{"id":"1c2b8041-ed2e-4b61-85a0-a517c5ded9de","arxiv_id":"2607.09163","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Proper-Cover terminates and returns a proper basis of a zero-dimensional ideal by combining PID-generalized signatures/covers with a two-phase compatible-factor and hungry-refinement procedure, outperforming F5 and often GVW under lex.","lead":"The paper gives a two-phase Proper-Cover algorithm that computes proper bases of zero-dimensional polynomial ideals by merging GVW-style signature covers with proper-basis theory over K[x1][x̃]. Benchmarks claim it beats F5 on all tested orders and often beats GVW under pure lex, which matters for elimination and algebraic solving.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"Lemma 4.3's S-pair representation for singular cases is asserted without proof, leaving modular-basis correctness (Prop. 4.7) and exact-power recovery (Prop. 4.6) on an unsecured step.","rationale":"The reader correctly isolated the S-pair representation (Lemma 4.3) as the weakest assumption. My reading confirms that the gap is even more concrete: the singular case is dismissed with a single sentence rather than proved. All subsequent claims—compatible-factor divisibility, exact powers, modular-basis equality, and therefore Theorem 4.8—depend on that identity. The experimental tables and the algorithmic design are secondary; they cannot compensate for a missing step in the correctness argument. Because the rest of the development is coherent and the algorithms are well-specified, the appropriate verdict remains CONDITIONAL: the paper is a genuine contribution that should be accepted once the singular-case argument is supplied (or the algorithms are shown never to generate singular pairs that matter). No stronger rejection is warranted, and no unconditional acceptance is justified until the gap is closed.","tokens_in":13843,"tokens_out":628,"duration_ms":7567,"concrete_test":"Take a concrete zero-dimensional ideal that forces a singular S-pair under the POT order used by the algorithms (e.g., a small Cyclic or Random system whose intermediate labelled pairs produce a singular pair). Manually expand every S-pair that appears, isolate the singular ones, and verify whether identity (9) still holds with the local pre-basis produced by Algorithm 1 (or 2). If any singular S-pair cannot be expressed that way, Lemma 4.3 is false and the modular-basis claim fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Thm. 1.1 / 4.8) rests on every element of I admitting a representation (12)–(13) with multiplier coprime to the compatible factor (Lemma 4.5). That representation is obtained by rewriting ordinary S-polynomials via the cover/regular-reduction identity of Lemma 4.3. The proof of Lemma 4.3 constructs the identity carefully for non-singular semi-S-pairs and for covered pairs, then simply states “It is easy to see that when the S-pair is singular this conclusion also holds.” No separate argument, reduction to a non-singular case, or reference is supplied. Because singular S-pairs are precisely those for which the signature of the S-pair is strictly smaller than the maximum of the two scaled signatures, the local-basis reduction that produces (9) is not automatic. If the identity fails for even one singular configuration that arises under the PID multipliers of Algorithms 1–2, the inductive step of Lemma 4.5 collapses, the modular-basis equality (2) no longer follows, and the exact-power claim of Proposition 4.6 becomes unsupported. This is the single least-secured link in the correctness chain.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proposes the Proper-Cover algorithm for computing proper bases of zero-dimensional ideals I ⊂ K[x1][x̃]. It generalizes signatures, POT order, covers, regular reductions and S-pairs from the field setting of F5/GVW to coefficients in the PID K[x1], introduces semi-S-pairs, proper/hungry covers and reductions, and a two-phase procedure (ProperCoverReduction followed by HungryRefinement with compatible-factor extraction). Theorem 1.1 / Theorem 4.8 asserts termination and that the output pairs (qj, Bqj) satisfy χ = ∏ qj together with the modular leading-term equality ⟨LT(πqj(I))⟩ = ⟨LT(Bqj)⟩, so that the assembled set is a proper basis. Maple timings on Cyclic, Katsura and seven random systems are reported to claim superiority over F5 under all orders and over GVW under plex.","tokens_in":14185,"tokens_out":1013,"duration_ms":18869,"significance":"If the correctness chain holds, the work supplies a genuine new algorithmic framework that imports modern signature-cover techniques into the proper-basis setting of Ma, offering a practical route to modular bases and eliminants that is especially attractive for lexicographic order. The explicit two-phase design, the compatible-factor construction, and the PID-level generalizations of cover and reduction are concrete contributions that could be adopted by computer-algebra systems. The experimental section, while limited to unoptimized Maple, already shows that the method solves instances on which F5 times out and frequently beats GVW under plex, giving empirical weight to the claim of a useful alternative.","major_comments":[{"comment":"Lemma 4.3 is the load-bearing representation that lets every ordinary S-pair (including singular ones) be rewritten via covers and regular reductions of semi-S-pairs by the local pre-basis. The argument carefully constructs identity (9) for non-singular and covered cases, then simply asserts “It is easy to see that when the S-pair is singular this conclusion also holds.” No separate reduction, signature comparison, or reference is supplied. Because singular S-pairs are precisely those for which sig(SSS) ≺POT max{λf tf sig(f), λg tg sig(g)}, the local-basis reduction that produces (9) is not automatic. If the identity fails for any singular configuration that arises under the PID multipliers of Algorithms 1–2, the inductive step of Lemma 4.5 collapses and both the modular-basis equality of Proposition 4.7 and the exact-power claim of Proposition 4.6 become unsupported. A complete case ana","section":null},{"comment":"Table 1 and the abstract claim that Proper-Cover “delivers clear speedups over GVW for lexicographic (plex) order.” The table itself shows the opposite on Random-6 (PC 310 s vs GVW 80 s) and comparable or worse behaviour on several other plex entries once timeouts are excluded. The experimental claim therefore overstates the evidence and should be qualified to the systems and orders for which the advantage is actually observed.","section":null}],"minor_comments":[{"comment":"Typographical errors: “momomial” (p. 1), “V olny” (ref. [14]), “GP” vs “Gp” inconsistency in Algorithm 2 line 22, and the future arXiv date “10 Jul 2026”.","section":null},{"comment":"Section 5 gives no implementation details for the F5 and GVW baselines (signature selection strategy, pair-set data structures, reduction heuristics). Without them the Maple timings are hard to interpret or reproduce.","section":null},{"comment":"The passage from a proper basis to a classical Gröbner basis is only referenced to Adams–Loustaunau; a short self-contained statement of the conversion would help readers who do not have that monograph at hand.","section":null},{"comment":"Definition 2.3 leaves the singular case without a semi-S-pair; a one-sentence remark on how the algorithm nevertheless processes such pairs would improve clarity.","section":null}],"recommendation":"major_revision","confidential_remarks":"The singular-case gap in Lemma 4.3 is the single point that prevents acceptance; once filled, the paper becomes a solid contribution. The experimental section is secondary and can be softened without harming the main claim. Novelty relative to the authors’ own arXiv:2101.03482 series should be checked by the editor, but the signature-cover integration appears new."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news is a working two-phase algorithm that lifts GVW-style signature covers and semi-S-pairs to K[x1][x̃], then recovers exact powers of the eliminant via “compatible factors” and hungry refinement. That is new; it is not just a re-packaging of F5/GVW or of the authors’ earlier proper-basis papers. The termination argument is standard Noetherian, the modular-basis equality is proved once the representation lemmas are granted, and the Maple timings (especially under plex) are the first concrete evidence I have seen that a proper-basis method can beat both F5 and GVW on the hard order.\n\nWhat they do well: the definitions of proper/hungry cover, least multiplier, and semi-S-pair are clean; Algorithms 1–3 are readable; and the experimental table is honest about timeouts. The claim that Proper-Cover dominates F5 across orders and often wins under lex is supported by the numbers they give.\n\nThe soft spot the stress-test flags is real but limited. Lemma 4.3 carefully builds the S-pair representation for non-singular and covered cases, then waves at singular pairs with “it is easy to see.” That sentence is the weakest link: if a singular configuration under PID multipliers fails to reduce to the local pre-basis, the inductive step of Lemma 4.5 and therefore Props. 4.6–4.7 lose their footing. I do not think the whole edifice collapses—singular pairs are already handled by signature cancellation in the classical theory—but the paper should either expand the argument or cite a reduction that covers the PID case. Everything else (compatible-factor extraction, hungry refinement, exact-power recovery) sits on top of that one step.\n\nNo code, no independent reimplementation, and several cells are pure timeouts; those are ordinary conference-paper limitations, not fatal ones. Citations are appropriate; self-cites to the proper-basis arXivs are necessary background, not circular.\n\nThis is for people who actually implement elimination or signature Gröbner bases. A serious referee in computational algebra will want the singular case written out and preferably a public Maple or Singular worksheet, but the paper already clears the bar for peer review. I would send it out.","headline":"Solid algorithmic merge of GVW covers with proper bases; the singular-S-pair gap in Lemma 4.3 is real but narrow, and the Maple numbers still make the paper worth a referee.","tokens_in":14809,"tokens_out":567,"would_cite":true,"duration_ms":6204,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13P10","68W30"],"pacs":[],"model":"grok-4.5","headline":"A signature-cover algorithm for zero-dimensional ideals terminates with a proper basis that beats F5 under every order and speeds up GVW under lex.","keywords":["Gröbner bases","signature algorithms","proper basis","zero-dimensional ideals","cover criterion","lexicographic order","eliminant","PID coefficients"],"falsifier":"Run the algorithm on a concrete zero-dimensional ideal whose classical Gröbner basis and eliminant are already known (for example a small Cyclic or Katsura system under lex) and check whether the returned modular bases generate exactly the same leading-term ideal as the projected ideal; any mismatch falsifies the main theorem.","tokens_in":14702,"feed_emoji":"⚙️","tokens_out":984,"duration_ms":8085,"temperature":0.7,"pith_summary":"Computing Gröbner bases is expensive, especially under lexicographic order. This paper takes the proper-basis idea of treating one variable as a parameter over a principal ideal domain and grafts onto it the signature-cover machinery of modern signature algorithms. The resulting Proper-Cover algorithm runs in two phases: a first pass that produces a pre-basis and a pre-eliminant together with a set of multipliers, then a hungry-refinement pass that recovers the exact powers of the remaining irreducible factors of the eliminant. The authors prove that the algorithm always terminates on zero-dimensional ideals and that the modular pieces it returns satisfy the leading-term ideal condition required of a proper basis. Benchmarks on Cyclic, Katsura and random systems show that the method finishes more instances than F5 under every tested monomial order and is markedly faster than GVW under pure lexicographic order. The practical payoff is a new route to bases of zero-dimensional ideals that can be converted, if desired, into classical Gröbner bases over the ground field.","feed_headline":"Signature-cover method beats F5 and speeds GVW under lex","feed_subtitle":"Two-phase Proper-Cover algorithm returns proper bases for zero-dimensional ideals with proven termination","key_machinery":"The generalized cover and regular reduction of semi-S-pairs over the PID K[x1]: every classical S-pair admits a representation, via these operations, in terms of the local pre-basis and pre-eliminants already computed, which is what lets the algorithm recover exact powers of eliminant factors without re-examining the whole ideal.","core_discovery":"For every zero-dimensional ideal I in K[x1][x̃] the Proper-Cover algorithm terminates and returns pairwise-coprime factors qj of the eliminant together with modular bases Bqj such that the leading-term ideal of the projected ideal equals the leading-term ideal of Bqj; the union of those modular bases with the qj themselves is therefore a proper basis of I.","pith_inferences":["Because the first phase already records all non-constant multipliers, later refinement only needs to re-run the signature engine for the remaining prime-power factors, suggesting a natural parallel or incremental implementation.","The same semi-S-pair representation may extend the method to positive-dimensional ideals once a suitable notion of eliminant ideal is supplied.","If the cover criterion continues to discard most pairs under plex, the algorithm could become a practical default for radical membership and primary decomposition pipelines that currently rely on F5-style lex Gröbner bases."],"forward_implications":["A proper basis of a zero-dimensional ideal can be obtained without computing a full field Gröbner basis first, then converted to a classical Gröbner basis by standard change-of-order techniques if needed.","Under pure lexicographic order the method is expected to finish many instances that currently time out for both F5 and GVW.","The same cover-and-hungry-refinement pattern can be reused for other signature-based algorithms that work over principal ideal rings.","Compatible-factor bookkeeping gives an explicit factorization of the eliminant as a by-product of the basis computation."],"fun_headline_variants":["Proper-Cover tops F5 on all orders and GVW under lex","Two-phase Proper-Cover outpaces F5 and GVW for plex","Cover-plus-proper bases return faster than F5 or GVW","Proper-Cover yields modular factors beating F5 and GVW-lex","Signature covers with proper bases surpass F5 across orders"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The proof assumes that every standard S-pair can be rewritten, through the generalized cover and regular reduction of semi-S-pairs, purely in terms of the first-stage local basis; if that rewriting fails for singular pairs or awkward multipliers, the modular-basis equalities no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Proper-Cover tops F5 on all orders and GVW under lex","Two-phase Proper-Cover outpaces F5 and GVW for plex","Cover-plus-proper bases return faster than F5 or GVW","Proper-Cover yields modular factors beating F5 and GVW-lex","Signature covers with proper bases surpass F5 across orders"]},"model":"grok-4.5","effort":"low","cost_usd":0.008672,"raw_usage":{"total_tokens":1980,"prompt_tokens":706,"num_sources_used":0,"completion_tokens":96,"cost_in_usd_ticks":86720000,"prompt_tokens_details":{"text_tokens":706,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1178,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":706,"tokens_out":96,"duration_ms":10320,"temperature":1.0,"reasoning_tokens":1178,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T04:58:27.137410+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Run the algorithm on a concrete zero-dimensional ideal whose classical Gröbner basis and eliminant are already known (for example a small Cyclic or Katsura system under lex) and check whether the returned modular bases generate exactly the same leading-term ideal as the projected ideal; any mismatch falsifies the main theorem.","supporting_citations":[],"review_version":1}