[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"vbNaUhL2Hq":3},"# seiberg-witten\n\nThe **Seiberg–Witten solution of N=2 SU(2) super-Yang–Mills**, formalized in **Lean 4**.\n\nNo rigorous proof of this result exists — it presupposes constructing the interacting\n4d quantum field theory. What a proof assistant *can* do is treat the physics argument\nrigorously: state the physical inputs as **explicit, named postulates** (H0–H7) and\ndictionary transcriptions carried in theorem types — never as axioms — and machine-check\nthat the Seiberg–Witten solution follows. The trusted base is then Lean's three logical\naxioms plus a short, named list of classical mathematics, tracked *per theorem* by a\ngolden `#print axioms` certificate. The companion paper is\n[`docs/paper1.tex`](docs/paper1.tex).\n\nThe library is **self-contained on pinned Mathlib** and `sorry`-free.\n\n## The claim\n\nFor SU(2)/genus one, the compiled development proves the listed Seiberg–Witten\nconsequences without `sorry` and without hidden physics axioms; physics enters through\nexplicit H0–H7 predicates, structures, definitions, and stated dictionary\ntranscriptions. The remaining trusted inputs beyond Lean's standard three are\nexplicitly named classical-mathematics axioms, tracked per theorem by\n[`audit/axiom-report.txt`](audit/axiom-report.txt). The higher-rank SU(N) headline is a\nproved skeleton modulo the coarse period-geometry axiom `periodRigidityAxiom`, whose\ndischarge is future work.\n\n## What is proved (highlights)\n\n- **Uniqueness up to Γ(2) duality** (`SU2.sw_su2_unique`, `sw_su2_unique_coulomb`):\n  footprint = standard-3 + the Γ(2) covering/lift pair — nothing else. The full chain\n  is formal: qualitative cusp data (`SWModulusData`) ⟹ the developing formula\n  `λ(τ(u)) = 2Λ²/(u+Λ²)` *including its normalization*\n  (`swModulusData_eq_crossRatio`, **axiom-free**) ⟹ uniqueness; and the cusp data\n  itself follows from an H4-style atlas plus parabolic cusp lifts with non-extension\n  (`swModulusData_of_atlas_and_lifts`).\n- **The explicit coupling** (`su2_coupling_exists`, special coordinates, `da_D/da = τ`,\n  the one-loop running): classical Jacobi inversion (`AX_elliptic_inversion`) as the\n  one extra input.\n- **Why exactly one monopole–dyon pair** (`singularity_count_pinch`): the Euler/mod-12\n  counting alone allows `n ≡ 1 (mod 6)` — the K3 loophole is *witnessed* — and the\n  R-spurion covariance (H7, through its curve transcription) pinches `n = 1`.\n  Axiom-free.\n- **SU(2) SQCD (matter)**: the coupling's uniqueness on the same classical basis\n  (`matter_coupling_rigidity`); the Argyres–Douglas locus nonempty **axiom-free**\n  (`matter_argyresDouglasLocus_nonempty`), via the development's first constructed\n  period chart and an exact discriminant factorization.\n- **The θ/λ layer**: `λ = θ₂⁴/θ₃⁴` proved Γ(2)-invariant, the S/T laws, `λ` omitting\n  `{0,1}`, `λ → 0` at the cusp — on two citable Jacobi-θ identities.\n\n## What is assumed\n\n- **Physics**: the named postulates **H0–H7** below (predicates and structures in\n  theorem types — there is no physical `axiom` anywhere) and their stated\n  curve/Kodaira transcriptions, each validated per\n  [`audit/FIDELITY_REVIEW.md`](audit/FIDELITY_REVIEW.md).\n- **Classical mathematics, as named axioms** (citable, numerically vetted): the Γ(2)\n  covering/lift pair, Jacobi inversion, and two Jacobi-θ identities.\n- **Higher rank only**: `periodRigidityAxiom` (Gauss–Manin period geometry) — the\n  declared future work.\n\nFull inventory: [`AXIOM_AUDIT.md`](AXIOM_AUDIT.md).\n\n### The postulates H0–H7\n\n| | Postulate | One-line content | Lean carrier |\n|---|---|---|---|\n| **H0** | The carrier | Coulomb-branch data: special coordinates `a, a_D`, coupling `τ`, and the Dirac-paired charge lattice, on the punctured moduli space | `PeriodBase` / `PeriodChart` |\n| **H1** | Special geometry | One holomorphic prepotential `F` with `a_D = ∂F/∂a`, `τ = ∂²F/∂a²`, and `Im τ ≻ 0` (unitarity) | `SpecialGeometry` |\n| **H2** | BPS singularities | A singularity of the effective theory is a charged BPS state going massless, `Z → 0` | `PeriodsDegenerateOnBoundary` |\n| **H3** | Monodromy | Around such a point the periods undergo the symplectic Picard–Lefschetz transvection in the light state's charge (one-loop log + Witten effect) | `PicardLefschetzAtGenericStratum` |\n| **H4** | EM duality | Charts glue by `Sp(2r,ℤ)` preserving the Dirac pairing and the central charge | `SymplecticReframing` / `PeriodAtlas` |\n| **H5** | R-symmetry | The anomaly-surviving discrete `U(1)_R` acts on the moduli with order dividing `2N` and permutes the singular locus | `HasFiniteOrderAutomorphism` |\n| **H6** | Weak-coupling asymptotics | The prepotential is one-loop exact up to a `Λ^{2N}`-weighted instanton series of prescribed scaling weights | `HasPrescribedAsymptotics` |\n| **H7** | Spurionic `U(1)_R` covariance of the Λ-family | Rescaling `Λ` is an RG transformation, `F(tΛ)(t·a) = t²F(Λ)(a)`; `Λ → ζΛ` with `ζ^{2N} = 1` (the θ-angle shift) changes `F` only by one integer EM frame shift `½aᵀBa`, constant across the family | `SpurionCovariantFamily` |\n\nH0–H6 constrain the theory at fixed `Λ` (bundled as `IsPolarizedPeriodChart`, with H4\nat the atlas level); H7 ties the `Λ`-family together. Physical justification: paper\n§4.2 and Appendix A; definitions: `SeibergWitten/Physics/Hypotheses.lean`.\n\n## Verify it yourself\n\n```bash\nlake build                                # pinned Mathlib; no other dependency\nbash audit/gen_axiom_report.sh --check    # the golden #print axioms certificate\npython3 audit/check_faithfulness.py       # paper-facing quotes match the source\nfor f in audit/numerical/validate_*.py; do python3 \"$f\"; done   # 9 oracle suites, 278 checks\n```\n\nReviewing the project? **Start at [`audit/REVIEWER.md`](audit/REVIEWER.md)** — the\nclaims, what each rests on, and the evidence per claim. The method follows\n[`math-commons/formalization-assurance`](https://github.com/math-commons/formalization-assurance):\nfaithfulness digests with machine-verified quotes, fidelity review with independent\nnumeric cross-checks, difficult points recorded, adversarial statement review before\nadoption.\n\n## Layout\n\n```\nSeibergWitten/          the library (physics layer: Physics/)\nRiemannPeriods/         weight-1 VHS bootstrap (Mathlib-only)\nHigherGenus/            higher-genus Riemann-surface layer — NOT built (see below)\naudit/                  certificate, checkers, oracles, V&V documents\ndocs/paper1.tex         the companion paper\n```\n\n## Higher genus and `jacobian-challenge`\n\nThe general-SU(N) story (genus `N−1` proved, Riemann-bilinear positivity\n`Im τ ≻ 0`) lives in [`HigherGenus/`](HigherGenus/), which builds against the external\nRiemann-surface / period / Jacobian library\n[`jacobian-challenge`](https://github.com/mrdouglasny/jacobian-challenge) and is kept\n**outside the certified build** — the main library and every footprint in the paper\nare Mathlib-only. `HigherGenus/README.md` has the re-enable recipe; discharging\n`periodRigidityAxiom` through that layer is the roadmap in\n[`audit/PERIOD_LAYER_DISCHARGE.md`](audit/PERIOD_LAYER_DISCHARGE.md).\n\n## Acknowledgments\n\nThe certified library depends only on Mathlib; it is the **higher-genus skeleton**\n(`HigherGenus/`, outside the build) that builds against\n[`jacobian-challenge`](https://github.com/mrdouglasny/jacobian-challenge), which in\nturn vendors **Rado Kirov**'s Dolbeault/Riemann-surface development. This project owes\nmuch to that library and its contributors — **Kevin Buzzard**, **Rado Kirov**, and the\n[other contributors](https://github.com/mrdouglasny/jacobian-challenge/graphs/contributors).\nSee the paper's acknowledgments for the full list of colleagues whose reviews and\ndiscussions shaped the project.\n\n## License\n\nCopyright 2026 Michael R. Douglas. Released under the\n[Apache License 2.0](LICENSE) (the Lean/Mathlib ecosystem convention).\n\n---\n\n**Comparator-verified** (Lean FRO [comparator](https://github.com/leanprover/comparator), commit `75bb75b`, 2026-07-07, via `lean4export@v4.30.0`; macOS fake-landrun): all no-`sorry`, kernel-replayed, axioms exactly as printed.\n- *Axiom-clean* (standard-3 only): `matter_argyresDouglasLocus_nonempty`, `betaFunction_weakCoupling`, and the SU(2)-from-postulate pair `SU2.sw_su2_unique_of_periodLayer` / `sw_su2_exists_of_periodLayer`.\n- *Concrete SU(2) headline family* (standard-3 + the classical inputs `AX_thrice_punctured_uniformization`, `AX_developing_map_rigidity`, `AX_elliptic_inversion`): `SU2.sw_su2_unique`, `su2_coupling_exists`, `su2_coupling_canonical`, `swAD_tendsto_zero_monopole` (H2), `swA_weakCoupling` (H6), `swAD_deriv_eq_swTau_mul_swA_deriv`, `swTau_logDeriv_weakCoupling`.\n",1783784691599]