This mathematical methods checkpoint completes a conditional, error-controlled adjoint calculation for the unchanged registered two-source/two-detector experiment in the fixed-background linear five-field Einstein–scalar model. The response determinant is enclosed by [2.12278154373842557755059, 2.53743323308364125340774]. Its strictly positive lower endpoint establishes rank two under the stated inherited model, background-root, Green/source and stability premises. The certificate joins one exact piecewise-polynomial candidate across all 176 closed characteristic rectangles, certifies its complete five-field residual for all three registered loads over the inherited background-root interval, and encloses both source functionals of that same candidate. It includes boundary-data error and the direct source contribution of the boundary-error lift. Exact whole-face matching and the characteristic gluing argument account for internal interfaces. A determinant-preserving detector contrast retains the small response difference without changing the registered sources or detectors. An initial reconstruction had a certified defect at a characteristic trace junction. Evolving the correction to the prescribed boundary polynomial directly resolved that problem and reduced the error sufficiently to certify positivity. The initial candidate and its inconclusive error budget are preserved for inspection. Earlier project versions already reported rank-two results through an analytic route; this version supplies a validated full-domain reconstruction route, not a first rank-two claim or a tighter determinant estimate. The delivered archive passed a clean-extraction, standard-library Python replay with 13 top-level stages, including a nested nine-stage inherited component replay. All 176 selected cells were recomputed, and the 447-file payload reproduced without byte mutations. Exact arithmetic and explicit remainder bounds determine the result. Internal independent-agent reviews and adversarial checks support verification but are not external peer review or proof-assistant formalization. The conclusion remains conditional on archived shooting-root membership, the registered background equations and signed Green/source identities, inherited global stability majorants, and the characteristic gluing theorem. A rigorous continuum error enclosure for the independent forward calculation remains open. Registered coefficient normalizations do not establish physical source-energy normalization, detector noise sensitivity or observational accessibility. Nonlinear persistence, agreement with cosmological observations and a higher-dimensional cause of the Big Bang remain NOT_ESTABLISHED; external mathematical novelty remains NOT_ASSESSED.