Prescribed digit sums in residue classes and a conjecture of Murthy (OEIS A077754)

We prove an equidistribution theorem for numbers with all-nonzero decimal digits and prescribed digit sum and residue class. As a corollary, we resolve Murthy's conjecture (OEIS A077754, 2002): an integer n admits a multiple with at least two nonzero digits and digit sum n iff n is not in {1, 2, 5, 11} ∪ 10ℕ.

The corollary is fully machine-verified in Lean 4 with mathlib (theorem A077754, zero sorry; toolchain leanprover/lean4:v4.35.0, mathlib v4.35.0). Version 1.1.0 upgrades the finite certificates to kernel-pure evaluation: lake build exits 0 with zero native_decide; #print axioms on the main theorem reports exactly [propext, Classical.choice, Quot.sound] — the whole development, including all finite certificates (witness tables witSmall/witCorner and stage coverage), is verified inside the Lean kernel.

The deposit (A077754-ancillary.tar.gz + A077754-paper.pdf) includes the paper, the complete Lean formalization (A077754.lean, A077754Cert.lean, CornerTable.lean, shared digit modules), the recorded axiom reports and blind oracle audits plus independent counter-verification scripts (VERIFICATIONS/), the Palomar challenge package, the independent Python verifier A077754-ANNEXE.py re-deriving the finite certificates, the working notes, and the pinned build files (lakefile.toml, lake-manifest.json, lean-toolchain) for long-term reproducibility.