Difficulty / Depth Metric for Mathematical Objects

[eugnes03]

Problem

MathAtlas currently lists objects without any indication of their relative difficulty or abstraction level.

However, mathematical results naturally form layers of increasing complexity.

For example:

Group
  ↓
Representation
  ↓
Irreducible Representation
  ↓
Schur's Lemma

Users studying from their notes often want to understand:

• which objects are foundational
• which results are more advanced
• the natural learning order of concepts

Right now the site does not expose this structural information.

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Proposed Solution

Compute a difficulty score for each object based on its position in the dependency graph.

A simple definition:

difficulty = length of the longest dependency chain ending at that object

Example:

Group → difficulty 1
Representation → difficulty 2
Irreducible Representation → difficulty 3
Schur's Lemma → difficulty 4

This metric reflects how many conceptual layers are required before reaching the object.

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Implementation

Extend mathatlas.graph to compute node depth.

The dependency graph is already a directed acyclic graph (DAG), so we can compute the longest path.

Suggested function:

compute-difficulty

Output should be a map:

object-id → depth

Example:

{"abc123" 3
 "def456" 1
 "ghi789" 4}

This value can then be attached to objects during site generation.

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UI

Display difficulty on object pages.

Example:

Difficulty: ★★★★☆

or alternatively:

Difficulty Level: 4

Objects could also be sortable by difficulty on index pages.

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Site Generation Changes

During site generation:

1. compute difficulty values for all objects
2. attach difficulty metadata when rendering object pages
3. optionally allow sorting objects by difficulty

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Acceptance Criteria

• a difficulty score is computed for every object
• difficulty values reflect dependency depth
• difficulty is displayed on object pages
• computation works across files and areas
• the feature does not slow down site generation significantly