fractal-array-fragility

• Builds the fractal sparse array F_r = {sum_(i<r) g_i b^i : g_i in G} of a generator G with hole-free coarray [-a, a], at the base M = 2a + 1 of Cohen and Eldar and at every compressed base a < b <= 2a + 1.
• Tests essentialness by the definition: remove the sensor, recompute the difference coarray, compare. A fast test by pair counts (a lag of weight 1 kills both ends, a lag of weight 2 kills the sensor its two pairs share) is checked equal to the definition on every array up to 250 sensors before it is used alone; dial builds the weight level by level as the convolution of w_G stretched by b^(r-1), checked against the counted weight up to 250 sensors.
• exact: every hole-free generator with L <= 6 and span a <= 13, one of each mirror pair, 119 generators, at r = 2, 3; checks the essential set of F_r against the digit rule of Theorem A (every digit in U(G)), checks the tightness criterion e_r = card E(G)^r iff E(G) = U(G), asserts the coarray hole-free, and lists the loose generators with card E(G), u and whether G is maximally economic.
• dial: the generators {0, 1, 2}, {0, 1, 4, 6}, {0, 1, 2, 3, 7}, {0, 1, 4, 7, 9}, {0, 1, 2, 3, 7, 11} at every base a < b <= 2a + 1, e_r for every r with at most 17000 sensors and 8 million lags, the hole-free coarray [-A, A], A = a(b^r - 1)/(b - 1), asserted on every row, and the affine law e_(r+1) = lambda e_r + c the row obeys from r = 2 or 3, or u^r; lambda >= 1 and c are fitted on two steps, taking lambda = 1 on a flat tail, no upper bound, and the line prints how many further steps check the law.

RUN

• uv run python research/lab/py/fractal-array-fragility/fragility.py exact from mrlyprod/; about 21 seconds.
• uv run python research/lab/py/fractal-array-fragility/fragility.py dial; about 21 seconds.
• Standard library and numpy; no argument runs both verbs.

WITNESSES

• arrays, Fragility is exact - Theorem A against the definition on 238 cases with 0 mismatches, and {0, 1, 2} at 4 of 9 and 8 of 27: verb exact.
• arrays, The two bounds - the tightness criterion with 0 failures (a consequence of the Theorem A check), {0, 1, 2, 3} at 4 of 16, the 21 loose generators, 14 of them maximally economic, and the excess card E(G) - u = 1: verb exact.
• arrays, The compressed base - the hole-free coarray, the table of e_r, the affine fits with their checks, and 6561 sensors with 87381 lags for {0, 1, 2} at base 4: verb dial.