carpet-nodal

• Builds the Sierpinski carpet as the Kronecker power at level of the 3 x 3 tile with the centre void, 8^level cells, asserted at 512 for level 3 and 4096 for level 4.
• Takes the cell graph on 4-neighbour adjacency between filled cells, checks it is one component, and diagonalises the combinatorial Laplacian D - A, degree minus adjacency, densely with numpy.linalg.eigh.
• For the k-th returned eigenvector counts strong nodal domains nu_k: connected components of the positive cells plus connected components of the negative cells, a cell with |v| < 1e-9 belonging to neither.
• Clusters the spectrum at gap tolerance 1e-8 into eigenvalue classes, reads the multiplicity r_k and the last index of each class, and checks every k against Courant nu_k <= k and against the discrete nodal domain theorem nu_k <= k + r_k - 1.
• Prints per graph: nu_k, r_k and the zero-cell count for k = 1..12, the maxima, the Courant and discrete-theorem fractions, the class and degeneracy counts, the five-bin histogram of nu_k / k, the top eigenvector's sign pattern, and the first double eigenvalue where the two returned vectors, their sum and their difference disagree in nu.
• Controls: the 22 x 22 and 64 x 64 grid graphs, once in numpy's basis and once in the separable cosine basis ordered by eigenvalue then by the mode pair (p, q), whose count is checked against (p + 1)(q + 1) derived from the one-dimensional sign changes.
• On a degenerate eigenvalue the nodal count depends on the basis, so every nu_k is a fact about the returned basis only; the theorem bound k + r_k - 1 is the only basis-free statement, and the study claims nothing about the eigenspace.
• Domain: level 3, 4, zero tolerance 1e-9, multiplicity tolerance 1e-8; level 5 has 32768 cells and is not attempted.

RUN

uv run python research/lab/py/carpet-nodal/nodal.py

Needs numpy and scipy. One pass, about 30 seconds, exits 0.

WITNESSES

• automata.md section 13, the nodal domain lines: nu_k for k = 1..12 at level 3 and level 4, the Courant fraction 1 on all 512 and all 4096 returned eigenvectors, the degenerate index counts 262 of 512 and 2062 of 4096, the maximal multiplicities 4 and 20, and the double eigenvalue at k = 6, 7 whose returned vectors have 4 domains while their sum and difference have 6.
• The control lines: the separable count (p + 1)(q + 1) on all 484 and 4096 grid eigenvectors, and the mean of nu_k / k over k >= 2, 0.605 and 0.536 on the carpet against 0.508 and 0.465 on the separable grid.
• The top eigenvector line: no edge joins two nonzero cells of one sign on any of the six graphs, and at level 4 twelve cells of that vector fall below the zero tolerance, so its printed count is 4084.