Audited Censuses of Two Generalized Fermat Families: Coverage Ledgers and Large Datasets

We report audited computational censuses of two families of the generalized Fermat equation, with machine-auditable coverage records. For the signature family {2,3,m}, m ≥ 7, we determine every numerical solution — proper and improper, under the canonical-anchor convention stated below. With high power a^m ≤ 10^16 and cube base x ≤ 10^9, exactly the seven known coprime solutions appear. For the Beal-signature family {3,3,m}, m ≥ 4 (all exponents at least three), a divisor identity makes the census exact and unconditional: all 193,776 solutions with high power at most 10^30 are determined, none coprime, with two structural features that are both theorems reproduced by the census: an empty 3 | m column (Euler's theorem on Fermat's cubic, appearing as a spectral gap) and a deep exponent tail consisting exactly of two one-parameter skeleton families, which we prove exhaustive for every m ≥ 43 at this ceiling and, more generally, for m > log X / log 5 at any ceiling X, rather than a finite-range regularity of the scanned window. Every improper solution admits a weighted sixth-power descent (the lift lemma); the resulting tower decomposition supplies internal closure audits, including an independently enumerable 28-member Catalan tower that exposed a coverage defect. We record three defects found after the scans were complete — a boundary defect and a coverage gap in the {2,3,m} scanner and its ledger, the latter having omitted 41% of the solutions, and an endpoint defect in the analysis script supplying the model mass — with their repairs, the restated coverage record, and the audits added in consequence. Both censuses have since been reproduced record for record by independent implementations sharing no code with the scanners: all 855 solutions of the {2,3,m} census and all 193,776 of the {3,3,m} census agree as individual records, with no record on either side alone. A companion paper treats the Hall near-miss census and its statistical claims.