Sparse arrays

• 2026-09-23 [Proved] For a generator G with hole-free coarray [-a, a], M = 2a + 1, every lag of the fractal array F_r has one balanced base-M expansion and its coarray weight is prod_i w_G(t_i), the digit product of the generator's weight. Witness: arrays, The weight is a product over digits.
• 2026-09-23 [Proved] Theorem A: for G hole-free on [-a, a], M = 2a + 1 and r >= 2, a sensor of F_r is essential exactly when every base-M digit lies in U(G), the sensors with a weight-1 difference partner, so the fragility of F_r is exactly (u/L)^r. Witness: arrays, Fragility is exact.
• 2026-09-23 [Verified] Theorem A holds against the remove-and-recompute definition on all 119 hole-free generators with L <= 6 and span a <= 13 at r = 2, 3, 238 cases, 0 mismatches. Witness: lab/py/fractal-array-fragility, verb exact.
• 2026-09-23 [Proved] For r >= 2, F_r is maximally economic exactly when G satisfies condition C1 of Cohen and Eldar, the converse of their Theorem 5, which fails at r = 1 on G = {0, 1, 2}; and exactly then their Theorem 6, fragility of F_r at most card E(G)/L, is attained, strictly loose otherwise, as {0, 1, 2, 3} shows at 1/4 against 1/2. Witness: arrays, The two bounds.
• 2026-09-23 [Proved] For r >= 2, the bound of Yang, Shen, Liu, Eldar and Cui, fragility of F_r at most (card E(G)/L)^r, is attained exactly when every essential sensor of G has a weight-1 difference partner, and is strictly loose otherwise. Witness: arrays, The two bounds.
• 2026-09-23 [Verified] Among the 119 hole-free generators with L <= 6 and span a <= 13, 21 make the bound (card E(G)/L)^r strictly loose and 14 of those are maximally economic, where both published bounds equal 1 while the fragility is (u/L)^r; every loose one has card E(G) = u + 1. Witness: lab/py/fractal-array-fragility, verb exact.
• 2026-09-23 [Proved] For a < b <= 2a + 1 the digit design with base b and digits G at level r has L^r sensors and the hole-free coarray [-A, A], A = a(b^r - 1)/(b - 1). Witness: arrays, The compressed base.
• 2026-09-23 [Verified] At compressed bases the essential count leaves the product law: {0, 1, 2} at base 4 has 2 essential sensors at every r from 2 to 8, {0, 1, 4, 6} at base 7 has 6 from r = 3 to 7 and at base 8 has 4, 11, 25, 53, 109, 221, 445, {0, 1, 2, 3, 7} at base 12 has 5, 17, 53, 161, 485, 1457, and {0, 1, 2, 3, 7, 11} at bases 12 to 14 has 6, 7, 7, 7, 7. Witness: lab/py/fractal-array-fragility, verb dial.
• 2026-09-23 [Conjecture] For the five dial generators {0, 1, 2}, {0, 1, 4, 6}, {0, 1, 2, 3, 7}, {0, 1, 4, 7, 9} and {0, 1, 2, 3, 7, 11} at every base a < b <= 2a + 1, from r = 2 or 3 on, the essential count e_r(G, b) obeys e_(r+1) = lambda e_r + c with an integer lambda between 1 and u, three regimes, bounded, geometric and u^r, and the u^r regime starts below 2a + 1 for {0, 1, 4, 6}, {0, 1, 2, 3, 7}, {0, 1, 4, 7, 9} and {0, 1, 2, 3, 7, 11} at bases 9, 14, 14 and 20. Witness: lab/py/fractal-array-fragility, verb dial, which fits lambda >= 1 and c on 2 steps, taking lambda = 1 on a flat tail, no upper bound, and checks the law on 1 to 4 further steps a row.