[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"Yt0K4pj4x7":3},"# Foundational theorems in local complex-analytic geometry\n\nThis repository formalizes four classical results in Lean 4:\n\n1. Rückert Noetherianity for local holomorphic-germ rings;\n2. algebraic and geometric finite projection for a nontrivial hypersurface\n   germ;\n3. the holomorphic constant-rank normal form;\n4. Rückert's finite-family local analytic Nullstellensatz.\n\nThe four compared declarations are:\n\n```text\nLocalComplexGeometry.holomorphicGerm_isNoetherian\nLocalComplexGeometry.hypersurface_finiteProjection\nLocalComplexGeometry.holomorphic_constantRank_normalForm\nLocalComplexGeometry.localAnalyticNullstellensatz\n```\n\nTheir proved public wrappers are in `LocalComplexGeometry/Palomar.lean`.\nDetailed mathematical formulations and fidelity notes are in\n`MATHEMATICAL_STATEMENTS.md`.\n\n## Mathematical content\n\nWrite\n\n$$\nE_n=\\mathbf C^n=\\operatorname{Fin}(n)\\to\\mathbf C,\n\\qquad\n\\mathcal G_n=\\operatorname{Germ}_{\\mathcal N(0)}(E_n,\\mathbf C),\n$$\n\nand let $\\mathcal O_n\\subseteq\\mathcal G_n$ be the subring of germs which\nhave a representative analytic at the origin.\n\n### 1. Noetherianity\n\nFor every natural number $n$, including $n=0$, the ring\n$\\mathcal O_n$ is Noetherian.\n\n### 2. Finite projection\n\nLet $f\\in\\mathcal O_{n+1}$ be nonzero and vanish at the origin. The theorem\nreturns a continuous complex-linear equivalence $L$, a positive natural\nnumber $d$, analytic representatives $H,F$, and a transformed germ $g$,\nwhere $H$ represents $f$, $F$ represents $H\\circ L$, and $g$ is\nrepresented by $F$.\n\nPut $I=(g)$, $Q=\\mathcal O_{n+1}/I$, and let $q$ be the quotient map.\nThe result supplies the natural base-coordinate inclusion\n$\\iota:\\mathcal O_n\\to\\mathcal O_{n+1}$ and the last-coordinate germ\n$w$. Every $y\\in Q$ has a unique expansion\n\n$$\ny=\\sum_{i=0}^{d-1}q\\!\\left(\\iota(c_i)w^i\\right),\n\\qquad c_i\\in\\mathcal O_n.\n$$\n\nThus, for the $\\mathcal O_n$-algebra structure induced by $q\\circ\\iota$,\nthe quotient is finite free of rank $d$, with explicit power basis\n$1,\\bar w,\\ldots,\\bar w^{d-1}$. The compared theorem states uniqueness\ndirectly rather than exposing the proof-bearing internal `Module.Basis`.\n\nThe same theorem gives an open preconnected neighborhood $U\\ni0$, a radius\n$R>0$, coefficient functions $a_i$ analytic near every point of $U$,\nand a factor $u$ analytic and nowhere zero on the vertical tube, such that\n\n$$\nF(z,w)=u(z,w)\\left(w^d+\\sum_{i=0}^{d-1}a_i(z)w^i\\right),\n\\qquad a_i(0)=0.\n$$\n\nFor every $z\\in U$ and $|w|=R$, it asserts separately that both the\nprepared polynomial and $F(z,w)$ are nonzero. It also gives finite fibers with\nat most $d$ distinct points and surjectivity and properness of the zero-locus\nprojection to $U$.\n\n### 3. Constant rank\n\nFor an analytic map $F:E_n\\to E_m$ at $a$, with\n$r\\le n,m$ and\n\n$$\n\\dim_{\\mathbf C}\\operatorname{range}(D F_x)=r\n$$\n\nthroughout a neighborhood of $a$, the theorem returns source coordinates\n$S,S^{-1}$ and target coordinates $T,T^{-1}$. It states their four\nbase-point equations, analyticity at the relevant points, all four eventual\ninverse identities, and\n\n$$\nT\\circ F\\circ S^{-1}=J_r\n$$\n\nnear zero, where $J_r(x)_j=x_j$ for $j\u003Cr$ and $0$ otherwise.\n\n### 4. Local analytic Nullstellensatz\n\nIf $f_0,\\ldots,f_{s-1},g:E_n\\to\\mathbf C$ are analytic at zero and, on one\nneighborhood of zero,\n\n$$\nf_0(x)=\\cdots=f_{s-1}(x)=0\\Longrightarrow g(x)=0,\n$$\n\nthen there are a positive natural number $N$ and functions $h_i$, each\nanalytic at zero, such that near zero\n\n$$\ng^N=\\sum_{i=0}^{s-1}h_i f_i.\n$$\n\nThe library also proves the separately audited arbitrary-ideal identity\n$I(V(I))=\\sqrt I$, plus submersion, immersion, and regular-level-set\ncorollaries. These additional results are not part of the four-declaration\nComparator surface.\n\n## Palomar trust boundary\n\n`NullstellensatzPalomarChallenge.lean` imports only Mathlib. Its transitive\nsource dependencies contain no substantive project module and no\n`ClassicalComplexWPT` source. It locally spells out only the elementary\nanalytic-germ ring needed by the first two statements.\n\n`NullstellensatzPalomarSolution.lean` imports `LocalComplexGeometry.Palomar`,\nwhose wrappers derive the same four declarations from the substantive library.\nThe Weierstrass-preparation project is therefore a Solution-only dependency,\npinned to the public commit\n`b4a7273fe5c9752753c52e10494097569089642d`. Mathlib is pinned through\n`lake-manifest.json` at `81a5d257c8e410db227a6665ed08f64fea08e997`.\n\nThe unique module names are intentional: the WPT dependency has its own generic\n`Challenge` and `Solution` modules, and Palomar resolves modules over the whole\nLake source path.\n\n## Reproduction\n\nWith the pinned Lean toolchain and normal Lake/Git access:\n\n```bash\nlake build\nlake build NullstellensatzPalomarChallenge NullstellensatzPalomarSolution\nlake env lean LocalComplexGeometry/AxiomAudit.lean\n./scripts/audit-lean-sources.sh\n./scripts/verify-comparator.sh\n```\n\nThe comparator script pins the current Palomar toolchain components, including\nComparator `575674928e239f5bc452aab72d1dd7b0f1326494`, and runs Comparator,\nNanoDa, and Lean's default kernel. `VERIFICATION.md` records the audit surface.\n\n## Repository map\n\n- `NullstellensatzPalomarChallenge.lean`: Mathlib-only statement module with\n  exactly four deliberate theorem holes.\n- `NullstellensatzPalomarSolution.lean`: proved comparison module.\n- `LocalComplexGeometry/Palomar.lean`: proof-development-to-public-surface\n  wrappers.\n- `LocalComplexGeometry/`: substantive proof development.\n- `comparator.json`: four theorem names, permitted axioms, and NanoDa setting.\n- `formalization.yaml`: Palomar editorial metadata and disclosure.\n- `MATHEMATICAL_STATEMENTS.md`: precise statement-by-statement interpretation.\n- `PALOMAR_SUBMISSION.md`: exact submission instructions.\n\n## Sources and disclosure\n\nThe principal references are Gunning–Rossi, *Analytic Functions of Several\nComplex Variables*; Łojasiewicz, *Introduction to Complex Analytic Geometry*;\nand Rückert, “Zum Eliminationsproblem der Potenzreihenideale.” Exact locations\nand relationships are recorded in `formalization.yaml`.\n\nCodex agents performed the Lean implementation, wrapper redesign, and automated\naudits under user direction. No external human mathematical or code review is\nclaimed. The repository is Apache-2.0 licensed; the cited sources retain their\nown copyrights.\n\nPalomar intake: \u003Chttps://submit.palomar-registry.org/>.\n",1788104606412]