Dimensionless, Fractal Governance — Entropy Formulation

A compact, implementation‑ready extension of the earlier sketch that makes entropy the first‑class currency of governance across logs, metrics, tickets, CDC streams, and more.

0) Setup: canonical atoms → empirical distributions

Represent any event as a canonical atom [ e=(a,o,v,m,t,c) ] actor a, object o, verb/type v, magnitude m, timestamp t, context c.
From a stream (E={e_i}), define empirical distributions at resolution (\Delta) over relevant alphabets (\mathcal{A}):

  • Type distribution: (P_V(v)).
  • Role/Context: (P_R(r)), (P_{V\mid R}(v\mid r)).
  • Edge flow: (P_{U\to W}(u\to w)) from the graph (G=(V,E_d)).
  • Path/loop: (P_{\text{path}}(\pi_L)) for paths of length (L); (P_{\circlearrowleft}(\ell)) for cycle lengths.
  • Time process: word distributions (P(X_{1:L})) for sequences of event types (or anomalies/controls) in a time window.

All quantities below are dimensionless via normalization by the relevant alphabet sizes.

1) Entropy primitives (dimensionless by construction)

Let (H(X)) be Shannon entropy (bits), (H(X\mid Y)) conditional entropy, (I(X;Y)) mutual information, and (D_{\mathrm{KL}}(P\Vert Q)) Kullback–Leibler divergence.

Define normalized, dimensionless entropies: [ \hat H(X)=\frac{H(X)}{\log \lvert\mathcal{X}\rvert},\quad \hat H(X\mid Y)=\frac{H(X\mid Y)}{\log \lvert\mathcal{X}\rvert},\quad \hat I(X;Y)=\frac{I(X;Y)}{\log \lvert\mathcal{X}\rvert}. ]

Process‑level quantities (per unit time or per symbol):

  • Entropy rate (h_\mu): limit of (H(X_t\mid X_{1:t-1})) (normalize by (\log\lvert\mathcal{X}\rvert)).
  • Predictive information / excess entropy: (\mathcal{E}=I(X_{\text{past}};X_{\text{future}})), normalized.
  • Transfer entropy (causal influence): (T_{X\to Y}=I(X_{\text{past}};Y_t\mid Y_{\text{past}})), normalized.
  • Multiscale entropy (MSE): (\mathrm{MSE}(\Delta)=\hat H) computed after coarse‑graining the series at scale (\Delta).

Irreversibility / entropy production for trajectories over window (\Delta): [ \sigma(\Delta)=\frac{1}{\Delta},D_{\mathrm{KL}}\big(P_\Delta(\text{traj}),\Vert,P_\Delta^{\mathrm{rev}}(\text{traj})\big) ] (bits per unit time). High (\sigma) indicates arrow‑of‑time asymmetry (e.g., incidents), low (\sigma) indicates near‑reversible “healthy” operation.

2) Entropy‑based invariants (scale‑aware, schema‑free)

All are dimensionless (normalized) and designed to be stable under relabeling and moderate coarse‑graining.

  • Type entropy: (\hat H_V=\hat H(V)) — diversity of event types.
  • Role‑conditioned entropy: (\hat H_{V\mid R}=\hat H(V\mid R)) — specificity of behavior by role/context.
  • Flow entropy: (\hat H_F(\Delta)=\hat H(U\to W)) — dispersion of traffic over edges at scale (\Delta).
  • Path entropy: (\hat H_{\text{path}}(L)=\hat H(\pi_L)) — diversity of causal routes.
  • Cycle entropy: (\hat H_{\circlearrowleft}=\hat H(\ell)) — richness of feedback loops.
  • Entropy rate (baseline vs. anomaly): (\hat h_\mu^{\text{base}},\ \hat h_\mu^{\text{anom}}).
  • Predictive information: (\hat{\mathcal{E}}) — memory/structure in the process.
  • Transfer entropies:
    • (\hat T_{\text{anom}\to\text{incident}}) — how strongly anomalies drive incidents,
    • (\hat T_{\text{control}\to\text{incident}}) — how strongly controls steer outcomes,
    • (\hat T_{\text{anom}\to\text{control}}) — whether the system learns to react to anomalies.
  • Entropy production: (\sigma(\Delta)) — irreversibility of local dynamics (incidents raise (\sigma)).

Design note: Normalizing by (\log) of alphabet size (or max attainable) makes these truly unitless and thus portable across domains.

3) Fractal / RG (renormalization) viewpoint via multiscale entropy

Let coarse‑graining (\mathcal{C}_\Delta) aggregate time into windows of size (\Delta) and optionally cluster nodes.

Define the RG operator on summary vector (\theta_H={\hat H_V,\hat H_{V\mid R},\hat H_F(\Delta),\hat H_{\text{path}}(L),\hat h_\mu,\hat{\mathcal{E}},\hat T_\cdot,\sigma(\Delta)}): [ \theta_H’ = \mathcal{R}\Delta(\theta_H). ] Fractality ⇔ either (\theta_H’\approx\theta_H) (fixed point) or follows stable power‑law scalings across (\Delta) (e.g., (\hat H_F(\Delta)\sim \Delta^{-\beta_H})).
Define multi‑scale consistency [ \kappa_I=\sup{\Delta\in[\Delta_{\min},\Delta_{\max}]}\lVert\theta_H-\mathcal{R}_\Delta(\theta_H)\rVert_1, ] small (\kappa_I) ⇒ fractal‑stable governance statistics.

4) “Maternal‑instinct” control as entropy shaping

We want to reduce harmful irreversibility while preserving useful complexity.

4.1 Potential function (dimensionless)

[ \Phi(x)= \alpha,\sigma(\Delta) +\beta,\hat T_{\text{anom}\to\text{incident}} -\gamma,\hat T_{\text{control}\to\text{incident}} +\delta,(1-\hat{\mathcal{E}}) +\eta,(1-\hat C^\star) ] with weights (\alpha,\beta,\gamma,\delta,\eta>0), and (\hat C^\star) the dimensionless “causal closure” fraction (from earlier sketch).

  • Lower (\sigma) ⇒ fewer irreversible cascades.
  • Lower (\hat T_{\text{anom}\to\text{incident}}), higher (\hat T_{\text{control}\to\text{incident}}) ⇒ controls dominate dynamics.
  • Higher (\hat{\mathcal{E}}) ⇒ structure/predictability without degeneracy.

4.2 Control law (policy‑agnostic)

[ u_{\text{protect}}(t)=-K,\nabla_x \Phi\big(x(t)\big)\quad \text{with}\quad x(t)\equiv\theta_H(t). ] Any actuator (throttling, isolation, routing, reprovisioning) that descends (\Phi) is “maternal.” If (\Phi) is Lyapunov‑like for the closed loop and (\kappa_I) is small, safety persists across scales.

5) Dimensionless “information numbers”

  • Information Reynolds (cascade risk): [ \mathrm{Re}I=\frac{\hat h\mu^{\text{anom}}\cdot A^\star}{\mathcal{C}{\text{ctrl}}} ] where (A^\star) is amplification index (dimensionless) and (\mathcal{C}{\text{ctrl}}) is effective control‑channel capacity (bits/time). Cascades likely when (\mathrm{Re}_I>1).

  • Governance efficacy (informational): [ \Gamma_I=\frac{\hat T_{\text{control}\to\text{incident}}\cdot \hat C^\star}{\hat T_{\text{anom}\to\text{incident}}+\varepsilon}. ]

  • Fractal‑stability: (\kappa_I) as above ((\downarrow) is better).

6) Universality claims (now explicitly entropic)

  1. Domain invariance: Systems with similar (\theta_H) behave similarly under shocks, regardless of whether events came from logs, tickets, or CDC.
  2. Scale portability: If (\kappa_I) is small, policies tuned at one (\Delta) remain effective at others.
  3. Safety condition (sketch): If (\mathrm{Re}_I<1) and (\dot\Phi\le 0) along trajectories, the state stays within or returns to a safe manifold.

7) Minimal PoC (entropy‑first)

  1. Unify four feeds → canonical atoms in ClickHouse (or equivalent).
  2. Compute distributions at multiple (\Delta) (1, 2, 4, 8… hours) and derive (\hat H_V), (\hat H_{V\mid R}), (\hat H_F), (\hat H_{\text{path}}), (\hat h_\mu), (\hat{\mathcal{E}}), transfer entropies, and (\sigma(\Delta)).
  3. Plot MSE curves and evaluate (\kappa_I). Check for power‑laws/fixed points.
  4. Deploy a simple controller (e.g., auto‑isolate top‑k anomaly fan‑out edges when (\mathrm{Re}_I>1) or when (\partial\Phi/\partial A^\star>0)).
  5. Acceptance: (\Gamma_I\uparrow), (\mathrm{Re}_I\downarrow), (\sigma\downarrow) across scales.

8) Implementation notes (pragmatic)

  • Normalization: keep alphabets explicit to normalize entropies ((\log\lvert\mathcal{X}\rvert)).
  • Streaming estimates: use count‑min sketches or reservoir sampling to keep distributions online.
  • Transfer entropy: start with plug‑in estimators over discretized symbols; graduate to kNN or model‑based estimators as needed.
  • Entropy production: approximate time‑reversal by flipping sequences within window (\Delta); use n‑gram models to estimate trajectory likelihoods.
  • Controls: expose actuators via a small policy engine; periodically recompute (\theta_H) and descend (\Phi).

TL;DR

Make entropy the common language across heterogeneous event streams, normalize it so it’s dimensionless, verify fractal stability via coarse‑graining, and define “maternal” control as entropy‑shaping that reduces irreversibility while preserving useful structure. This yields portable policies, measurable safety, and a plausible bridge to Shannon‑style theory.