`ComparatorChallenges/TraceIdealTransportSupport.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. 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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

47 declarations, 17 contain sorry, 0 declared as axioms

abbrev OAI.TraceConeClassification.Positive line 12
Also declared: OAI/Analysis/TraceCone/Basic.lean:29 (same declaration text).
def OAI.TraceConeClassification.Positive.zero line 16
Also declared: OAI/Analysis/TraceCone/Basic.lean:35 (same declaration text).
def OAI.TraceConeClassification.Positive.add line 17
Also declared: OAI/Analysis/TraceCone/Basic.lean:36 (same declaration text).
def OAI.TraceConeClassification.Positive.scale line 18
Also declared: OAI/Analysis/TraceCone/Basic.lean:37 (same declaration text).
def OAI.TraceConeClassification.Positive.starMul line 20
Also declared: OAI/Analysis/TraceCone/Basic.lean:39 (same declaration text).
def OAI.TraceConeClassification.Positive.mulStar line 21
Also declared: OAI/Analysis/TraceCone/Basic.lean:40 (same declaration text).
def OAI.TraceConeClassification.Positive.cut line 22
Also declared: OAI/Analysis/TraceCone/Basic.lean:43 (same declaration text).
structure OAI.TraceConeClassification.ExtendedTrace line 27
Also declared: OAI/Analysis/TraceCone/Basic.lean:50 (same declaration text).
instance OAI.TraceConeClassification.(instance at line 36) line 36
def OAI.TraceConeClassification.CanonicalConvergence line 39
Distinct definitions of this name: OAI/Analysis/TraceCone/Basic.lean:65 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.CanonicalConvergence@path.
def OAI.TraceConeClassification.ExtendedTrace.scale line 48
Also declared: OAI/Analysis/TraceCone/Basic.lean:314 (same declaration text).
theorem OAI.TraceConeClassification.closed_ideal_weight_exists Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 61
Source comment
The proof obligations below make the ideal and equivalence constructions well typed.
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:106 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.closed_ideal_weight_exists@path.
```
theorem closed_ideal_weight_exists
    (I : NonUnitalStarSubalgebra ℂ A) (hI : IsClosed (I : Set A))
    (hleft : ∀ (x y : A), y ∈ I → x * y ∈ I)
    (hright : ∀ (x y : A), x ∈ I → x * y ∈ I) :
    ∃ τ : ExtendedTrace A, ∀ a : Positive A, τ a = if a.1 ∈ I then 0 else ∞ := by
…
```
Mentions: OAI.TraceConeClassification.ExtendedTrace, OAI.TraceConeClassification.Positive
def OAI.TraceConeClassification.ExtendedTrace.finiteSquareSet line 70
Also declared: OAI/Analysis/TraceCone/FiniteIdeal.lean:172 (same declaration text).
lemma OAI.TraceConeClassification.ExtendedTrace.zero_mem_finiteSquareSet Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 71
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:174 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.zero_mem_finiteSquareSet@path.
```
lemma zero_mem_finiteSquareSet : (0 : A) ∈ τ.finiteSquareSet := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.star_mem_finiteSquareSet Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 72
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:178 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.star_mem_finiteSquareSet@path.
```
lemma star_mem_finiteSquareSet {x : A} (hx : x ∈ τ.finiteSquareSet) :
    star x ∈ τ.finiteSquareSet := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.smul_mem_finiteSquareSet Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 74
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:187 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.smul_mem_finiteSquareSet@path.
```
lemma smul_mem_finiteSquareSet (c : ℂ) {x : A} (hx : x ∈ τ.finiteSquareSet) :
    c • x ∈ τ.finiteSquareSet := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.add_mem_finiteSquareSet Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 76
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:202 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.add_mem_finiteSquareSet@path.
```
lemma add_mem_finiteSquareSet {x y : A}
    (hx : x ∈ τ.finiteSquareSet) (hy : y ∈ τ.finiteSquareSet) :
    x + y ∈ τ.finiteSquareSet := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteSquareSet_left Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 79
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:218 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteSquareSet_left@path.
```
lemma mul_mem_finiteSquareSet_left (x : A) {y : A} (hy : y ∈ τ.finiteSquareSet) :
    x * y ∈ τ.finiteSquareSet := by sorry
```
def OAI.TraceConeClassification.ExtendedTrace.finiteSquareAlgebra line 82
Also declared: OAI/Analysis/TraceCone/FiniteIdeal.lean:235 (same declaration text).
def OAI.TraceConeClassification.ExtendedTrace.finiteIdeal line 90
Also declared: OAI/Analysis/TraceCone/FiniteIdeal.lean:244 (same declaration text).
lemma OAI.TraceConeClassification.ExtendedTrace.isClosed_finiteIdeal Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 91
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:246 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.isClosed_finiteIdeal@path.
```
lemma isClosed_finiteIdeal : IsClosed (τ.finiteIdeal : Set A) := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteIdeal_left Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 92
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:249 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteIdeal_left@path.
```
lemma mul_mem_finiteIdeal_left (x : A) {y : A} (hy : y ∈ τ.finiteIdeal) :
    x * y ∈ τ.finiteIdeal := by sorry
```
lemma OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteIdeal_right Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 94
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/FiniteIdeal.lean:256 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.mul_mem_finiteIdeal_right@path.
```
lemma mul_mem_finiteIdeal_right {x : A} (hx : x ∈ τ.finiteIdeal) (y : A) :
    x * y ∈ τ.finiteIdeal := by sorry
…
```
def OAI.TraceConeClassification.ExtendedTrace.add line 105
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:32 (same declaration text).
def OAI.TraceConeClassification.ExtendedTrace.zero line 113
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:44 (same declaration text).
def OAI.TraceConeClassification.ClosedIdeal line 123
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:66 (same declaration text).
def OAI.TraceConeClassification.ExtendedTrace.finIdeal line 130
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:134 (same declaration text).
lemma OAI.TraceConeClassification.ExtendedTrace.lowerSemicontinuous_supportWeight Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 134
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:280 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.lowerSemicontinuous_supportWeight@path.
```
lemma lowerSemicontinuous_supportWeight (τ : ExtendedTrace A) :
    LowerSemicontinuous (fun a : Positive A => if τ a = 0 then (0 : ℝ≥0∞) else ∞) := by sorry
```
Mentions: OAI.TraceConeClassification.ExtendedTrace, OAI.TraceConeClassification.Positive
def OAI.TraceConeClassification.ExtendedTrace.supportWeight line 137
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:296 (same declaration text).
def OAI.TraceConeClassification.ExtendedTrace.zeroIdeal line 149
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:320 (same declaration text).
instance OAI.TraceConeClassification.ClosedIdeal.(instance at line 154) line 154
def OAI.TraceConeClassification.ClosedIdeal.weight line 155
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:76 (same declaration text).
theorem OAI.TraceConeClassification.ClosedIdeal.weight_idempotent Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 157
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:83 (different declaration text). Qualify references as OAI.TraceConeClassification.ClosedIdeal.weight_idempotent@path.
```
theorem weight_idempotent (I : ClosedIdeal A) : I.weight.add I.weight = I.weight := by sorry
```
Mentions: OAI.TraceConeClassification.ClosedIdeal
theorem OAI.TraceConeClassification.ClosedIdeal.finiteIdeal_weight Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 158
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:89 (different declaration text). Qualify references as OAI.TraceConeClassification.ClosedIdeal.finiteIdeal_weight@path.
```
theorem finiteIdeal_weight (I : ClosedIdeal A) : I.weight.finiteIdeal = I.1 := by sorry
```
Mentions: OAI.TraceConeClassification.ClosedIdeal
theorem OAI.TraceConeClassification.ClosedIdeal.le_iff_add_weight Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 159
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:108 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ClosedIdeal.le_iff_add_weight@path.
```
theorem le_iff_add_weight (I J : ClosedIdeal A) :
    I ≤ J ↔ I.weight.add J.weight = I.weight := by sorry
…
```
Mentions: OAI.TraceConeClassification.ClosedIdeal
theorem OAI.TraceConeClassification.ExtendedTrace.idempotent_eq_weight Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 164
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:151 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.ExtendedTrace.idempotent_eq_weight@path.
```
theorem idempotent_eq_weight {τ : ExtendedTrace A} (hτ : τ.add τ = τ) :
    τ.finIdeal.weight = τ := by sorry
…
```
Mentions: OAI.TraceConeClassification.ExtendedTrace
def OAI.TraceConeClassification.IdempotentTrace line 168
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:166 (same declaration text).
def OAI.TraceConeClassification.closedIdealEquivIdempotent line 171
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:170 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.closedIdealEquivIdempotent@path.
def OAI.TraceConeClassification.idempotentTraceEquiv line 179
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:195 (same declaration text).
def OAI.TraceConeClassification.closedIdealEquivOfAdditive line 188
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:205 (same declaration text).
lemma OAI.TraceConeClassification.closedIdealEquivOfAdditive_weight Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 193
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:210 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.closedIdealEquivOfAdditive_weight@path.
```
lemma closedIdealEquivOfAdditive_weight (e : ExtendedTrace A ≃ ExtendedTrace B)
    (he : ∀ τ σ, e (τ.add σ) = (e τ).add (e σ)) (I : ClosedIdeal A) :
    (closedIdealEquivOfAdditive e he I).weight = e I.weight := by sorry
```
Mentions: OAI.TraceConeClassification.ExtendedTrace, OAI.TraceConeClassification.ClosedIdeal, OAI.TraceConeClassification.closedIdealEquivOfAdditive
def OAI.TraceConeClassification.closedIdealOrderEquivOfAdditive line 197
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:217 (same declaration text).
def OAI.TraceConeClassification.PreservesCanonicalConvergence line 206
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:245 (same declaration text).
structure OAI.TraceConeClassification.TraceConeEquiv line 210
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:369 (same declaration text).
def OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso line 222
Also declared: OAI/Analysis/TraceCone/IdealTransport.lean:391 (same declaration text).
theorem OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_finIdeal Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 225
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:400 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_finIdeal@path.
```
theorem idealOrderIso_finIdeal (τ : ExtendedTrace A) :
    F.idealOrderIso τ.finIdeal = (F.toEquiv τ).finIdeal := by
…
```
Mentions: OAI.TraceConeClassification.ExtendedTrace
theorem OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_zeroIdeal Challenge statement: Trace cones and Razak–Jacelon stabilization sorry line 229
L4DEL explanation
The paper classifies separable nuclear complex $C^*$-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their cones of extended lower-semicontinuous traces. The linked formalization proves the ideal-transport part: a trace-cone equivalence induces an order isomorphism of closed two-sided ideals, carrying the ideals determined by finiteness and vanishing of each trace to those of its image. For each closed ideal $I$, it also constructs the trace that is zero on positive elements of $I$ and infinite elsewhere. These traces are exactly the additive idempotents, meaning $\tau+\tau=\tau$, and their addition recovers ideal inclusion. Spatial classification, existence of the stabilization models, norm uniqueness, and final intertwining are outside this supporting comparison.
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.Analysis.TraceCone.IdealTransport as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TraceCone/IdealTransport.lean:411 (same statement text; different proof or body). Qualify references as OAI.TraceConeClassification.TraceConeEquiv.idealOrderIso_zeroIdeal@path.
```
theorem idealOrderIso_zeroIdeal (τ : ExtendedTrace A) :
    F.idealOrderIso τ.zeroIdeal = (F.toEquiv τ).zeroIdeal := by
…
```
Mentions: OAI.TraceConeClassification.ExtendedTrace