{"id":"6323785d-a0f4-4a69-a83f-04618cca0764","arxiv_id":"2607.25861","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For local complete intersection singularities, the new Hodge invariant HRH is bounded by integer Bernstein–Sato roots and minimum integer spectral numbers, and in the hypersurface case it is exactly determined by them.","lead":"Singular spaces can secretly behave like smooth manifolds in a coarse sense, and this paper connects a new fine-grained measure of that behavior—called HRH—to older invariants like Bernstein–Sato polynomials and singularity spectra. In the hypersurface case the new measure is exactly determined by those old invariants, while complete intersections obey useful inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main results depend on the unpublished companion [DOR26]: HRH is not defined here and Prop. 6.1/Theorem C invoke [DOR26, Cor. G]; if that corollary is absent or false, the central spectrum-to-HRH inequalities collapse.","rationale":"The reader's weakest assumption is exactly the dependence on [DOR26] for HRH and Cor. G. My review confirms this is the single most load-bearing point: Theorem A's equality uses HRH via [DOR26, Thm. D(1)], and Theorem C's proof is entirely mediated by Proposition 6.1, which relies on Cor. G. The paper's own contributions—the general linear combination comparison of Section 5, the duality in Proposition 6.4, the examples—are internally consistent as far as I can check. It may well be that Cor. G is correct and the results hold, but the manuscript as submitted does not contain that foundation. I do not see a reason to raise or lower the reader's confidence; CONDITIONAL remains appropriate. The suggested check (obtaining [DOR26] and re-deriving Prop. 6.1) would settle whether the concern lands.","tokens_in":23914,"tokens_out":11263,"duration_ms":90885,"concrete_test":"Obtain [DOR26] and verify: (1) HRH is defined and satisfies the V-filtration characterization used in §4.2; (2) Cor. G states HRH(Z)≥k iff ℓ_{p,q}=0 for p,q≤k, with a complete proof. Then re-derive Proposition 6.1 line by line without invoking Cor. G, starting only from the definition of HRH and [FL24, Props. 2.11, 2.12]. If the derivation cannot be completed, the main theorems remain unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claims are meaningful only if the invariant HRH and its key property [DOR26, Cor. G] are established in the authors' 'In preparation' predecessor. Section 3 immediately uses '[DOR26, Thm. D(1)]' to identify HRH with a filtered equality, and Section 6's Proposition 6.1 uses '[DOR26, Cor. G]' twice to translate HRH≥k into vanishing of link Hodge numbers ℓ_{p,q}. Since no definition of HRH or proof of Cor. G appears here, the chain Theorem A ↔ V-filtration characterization ↔ Proposition 6.1 ↔ Theorem C is conditional on an external document. This is not an internal inconsistency—the internal arguments appear coherent—but it is a genuine verification gap in the load-bearing premise. The rest of the paper (Section 5) partially defines and proves the invariants from scratch, but the HRH side of the story does not.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the invariant HRH (\"Hodge rational homology\" manifold level) for local complete intersection subvarieties of smooth complex varieties. In the hypersurface case, Theorem A asserts for each x in Z that eαbar_{Z,x}(f)-2 ≤ HRH_x(Z) = p(φ_{f,1}(O_X)_x,F)+n-2 ≤ Sp_{min,Z}(Z,x)-2, with equality of the second inequality at isolated singularities. For general local complete intersections, Theorem B bounds HRH(Z) below by p(Q_Z,F)+n-1 and bounds the integral spectrum below by p(Q^Z_x,F)+n+1; Theorem C gives HRH_x(Z) ≥ Sp_{min,Z}(Z,x)-2 for isolated LCI singularities; Theorem D characterizes vanishing of Q_Z in terms of the rational homology manifold property of the general linear combination hypersurface V(g|_U). The proofs use the V-filtration, Verdier specialization, Fourier-Laplace transforms, and results from the authors' companion paper [DOR26], which is listed as \"In preparation\".","tokens_in":24216,"tokens_out":5115,"duration_ms":47588,"significance":"If the main results are correct, they give a striking Hodge-theoretic description of rational homology manifold behavior: for hypersurfaces, HRH is completely determined by the first Hodge-filtration jump of unipotent vanishing cycles, hence by the DMS spectrum; for complete intersections, the paper provides new inequalities linking HRH, the minimal integer spectral number, and the reduced Bernstein-Sato polynomial. The paper contains no fitted parameters, and the comparisons are genuine inequalities rather than definitions in disguise. The explicit examples, including the Thom-Sebastiani examples and a Macaulay2 computation in Example 5.21, are useful. However, the central claims are currently conditional on the unpublished companion paper [DOR26], which supplies the definition of HRH and the key link-Hodge-number criterion used in Proposition 6.1 and Theorem C; this is a substantial verification gap that must be addressed before the results can be considered established.","major_comments":[{"comment":"The invariant HRH is never defined in this paper. Section 3 identifies k-Hodge rational homology with the filtered equality F^{k-n}W_{n+1}H^1_Z(O_X)=F^{k-n}H^1_Z(O_X) by quoting [DOR26, Thm. D(1)], and Proposition 6.1 uses [DOR26, Cor. G] to translate HRH(Z)≥k into vanishing of link Hodge numbers ℓ_{p,q}. Since [DOR26] is listed as \"In preparation\", the main comparisons to HRH are conditional on an external document. Please state the precise needed results from [DOR26] or include proofs, so that the paper is self-contained on this load-bearing point.","section":"§3 and §6 (HRH via [DOR26])"},{"comment":"The proof of Theorem A is too compressed at its most crucial step. The filtered weight-filtration formula F^pW_{n-1+i}Gr^1_V(B_f)=Σ_{ℓ≥max{0,-i}}(t∂_t)^ℓ F^{p-ℓ}ker((t∂_t)^{1+i+2ℓ}) is said to be \"an easy exercise\" following strictness, and the second half of the proof invokes \"By induction on k\" without spelling out the induction parameter or the base case. Since Theorem A is the central hypersurface characterization, these steps need a complete and detailed proof.","section":"§3, proof of Theorem A"},{"comment":"Proposition 6.1 is the bridge from the Milnor-fiber numbers s_p to HRH, and Theorem C depends entirely on it. Its proof invokes [DOR26, Cor. G] in both directions, but the exact statement of that corollary is not given. Even accepting [DOR26], the induction in the converse direction is only sketched: after proving s_{d-p}-s_p=0 for p≤k, the implication ℓ_{d-k,k}=0 ⇒ HRH(Z)≥k requires precisely the missing form of Cor. G. This is a load-bearing gap, not a presentation issue.","section":"§6, Proposition 6.1 and Theorem C"}],"minor_comments":[{"comment":"The plan refers to \"Section B\", \"Section C\", and \"Section D\", while the actual sections are numbered 1-8; use consistent numbering.","section":"Plan of the paper"},{"comment":"The inequality HRH(Z)<+∞ ⇒ HRH(Z)≤(d−3)/2 is stated without reference or proof. Add a citation or a short justification.","section":"Remark 5.15"},{"comment":"The Macaulay2 computation would be more reproducible if the code and version were included or mentioned in a footnote.","section":"Example 5.21"},{"comment":"The notation D(Q_{≠Z}) in the proof is not defined; clarify that it means the graded-dual construction on the monodromic pieces.","section":"§6, Proposition 6.4"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue for me is the dependence on the unpublished companion [DOR26]. The internal arguments in this paper appear coherent and the theorems are plausible, but the reader cannot verify the definition of HRH or the key property in Proposition 6.1 without access to that companion. If [DOR26] is expected to appear soon, the authors should also consider posting the relevant statements; otherwise the paper should be revised to include the necessary background. The compressed proof of Theorem A is a second issue that should be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jason — quick take on arXiv:2607.25861. The paper gives a real new dictionary: for hypersurfaces, HRH_x(Z) is sandwiched between the reduced Bernstein–Sato integer root and the minimal integer spectral number, with equality of the second bound in the isolated case. The LCI inequalities involving p(Q_Z,F), the DMS spectrum, and the unipotent Verdier specialization module Q_Z are new, and the characterization of Q_Z=0 via the general linear combination g|_U being a rational homology manifold is a clean statement. The 'missed jumps' language in Section 5 is a useful way to organize the comparisons. The internal arguments read as coherent; I did not find an obvious contradiction.\n\nThe soft spots are real but not disqualifying. First, the central invariant HRH is not defined in this paper; Section 3 starts by recalling [DOR26, Thm. D(1)] to identify it, and Proposition 6.1/Theorem C use [DOR26, Cor. G] twice to pass between HRH and vanishing of link Hodge numbers. If that corollary is missing or wrong, the spectrum-to-HRH inequalities lose their support. That is a load-bearing external dependency, and it is not a matter of style. Second, the proof of Theorem A contains an 'easy exercise' and an 'induction on k' on exactly the filtered weight-filtration formula that the argument turns on. Both steps are probably correct—I suspect an expert can fill them in—but they are compressed enough that a referee will need to ask for details. Third, the one Macaulay2 computation (Example 5.21) is stated without attached code or even a certificate; minor, but easy to fix.\n\nOn the other hand, Section 5 does define and prove the non-HRH invariants from scratch, and the reliance on same-group papers like [Dir23] and [CDMO24] is not circular in a bad way—those hold established results. The stress-test note about [DOR26] is on target. This is a paper for singularity theorists and Hodge theory people. If Part I appears and the compressed proofs are expanded, the results will be worth citing. As it stands, I would not build on it yet, but it deserves a serious referee. Recommendation: send to peer review, with explicit instruction to the authors to either include a full proof of Cor. G or restructure so the paper is self-contained enough to be evaluated.","headline":"Genuinely new comparison inequalities for HRH on local complete intersections, but the results are built on the unpublished [DOR26]; the math looks coherent and deserving of refereeing, not desk rejection.","tokens_in":24633,"tokens_out":2935,"would_cite":false,"duration_ms":26395,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14F10","32S35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For hypersurfaces, HRH equals the first Hodge-filtration jump of unipotent vanishing cycles, shifted by n−2; for complete intersections that same jump bounds HRH, with the gap measured by 'missed jumps'.","keywords":["Hodge rational homology manifold","local complete intersection","Bernstein–Sato polynomial","spectrum","V-filtration","Verdier specialization","vanishing cycles","minimal exponent"],"falsifier":"For a cone over a smooth complete intersection of degree (a_1,...,a_r) in P^{n-1}, the Milnor-fiber spectral numbers and link Hodge numbers are known; check whether HRH computed from the V-filtration (via the hypersurface formula) satisfies HRH = Sp_min − 2. A mismatch in any such example would refute the isolated-case equalities. More directly, if an isolated LCI singularity is found where HRH_x(Z) < Sp_min,Z(Z,x) − 2, then Theorem C fails.","tokens_in":23815,"feed_emoji":"🕳️","tokens_out":9672,"duration_ms":74547,"temperature":0.7,"pith_summary":"HRH(Z) is a proposed Hodge-theoretic refinement of the rational homology manifold condition: instead of asking whether the link has the homology of a sphere, it asks how deep into the Hodge filtration on local cohomology a nontrivial class survives. The paper proves that in the hypersurface case this level is completely pinned down by classical invariants: HRH_x(Z) equals the first Hodge filtration jump of the unipotent vanishing cycles φ_{f,1}(O_X) shifted by n−2, and in isolated singularities this equals the smallest integer spectral number minus two. For local complete intersections the authors show that the same jump, computed instead on the unipotent Verdier specialization module Q_Z, gives a lower bound HRH(Z) ≥ p(Q_Z,F)+n−1 and that the DMS spectrum gives HRH_x(Z) ≥ Sp_min,Z(Z,x)−2 in the isolated case. The gap between these bounds is controlled by a count of 'missed jumps' in the Hodge filtrations of the monodromic pieces of Q_Z, and the vanishing Q_Z = 0 is characterized by the rational homology manifold property of the generic linear combination hypersurface.","feed_headline":"One Hodge jump sets a hypersurface's singularity level","feed_subtitle":"In isolated singularities this level equals the smallest spectral number minus two; in higher codimension it is bounded by the same jump.","key_machinery":"The engine of the paper is the V-filtration of Kashiwara–Malgrange on the graph-embedded structure module B_f = Γ_*(Q_X^H[n]), and its associated Verdier specialization Sp(B_f) along Z ⊆ X. The module Q_Z is the cokernel in 0 → L → Sp(B_f)^Z → Q_Z → 0, where L is the trivial Hodge module on Z × A^r; its first Hodge filtration jump p(Q_Z,F) is the key numerical invariant. A secondary but equally important object is the number j(Z) of 'missed jumps' — the count of ℓ ∈ [1,r−1] for which p(Q_{r−ℓ}, F) = p(Q_{r−ℓ+1}, F) — which measures when the Hodge filtrations on consecutive monodromic pieces align, and which turns the general inequalities into equalities.","core_discovery":"The central discovery is that the invariant HRH, defined as the largest k for which F^{k−n}W_{n+1}H^1_Z(O_X) = F^{k−n}H^1_Z(O_X), is completely determined by the Hodge filtration of unipotent nearby/vanishing cycles in the hypersurface case: HRH_x(Z) = p(φ_{f,1}(O_X)_x,F) + n − 2 (Theorem A); if x is isolated this equals Sp_min,Z(Z,x) − 2. In codimension r > 1, the same role is played by the unipotent Verdier specialization module Q_Z, giving p(Q_Z,F)+n−1 ≤ HRH(Z) and, for isolated singularities, HRH_x(Z) ≥ Sp_min,Z(Z,x) − 2 (Theorems B and C). Equality is controlled by the absence of 'missed jumps' among the Hodge filtrations on the monodromic pieces Q_k. The vanishing Q_Z = 0 is characteri","pith_inferences":["The missed-jump count j(Z) likely measures the failure of the Hodge filtration on Q_Z to be 'as spread out as possible'; if so, it should be computable from the dimensions s_p of the Milnor-fiber Hodge pieces, and one could test whether j(Z) equals the number of p < r for which s_{d−p} = s_p in isolated singularities.","The general linear combination trick opens a practical route to compute HRH for complete intersections: pick a generic linear combination g, compute the spectrum (or Bernstein–Sato polynomial) of g|_U, and correct by r and j(Z). This could turn HRH into a computable invariant for concrete families, e.g., cones over smooth complete intersections.","Conjecture 5.20, if true, would give a purely combinatorial criterion: Q_Z = 0 (and hence HRH maximal) can be read off from whether the Bernstein–Sato polynomial of the tuple has extra factors beyond (s+r); testing this on the Thorrelli-type product examples would be a concrete check.","The distinction the paper draws between Q_Z = 0 and rational-homology-manifold-ness suggests that in codimension > 1 the Hodge-theoretic boundary is not the ordinary link condition but a finer one; one might expect HRH to relate to the vanishing of the Du Bois complex or to k-rationality thresholds in a way that Q_Z does not."],"forward_implications":["For any hypersurface singularity, HRH_x(Z) equals the first Hodge jump of unipotent vanishing cycles shifted by n−2; in particular, the reduced Bernstein–Sato polynomial determines HRH_x(Z) via eαbar_{Z,x}(f).","In isolated hypersurface singularities, HRH_x(Z) = Sp_min,Z(Z,x) − 2, so the classical Steenbrink spectrum fully determines the Hodge rational homology level.","For local complete intersections, HRH(Z) ≥ p(Q_Z,F) + n − 1; thus the first jump on the unipotent Verdier specialization forces a quantitative bound on how close Z is to a rational homology manifold.","The equality conditions in the bounds are governed by the missed-jump count j(Z); when j(Z) = 0 the invariants HRH(Z), p(Q_Z,F), and HRH(g|_U) differ only by explicit shifts in r.","If Z has rational singularities and eαZ(Z) = +∞ (i.e., the reduced Bernstein–Sato polynomial has no integer roots), then Z is a rational homology manifold; the converse fails, as Torrelli's example shows."],"fun_headline_variants":["Hypersurface singularity level: one Hodge jump decides","Hodge invariant pins down hypersurface singularities","Singularity level equals minimal spectral shift minus two","For hypersurfaces, Hodge filtration fixes singularity level","Codimension >1: Hodge invariant only bounds singularity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper relies on an unpublished companion paper for the definition of HRH and, crucially, for the equivalence (Corollary G there) between HRH ≥ k and the vanishing of the link Hodge numbers ℓ_{p,q} used in Proposition 6.1; if that equivalence is wrong or unavailable, Theorem C and the spectrum-to-HRH inequalities lose their support.","fun_headline_variants_meta":{"raw":{"variants":["Hypersurface singularity level: one Hodge jump decides","Hodge invariant pins down hypersurface singularities","Singularity level equals minimal spectral shift minus two","For hypersurfaces, Hodge filtration fixes singularity level","Codimension >1: Hodge invariant only bounds singularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1295,"prompt_tokens":706,"completion_tokens":589,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":511}},"tokens_in":450,"tokens_out":589,"duration_ms":5656,"temperature":1.0,"reasoning_tokens":511,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:16:22.752126+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a cone over a smooth complete intersection of degree (a_1,...,a_r) in P^{n-1}, the Milnor-fiber spectral numbers and link Hodge numbers are known; check whether HRH computed from the V-filtration (via the hypersurface formula) satisfies HRH = Sp_min − 2. A mismatch in any such example would refute the isolated-case equalities. More directly, if an isolated LCI singularity is found where HRH_x(Z) < Sp_min,Z(Z,x) − 2, then Theorem C fails.","supporting_citations":[],"review_version":1}