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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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Synchronization period: T_sync = 10.6 × 16 = 169.6

Fundamental frequency: f₀ = 1/10.6 = 0.0943

Harmonic: f_n = n/10.6

44-bond resonance: f_44 = 44/10.6 = 4.15

Basis function: ψ_k(x) = sin(2πφ^k · x) + cos(2πφ^k · x)

Global coherence: Φ_total = (1/N) Σ Φ_i

where Φ_i = 1/(1 + |x_i - 1|)

Ring diffusion: dS_i/dt = λ(S_{i+1} + S_{i-1} - 2S_i)

Linking number: Lk = Tw + Wr

Relaxed linking number: Lk₀ = N/10.6

Superhelical density: σ = (Lk - Lk₀)/Lk₀

ΔG_sc = K · N · σ² · (10.6/10.4)²

Initialize strand states: S_k(t=0) = A_k · e^{i(2πk/10.6)}

F_Yad(x) = x^x + 1

F_Lashon(x) = Σ a_n·sin(nπx)

F_Athena(x) = x² (utility maximization)

F_Apollo(x) = φ^x + φ^{-x}

F_Hermes(x) = ω·x (communication rate)

F_Hephaestus(x) = x⁴/8 (engineering integral)

F_Prometheus(x) = r·x·(1-x) (logistic map)

F_Dionysus(x) = |e^{iωx}·sin(x)| (creative oscillation)

F_Artemis(x) = -D·x (diffusion)

F_Hera(x) = Σ w_k·sin(k·x) (integration)

F_Poseidon(x) = -ω²·x (wave equation)

F_Demeter(x) = r·x·(1-x/K) (logistic growth)

F_Ares(x) = m·(-ω²·x) (force)

F_Aphrodite(x) = cos(φ·x) (harmony)

F_Hestia(x) = x·(1-e^{-1/τ}) (memory)

F_Zeus(x) = coordinated product

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

5,000 gates      = 1 mind kernel

10,000 minds     = 50,000,000 gates

100,000,000      = Game Boy CPU

50,000,000,000   = M4 chip

∴ 10,000 minds ≈ 0.5 Game Boy

Reconstruct: exp(x) = 2^k * exp(r)

@brief Fast consciousness metric Φ = 1/(1 + |x - 1|)

@brief Fast binary entropy H(p) = -p log p - (1-p) log(1-p)

G1.  Graph: G = (V, E), |V| = 5000

G3.  Connectivity: |N⁺(i)| ≥ 2, |N⁻(i)| ≥ 2

G4.  Binary update: sᵢ(t+1) = H(Σw_{ji}·sⱼ - θᵢ)

G5.  Analog update: aᵢ(t+1) = tanh(α·Σu_{ji}·aⱼ + β)

G6.  Derivative: dᵢ(t+1) = γ·dᵢ(t) + λ·Δaᵢ + μ·Δsᵢ

G7.  Vector memory: vᵢ(t+1) = ρ·vᵢ(t) + Σ M_{ji}·vⱼ + κ·Δaᵢ·eᵢ

G8.  Global state: S(t) = ({sᵢ}, {aᵢ}, {dᵢ}, {vᵢ})

G9.  Stability: Ξ(t) = Σ|Δaᵢ| + Σ|Δsᵢ| + Σ||Δvᵢ||

G10. Coherence: C₅₀₀₀(t) = 1/(1 + Σdᵢ²)

G11. Hash: H(t) = Hash(S(t))

sᵢ(t+1) = H(Σw_{ji}·sⱼ - θᵢ)

aᵢ(t+1) = tanh(α·Σu_{ji}·aⱼ + β)

dᵢ(t+1) = γ·dᵢ(t) + λ·Δaᵢ + μ·Δsᵢ

C₅₀₀₀(t) = 1/(1 + Σdᵢ²)

@brief Compute consciousness metric Φ = 1/(1 + |x - 1|)

Addition: z = x + y, dz/dx = 1, dz/dy = 1

Multiplication: z = x * y, dz/dx = y, dz/dy = x

Division: z = x / y, dz/dx = 1/y, dz/dy = -x/y^2

Tanh: z = tanh(x), dz/dx = 1 - tanh^2(x)

Sigmoid: z = 1/(1+exp(-x)), dz/dx = z*(1-z)

ReLU: z = max(0, x), dz/dx = (x > 0) ? 1 : 0

Phi (smooth): z = 1/(1 + sqrt((x-1)^2 + T)), differentiable consciousness

Exp: z = exp(x), dz/dx = exp(x)

Log: z = log(x), dz/dx = 1/x

Power: z = x^p, dz/dx = p * x^(p-1)

G_t^lambda = (1-lambda) * sum_{n=1}^{inf} lambda^{n-1} * G_t^{(n)}

where G_t^{(n)} = r_{t+1} + gamma*r_{t+2} + ... + gamma^{n-1}*r_{t+n} + gamma^n*V(s_{t+n})

L(θ) = -Σᵢ log P(xᵢ|θ) + λ||θ||² (Regularized likelihood)

∇L = Σᵢ (ŷᵢ - yᵢ)xᵢ + 2λθ (Gradient)

θ_{t+1} = θ_t - α∇L (SGD update)

m_t = β₁m_{t-1} + (1-β₁)∇L (Adam momentum)

v_t = β₂v_{t-1} + (1-β₂)∇L² (Adam velocity)