{"id":"f850c976-bfe5-42d4-bd4f-b499aaf89a31","arxiv_id":"1907.11802","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Defines shifted R-polynomials for weighted Bruhat path counting in Coxeter groups, proves a new irregularity criterion for lower intervals, and bounds the polynomials by Jacobsthal numbers.","lead":"The paper introduces shifted R-polynomials (Bruhat weights) for Bruhat intervals in finite Coxeter groups and applies them to weighted counting of Bruhat paths. A smart generalist might read it to learn new combinatorial tools for analyzing symmetries and paths in group structures that appear in algebra and geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption is precisely the point that would need to fail for the claims to collapse, yet the supplied text contains an explicit, self-contained definition together with direct verifications that remove the need for further hidden restrictions. No internal inconsistency or missing step is visible.","tokens_in":1653,"tokens_out":239,"duration_ms":20693,"concrete_test":"Recompute the shifted R-polynomial for the longest element interval in the dihedral group I_2(5) using the explicit definition in §2; confirm that the resulting polynomial satisfies the claimed recurrence and yields the stated Jacobsthal upper bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims rest on a new but explicitly defined family of polynomials on Bruhat intervals. The manuscript supplies the definition, verifies the necessary recurrence and positivity properties directly from the Bruhat order, derives the weighted path enumeration, states the irregularity criterion as a direct consequence, and proves the Jacobsthal bound by induction on interval length. All steps are internal to the given Coxeter system and require no external unstated restrictions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces shifted R-polynomials (also called Bruhat weights) for all Bruhat intervals in finite Coxeter groups. It applies these polynomials to obtain weighted counts of Bruhat paths, proves a new criterion for irregularity of lower intervals that is analogous to the results of Carrell-Peterson and Dyer, and establishes an upper bound on the shifted R-polynomials for intervals of fixed length in terms of Jacobsthal numbers.","tokens_in":1726,"tokens_out":390,"duration_ms":27412,"significance":"If the central claims hold, the work supplies a new family of polynomials on Bruhat intervals together with recurrence relations, positivity properties, a weighted path enumeration, an irregularity criterion, and a Jacobsthal bound, all derived internally from the Bruhat order. These tools extend existing results on R-polynomials and may be useful for further combinatorial and algebraic study of Coxeter groups.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the shifted R-polynomials are defined 'for all Bruhat intervals in finite Coxeter groups,' but the introduction should explicitly record the precise recurrence or initial conditions used to define them (e.g., the base case for length-0 intervals and the covering-relation step).","section":"Introduction"},{"comment":"The statement of the irregularity criterion (analogous to Carrell-Peterson and Dyer) should include a short reminder of the original statements so that the analogy is immediately verifiable by the reader.","section":"Section on irregularity criterion"},{"comment":"The induction proof of the Jacobsthal bound would benefit from an explicit statement of the inductive hypothesis on the length of the interval.","section":"Section containing the Jacobsthal bound"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the paper's contributions on shifted R-polynomials, weighted Bruhat path counting, the irregularity criterion, and the Jacobsthal bound.","responses":[],"tokens_in":1090,"tokens_out":71,"duration_ms":10986,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper defines shifted R-polynomials on Bruhat intervals in finite Coxeter groups and applies them to weighted counting of Bruhat paths. It also gives a new irregularity criterion for lower intervals modeled on Carrell-Peterson and Dyer, plus an upper bound by Jacobsthal numbers for fixed-length intervals.","headline":"The paper defines shifted R-polynomials on Bruhat intervals, derives weighted path counts from them, and proves an irregularity criterion plus a Jacobsthal bound.","tokens_in":2177,"tokens_out":140,"would_cite":false,"duration_ms":11628,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Combinatorial enumeration on Bruhat intervals unrelated to recognition cost or distinction forcing","alignment":"orthogonal","rationale":"The paper's central objects are shifted R-polynomials (Ruw(q+1)), Bruhat weights ρ(Γ)=(q+1)^((ℓ(Γ)−a(Γ))/2) q^a(Γ), Jacobsthal/dihedral polynomials dn(q), and irregularity criteria on lower intervals in finite Coxeter groups. These are derived from the recurrence relations of R-polynomials and EL-shellability of Bruhat order. No connection exists to the RS forcing chain: the single-distinction axiom, the reciprocal cost J(x)=½(x+x⁻¹)−1 (Cost/FunctionalEquation.washburn_uniqueness_aczel), the φ-ladder, the 8-tick period, or the derivation of D=3 (Foundation/DimensionForcing, AlexanderDuality). The paper operates entirely within algebraic combinatorics on posets and contains none of the RS structural signatures (cosh-cost, ratio symmetry, parameter-free constants).","tokens_in":58442,"confidence":"high","tokens_out":236,"duration_ms":5919,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Shifted R-polynomials weight Bruhat paths and bound them by Jacobsthal numbers in finite Coxeter groups.","keywords":["shifted R-polynomials","Bruhat paths","R-polynomials","Coxeter groups","Jacobsthal numbers","Bruhat intervals","interval irregularity"],"falsifier":"An explicit Bruhat interval of length n in some finite Coxeter group where the shifted R-polynomial either fails to equal the weighted path sum or exceeds the nth Jacobsthal number.","tokens_in":2548,"feed_emoji":"","tokens_out":629,"duration_ms":19519,"temperature":0.7,"pith_summary":"The paper defines shifted R-polynomials, also called Bruhat weights, for every Bruhat interval in a finite Coxeter group. These polynomials are then applied to count Bruhat paths with explicit weights. The construction produces a criterion that detects when a lower interval is irregular, modeled on earlier tests by Carrell-Peterson and Dyer. For any interval of fixed length the shifted R-polynomials are shown to lie below the corresponding Jacobsthal number.","feed_headline":"Shifted R-polynomials weight Bruhat paths by Jacobsthal bounds","feed_subtitle":"The polynomials also supply a new irregularity test for lower intervals in any finite Coxeter group.","key_machinery":"Shifted R-polynomials, which assign a Bruhat weight to each interval and thereby encode the weighted sum over all paths in that interval.","core_discovery":"Defining shifted R-polynomials on Bruhat intervals supplies a uniform mechanism for weighted enumeration of paths between any two elements in the Bruhat order of a finite Coxeter group. The same definition immediately yields an irregularity criterion for lower intervals and an explicit upper bound, expressed by Jacobsthal numbers, once the interval length is fixed.","pith_inferences":["Explicit algorithms for computing the weighted path counts could now be written for concrete groups such as the symmetric group.","The appearance of Jacobsthal numbers may link Bruhat-path statistics to other combinatorial sequences counted by the same numbers.","If the definition of shifted R-polynomials extends beyond finite groups, the irregularity criterion could be tested in affine or infinite Coxeter systems."],"forward_implications":["Weighted counting of Bruhat paths becomes available for every pair of elements in any finite Coxeter group.","A new test, analogous to Carrell-Peterson and Dyer, identifies irregular lower intervals.","Shifted R-polynomials on intervals of fixed length are bounded above by Jacobsthal numbers.","The entire apparatus applies uniformly without further restrictions on the Coxeter group or the interval."],"fun_headline_variants":["Shifted R-polynomials weight Bruhat paths with Jacobsthal bounds","Bruhat paths weighted by shifted R-polynomials and Jacobsthal bounds","Shifted R-polynomials bound Bruhat paths via Jacobsthal numbers","Shifted R-polynomials yield irregularity test for Bruhat intervals","Weighted Bruhat paths counted via shifted R-polynomials bounds"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Shifted R-polynomials are well-defined and consistent on every Bruhat interval of every finite Coxeter group so that path weights and the irregularity test follow directly from the definition.","fun_headline_variants_meta":{"raw":{"variants":["Shifted R-polynomials weight Bruhat paths with Jacobsthal bounds","Bruhat paths weighted by shifted R-polynomials and Jacobsthal bounds","Shifted R-polynomials bound Bruhat paths via Jacobsthal numbers","Shifted R-polynomials yield irregularity test for Bruhat intervals","Weighted Bruhat paths counted via shifted R-polynomials bounds"]},"model":"grok-4.3","cost_usd":0.008942,"raw_usage":{"total_tokens":3946,"prompt_tokens":523,"num_sources_used":0,"completion_tokens":86,"cost_in_usd_ticks":89424500,"prompt_tokens_details":{"text_tokens":523,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3337,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":523,"tokens_out":86,"duration_ms":39447,"temperature":1.0,"reasoning_tokens":3337,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T15:08:17.497480+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit Bruhat interval of length n in some finite Coxeter group where the shifted R-polynomial either fails to equal the weighted path sum or exceeds the nth Jacobsthal number.","supporting_citations":[],"review_version":1}