Logarithmic Bray sampling in symplectic and orthogonal groups

Abstract

For every odd prime-power field order q and all sufficiently large natural dimensions n, for every actual fixed strong involution t in Sp(V), O(V), SO(V) or Ω(V), each separately specified t-eigenspace has probability at least c/log n of receiving a strong restriction of the even Bray output inv(tt^g), with g independent and uniformly distributed in the actual group. On a d-dimensional space, strong means both eigenspace dimensions lie in the closed interval [d/3, 2d/3]. The manuscript proves unconditional graph-chart masses for self-adjoint operators, all four orthogonal determinant-character sectors including q=3 and zero-dimensional complements, and an exact half-mass square-spinor refinement. A 192-prescribed-class generation argument using the stated published Burness and Liebeck–Pyber–Shalev results, together with a natural-central-cover and simple-quotient graph argument using 385 disjoint first-success blocks, shows that O(log n·(1+log(1/δ))) independent uniform samples generate a subgroup containing C_G(t)′′ with probability at least 1−δ, for every 0<δ<1. This addresses the symplectic and orthogonal parts of the fixed-strong-involution Bray sampling problem reported in Oberwolfach Report37/2016, pp2160–2161 (repository problem30003185, PR1330). The linear and unitary marginal/generation results are existing work of Dixon–Praeger–Seress and Glasby–Praeger–Roney-Dougal; their distinguished2-part method and independent diagonal-escape strategy are credited. The 2026 stingray random-power construction has a different target. Bounded priority comparisons do not establish global firstness. The theorem is asymptotic in natural dimension and concerns sample count at prescribed confidence. It makes no all-dimension, characteristic-two, projective-involution, order-four-lift, similitude or total-running-time assertion, and no claim that a fixed multiplier gives vanishing error. Sp4 zero marginal and arbitrarily large 2:1 non-simultaneous-strength obstructions are explained. The two eigenspace events are separate, not a simultaneous bound. The deposit contains the paper PDF and a source-and-verification archive with complete editable proofs, exact standard-library finite diagnostics and expected results, dependency and priority limits, provenance, licenses, a source manifest and an offline deterministic packaging builder. Finite diagnostics corroborate selected interfaces and do not prove universal mathematics. No third-party article/book body, account capture or operational service tool is distributed. AI tools were extensively involved in the mathematical research, proof development, computational verification, manuscript preparation and auditing. This is an unrefereed preprint and has not undergone formal independent human peer review.