The first prime beyond Sun's table violates both sign characterizations in part (ii).

Part (ii) of Zhi-Wei Sun's Conjecture 4.6 is false as stated at p = 29, where both predicted signs fail simultaneously. Exact computations across four algorithmically independent layers and five implementations certify the two integer values; the result leaves the conjecture's divisibility assertion in part (i) untouched.

Conjecture 4.6 appears in Arithmetic properties of some permanents (arXiv:2108.07723v7). It concerns the integer-normalized sine and cosecant permanents s_n and s′_p defined in Theorem 1.6: part (i) gives a divisibility assertion for odd composite n, while part (ii) proposes two sign characterizations for odd primes.

The conjecture was posed in 2021, remained unchanged through v7 in 2022, and that version was still latest on 12 June 2026. Sun's table stops at p = 23, and neither sequence is in OEIS. The author was notified privately.

The frozen statement

The TeX source at paper.tex, line 1109, states: “(i) If n>1 is odd and composite, then s_n ≡ 0 (mod n). (ii) Let p be an odd prime. Then s_p < 0 ⇔ p ≡ 5 (mod 12), and s′_p < 0 ⇔ p ≡ 7 (mod 8).”

The result

At p = 29, s_29 = 1,053,859 > 0 although 29 ≡ 5 (mod 12), while s′_29 = −4,806,838,304 < 0 although 29 ≡ 5 (mod 8) ≠ 7. Thus both clauses of part (ii) fail as stated.

Further failures occur at p = 41 and 53 for the mod-12 clause and at p = 61 for the mod-8 clause. Every computed value satisfies Sun's proved congruences s_p ≡ (−1)^((p+1)/2) and s′_p ≡ 1 (mod p).

Artifacts

Limitations

Part (i) is not refuted: it was tested for all 15 odd composite n ≤ 65 and held in every case, and it remains open. No replacement sign law is claimed; the certified range ends at p ≤ 61 for primes and n ≤ 65 for odd composites.

The result has not been externally refereed. Corrections will be logged, dated, and never silent.