Zero patterns and unitary similarity: the cell An and Đoković could not compute, and their Conjecture 3.6 through n = 12

Compute-first gate: J_(k,n) really lies in P'ₙ for every 2 <= n <= 12 and is doubly laced at (10,5); and the two closed forms the source states or proves are reproduced from an independent implementation – <chiₙ,chiₙ> = n! and <chi_(Piₙ),chiₙ> = n!! for 2 <= n <= 10

Two independent implementations of <chi_I, chiₙ> agree on J_(k,n), NEₙ and Piₙ at every n <= 7: the unfolded sparse-polynomial definition and the block-closing dynamic program that carries n >= 8

Conjecture 3.6 reproduced at all 36 cells with 2 <= n <= 9 – the range the source reports verified – from an independent implementation

The cell the source's program ran out of memory on: <chi_(J_(5,10)), chi₁0> = -17640 = (-1)⁹ C(10,5) C(8,4), so Conjecture 3.6 holds at (n,k) = (10,5)

Consequences: Conjecture 3.6 now holds at every one of the 45 cells with n <= 10 with no exception; and <chi_(J_(5,10)), chi₁0>!= 0, so J_(5,10) is 10-nonsingular and hence 10-universal – every traceless complex 10x10 matrix is unitarily similar to one vanishing at all 45 of its positions

One row past the source's verified range: <chi_(J_(k,11)), chi₁1> = C(11,k) C(9,k-1) for every k in Z₁0, i.e. 11, 495, 5940, 27720, 58212, 58212, 27720, 5940, 495, 11 – Conjecture 3.6 holds at n = 11, and every J_(k,11) is 11-nonsingular hence 11-universal

The conjectured right-hand side is n times a Narayana number: (n-1) C(n,k) C(n-2,k-1) = n C(n-1,k-1) C(n-1,k) for 2 <= n <= 20, so the row at n is row n-1 of OEIS A118963 (Grand Dyck paths of semilength n-1 by double rises) and the row sums are the central binomial coefficients C(2n-2,n-1)

Negative controls: a strictly larger claim (C(n-1,k-1) in place of C(n-2,k-1)) and a strictly smaller one (C(n-3,k-1)) are both refuted at n = 9; the conjectured row is not a property of every pattern in P'ₙ (Pi₉ sits off it); and the (10,5) value is none of -17639, -17641 or +17640

Measurement: the block-closing DP's cost depends on the VERTEX ORDER by an order of magnitude – at (10,5) the identity order is the second worst of 65 relabellings (1377615 live states) and the best is 151208 (9.1x), and at n=12 a good relabelling takes the largest cell from 8.4e7 states / 8.40 GB / 351 s to 8.3e6 / 1.06 GB / 36 s, which is what brought n=12 inside the cap

Compute-first gate for the source's Problems 3.4 and 3.5: both hold over EVERY pattern of P'₄ (924) and P'₅ (184756) individually, enumerated as bitmasks with no multiset or symmetry reduction – the range the source reports verified

The multiset/Sₙ-orbit certificate returns the same answer as the reduction-free scan at n = 4 and n = 5, where both are affordable

Problems 3.4 and 3.5 hold at n = 6, the first case the source leaves open: <chi_I,chi₆> <= 720 and <chi_I,chi_I> >= 720 for every I in P'₆, with equality in either exactly when chi_I = +-chi₆

Negative controls for the n = 6 certificate: equality is attained exactly once on each side (so neither <= 719 nor >= 721 holds), the all-ones multiset is present in the list, dropping one representative breaks the covering count, the same codes are rejected at n = 5, and the same codes in reverse order are rejected

The source's relabelling identity <chi_(sigma(I)),chiₙ> = sgn(sigma) <chi_I,chiₙ> checked, not assumed: at every cell with 3 <= n <= 8 against all n rotations and the reversal (193 cell/permutation pairs), and at n = 10 against three permutations across all nine cells; and the relabelled pattern is still an element of P'₁2

Two rows past the source's range: <chi_(J_(k,12)), chi₁2> = -C(12,k) C(10,k-1) for every k in Z₁1, i.e. -12, -660, -9900, -59400, -166320, -232848, -166320, -59400, -9900, -660, -12 – Conjecture 3.6 holds at n = 12, and every J_(k,12) is 12-nonsingular hence 12-universal