[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"rkHpf8jQ0U":3},"# The Escauriaza–Seregin–Šverák theorem, formalized in Lean 4\n\n[![Build and verify](https://github.com/scottnarmstrong/EscauriazaSereginSverak/actions/workflows/build.yml/badge.svg?branch=main)](https://github.com/scottnarmstrong/EscauriazaSereginSverak/actions/workflows/build.yml)\n[![Comparators](https://github.com/scottnarmstrong/EscauriazaSereginSverak/actions/workflows/comparators.yml/badge.svg?branch=main)](https://github.com/scottnarmstrong/EscauriazaSereginSverak/actions/workflows/comparators.yml)\n\nA complete, machine-checked proof of the Escauriaza–Seregin–Šverák theorem\nfor the three-dimensional incompressible Navier–Stokes equations on\n$\\mathbb{R}^3$, together with the Ladyzhenskaya–Prodi–Serrin theorem:\n\n- **The Escauriaza–Seregin–Šverák theorem.** A Leray–Hopf solution whose\n  velocity is bounded in $L^\\infty(0,T;L^3(\\mathbb{R}^3))$ has no singular\n  points, belongs to $L^5(\\mathbb{R}^3\\times(0,T))$, is the only\n  Leray–Hopf solution with its initial datum, and is smooth on\n  $\\mathbb{R}^3\\times(0,T]$. This is Theorem 1.3 of the paper in full.\n- **The Ladyzhenskaya–Prodi–Serrin theorem.** A Leray–Hopf solution on\n  $[0,T]$ that lies in $L^\\ell(0,T;L^s(\\mathbb{R}^3))$ with\n  $3/s+2/\\ell=1$ and $3\u003Cs\\le\\infty$ is the only Leray–Hopf solution with its\n  initial datum and agrees almost everywhere with a function that is\n  $C^\\infty$ on $\\mathbb{R}^3\\times(0,T]$ (one-sided at $T$).\n- **The regularity criterion for $3\\le s\\le\\infty$.** The two theorems\n  combined: a Leray–Hopf solution in $L^\\infty(0,T;L^3)$, in\n  $L^\\ell(0,T;L^s)$ with $3\u003Cs\u003C\\infty$ and $\\ell=2s/(s-3)$, or in\n  $L^2(0,T;L^\\infty)$ is unique and has a $C^\\infty$ representative on\n  $\\mathbb{R}^3\\times(0,T]$.\n\nThe formalization is written in Lean 4 on top of Mathlib and of the\n[Caffarelli–Kohn–Nirenberg formalization](https://github.com/scottnarmstrong/CaffarelliKohnNirenberg)\n(CKN below). CKN supplies the setting: suitable weak solutions and regular\npoints, Leray–Hopf solutions, Leray's global existence theorem in suitable\nform, the associated pressure, and the Caffarelli–Kohn–Nirenberg theorems.\nThis project proves the rest of the argument: the local regularity theorem\nand its blow-up proof, the Carleman inequalities, unique continuation and\nbackward uniqueness, the $L^5$ and uniqueness theorem, and the\nLadyzhenskaya–Prodi–Serrin uniqueness and smoothing theorem. The library uses\nno axioms beyond Lean's standard three and contains no unfinished proofs.\n\nThe proof is written out in full in the accompanying\n[manuscript](paper/ess.pdf) ([source](paper/ess.tex)). The manuscript and the\nLean development were produced together: the main theorems are stated in Lean\nwith the manuscript's hypotheses and conclusions, and the manuscript was\ncorrected as the formalization progressed.\n\n## Results formalized\n\n- **L. Escauriaza, G. A. Seregin and V. Šverák, \"$L_{3,\\infty}$-solutions of\n  Navier–Stokes equations and backward uniqueness\", *Russian Math. Surveys*\n  58:2 (2003), 211–250.** Formalized, with the paper's numbering:\n  - Theorem 1.4 (local regularity: a solution in $L^\\infty L^3$ on the unit\n    cylinder is Hölder continuous on the closure of the half cylinder);\n  - Theorem 1.3 (a Leray–Hopf solution in $L^\\infty(0,T;L^3)$ belongs to\n    $L^5$, is unique among Leray–Hopf solutions with its datum, and is\n    smooth on $\\mathbb{R}^3\\times(0,T]$), in full;\n  - Theorem 1.2 (the Ladyzhenskaya–Prodi–Serrin theorem, see the next\n    item), in the whole-space form used for Theorem 1.3;\n  - Theorem 4.1 (unique continuation across spatial boundaries);\n  - Theorem 5.1 (backward uniqueness for the heat operator with lower-order\n    terms on a half-space), in dimension three;\n  - Propositions 6.1 and 6.2 (the Carleman inequalities (6.1) and (6.12)).\n\n  The paper's Theorem 1.1 (existence of suitable Leray–Hopf solutions) and\n  the associated pressure used in its §3 are taken from CKN, where the\n  pressure is constructed by Riesz transforms.\n- **The Ladyzhenskaya–Prodi–Serrin theorem.** G. Prodi, \"Un teorema di\n  unicità per le equazioni di Navier–Stokes\", *Ann. Mat. Pura Appl.* (4) 48\n  (1959), 173–182; J. Serrin, \"The initial value problem for the\n  Navier–Stokes equations\", in *Nonlinear Problems* (R. E. Langer, ed.),\n  Univ. of Wisconsin Press, Madison, 1963, 69–98 (and, for interior\n  regularity, *Arch. Rational Mech. Anal.* 9 (1962), 187–195); O. A.\n  Ladyzhenskaya, \"On uniqueness and smoothness of generalized solutions to the\n  Navier–Stokes equations\", *Zap. Nauchn. Sem. LOMI* 5 (1967), 169–185.\n  Formalized in the whole-space form of Theorem 1.2 of Escauriaza, Seregin and\n  Šverák, following J. C. Robinson, J. L. Rodrigo and W. Sadowski, *The\n  Three-Dimensional Navier–Stokes Equations* (CUP, 2016), Theorems 8.17 and\n  8.19, for the whole range $3\u003Cs\\le\\infty$ (with $\\ell=2s/(s-3)$, and\n  $L^2(0,T;L^\\infty)$ at $s=\\infty$). Together with the\n  Escauriaza–Seregin–Šverák theorem, which is the case $s=3$, it gives the\n  regularity criterion for $3\\le s\\le\\infty$.\n- **L. Escauriaza, G. A. Seregin and V. Šverák, \"Backward uniqueness for\n  the heat operator in half-space\", *Algebra i Analiz* 15:1 (2003), 201–214;\n  English translation in *St. Petersburg Math. J.* 15 (2004), 139–148.** This\n  paper concerns the same half-space backward-uniqueness problem, whose\n  theorem appears with its proof as Theorem 5.1 of the paper above. The\n  formalization follows the *Russian Math. Surveys* paper; it has not been\n  compared line by line with this one.\n\nNot formalized:\n\n- the exterior-domain backward uniqueness theorem of L. Escauriaza,\n  G. A. Seregin and V. Šverák, \"Backward uniqueness for parabolic equations\",\n  *Arch. Ration. Mech. Anal.* 169 (2003), 147–157, which the half-space\n  theorem extends;\n- the general Carleman and unique-continuation theory of L. Escauriaza\n  (*Duke Math. J.*, 2000) and L. Escauriaza and F. J. Fernández (*Ark. Mat.*,\n  2003) beyond the statements listed above.\n\n## Sources\n\n- L. Escauriaza, G. A. Seregin and V. Šverák, \"$L_{3,\\infty}$-solutions of\n  Navier–Stokes equations and backward uniqueness\", *Russian Math. Surveys*\n  58:2 (2003), 211–250.\n- J. C. Robinson, J. L. Rodrigo and W. Sadowski, *The Three-Dimensional\n  Navier–Stokes Equations: Classical Theory*, Cambridge Studies in Advanced\n  Mathematics 157, CUP (2016).\n- G. Prodi, \"Un teorema di unicità per le equazioni di Navier–Stokes\", *Ann.\n  Mat. Pura Appl.* (4) 48 (1959), 173–182.\n- J. Serrin, \"On the interior regularity of weak solutions of the\n  Navier–Stokes equations\", *Arch. Rational Mech. Anal.* 9 (1962), 187–195,\n  and \"The initial value problem for the Navier–Stokes equations\", in\n  *Nonlinear Problems* (R. E. Langer, ed.), Univ. of Wisconsin Press (1963),\n  69–98.\n- O. A. Ladyzhenskaya, \"On uniqueness and smoothness of generalized solutions\n  to the Navier–Stokes equations\", *Zap. Nauchn. Sem. LOMI* 5 (1967),\n  169–185.\n- J. Leray, \"Sur le mouvement d'un liquide visqueux emplissant l'espace\",\n  *Acta Math.* 63 (1934), 193–248.\n- L. Caffarelli, R. Kohn and L. Nirenberg, \"Partial regularity of suitable\n  weak solutions of the Navier–Stokes equations\", *Comm. Pure Appl. Math.* 35\n  (1982), 771–831.\n- P. G. Lemarié-Rieusset, *The Navier–Stokes Problem in the 21st Century*,\n  CRC Press (2016).\n- S. Armstrong and V. Vicol, *The Caffarelli–Kohn–Nirenberg theorem,\n  formalized in Lean 4* (2026),\n  [github.com/scottnarmstrong/CaffarelliKohnNirenberg](https://github.com/scottnarmstrong/CaffarelliKohnNirenberg).\n\nThe [sources](docs/SOURCES.md) page gives the full bibliography with DOIs and\nsays what each work is used for.\n\n## What is proved\n\nSpace is $\\mathbb{R}^3$ and time is $\\mathbb{R}$. The space $J$ of initial\ndata is the $L^2$ closure of smooth, compactly supported, divergence-free\nvector fields. A Leray–Hopf solution on $[0,T]$ with datum $a\\in J$\n(Definition 3.3 of the manuscript) is a velocity $u$ with an explicit weak\nspatial gradient $Du$ that has finite energy and dissipation, is weakly\ndivergence free, is weakly continuous in time on the closed interval,\nsatisfies the weak Navier–Stokes equation against divergence-free tests,\nsatisfies the energy inequality at every time, and converges to $a$ in $L^2$\nas $t\\downarrow 0$. These definitions, like suitable weak solutions, regular\npoints and singular points, are those of CKN, which also proves that every\n$a\\in J$ has a global Leray–Hopf solution that is suitable\n(`CKN.leray_existence`).\n\n**Local regularity** (Theorem 3.12 of the manuscript; Theorem 1.4 of\nEscauriaza, Seregin and Šverák). If $u$, $Du$ and $p$ on $B_1\\times(-1,0)$\nhave finite energy, $p\\in L^{3/2}$, $u\\in L^\\infty(-1,0;L^3(B_1))$, and\nsatisfy the divergence-free condition and the momentum equation weakly (no\nenergy inequality is assumed), then $u$ agrees almost everywhere on\n$B_{1/2}\\times(-1/4,0)$ with a parabolically Hölder continuous function on\nthe closure of that cylinder.\n\n**Global regularity, $L^5$ and uniqueness** (Theorems 3.13 and 16.3; Theorem\n1.3 of Escauriaza, Seregin and Šverák). If $(u,Du)$ is a Leray–Hopf solution\non $[0,T]$ and $\\operatorname{ess\\,sup}_{t\\in(0,T)}\\|u(\\cdot,t)\\|_{L^3}\u003C\\infty$,\nthen $u$ has no singular point in $\\mathbb{R}^3\\times(0,T)$,\n$u\\in L^5(\\mathbb{R}^3\\times(0,T))$, and every Leray–Hopf solution on\n$[0,T]$ with the same datum equals $u$ almost everywhere.\n\n**Ladyzhenskaya–Prodi–Serrin** (Theorem 17.1 and Corollary 17.2; Theorems 1.2\nand 1.3 of Escauriaza, Seregin and Šverák). If $(u,Du)$ is a Leray–Hopf\nsolution on $[0,T]$ with datum $a$ and either\n$u\\in L^{\\ell}(0,T;L^s(\\mathbb{R}^3))$ with $3\u003Cs\u003C\\infty$ and\n$\\ell=2s/(s-3)$, or $u\\in L^2(0,T;L^\\infty(\\mathbb{R}^3))$, then every\nLeray–Hopf solution on $[0,T]$ with datum $a$ equals $u$ almost everywhere,\nand $u$ agrees almost everywhere with a function that is $C^\\infty$ on\n$\\mathbb{R}^3\\times(0,T]$, with derivatives at $T$ taken within that set.\nConsequently a Leray–Hopf solution with\n$\\operatorname{ess\\,sup}_t\\|u(t)\\|_{L^3}\u003C\\infty$ lies in $L^5$, is unique\nand is smooth in this sense; this is Theorem 1.3 of Escauriaza, Seregin and\nŠverák in full. Equality of solutions is almost everywhere, since the\nprescribed time slices of a Leray–Hopf solution need not agree at every\npoint.\n\n**Regularity criterion** (Corollary 17.3). If $(u,Du)$ is a Leray–Hopf\nsolution on $[0,T]$ with datum $a$ and $u$ lies in $L^\\infty(0,T;L^3)$, or\nin $L^\\ell(0,T;L^s)$ with $3\u003Cs\u003C\\infty$ and $\\ell=2s/(s-3)$, or in\n$L^2(0,T;L^\\infty)$, then $u$ is the only Leray–Hopf solution on $[0,T]$\nwith datum $a$ (almost everywhere) and agrees almost everywhere with a\nfunction that is $C^\\infty$ on $\\mathbb{R}^3\\times(0,T]$, one-sided at $T$.\n\n**Linear continuation** (Propositions 4.1 and 4.2, Theorems 5.2 and 6.7;\nPropositions 6.1 and 6.2 and Theorems 4.1 and 5.1 of Escauriaza, Seregin and\nŠverák). Unique continuation across a spatial boundary and backward\nuniqueness on a half-space for vector fields satisfying the differential\ninequality $|\\partial_t w+\\Delta w|\\le c_1(|\\nabla w|+|w|)$, and the two\nCarleman inequalities behind them. They are stated for fields with explicit\nspace-time weak derivatives in the class $W^{2,1}_2$, which is the regularity\navailable when they are applied to the vorticity of a suitable weak\nsolution.\n\n## The Lean statements\n\nThe main statements are in [ESS/Statements](ESS/Statements), one declaration\nper file; their proofs are assembled in [ESS/Main](ESS/Main). As in CKN,\n`Vec3` is `Fin 3 → ℝ`, a `ParabolicPoint` is a pair of a point and a time,\nand the weak spatial gradient `Du` is explicit data with `Du z i j` the\nderivative $\\partial_j u_i$. The definitions `IsLerayHopfSolution`,\n`SingularSet` and `parabolicHausdorffMeasure` are those of\n[CKN](https://github.com/scottnarmstrong/CaffarelliKohnNirenberg/tree/main/CKN/Statements).\nThe [design notes](docs/DESIGN_NOTES.md) explain the choices behind them.\n\nGlobal regularity, [`ESS.essGlobal`](ESS/Statements/EssGlobal.lean):\n\n```lean\ntheorem essGlobal :\n    ∀ T : ℝ, ∀ a : Vec3 → Vec3,\n    ∀ u : ParabolicPoint → Vec3,\n    ∀ Du : ParabolicPoint → Fin 3 → Vec3,\n      IsLerayHopfSolution T a u Du →\n      essSup\n        (fun t : ℝ => ∫⁻ x : Vec3,\n          ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ (3 : ℝ))\n        (volume.restrict (Ioo 0 T)) \u003C ⊤ →\n      SingularSet (Set.univ : Set Vec3) (Ioo 0 T) u = ∅\n```\n\n$L^5$ and uniqueness, [`ESS.essL5Unique`](ESS/Statements/EssL5Unique.lean):\n\n```lean\ntheorem essL5Unique :\n    ∀ T : ℝ, ∀ a : Vec3 → Vec3,\n    ∀ u : ParabolicPoint → Vec3,\n    ∀ Du : ParabolicPoint → Fin 3 → Vec3,\n      IsLerayHopfSolution T a u Du →\n      essSup\n        (fun t : ℝ => ∫⁻ x : Vec3,\n          ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ (3 : ℝ))\n        (volume.restrict (Ioo 0 T)) \u003C ⊤ →\n      MemLp u (ENNReal.ofReal (5 : ℝ))\n        (volume.restrict (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))) ∧\n      ∀ v : ParabolicPoint → Vec3,\n        ∀ Dv : ParabolicPoint → Fin 3 → Vec3,\n          IsLerayHopfSolution T a v Dv →\n            v =ᵐ[volume.restrict\n              (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u\n```\n\nLadyzhenskaya–Prodi–Serrin,\n[`ESS.ladyzhenskayaProdiSerrin`](ESS/Statements/LadyzhenskayaProdiSerrin.lean)\n(the hypothesis is the Serrin condition, a disjunction of the finite-$s$ mixed\nnorm with $\\ell=2s/(s-3)$ and the $L^2_tL^\\infty_x$ endpoint; the smooth\nrepresentative is a function $\\text{uSmooth}$ with\n`ContDiffOn ℝ (⊤ : ℕ∞)` on `univ ×ˢ Ioc 0 T`):\n\n```lean\ntheorem ladyzhenskayaProdiSerrin :\n    ∀ T : ℝ, ∀ a : Vec3 → Vec3,\n    ∀ u : ParabolicPoint → Vec3,\n    ∀ Du : ParabolicPoint → Fin 3 → Vec3,\n      IsLerayHopfSolution T a u Du →\n      ((∃ s : ℝ, 3 \u003C s ∧\n          (∫⁻ t in Ioo (0 : ℝ) T,\n            (∫⁻ x : Vec3,\n              ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ s) ^\n                ((2 * s / (s - 3)) / s)) \u003C ⊤) ∨\n        (∫⁻ t in Ioo (0 : ℝ) T,\n          (essSup\n            (fun x : Vec3 => ENNReal.ofReal (vec3EuclideanNorm (u (x, t))))\n            (volume : Measure Vec3)) ^ (2 : ℝ)) \u003C ⊤) →\n      (∀ v : ParabolicPoint → Vec3,\n        ∀ Dv : ParabolicPoint → Fin 3 → Vec3,\n          IsLerayHopfSolution T a v Dv →\n            v =ᵐ[volume.restrict\n              (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u) ∧\n      (∃ uSmooth : ParabolicPoint → Vec3,\n        uSmooth =ᵐ[volume.restrict\n          (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u ∧\n        ContDiffOn ℝ (⊤ : ℕ∞) (fun z : Vec3 × ℝ => uSmooth z)\n          ((Set.univ : Set Vec3) ×ˢ Ioc (0 : ℝ) T))\n```\n\nThe corollary [`ESS.essSmooth`](ESS/Statements/EssSmooth.lean) has the\nhypothesis of `essL5Unique` and concludes with the $L^5$ membership, the\nuniqueness clause and the smooth representative of the previous statement:\n\n```lean\ntheorem essSmooth :\n    ∀ T : ℝ, ∀ a : Vec3 → Vec3,\n    ∀ u : ParabolicPoint → Vec3,\n    ∀ Du : ParabolicPoint → Fin 3 → Vec3,\n      IsLerayHopfSolution T a u Du →\n      essSup\n        (fun t : ℝ => ∫⁻ x : Vec3,\n          ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ (3 : ℝ))\n        (volume.restrict (Ioo 0 T)) \u003C ⊤ →\n      MemLp u (ENNReal.ofReal (5 : ℝ))\n        (volume.restrict (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))) ∧\n      (∀ v : ParabolicPoint → Vec3,\n        ∀ Dv : ParabolicPoint → Fin 3 → Vec3,\n          IsLerayHopfSolution T a v Dv →\n            v =ᵐ[volume.restrict\n              (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u) ∧\n      (∃ uSmooth : ParabolicPoint → Vec3,\n        uSmooth =ᵐ[volume.restrict\n          (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u ∧\n        ContDiffOn ℝ (⊤ : ℕ∞) (fun z : Vec3 × ℝ => uSmooth z)\n          ((Set.univ : Set Vec3) ×ˢ Ioc (0 : ℝ) T))\n```\n\nThe combined criterion for $3\\le s\\le\\infty$,\n[`ESS.serrinCriterion`](ESS/Statements/SerrinCriterion.lean):\n\n```lean\ntheorem serrinCriterion :\n    ∀ T : ℝ, ∀ a : Vec3 → Vec3,\n    ∀ u : ParabolicPoint → Vec3,\n    ∀ Du : ParabolicPoint → Fin 3 → Vec3,\n      IsLerayHopfSolution T a u Du →\n      ((essSup\n          (fun t : ℝ => ∫⁻ x : Vec3,\n            ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ (3 : ℝ))\n          (volume.restrict (Ioo 0 T)) \u003C ⊤) ∨\n        (∃ s : ℝ, 3 \u003C s ∧\n          (∫⁻ t in Ioo (0 : ℝ) T,\n            (∫⁻ x : Vec3,\n              ENNReal.ofReal (vec3EuclideanNorm (u (x, t))) ^ s) ^\n                ((2 * s / (s - 3)) / s)) \u003C ⊤) ∨\n        (∫⁻ t in Ioo (0 : ℝ) T,\n          (essSup\n            (fun x : Vec3 => ENNReal.ofReal (vec3EuclideanNorm (u (x, t))))\n            (volume : Measure Vec3)) ^ (2 : ℝ)) \u003C ⊤) →\n      (∀ v : ParabolicPoint → Vec3,\n        ∀ Dv : ParabolicPoint → Fin 3 → Vec3,\n          IsLerayHopfSolution T a v Dv →\n            v =ᵐ[volume.restrict\n              (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u) ∧\n      (∃ uSmooth : ParabolicPoint → Vec3,\n        uSmooth =ᵐ[volume.restrict\n          (spaceTimeSet (Set.univ : Set Vec3) (Ioo 0 T))] u ∧\n        ContDiffOn ℝ (⊤ : ℕ∞) (fun z : Vec3 × ℝ => uSmooth z)\n          ((Set.univ : Set Vec3) ×ˢ Ioc (0 : ℝ) T))\n```\n\nThe remaining main theorems are stated in the same directory:\n\n| Manuscript | Lean declaration |\n|---|---|\n| Theorem 3.12 (local regularity) | [`ESS.essLocal`](ESS/Statements/EssLocal.lean) |\n| Theorem 5.2 (unique continuation) | [`ESS.uniqueContinuation`](ESS/Statements/UniqueContinuation.lean) |\n| Theorem 6.7 (backward uniqueness) | [`ESS.backwardUniqueness`](ESS/Statements/BackwardUniqueness.lean) |\n| Proposition 4.1 (Gaussian Carleman inequality) | [`ESS.carlemanGaussian`](ESS/Statements/CarlemanGaussian.lean) |\n| Proposition 4.2 (half-space Carleman inequality) | [`ESS.carlemanHalfSpace`](ESS/Statements/CarlemanHalfSpace.lean) |\n\nTogether with `essGlobal`, `essL5Unique`, `ladyzhenskayaProdiSerrin`,\n`essSmooth` and `serrinCriterion` above, these are the ten main theorems.\n\nBecause a formal statement is only as good as the definitions inside it,\nthe repository also contains two independent Challenge/Solution pairs in\n[comparators](comparators/README.md): `Linear` (unique continuation, backward\nuniqueness and the two Carleman inequalities) and `Regularity` (`essLocal`,\n`essGlobal`, `essL5Unique`, `ladyzhenskayaProdiSerrin`, `essSmooth` and\n`serrinCriterion`). Each Challenge imports only Mathlib and defines every\nnotion it uses from Mathlib (`EuclideanSpace ℝ (Fin 3)`, `fderiv`, Mathlib\nmeasures), so it can be read without reading this library or CKN; it states\nits theorems in the namespace `ESSChallenge` with one intentional proof\nplaceholder each. The corresponding Solution proves the identical statements\nfrom the library, with transport lemmas between the Mathlib-native notions\nand the library's. [Comparator](comparators/README.md) checks their\nstatement dependency closures and proofs, including an independent NanoDa\nkernel replay. Reading the Challenges is the quickest way to inspect the\nprecise mathematical claims.\n\n## How the formalization relates to the manuscript and to CKN\n\nThe manuscript is the paper being formalized, not a description written\nafter the fact. Where it departs from the published sources it says so in a\nremark next to the result concerned; the [deviations](docs/DEVIATIONS.md)\ndocument collects these departures. Examples are the finite vorticity\nbootstrap that replaces all-orders Stokes estimates in the blow-up argument,\nthe manuscript's own proof of the short-time $L^5$ bound, and, for the\nLadyzhenskaya–Prodi–Serrin theorem, an energy equality proved almost\neverywhere in time from the Serrin condition and a smoothing argument by a\nregularity ladder.\n\nThe solution classes are CKN's. A Leray–Hopf solution is one specified\nfunction whose every time slice is constrained, and the same function is the\nsuitable weak solution to which the Caffarelli–Kohn–Nirenberg theorems\napply; the local regularity proof uses CKN's Theorem A, as the\n[design notes](docs/DESIGN_NOTES.md) explain. The repository also contains\nexplicit examples showing that the definitions used in the statements are\ninhabited; the [witnesses](docs/WITNESSES.md) page lists them.\n\n## Building and checking it yourself\n\nThe project pins Lean 4 and Mathlib at v4.35.0-rc2 and CKN at the exact\ncommit `381d658ead0f03a18361965cc0427ce3fa5844ab`. With `elan` and Python 3 installed:\n\n```sh\nelan toolchain install leanprover/lean4:v4.35.0-rc2\nlake exe cache get\nESS_IGNORE_PACKAGE_BUILD=1 python3 scripts/build.py ESS\n```\n\nThe library contains about 205,000 lines of Lean (1,008 files). The Mathlib\ncache does not include CKN, so the first build compiles it from source as well\n(about 410,000 further lines); `ESS_IGNORE_PACKAGE_BUILD=1` lets the build\ncreate CKN's compiled files, while the build script still checks that the\nsources of CKN and Mathlib are unchanged. Build time depends on the machine. Keep the\ncommitted dependency manifest; avoid `lake update` or `lake clean` when\nverifying this version.\n\nTo confirm the axioms used by the main theorems, or to run the source and\ncomparator checks, follow the [verification guide](docs/VERIFICATION.md).\nEach of the ten main theorems depends exactly on `propext`,\n`Classical.choice` and `Quot.sound`.\n\n## Repository layout\n\n- [ESS/Statements](ESS/Statements): the ten main theorem statements, one declaration per file.\n- [ESS/Main](ESS/Main): the assembly of each main theorem from its proof.\n- [ESS/Linear](ESS/Linear): the Carleman inequalities, unique continuation and backward uniqueness (Part I of the manuscript).\n- [ESS/Endpoint](ESS/Endpoint): the local energy equality, the smallness criterion, the blow-up argument, the vorticity bootstrap and the proofs of the local and global regularity theorems (Part IV).\n- [ESS/PartV](ESS/PartV): the short-time $L^5$ solution, the heat and Stokes estimates, and weak–strong uniqueness (Part V).\n- [ESS/LPS](ESS/LPS): Serrin mixed norms, the energy equality under the Serrin condition, the local strong solution, the $H^1$ estimate and continuation, general-exponent uniqueness and smoothing up to the final time (Part VI).\n- [ESS/Witnesses](ESS/Witnesses): explicit examples showing the definitions are inhabited.\n- [comparators](comparators): the two Mathlib-only Challenge/Solution pairs described above.\n- [paper](paper): the manuscript source and PDF.\n- [docs](docs): design notes, deviations, witnesses, verification guide, and the [bibliography](docs/SOURCES.md).\n- [scripts](scripts): the guarded build, checking, comparison, and release-verification tools.\n\n## How this was made\n\nThe Lean development was written between 2026-09-26 and 2026-09-29 using AI\ncoding agents under the author's supervision. Claude Opus 5.5 coordinated\nagents using GPT-6 Luna, GPT-6 Sol and Claude Sonnet 5.5. The author reviewed\nthe theorem statements before proof development and decided the mathematics\nand the corrections to the manuscript. Separate reviews checked the statements\nand the use of intermediate results in the main proofs. Lean checks the\nproofs; the comparator files make their mathematical statements available for\nindependent inspection.\n\n## Contributing, author and license\n\nSee [Contributing](CONTRIBUTING.md) for the source rules and checks, and\n[CITATION.cff](CITATION.cff) for how to cite this work.\n\nThe Lean development is by:\n\n- **Scott Armstrong**, CNRS and Laboratoire Jacques-Louis Lions, Sorbonne\n  Université; Courant Institute School of Mathematics, Computing, and Data\n  Science, New York University. Supported by the European Research Council\n  under the European Union's Horizon Europe programme, grant agreement\n  No. 101200828.\n\nThe Lean library, software, documentation and included manuscript are\ncopyright © 2026 Scott Armstrong and distributed under the\n[Apache License 2.0](LICENSE). Cited third-party works and dependencies retain\ntheir own licenses.\n",1791060113783]