Certifying conditional dependence among environmental records from Bell measurements

Environmental records can be individually accurate while sharing the same errors. This preprint derives a sharp relation between retained Bell coherence, local record accuracy and conditional dependence among designated environmental readouts. For a balanced two-qubit state in the subspace spanned by |00⟩ and |11⟩, the full extremum reduces to a convex problem over R + 1 Hamming-weight probabilities and has an attaining quantum realization with an unobserved remainder. Its inverse gives the least measured conditional total correlation compatible with a CHSH observation and a common upper bound on both conditional errors of every record. The two-record inverse is explicit. At fixed accuracy, a fixed positive Bell excess requires dependence growing at least linearly with record count. Independent monitoring reaches the zero-dependence boundary in a Lindblad dephasing limit. A shared-memory Lindblad process improves record accuracy at constant Bell value and yields an exact onset of detectable dependence. The record contains a standalone PDF preprint and a supporting archive with the PDF, TeX and Markdown manuscript sources, computational supplement, reproducible Python source, calculated data, figures and build instructions. All results concern the stated theoretical models; no experimental dataset is used. The preprint has not undergone journal peer review. AI assistance is disclosed in the manuscript.