Odd weight and the dimension of an abelian bicycle code: k ≠ 2, k=4 forces 3 | Exp G, and k=6 forces 7 | Exp G

The model is checked, not assumed: CSS orthogonality H_X H_Z^T = 0 entrywise on all eight codes of the source's Table 1; the literal check-matrix k = 2N - rank H_X - rank H_Z agrees with 2 dim F₂[G]/(A,B), and rank H_X = rank H_Z, on all 196196 normalised weight-3 pairs of eight tori; and normalising both A and B to contain the identity preserves the whole k-value set, against the full unnormalised sweep on five tori (396202 pairs)

All eight rows of arXiv:2603.17703v1 Table 1 – the two BB reference codes, the three k = 6 ITB codes, the [[54,8,6]] code and the two even-weight self-dual codes – reproduce their printed k (12, 12, 6, 6, 6, 8, 14, 20) from the paper's own definition of H_X, H_Z and k, with rank H_X = rank H_Z on every row

The exhaustive weight-3 census of 30 finite abelian groups up to N = 56: for each group the exact set of attained k > 0, over all 9959476 normalised pairs (standing for 2770442265 unnormalised ones), with the exponent column checked against lcm(dims)

On all 30 swept groups: k = 6 occurs only where 7 | exp G and k = 4 only where 3 | exp G, and k = 2 occurs nowhere; both converses are FALSE, on exactly one group each – Z₂ x Z₂ x Z₂ x Z₇ is seven-divisible and attains no k = 6 while Z₂ x Z₄ x Z₇ and Z₈ x Z₇ of the same order 56 do, and Z₂ x Z₂ x Z₆ is three-divisible and attains no k = 4. Sharper: every attained k lies in the set the odd-weight mechanism predicts from the group's 2-cyclotomic class degrees alone – and that prediction, computed with no code in it, gives all three statements as equivalences, so it says which k are allowed and not which are attained

The cyclic case of the source's open sentence, kernel-clean and for every n at once: a divisor g of Xⁿ - 1 over F₂ with g(1)!= 0 never has degree 1, has degree 2 only if 3 | n, and has degree 3 only if 7 | n – i.e. through k = 2 deg gcd(a, b, Xⁿ - 1) a cyclic bicycle code whose A has odd weight never has k = 2, has k = 4 only if 3 | n, and has k = 6 only if 7 | n

The finite core of the general abelian argument, replacing Artinian structure theory by enumeration: over all commutative unital associative F₂-algebra structures on F₂ᵈ with a distinguished basis starting at 1 (1, 4 and 64 of them for d = 1, 2, 3), those admitting no F₂-algebra homomorphism to F₂ number 0, 1 and 8; the one at d = 2 has every element other than 0 and 1 of multiplicative order exactly 3, and each of the eight at d = 3 has order exactly 7

For every finite abelian G and all A, B in F₂[G] with A of odd weight, the bicycle code H_X = (A | B), H_Z = (B^T | A^T) satisfies: k!= 2; k = 4 forces 3 | exp G; and k = 6 forces 7 | exp G. The last is the sentence arXiv:2603.17703v1 prints as unproven, for every abelian G and every odd weight rather than weight-3 pairs on a 3-torus

Negative controls: the odd-weight hypothesis is neither vacuous nor removable (on the 2-group Z₄³ every weight-3 pair gives k = 0 while the source's own weight-4 code there has k = 20; on Z₃³ every weight-3 pair gives 4 | k while the source's weight-4 code there has k = 14, an odd quotient dimension), the k = 6 question is not vacuous (Z₇ attains it) and not trivial (Z₂ x Z₃ x Z₅ does not), the predicted set is strictly larger than the census on 27 of the 30 groups, each of the two dichotomies fails as an equivalence on exactly one group, and the pointlessness hypothesis of BOK6 cannot be dropped