Circular sorting in the alternating group: h(99) = 49 by an explicit witness, a (-1)-witness for 33 (hence h(33k) = (33k-1)/2 for every odd k), and the by-sign census of witness-type permutations of Zₙ for n <= 9

h(99) = 49: an explicit witness for n = 99, the smallest COMPOSITE integer whose circular sorting number arXiv:2608.06338v1 names as unknown. The permutation is checked against Definition 5.1 directly – it is a permutation of Z₉9, it is even, and each of its 99 rotations cᵏ pi has exactly one fixed point with every other point in a cycle of even length – and h([pi]) = 49 is computed from the Herzog-Reid cycle formula over all 99 rotations. With h(n) <= (n-1)/2 for odd n (source Lemma 2.1(2)) this is h(99) = 49, confirming Conjecture 2.10 at n = 99

A (-1)-witness for n = 33: an explicit witness-type permutation of Z₃3 of sign -1 (cycle type 1+4+6+6+8+8). Together with the witness x |-> -x for 33 (33 = 1 mod 4) this puts BOTH signs in S₃3, which is the hypothesis of the source's Corollary 5.3(2) and therefore gives h(33k) = (33k-1)/2 for EVERY odd k – an infinite family whose previously open members include n = 99 = 3²*11 and n = 3*11ᵇ for even b >= 2 (363, 43923,...)

The by-sign census of witness-type permutations of Zₙ, exhaustive over the full symmetric group for n = 3, 5, 7, 9: 3 (0 even / 3 odd), 15 (5/10), 119 (0/119), 81 (81/0). The two zeros are the content: S₇ contains no witness (so h(7) = 2) and S₉ contains no (-1)-witness, which is exactly what closes the 9 x 11 route to a witness for 99 and forces the 3 x 33 route

h(3) = 0, h(5) = 2, h(7) = 2, h(9) = 4 recomputed from Definition 1.1 by brute force over the whole alternating group, together with the distribution of h([pi]) over A₇ and A₉ (A₇: 7/490/2023 at h = 0/1/2; A₉: 9/1512/57609/122229/81 at h = 0..4)

The source's own published witnesses re-verified against Definition 5.1: Example 4.5 (n = 19, from K = <8> <= Z₁9ˣ), Example 5.6 (n = 31, whose verification the source leaves to the reader), and the worked Example 1.2 (h([pi]) = 2 for pi = (0,1,4,5,6,3,2) in A₇)

A second, independent implementation of [FSW, Definition 5.1] with the bit width a parameter, checkable at n ≈ 1000. The permutation test is O(n) (bitmask of the images) instead of O(n²); the coset check is split by rotation range (goodFrom / goodFromₐdd / goodFromₛplit) so no single kernel computation exceeds 50,000 rotation-points; and h(cᵏ π) = (n-1)/2 is derived from witness-type by arithmetic on the cycle census (Cen.hₑq) rather than recomputed, giving the stronger statement that *every* element of the coset needs (n-1)/2 three-cycles. Validated against the landed 7-bit model on w19, w31 (the source's Examples 4.5 and 5.6), w33 and w99 — including a re-derivation of w99's witness-type from two half-ranges that lands on the identical statement — plus four refutation controls (= false) and two non-vacuity controls

h(251) = 125, h(311) = 155, h(431) = 215, h(491) = 245 — four primes ≡ 3 (mod 4) whose 3-cycle sorting number in Aₙ arXiv:2608.06338v1 leaves open. Each witness is the generalized cyclotomic map of index 5 on Zₚ (π(0) = 0, π(x) = aᵢ x on the cosets of K = ⟨g⁵⟩), verified in the kernel against Definition 5.1 over all p rotations: a permutation of Zₚ, even, and every rotation with a unique fixed point and all other points in even cycles; and every element of the coset needs exactly (p-1)/2 three-cycles by the Herzog–Reid formula. With h(n) ≤ (n-1)/2 for odd n [FSW, Lemma 2.1(2)] this is h(p) = (p-1)/2, confirming Conjecture 2.10 at four more primes. x ↦ -x is separately checked to be a (-1)-witness for 251, so [FSW, Corollary 5.3(2)] gives h(251k) = (251k-1)/2 for every odd k

h(911) = 455 and h(971) = 485, by the same index-5 construction at 10 bits per entry (9,710-bit packed Nat) and 16/18 rotation ranges. With CS7 this closes the index-5 class below 1000: every prime p ≡ 3 (mod 4) below 1000 with 5 | (p-1)/2 now has h(p) = (p-1)/2 — the index-5 analogue of what the source's two theorem lists do for index 3. What remains open below 1000 is the index-7/11/13/17 cases 239, 419, 443, 599, 647, 659, 683, 743, 827, 947 and the 21 primes with (p-1)/2 prime, where no index is available at all [2026-08-28 CORRECTION: the class statement holds for 13 <= p < 1000 only; p = 11 has h(11) = 4]