Topo-cognitive event horizon — coupled network–operator resilience

Parent: README.md
Formal index: ../FORMAL_LEDGER.md
Prerequisite synthesis: THREE_ARENAS_SYNTHESIS.md
Layer C policy: ../lean/Lean/ForkRaceFoldTheorems/EntropyBridge/README.md

This note extends the three-arena synthesis with a fourth interface: socio-technical dynamics where bottleneck capacity depends on operator state. It is expository engineering math—not a clinical or neuroscience claim, and not a promoted Lean theorem until separately mechanized with explicit Layer C axioms.

Ground truths (recap)

Logic wall: Verification ends where Layer C (hardware, measurement, certificates) begins.
Fiedler trap: Survival is governed by backlog drift and bottlenecks, not β₁ alone;
(\frac{dX}{dt} = D(t) - \min(R(t), C_B)) is the fluid backbone.
Somatic override: Under flooding, analytical bandwidth collapses; intensity must drop before deliberation returns.

Synthesis: In human-in-the-loop or human-aligned control, (C_B) is not fixed silicon—it couples to cognitive arousal (A(t)) (or a proxy). Alert storms can push (A) up; as (C_B(A)) falls, the same network topology can become non-viable because repair authority disappears.

1. Coupled dynamics

Keep scalar backlog (X(t)). Separate network damage from alert-induced load on the operator:

[ \frac{dX}{dt} = D_{\mathrm{net}}(t) - \min!\bigl(R_{\mathrm{sys}}(t),, C_B(A(t))\bigr). ]

Model cognitive stressor input (e.g. alert rate × severity) as (D_{\mathrm{cog}}(t)) feeding (A). A minimal autonomic-style sketch:

[ \frac{dA}{dt} = \alpha, D_{\mathrm{cog}}(t) - \beta, \mathrm{cool}(t) - \gamma,(A - A_{\mathrm{rest}}), ]

where cool(t) is the automation control: silence, batching, cadence slowdown, deferral—analogous to the behavioral override, but implemented as policy.

Scope: No claim that this ODE matches biology. It is an interface for guarded automation: if a class of policies bounds (\dot{A}), compare trajectories of (X).

2. Joint metrics: network homology slack and “cognitive β₁”

2.1 Network side

On the effective graph (G(t)):

β₁(G) — cycle rank (redundancy shape).
Add bottleneck / cohesion slack (β₁ is insufficient alone): min-cut margin, algebraic connectivity (\lambda_2), or load-scaled surrogates.

Example scalar network health:

[ H_{\mathrm{net}}(t) = \min(S_{\mathrm{topo}}, S_{\mathrm{cut}}, S_{\lambda_2}) ]

with definitions fixed per product (normalized ratios vs absolute floors).

2.2 Operator side (formal analogue)

There is no literal homology of a brain in this stack. Use finite concurrent repair capacity:

K — max serious contexts trackable without quality collapse (design + training).
L(t) — active critical contexts (incidents, decisions, alert families).

Cognitive slack:

[ S_{\mathrm{cog}}(t) = \mathrm{clamp}_{[0,1]}\left(\frac{K - L(t)}{K}\right). ]

When (L \to K), effective human repair (\to 0) — the “cognitive β₁ collapse”: attention graph loses parallel cycles; work serializes.

2.3 Joint deficit

StructuralErrorgle scalar for automation (examples):

[ \mathcal{M}(t) = \min\bigl(H_{\mathrm{net}}(t),, S_{\mathrm{cog}}(t)\bigr), \qquad \mathcal{D}(t) = \max\bigl(0,, 1 - H_{\mathrm{net}},, 1 - S_{\mathrm{cog}},, \mathbf{1}{X > X{\mathrm{warn}}}\bigr). ]

The stack is jointly fragile when either structural or cognitive slack hits the floor.

3. Checkpoint: topo-cognitive event horizon (control trigger)

Define thresholds ((A^*, \dot{A}^*, s^*, X^*)). A cooling policy (\pi_{\mathrm{cool}}) engages when:

[ \mathcal{T}(t) \equiv (A \ge A^*) ;\vee; (\dot{A} \ge \dot{A}^*) ;\vee; (S_{\mathrm{cog}} \le s^*) ;\vee; (X \ge X^*). ]

Mandatory actions when (\mathcal{T}):

Cap alert rate (dedupe, severity gating, suppress non-actionable noise).
Lengthen control cadence (reduce controller thrash).
Prefer autonomous repair within pre-authorized bounds while (A) is high.

Hysteresis: require (\mathcal{T}) for duration (h) to enter cooling; exit only when (A \le A_{\mathrm{clear}} < A^*) and (X \le X_{\mathrm{clear}}) (and optionally (H_{\mathrm{net}}) recovered)—prevents flapping.

This is the mathematical sense of stopping alerts so (C_B(A)) does not crash.

4. Viability set (“resilience” as invariant, not hype)

[ \mathcal{V} = {(X, A, G, \ldots) \mid X \le X_{\mathrm{safe}},; A \le A_{\mathrm{safe}},; H_{\mathrm{net}} \ge h_{\min},; S_{\mathrm{cog}} \ge s_{\min}}. ]

A certified claim has the shape: policy (\pi) makes (\mathcal{V}) positively invariant under the coupled dynamics, given explicit Layer C bounds (below).

4.1 Implementation hooks (TypeScript)

src/cooling-policy.ts — cognitiveSlack(K,L), coolingTrigger, mergeAllowedWhenCooling (merge promotion should respect cooling mode).
src/socio-topology/ — EffectiveGraphSnapshot, betti1Proxy, RollbackController for metric streams and rollback.

5. Lean encoding: theorems vs Layer C

Provable inside Lean (conditional, A/B style)

Monotonicity: If (C_B) is decreasing in (A) and cooling increases cool, then upper bounds on (\dot{X}) improve vs a baseline policy (comparison lemmas, discrete or ODE).
Invariant proofs: CoolingInvariant — if guard (\mathcal{T} \Rightarrow \pi_{\mathrm{cool}}) and the model satisfies stated inequalities, then (X \le X_{\mathrm{budget}}).

Must remain Layer C (named axioms / opaque interfaces)

C_B : ℝ → ℝ monotone decreasing on the operational band (or parameterized family validated externally).
alert_rate ↦ ΔA bounds from human-factors / ops data, not from Peano.
measureA : OperatorState → ArousalProxy — opaque proxy type; real systems use heart-rate variability, self-report, task errors, or engineered load scores.

Example discipline (names only):

axiom C_B_monotone : ∀ a₁ a₂ : ℝ, a₁ ≤ a₂ → C_B a₂ ≤ C_B a₁
axiom alert_impact_bounded : ∀ r : ℝ, r ≤ r_safe → ΔA_per_step ≤ f(r)

No sorry on promoted surfaces—finish the proof or expose the assumption.

Suggested module layout (future work): SocioTechnicalState, policy mode Normal | Cooling, networkHomologySlack, cognitiveSlack as real parameters fed by instrumentation.

6. Honest closure

A defensible safety story is:

There exists a cooling controller (\pi_{\mathrm{cool}}) such that, assuming validated bounds linking alerts to (A) and (A) to (C_B), the closed-loop system maintains invariant (\mathcal{V}) and avoids escape of (X).

That is certified control under explicit external assumptions—the same epistemic posture as hardware entropy in Layer C, extended to operator coupling.

Wording discipline: This architecture is load-aware automation under explicit human-capacity constraints—it models stress on the maintainer as a bounded resource and optimizes machine-side behavior (alert policy, cadence, autonomy) under that constraint. It does not assert folk-psychology “emotional intelligence,” qualia, or empathy in the machine. The control variable (\mathrm{cool}(t)) reduces endogenous cognitive load so that non-zero repair authority (C_B(A)) can apply; falsification lives in ops metrics (incident concurrency, alert rates, time-to-mitigate, error rates), not in claims about internal states.

forkjoin-ai/gnosis