Angle–Slack Accounting and Minimality Heuristics for Triangle-to-Square Dissections

This note introduces an angle–slack bookkeeping framework for planar dissections, encoding local angle constraints via a typed cut graph. Interior “slack” vertices represent degrees of freedom that cannot be eliminated under local moves but may be transported through junctions. The framework explains persistent obstructions in three-piece dissections of an equilateral triangle to a square and clarifies why four pieces form the minimal entropy sink. Beyond classical dissection puzzles, the method illustrates a general invariant-based approach relevant to rigidity, obstruction theory, curvature accounting, and scissors congruence. This work was developed through interactive collaboration between William M. Bailey III and ChatGPT (GPT-5.2 Thinking). The framework and exposition emerged via iterative human–AI co-reasoning.