{"id":"f01b1145-6ef2-4b98-a1cf-d6366aafb50b","arxiv_id":"2606.03757","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes Ulam stability for several classes of nuclear C*-algebras with von Neumann targets and derives rigidity results for corona algebras.","lead":"The paper examines Ulam stability for approximate *-homomorphisms of nuclear C*-algebras into von Neumann algebras. It claims stability results for abelian cases and Elliott-classified algebras, plus applications to corona algebra rigidity.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment is based solely on the abstract; the stress-test cannot locate an internal inconsistency or unsupported step because the argument itself is unavailable. Honest non-finding is therefore the appropriate output. No adjustment to the UNVERDICTED verdict is warranted on the basis of the accessible material.","tokens_in":1530,"tokens_out":247,"duration_ms":9670,"concrete_test":"Obtain the full manuscript and verify whether the stability statements in the main theorems hold under the stated nuclearity hypothesis alone, or whether an additional regularity condition (e.g., on the approximate homomorphism or on the target) is tacitly used in the proofs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The provided information consists only of the abstract and a note that the full manuscript is not accessible. Without the detailed argument, definitions of approximate *-homomorphisms, the precise stability notion, or the proofs for the nuclear classes (abelian or Elliott-program), no load-bearing technical concern in the central claim can be isolated. The modeling choice of nuclear domain + von Neumann codomain is explicitly flagged by the reader as necessary for the result as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies Ulam stability for approximate *-homomorphisms of C*-algebras. It proves stability results for several classes of nuclear C*-algebras with respect to von Neumann algebra targets, including abelian C*-algebras and large classes arising in the Elliott classification program. It also discusses permanence properties, counterexamples, and related stability phenomena. As applications, it obtains rigidity and independence results for corona algebras.","tokens_in":1584,"tokens_out":217,"duration_ms":10990,"significance":"If the results hold, the work would provide new stability theorems in operator algebras for nuclear C*-algebras targeting von Neumann algebras, with potential implications for the Elliott classification program and corona algebra rigidity. The modeling choice of nuclear domains and von Neumann codomains is explicitly required for the stated stability.","major_comments":[],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":"Full manuscript text was not accessible beyond the abstract for technical verification of proofs or definitions; this limits assessment of soundness to low confidence with no load-bearing issues identifiable from available material."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and summary of our manuscript on Ulam stability for classes of nuclear C*-algebras. The referee's description accurately reflects the scope of the results, including stability theorems for nuclear domains with von Neumann targets, permanence properties, counterexamples, and applications to corona algebra rigidity. No specific major comments were provided in the report, so we offer no point-by-point responses below. We remain available to address any concrete questions or concerns the referee may have.","responses":[],"tokens_in":1026,"tokens_out":107,"duration_ms":9757,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this paper establishes Ulam stability for approximate *-homomorphisms from several classes of nuclear C*-algebras to von Neumann algebra targets, covering abelian cases and large classes tied to the Elliott program, along with permanence properties, counterexamples, and applications to corona algebra rigidity.\n\nWhat stands out is the choice of setting: nuclear domains paired with von Neumann codomains lets the stability hold, and linking it to classification-relevant classes is a reasonable move if the details work. They also map out related phenomena, which helps clarify the boundaries.\n\nThe soft spot is straightforward: only the abstract is here, so there is no way to inspect the definitions of approximate homomorphisms, the exact stability notion, or the arguments for the specific classes. Without that, it is impossible to tell whether the results are genuinely new or rest on prior work in a way that limits the advance. The von Neumann target choice is flagged as necessary, which is fine but narrows the scope.\n\nThis is for specialists in operator algebras who care about stability questions or the Elliott program. A reader already following those topics could extract value from the stated results and applications. It deserves a serious referee because the topic is central enough that the claims warrant technical review even if revisions are needed.\n\nI would send it to peer review to get the full proofs evaluated.","headline":"The paper claims Ulam stability results for nuclear C*-algebras into von Neumann targets including Elliott classes, but only the abstract is available so the proofs cannot be checked.","tokens_in":2025,"tokens_out":353,"would_cite":false,"duration_ms":15144,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Nuclear C*-algebras are Ulam stable when mapping approximately into von Neumann algebras.","keywords":["Ulam stability","nuclear C*-algebras","von Neumann algebras","Elliott classification","corona algebras","approximate *-homomorphisms","rigidity","permanence properties"],"falsifier":"An explicit nuclear C*-algebra, a von Neumann algebra target, and a sequence of approximate *-homomorphisms whose distance to every true *-homomorphism stays bounded away from zero would falsify the stability claim.","tokens_in":2438,"feed_emoji":"","tokens_out":596,"duration_ms":19309,"temperature":0.7,"pith_summary":"The paper proves that approximate star-homomorphisms from several classes of nuclear C-star algebras into von Neumann algebras stay close to genuine star-homomorphisms. It includes all abelian C-star algebras and many classes covered by the Elliott classification program. A reader would care because the results turn approximation questions into exact algebraic statements and produce rigidity and independence statements for corona algebras. The work also records permanence properties under standard operations and notes some counterexamples outside the stated classes.","feed_headline":"Nuclear C*-algebras exhibit Ulam stability to von Neumann targets","feed_subtitle":"Covers abelian cases and Elliott classes and yields rigidity results for corona algebras.","key_machinery":"Ulam stability for approximate *-homomorphisms from nuclear C*-algebras into von Neumann algebras","core_discovery":"We prove stability results for several classes of nuclear C*-algebras with respect to von Neumann algebra targets, including abelian C*-algebras and large classes arising in the Elliott classification program. Approximate *-homomorphisms from these sources into von Neumann algebras are close to true *-homomorphisms. The paper discusses permanence properties, counterexamples, and related stability phenomena, and obtains rigidity and independence results for corona algebras as applications.","pith_inferences":["The same stability might be tested on concrete examples such as the Cuntz algebra or irrational rotation algebras to extract explicit distance bounds.","If the nuclearity assumption can be weakened while keeping von Neumann targets, the results could apply to a larger family of C*-algebras.","The corona-algebra rigidity statements may interact with existing classification theorems to produce new uniqueness results for extensions."],"forward_implications":["Rigidity results hold for the corona algebras of the covered C*-algebras.","Independence results hold for the corona algebras of the covered C*-algebras.","The stability passes to certain constructions that preserve nuclearity.","Outside the listed classes, stability can fail even for nuclear sources and von Neumann targets."],"fun_headline_variants":["Ulam stability holds for nuclear C*-algebras","Stability for nuclear C*-algebras to von Neumann targets","Abelian and Elliott C*-algebras show Ulam stability","Corona algebras rigid from C*-stability phenomena"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The source algebras must be nuclear and the target algebras must be von Neumann algebras.","fun_headline_variants_meta":{"raw":{"variants":["Ulam stability holds for nuclear C*-algebras","Stability for nuclear C*-algebras to von Neumann targets","Abelian and Elliott C*-algebras show Ulam stability","Corona algebras rigid from C*-stability phenomena"]},"model":"grok-4.3","cost_usd":0.009484,"raw_usage":{"total_tokens":4072,"prompt_tokens":503,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":94840500,"prompt_tokens_details":{"text_tokens":503,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3505,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":503,"tokens_out":64,"duration_ms":25553,"temperature":1.0,"reasoning_tokens":3505,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T07:11:56.384303+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit nuclear C*-algebra, a von Neumann algebra target, and a sequence of approximate *-homomorphisms whose distance to every true *-homomorphism stays bounded away from zero would falsify the stability claim.","supporting_citations":[],"review_version":1}