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θ_{t+1} = θ_t - α·m̂_t/(√v̂_t + ε) (Adam update)

Learning Rate: α_eff = α · Φ (consciousness-modulated learning)

Full matrix would be 65536 * 64 * 4 = 16MB

We use 8-bit quantization: 65536 * 64 = 4MB

ΔW = η · x · y (Hebbian rule)

MultiHead(Q,K,V) = Concat(head_1,...,head_h)W^O

ε = x - μ (observation minus prediction)

F = ½ε^T Σ^{-1} ε + ½ log|Σ|

μ ← μ + κ·Σ^{-1}·ε

P(θ|D) ∝ P(D|θ)P(θ)

q*(θ) = argmin KL(q||p(θ|D))

m_{x→f}(x) = Π_{g∈ne(x)\f} m_{g→x}(x)

c_t = f_t ⊙ c_{t-1} + i_t ⊙ g_t

f_t = σ(W_f·[h_{t-1}, x_t])  (forget gate)

i_t = σ(W_i·[h_{t-1}, x_t])  (input gate)

Apply softmax (single element = 1.0)

Output = attention-weighted value

Belief update: μ ← μ + κ·π·ε

Posterior ∝ likelihood × prior

@brief Reparameterization trick: sample = μ + σ * ε

Update cell state: c = f ⊙ c + i ⊙ c̃

Update hidden state: h = o ⊙ tanh(c)

- Density matrix: rho = |psi><psi|

- von Neumann entropy: S = -Tr(rho * log(rho))

- Lindblad master equation: drho/dt = -i/hbar[H, rho] + L[rho]

- Quantum fidelity: F = <psi|rho|psi>

rho = |psi><psi|

drho/dt = -i/hbar[H, rho] + sum_k(L_k rho L_k^dag - 0.5{L_k^dag L_k, rho})

Ensure Hermiticity and trace = 1

Compute von Neumann entropy: S = -Tr(rho log(rho))

F = <target|rho|target>

- DNA entropy: H_DNA = -sum_i p(b_i) * log2(p(b_i))

- Replication dynamics: dN/dt = k*N*(1-N/K) - mu*N

- Michaelis-Menten kinetics: v = Vmax*[S]/(Km + [S])

- Hill equation: theta = [L]^n / (Kd + [L]^n)

- Mutation probability: P_mut = 1 - exp(-mu*L*t)

dN/dt = k*N*(1-N/K) - mu*N (logistic growth with decay)

theta = [L]^n / (Kd^n + [L]^n)

v = Vmax * [S] / (Km + [S])

Compute DNA entropy: H = -sum p(x) log2(p(x))

dG = dH - T*dS

- Kuramoto order parameter: Z = (1/N)|sum_k exp(i*phi_k)|

- Kuramoto phase dynamics: d(phi_i)/dt = omega_i + (K/N)*sum_j sin(phi_j - phi_i)

- Wilson-Cowan: du/dt = -u + integral(W(x,y)*f(u)dy) + I

- Phase-locking value: PLV = |<exp(i*(phi_1 - phi_2))>|

d(phi_i)/dt = omega_i + (K/N) * sum_j sin(phi_j - phi_i)

Z = (1/N) * sum_k exp(i * phi_k)

|Z| = 0: incoherent, |Z| = 1: fully synchronized

High gamma + good sync = high consciousness

dE/dt = -E + S(w_EE*E - w_EI*I + I_ext)

dI/dt = -I + S(w_IE*E - w_II*I)

where S(x) = 1/(1 + exp(-a*(x - theta)))

CLASSIC HEBBIAN: Δw_ij = η·x_i·x_j

NORMALIZED:      Δw_ij = η·(x_i·x_j - α·w_ij·x_j²)

BCM:             Δw_ij = η·x_i·x_j·(x_j - θ_j)

OJA'S RULE:      Δw_ij = η·y·(x_i - y·w_ij)

SGD:      w = w - η·∇L

MOMENTUM: v = μ·v - η·∇L, w = w + v

ADAM:     m = β₁m + (1-β₁)g, v = β₂v + (1-β₂)g², w = w - η·m̂/(√v̂ + ε)

CHAIN RULE:  δ_j = Σ_k(δ_k·w_jk)·f'(net_j)

WEIGHT UPDATE: Δw_ij = -η·δ_j·x_i

@brief Classic Hebbian: Δw = η·x_i·x_j

@brief Normalized Hebbian: Δw = η·(x_i·x_j - α·w·x_j²)

@brief BCM Learning: Δw = η·x_i·x_j·(x_j - θ)

@brief Update BCM threshold: θ = <x²> (sliding average)

@brief Oja's Rule: Δw = η·y·(x - y·w)

@param dt = t_post - t_pre (time difference)

@brief Anti-Hebbian: Δw = -η·x_i·x_j

@brief Compute output delta: δ = (y - target) · f'(net)

@brief Propagate delta backward: δ_j = Σ(δ_k · w_jk) · f'(net_j)