A spectrally calibrated retained quantum cell

A retained process carries its constitution into the next transformation. We construct a two-level model in which one calibrated Hamiltonian family determines a constitutive record law and an exact control-response curvature. The resulting two-Kraus channel has transition probabilities pₘ, population contrast pole λ = 1 − p₀ − p₁, and normalized Choi negativity N = |λ|/2. Its moving-frame lower eigenvalue gives μₘ = ℏ²/(2Δₘ). The selected gap law Δₘ = 2gₘD/τ therefore enforces pₘ = sin²(ℏτ/(4μₘ)) and μₘgₘ = ℏ²τ/(4D). Positive integer gap shifts preserve the whole endpoint unitary and change the curvature. Endpoint data thus admit distinct continuous responses. A separate finite compatibility theorem characterizes common displacement along three independent rays. The matrix-exponential, channel, trace-modulus, moving-frame, curvature and alias calculations are connected in a Lean formalization with 135 authored public propositions. The contribution is a constructive compression: several observables become consequences of one specified implementation. The accompanying Lean release RC-QCELL-1.0 includes the paper-to-theorem map, pinned dependencies and verification records. The paper’s five-module proof scope is distinguished from the larger 27-module, 948-proposition checkpoint included in the archive.