Dependent bases

• 2026-09-14 [Proved] The collapse theorem: for one root r >= 2, bases base_i = r^(e_i) with i = 1..m, any dimension dim >= 1 and any digit sets A_i inside {0,...,base_i - 1}^dim each containing 0, put M = lcm(e_1,...,e_m) and B = r^M; then cap_i F(base_i, A_i) = F(B, A) exactly, where A = (cap_i F(base_i, A_i)) cap [0, B)^dim is the joint set's own bottom block, the count law is exact at card {n in [0, B^level)^dim : n in F(B, A)} = (card A)^level for every level >= 0, and the attractor of {x -> (x + a)/B : a in A} has Hausdorff and box dimension log card A / (M log r). Proof in four steps: e_i dividing M makes every e_i-group of base-r positions sit inside one M-group, so no group straddles and lcm is forced rather than chosen; each digit at base base_i is then a function of one base-B digit, so the constraint is per base-B digit with nothing carried between them; the top base-B digit is read with its full M / e_i base-base_i digits while the base-base_i expansion of n stops at its top nonzero digit, so the two readings agree exactly when 0 in A_i; and membership one digit at a time gives the count with no error term while the maps send the unit cube to boxes with disjoint interiors, so the open set condition gives the dimension. No step uses dim = 1 and no step asks A_i to be a product across the dim axes, so dim >= 2 is covered with no extra hypothesis, compound designs included. Across two dependence classes the theorem says nothing. Witness: lab/py/base-collapse verbs blocks, dim, check; bases.md:92, bases.md:101-107.
• 2026-09-14 [Proved] Multiplicative dependence is an equivalence relation on bases >= 2 and each class is the set of integer powers of its least member: reading a base by its vector of prime exponents turns p^a = q^base into parallel vectors, so a class is the set of integer points on one ray through the origin, its least member r is the primitive vector on that ray and every member is r^e for one integer e >= 1; transitivity is one line, p^a = q^base and q^c = s^d give p^(ac) = s^(bd). The half of the rule that follows is collapse first: partition the bases into dependence classes and replace each class by the single design the collapse theorem gives it, with its own base r^M, its own digit set and its exact dimension; a transversality budget inside a class is not allowed, by the Refuted row below. Witness: lab/py/base-collapse verb blocks; bases.md:88, bases.md:126.
• 2026-09-14 [Verified] Four dependent cells rebuilt from their digit sets alone, root, exponents, M, block set and exact dimension: base 4 on {0,1} with base 8 on {0,1,2,3} gives r = 2, M = 6, A = {0,1,16,17} and dimension exactly 1/3; base 4 on {0,1} with base 16 on {0,1,4,5} gives M = 4, A = {0,1,4,5} and exactly 1/2; base 9 on {0,1,2} with base 27 on {0,...,8} gives r = 3, M = 6, nine blocks and exactly 1/3; bases 4, 8, 16 on {0,1}, {0,1,2,3}, {0,...,7} give M = 12, sixteen blocks and exactly 1/3. The block set is built twice per cell and asserted equal, once by sieving all r^M base-r words against the per-group constraint and once by testing every integer below r^M for membership in each original design. Witness: lab/py/base-collapse verbs blocks, dim; bases.md:115-120.
• 2026-09-14 [Verified] The exact count law survives brute force to 10^13 on all four cells, three checks each: every element of F(B, A) below 10^13 passes a digit test in each original base; the joint count from enumerating the lowest-dimension original design (2^22 - 1 elements at three cells, 3^14 - 1 at the base-9 cell) and filtering it by the others equals the count of F(B, A) below 10^13, which with the first check gives set equality, the four counts reading 32767, 4194303, 19682, 32767; and the joint count below B^level is exactly (card A)^level at every level with B^level <= 10^13, reaching level = 7 and 16384, level = 10 and 1048576, level = 4 and 6561, level = 3 and 4096. Witness: lab/py/base-collapse verb check; bases.md:124.
• 2026-09-14 [Verified] A dependence class can give a block count that is not a power of its root, so the exact dimension can be irrational: base 4 on {0,1,2} with base 16 on the full digit set gives r = 2, exponents 2, 4, M = 4 and card A = 9, dimension log_2(9) / 4 = log_2(3) / 2. The dimension is log_r(card A) / M always, and for the least base r of the class it is rational exactly when card A is a power of r, since a primitive r is no perfect power. Witness: lab/py/base-collapse verb dim cell I; bases.md:107.
• 2026-09-14 [Verified] The hypothesis 0 in A_i is sharp and not decoration: at r = 2 with base_1 = 2 on A_1 = {1} and base_2 = 4 on the full digit set, M = 2 and the bottom block is A = {0, 1, 3}, yet F(4, A) holds 4, 5, 12 and 13, whose base-2 words carry a digit outside A_1, so the collapse strictly over-counts the joint set. Witness: lab/py/base-collapse verb blocks; bases.md:105.
• 2026-09-14 [Conjecture] Budgeting across the collapsed classes alone, with m the number of classes and not the number of bases, is the right second step: nothing here shows a collapsed class behaves in a cross-class budget like an ordinary design of the same dimension, so the step is assumed. Witness: bases.md:126; no proof and no lab verb.
• 2026-09-14 [Refuted] The naive budget sum_i dim A_i - (m - 1) is not an upper bound on a multiplicatively dependent cell and is low on every one tested: base 4 on {0,1} with base 8 on {0,1,2,3} has exact dimension 1/3 against a budget of 1/2 + 2/3 - 1 = 1/6; base 4 on {0,1} with base 16 on {0,1,4,5} has exact dimension 1/2 against a budget of 1/2 + 1/2 - 1 = 0; base 9 on {0,1,2} with base 27 on {0,...,8} reads 1/3 against 1/6; and the three-base cell 4, 8, 16 reads 1/3 against max(0, 1/2 + 2/3 + 3/4 - 2) = 0, the budget being read at zero because no dimension is negative and the raw sum there is -1/12. The second cell is the clean failure: F(16, {0,1,4,5}) = F(4, {0,1}) as sets, since a base-16 digit lies in {0,1,4,5} exactly when both of its base-4 digits lie in {0,1}, so the budget prices the intersection of a set with itself at dimension 0 while the truth is that set, of dimension 1/2. A budget built on transversality cannot be applied inside a dependence class. Witness: lab/py/base-collapse verbs dim, check; bases.md:113, bases.md:115-120.