# 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](../notes/arrays.md), 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](../notes/arrays.md), 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](../notes/arrays.md), 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](../notes/arrays.md), 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](../notes/arrays.md), 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.