Binary Jacobian Quotient Representatives and H/G Common-Zero Exclusion: Lean Stage 3 Algebraic Certificates

Stage 3 algebraic note, version 1.1, companion to the Stage 1 (10.5281/zenodo.24082011) and Stage 2 (10.5281/zenodo.24082012) notes. We record thirteen Lean 4 theorems in Hodge.ExplicitJacobianQuotient. Over the polynomial ring K[x,y], with f_k = kx^5y + y^6, the file identifies the Jacobian ideal generated by the two formal partial derivatives, proves the quotient relations (including x^4y = 0 and y^6 = 0 when k ≠ 0), identifies the section Wronskian W with the Stage 2 coefficient formulas, proves the four-monomial identity [HW] = u_0y^5 + u_1xy^4 + u_2x^2y^3 + u_3x^3y^2 in the quotient at every root of the degree-24 modulus, proves G^2 − H^3 = f_k in K[x,y], and proves that any common evaluation zero of H and G is the origin. What Lean checks. All thirteen theorems (hypotheses: commutative ring or field, k ≠ 0, characteristic zero with ℚ-algebra structure, modulus root, as tabulated in NOTE3_README.md; no mismatch found). Non-vacuity: a complex modulus root exists, and one with [HW] ≠ 0 (degree-five milestone). Not proved: the Sylvester-resultant identification, a basis or dimension of the whole quotient, schemes. Not claimed. No residue, cohomology, Hodge-type, cycle-class, descent or Hodge-conjecture statement is proved or claimed. In the HODGE repository the geometric bridges (for example ChainSexticCharacterBridge and the cycle class map) remain explicit hypotheses, not theorems; this record does not change that. Reproducibility. Full Lean source: BenFrohman/HODGE at 4c768890dc88bc180bf10887e4ba1b426f33a41a (tip of main on 11 November 2026; a git archive zip of that commit is attached). The paper cites code commit 52325b0; the Lean sources, scripts, lakefile, manifest and toolchain are byte-identical between that commit and 4c76889. Lean leanprover/lean4:v4.22.0, Mathlib 79e94a093aff4a60fb1b1f92d9681e407124c2ca. One-command rebuild from the repository root: lake exe cache get && lake build && python3 scripts/check_residue_axioms.py. GitHub Actions on main at 4c76889 passed (Lean certificate 42935333059, Lean 42935333161, Paper 42935333072). On 11 November 2026 the pinned commit was rebuilt locally from scratch for the project modules (attached build.log); the attached axiom output (axiom-output-residue.txt: PASS, 78 incremental declarations; axiom-output-root.txt: 699 of 699 declarations) shows only propext, Classical.choice and Quot.sound (or a subset), and the attached scan found no sorry, admit, native_decide or axiom declarations. SHA256SUMS lists every attached file. Limits. A kernel check shows that the stated Lean propositions follow from the stated Lean definitions. It does not show that those definitions model any geometric or physical object.