BUT-Q Before Classicality: Exact-Cut Fragility and Granular Stability in Quantum-to-Classical Genesis

BUT-Q is a finite-dimensional quantum instance of the Before Unity Theory (BUT v1.2) framework, addressed to the question of when a quantum multiplicity becomes eligible to be treated as a stabilized classical unit. It operates over the preunitary domain X := D(H_S ⊗ H_E), the partial-trace cut c_env := Tr_E, trace distance as the default post-cut metric, and a declared threshold ε_env. Two structural theorems are established. Theorem Q2 (exact-cut fragility) shows that under ε_env = 0 with a perturbation class containing arbitrarily small attenuated mixtures with a CPTP separator, the stability margin μ_env is zero. Theorem Q3 (granular stability) shows that under ε_env > 0 and diam_c(Ξ) ≤ ε_env − m for a declared slack m > 0, the margin admits the certified lower bound μ_env ≥ m. The bound is sharpened to an exact characterization (Q3⋆): the margin equals the diamond-norm distance at which the perturbed post-cut diameter first crosses the threshold along a convex exposure curve. BUT-Q Genesis names the configured-eligibility regime; a frame-auditability corollary (Q5) and a graded componentwise-necessity proposition specify which frame components must be declared for a classicality claim to be inspectable. Maximum claim: BUT-Q provides an auditability layer for quantum-to-classical emergence by formalizing when a quantum multiplicity becomes eligible to be treated as a stabilized classical unit under declared cut, threshold, residue, perturbation class, stability margin, and frame; within this layer the stability margin is characterized exactly, and each core frame component is shown to be non-redundant. Non-claims: BUT-Q does not derive decoherence, does not solve the measurement problem, does not replace einselection / Quantum Darwinism / consistent histories, does not predict new quantum dynamics, and does not assert that standard quantum mechanics is incomplete. Extension to infinite-dimensional / continuous-variable settings is an extension of the framework, not a completeness gap. A Lean 4 / Mathlib v4.29.1 certificate accompanies the paper (Appendix E), with a primary abstract certificate and a secondary concrete (PHYSLIB) certificate.