{"id":"e0506b8f-cad7-4c54-bdde-423f7dc3875f","arxiv_id":"2608.02025","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Assuming OpenAI's soficity criterion, this paper constructs a finitely presented torsion-free non-sofic group.","lead":"OpenAI has announced a non-sofic group, and this paper presents a different example that is additionally finitely presented and torsion-free. If the announced technical criterion it relies on is correct, the result sharpens the resolution of a long-standing classification question about finite approximability of groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 rests entirely on the unproven external Proposition 1.2, cited to a redacted OpenAI announcement; no proof is given in the paper.","rationale":"One good-faith reading: the paper is an expository note showing that, conditional on Proposition 1.2, one can construct a finitely presented torsion-free non-sofic group without Leavitt algebras. Read this way, the paper succeeds: the construction is transparent, uses standard tools, and I found no error in the group-theoretic steps. The proof that Γ∩J=1 (Section 2) is particularly clean: the normality of N in P_1×S and the injectivity of π on S force the S-coordinate to be central in S, hence trivial. The acylindrical hyperbolicity argument for the double HNN extension is also standard, given that P_1∩P_2=1.\n\nHowever, the theorem as stated is unconditional: 'There exists a finitely presented torsion-free non-sofic group.' That existence is not established without Proposition 1.2. The proposition is the entire engine; the paper even calls it 'the key technical ingredient' and 'the key novelty.' No proof is supplied here, and the only source is an OpenAI announcement that has already had a claim redacted. This is exactly the kind of load-bearing external dependency that warrants a conditional verdict.\n\nThe reader's weakest_assumption identifies the same point. I agree. I would not escalate to REJECT, because the construction is sound and the proposition is plausible in light of [Kun16, KT19]; but I would not accept it unconditionally until Proposition 1.2 is verified in the literature or a proof is provided. Hence verdict stays CONDITIONAL.","tokens_in":3892,"tokens_out":26211,"duration_ms":279341,"concrete_test":"Independently reconstruct the proof of Proposition 1.2 from [Kun16] and [KT19]. Concretely: take a sofic approximation σ_n: G → Sym(Ω_n); using property (T) of Γ, show that for every finite F ⊆ J the restrictions σ_n|_F are asymptotically injective, and that the conjugation hypothesis t_1 J t_1^{-1} ≤ Γ converts the asymptotic centrality of Γ into an asymptotic embedding of J. If this derivation fails, or if it requires additional assumptions beyond the stated hypotheses, Theorem 1.3 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.3) does not stand alone: it is a corollary of Proposition 1.2, which is neither proved in this paper nor shown to follow from the cited works [Kun16, KT19]. The proposition is quoted from [Ope26, Proposition 2.3], an OpenAI website announcement that the paper itself notes has already issued a redaction (footnote 2). If Proposition 1.2 is false, or if its proof requires hypotheses not stated here (e.g., that G be finitely presented, or that J be finitely presented rather than merely finitely generated), then the construction produces a group satisfying the hypotheses but yields no contradiction: G could be sofic and J would not be forced to be LEF. The internal group-theoretic steps—universal embedding, the HNN extension over P_1, P_2, the small-cancellation quotient using [Hul16], and the Γ∩J=1 argument—are all coherent and appear to satisfy the hypotheses of Proposition 1.2. So the soft spot is exclusively the external criterion, not the construction itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper announces the existence of a finitely presented torsion-free non-sofic group, conditional on an external soficity criterion. The construction begins with a universal finitely presented torsion-free group U, embeds it into a torsion-free property (T) group P by small cancellation, and then forms a double HNN extension E with stable letters conjugating P to two copies P_1, P_2 inside P, while also containing a finitely presented simple torsion-free subgroup S. A further Hull-type small-cancellation quotient yields a property (T) group G in which S survives injectively. Setting Γ = π(P), t_i = π(u_i), and J = t_1^{-1}π(S)t_1, the author verifies the commutation, intersection, and inclusion conditions of Proposition 1.2. If Proposition 1.2 holds, soficity of G would force J to be LEF, contradicting the fact that J ≅ S is an infinite finitely presented simple group. Hence Theorem 1.3.","tokens_in":4110,"tokens_out":10704,"duration_ms":129116,"significance":"Conditional on Proposition 1.2, this would be a genuine strengthening of the announced existence of a non-sofic group: the example is both finitely presented and torsion-free, and the construction avoids Leavitt algebras entirely, showing the flexibility of the underlying criterion. The internal group-theoretic work is coherent and elegant: the Γ∩J computation is correct, and the use of acylindrical hyperbolicity and small cancellation is standard. The paper also makes a fair point that non-soficity is open in the space of marked groups while torsion-freeness is not, so the direct construction is necessary. However, the paper is explicit that it relies on the same technical criterion as the OpenAI announcement, and that criterion is not proven here; as it stands, the central claim is conditional rather than established.","major_comments":[{"comment":"The entire theorem rests on Proposition 1.2, which is quoted from [Ope26, Proposition 2.3], a non-peer-reviewed website announcement. The paper neither proves this proposition nor shows it follows from the cited works [Kun16, KT19]. Footnote 2 even notes that the OpenAI announcement has issued a redaction. Since Proposition 1.2 is the only bridge from the explicit construction to non-soficity, this is a load-bearing gap, not a presentation issue. A revision must include a complete proof of Proposition 1.2, or a precise derivation from published peer-reviewed results, together with the exact hypotheses. Otherwise Theorem 1.3 should be stated only as conditional on [Ope26].","section":"Section 1, Proposition 1.2 and Theorem 1.3"},{"comment":"The verification of the hypotheses of Proposition 1.2 is mostly sound: Γ and G are infinite property (T) groups, t_iΓt_i^{-1} ≤ Γ, and the computation of [Γ,J]=1 and Γ∩J=1 is correct. However, because Proposition 1.2 is not established in the manuscript, the application inherits any hidden or misstated hypothesis in the external announcement. For instance, the author notes that [Ope26] insists Γ and G be finitely generated and that property (T) supplies this, but the possibility of other unstated hypotheses (e.g., finite presentability of G or J) cannot be excluded without a self-contained statement and proof. This reinforces the need for an independent proof of Proposition 1.2.","section":"Section 2, application of Proposition 1.2"}],"minor_comments":[{"comment":"The sentence 'all property (T) groups are [Kaž67]' is incomplete; it should read 'all property (T) groups are finitely generated'.","section":"Footnote 3"},{"comment":"The phrase 'Since P_1 and P_2 are disjoint edge groups' would be clearer if it explicitly said P_1∩P_2={1} in their identification inside P.","section":"Section 2"},{"comment":"The assertion that 'By universality, P contains a subgroup of the form P_1×P_2×S' deserves one explanatory sentence: P contains a copy of U, and U contains every finitely presented torsion-free group, including P×P×S.","section":"Section 2"},{"comment":"The manuscript contains no equation numbers; numbering the main displayed statements would make refereeing and citation easier.","section":"General"},{"comment":"The historical and speculative remarks about the OpenAI announcement, especially the statement about what 'experts aware of [KT19]' would have proved, are informal and could be removed or significantly softened in a journal version.","section":"Introduction"},{"comment":"References [BSr16], [HO17], and [Par11] are mentioned only as background on the Leavitt algebra connection and are not used in the construction; consider cutting them or making the connection explicit.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the mathematical construction is sound conditional on Proposition 1.2, and the author is transparent about this dependence. The sole blocker is that Proposition 1.2 is quoted from an unreviewed, mutable website source and is not proved or derived from published work. If the author can supply a complete proof, I expect the paper would be publishable. I would also ask the author to treat the OpenAI announcement more neutrally, avoiding speculation about what experts would have found, and to state Theorem 1.3 as conditional if Proposition 1.2 remains external."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does one thing and does it well: it shows that, assuming the OpenAI team's Proposition 1.2, there is a finitely presented torsion-free non-sofic group. That is a strictly sharper counterexample than the announced non-sofic group, since the Leavitt algebra unit group carries torsion. The author is upfront that the novelty is the application of the criterion, not the criterion itself.\n\nThe internal group theory checks out. The mix of a universal f.p. torsion-free group, a Burger–Mozes simple torsion-free group, a random hyperbolic property (T) group, and small cancellation is standard but assembled cleanly. I traced the HNN extension, the acylindrical hyperbolicity step, and the Γ∩J=1 argument; the normal-subgroup trick with N inside P1×S is neat and correct. The paper is also honest about the provenance of Proposition 1.2, flagging the redaction in the OpenAI announcement.\n\nThe soft spot is exactly where the stress-test note puts it: Proposition 1.2 is the load-bearing external input, and it is not proved here or independently verified. The proposition is cited to a website announcement that has already redacted at least one claim (footnote 2). If the proposition is false, or if its proof requires hypotheses not stated in the paper, then Theorem 1.3 is unsupported. That is a genuine limitation, not a manufactured one. However, the author does not hide it, and the paper is best read as a conditional contribution: if the proposition gets a real proof elsewhere, this construction will stand as a useful strengthening.\n\nMinor issues: the self-citation [FF25] for a standard embedding is unnecessary when Osin's original result is in the bibliography; and the phrasing \"random group at suitable density\" is a bit quick for a proof. Neither affects the argument.\n\nWho should read this? Anyone tracking the soficity question or recent OpenAI announcements. It clarifies how much of the non-soficity statement is new machinery versus a transfer argument. I would bring it to a reading group and would cite it in any follow-up on non-sofic groups, even with the caveat. It deserves a serious referee—someone should check the small-cancellation details and, ideally, prompt the author to include a proof or a precise reference for Proposition 1.2. I would send it out rather than desk-reject.","headline":"Conditional on the unproven OpenAI criterion, this is a clean construction of a f.p. torsion-free non-sofic group—worth refereeing, but the dependence on the redacted external proposition is real and cannot be waved off.","tokens_in":4594,"tokens_out":1815,"would_cite":true,"duration_ms":23801,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","20F06","20E26"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finitely presented torsion-free non-sofic group exists, constructed through property (T) and small cancellation, assuming a recently announced soficity-to-LEF criterion.","keywords":["non-sofic groups","soficity conjecture","torsion-free groups","property (T)","LEF groups","finitely presented groups","small cancellation theory","acylindrically hyperbolic groups"],"falsifier":"Construct a sofic approximation for the group G defined in Section 2, or produce any sofic group satisfying the hypotheses of Proposition 1.2 whose subgroup J is a finitely presented infinite simple group.","tokens_in":3756,"feed_emoji":"","tokens_out":7303,"duration_ms":70894,"temperature":0.7,"pith_summary":"This paper establishes that there exists a finitely presented torsion-free group that is not sofic. The construction avoids the Leavitt-algebra framework of the recent non-sofic example and instead builds the group from a universal torsion-free group, a simple torsion-free group, and a property (T) group, using small cancellation. The argument hinges on a technical criterion (Proposition 1.2) quoted from a recent announcement: under certain commutation conditions, soficity of a property (T) group forces a certain subgroup to be LEF. Since the constructed simple subgroup cannot be LEF, the group must be non-sofic. The result strengthens the prior counterexample by removing torsion, and it shows the soficity question has counterexamples even among torsion-free finitely presented groups.","feed_headline":"Finitely presented torsion-free non-sofic group exists","feed_subtitle":"New construction avoids Leavitt algebras and removes torsion, tightening the recent non-sofic counterexample.","key_machinery":"The load-bearing mechanism is Proposition 1.2, a soficity-to-LEF criterion (cited from a recent announcement): if Γ ≤ G are infinite property (T) groups, G is generated by Γ and elements tᵢ with tᵢΓtᵢ⁻¹ ≤ Γ, and there exists a finitely generated J ≤ G with [Γ,J] = Γ∩J = 1 and t₁Jt₁⁻¹ ≤ Γ, then soficity of G forces J to be LEF. The paper's contribution is a torsion-free instance of these hypotheses: it uses the universal finitely presented torsion-free group to embed a simple torsion-free group S into a property (T) group P, then a double HNN extension with Bass–Serre tree provides the commuting conjugate in an acylindrically hyperbolic group, and a small-cancellation quotient supplies proper","core_discovery":"The paper proves Theorem 1.3: there exists a finitely presented torsion-free non-sofic group G. The construction starts with a universal finitely presented torsion-free group U and a finitely presented simple torsion-free group S, embeds U into a property (T) group P, and forms a double HNN extension E in which two copies of P are attached so that a conjugate of S commutes with P and intersects it trivially. A small-cancellation quotient G of E is then taken that preserves property (T), remains torsion-free, and embeds S injectively. Defining Γ = π(P) and J = t₁⁻¹π(S)t₁, the group G satisfies the hypotheses of Proposition 1.2, so if G were sofic, J would be LEF; but J is a finitely presented","pith_inferences":["The same template could be used to seek non-sofic groups with other properties that are not open in the space of marked groups, such as left-orderability or unique product, by choosing different simple torsion-free input groups—though the small-cancellation quotient may not preserve those properties automatically.","A fully self-contained proof of Proposition 1.2 in the literature would remove the dependence on the recent announcement and its redacted passage; the present paper shows only that the theorem follows from that criterion.","If Proposition 1.2 is false, Theorem 1.3 collapses, since the rest of the paper's construction is standard; the paper's exposition effectively isolates the entire risk in that one unproven criterion."],"forward_implications":["If Theorem 1.3 is correct, the soficity question has a negative answer among finitely presented torsion-free groups, not just groups with torsion.","The construction demonstrates that Proposition 1.2 is a flexible tool: it applies beyond Leavitt algebras to a small-cancellation setting, yielding new freedom in choosing the non-sofic group's algebraic properties.","Because the example is finitely presented, it is also a limit of marked groups, so the usual open-property argument for passing from a non-sofic group to a finitely presented one is not needed here.","The argument rules out soficity for the specific group G, and thus any future positive result on soficity of torsion-free groups would have to exclude this example."],"fun_headline_variants":["New torsion-free group is non-sofic","Finitely presented torsion-free non-sofic group","Torsion-free non-sofic group via new construction","A non-sofic group that stays torsion-free"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole argument rests on Proposition 1.2, a soficity-to-LEF criterion that this paper quotes from a recent announcement and does not prove; if that criterion is false, the theorem is unproven.","fun_headline_variants_meta":{"raw":{"variants":["New torsion-free group is non-sofic","Finitely presented torsion-free non-sofic group","Torsion-free non-sofic group via new construction","A non-sofic group that stays torsion-free"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000695,"raw_usage":{"total_tokens":2889,"prompt_tokens":565,"completion_tokens":2324,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":309,"completion_tokens_details":{"reasoning_tokens":2264}},"tokens_in":309,"tokens_out":2324,"duration_ms":20258,"temperature":1.0,"reasoning_tokens":2264,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T16:24:42.334745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a sofic approximation for the group G defined in Section 2, or produce any sofic group satisfying the hypotheses of Proposition 1.2 whose subgroup J is a finitely presented infinite simple group.","supporting_citations":[],"review_version":1}