Waves

• 2026-09-08 [Proved] Riesz product: the DFT of a base-2 design mask popped at level level at frequency xi on an N torus is the product over j = 0..level-1 of the level-1 tile polynomial P(side^j xi / N) minus the tile centre value, since the offsets are digit sums and the exponential sum factorises; checked at codes 7, 6, 9 and levels 2 to 4 over all 256 x 256 frequencies, largest error 2.5e-12, the subtracted term 1 for code 9 and 0 for codes 7 and 6. Witness: lab/py/kernel-waves.
• 2026-09-08 [Verified] From the density-0.5 soup on the 256 torus, 64 steps at one seed, the Bugs, wide-survive and narrow-birth windows reach no ring still on the box r = 5, 13 masks or the design masks code 7 at levels 2 to 4 and codes 6 and 9 at level 3: 0 of 21 cells, 10 dead, 6 flat frozen soups, 5 active; a wide survive on a mask of 512 ones or more freezes the soup unchanged. Witness: lab/py/kernel-waves.
• 2026-09-08 [Verified] On box r = 5, carpet code 7 at levels 2 and 3 and diagonal code 9 at level 3 every one of 130 ring stills has its peak ring k* inside the mask first negative band of the ring-mean signed DFT, k*/k_2 from 0.77 to 1.17, k* read on the rings k <= 128 which drop a fifth of the frequencies; as a universal law it fails on 3 of 176: box r = 13 at k* = 10, code 7 level 4 at k* = 2, code 6 level 3 at k* = 8. Witness: lab/py/kernel-waves.
• 2026-09-08 [Verified] The carpet family median still wavelength over mask side is 0.729 at levels 2 and 3, equal to 256/k_2/side at both; level 4 reads 0.790 against 0.632 where the 256 torus holds 3.2 mask widths. Witness: lab/py/kernel-waves.
• 2026-09-08 [Verified] On the carpet cell graph at level = 3, 512 cells on 4-neighbour adjacency with Laplacian D - A, the strong nodal domain counts of the returned eigenvectors are 1, 2, 2, 4, 4, 4, 4, 5, 8, 8, 8, 8 for k = 1..12 with multiplicities 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, Courant nu_k <= k holds on all 512 at every zero tolerance from 1e-12 to 1e-6, the spectrum has 380 classes at gap 1e-8 of which 130 are degenerate covering 262 indices with multiplicity up to 4, and nu_k / k falls in five bins of width 0.2 as 0, 42, 217, 193, 60 at tolerance 1e-9; counts for the returned basis only. Witness: lab/py/carpet-nodal.
• 2026-09-08 [Verified] At level = 4, 4096 carpet cells, the counts are 1, 2, 2, 4, 4, 4, 4, 5, 4, 4, 8, 8 with the same multiplicity pattern, where the 4, 4 at k = 9, 10 rests on 64 cells at |v| = 1.4e-7 in each returned vector and reads 8, 8 at tolerance 1e-6 as at level = 3; Courant holds on all 4096, the spectrum has 3056 classes of which 1022 are degenerate covering 2062 indices with multiplicity up to 20, the bins are 24, 797, 2061, 971, 243, and the top eigenvector has 12 cells below the zero tolerance so its printed count is 4084 while no edge joins two nonzero cells of one sign. Witness: lab/py/carpet-nodal.
• 2026-09-08 [Verified] The nodal count on a degenerate eigenvalue is basis-dependent: at the double eigenvalue k = 6, 7 of the carpet cell graph the two returned vectors have 4 strong nodal domains each and their normalised sum and difference have 6 each, at level = 3 and level = 4 alike, and such a disagreement occurs at 108 of the 129 double eigenvalues at level = 3 and 971 of 1021 at level = 4. Witness: lab/py/carpet-nodal.
• 2026-09-08 [Proved] By the discrete nodal domain theorem every eigenvector of an eigenvalue of index k and multiplicity r on a connected graph has at most k + r - 1 strong and at most k weak nodal domains, so at the double eigenvalue k = 6, 7 of the carpet cell graph at level = 3 and level = 4, index 6 and multiplicity 2 read at gap 1e-8, no vector of the eigenspace has more than 7 strong domains, and the returned basis sum and difference reach 6. Witness: lab/py/carpet-nodal.
• 2026-09-08 [Verified] On the control grids 22 x 22 and 64 x 64 the separable cosine basis has exactly (p + 1)(q + 1) strong nodal domains on all 484 and all 4096 eigenvectors, the returned basis satisfies Courant on all of them with degenerate index fractions 0.9566 and 0.9846 and multiplicity up to 21 and 63, and the mean of nu_k / k over k >= 2 at tolerance 1e-9 is 0.605 and 0.536 on the carpet (0.602 to 0.610 at level = 3 across tolerances 1e-12 to 1e-6) against 0.508 and 0.465 on the separable grid and 0.433 and 0.330 on the returned grid basis. Witness: lab/py/carpet-nodal.
• 2026-09-08 [Conjecture] A Larger-than-Life still sits where the kernel ring-mean transform is negative because a pattern there has its count anti-correlated with its state, which a birth window below the mean and a survive window above it hold fixed; the correlation is -0.52 to -0.85 on the strongest stills of the four lobe masks; a proof, a 1024 torus for the side-81 mask and the cross mask lobe are open. Witness: lab/py/kernel-waves.
• 2026-09-08 [Refuted] The wavelength of a ring still is the mask width, k* at the first minimum k_min of the mask ring-mean transform: over 180 window rules at two seeds, 2520 runs and 176 ring stills, k* sits within one ring of k_min in 25, 21 of them on the under-resolved side-81 mask; witness box r = 5, all 25 ring stills at k* = 30..36 against k_min = 24 and k_2 = 32, wavelength 7.1 to 8.5 cells for a mask 11 wide. Witness: lab/py/kernel-waves.
• 2026-09-08 [Refuted] One k* per mask across rules: box r = 5 spreads k* = 30..36 over the grid rules and seeds, one rule alone reading 32 at one seed and 36 at the other; the spread is up to two fifths (10 to 14 on carpet level 3) and never leaves the mask negative lobe on the four masks where the lobe law holds. Witness: lab/py/kernel-waves.
• 2026-09-19 [Verified] The FFT count step on a power-of-two torus agrees cell for cell with the direct neighbour count under a wrap boundary, on the Moore mask and the level-2 carpet mask over 8 steps. Witness: mrlydemo::chladni against mrlymath::life::next_grid.
• 2026-09-19 [Verified] Under the wide-survive window rule, the Bugs birth with S[0.28, 0.60], from a density-0.5 soup on a 256 torus, the three masks of 512 cells or more in the sweep, the box r = 13 at 728 cells and code 7 at levels 3 and 4, freeze the soup unchanged. Witness: lab/py/kernel-waves.