[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"Fc4xDjwo34":3},"# Hille-Yosida and BCR Semigroup Theory\n\nA Lean 4 formalization of contraction semigroups (Hille-Yosida), Bernstein's theorem for completely monotone functions, and the BCR Bochner theorem (Theorem 4.1.13) for positive-definite functions on involutive semigroups.\n\n## Main Results\n\n### Hille-Yosida Resolvent Theorem (fully proved, 0 axioms)\n\nFor a contraction semigroup $\\{S(t)\\}_{t \\geq 0}$ on a Banach space, the resolvent $R(\\lambda) = \\int_0^\\infty e^{-\\lambda t} S(t) dt$ satisfies $(\\lambda I - A) R(\\lambda) = I$ and $\\|R(\\lambda)\\| \\leq 1/\\lambda$.\n\n### Bernstein's Theorem (fully proved, 0 axioms)\n\nA completely monotone function on $[0,\\infty)$ is the Laplace transform of a unique finite positive measure supported on $[0,\\infty)$.\n\n### BCR Theorem 4.1.13 -- Semigroup Bochner (fully proved, 0 axioms)\n\nBounded continuous positive-definite functions on $[0,\\infty) \\times \\mathbb{R}^d$ are Fourier-Laplace transforms of a unique finite positive measure:\n\n$$F(t, a) = \\int_{[0,\\infty) \\times \\mathbb{R}^d} e^{-tp} \\, e^{i\\langle a, q\\rangle} \\, d\\mu(p, q) \\quad \\text{for } t \\geq 0$$\n\n**Existence** (`semigroupGroupBochner`): decomposes into spatial Bochner + temporal BCR d=0 + product measure assembly.\n\n**Uniqueness** (`laplaceFourier_unique`): finite measures on $[0,\\infty) \\times \\mathbb{R}^d$ with equal Laplace-Fourier transforms are equal. Proved via Fourier uniqueness on spatial slices, Laplace uniqueness on temporal slices, and rectangular measure extension.\n\n## File Structure\n\n| File | Contents |\n|------|----------|\n| [StronglyContinuousSemigroup.lean](HilleYosida/StronglyContinuousSemigroup.lean) | [C₀-semigroups, generators, resolvent, Hille-Yosida](summary/HilleYosida/StronglyContinuousSemigroup.md) |\n| [BernsteinBasic.lean](HilleYosida/BernsteinBasic.lean) | [`IsCompletelyMonotone`, `taylor_integral_remainder`](summary/HilleYosida/BernsteinBasic.md) |\n| [BernsteinMeasures.lean](HilleYosida/BernsteinMeasures.lean) | [Density, IBP, kernel, packaging](summary/HilleYosida/BernsteinMeasures.md) |\n| [BernsteinChafai.lean](HilleYosida/BernsteinChafai.lean) | [Chafai identity, Prokhorov extraction](summary/HilleYosida/BernsteinChafai.md) |\n| [Bernstein.lean](HilleYosida/Bernstein.lean) | [`bernstein_theorem`](summary/HilleYosida/Bernstein.md) |\n| [FourierPD.lean](HilleYosida/FourierPD.lean) | [Fourier PD: `pd_quadratic_form_of_measure`](summary/HilleYosida/FourierPD.md) |\n| [BCR_d0.lean](HilleYosida/BCR_d0.lean) | [BCR 4.1.13 for d=0: `semigroup_pd_laplace`](summary/HilleYosida/BCR_d0.md) |\n| [BCR_General.lean](HilleYosida/BCR_General.lean) | [BCR 4.1.13: `semigroupGroupBochner_proof` + `laplaceFourier_unique`](summary/HilleYosida/BCR_General.md) |\n| [SemigroupGroupExtension.lean](HilleYosida/SemigroupGroupExtension.lean) | [`semigroupGroupBochner` + group extension](summary/HilleYosida/SemigroupGroupExtension.md) |\n| [SemigroupGroupDefs.lean](HilleYosida/SemigroupGroupDefs.lean) | [`IsSemigroupGroupPD` definition](summary/HilleYosida/SemigroupGroupDefs.md) |\n| [Future/GenerationTheorem.lean](HilleYosida/Future/GenerationTheorem.lean) | [HY converse (Lumer-Phillips)](summary/HilleYosida/Future/GenerationTheorem.md) |\n\n## Axiom Inventory\n\n**All main theorems are fully proved with 0 axioms and 0 sorry's — the library is now\naxiom-free.** (`Future/GenerationTheorem.lean`, the converse Hille–Yosida / Lumer–Phillips\ndirection, holds only scaffolding — the `IsDissipative` setup — and no longer declares\naxioms.) The kernel-authoritative audit is in [`AXIOM_AUDIT.md`](AXIOM_AUDIT.md); generate the\nlive trace with `lake env lean audit/axiom_report.lean`.\n\n## Assurance conventions\n\nThis project follows\n[`math-commons/formalization-assurance`](https://github.com/math-commons/formalization-assurance)\n(verification / validation / faithfulness, axiom vetting, `formalization.yaml`, object\ncontracts). Local settings:\n\n| Setting | Where |\n|---|---|\n| Project card | [`formalization.yaml`](formalization.yaml) |\n| Axiom audit | [`AXIOM_AUDIT.md`](AXIOM_AUDIT.md) — **0 axioms** |\n| Vetting strictness | [`audit/vetting/policy.yml`](audit/vetting/policy.yml) — `L1` |\n| Faithfulness (informal↔formal) | [`audit/FAITHFULNESS.md`](audit/FAITHFULNESS.md) |\n| Acceptance / characterization | [`audit/VALIDATION.md`](audit/VALIDATION.md) |\n| Object contracts | [`audit/contracts/`](audit/contracts/) — `known_values` cards (`resolvent`, `completelyMonotone`) |\n| Kernel axiom report | [`audit/axiom_report.lean`](audit/axiom_report.lean) (generator) |\n| CI assurance gate | [`.github/workflows/assurance.yml`](.github/workflows/assurance.yml) → the hub's reusable `assure.yml` (build + axiom-report-in-sync + sorry-confinement; warn-only at `L1`) |\n\n> Machine-gate TODO: commit the golden `audit/axiom-report.txt` from a build\n> (`lake env lean audit/axiom_report.lean > audit/axiom-report.txt`) and raise the policy to\n> `L2` to enforce.\n\n## BCR d=0: Proof Architecture (1503 lines)\n\nThe core engine `semigroup_pd_laplace` in `BCR_d0.lean` proves that bounded continuous semigroup-PD functions on $[0,\\infty)$ are Laplace transforms. The proof:\n\n1. **PD to alternating differences:** Vandermonde convolution for even order, convexity bootstrap for odd order\n2. **Iterated integral bridge:** The n-th forward difference equals the n-th iterated integral of the n-th derivative (bypasses Widder IV.12a)\n3. **Mollifier trick:** Smooth approximation inherits alternating differences, hence is `IsCompletelyMonotone`\n4. **Bernstein to measures:** Apply `bernstein_theorem` to each mollified function\n5. **Prokhorov extraction:** Uniform mass bound + tightness gives weak limit\n6. **Verification:** Pointwise convergence + Laplace convergence gives the representation\n\n## Application: QFT Reconstruction\n\nThis project provides the `semigroupGroupBochner` theorem needed by [OSreconstruction](https://github.com/xiyin137/OSreconstruction) for the E-to-R direction of Osterwalder-Schrader reconstruction. The Fourier-Laplace measure representation connects Euclidean contraction semigroups $e^{-tH}$ to Lorentzian unitary groups $e^{itH}$ via analytic continuation.\n\n## Dependencies\n\n- **Mathlib** (v4.29.0-rc6)\n- **[BochnerMinlos](https://github.com/mrdouglasny/bochner)** -- Bochner's theorem for finite-dimensional PD functions\n\n## References\n\n- Berg-Christensen-Ressel, *Harmonic Analysis on Semigroups* (1984), Theorem 4.1.13\n- Engel-Nagel, *One-Parameter Semigroups for Linear Evolution Equations*, GTM 194 (2000)\n- Widder, *The Laplace Transform* (1941), Chapter IV\n- Reed-Simon I-II (1972-1975)\n\n## Author\n\nMichael R. Douglas\n\n## License\n\nCopyright (c) 2026 Michael R. Douglas. Released under the Apache 2.0 license.\n",1782661975640]