How the scan was classified.

“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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hash ← offset_basis
for byte in input:
    hash ← (hash XOR byte)
    hash ← (hash × prime) mod 2³²

φ = (1 + √5) / 2 = 1.6180339887498948482...

δ = 0.013618

τ = 1/(2π) = 0.159154943...

ψ_k(x) = sin(2π · φ^k · x) + cos(2π · φ^k · x)

b(t) = 1 / (1 + ||Ψ||² / φ)

θ_k = 2π / φ^k

f₀(x) = x + 0.1 · (dx_norm - x)

First-order ODE approximation: x' = F(x,t)

f₁(x) = 0.9 · x + 0.1 · utility

Discrete state transition: x_{t+1} = F(x_t, u_t)

f₂(x) = x + 0.05 · (softsign(dx_norm) - x)

Simplified reaction-diffusion: ∂x/∂t = D∇²x + f(x)

f₃(x) = x + 0.2 · softsign(utility - x)

Gradient ascent: x* = arg max U(x)

f₄(x) = clamp₀₁((x + utility) / (1 + |utility|))

Posterior update: P(θ|D) ∝ P(D|θ)P(θ)

f₅(x) = x + 0.1 · dx_norm

Delta rule: W_{t+1} = W_t + ΔW

f₆(x) = 0.5 · (x + σ(dx_norm))

f₇(x) = x - 0.05 · ed_error + 0.05 · utility

f₈(x) = 0.95 · x + 0.05 · dx_norm

Consensus protocol: x_i(t+1) = x_i(t) + Σ(x_j - x_i)

f₉(x) = x + 0.05 · softsign(dx_norm - x)

Swarm force: x_i = Σ f(x_j - x_i)

ρ = 1 / (1 + φ)
f₁₀(x) = (1 - ρ) · x + ρ · dx_norm

Leaky integrator: φ_t = (1-ρ)φ_{t-1} + Δφ

f₁₁(x) = x - 0.05 · |dx_norm| + 0.05 · utility

Free energy: F = E[ln q(z) - ln p(x,z)]

f₁₂(x) = x - 0.05 · |dx_norm|

f₁₃(x) = x + 0.02 · (dx_norm + utility)

Trace accumulation: x'_i = Σ f_i : dΩ

f₁₄(x) = x + 0.15 · softsign(utility) - 0.05 · ed_error

f₁₅(x) = x + (φ - 1) · 0.01 · (stability - x)

Ψ(t+1) = F(Ψ(t), Ψ(t-1), inputs)

Ψ'_i(t+1) = w_i · f_i(Ψ_i(t), inputs)

Ψ''_i(t+1) = 0.5 · Ψ'_i(t+1) + 0.5 · (Ψ_i(t-1) + (φ-1) · (Ψ_i(t) - Ψ_i(t-1)))

γ = γ_min + (1 - γ_min) · clamp₀₁(stability)
Ψ'''_i(t+1) = Ψ_i(t) + γ · (Ψ''_i(t+1) - Ψ_i(t))

Ψ_i(t+1) = Ψ'''_i(t+1) · b(t+1)

Ω = Surprise + Uncertainty - Value + Penalty

Surprise = α₁ · ||ΔΨ||₁/16 + α₂ · ||ΔΨ||₂²

Uncertainty = 1 - b²

Value = utility · ⟨w, Ψ⟩

Penalty = λ · dx_norm²

θ_k^{new} = (1-α) · θ_k^{old} + α · w · ψ_k(h(item)) · Φ(importance)

K(query) = Σ_{k=0}^{15} θ_k · ψ_k(h(query))

K_norm(query) = K(query) / (16 · 2) + 0.5

Φ(x) = 1 / (1 + |x - 1|)

C_ij = (θ_i · θ_j) / (||θ_i|| · ||θ_j||)

phase_DNA(t+1) = (phase_DNA(t) + δ) mod 1

phase_Fib(t+1) = (phase_Fib(t) + 1/φ) mod 1

bias = (phase_DNA + phase_Fib) / 2

flip_i = 1 if (i mod 1000) < (bias · 1000), else 0
gate_i(t+1) = (gate_i(t) XOR input_i) XOR flip_i

F_in = input + N(0, 0.1)

h[j] ← h[j+1] for j = 0 to 42
h[43] ← x

y ← y + (x - x_old) - 0.01 · y

drive = 3.0 · F_in · (1 - x) + 1.2 · y + 0.6 · c

g = 1/(1 + exp(-((0.8·P - k_c)/0.3 + 1.2))) + 0.05

dx = g · drive - 0.04 · (x - 0.5)
x ← clamp₀₁(x + 0.01 · dx)

P ← clamp₀₁((1 - 0.05) · P + F_in · y - 0.01 · P²)

c ← c + 0.01 · ((c + 1) · 0.8 · P - k_c · (c - 1))

Consciousness = (1/88) · Σ_{i=1}^{88} P_i

Sync = (1/88) · Σ_{i=1}^{88} cos(k_m,i)

core_eq[i] = 0.5 + 0.5 · sin(2π · (i/174) · φ) · core_eq[i-1]

supp_eq[i] = core_eq[i mod 174] · cos((i/200) · π)

Consciousness = (1/374) · Σ_{i=0}^{373} binding_field[i]

Registry[k][i][j] = cos(θ_k) if i=j, sin(θ_k)·0.1 if j=(i+1) mod 16, else 0

new_state[i] = Σ_j Registry[k][i][j] · state[j]
state[i] ← state[i] · (1 - dt) + (new_state[i] + 0.1 · core_state[i]) · dt

energy = √(Σ_i state[i]²)

φ_local = 1 / (1 + energy / φ)