Independently Reproducing Mishra's Official 0.8344 Certificate, the B7 Candidate Remains Only a Search Prior: Threshold-Conditioned Minimizer Atlas and the Witness-Switching Lower Bound

Round 07 (AMRAL-LUC-FC-R07, 2026-09-18) first carries out a data audit aimed at the external literature, and then builds new theory on top of it: directly checking the Mishra 2026 paper (arXiv:2608.30538) itself and the official source repository Ujjwal238/universal-cover-problem's domain.py, geom.py, verifyB.py, ceilings.py, phase1.py, phase2.py, certgen.py, certemitB.py and their corresponding logs, independently reproducing its already-published three-witness family F₃={D,B₃,B₅}, the five-dimensional normalized placement domain (x₃,y₃,φ₅,x₅,y₅), the certified lower bound 0.8344, the 486,799,600-node certificate, the worst leaf slack 1.240900×10⁻⁵, the rigorous floating-point error bound 1.72×10⁻⁹, and the official best-exhibited near-minimizer seed coordinates (exact-kernel family area 0.834780945912, rigorous outer polygon 0.834781190917). On this basis, this round proposes the next improvement milestone T₁=0.8350 (corresponding to the sharper translation domain |t₃|≤0.194856180909, |t₅|≤0.197820670401), and proves three theorems: Threshold-Conditioned Witness Lifting — a new witness's dimensions need only be activated locally on individual cells of the base sublevel atlas (Lazy Witness Dimension Activation), without a Cartesian-product-style expansion over the entire base domain; the Witness-Switching lower bound theorem — different cells can be closed by different witnesses, giving Λ(F∪B)≥min_q max_{K∈B} J_K(H_F(q)); and, based on the Steiner formula, Robust Cell-Lift Transfer, which allows a single cell to represent a hull rather than rerunning erosion point by point. The single point emphasized most throughout is discipline, not results: a small-scale SEARCH-ONLY numerical probe, fixed at the official seed and using boundary sampling, ranks the regular Reuleaux heptagon B₇ first among candidates (about 0.83713, above B₉'s 0.83563, B₁₁'s 0.83613, and B₁₃'s 0.83597), but the document uses its own data to directly prove that this ranking cannot be taken as a four-body lower bound — the official full 8-dimensional numerical search ceiling is only 0.836494901, lower than this fixed-basepoint B₇ value, showing that once a new witness is added, B₃ and B₅ themselves reconfigure, and the old minimizer point no longer represents the entire sublevel region; B₇ is therefore labeled only SEARCH-PRIOR, and the official raw estimate for naive full 8-dimensional certification up to T=0.835 (about 5.527×10¹² boxes, about 5635.8 days) is itself explicitly stated to be only an empirical cost extrapolation, not a proof of mathematical impossibility. Research direction was chosen and led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol.