Research-Stack/6-Documentation/docs/papers/EQUATION_10_MOE_COGNITIVE_CONTROL.md

MoE Cognitive Control: State-Dependent Expert Blending

Equation ID: 10
Family: Cognitive Load
Status: ✅ REFINED
Cross-refs: 0.1, 1, 6, 7, 29

---

The Equation

P_{\text{control}} = \sum_{k=1}^{K} g_k \cdot E_k

Where the gating weights follow:

\boxed{g_k = \sigma\left(\lambda_1 \rho + \lambda_2 c - \lambda_3 s + \lambda_4 u\right)}

---

Variables

Symbol	Meaning
P_{\text{control}}	Control decision policy
g_k	Gating weight for expert k
E_k	Expert evaluator k (cognitive or affective)
\sigma	Sigmoid function (smooth blending)
\rho = \frac{L_{\text{total}}}{C_{\text{cognitive}}}	Load-to-capacity ratio
c	Cortisol (stress chemistry)
s	Serotonin (stabilization)
u	Urgency / threat salience
\lambda_{1,2,3,4}	Weighting coefficients

---

Purpose

Models cognitive control as a blended mechanism under capacity constraints:

1. Not a binary switch: No hard transition from "cognition" to "emotion"
2. Continuous mixture: Expert weights deform smoothly with load
3. Neurochemical modulation: Stress chemistry biases gating
4. Efficiency-optimal: Seeks blend that maximizes \eta_{\text{MoE}}

---

Two-Expert Cognitive Case

For the cognitive/affective system:

\boxed{D(x) = \alpha \cdot U_{\text{cog}}(x) + \beta \cdot U_{\text{aff}}(x)}

Where:

• \alpha = g_{\text{cog}} = deliberative weight
• \beta = g_{\text{aff}} = affective weight
• \alpha + \beta = 1

---

Gating Dynamics

Low Load Regime

• $\rho \approx 0.1$-0.3
• $\beta \approx 0.1$-0.2
• Result: Cognition dominates, high precision

Medium Load Regime

• $\rho \approx 0.5$-0.8
• $\beta \approx 0.3$-0.5
• Result: Balanced, adaptive mixture

High Load Regime

• $\rho \approx 1.0$-1.5
• $\beta \approx 0.6$-0.9
• Result: Affective heuristics dominate, fast but lossy

---

Connection to ηMoE

The MoE control feeds directly into the efficiency equation:

\eta_{\text{MoE}}(x) = \frac{\sum_k g_k \left(\frac{w_k h_k}{\ln N_k} - \frac{v_k p_k}{\ln N_k}\right)}{\sum_k g_k(a_k \ln N_k + c_k) + k_B T I_{\text{discarded}} + C_{\text{platform}}}

Key insight: Cognitive load doesn't directly choose actions—it deforms the efficiency landscape, and the system seeks gate weights that maximize \eta_{\text{MoE}} under current constraints.

---

The "Topological Throat" Interpretation

The throat is not a moment where emotion replaces cognition. It is:

\boxed{\text{Capacity-constrained gating collapse}}

When \sum_k g_k C_k > C_{\text{cognitive}}, the system must:

• Drop high-cost experts
• Simplify routing
• Shift weight toward low-cost heuristics

---

Physical Safeguards

From MathMan's stress-testing:

1. g_k \in [0, 1]: Bounded weights
2. \sum_k g_k = 1: Conservation of control
3. C_k > 0: Per-expert cost prevents free lunch
4. N_k \geq 2: Minimum complexity prevents singularity

---

Theorem (Load-Triggered Phase Transition)

Statement: Under rising cognitive load, the system undergoes a continuous control regime shift from deliberative to affective weighting.

Proof sketch:

1. Define \beta = \sigma(k_1 \rho + k_2 c - k_3 s + k_4 u)
2. As \rho \uparrow, argument to \sigma increases
3. Since \sigma is monotone, \beta \uparrow
4. Therefore \alpha = 1 - \beta \downarrow
5. Control shifts from cognitive to affective dominance

QED

---

Implementation

Lean module: MoECognitiveControl.lean
Python shim: cognitive_moe_blender.py

---

References

• Shazeer et al. (2017): Outrageously Large Neural Networks (MoE)
• Kahneman (2011): Thinking, Fast and Slow (Dual-process theory)
• Sapolsky (2017): Behave (Stress neurobiology)
• OTOM Cognitive Load Integration (v3)

---

Status: ✅ Ready for Triumvirate Warden review