BRPT Galois Certificates and Exact Chebotarev Densities for Search Rings 1–20

This dataset contains the exact Galois-theoretic certificates, cumulative cubic-field classifications, and Chebotarev densities used to calibrate the deterministic semiperimeter search window of the Boundary-Ring Probable-prime Test (BRPT). The calculations concern the two-parameter family of depressed cubics X^3 + aX + b, with the coefficient pairs ordered according to the BRPT semiperimeter search. The dataset covers search rings 1–20, corresponding to 800 cumulative coefficient pairs. For each ring, the deposited files provide the classification of the associated cubic fields and the exact cumulative density of primes for which none of the coefficient pairs considered up to that ring yields the required irreducible cubic. The computation accounts explicitly for isomorphic cubic fields, dependencies between S3 extensions sharing quadratic resolvents, cyclic cubic (A3) fields, and dependencies among quadratic characters. The certificate generation uses exact field isomorphism tests and exact compositum-degree computations provided by SymPy, together with sparse binary variable elimination for the quadratic-character partition function. Exact rational fractions are retained in the machine-readable outputs; decimal values are provided for convenience. For ring r, let δ_r denote the cumulative density of unramified primes for which no pair in rings 1,...,r gives an irreducible cubic of the required type. The certified values decrease from δ_1 = 4/9 to approximately δ_20 = 4.0072428457 × 10^−110. Across rings 1–20 the values satisfy the conservative finite envelope δ_r ≤ exp(−r^2 / 1.3^2). These certified densities provide the quantitative calibration for the BRPT search radius. The accompanying derived tables also verify directly that, for the effective BRPT window R_eff(L), where L is the bit length of the candidate integer, δ_{R_eff(L)} < 1 / (2^L − 1) for every bit length 3 ≤ L ≤ 282. At L = 283 the implemented search window first requires ring 21, which lies outside the present certificate. The deposit includes: cumulative classifications through rings 5, 10, 15, and 20; exact density tables for rings 1–5, 6–10, 11–15, and 16–20; four machine-readable Galois certificates; the Python generator used to construct and verify the certificates; derived window-calibration tables; SHA-256 checksums, provenance information, and an independent package verifier; reproducibility instructions. The final ring-20 certificate contains 666 distinct S3 cubic fields, total S3 cubic rank 646, 600 distinct quadratic resolvents, squareclass rank 287, and four independent cyclic cubic fields. These results support the Galois-density analysis and search-window calibration in the associated BRPT manuscript. They should not be interpreted as a probability of failure for an individual fixed prime, as a uniform proof that every prime is covered by the finite search window, or as an error bound for composite inputs. Universal window coverage remains a separate mathematical question. The deposited generator and CSV results correspond to the computational artifacts described in the manuscript. Provenance information in this record documents the execution environments and SHA-256 digests required for independent verification and reproduction.