Paley graphs of capture depth seven: the two-basepoint capture depth beyond p=250

Abstract

To a prime p and a subgroup E ≤ ℤₚ^(×) containing -1, Marashdeh attaches a nested chain of subspaces V₀ ⊆ V₁ ⊆ … of ℂᵖ, generated from the two point masses δ₀,δ₁ by the class matrices of the cyclotomic scheme and the diagonal idempotents of the basepoint partition, and calls the least d for which V_d contains a point mass δ_z with z ∉ {0,1} the capture depth d(p,E); one capture, at any depth, forces every circulant graph of order p with multiplier group E to have no quantum symmetry. His table of depths covers all 214 pairs with p ≤ 250; along the Paley graphs Pₚ it reads 1,2,3,4,5,6 on p=5, 13, 17, 29–61, 73–181, 193–241, and he asks whether the Paley depth is unbounded. We compute the first values beyond that range: d(P₅₂₁)=7, the first capture depth seven on record, and d(P₅₀₉)=6; every Paley prime 257 ≤ p ≤ 509 has depth at most 6; and 521 is the least Paley prime of depth exceeding six, the list of the 45 Paley primes below 521 being itself part of the theorem. Each statement is certified two-sidedly and exactly over ℚ: membership in V_d by an explicit integer combination of generator-word vectors, non-membership in V_(d-1) by a graded spanning certificate for V₄ together with one integer functional whose adjoint-word images annihilate it. The certificate checkers are proved sound against the definition of the filtration and the checks are carried out by a compiled evaluator; everything is machine-checked in Lean 4 with no sorry. Outside the formal development, and labelled as such, a modular computation reproduces the source's whole table, extends the Paley row to p ≤ 1621 (depth 7 on 521–1453, depth 8 from 1481), and records that dim Vⱼ=2 · 3ʲ⁻¹+4 independently of p wherever the filtration has not saturated, so that the depth grows like log₃p.

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