Entropy fields

  • Each metric stream: $h_i(t)$ = rolling Shannon entropy of metric $i$.
  • Stack into vector: $\mathbf{h}(t) \in \mathbb{R}^n$.
  • Covariance field: $\Sigma(t) = \mathrm{Cov}[\mathbf{h}(t)]$.

C1. Continuity (balance) of entropy

$$ \dot h_i = s_i - \kappa_i h_i - \sum_{j}\nabla!\cdot J_{ij} + \eta_i $$

Sources $s_i$, damping $\kappa_i \ge 0$, fluxes $J_{ij}$, noise $\eta_i$.

C2. Constitutive law (flux response)

$$ J_{ij} = -D_{ij},(h_j - h_i) \quad\Longrightarrow\quad \dot{\mathbf h} = -\alpha,\mathbf h - \beta,L,\mathbf h + \mathbf s + \boldsymbol\eta $$

Here $L$ is a graph Laplacian; $\alpha,\beta \ge 0$.

C3. Correlation evolution (Lyapunov dynamics)

$$ \dot{\Sigma} = A\Sigma + \Sigma A^\top + Q - \Gamma(\Sigma) $$

with $A = -(\alpha I + \beta L)$, $Q = \mathrm{Cov}[\boldsymbol\eta]$, and $\Gamma(\Sigma)$ = control.

Discrete time: $$ \mathbf h_{t+1} \approx A_t \mathbf h_t + \boldsymbol\varepsilon_t,\qquad \Sigma_{t+1} = A_t \Sigma_t A_t^\top + Q_t - \Gamma_t $$

C4. Alignment law (Gauss-style)

$$ \rho_{\text{align}}(t) = \sum_{i\ne j} w_{ij},\Sigma_{ij}(t), \qquad \lambda_1(t) = \lambda_{\max}!\big(\Sigma(t)\big) $$

Alignment accumulates from couplings and noise, and is drained by control: $$ \frac{d}{dt}\rho_{\text{align}} = \Phi_{\text{coupling}}(A,\Sigma) + \mathrm{tr}(WQ) - \mathrm{tr}(W\Gamma) $$

Eigenmode monitor (operational early warning)

Let $u_1(t)$ be the eigenvector for $\lambda_1(t)$. From the Lyapunov dynamics: $$ \dot\lambda_1 \approx u_1^\top!\big(A\Sigma + \Sigma A^\top + Q - \Gamma\big),u_1 $$

If the symmetric part $\mathrm{Sym}(A)=(A{+}A^\top)/2$ loses damping (critical slowing), $\lambda_1$ rises — the measurable “karma spike.”

Fitting recipe (discrete-time, practical)

  1. Compute $h_i(t)$: rolling Shannon entropy per metric.
  2. Fit VAR(1): $\mathbf h_{t+1} \approx A_t \mathbf h_t + \varepsilon_t$.
  3. Estimate $Q_t = \mathrm{Cov}[\varepsilon_t]$.
  4. Propagate covariance: $\Sigma_{t+1} = A_t \Sigma_t A_t^\top + Q_t$.
  5. Monitor $\lambda_1(t)$, $\mathrm{tr}(\Sigma)$, and off-diagonal mass.
  6. Trigger alerts or protective bias when $\lambda_1$ spikes above baseline.

Summary

This appendix encodes Ted’s Law of Karma as a dynamical system: entropy streams behave like fields, their covariances evolve by Lyapunov dynamics, and eigenvalue spikes precede shared-fate events — just as Maxwell encoded Faraday’s lines of force into equations.