Twelve-element countermodels to Wilkie's identity: rigidity, two errata, and the 2026 classification

the Hajdari-Niederhauser twelve-element algebra, decoded from the SAT assignment in the paper's own Zenodo record, satisfies all eleven High School Identities and its three tables are operations on the carrier

Wilkie's identity fails in that algebra at (4,5), with the two sides taking the values 11 and 12, and at no other pair of the twelve elements

the integer chain of that algebra is 1, 2, 3, 11 and then constant; the witness pair 4, 5 consists of non-integers and 4 does not divide 5

Table 1 of arXiv:2608.16406v1 is not an HSI-algebra: as printed it fails xʸ⁺ᶻ = xʸ * xᶻ at exactly 144 triples and (xʸ)ᶻ = x^(y*z) at exactly 35, 179 in all, while the other nine identities hold; the smallest witness is 7¹⁺¹ = 7 against 7*7 = 11

the + and. tables printed in Table 1 of arXiv:2608.16406v1 agree entry for entry with the paper's SAT assignment; the printed exp table differs from it in exactly thirteen entries – the whole of row 7 except its first entry, and the two entries exp(8,4) and exp(8,8) – where the paper prints 7 for 11 and 11 for 12

the repair is forced: at each of the thirteen misprinted cells, no other element of the carrier restores all eleven High School Identities, so Table 1 determines the algebra uniquely despite being wrong

the full rigidity profile: of the 1,584 ways to change a single entry of the exponentiation table of that algebra, exactly eighteen again give an HSI-algebra, and they sit at exactly nine of the 144 positions

the Burris-Yeats Figure 4 algebra of 2005 is an HSI-algebra with closed tables and integer chain 1, 2, 3, 4

Wilkie's identity fails in the Burris-Yeats algebra at the pair (a, e), with the two sides taking the values 4 and h, and at no other pair

the Hajdari-Niederhauser and Burris-Yeats twelve-element algebras are not isomorphic – no bijection of the carriers carries +,. and exp across

the multiplicative reducts of the two algebras ARE isomorphic, by an explicit permutation of the twelve elements, and that same permutation does not carry + across

both twelve-element algebras are instances of the classification template of arXiv:2608.08421v1 section 6, with explicit relabellings and parameter values (both at alpha-option 3 and gamma-option 2 with every delta equal to 12, differing in the beta- and eps-cells); both instances are Gurevic-Burris algebras failing Wilkie's identity only at (3,4)

the thirteen-element algebra of arXiv:2608.08421v1 Appendix A is an HSI-algebra with closed tables, 2 = 1+1, 2⁴ = 5, and Wilkie's identity failing at (4,5) and at no other pair

Gurevic's univariate identity – Wilkie's identity with y:= 2ˣ – fails in that thirteen-element algebra at exactly one point, and the sibling identity with y:= 3ˣ holds at every point of the same algebra

controls: Wilkie's identity is valid over the natural numbers throughout 1..5² while a one-off perturbation of the same expression fails at all 25 points; the one-element algebra satisfies HSI and satisfies Wilkie's identity; the integer chain 1,2,3 of the twelve-element algebra is not closed under +

the non-isomorphism check shipped with arXiv:2608.16406 (iso.c in Zenodo 10.5281/zenodo.18568303) compares the printed, non-HSI exp table against a structure that is not the Burris-Yeats algebra: its + table is Burris-Yeats' but its x table differs in eleven entries and its ^ table in twenty-four, and the result fails six of the eleven identities (H3, H4, H5, H7, H8, H9) at 1,701 triples and refutes Wilkie's identity nowhere

the nine free positions of the exponentiation table are exactly the delta- and eps-cells of the classification template; the eleven gamma-cells are rigid to single-entry change; and of the eighteen changes that keep HSI, thirteen leave a Gurevic-Burris algebra (still failing Wilkie only at (4,5)) while five leave a twelve-element HSI-algebra in which Wilkie's identity holds – exactly the five single-cell violations of the classification's eps-condition