`ComparatorChallenges/CirculantHadamard.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

6 declarations, 1 contain sorry, 0 declared as axioms

def OAI.CirculantHadamard.IsSign line 9
Also declared: ComparatorChallenges/EvenBarker.lean:9 (same declaration text), OAI/LinearAlgebra/CirculantHadamard/Model.lean:9 (same declaration text).
abbrev OAI.CirculantHadamard.RealMatrix line 12
Also declared: OAI/LinearAlgebra/CirculantHadamard/Model.lean:12 (same declaration text).
def OAI.CirculantHadamard.IsCirculant line 14
Also declared: OAI/LinearAlgebra/CirculantHadamard/Model.lean:14 (same declaration text).
def OAI.CirculantHadamard.IsSignHadamard line 17
Also declared: OAI/LinearAlgebra/CirculantHadamard/Model.lean:17 (same declaration text).
def OAI.CirculantHadamard.ExistsRealCirculantHadamard line 21
Also declared: OAI/LinearAlgebra/CirculantHadamard/Model.lean:21 (same declaration text).
theorem OAI.CirculantHadamard.exists_iff_order_one_or_four Challenge statement: Real circulant Hadamard matrices have order one or four sorry line 24
L4DEL explanation
a real circulant Hadamard matrix has order 1 or 4
— OpenAI circulant-Hadamard manuscript (abstract)
It will be shown here that for odd n, these are all the sequences with the property |cₖ| ≤ 1
— Richard Turyn and James Storer (1961), original paper
A positive-order real matrix whose entries are signs, whose rows are cyclic shifts, and whose rows are mutually orthogonal exists exactly in orders one and four.
1. IsCirculant H means that one function h supplies every entry as H i j = h (j-i), so each row is a cyclic shift of the same sign pattern.
2. IsSignHadamard H requires every entry to equal 1 or -1 and requires H * Hᵀ = nI. ExistsRealCirculantHadamard n existentially combines these conditions.
3. For every positive n, the theorem proves an equivalence: such a matrix exists if and only if n=1 or n=4. Both existence directions and every nonexistence case are part of the same statement.
4. This declaration concerns real sign-valued circulant Hadamard matrices only. The Barker-sequence corollary discussed by the manuscript is a separate comparator statement. The body is sorry, paired with OAI.LinearAlgebra.CirculantHadamard.Main.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI circulant-Hadamard manuscript (abstract), Pinned OpenAI formalization note 179, Richard Turyn and James Storer (1961), original paper
Distinct definitions of this name: OAI/LinearAlgebra/CirculantHadamard/Main.lean:21 (same statement text; different proof or body). Qualify references as OAI.CirculantHadamard.exists_iff_order_one_or_four@path.

circulant matrix: a matrix in which every row is a cyclic shift of one common row
Hadamard matrix: here, a sign-valued square matrix whose product with its transpose is its order times the identity
```
theorem exists_iff_order_one_or_four (n : ℕ) (hn : 0 < n) :
    ExistsRealCirculantHadamard n ↔ n = 1 ∨ n = 4 := by
…
```
Mentions: OAI.CirculantHadamard.ExistsRealCirculantHadamard