Cognitive Load Theory: Information-Theoretic Compression Routing

Authors: Research Stack Team
Date: April 2026
Domain: TTM Layer A (Compression/Routing)
OTOM Version: 2.2

Abstract

This paper presents the foundational routing engine for the OTOM framework. Every input byte sequence is evaluated across five load dimensions, enabling optimal compression routing through information-theoretic analysis.

1. Introduction

The Cognitive Load Theory (CLT) provides the mathematical foundation for routing decisions in the OTOM framework. It extends Sweller's cognitive load theory into the information-theoretic domain, treating compression as a cognitive process performed by computational agents.

2. Core Equations

2.1 Intrinsic Load (Shannon Entropy)

L_I(x) = -\sum p(b|x) \log_2 p(b|x)

The irreducible complexity of input sequence x, measured as Shannon entropy of its byte distribution.

2.2 Extraneous Load (Suboptimal Policy Cost)

L_E(x) = \text{BPB}(x, w_{\text{prior}}) - \text{BPB}^*(x)

The cost of using a suboptimal routing policy versus the optimal achievable.

2.3 Germane Load (Learning Effort)

L_G(x,t) = \sum \gamma^s \cdot \Delta L_E(x_s, t+1)

Productive learning effort that improves future routing decisions.

2.4 Routing Load (Decision Cost)

L_R(x) = \sum c_j \cdot \mathbb{1}[f_j] + \sum \log_2|M_l|

Classification cost plus decision tree traversal cost.

2.5 Memory Load (Storage Burden)

L_M(x) = \log_2|E| + \alpha \cdot \mathbb{1}[\text{hit}] + \beta + \lambda \cdot \frac{|E|}{|E_{\text{max}}|}

Engram storage, retrieval, and update burden.

2.6 Total Load

L_{\text{total}} = \lambda_I \cdot \hat{l}_I + \lambda_E \cdot \hat{l}_E - \lambda_G \cdot \hat{l}_G + \lambda_R \cdot \hat{l}_R + \lambda_M \cdot \hat{l}_M

3. 9D Feature Vector

Dimension	Symbol	Description
1	byteEntropy	Shannon entropy of byte distribution
2	repetitionRate	Frequency of repeated sequences
3	dictPotential	Dictionary compression opportunity
4	periodicityLag1	First-order periodicity measure
5	residualSparsity	Sparsity after transformation
6	matchDensity	Density of matching contexts
7	longestMatch	Length of longest repeated sequence
8	bitplaneBias	Correlation across bit planes
9	bestStrideCorr	Best stride correlation

4. Mixture-of-Experts Predictor

P_w(x_i) = \sum_j w_j \cdot P_{m_j}(x_i | x_{<i})

Weighted ensemble prediction across expert models m_j.

5. Implementation

Location: core/intrinsic/specs/COGNITIVE_LOAD_FUNCTIONS_SPEC.md

Lean 4 Module: CognitiveLoad.lean (TODO: Create)

6. References

Sweller, J. (1988). Cognitive load during problem solving.
Shannon, C.E. (1948). A mathematical theory of communication.
Research Stack, OTOM Ontology v2.2.

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