`ComparatorChallenges/UniformFourier.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

58 declarations, 2 contain sorry, 0 declared as axioms

Exact scalar and index RAM used for the algorithm

Finite, first-order, expression/program syntax. Arrays below are random-access arrays of registers (an array-valued register holds its address). An expression building an array uses a fresh array; sow builds by a loop of writes instead of a loop supplying consecutive entries. Programs have their usual sequential stack evaluation. Evaluation here records work (including stack, index and array work), the high water mark on produced/logically looped-over integer values (addresses within arrays are integers too), the value, and the side condition required along the executed path (in particular no division by zero). The charged units of work are constant time; as usual constant rescalings between ways to charge a fixed number of scalar or address operations are immaterial.

Enough address bits must be provided BOTH for the integer high water mark (and inputs), AND for the work bound together with input size, as will be required in the theorem. Indeed we use the particularly simple RAM convention where temporaries aren't reused: one work unit can leave at most a constant number of cells (stack, pairs or array cells); whenever many array entries are made their number is charged. So a constant number of words of those sizes also suffices for arena addresses. Nothing charges constant work for changing an already published array to a different array as a whole. Register and stack layouts, and the convention whether registers hold pairs or addresses of pairs, only have constant factors (the program is fixed).

Scalar and data field registers are separated. Tests and loop/request sizes are on nonnegative integer registers. There is no instruction for an arbitrary complex constant. A supplied specified root will be an input. Divisions are on prepared scalars; valid runs require every such denominator nonzero; the side condition is not a test performed by the algorithm. The Bool mode on the syntax allows or prohibits general products (two-input convolution versus the transform); even when allowed we use separate register kinds before and after the bilinear products.

Definitions are mathematical operation-count semantics, not a model of running Lean code. E.g. a function on Fin n represents the n cells of a random-access array, not a function closure at runtime.

def OAI.ExactFourier.zeta line 6
Also declared: ComparatorChallenges/ExactFourier.lean:44 (same declaration text), OAI/Computability/FourierCircuit/Core.lean:40 (same declaration text).
structure OAI.PowerSaving.Tape Register arrays line 51
Source documentation
Register arrays.
L4DEL explanation
Register arrays.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:47 (same declaration text).
```
structure Tape (α : Type u) where
…
```
def OAI.PowerSaving.Tape.empty line 57
Also declared: OAI/Computability/FourierTransform/RAM.lean:53 (same declaration text).
def OAI.PowerSaving.Tape.tab Materialized array line 60
Source documentation
Materialized array.
L4DEL explanation
Materialized array.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:56 (same declaration text).
```
def tab (n : ℕ) (f : ℕ → α) : Tape α := ⟨n, fun i => f i.val⟩
```
Mentions: OAI.PowerSaving.Tape
def OAI.PowerSaving.Tape.look Read with fixed out of bounds default line 63
Source documentation
Read with fixed out of bounds default.
L4DEL explanation
Read with fixed out of bounds default.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:59 (same declaration text).
```
def look (v : Tape α) (i : ℕ) (z : α) : α :=
…
```
Mentions: OAI.PowerSaving.Tape
theorem OAI.PowerSaving.Tape.look_of_lt line 66
Also declared: OAI/Computability/FourierTransform/RAM.lean:62 (same declaration text).
theorem OAI.PowerSaving.Tape.look_of_le line 69
Also declared: OAI/Computability/FourierTransform/RAM.lean:65 (same declaration text).
def OAI.PowerSaving.Tape.set Semantics of a write into fresh storage; out-of-range writes are ignored line 74
Source documentation
Semantics of a write into fresh storage; out-of-range writes are ignored. Only used during construction, never a whole-tape functional update instruction.
L4DEL explanation
Semantics of a write into fresh storage; out-of-range writes are ignored. Only used during construction, never a whole-tape functional update instruction.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:70 (same declaration text).
```
def set (v : Tape α) (i : ℕ) (x : α) : Tape α :=
…
```
Mentions: OAI.PowerSaving.Tape
def OAI.PowerSaving.Tape.sow line 77
Also declared: OAI/Computability/FourierTransform/RAM.lean:73 (same declaration text).
def OAI.PowerSaving.Tape.map line 84
Also declared: OAI/Computability/FourierTransform/RAM.lean:80 (same declaration text).
inductive OAI.PowerSaving.RAM.Paint line 90
Also declared: OAI/Computability/FourierTransform/RAM.lean:86 (same declaration text).
inductive OAI.PowerSaving.RAM.Ty line 96
Also declared: OAI/Computability/FourierTransform/RAM.lean:92 (same declaration text).
abbrev OAI.PowerSaving.RAM.sc line 104
Also declared: OAI/Computability/FourierTransform/RAM.lean:100 (same declaration text).
def OAI.PowerSaving.RAM.Ty.T line 106
Also declared: OAI/Computability/FourierTransform/RAM.lean:102 (same declaration text).
def OAI.PowerSaving.RAM.Ty.blank line 112
Also declared: OAI/Computability/FourierTransform/RAM.lean:108 (same declaration text).
structure OAI.PowerSaving.RAM.Bill One basic work unit below pays for O(1) primitive RAM operations; includes loop and stack overhead line 121
Source documentation
One basic work unit below pays for O(1) primitive RAM operations; includes loop and stack overhead. peak tracks produced/needed integer quantities beyond inputs (total allocation for stack and array bases is covered separately by work).
L4DEL explanation
One basic work unit below pays for O(1) primitive RAM operations; includes loop and stack overhead. peak tracks produced/needed integer quantities beyond inputs (total allocation for stack and array bases is covered separately by work).
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:117 (same declaration text).
```
structure Bill (α : Type u) where
…
```
def OAI.PowerSaving.RAM.Bill.pay Charge some work and optionally a new word bound line 130
Source documentation
Charge some work and optionally a new word bound.
L4DEL explanation
Charge some work and optionally a new word bound.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:126 (same declaration text).
```
def pay (b : Bill α) (n : ℕ) (m : ℕ) : Bill α :=
…
```
Mentions: OAI.PowerSaving.RAM.Bill
def OAI.PowerSaving.RAM.Bill.pass Both computations are evaluated line 134
Source documentation
Both computations are evaluated.
L4DEL explanation
Both computations are evaluated.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:130 (same declaration text).
```
def pass (b : Bill α) (f : α → Bill β) : Bill β :=
…
```
Mentions: OAI.PowerSaving.RAM.Bill
def OAI.PowerSaving.RAM.Bill.one line 138
Also declared: OAI/Computability/FourierTransform/RAM.lean:134 (same declaration text).
def OAI.PowerSaving.RAM.Bill.word line 139
Also declared: OAI/Computability/FourierTransform/RAM.lean:135 (same declaration text).
def OAI.PowerSaving.RAM.Bill.steps Execute f 0 through f (n-1), iterating an accumulator line 142
Source documentation
Execute f 0 through f (n-1), iterating an accumulator.
L4DEL explanation
Execute f 0 through f (n-1), iterating an accumulator.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:138 (same declaration text).
```
def steps (init : α) (f : ℕ → α → Bill α) : ℕ → Bill α
  | 0 => Bill.one init
  | i+1 => ((steps init f i).pass (f i)).pay 1 (i+1)

/-- Build Tape len by count writes (initialized with a blank z). -/
```
Mentions: OAI.PowerSaving.RAM.Bill, OAI.PowerSaving.RAM.Bill.one, OAI.PowerSaving.RAM.Bill.pass, OAI.PowerSaving.RAM.Bill.pay, OAI.PowerSaving.Tape
def OAI.PowerSaving.RAM.Bill.sow Build Tape len by count writes (initialized with a blank z) line 147
Source documentation
Build Tape len by count writes (initialized with a blank z).
L4DEL explanation
Build Tape len by count writes (initialized with a blank z).
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:143 (same declaration text).
```
def sow (len count : ℕ) (z : α) (f : ℕ → Bill (ℕ × α)) : Bill (Tape α) :=
…
```
Mentions: OAI.PowerSaving.RAM.Bill, OAI.PowerSaving.Tape
def OAI.PowerSaving.RAM.Bill.tab Build Tape len line 152
Source documentation
Build Tape len. As for sow we can first allocate then perform writes.
L4DEL explanation
Build Tape len. As for sow we can first allocate then perform writes.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:148 (same declaration text).
```
def tab (len : ℕ) (z : α) (f : ℕ → Bill α) : Bill (Tape α) :=
…
```
Mentions: OAI.PowerSaving.RAM.Bill, OAI.PowerSaving.Tape
inductive OAI.PowerSaving.RAM.NOp line 157
Also declared: OAI/Computability/FourierTransform/RAM.lean:153 (same declaration text).
inductive OAI.PowerSaving.RAM.Atom line 160
Also declared: OAI/Computability/FourierTransform/RAM.lean:156 (same declaration text).
def OAI.PowerSaving.RAM.NOp.run line 177
Also declared: OAI/Computability/FourierTransform/RAM.lean:173 (same declaration text).
def OAI.PowerSaving.RAM.Atom.run line 185
Also declared: OAI/Computability/FourierTransform/RAM.lean:181 (same declaration text).
abbrev OAI.PowerSaving.RAM.Port line 201
Also declared: OAI/Computability/FourierTransform/RAM.lean:197 (same declaration text).
inductive OAI.PowerSaving.RAM.Code call is internal for bounded recursion; a program has Port = none line 206
Source documentation
call is internal for bounded recursion; a program has Port = none. loop, tab and sow retain the input to access parameters in the body. descend sets up bounded recursion, reading fuel and initial argument.
L4DEL explanation
call is internal for bounded recursion; a program has Port = none. loop, tab and sow retain the input to access parameters in the body. descend sets up bounded recursion, reading fuel and initial argument.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Computability/FourierTransform/RAM.lean:202 (same declaration text).
```
inductive Code (allow : Bool) : Port → Ty → Ty → Type
  | atom {r s t} (op : Atom allow s t) : Code allow r s t
  | comp {r s t u} (f : Code allow r s t) (g : Code allow r t u) : Code allow r s u
  | fork {r s t u} (f : Code allow r s t) (g : Code allow r s u) : Code allow r s (p t u)
  | ifz {r s t} (q : Code allow r s w) (f g : Code allow r s t) : Code allow r s t
  | loop {r s t} (n : Code allow r s w) (init : Code allow r s t)
      (body : Code allow r (p s (p w t)) t) : Code allow r s t
  | tab {r s t} (n : Code allow r s w)
      (body : Code allow r (p s w) t) : Code allow r s (a t)
  | sow {r s t} (n count : Code allow r s w)
      (body : Code allow r (p s w) (p w t)) : Code allow r s (a t)
  | importClosed {r s t} (f : Code allow none s t) : Code allow r s t
…
```
Mentions: OAI.PowerSaving.RAM.Port, OAI.PowerSaving.RAM.Ty, OAI.PowerSaving.RAM.Atom
def OAI.PowerSaving.RAM.Handler line 222
Also declared: OAI/Computability/FourierTransform/RAM.lean:218 (same declaration text).
def OAI.PowerSaving.RAM.depthRun line 226
Also declared: OAI/Computability/FourierTransform/RAM.lean:222 (same declaration text).
def OAI.PowerSaving.RAM.Code.run line 231
Also declared: OAI/Computability/FourierTransform/RAM.lean:227 (same declaration text).
abbrev OAI.PowerSaving.RAM.Prog line 251
Also declared: OAI/Computability/FourierTransform/RAM.lean:247 (same declaration text).
abbrev OAI.PowerSaving.RAM.run line 253
Also declared: OAI/Computability/FourierTransform/RAM.lean:249 (same declaration text).
def OAI.PowerSaving.paperM line 264
Also declared: OAI/Computability/FourierTransform/Goal.lean:25 (same declaration text).
def OAI.PowerSaving.paperW line 265
Also declared: OAI/Computability/FourierTransform/Goal.lean:26 (same declaration text).
def OAI.PowerSaving.paperDelta line 266
Also declared: OAI/Computability/FourierTransform/Goal.lean:27 (same declaration text).
def OAI.PowerSaving.paperMultiplier line 267
Also declared: OAI/Computability/FourierTransform/Goal.lean:28 (same declaration text).
def OAI.PowerSaving.theta line 268
Also declared: OAI/Computability/FourierTransform/Goal.lean:29 (same declaration text).
def OAI.PowerSaving.decimalExponent line 269
Also declared: OAI/Computability/FourierTransform/Goal.lean:30 (same declaration text).
def OAI.PowerSaving.paperTime line 271
Also declared: OAI/Computability/FourierTransform/Goal.lean:32 (same declaration text).
def OAI.PowerSaving.decimalTime line 274
Also declared: OAI/Computability/FourierTransform/Goal.lean:35 (same declaration text).
def OAI.PowerSaving.nlogn line 276
Also declared: OAI/Computability/FourierTransform/Goal.lean:37 (same declaration text).
abbrev OAI.PowerSaving.root line 278
Also declared: OAI/Computability/FourierTransform/Goal.lean:12 (same declaration text).
abbrev OAI.PowerSaving.Vec line 280
Also declared: OAI/Computability/FourierTransform/Goal.lean:14 (same declaration text).
def OAI.PowerSaving.dft line 282
Also declared: OAI/Computability/FourierTransform/Goal.lean:18 (same declaration text).
def OAI.PowerSaving.conv line 285
Also declared: OAI/Computability/FourierTransform/Goal.lean:21 (same declaration text).
abbrev OAI.PowerSaving.dftInKind line 288
Also declared: OAI/Computability/FourierTransform/Goal.lean:39 (same declaration text).
abbrev OAI.PowerSaving.dftOutKind line 289
Also declared: OAI/Computability/FourierTransform/Goal.lean:40 (same declaration text).
abbrev OAI.PowerSaving.convInKind line 290
Also declared: OAI/Computability/FourierTransform/Goal.lean:41 (same declaration text).
abbrev OAI.PowerSaving.convOutKind line 291
Also declared: OAI/Computability/FourierTransform/Goal.lean:42 (same declaration text).
def OAI.PowerSaving.DFTProgram line 293
Also declared: OAI/Computability/FourierTransform/Goal.lean:48 (same declaration text).
def OAI.PowerSaving.ConvProgram line 305
Also declared: OAI/Computability/FourierTransform/Goal.lean:62 (same declaration text).
def OAI.PowerSaving.TimeBounds line 317
Also declared: OAI/Computability/FourierTransform/Goal.lean:75 (same declaration text).
def OAI.PowerSaving.DFTGoal line 321
Also declared: OAI/Computability/FourierTransform/Goal.lean:79 (same declaration text).
def OAI.PowerSaving.ConvGoal line 323
Also declared: OAI/Computability/FourierTransform/Goal.lean:81 (same declaration text).
theorem OAI.PowerSaving.transform_main Challenge statement: Uniform exact DFT and convolution with a power saving sorry line 326
Source documentation
One fixed program computes every positive-length canonical DFT, including preparation.
L4DEL explanation
One fixed program computes every positive-length canonical DFT, including preparation.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Computability.FourierTransform.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/FourierTransform/Main.lean:137 (same statement text; different proof or body). Qualify references as OAI.PowerSaving.transform_main@path.
```
theorem transform_main : DFTGoal := by
…
```
Mentions: OAI.PowerSaving.DFTGoal
theorem OAI.PowerSaving.convolution_main Challenge statement: Uniform exact DFT and convolution with a power saving sorry line 330
Source documentation
One fixed bilinear program computes convolution with the same asymptotic bounds.
L4DEL explanation
One fixed bilinear program computes convolution with the same asymptotic bounds.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Computability.FourierTransform.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/FourierTransform/Main.lean:141 (same statement text; different proof or body). Qualify references as OAI.PowerSaving.convolution_main@path.
```
theorem convolution_main : ConvGoal := by
…
```
Mentions: OAI.PowerSaving.ConvGoal