A bounded null result on integer relations in a hybrid basis built from Khinchin's constant K₀, its low-order powers, log K₀, and bilinear products of K₀ with seven named constants pi, e, ln 2, gamma, zeta (2), zeta (3), G. Sixty-five sub-bases are tested under a two-tier predicate: a rigorous Euclidean-norm bound Hᵣig = 1. 0361 * 10⁷2 derived from Ferguson-Bailey-Arno 1999 Theorem 1 + Corollary 2 at verification class provencorollary, and an empirical operational bound Hₑmpop ~ 8 * 10⁶9 at the primary basis under verification class fieldₛtandardₚractice per Bailey 1998 section 2's general rule. The M3. 1 harness uses mpmath PSLQ with cascading-precision triage (P, 2P, 4P at P = 2160 digits) and an independent PARI/GP lindep second leg. Every sub-basis returns cascade-stable null. The mission also documents a fundamental capability gap at the symbolic-closure stage M4 (K₀'s transcendence status is the major open question; seven candidate structural arguments and a four-path SymPy probe each fail), and a Lean 4 statement-shape encoding (~385 lines, Mathlib v4. 14. 0) which type-checks against an explicit auxiliary-axiom set. The work is conducted under an explicit-evidence-acceptance-and-logging (AEAL) discipline with eleven named heuristics, six audited halt events, and twelve mutation-log entries.
papanokechi papanokechi (Sun,) studied this question.