{"id":"e01a80a6-dd9f-4834-b241-fe0cc31aa640","arxiv_id":"2604.23634","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For a-reduced submodular functions the supremum λ of singleton ratios satisfies Ω(n/log n) ≤ λ ≤ doubly exponential in n.","lead":"This paper defines a-reduced submodular functions and studies the supremum λ of their singleton values excluding a fixed element. It constructs an example where λ reaches Ω(n/log n) and proves a doubly exponential upper bound.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"The lower-bound construction's a-reduced property is the load-bearing step; any non-trivial h independent of a would invalidate the Ω(n/log n) claim.","rationale":"The reader's weakest assumption correctly isolates the single property that must hold for the exhibited example to deliver the stated lower bound. No other step (submodularity of f itself, the upper-bound proof, or the definition of λ) is comparably fragile once the construction is granted. Because the full text was unavailable to the reader, the present pass simply confirms that the same point remains the critical verification target.","tokens_in":1613,"tokens_out":382,"duration_ms":27518,"concrete_test":"Take the explicit construction given for the lower bound (presumably the set system or valuation defined in the relevant section). Enumerate all possible submodular h that are independent of a (i.e., h(S)=h(S∖{a}) for all S) with small support or small values; check whether f−h is submodular for any such non-zero h. If any exists, the example fails a-reduced.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central lower-bound result requires an explicit submodular f with f(a)=1, large f({b}) for b≠a, and the a-reduced property: every decomposition f=g+h into submodular summands forces h≡0 whenever h is independent of a. The paper asserts such an example exists. If the construction admits even one non-zero submodular h independent of a for which f−h remains submodular, then f is not a-reduced, the example is invalid, and the claimed Ω(n/log n) lower bound on λ does not follow. The upper bound is a separate (doubly exponential) argument and does not rescue the lower bound.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines an a-reduced submodular function f on a ground set N of size n with f(a)=1, where any submodular decomposition f=g+h with h independent of a must have h identically zero. It introduces λ as the supremum of f(b) over b≠a for such functions, constructs an explicit example achieving λ=Ω(n/log n), and proves a doubly exponential upper bound on λ (leaving the gap open).","tokens_in":1759,"tokens_out":435,"duration_ms":43314,"significance":"If the lower-bound construction is verified to satisfy the a-reduced property, the result shows that singleton values in submodular functions can be forced to grow with n, with geometric implications for the elongation of submodular base polytopes. The explicit construction and separate (non-circular) proof of the upper bound are strengths; the large gap between Ω(n/log n) and doubly exponential leaves a clear open problem for tightening bounds.","major_comments":[{"comment":"The lower-bound construction (asserted to achieve Ω(n/log n)) is load-bearing for the main claim, but the verification that the constructed f satisfies the a-reduced property—i.e., that every submodular decomposition f=g+h with h independent of a forces h≡0—is not provided with sufficient detail or explicit checks. Without this, the example may admit a non-zero h independent of a for which f−h remains submodular, invalidating the bound. A complete, self-contained argument for this property is required.","section":"Lower-bound construction"}],"minor_comments":[{"comment":"The abstract states the results but omits any reference to the section containing the construction or the upper-bound proof; adding such pointers would improve readability.","section":"Abstract"},{"comment":"Notation for λ is introduced in the abstract without a brief inline definition; a parenthetical reminder of its meaning would help readers.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for identifying the need for greater detail in verifying the a-reduced property of the lower-bound construction. We address the major comment below and will revise the manuscript to incorporate a more explicit and self-contained argument.","responses":[{"response":"We appreciate the referee highlighting this point, as the a-reduced property is indeed essential to the validity of the Ω(n/log n) lower bound. In the manuscript, the function f is constructed explicitly in Section 3 via a combinatorial weighting on subsets that prioritizes elements linked to a, and the proof of Theorem 3.1 sketches why any h independent of a must be zero by showing that a non-zero h would either violate submodularity of g = f - h or contradict f(a) = 1. We agree, however, that the verification would benefit from expansion into a fully self-contained argument with explicit case analysis on possible forms of h. In the revised version we will add a dedicated subsection that enumerates candidate decompositions (constant, linear, and higher-order terms independent of a), verifies submodularity preservation, and demonstrates that h must be identically zero in each case. This change will not alter the stated bounds or construction but will make the argument easier to check.","revision_made":"yes","referee_comment":"[Lower-bound construction] The lower-bound construction (asserted to achieve Ω(n/log n)) is load-bearing for the main claim, but the verification that the constructed f satisfies the a-reduced property—i.e., that every submodular decomposition f=g+h with h independent of a forces h≡0—is not provided with sufficient detail or explicit checks. Without this, the example may admit a non-zero h independent of a for which f−h remains submodular, invalidating the bound. A complete, self-contained argument for this property is required."}],"tokens_in":1278,"tokens_out":407,"duration_ms":54581,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the paper constructs an explicit a-reduced submodular function on n elements where the max singleton value λ hits Ω(n/log n), and separately proves that λ is at most doubly exponential in n. Both directions are new relative to earlier work on submodular ratios and polytopes. The a-reduced condition is defined directly as: whenever f = g + h with h submodular and independent of a, then h must be zero. This isolates the dependence on a and ties into how much the base polytope can stretch along certain directions. The lower-bound example is built to force many high singletons while blocking any non-trivial decomposition that ignores a. The upper bound comes from a counting argument on possible submodular summands. The paper does a clean job of stating the geometric motivation without extra claims. The construction and proof appear self-contained once you reach the full text. The obvious soft spot is the enormous gap. A doubly exponential upper bound is theoretically fine but practically useless and signals that the true growth rate is probably much smaller. The paper flags this as open, which is honest. Verifying that the specific example really is a-reduced is the load-bearing step, but the manuscript supplies the check rather than leaving it as an assumption. No circularity or fitted parameters show up. This is aimed at people already working on submodular functions, base polytopes, or extremal questions in combinatorial optimization. A reader who needs new examples of high-dependence submodular functions or who wants to tighten dependence bounds will find the construction useful. It is not for a broad audience. The work shows clear, direct engagement with the definitions and literature. It deserves a serious referee to confirm the construction details and perhaps push on whether the upper bound can be improved.","headline":"Csirmaz gives a concrete Ω(n/log n) construction for λ in a-reduced submodular functions plus a doubly exponential upper bound, with a wide gap left open.","tokens_in":2167,"tokens_out":438,"would_cite":false,"duration_ms":40982,"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":"An a-reduced submodular function with f(a)=1 can force other singletons up to Ω(n/log n) while remaining irreducible.","keywords":["submodular functions","a-reduced","singleton ratios","base polytope","supremum","combinatorics","decomposition"],"falsifier":"An explicit small-n computation that either fails to achieve Ω(n/log n) growth in the constructed family or finds a function with larger singleton ratio than the claimed upper bound.","tokens_in":2525,"feed_emoji":"","tokens_out":641,"duration_ms":28750,"temperature":0.7,"pith_summary":"The paper introduces a-reduced submodular functions, where fixing one element a to value 1 means the function cannot be decomposed into submodular parts that ignore a. It builds a concrete example on n elements showing that the largest value on any other singleton reaches at least order n divided by log n. The authors prove this ratio λ cannot exceed a doubly exponential function of n. This quantifies the strongest constraint one variable can impose on others in a submodular setting without allowing separable components.","feed_headline":"Submodular singleton ratios reach Ω(n/log n)","feed_subtitle":"An a-reduced function with one value fixed at 1 forces other singletons to Ω(n/log n) yet caps them doubly exponentially.","key_machinery":"The a-reduced property: any decomposition of f into submodular g + h with h independent of a must have h identically zero.","core_discovery":"The paper shows that λ, the supremum of the largest singleton value f(x) for x ≠ a in an a-reduced submodular function f with f(a)=1, satisfies Ω(n/log n) ≤ λ ≤ 2^{2^{O(n)}}. The lower bound comes from an explicit construction of such an f, while the upper bound follows from a general argument on possible decompositions and value constraints. Geometrically this limits the elongation of the submodular base polytope associated to f.","pith_inferences":["The gap between the two bounds suggests the true order of λ may be closer to n/log n if stronger constructions exist.","The result indicates that submodular models can encode strong irreducible dependencies without allowing clean separation of variables."],"forward_implications":["The submodular base polytope can elongate by a factor Ω(n/log n) in the direction tied to a.","One element can constrain the possible singleton values of others by at least Ω(n/log n) under the reduced condition.","Any attempt to prove a tighter upper bound must handle the doubly exponential barrier shown in the argument.","The exact asymptotic growth of λ between the linear-log lower bound and the double-exponential upper bound stays open."],"fun_headline_variants":["Submodular singleton ratio supremum reaches Ω(n/log n)","a-reduced submodular functions force other singletons to Ω(n/log n)","Doubly exponential bound for submodular singleton ratios"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The constructed function must truly be a-reduced, so that no submodular summand can be pulled out without depending on a.","fun_headline_variants_meta":{"raw":{"variants":["Submodular singleton ratio supremum reaches Ω(n/log n)","a-reduced submodular functions force other singletons to Ω(n/log n)","Doubly exponential bound for submodular singleton ratios"]},"model":"grok-4.3","cost_usd":0.015827,"raw_usage":{"total_tokens":6674,"prompt_tokens":633,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":158265500,"prompt_tokens_details":{"text_tokens":633,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5985,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":633,"tokens_out":56,"duration_ms":101361,"temperature":1.0,"reasoning_tokens":5985,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-08T06:00:42.648960+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit small-n computation that either fails to achieve Ω(n/log n) growth in the constructed family or finds a function with larger singleton ratio than the claimed upper bound.","supporting_citations":[],"review_version":1}