Research-Stack/6-Documentation/docs/papers/EQUATION_01_ETA_MOE_EFFICIENCY.md

ηMoE: Mixture-of-Experts Cognitive Efficiency

Equation ID: 0.1
Family: Field Optimization
Status: ✅ REFINED
Cross-refs: 0, 1.1, 1.2, 6, 29

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The Equation

\eta_{\text{MoE}}(x) = \frac{\sum_{k=1}^{K} g_k(x) \left(\frac{w_k h_k}{\ln N_k} - \frac{v_k p_k}{\ln N_k}\right)}{\sum_{k=1}^{K} g_k(x)(a_k \ln N_k + c_k) + k_B T \cdot I_{\text{discarded}}(x) + C_{\text{platform}}(x)}

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Variables

Symbol	Meaning
g_k(x)	Gating weight for expert k (state-dependent)
w_k	Quality weight for expert k
h_k	Structural coherence / benefit from expert k
v_k	Penalty weight for expert k
p_k	Distortion / error from expert k
N_k	Effective arity / complexity of expert k
a_k	Cost coefficient for expert k
c_k	Per-expert overhead (fixed cost)
I_{\text{discarded}}	Irreversible information loss (Landauer)
C_{\text{platform}}	Substrate / hardware overhead

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Purpose

The ηMoE equation bridges microscopic cognitive control to macroscopic universal efficiency:

1. Mechanism: Uses Mixture-of-Experts (MoE) architecture with state-dependent gating
2. Load-dependence: Gating weights deform under cognitive load (\rho = L_{\text{total}}/C_{\text{cognitive}})
3. Neurochemistry: Gating modulated by cortisol (c), serotonin (s), and urgency (u)
4. Thermodynamic bounds: Includes Landauer cost k_B T I_{\text{discarded}} and platform overhead C_{\text{platform}}

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Gating Function

The expert selection weights follow a sigmoid-based load response:

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

Where:

• \rho = load-to-capacity ratio
• c = cortisol (stress drive)
• s = serotonin (stabilization)
• u = urgency / threat salience
• \sigma = sigmoid function

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Physical Constraints (Anti-Paradox Safeguards)

From MathMan's stress-testing:

1. $N_k \geq 2$: Minimum arity prevents singularity
2. $c_k > 0$: Per-expert overhead prevents vanishing cost
3. $C_{\text{platform}} > 0$: Baseline cost prevents infinite efficiency
4. $I_{\text{discarded}} \geq 0$: Landauer limit respected

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Two-Expert Cognitive Case

For the cognitive/affective blend:

\eta_{\text{cognitive}}(x) = \frac{\alpha\left(\frac{w_c h_c}{\ln N_c} - \frac{v_c p_c}{\ln N_c}\right) + \beta\left(\frac{w_a h_a}{\ln N_a} - \frac{v_a p_a}{\ln N_a}\right)}{\alpha(a_c \ln N_c + c_c) + \beta(a_a \ln N_a + c_a) + k_B T I_{\text{discarded}} + C_{\text{neural}}}

Where:

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

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Connection to Φ_universal

The ηMoE is the micro-mechanism that realizes the macroscopic field equation:

Layer	Role
Φ_universal	Macro field equation (ground truth)
ηMoE	Local efficiency scoring (mechanism)
MoE Control	Gating dynamics (implementation)

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Theorem (Efficiency-Load Tradeoff)

Statement: Under rising cognitive load, the efficiency-maximizing gate shifts from deliberative to affective experts.

Proof sketch:

• Low load: \rho \approx 0 \Rightarrow \beta \approx 0 \Rightarrow cognition dominates
• Rising load: \rho \uparrow \Rightarrow \beta \uparrow \Rightarrow affective weighting increases
• High load: \beta \gg \alpha \Rightarrow heuristic control dominates

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Implementation

Lean module: EtaMoE.lean
Python shim: cognitive_efficiency_moe.py

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References

• Landauer (1961): Irreversibility and heat generation
• Shazeer et al. (2017): Outrageously Large Neural Networks (MoE)
• OTOM Cognitive Load Integration (v3)

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Status: ✅ Ready for Triumvirate Warden review