Exceptional primes for Katre's MDS conjecture on the Jacobi-sum matrix D, and a rank-drop criterion free of Jacobi sums

Compute-first gate: the source's own printed data rebuilt from our definitions – the worked p = 61 example (J(1,1)₅ = -6 zeta² + 3 zeta³ + 2 zeta⁴, G = Dᵗ = [[9,1,0,7],[2,3,55,0]], Y = [1,53,59,59]), the general l = 5 matrix and its equations (I)-(IV) over every conjugate of every p = 1 mod 5 up to 400, the general l = 3 matrix and its two congruences up to 300, Remark 5.1's conjugate permutation N^((bᵗ))ᵢ = N^((b))ₜᵢ for l = 5, 7 up to p = 150, and the consistency identity D*(b, b²,..., bᵐ) + Y = 0 mod p for l = 3, 5, 7, 11 up to p = 150

The source's l = 3 and l = 5 theorems as a finite census: every conjugate of every prime p = 1 mod l with p <= 600 gives an MDS code, no exceptional prime (kernel-clean); the same for p < 10⁵ in C

Negative controls: a too-large claim ('every conjugate at every p = 1 mod 13 is MDS') is refuted by isMDS 79 13 (Dmat 79 13 62) = false; a too-small claim ('the prime 79 fails outright') is refuted by ten of the twelve conjugates having full rank and b = 18 being MDS; the MDS test is sensitive to the data in both directions (a single-entry perturbation of the exceptional matrix restores rank 6 at two different positions; adding 1 to one entry of a GOOD matrix creates seven singular minors and breaks MDS – 71 of the 72 single-entry perturbations do); two different 6-subsets of rows at a good conjugate are nonsingular; and the hypothesis is not vacuous (62 is a primitive 13th root of unity mod 79, 1 is not, there are exactly twelve)

The source's reported (p, l) = (79, 13) exception, certified with data the source does not give: exactly two of the twelve conjugates fail, b = 62 and b = 65 (an inverse pair, 62*65 = 1 mod 79), and each fails by a rank collapse – rank D = 5 < 6, so all C(12,6) = 924 maximal minors are singular (enumerated in the kernel), with an explicit nonzero right kernel vector of D and an explicit nonzero left kernel vector of its first six rows

Two exceptional pairs not in the source, certified kernel-clean: (p, l) = (277, 23) with rank D = 10 < 11 at b = 164, 201, and (p, l) = (659, 47) with rank D = 22 < 23 at b = 14, 612; in both cases exactly two of the l-1 conjugates fail and they are an inverse pair, and the rank certificate is what makes the claim checkable at all (C(22,11) = 705,432 and C(46,23) = 8.2*10¹2 minors)

Eleven further exceptional pairs, certified by native_decide: (3067, 73), (12547, 41), (14447, 31); (14551, 97), (21727, 71), (73877, 73); and (107, 53), (167, 83), (293, 73), (809, 101), (593141, 47). In each the rank of the (l-1) x (l-1)/2 matrix D is exactly one short of full at exactly two conjugates b, b⁻¹, with explicit right and left kernel vectors. With KM4 and KM5 this certifies ALL fourteen exceptional pairs found anywhere in the family

Katre's conjecture verified for the next three orders by exhaustive minor enumeration: l = 7 for every prime p = 1 mod 7 with p <= 5000 (110 primes, 660 conjugates, C(6,3) = 20 minors each), l = 11 for p <= 3000 (42 primes, 420 conjugates, 252 minors each), and l = 13 for p <= 3000 (35 primes, 420 conjugates, 924 minors each) where p = 79 is the only exceptional prime; with the consistency control over the same windows

The rank-drop criterion agrees with the Jacobi-sum rank on every conjugate of every prime in a census window: rankDrops l B = deltaZeros l B for l = 7, 11, 13 with p <= 2000 and for l = 17, 19, 23, 29, 31 with p <= 1000, where the left side is computed from the Jacobi sum J(1,1)ₗ and the right side from a determinant in (p, l, b) alone; both sides are nonempty (they find (79, 13) and (277, 23) respectively)

The complete exceptional-pair list by the criterion for p <= 20000: for 3 <= l <= 23 exactly (79, 13) and (277, 23), each at exactly the two conjugates b, b⁻¹; for 29 <= l <= 47 exactly (14447, 31), (12547, 41) and (659, 47), likewise

Theorem (prose, journal section 3.2): for q = p, rank_(Fₚ) D < (l-1)/2 if and only if Delta(b) = 0, where S = k⁻¹ mod l: 1 <= k <= (l-1)/2 is a half-system mod l, Fⱼ(T) = prod_(u in S) (T - bʲᵘ) in Fₚ[T], and Delta(b) = det[[Tⁱ] Fⱼ(T)]_(1 <= j, i <= (l-1)/2). Delta involves no Jacobi sum: it is a function of (p, l, b) alone, computable in O(l³) against O(p log p) for one Jacobi sum

The wide C censuses, beyond what fits in the kernel: exhaustive minor enumeration finds (79, 13) as the ONLY MDS failure for l = 7, 11, 13 over every prime p < 10⁶ (27,460 pairs, 235,426 conjugates) and no failure at all for l = 17, 19 over every p < 10⁵ (1,128 pairs, 19,098 conjugates) or for l = 3, 5 over every p < 10⁵; and the criterion sweep is complete for 3 <= l <= 47 with p < 10⁷ (950,317 pairs, 12 rank drops = 6 exceptional pairs including (593141, 47)), for 53 <= l <= 101 with p < 3*10⁵ (3,981 pairs, 16 rank drops = 8 pairs), and – unified – for 3 <= l <= 101 with p < 10⁶ (124,131 pairs, 1,954,784 conjugates, 28 rank drops = exactly the same 14 pairs and no fifteenth). Fourteen exceptional pairs in all, thirteen of them not in the source. In every single failure the rank drops by exactly one and exactly the inverse pair b, b⁻¹ fails; no failure at full rank has ever been observed

Heuristic with measured agreement: since Delta(b) is a single determinant in Fₚ, a conjugate should fail with probability about 1/p, so the expected number of failing conjugates with p <= N is sum_(p = 1 mod l, p <= N) (l-1)/p, which is about log log N + Cₗ and DIVERGES. Measured on three sweeps: 21.2 predicted against 12 observed for l <= 47 and p < 10⁷ (-1.4 sigma), 10.7 against 16 for 53 <= l <= 101 and p < 3*10⁵ (+1.2 sigma), and 30.70 against 28 for the unified window 3 <= l <= 101, p < 10⁶ (-0.4 sigma, the largest and the only uniform window). If the heuristic is right, Katre's 'except possibly for finitely many primes p' is false for every l >= 7, though only just: one expects about one new exceptional prime per l per factor eᵉ in N