First-return defect for the same actual quotients
Abstract
The first signed word-return defect separates two actual markings of one finite quotient group.
Theorem 1.1 (Equal short signed word fibers and the first two extra returns).
Lean statement: D5/S3/FiniteGroups/BajnokBeattyFirstReturnDefect.first_return_defect
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/BajnokBeattyFirstReturnDefect.first_return_defect (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. B. Bajnok and W. K. Beatty (2024). On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups. URL: https://arxiv.org/abs/2406.04045v1.
Commentary.
Let p and t be integers with p >= 3 odd and t >= 2. Put m = 6t - 1, D = (m^2 - 9)/2, and n = p(m - 3). The two quotient maps are the actual maps from Z^2 onto the quotient groups by the lattices generated by (p(m-1)/2,p(m+1)/2),(p(m-9)/2,-p(m+9)/2) and by (p(m-3)/2,p(m+3)/2),(p(m-3)/2,-p(m+3)/2). The previously established coordinate equivalences identify both quotients with H = Z/pZ x Z/(pD)Z; the first rows are (-1,1) and (t,1-t), and the second rows are (-3t,3t-1) and (-3t-1,3t-2).
A word of length l is a function from Fin l to the four signed coordinate steps (+1,0),(-1,0),(0,+1),(0,-1). Its endpoint is the sum of its steps in Z^2. W19(l) and W33(l) are the cardinalities of the actual endpoint fibers that return to zero under the two quotient maps. The statement concerns these fibers directly; it does not replace them by a ball-cardinality or covering hypothesis.
If a quotient return has endpoint z, the first residue coordinate and the second residue coordinate reduced modulo p show that both coordinates of z are divisible by p. Writing z = p(X,Y), the second residue gives D dividing the corresponding integer row. For the (1,9) quotient the row is -3tX+(3t-1)Y. Its absolute value is at most 3t(|X|+|Y|). A word of length l has Lee norm at most l, so for l < n this is strictly below D = (3t+1)(m-3). The row therefore vanishes. Its coprime coefficients force X=(3t-1)k and Y=3tk; the Lee norm is m|k|, and |X|+|Y| <= m-3 forces k=0.
For the (3,3) quotient the row is -(3t+1)X+(3t-2)Y. At Lee norm at most m-3 its absolute value is at most D, with the sharper estimate D-3|Y|. If the row is zero, the determinant-one inverse row gives k=tX+(1-t)Y and the solutions X=(3t-2)k, Y=(3t+1)k, again costing m|k|. If its absolute value is D, the sharper estimate forces Y=0 and then X = plus or minus (m-3). Thus below n both kernels contain only zero, while at n the second kernel contains exactly 0, (n,0), and (-n,0), and the first still contains only zero.
The Lee norm bound for a word endpoint is obtained by summing the absolute values of its two coordinate sums. Equality at (n,0) forces every horizontal coordinate step to be +1, hence the unique constant positive-horizontal word. Equality at (-n,0) forces every horizontal coordinate step to be -1, hence the unique constant negative-horizontal word. These words are distinct and neither belongs to the zero-endpoint fiber for the first quotient.
For every l<n, the identical zero kernels give an explicit identity equivalence between the two return-word subtypes, so W19(l)=W33(l). At l=n, partitioning the second return subtype into its zero endpoint fiber and the two extremal endpoint fibers gives an equivalence with the disjoint sum of the first return subtype and Fin 2. Consequently W33(n)=W19(n)+2. The proof uses the actual quotient maps and the previously compiled coordinate equivalences, without assuming a short-kernel classification or any covering statement.
For p=3,t=2, the first difference is at n=24; for p=5,t=2, it is at n=40. In both cases the return counts agree at every smaller length and differ by exactly two at the displayed length. These complete-telescope applications also give strict inequality. The result is a first-return statement for the signed four-step word model. It does not assert a graph adjacency theorem, a spectral consequence, noncospectrality, or a literature novelty claim. The cited Bajnok–Beatty source supplies the Cayley-diameter context; the kernel classification and the two unique extremal word fibers are the repository derivation.