max of utility function U(x,y) = -y² + 2xy

argmax_y: dU/dy = -2y + 2x = 0 => y* = x

U(x, x) = -x² + 2x² = x²

FApollo: F_Apollo(x) = φ^x + φ^{-x} (Mathematics - golden ratio)

FHermes: F_Hermes(x) = dx/dt approximation (Communication - rate of change)

FHephaestus: F_Hephaestus(x) = ∫∫ K(x,y) dy dx (Engineering - double integral)

Simplified: K(x,y) = xy, ∫∫ xy dy dx from 0 to x

= ∫ x·(x²/2) dx = x⁴/8

FPrometheus: F_Prometheus(x) = x·(1-x)·r (Innovation - logistic map)

FDionysus: F_Dionysus(x) = e^{iωt}·A(x) (Creativity - complex exponential)

FArtemis: F_Artemis(x) = ∇·(D∇x) (Perception - diffusion operator)

FHera: F_Hera(x) = Σ_k w_k·x_k (Integration - weighted sum)

FPoseidon: F_Poseidon(x) = ∂²x/∂t² + ω²x (Depth/Emotion - wave equation)

FDemeter: F_Demeter(x) = x·(1-x/K) (Growth - logistic growth)

FAres: F_Ares(x) = m·d²x/dt² (Action - force/acceleration)

F = ma, assume simple harmonic: a = -ω²x

FAphrodite: F_Aphrodite(x) = cos(φ·x) (Harmony - golden cosine)

FHestia: F_Hestia(x) = ∫_{-∞}^{t} x(τ)·e^{-(t-τ)/τ_m} dτ (Memory - exponential decay)

FZeus: F_Zeus(x) = Π_k F_k(x_k) (Coordination - product of all)

ComplementaryCoupling calculates J_complement(k, k') = J_0 × [1 + ρ·δ(k, k'_complement)]

T = Φ · ∫[M(t')·exp((QK^T/√d_k)V)] dt'

Integrate computes T = Φ · ∫[M(t')·exp((QK^T/√d_k)V)] dt'

VerticalOffset returns δ_k = k·h/16

GoldenPhaseOffset returns θ_k = 2πk·φ⁻¹ mod 2π

x_k(θ) = r·cos(θ + 2πk/16)

y_k(θ) = r·sin(θ + 2πk/16)

z_k(θ) = (p/2π)·θ + δ_k

Curvature returns κ = r/(r² + (p·h/2π)²)

Torsion returns τ = (p·h/2π)/(r² + (p·h/2π)²)

TotalRotation returns θ_total = 16 × (2π/10.6) = 9.49 rad

VerifyBondCount verifies Σ A_{ij}/2 = 44

BondArcLength returns s_bond = 2πr/10.6 = 0.593 nm per bp

VerticalRisePerBond returns Δz_bond = P/44 = 0.0819 nm

AdjacentBondLength returns L_adj = √[2r²(1 - cos(2π/10.6)) + h²] = 0.54 nm

FibonacciScaledStrand returns D_k = φ^k · F_k

HelixSum16 returns Helix₁₆ = Σ D_k·sin(kφ)

BondFormationG16 returns Bond_G16 = Π(1 + D_k/Helix₁₆)

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

Transform: T_k = S_k · (cos(k) + i·sin(k))

ModularExtraction returns X_k = |T_k| mod φ^k

FusionIntegral returns Fusion = Σ(|T_k| · φ⁻ᵏ)

FullG16Transform returns G¹⁶ = Π R_k

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

OffsetDistribution returns offset_i = i · φ⁻¹ · 10.6

Harmonic returns f_n = n/10.6

Psi returns ψ_k(x) = sin(2πφ^k · x) + cos(2πφ^k · x)

GlobalCoherence returns Φ_total = (1/N) Σ Φ_i where Φ_i = 1/(1 + |x_i - 1|)