Quadratic carriers with a cyclic signed quotient
Abstract
Signed odd residue classes realize the complete original support family.
Theorem 1.1 (Even padding supplies cyclic carriers at every odd order).
Lean statement: D5/S3/Combinatorics/Sequenceability/OriginalSignedCyclicQuotientSharpness.sharpness
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Sequenceability/OriginalSignedCyclicQuotientSharpness.sharpness (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every natural s at least one and every natural padding parameter k, put m = 2s+1 and n = 4m-6+2k. The carrier G = ZMod m times ZMod n is finite and has exactly m(4m-6+2k) elements. For every original coordinate subset C, nonemptiness together with vanishing of its label sum is equivalent to C being precisely one original triple {q,a_i,b_i}. The q label has additive order exactly m. The same theorem supplies cyclicity of G whenever m and n are coprime. It assumes no label injectivity, nonzero-label condition, selected-support restriction or zero-support classification.
Retain the original I = Fin(m-1) and Coordinate = Option I summed with Option I. For i from zero through m-2, set A_i = 2i+1 in ZMod n. Label q by (1,0), p by (-1,2), a_i by (s,A_i), and b_i by (s,-A_i). The A_i run through the positive odd representatives up to (4m-6)/2, which is at most n/2. They are distinct, and A_i = -A_j implies i = j: equality of their positive representatives or complementary representatives gives this directly. At k = zero the final representative n/2 is self-negative; at positive k the representatives lie strictly below n/2. The proof allows the self-negative endpoint.
For an arbitrary original coordinate subset C, let U be the indices whose a-coordinate is selected and V those whose b-coordinate is selected. These sets may overlap. Write epsilon for the membership indicator of q and delta for that of p. Reduction of the second coordinate modulo two sends each selected a_i or b_i to one and p to zero. A zero sum therefore forces card(U)+card(V) = 2*r. Since each set has at most m-1 elements, zero is at most r and r is at most m-1.
The first coordinate of the label sum is epsilon-delta+s*(card(U)+card(V)). Since 2*s = -1 in ZMod m, it reduces to epsilon-delta-r. If p is absent, comparison of residues in the interval from zero through m-1 gives r = epsilon. Nonemptiness excludes r = epsilon = zero. Thus q is present and exactly two paired coordinates are selected. If both were a-coordinates or both b-coordinates, vanishing of the second coordinate would identify two different negation orbits. If one is a_i and the other b_j, vanishing forces A_i = A_j and hence i = j. The support is therefore one prescribed triple.
If p is present, the first-coordinate equation permits only r = zero with epsilon = one, or r = m-1 with epsilon = zero. The first case selects no paired coordinates. The second selects every a_i and every b_i, so their signed second coordinates cancel pairwise. In both cases the second coordinate of the sum is the nonzero class two in ZMod n; here n is at least six. Hence no zero support contains p. Conversely, every original triple has first-coordinate sum 1+2*s = m and second-coordinate sum A_i-A_i = zero. This proves the classification for all subsets of the full original coordinate set.
For coprime m and n the standard Chinese remainder equivalence identifies their product carrier with ZMod(mn). At k = zero, every common divisor of m and n divides six, so coprimality of m and six suffices. At k = one, n = 4(m-1), which is coprime to every odd m: m is coprime both to four and to m-1. This gives a cyclic carrier of size 4m(m-1) for every odd m at least three. The unpadded carrier, whenever m is coprime to six, is smaller by 2m elements. These cyclic carrier sizes improve the prior construction m(6*m-7); no abelian size improvement over affine carriers is asserted.
At k = zero the formula gives seventy elements at m = five and 154 at m = seven; both parameters are coprime to six. At k = one it gives a cyclic carrier of 16ss+8s elements for every s at least one, equivalently 4m*(m-1). The parameter-five value agrees with the separately established cyclic minimum. This theorem alone makes no optimality claim at m = seven or at other parameters. Its proof is uniform in s and uses no bounded support enumeration or standalone numerical instance. The construction is presented with repository provenance and without a literature-priority claim.