{"id":"cf01e464-0fec-4bd1-8c93-79965887ae9a","arxiv_id":"2607.20849","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"All three simples of a three-vertex Gap-A failure cannot all have infinite projective dimension; the two-infinite case is forced to be a 'mutual dumbbell', and a protected corner forces Ext^n(S_x,S_x) nonzero in every degree.","lead":"Han's conjecture says an algebra with eventually vanishing Hochschild homology must have finite global dimension. This paper proves that any 'Gap-A' counterexample on three strongly connected vertices can only be one of two very specific shapes, and that a new protected-corner theorem forces endless self-extensions at certain vertices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Protection hypothesis is stated with reversed Peirce components: as written, yΛx=0 for an external in-neighbour y is impossible because the arrow y→x lies in yΛx. The intended condition (no path x→y) is not used consistently in Lemma 9.11/Theorem 9.12.","rationale":"The reader's weakest_assumption correctly identified the internal Peirce-decomposition induction as fragile, but did not isolate the Peirce-component reversal that makes the main hypothesis literally unsatisfiable under the paper's own definitions. This is the most load-bearing concern because Theorem 9.12 is the engine of the whole paper: it powers the protected-corner theorem, the contradiction in the all-infinite v=3 case (Theorem 9.14(1)), and the dumbbell classification (Theorem 9.14(2)). If the hypothesis is vacuous, the central claim as stated proves nothing; if it is corrected to e_xΛe_y=0 for external in-neighbours y, the intended proof may be sound but the manuscript does not state this consistently and Lemma 9.11's written proof applies the vanishing to the wrong Peirce space. The concrete test—re-deriving Lemma 9.11 and the induction under an explicit convention—settles whether this is a typographical issue or a substantive gap. An external dependence on [9] is also present, as the reader noted, but that is a normal citation risk; the internal notational inconsistency is a correctness risk that the reader can act on. The verdict stays CONDITIONAL: the result may be true once the convention is fixed and the proof rechecked, but as written the central theorem is not verifiable.","tokens_in":30506,"tokens_out":34451,"duration_ms":285731,"concrete_test":"Fix a single convention: write e_aΛe_b for paths a→b, so an arrow y→x is a nonzero element of e_yΛe_x. Re-state the protected-corner hypothesis as: e_xΛe_y=0 for every y with e_yΛe_x≠0 (arrow y→x). Re-derive Lemma 9.11 and the induction of Theorem 9.12 in this notation, checking in particular that the factor p′q in Lemma 9.11 belongs to e_xΛe_{y0} and vanishes by the corrected hypothesis e_xΛe_{y0}=0. If the re-derivation goes through without any additional hypothesis, the concern is a typographical notational issue; if a step needs e_yΛe_x=0 or another condition not stated, then the protected corner theorem, and with it the exclusion of the all-infinite v=3 configuration, is not proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Under the paper's own definition, an external in-neighbour of x is a vertex y≠x with an arrow y→x. In standard path-algebra notation that arrow is an element of e_yΛe_x = yΛx, so yΛx≠0. Thus the hypothesis 'yΛx=0 for every external in-neighbour y' in Theorems 9.7, 9.12, and Corollary 9.16 is unsatisfiable unless the notation is reversed. The intended condition, from the gloss ('no path from x back to a vertex pointing at x'), is e_xΛe_y=0, i.e. xΛy=0. But the manuscript does not consistently use this convention: Lemma 2.2 writes dime_zP_y = dim zΛy = C_{yz} (standard: e_zΛe_y), while Lemma 9.11 and Example 9.18 treat yΛx as paths x→y. This ambiguity is not cosmetic. In Lemma 9.11, the proof of e_xΛe_y·e_yΛe_x=0 factors p:y→x as p=sαp′ with p′:y→y0, α:y0→x, s∈Γ; the key factor is then p′q for q:x→y. This factor lies in e_xΛe_{y0}, not in e_{y0}Λe_x, so the stated vanishing y0Λx=0 (or xΛy0=0, depending on convention) is not the one that applies. Observation (O2) of Theorem 9.12 likewise needs the cross-term (e_xΛe_y)(e_yΛe_x) to vanish for y≠x; with the literal statement it is the in-arrow itself that would have to vanish. The admitted gap of Remark 9.10 is therefore closed only if a consistent, corrected Peirce convention is imposed; as written, the central theorem is vacuous or its proof is not checkable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Han's conjecture through the τ-Hochschild framework of Cibils–Lanzilotta–Marcos–Solotar. It proves a K0 identity relating syzygy dimension matrices to Ext and Cartan data, a syzygy-return criterion (Theorem 3.1) equivalent to infinite+ global dimension, a reduction of Gap A to strongly connected quivers, and a χ-law for Hochschild Euler characteristics under eventual vanishing. The main new results are: the protected corner theorem (Theorem 9.12), giving dim_k Ext^n_Λ(S_x,S_x) ≥ dim_k Ext^n_Γ(k,k) ≥ 1 for all n whenever Γ_x=e_xΛe_x≠k and yΛx=0 for every external in-neighbour y of x; the exclusion of the all-infinite three-vertex Gap-A failure (Theorem 9.14(1)); the classification of the two-infinite three-vertex failure as a mutual dumbbell (Theorem 9.14(2)); and a Hattori–Stallings/Cartan analysis of the dumbbell, including eventual vanishing of Hochschild homology for any Gap-A failure with Euler characteristic pinned at v (Proposition 10.2).","tokens_in":30932,"tokens_out":41231,"duration_ms":407001,"significance":"If the proofs are correct, this is a significant step on a hard open problem. The paper gives an explicit, checkable algebraic engine (K0 identity, syzygy-return criterion, χ-law), produces genuine unconditional instances of the Liu–Morin extension conjecture beyond monomial and special biserial algebras, and sharply constrains hypothetical three-vertex counterexamples. The author is also commendably explicit about an earlier gap in the proof of Theorem 9.7 and patches it via Lemma 9.11. The main caveat is that the entire notion of Gap-A failure and the equivalences in Theorem 3.1 are imported from the external paper [9], which is still 'to appear'; the internal chain from those hypotheses to the trichotomy is then what needs careful verification.","major_comments":[{"comment":"I specifically checked the objection that the protection hypothesis yΛx=0 is vacuous for an external in-neighbour y. Under the paper's convention C_{ab}=dim bΛa, the space yΛx=e_yΛe_x consists of paths x→y, while an arrow y→x lies in xΛy. Thus yΛx=0 is exactly the intended no-return condition, and the factorization in Lemma 9.11 is consistent: p'q lies in e_{y0}Λe_x=y0Λx, which is killed by the hypothesis applied to the external in-neighbour y0. The stress-test concern does not land. Nevertheless, because this convention is easy to misread and the central theorem depends on it, an explicit notation warning after Definition 2.1 would materially improve the paper.","section":"Lemma 9.11 / Theorem 9.12"},{"comment":"The induction in Theorem 9.12 is the load-bearing step for the whole trichotomy, and in its present form it is not fully verifiable. In the induction step, after defining τ'_j as lifts of minimal generators of Z_{n+1}, the proof asserts that one can 'normalize the remaining generators as in the base to secure (b)' without proving that this operation preserves (i) the defining s-tuple property (c) for all elements of Ω^{n+1}, (ii) the fact that the normalized elements remain in Ω^{n+1}, and (iii) the independence of the K_{n+1} classes modulo rΩ^{n+1}. The base case does not automatically cover the general step because off-diagonal Peirce contributions are controlled by clause (b) in a way that changes after replacement. Please expand this into a complete induction, or replace it by an explicit construction of a partial Λ-resolution of S_x embedding the minimal Γ-resolution of k.","section":"Theorem 9.12, proof"},{"comment":"In the case z=u, the phrase 'Γ_x=Γ_{σ(u)}' uses an undefined symbol σ; from the context it should state 'Γ_x≠k by Theorem 9.1(4) applied at u, whose dead zone is x'. Also, after ruling out an arrow x→u, the text does not explicitly rule out an arrow x→w; without that, there is a surviving path x→w→u, which would give uΛx≠0 and contradict the conclusion Z(u)={x}. These are easy repairs, but the case analysis is currently incomplete as written.","section":"Theorem 9.14(2), proof"}],"minor_comments":[{"comment":"Add a warning that yΛx=e_yΛe_x is the space of paths x→y, so an arrow y→x lies in xΛy. This would prevent the most natural misreading of the protection hypothesis.","section":"Definition 2.1"},{"comment":"Fix the undefined σ in 'Γ_x=Γ_{σ(u)}' and spell out the contradiction from a hypothetical arrow x→w in the z=u case.","section":"Theorem 9.14(2)"},{"comment":"The notation 'Λ_w \\bar{e}_w' is left undefined; it should be the image of the vertex idempotent e_w in Λ_w=Λ/J_x.","section":"Proposition 10.5"},{"comment":"In (1), the exact sequence should be written as 0→e_xP_{N−1}→...→e_xP_0→k→0, with the explicit remark that e_xΩ^N(S_x)=0; the current text could be misread as including a missing first term.","section":"Theorem 10.6"},{"comment":"Clarify the orientation of the 'arrow-count matrix': C^{(1)}_{xy} counts arrows y→x, not x→y. The convention is clear from the preceding definitions but deserves an explicit sentence.","section":"Definition 2.1(ii)"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core appears defensible, but the proof of Theorem 9.12 must be written out in full before I can certify the trichotomy. The disclosure of substantial AI assistance in drafting proofs makes this verification even more important. Also, since the paper leans so heavily on [9], which is 'to appear', the editorial office should confirm that the relevant theorems of [9] are final and that the quoted equivalences are in force."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe short version: this paper deserves a serious referee. The protected corner theorem (Thm 9.12) and the v=3 trichotomy (Thm 9.14) are new, substantial structural results about hypothetical counterexamples to Han's conjecture, and the supporting machinery (K0 identity, syzygy-return criterion, SCC reduction) is mostly clean and checkable.\n\nI checked the stress-test note carefully, and I think it is wrong. The paper consistently uses yΛx = e_yΛe_x, meaning paths from x to y, not the reverse. Under that convention the arrow y→x lies in xΛy, so the protection hypothesis yΛx=0 exactly says no path returns from x to an in-neighbour. Example 9.18 is a concrete instance where the hypothesis is satisfiable, and the factorization in Lemma 9.11 is consistent with that convention. So the central theorem is not vacuous.\n\nWhat is genuinely new: the lower bound dim Ext^n_Λ(S_x,S_x) ≥ dim Ext^n_Γ(k,k) under a dead-return condition, and the proof that an all-infinite three-vertex Gap-A failure cannot exist, with the two-infinite case pinned down as a mutual dumbbell. The paper is also honest about its own missteps — Remark 9.10 flags an incomplete step in an earlier proof, patched by Lemma 9.11. That kind of self-correction is a good sign.\n\nSoft spots, in proportion. The abstract oversells the extension-conjecture scope: 'settling' is too strong; the body (Remark 9.13) correctly says 'new instances.' The heavy reliance on the τ-Hochschild framework of [9] is cited rather than reproved; if those equivalences are wrong, the Gap-A/Gap-B split changes. That is an external dependency, not a flaw in the paper, but it means the main theorems inherit a lot from [9]. The induction in Theorem 9.12 is intricate — the admitted gap and patch are in that argument — so independent verification would be valuable.\n\nWho this is for: anyone working on Han's conjecture, the extension conjecture, or homological properties of finite-dimensional algebras. It won't settle the conjecture, but it genuinely narrows the search space for counterexamples on three vertices.\n\nRecommendation: send it to peer review. The referee should be someone comfortable with τ-Hochschild and syzygy computations, and should spend time on the Peirce-decomposition induction. If the external framework holds up, this is a solid, citable contribution.\n\nBest,","headline":"The Peirce-convention worry in the stress-test is a false alarm; the protected corner theorem and the v=3 trichotomy are real contributions, the abstract overclaims a bit, but this paper deserves serious refereeing.","tokens_in":31508,"tokens_out":7038,"would_cite":true,"duration_ms":61788,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","16E10","16G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A protected corner in a quiver algebra forces self-extensions in every degree, and on three strongly connected vertices it leaves only two possible Gap-A failure shapes for Han's conjecture.","keywords":["Han's conjecture","Hochschild homology","global dimension","tau-Hochschild homology","extension conjecture","syzygy","quiver algebras","protected corner"],"falsifier":"One concrete falsifier: build a bound quiver algebra on a strongly connected three-vertex quiver whose three simple modules all have infinite projective dimension, with HH_n = 0 for n > N and b_n = 0 for n ≥ N — i.e., realize the Gap-A template of Definition 9.5 with at least two loops per vertex and non-uniserial corners. Theorem 9.14(1) asserts no such algebra exists. The reverse test: compute the mixed-necklace homology of the mutual dumbbell; if it does not vanish in high degrees, the dumbbell is a live Han counterexample.","tokens_in":30290,"feed_emoji":"🧩","tokens_out":8274,"duration_ms":63711,"temperature":0.7,"pith_summary":"The paper attacks Han's conjecture, which predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology must have finite global dimension. Its target is the persistence half of the conjecture (Gap A): the claim that infinite global dimension forces infinite τ-Hochschild homology. The paper proves a protected corner theorem: at a vertex whose corner algebra is nontrivial and whose external in-neighbours have no surviving return paths to it, the simple module has self-extensions in every positive degree, so the extension conjecture holds there and the algebra is of infinite+ global dimension. From this it derives a trichotomy for three strongly connected vertices: a Gap-A failure has one or two simples of infinite projective dimension, never three, and the two-infinite case is a specific 'mutual dumbbell' that glues two two-vertex algebras at a finite-dimensional hub. If correct, this pins down the minimal anatomy of every potential counterexample and shows unconditionally that Gap-A failures satisfy the full Han conclusion with Euler characteristic equal to the number of vertices.","feed_headline":"Two shapes remain for three-vertex Han counterexamples","feed_subtitle":"A protected corner theorem rules out the all-infinite case, leaving a single-infinite shape or the mutual dumbbell and pinning the Euler cha","key_machinery":"The central identity is the K_0 relation S_n + S_{n+1} = C^{(n)}C between syzygy-dimension matrices and Ext matrices of a bound quiver algebra; its trace gives b_n = tr S_n + tr S_{n+1}, the dimension of the chain spaces of the minimal bimodule resolution. This turns 'infinite τ-Hochschild homology' into a syzygy-return criterion (Theorem 3.1): Λ is of infinite+ global dimension iff some diagonal entry e_xΩ^n(S_x) is nonzero infinitely often. The protected corner theorem (9.12) forces such diagonal return by embedding the corner's minimal resolution into Λ's; the one-way law (8.1) states that a Gap-A failure must eventually send every syzygy top only into vertices from which no surviving pat","core_discovery":"The paper's central claim is Theorem 9.12: if a vertex x of a bound quiver algebra Λ has a nontrivial corner algebra Γ = e_xΛe_x ≠ k, and every external in-neighbour y of x satisfies yΛx = 0 (no surviving path from x back to any vertex pointing at x), then the trivial module k over Γ embeds as a 'protected thread' in the minimal resolution of the simple S_x. The quantitative consequence is dim_k Ext^n_Λ(S_x,S_x) ≥ dim_k Ext^n_Γ(k,k) ≥ 1 for every n ≥ 1. Thus an algebra with a protected corner is automatically of infinite+ global dimension, and the extension conjecture holds there. From this, with the one-way law and the strong no-loop theorem, the paper derives Theorem 9.14: a three-vertex s","pith_inferences":["A practical prefilter for the counterexample search emerges: on any strongly connected bound quiver algebra, scan for protected vertices; if one exists, the algebra is certified infinite+ and cannot be a Gap-A failure. The exponential lower bound at a radical-square-zero corner with ℓ loops (≥ ℓ^n in every degree) makes this check numerically effective even when full resolutions are out of reach.","The trichotomy funnels the entire three-vertex search into the dumbbell: the paper reduces its fate to the vanishing of 'mixed-necklace' homology built from Tor of the two sides against the hub. If that homology dies in high degrees, the dumbbell is eliminated and Gap A at three vertices collapses to the single-infinite configuration.","The χ-law and the no-go theorem suggest that no dimension-level invariant can expose a Gap-B failure: any counterexample is numerically indistinguishable from a smooth algebra at the level of Hochschild homology dimensions. The remaining hope for Gap B lies in rank-theoretic or operator-level invariants, where differentials rather than spaces decide survival."],"forward_implications":["The extension conjecture holds at every protected vertex; in particular, a Gap-A failure cannot contain a protected vertex, so any three-vertex all-infinite configuration is impossible.","On three strongly connected vertices, a Gap-A failure has exactly one or two infinite simples; the two-infinite case is the mutual dumbbell, a fibre product of two two-vertex algebras glued at a finite-dimensional hub.","Every Gap-A failure has HH_n(Λ) = 0 for all n ≫ 0 and the alternating-dimension sum of its Hochschild homology equals the number of vertices, in every characteristic.","The dumbbell's linking bimodules are non-projective over the nontrivial corners, placing the configuration outside the null-square projective and bounded-extension classes for which Han's conjecture is known to propagate.","The Cartan determinant of the dumbbell is nonzero and divides five explicit products, giving rigid numerical constraints that any realization must satisfy."],"fun_headline_variants":["Three-vertex Han failures: all-infinite ruled out, two shapes remain","No all-infinite Han counterexample; mutual dumbbell or single-infinite left","Protected corner theorem: all-infinite Han case impossible on three vertices","Trichotomy for Han: one-infinite or mutual dumbbell are the only failures"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire classification imports the τ-Hochschild equivalences of a companion paper: infinite+ global dimension is identified with infinite τ-Hochschild homology, and vanishing τ-Hochschild homology with eventual vanishing of the minimal-resolution chain spaces; these are cited, not reproved, so if they fail, the notion of Gap-A failure and the trichotomy shift.","fun_headline_variants_meta":{"raw":{"variants":["Three-vertex Han failures: all-infinite ruled out, two shapes remain","No all-infinite Han counterexample; mutual dumbbell or single-infinite left","Protected corner theorem: all-infinite Han case impossible on three vertices","Trichotomy for Han: one-infinite or mutual dumbbell are the only failures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3157,"prompt_tokens":684,"completion_tokens":2473,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":2387}},"tokens_in":428,"tokens_out":2473,"duration_ms":15878,"temperature":1.0,"reasoning_tokens":2387,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:08:44.630630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete falsifier: build a bound quiver algebra on a strongly connected three-vertex quiver whose three simple modules all have infinite projective dimension, with HH_n = 0 for n > N and b_n = 0 for n ≥ N — i.e., realize the Gap-A template of Definition 9.5 with at least two loops per vertex and non-uniserial corners. Theorem 9.14(1) asserts no such algebra exists. The reverse test: compute the mixed-necklace homology of the mutual dumbbell; if it does not vanish in high degrees, the dumbbell is a live Han counterexample.","supporting_citations":[],"review_version":1}