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Information Cosmology operators

17 operators in the information_cosmology category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
ICO1Shannon information entropy H(X) quantifying the average surprise or information content of a discrete random source.H(X) = -\sum_{i=1}^n p(x_i) \log_2 p(x_i), \quad 0 \leq H(X) \leq \log_2 n
ICO10Critical exponents characterizing universal power-law behavior of physical quantities near continuous phase transitions.\xi \sim |T - T_c|^{-\nu}, \quad C \sim |T - T_c|^{-\alpha}, \quad m \sim |T_c - T|^\beta
ICO11Information cascade operator modeling logistic adoption dynamics where agents follow predecessors rather than private signals.P(\text{adopt}|k \text{ adopters}) = \frac{1}{1 + e^{-\beta(k - \theta)}}
ICO12Power-law distribution P(x) ~ x^(-alpha) describing scale-free phenomena from wealth to city sizes.P(x) = C x^{-\alpha}, \quad \text{for } x \geq x_{min}, \quad \alpha > 1
ICO13Small-world network model exhibiting high clustering and short path lengths, bridging regular lattices and random graphs.L \sim \frac{\log N}{\log \langle k \rangle}, \quad C = \frac{3(k-1)}{2(2k-1)} \quad \text{for regular graphs}
ICO14Partition function Z = sum(e^(-betaE_i)) and Helmholtz free energy F = -k_BT*ln(Z) for statistical ensemble thermodynamics.F = -k_B T \ln Z, \quad Z = \sum_{\text{configurations}} e^{-E/k_B T}
ICO15Percolation theory modeling the emergence of connected clusters as occupation probability crosses a critical threshold.P_\infty(p) \sim (p - p_c)^\beta, \quad \xi(p) \sim |p - p_c|^{-\nu}
ICO16Kuramoto synchronization model d(theta_i)/dt = omega_i + (K/N)*sum(sin(theta_j - theta_i)) for coupled oscillator phase locking.\frac{d\theta_i}{dt} = \omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j - \theta_i) + \eta_i(t)
ICO17Replicator dynamics dx_i/dt = x_i*(f_i - f_avg) modeling evolutionary selection of strategies based on relative fitness.\frac{dx_i}{dt} = x_i \left[(A\vec{x})_i - \vec{x} \cdot A\vec{x}\right] + \mu \left(1 - Nx_i\right)
ICO18Collective intelligence operator aggregating distributed agent computations into emergent group-level problem solving.I_{collective} = \alpha \sum_i I_i + \beta \sum_{i\neq j} C_{ij} I_i I_j + \gamma \cdot \text{diversity}
ICO2Algorithmic (Kolmogorov) complexity K(x) measuring the length of the shortest program that produces string x.K(x) = \min{|p| : U(p) = x}, \quad \text{for universal Turing machine } U
ICO3Landauer's principle E >= k_BTln(2) setting the minimum energy cost of erasing one bit of information.E \geq k_B T \ln 2 \quad \text{per bit erased}, \quad \text{at temperature } T
ICO4Bekenstein-Hawking entropy S_BH proportional to black hole horizon area, linking gravity to information content.S_{BH} = \frac{k_B c^3 A}{4G\hbar} = \frac{k_B A}{4l_P^2}, \quad l_P = \sqrt{\frac{G\hbar}{c^3}}
ICO5Computational complexity class O(f(n)) characterizing the asymptotic resource requirements of algorithms.T(n) = O(f(n)), \quad \text{for input size } n, \text{ time/space complexity } f(n)
ICO6Betweenness centrality measuring the fraction of shortest paths passing through a node in a network.C_B(v) = \sum_{s\neq v\neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}, \quad C_C(v) = \frac{1}{\sum_{u} d(u,v)}
ICO7Fractal dimension D = log(N)/log(s) quantifying the self-similar scaling complexity of fractal geometries.D = \frac{\log N}{\log s}, \quad \text{for } N \text{ self-similar pieces, scaling factor } s
ICO9Self-organization via reaction-diffusion equations modeling spontaneous pattern formation from homogeneous initial conditions.\frac{\partial u}{\partial t} = D\nabla^2 u + f(u) + g(u,v), \quad \frac{\partial v}{\partial t} = D_v\nabla^2 v + h(u,v)

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["ICO1"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​