Cognitive Load Theory: Information-Theoretic# Cognitive Load Theory

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

References: See 00_Master_References.md for complete source mapping

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. The load dimensions are now extended with geometric structure folding (Torus-Menger-Horn) and Mass Number admissibility gates for manifold merging.

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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