The Number of Distinct Terms of the q-Derivatives of y₀ in the Grammar G_Sec

Guo-Niu Han, Kathy Q. Ji and Huan Xiong, q-Derivative Grammar, arXiv:2604.23959v2, Appendix III, Conjecture III.7. The grammar G_Sec has rules x_j ↦ q^j(1 + x_j x_{j+1}) and y_j ↦ q^j y_j x_{j+1}, and its order sorts the variables of a word by descending index, x before y at equal index. The q-derivative is D(w₁⋯wₙ) = Σ_j ρ(w₁⋯w_{j−1} R(w_j) ↑(w_{j+1}⋯wₙ)), where ↑ raises every index by one, extended linearly. With Ω(E) the number of distinct words of E with nonzero coefficient, the conjecture states Ω(Dⁿ(y₀)) = 1 for n = 1, 3 for n = 2, (20k³ + 33k² + k − 6)/6 for n = 2k + 1 with k ≥ 1, and (20k − 17)(k + 1)k/6 for n = 2k with k ≥ 2.

The theorem D5/S3/Combinatorics/QGrammar/SecGrammar.result establishes the formula for every n ≥ 1.

Pre-registration issue 12835 records the literature screen: the paper lists the initial values and states the formula as a conjecture, and its citing paper arXiv:2607.17130 does not treat the grammar G_Sec. This is a bounded negative finding.

1. Every coefficient of Dⁿ(y₀) is a nonzero polynomial in q with nonnegative coefficients, so no cancellation occurs and the support of the next derivative is the set of normalized successors of the support.
2. Every word of Dⁿ(y₀) contains exactly one y, and its indices lie in a strip of width at most one; such words are encoded by small integer tuples in two families.
3. The derivative acts on these tuples by explicit transitions; the set of tuples reachable at step n is an explicit region, established by an invariant for inclusion and explicit predecessors for every tuple of the region.
4. Counting the region gives the stated quasipolynomial.

The statement would fail if two different derivative histories cancelled a word, or if some tuple of the region had no predecessor at the previous step.

An independent referee implementation computed the supports from the definitions for every n ≤ 26 and checked the region in both directions and every transition on the computed data; the counts at n = 25, 26 are 6553 and 7371.

theorem; the statement is Conjecture III.7 of arXiv:2604.23959v2, quantified over every n ≥ 1.

• Proved (formalized): the support of Dⁿ(y₀) is exactly the region described above, and its size is the stated quasipolynomial.
• Open: the companion sequence for the grammar G_Sec′ of the same appendix, which the paper records as data without a conjectured formula.

The literature screen is limited to the arXiv record, its citing paper arXiv:2607.17130, web and GitHub searches and the repository checks.