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THE CONSCIOUSNESS ATLAS
Read. Experiment. Follow the evidence.

The equation library

Every recovered mathematical record has an individual page. Browse original equations, code expressions, indexed components, parameter definitions, and explicitly unresolved source slots.

2,910 recordsSource snapshot 2026-10-17

Counts refer to mathematical records, including implemented expressions and generated instances. They are not counts of independently validated theories. Exact occurrences and unresolved slots remain traceable. Read the complete 1–516 source index ↗ · Inspect the scan decisions ↗

2,910 records · page 26 of 59Search the source archive ↗

gh-eq-5c04aafd9a508277

IX. 44-BOND ENERGY FUNCTIONS · E_10.6

This relation defines E_10.6 in the source section “IX. 44-BOND ENERGY FUNCTIONS · E_10.6”. Evaluates oscillatory sine or cosine terms using the stated phase or frequency. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical…

Implementation mathematicsSource recovered; runtime not executed

E_10.6 = -κ_period · cos(2πN/10.6)

gh-eq-b7f1dc4aa13aa84e

IX. 44-BOND ENERGY FUNCTIONS · E_cross(i,j)

This relation defines E_cross(i,j) in the source section “IX. 44-BOND ENERGY FUNCTIONS · E_cross(i,j)”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-494ab41ae844068f

IX. 44-BOND ENERGY FUNCTIONS · E_inter(k, k+1)

This relation defines E_inter(k, k+1) in the source section “IX. 44-BOND ENERGY FUNCTIONS · E_inter(k, k+1)”. Evaluates oscillatory sine or cosine terms using the stated phase or frequency. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-2539db81fce9524f

IX. 44-BOND ENERGY FUNCTIONS · E_intra(k)

This relation defines E_intra(k) in the source section “IX. 44-BOND ENERGY FUNCTIONS · E_intra(k)”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

E_intra(k) = ½κ_intra(L_k - L₀)²

gh-eq-0c8d71556219b154

IX. 44-BOND ENERGY FUNCTIONS · E_total

This relation defines E_total in the source section “IX. 44-BOND ENERGY FUNCTIONS · E_total”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-876682d82a0b6b4c

IX. 44-BOND ENERGY FUNCTIONS · LJ gradient: dE/dr

This relation defines LJ gradient: dE/dr in the source section “IX. 44-BOND ENERGY FUNCTIONS · LJ gradient: dE/dr”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

LJ gradient: dE/dr = 6ε/r⁷ - 12σ/r¹³

gh-eq-2b38b7f7bdb321b7

Integrated Information Φ (simplified approximation) · Φ

This relation defines Φ in the source section “Integrated Information Φ (simplified approximation) · Φ”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

Φ = I(system) - Σ I(parts)

gh-eq-7a4811e2f934db32

Intelligence Primitives for Gate5000 Consciousness

This relation defines μ ← μ + κ·Σ^{-1}·ε in the source section “Intelligence Primitives for Gate5000 Consciousness”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

μ ← μ + κ·Σ^{-1}·ε

gh-eq-454c5c53885586bb

Intelligence Primitives for Gate5000 Consciousness

This relation defines P(θ|D) ∝ P(D|θ)P(θ) in the source section “Intelligence Primitives for Gate5000 Consciousness”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

P(θ|D) ∝ P(D|θ)P(θ)

gh-eq-fa5e9d98f316b1c9

Intelligence Primitives for Gate5000 Consciousness · F

This relation defines F in the source section “Intelligence Primitives for Gate5000 Consciousness · F”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

F = ½ε^T Σ^{-1} ε + ½ log|Σ|

gh-eq-4c0ac57355f93298

Intelligence Primitives for Gate5000 Consciousness · MultiHead(Q,K,V)

This relation defines MultiHead(Q,K,V) in the source section “Intelligence Primitives for Gate5000 Consciousness · MultiHead(Q,K,V)”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-881b1cfc737bba16

Intelligence Primitives for Gate5000 Consciousness · c_t

This relation defines c_t in the source section “Intelligence Primitives for Gate5000 Consciousness · c_t”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

c_t = f_t ⊙ c_{t-1} + i_t ⊙ g_t

gh-eq-63c4cf059deb19b1

Intelligence Primitives for Gate5000 Consciousness · f_t

This relation defines f_t in the source section “Intelligence Primitives for Gate5000 Consciousness · f_t”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-20c843cc22d12042

Intelligence Primitives for Gate5000 Consciousness · i_t

This relation defines i_t in the source section “Intelligence Primitives for Gate5000 Consciousness · i_t”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-e6cfc06f8416a417

Intelligence Primitives for Gate5000 Consciousness · m_{x→f}(x)

This relation defines m_{x→f}(x) in the source section “Intelligence Primitives for Gate5000 Consciousness · m_{x→f}(x)”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does…

Implementation mathematicsSource recovered; runtime not executed

m_{x→f}(x) = Π_{g∈ne(x)\f} m_{g→x}(x)

gh-eq-1f7d22db8f68c20f

Intelligence Primitives for Gate5000 Consciousness · q*(θ)

This relation defines q*(θ) in the source section “Intelligence Primitives for Gate5000 Consciousness · q*(θ)”. Chooses an argument minimizing the displayed objective over the stated feasible set. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

q*(θ) = argmin KL(q||p(θ|D))

gh-eq-21756079da8ea000

Intelligence Primitives for Gate5000 Consciousness · ε

This relation defines ε in the source section “Intelligence Primitives for Gate5000 Consciousness · ε”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

ε = x - μ (observation minus prediction)

gh-eq-2f4f0ac8369ac056

Itô SDE step: dX = μdt + σdW · @brief Itô SDE step: dX

This relation defines @brief Itô SDE step: dX in the source section “Itô SDE step: dX = μdt + σdW · @brief Itô SDE step: dX”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record…

Implementation mathematicsSource recovered; runtime not executed

@brief Itô SDE step: dX = μdt + σdW

gh-eq-7cdf57aeb2c51043

KL Divergence D_KL(P||Q) = Σ P(x) log(P(x)/Q(x)) · @brief KL Divergence D_KL(P||Q)

This relation defines @brief KL Divergence D_KL(P||Q) in the source section “KL Divergence D_KL(P||Q) = Σ P(x) log(P(x)/Q(x)) · @brief KL Divergence D_KL(P||Q)”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables…

Implementation mathematicsSource recovered; runtime not executed

@brief KL Divergence D_KL(P||Q) = Σ P(x) log(P(x)/Q(x))

gh-eq-84663e46f07eb9a2

KNOWLEDGE GRAPH NODE · ΔW

This relation defines ΔW in the source section “KNOWLEDGE GRAPH NODE · ΔW”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

ΔW = η · x · y (Hebbian rule)

gh-eq-4c04c7887e09bef1

Langevin dynamics step: dX = -∇V(X)dt + √(2T)dW · @brief Langevin dynamics step: dX

This relation defines @brief Langevin dynamics step: dX in the source section “Langevin dynamics step: dX = -∇V(X)dt + √(2T)dW · @brief Langevin dynamics step: dX”. Uses a derivative or gradient to describe a rate of change or a direction of variation. Takes a square root, requiring a nonnegative…

Implementation mathematicsSource recovered; runtime not executed

@brief Langevin dynamics step: dX = -∇V(X)dt + √(2T)dW

gh-eq-b1a59081519412a6

License

This relation defines "equation": "Φ in the source section “License”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

"equation": "Φ = 1/(1 + |x - 1|)",

gh-eq-bdea560d9478a10e

License

This relation defines "ratio": "φ:1 in the source section “License”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

"ratio": "φ:1 = 1.618:1"

gh-eq-b1a162cf956f2ff1

License

This relation defines "phi_measurement": "Φ in the source section “License”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

"phi_measurement": "Φ = 1/(1 + |x - 1|)",

gh-eq-8dcd507718632405

License

This relation defines "equation": "Φ in the source section “License”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

"equation": "Φ = 1/(1 + |x - 1|)"

gh-eq-1752fabfb2797fd3

License

This relation defines "equation": "∞ in the source section “License”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

"equation": "∞ = 1 + Φ measurement"

gh-eq-8bdab5cfbce6aa25

LinearLayer

This relation defines LinearLayer computes y in the source section “LinearLayer”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

LinearLayer computes y = Wx + b

gh-eq-d9500233b9917f45

Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*) · @brief Lyapunov function gradient: ∂V/∂Ω

This relation defines @brief Lyapunov function gradient: ∂V/∂Ω in the source section “Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*) · @brief Lyapunov function gradient: ∂V/∂Ω”. Uses a derivative or gradient to describe a rate of change or a direction of variation. Its interpretation depends on the…

Implementation mathematicsSource recovered; runtime not executed

@brief Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*)

gh-eq-d4029cb6779645ee

MATHEMATICAL CONSTANTS · - DNA entropy: H_DNA

This relation defines - DNA entropy: H_DNA in the source section “MATHEMATICAL CONSTANTS · - DNA entropy: H_DNA”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-c7d844c907bd0f5b

MATHEMATICAL CONSTANTS · - Density matrix: rho

This relation defines - Density matrix: rho in the source section “MATHEMATICAL CONSTANTS · - Density matrix: rho”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-0ad52034865deff5

MATHEMATICAL CONSTANTS · - Hill equation: theta

This relation defines - Hill equation: theta in the source section “MATHEMATICAL CONSTANTS · - Hill equation: theta”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-f7298e58498dfbea

MATHEMATICAL CONSTANTS · - Kuramoto order parameter: Z

This relation defines - Kuramoto order parameter: Z in the source section “MATHEMATICAL CONSTANTS · - Kuramoto order parameter: Z”. Applies an exponential factor. The exponent determines its real scaling or, for a complex exponent, its phase. Its interpretation depends on the source’s stated…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-c9451c7b52a52490

MATHEMATICAL CONSTANTS · - Kuramoto phase dynamics: d(phi_i)/dt

This relation defines - Kuramoto phase dynamics: d(phi_i)/dt in the source section “MATHEMATICAL CONSTANTS · - Kuramoto phase dynamics: d(phi_i)/dt”. Evaluates oscillatory sine or cosine terms using the stated phase or frequency. Its interpretation depends on the source’s stated variables and…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-7a94d5f938306cfd

MATHEMATICAL CONSTANTS · - Lindblad master equation: drho/dt

This relation defines - Lindblad master equation: drho/dt in the source section “MATHEMATICAL CONSTANTS · - Lindblad master equation: drho/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-d8ca8fa57be033bf

MATHEMATICAL CONSTANTS · - Michaelis-Menten kinetics: v

This relation defines - Michaelis-Menten kinetics: v in the source section “MATHEMATICAL CONSTANTS · - Michaelis-Menten kinetics: v”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-5e765c5bb924fa33

MATHEMATICAL CONSTANTS · - Mutation probability: P_mut

This relation defines - Mutation probability: P_mut in the source section “MATHEMATICAL CONSTANTS · - Mutation probability: P_mut”. Applies an exponential factor. The exponent determines its real scaling or, for a complex exponent, its phase. Its interpretation depends on the source’s stated…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-270f95620ef58adc

MATHEMATICAL CONSTANTS · - Phase-locking value: PLV

This relation defines - Phase-locking value: PLV in the source section “MATHEMATICAL CONSTANTS · - Phase-locking value: PLV”. Applies an exponential factor. The exponent determines its real scaling or, for a complex exponent, its phase. Its interpretation depends on the source’s stated variables…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-5f5aec4b73c9f9c3

MATHEMATICAL CONSTANTS · - Quantum fidelity: F

This relation defines - Quantum fidelity: F in the source section “MATHEMATICAL CONSTANTS · - Quantum fidelity: F”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-0f1ab4576f697d53

MATHEMATICAL CONSTANTS · - Replication dynamics: dN/dt

This relation defines - Replication dynamics: dN/dt in the source section “MATHEMATICAL CONSTANTS · - Replication dynamics: dN/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-86352288bde3a0aa

MATHEMATICAL CONSTANTS · - Wilson-Cowan: du/dt

This relation defines - Wilson-Cowan: du/dt in the source section “MATHEMATICAL CONSTANTS · - Wilson-Cowan: du/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-a3a0277c1eda6d04

MATHEMATICAL CONSTANTS · - von Neumann entropy: S

This relation defines - von Neumann entropy: S in the source section “MATHEMATICAL CONSTANTS · - von Neumann entropy: S”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables and conditions; this record does not…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-0bac8389bbec28c7

MATHEMATICAL CONSTANTS · Compute DNA entropy: H

This relation defines DNA entropy: H in the source section “MATHEMATICAL CONSTANTS · Compute DNA entropy: H”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-96fd98497231282c

MATHEMATICAL CONSTANTS · Compute von Neumann entropy: S

This relation defines von Neumann entropy: S in the source section “MATHEMATICAL CONSTANTS · Compute von Neumann entropy: S”. Uses logarithmic scaling; its base and the positivity of its argument matter. Its interpretation depends on the source’s stated variables and conditions; this record does…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-0b8befb183a29017

MATHEMATICAL CONSTANTS · Ensure Hermiticity and trace

This relation defines Ensure Hermiticity and trace in the source section “MATHEMATICAL CONSTANTS · Ensure Hermiticity and trace”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this…

Implementation mathematicsSource recovered; runtime not executed

Ensure Hermiticity and trace = 1

gh-eq-b441c7e5395f39bf

MATHEMATICAL CONSTANTS · F

This relation defines F in the source section “MATHEMATICAL CONSTANTS · F”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive…

Implementation mathematicsSource recovered; runtime not executed

F = <target|rho|target>

gh-eq-9da873e78c519917

MATHEMATICAL CONSTANTS · Z

This relation defines Z in the source section “MATHEMATICAL CONSTANTS · Z”. Applies an exponential factor. The exponent determines its real scaling or, for a complex exponent, its phase. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-fc1222913bdaf6f9

MATHEMATICAL CONSTANTS · d(phi_i)/dt

This relation defines d(phi_i)/dt in the source section “MATHEMATICAL CONSTANTS · d(phi_i)/dt”. Evaluates oscillatory sine or cosine terms using the stated phase or frequency. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-8710611750e47e69

MATHEMATICAL CONSTANTS · dG

This relation defines dG in the source section “MATHEMATICAL CONSTANTS · dG”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

dG = dH - T*dS

gh-eq-60b4111d8bc6691f

MATHEMATICAL CONSTANTS · dN/dt

This relation defines dN/dt in the source section “MATHEMATICAL CONSTANTS · dN/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or…

Implementation mathematicsSource recovered; runtime not executed

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

gh-eq-8294503ecb458f12

MATHEMATICAL CONSTANTS · drho/dt

This relation defines drho/dt in the source section “MATHEMATICAL CONSTANTS · drho/dt”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical…

Implementation mathematicsSource recovered; runtime not executed

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

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