README.md

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laplacian-degeneracy

  • Builds the level-level Sierpinski triangle as the design-code-7 base-2 tile Kronecker-substituted into itself, level = 1..8, and checks the array against the i AND j = 0 characterisation instead of assuming it.
  • Takes the nearest-neighbour graph on the filled cells, checks it is one component, and diagonalises the normalised Laplacian I - D^-1/2 A D^-1/2 densely, with A the adjacency matrix and D the degree matrix.
  • Clusters the sorted spectrum by consecutive gaps above 1e-9 and reports, per level, the distinct and degenerate class counts, the repeated fraction, and the multiplicities of eigenvalue 1 and of 1 -/+ sqrt(30)/6.
  • Tracks the third family at 1 -/+ 0.988332421566, the two Fibonacci counting fits, the isolation of the class at eigenvalue 1, the widest degenerate class, and the tolerance sweep at level = 8.
  • Domain: level = 1..8, the same range the page states; level = 9 is not attempted and the program prints why.

RUN

uv run python research/lab/py/laplacian-degeneracy/degeneracy.py

Needs numpy and scipy. One pass, about 35 seconds, exits 0.

WITNESSES

  • complexity.md:371 | 1 | 3 | 3 | 0 | 0.0000 | 1 | 0 | through complexity.md:378 | 8 | 6561 | 1975 | 465 | 0.7699 | 2187 | 244 |, all eight table rows
  • complexity.md:388 4 * 10^-15, the worst deviation from 1 -/+ sqrt(30)/6 over level = 2..8, measured as 3.61e-15
  • complexity.md:393 1, 3, 9, 27, 81, 243, 729, 2187, the multiplicity of eigenvalue 1 against 3^(level-1)
  • complexity.md:397 2, 4, 10, 28, 82, 244, the second family against 3^(level-3) + 1
  • complexity.md:400 1 -/+ 0.988332421566 and complexity.md:401 2, 4, 10, 28, 82, the third family against 3^(level-4) + 1
  • complexity.md:402 Six distinct classes at level 8 carry multiplicity 82
  • complexity.md:406 2*Fibonacci(2 level) + 1 and complexity.md:407 2*Fibonacci(2 level - 3) - 1, fitting eight and seven levels
  • complexity.md:417 5.4 * 10^-14, the widest degenerate class at level = 8, measured as 5.42e-14
  • complexity.md:418 isolated by 0.442 on both sides, the neighbours being 0.5578103942 and 1.4421896058
  • complexity.md:419 flat across seven decades of clustering tolerance, the sweep from 1e-12 to 1e-5
  • complexity.md:420 19683 nodes and complexity.md:421 3.1 GB dense matrix
  • complexity.md:619 Laplacian degeneracy at level 8, the eight-point conjecture
  • complexity.md:636 the tolerance sweep and complexity.md:637 the reason level 9 stops