This repository explores the divergence between cognitive topography and verifier-induced structure in tractable problems—a concept we call Δ(structure).
Our canonical example: a 3x3 binary grid governed by overlapping 2x2 XOR constraints. Despite being solvable via linear algebra, naive bit-flip heuristics yield a maximally fragmented solution space in Hamming geometry.
xor_puzzle.py: Generates valid 3x3 XOR grids and checks verifier constraints.visualize_hamming_graph.py: Visualizes the solution space’s fragmentation in Hamming space.delta_tools.py: (Coming Soon) Tools for scoring and visualizing Δ(structure).notebooks/: Jupyter notebooks for experiments, metrics, and solver comparisons.
- Verifier Geometry: The hidden structure imposed by a problem’s constraints.
- Cognitive Topography: The solver’s perceived landscape (e.g., bit-flip adjacency).
- Δ(structure): The divergence between the two—high Δ correlates with high perceived difficulty.
Even in low-complexity problems, naive solvers can fail due to misaligned geometry. Understanding Δ(structure) helps us design better puzzles, hint systems, interpretable AI, and insight-driven curricula.
- Δ puzzle generator templates
- Δ metrics for solvers
- Insight-triggering scaffold design tools
- Formal write-up and arXiv preprint
If you use this work, please cite the forthcoming paper:
Law, S. (2025). Toward a Unified Theory of Problem Geometry.
MIT License
Built with nerd-fueled geometric rage.