When is the Asgarli–Ghioca–Yip curve blocking? A sharpness belief that fails at q=41, and the exact ranges for r ≤ 5
Abstract
Let r ≥ 2, let q ≡ 1 (mod r) be a prime power with p ∤ (r²-1), put m=(q-1)/r, and let k ∈ 𝔽_q^(*) be such that -k is not an r-th power. Asgarli, Ghioca and Yip proved that the plane curve -(ky+z)xᵐ+zyᵐ+kyzᵐ=0 is smooth and nontrivially blocking as soon as q>r⁴, and remarked on the sharpness of that hypothesis: for r=4 they wrote that they believe the conclusion holds for all q ≥ 37, and for r=5 that they believe it holds for all q ≥ 131. We show that the first belief is false and the second is true. At q=41 — a prime with 41 ≥ 37, 41 ≡ 1 (mod 4) and 41 ∤ 15 — the line x+y+z=0 carries none of the 104 points of the curve with k=2, and in fact the curve is blocking for no admissible k; q=41 is the only prime power of the range 37 ≤ q ≤ 256 admitted by the remaining hypotheses where this happens, so the sharp threshold at r=4 is q ≥ 49 and not q ≥ 37. At r=5 the conclusion does hold at every admitted prime power of the declared range 131 ≤ q ≤ 625, and indeed already from q ≥ 121. The mechanism is classical: the hypothesis q>r⁴ enters the proof only through a corollary asserting that -1 is a sum by+cz of prescribed power residues, and the failure of that corollary is the vanishing of a cyclotomic number; for r=4 the classical formula 16(0,0)₄=p-11-6a leaves exactly p=17 and p=41. We determine the exact set of prime powers where the corollary fails for every r ≤ 5. Every theorem and proposition below is machine-checked in Lean 4, with the scope limitation recorded in Remark [rem:scope]; the computations outside that development are labelled where they appear.
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Claim ledger
Stated results
BR1known2026-09-03
Corollary 2.2's conclusion at (q,m) implies that the curve −(ky+z)xᵐ + zyᵐ + kyzᵐ = 0 of Theorem 6.3 of arXiv:2208.13299v2 is blocking, for every k and with no hypothesis on m — the source's Proposition 6.1 with q > r⁴ replaced by the statement it is used to reach
BR2correction2026-09-03
At q = 41, r = 4, k = 2 the F₄1-line x + y + z = 0 carries none of the 104 F₄1-points of the curve of Theorem 6.3, so that curve is not blocking — although 41 is prime, 41 ≥ 37, 41 ≡ 1 (mod 4) and 41 ∤ 15
BR3correction2026-09-03
The r = 4 belief of Remark 6.4 of arXiv:2208.13299v2, "We believe that the conclusion holds for all q ≥ 37", is false: its consequence for prime q fails at q = 41
BR4known2026-09-03
The source's own datum reproduced: at q = 29, r = 4, k = 1 (admissible, −1 not a fourth power mod 29) the line x + y + z = 0 misses the curve, so the conclusion of Theorem 6.3 fails at q = 29
BR5routine2026-09-03
Controls: [1:1:1] lies on the curve for every q, m, k; the line x + y + 2z = 0 does meet C₂ over F₄1; exactly 30 of the 41 values of k are admissible at q = 41; the line x + y + z = 0 does *not* miss the curve for the admissible k = 11; and none of x + y + cz = 0, c ∈ 1,4,10, misses the curve at the source's positive data point q = 37
BR6routine2026-09-03
The coset reduction: PowerPairs q m (the conclusion of Corollary 2.2 of the source, in its equivalent m-th-root-of-unity form) follows from a covering check on r coset representatives plus the r² representative pairs — q·r·m steps instead of q³
BR7routine2026-09-03
Corollary 2.2 of arXiv:2208.13299v2 itself fails at q = 41, r = 4: there are no fourth powers y, z of F₄1^* with y + z = −1
BR8candidate2026-09-03
The positive half of the classification: at every prime q ≡ 1 (mod 4) with 37 ≤ q ≤ 256 other than q = 41, the curve of Theorem 6.3 at r = 4 is blocking for every k — so q = 41 is the only prime counterexample to the belief, and there is no residual gap, q > 256 being Theorem 6.3 itself
BR9routine2026-09-03
Control above the theorem's own threshold: at q = 257, 269, 277, 281, 293 — all > 4⁴ = 256, where Theorem 6.3 is proved — the model also says the curve is blocking for every k
BR10correction2026-09-03
At q = 41, r = 4 the curve of Theorem 6.3 is blocking for no admissible k: one of the three lines x + y + z = 0, x + y + 4z = 0, x + y + 10z = 0 misses it for each of the 30 admissible values
BR11candidate2026-09-03
The other belief of Remark 6.4 is true: at r = 5, for every prime q ≡ 1 (mod 5) with 131 ≤ q ≤ 625 the curve of Theorem 6.3 is blocking for every k; above 625 the conclusion is Theorem 6.3 itself, so the belief "the conclusion holds for all q ≥ 131" holds over its whole declared range
BR12prose2026-09-03
The exact range of validity of Corollary 2.2 of arXiv:2208.13299v2 over all prime powers, for r = 2, 3, 4, 5: it fails exactly at q = 3, 5 (r=2); q = 4, 7, 13, 16 (referee 2026-09-03: P(4,3) fails – mu₁ = 1, b + c = 1 fails at b = c = 1; q = 4 was omitted) (r=3); q = 5, 9, 13, 17, 25, 29, 41 (r=4); q = 11, 16, 31, 41, 61, 71, 101 (r=5) — complete, because each search range passes r⁴, above which the source proves it
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- Founded 2026-09-02/03. A correction family: a belief stated in a remark of a published paper is false at exactly one prime power, and the same machinery then determines the exact range of validity of the theorem the belief is about, with no residual gap.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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