Mathematical sketch: Dimensionless, fractal governance

1) Canonical atoms and measures

  • Let raw events from any source (logs, metrics, tickets, CDC, etc.) be mapped into a canonical atom: [ e=(a,o,v,m,t,c) ] actor (a), object (o), verb/type (v), magnitude (m) (possibly unitful), timestamp (t), context (c).
  • A stream (E={e_i}) induces measures over a graph (G=(V,E_d)) where (V) are entities and (E_d) are directed relations; measures are time-varying flows (F_{uv}^v(t)) (rate of type-(v) events on edge (u!\to!v)) and node intensities (I_u^v(t)).

2) Dimensionless normalization (Buckingham-style)

Pick characteristic scales from the data (not the domain):

  • time scale (T_0) (e.g., median inter-arrival),
  • magnitude scale (M_0) (e.g., robust median |m|),
  • population scale (N_0) (e.g., active nodes),
  • risk/impact scale (A_0) (allowed harm per unit time).

Define dimensionless variables: [ \tilde t=\frac{t}{T_0},\quad \tilde m=\frac{m}{M_0},\quad \tilde F=\frac{F}{M_0/T_0},\quad \tilde I=\frac{I}{M_0/T_0},\quad \tilde A=\frac{A}{A_0}. ] Everything downstream is computed in tildes. This is what makes the theory dimensionless and portable across domains.

3) Transformation group and invariants

Let (\mathcal{T}) be transformations that should not change “what the system is”:

  • unit re-scalings (already removed by tildes),
  • schema isomorphisms (renaming fields/types),
  • node relabelings (permutations of (V)),
  • coarse-grainings (\mathcal{C}_\Delta): merge time into windows of size (\Delta), optionally cluster nodes into super-nodes.

A governance invariant is any statistic (J) with [ J(\mathcal{T}(E))=J(E)\quad \text{and}\quad J(\mathcal{C}_\Delta(E))\approx J(E)\ \ \text{(within bounded distortion)}. ]

Concrete, computable candidates (all dimensionless):

  • Response ratio (R=\frac{\Pr(\text{corrective} \mid \text{anomaly})}{\Pr(\text{incident} \mid \text{anomaly})}).
  • Amplification index (A^\star=\frac{\text{avg downstream fan-out of anomaly edges}}{\text{avg baseline fan-out}}).
  • Causal closure (C^\star=) fraction of anomaly paths that terminate in a corrective action class within (\tilde{\tau}) time.
  • Cycle entropy (H_\circlearrowleft=) normalized entropy of cycle lengths in (G) weighted by (\tilde F) (captures feedback richness).
  • Assortativity by role (r^\star) (role labels from (v) or (c)), revealing whether anomalies remain local vs. cross subsystems.

These are schema-free (dimensionless, label-invariant) and designed to be scale-stable under (\mathcal{C}_\Delta).

4) Fractal/RG (renormalization) picture

Define an RG operator (\mathcal{R}\Delta) that maps parameter summaries at base resolution to summaries after coarse-graining by (\Delta): [ \theta’=\mathcal{R}\Delta(\theta),\quad \theta={R, A^\star, C^\star, H_\circlearrowleft, r^\star,\ldots}. ] A fractal law means there exist exponents (\beta) such that [ \theta’ \approx \Delta^{\beta}\odot \theta \quad \text{(componentwise)},\qquad \text{or fixed points }\theta^\ast\ \text{with}\ \mathcal{R}\Delta(\theta^\ast)=\theta^\ast. ] Empirically: if (R, A^\star, C^\star, H\circlearrowleft, r^\star) are stable (or follow power-laws) across (\Delta), your governance structure is self-similar (fractal).

5) “Maternal-instinct” bias as a control law

Let (x(t)) be a state vector of invariants (rolling estimates). Define a dimensionless risk potential [ \Phi(x)=\lambda_1(1-C^\star)+\lambda_2 A^\star+\lambda_3 H_\circlearrowleft-\lambda_4 R + \lambda_5(1-r^\star_{\text{healthy}}) ] with (\lambda_i>0) chosen by policy. The safe manifold is (\mathcal{S}={x:\Phi(x)\le \Phi_0}).

A generic protective control (“maternal instinct”) augments any operational policy (\pi) with: [ u_{\text{protect}}(t)=-K,\nabla_x \Phi\big(x(t)\big) ] (i.e., act in the steepest direction that reduces potential). If (\Phi) is a Lyapunov-like function for the closed loop, trajectories are driven toward (\mathcal{S}) at every scale, preserving fractal structure.

6) Dimensionless “governance numbers” (like Reynolds)

With (T_0,M_0,A_0) defined from data, form unit-free ratios that predict regimes:

  • Stability Reynolds: [ \mathrm{Re}s=\frac{\text{throughput gain}\times A^\star}{\text{damping capacity}} =\frac{\tilde F{\text{anom}}\cdot A^\star}{\tilde F_{\text{protect}}}. ] Expect cascading failures when (\mathrm{Re}_s>1).

  • Governance Efficacy: [ \Gamma=\frac{R\cdot C^\star}{A^\star}\quad (\uparrow \text{ is better}) ] high when responses dominate incidents, closures are fast, and amplification is low.

  • Multi-scale consistency: [ \kappa=\sup_{\Delta\in[\Delta_{\min},\Delta_{\max}]}| \theta-\mathcal{R}_\Delta(\theta)|_1 ] small (\kappa) ⇒ invariants are fractal-stable.

7) What “dimensionless + fractal” buys you (testable claims)

  1. Universality: If two systems have similar (\theta) (dimensionless invariants), they will respond similarly to shocks—even if the domains differ (tickets vs. logs).
  2. Scale portability: Policies tuned to (\theta) at one scale (\Delta) remain effective at other scales when (\kappa) is small (RG-stability).
  3. Safety guarantee (sketch): If (\Phi) decreases along trajectories under (u_{\text{protect}}) and (\mathrm{Re}_s<1), the system remains in or returns to (\mathcal{S}).

8) Minimal PoC to ground this

  • Ingest four sources (logs, metrics, tickets, CDC) → canonical atoms (e).
  • Estimate (T_0,M_0,A_0) from data; compute (\theta) and (\Gamma,\mathrm{Re}_s,\kappa).
  • Run coarse-graining for (\Delta\in{1,2,4,8,\ldots}) hours; verify RG stability ((\kappa) small) and power-law behavior.
  • Implement (u_{\text{protect}}) as an automated routing/throttling rule that steepest-descents (\Phi); observe whether (\Gamma\uparrow) and (\mathrm{Re}_s\downarrow) across scales.

This keeps your core insight dimensionless (unit-free, schema-free) and fractal (RG-stable across scales), while staying implementable: every quantity above can be computed from your Mongo→Kafka→CH fabric and evaluated in Grafana/CH queries.