These documents presents a four-run analytical programme applied to the twelve gemstones of the biblical High Priest's Breastplate. Each stone was translated into its modern mineral identity and analysed using host chemistry, impurity chemistry, and anomaly chemistry layers. A progressive elimination grammar — in which candidate materials survived only by reducing interpretive ambiguity rather than broadening it — was applied across seventeen stages in Run 1 (blind compression), nine stages in Run 2 (directed replication with prior hypotheses), six stages in Run 3 (composite reframe incorporating the gold matrix setting), and nineteen stages across four substitution priorities in Run 4 (falsification testing against alternative stone identifications). Across Runs 1 through 3, the surviving compact explanatory core converged on the same exact result: the uranium-beryllium host family UBe₁₃ → U₁₋ₓThₓBe₁₃. This family is a known heavy-fermion superconductor whose double superconducting transition at x ≈ 0.019–0.033 remains unresolved after four decades of study. Run 4 tested the two most critical stone identifications directly: replacing zircon at position 7 with amber broke the machine across six stages; replacing beryl at position 10 with yellow topaz broke the machine across five stages. Two further substitutions at contextual positions (positions 5 and 2) left the machine unchanged, confirming the grammar correctly distinguishes load-bearing from contextual identifications. The programme makes no claim of deliberate ancient encoding. It claims that a disciplined compression analysis, applied across four independent analytical regimes of increasing stringency, converges on a physically serious and unresolved system in condensed matter physics, and that the result is identification-specific at its two critical positions in the way that a real materials recipe would be. The programme generates a specific experimental proposal: systematic synthesis and characterisation of the RBe₁₃ series from uranium to thulium with consistent gold-contact geometry. The complete stage-by-stage record is included as supplementary material. Biblical Breastplate Decode: The Result Was Unexpected