`ComparatorChallenges/ContinuumCoulombHardness.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

101 declarations, 2 contain sorry, 0 declared as axioms

abbrev OAI.ContinuumCoulomb.BinaryEncoding.BitString line 12
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:15 (same declaration text).
structure OAI.ContinuumCoulomb.BinaryEncoding.Codec line 14
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:17 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.encodeDigits line 19
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:28 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.readDigits line 23
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:32 (same declaration text).
theorem OAI.ContinuumCoulomb.BinaryEncoding.readDigits_encode line 28
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:37 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.decodeDigits line 34
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:43 (same declaration text).
theorem OAI.ContinuumCoulomb.BinaryEncoding.decodeDigits_bits line 38
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:47 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.natural line 43
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:52 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.encodeUnary line 48
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:69 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.readUnary line 52
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:73 (same declaration text).
theorem OAI.ContinuumCoulomb.BinaryEncoding.readUnary_encode line 57
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:78 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.unary line 63
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:84 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.integer line 68
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:95 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.pair line 79
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:106 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.readCount line 86
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:113 (same declaration text).
theorem OAI.ContinuumCoulomb.BinaryEncoding.readCount_encode line 90
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:117 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.list line 97
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:124 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.equiv line 103
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:130 (same declaration text).
def OAI.ContinuumCoulomb.BinaryEncoding.sum line 108
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/BinaryEncoding.lean:135 (same declaration text).
abbrev OAI.Coulomb.Space line 125
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:12 (same declaration text).
abbrev OAI.Coulomb.Configuration line 126
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:13 (same declaration text).
abbrev OAI.Coulomb.Spins line 127
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:14 (same declaration text).
structure OAI.Coulomb.Nuclei line 129
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:16 (same declaration text).
def OAI.Coulomb.position line 136
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:25 (same declaration text).
def OAI.Coulomb.coulombKernel line 139
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:28 (same declaration text).
def OAI.Coulomb.attraction line 141
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:30 (same declaration text).
def OAI.Coulomb.permute line 144
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:33 (same declaration text).
structure OAI.Coulomb.H1Vector line 148
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:37 (same declaration text).
def OAI.Coulomb.Antisymmetric line 158
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:47 (same declaration text).
def OAI.Coulomb.mass line 163
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:52 (same declaration text).
def OAI.Coulomb.kinetic line 166
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:55 (same declaration text).
def OAI.Coulomb.nuclearEnergy line 169
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:58 (same declaration text).
def OAI.Coulomb.pairEnergy line 172
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:61 (same declaration text).
def OAI.Coulomb.form line 177
Also declared: OAI/Analysis/CoulombRadii/FieldAnalysis/Space.lean:66 (same declaration text).
abbrev OAI.ContinuumCoulomb.SourceSpinBasis line 187
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourceHamiltonian.lean:26 (same declaration text).
def OAI.ContinuumCoulomb.sourceTensor line 189
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourcePauli.lean:13 (same declaration text).
theorem OAI.ContinuumCoulomb.bellScale_atLeastTwo line 194
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/ManyBody/GlobalSinglet.lean:17 (same declaration text).
def OAI.ContinuumCoulomb.bellScale line 199
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/ManyBody/GlobalSinglet.lean:22 (same declaration text).
inductive OAI.ContinuumCoulomb.QMAGate line 203
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:18 (same declaration text).
def OAI.ContinuumCoulomb.qmaGateEquiv line 209
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:24 (same declaration text).
def OAI.ContinuumCoulomb.qmaGateCodec line 221
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:36 (same declaration text).
structure OAI.ContinuumCoulomb.QMACircuit line 224
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:39 (same declaration text).
def OAI.ContinuumCoulomb.qmaCircuitEquiv line 230
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:46 (same declaration text).
def OAI.ContinuumCoulomb.qmaCircuitCodec line 236
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:52 (same declaration text).
def OAI.ContinuumCoulomb.QMAGate.WellFormed line 239
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:55 (same declaration text).
def OAI.ContinuumCoulomb.QMACircuit.WellFormed line 244
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:60 (same declaration text).
def OAI.ContinuumCoulomb.qmaHadamard line 247
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:66 (same declaration text).
def OAI.ContinuumCoulomb.qmaPhaseT line 250
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:70 (same declaration text).
def OAI.ContinuumCoulomb.qmaQubit line 253
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:73 (same declaration text).
def OAI.ContinuumCoulomb.qmaControlledNot line 256
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:76 (same declaration text).
def OAI.ContinuumCoulomb.qmaGateMatrix line 260
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:80 (same declaration text).
def OAI.ContinuumCoulomb.qmaCircuitMatrix line 269
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:89 (same declaration text).
def OAI.ContinuumCoulomb.qmaInitialState line 273
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:95 (same declaration text).
def OAI.ContinuumCoulomb.qmaAcceptance line 280
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:103 (same declaration text).
structure OAI.ContinuumCoulomb.QuantumVerifier line 286
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:109 (same declaration text).
structure OAI.ContinuumCoulomb.PromiseProblem line 293
Distinct definitions of this name: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:116 (same statement text; different proof or body). Qualify references as OAI.ContinuumCoulomb.PromiseProblem@path.
theorem OAI.ContinuumCoulomb.inQMA_two_atLeastTwo line 312
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:123 (same declaration text).
theorem OAI.ContinuumCoulomb.inQMA_three_atLeastTwo line 315
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:126 (same declaration text).
def OAI.ContinuumCoulomb.InQMA line 320
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:132 (same declaration text).
structure OAI.ContinuumCoulomb.PolynomialManyOne line 329
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:142 (same declaration text).
def OAI.ContinuumCoulomb.QMAHard line 336
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Quantum/QuantumVerifier.lean:149 (same declaration text).
structure OAI.ContinuumCoulomb.BinaryRational line 339
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourcePromise.lean:15 (same declaration text).
def OAI.ContinuumCoulomb.BinaryRational.value line 344
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourcePromise.lean:20 (same declaration text).
def OAI.ContinuumCoulomb.binaryRationalEquiv line 347
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourcePromise.lean:23 (same declaration text).
def OAI.ContinuumCoulomb.binaryRationalCodec line 353
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/SourcePromise.lean:29 (same declaration text).
abbrev OAI.ContinuumCoulomb.Position line 355
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:22 (same declaration text).
abbrev OAI.ContinuumCoulomb.Configuration line 356
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:23 (same declaration text).
abbrev OAI.ContinuumCoulomb.SpinConfiguration line 357
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:24 (same declaration text).
structure OAI.ContinuumCoulomb.NuclearData line 359
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:27 (same declaration text).
structure OAI.ContinuumCoulomb.UnitNuclearData line 369
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:38 (same declaration text).
def OAI.ContinuumCoulomb.UnitNuclearData.toNuclearData line 377
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:46 (same declaration text).
theorem OAI.ContinuumCoulomb.realPosition_atLeastTwo line 389
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:58 (same declaration text).
def OAI.ContinuumCoulomb.realPosition line 394
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:63 (same declaration text).
theorem OAI.ContinuumCoulomb.realPosition_injective line 399
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:68 (same declaration text).
def OAI.ContinuumCoulomb.NuclearData.toNuclei line 406
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:76 (same declaration text).
abbrev OAI.ContinuumCoulomb.FormState line 415
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:86 (same declaration text).
abbrev OAI.ContinuumCoulomb.normSq line 417
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:88 (same declaration text).
abbrev OAI.ContinuumCoulomb.kineticEnergy line 418
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:89 (same declaration text).
def OAI.ContinuumCoulomb.coulombForm line 420
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:92 (same declaration text).
def OAI.ContinuumCoulomb.unitCoulombForm line 423
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:95 (same declaration text).
def OAI.ContinuumCoulomb.groundEnergy line 426
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:99 (same declaration text).
def OAI.ContinuumCoulomb.unitGroundEnergy line 430
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Model.lean:103 (same declaration text).
structure OAI.ContinuumCoulomb.BinaryPosition line 434
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:14 (same declaration text).
def OAI.ContinuumCoulomb.BinaryPosition.value line 440
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:20 (same declaration text).
def OAI.ContinuumCoulomb.BinaryPosition.Valid line 442
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:22 (same declaration text).
def OAI.ContinuumCoulomb.binaryPositionEquiv line 445
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:25 (same declaration text).
def OAI.ContinuumCoulomb.binaryPositionCodec line 451
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:31 (same declaration text).
structure OAI.ContinuumCoulomb.BinaryCoulomb line 454
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:34 (same declaration text).
def OAI.ContinuumCoulomb.binaryCoulombEquiv line 461
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:41 (same declaration text).
def OAI.ContinuumCoulomb.binaryCoulombCodec line 468
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:48 (same declaration text).
structure OAI.ContinuumCoulomb.BinaryCoulomb.Valid line 472
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:52 (same declaration text).
def OAI.ContinuumCoulomb.BinaryCoulomb.toData line 482
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:62 (same declaration text).
structure OAI.ContinuumCoulomb.UnitCoulomb line 492
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:72 (same declaration text).
def OAI.ContinuumCoulomb.unitCoulombEquiv line 499
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:79 (same declaration text).
def OAI.ContinuumCoulomb.unitCoulombCodec line 505
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:85 (same declaration text).
structure OAI.ContinuumCoulomb.UnitCoulomb.Valid line 509
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:89 (same declaration text).
def OAI.ContinuumCoulomb.UnitCoulomb.toData line 518
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:98 (same declaration text).
def OAI.ContinuumCoulomb.binaryCoulombPromise line 526
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:108 (same declaration text).
def OAI.ContinuumCoulomb.unitCoulombPromise line 536
Also declared: OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/NuclearPromise.lean:118 (same declaration text).
theorem OAI.ContinuumCoulomb.unit_coulomb_qmaHard Challenge statement: QMA-hardness of continuum Coulomb energy sorry line 546
L4DEL explanation
The formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space when positive integer nuclear charges are encoded in binary. The instances have distinct rational nuclear positions and a unary electron count, and the energy ranges over antisymmetric continuum states and all spin sectors. A deterministic polynomial-time many-one reduction produces the promise gap with threshold separation at least one, while keeping the complete output length polynomial. The same Comparator file also contains the companion unit-charge result. The formalization proves QMA-hardness of approximating the electronic ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum. The reduction is deterministic and polynomial time, with rational nuclear positions and separated rational energy thresholds. No orbital basis, magnetic field, additional external potential, or binding premise is part of the input. The same Comparator file also includes the companion hardness result when positive integer nuclear charges are encoded in binary.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.MathematicalPhysics.ContinuumCoulomb.Reduction.Hardness as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Hardness.lean:14 (same statement text; different proof or body). Qualify references as OAI.ContinuumCoulomb.unit_coulomb_qmaHard@path.
```
theorem unit_coulomb_qmaHard : QMAHard unitCoulombCodec.encode unitCoulombPromise := by
…
```
Mentions: OAI.ContinuumCoulomb.QMAHard, OAI.ContinuumCoulomb.unitCoulombPromise
theorem OAI.ContinuumCoulomb.binary_coulomb_qmaHard Challenge statement: QMA-hardness of continuum Coulomb energy sorry line 549
L4DEL explanation
The formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space when positive integer nuclear charges are encoded in binary. The instances have distinct rational nuclear positions and a unary electron count, and the energy ranges over antisymmetric continuum states and all spin sectors. A deterministic polynomial-time many-one reduction produces the promise gap with threshold separation at least one, while keeping the complete output length polynomial. The same Comparator file also contains the companion unit-charge result. The formalization proves QMA-hardness of approximating the electronic ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum. The reduction is deterministic and polynomial time, with rational nuclear positions and separated rational energy thresholds. No orbital basis, magnetic field, additional external potential, or binding premise is part of the input. The same Comparator file also includes the companion hardness result when positive integer nuclear charges are encoded in binary.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.MathematicalPhysics.ContinuumCoulomb.Reduction.Hardness as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Hardness.lean:17 (same statement text; different proof or body). Qualify references as OAI.ContinuumCoulomb.binary_coulomb_qmaHard@path.
```
theorem binary_coulomb_qmaHard : QMAHard binaryCoulombCodec.encode binaryCoulombPromise := by
…
```
Mentions: OAI.ContinuumCoulomb.QMAHard, OAI.ContinuumCoulomb.binaryCoulombPromise