{"id":"fb3cea91-e87a-495f-a6f3-db8584e8dc09","arxiv_id":"2606.00246","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove that δ-tribes functions, monotone Boolean functions with the tribe separation property, and Boolean functions with the semi-separation property satisfy the FEI conjecture using a stopping binary tree and a key entropy-influence inequality.","lead":"The paper identifies new classes of Boolean functions, such as δ-tribes and monotone functions with separation properties, that satisfy the Fourier Entropy-Influence conjecture via a stopping binary tree framework and a key inequality. A smart generalist might read it to track incremental progress on a conjecture linking entropy and influence measures in discrete structures with potential ties to learning and complexity.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the point at which the new framework could fail to deliver the claimed verifications. Because the manuscript states that it demonstrates the inequality for the listed classes, and the reduction itself contains no evident logical gap, the central claim stands on its own terms; the low-confidence UNVERDICTED rating is attributable to the abstract-only review rather than to any identified flaw in the argument.","tokens_in":1757,"tokens_out":319,"duration_ms":19097,"concrete_test":"For the δ-tribes functions, explicitly construct the stopping binary tree used in the paper, recompute the key inequality at each branching node using the explicit Fourier expressions for the tribes, and verify that the inequality holds with the claimed constant; if it fails for any node, the reduction does not apply to that class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper defines a stopping binary tree framework under which the FEI conjecture follows once a key inequality (controlling H(f) − (H(f+) + H(f−))/2 by Inf_m(f)) holds at branching nodes and the conjecture holds at stopping nodes. It then asserts that δ-tribes functions, monotone functions with the tribe separation property, and functions with the semi-separation property satisfy these conditions for suitably chosen trees. No internal inconsistency, circularity, or unstated assumption in the reduction itself is visible from the given description; the argument is a direct verification for restricted classes rather than a general proof.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces a stopping binary tree framework for the Fourier Entropy-Influence (FEI) conjecture: the conjecture holds for a Boolean function if a key inequality (bounding the difference H(f) − (H(f+) + H(f−))/2 by Inf_m(f)) holds at branching nodes and the conjecture holds at stopping nodes. The authors identify three classes—δ-tribes functions, monotone Boolean functions with the tribe separation property, and Boolean functions with the semi-separation property—that fit suitably chosen trees in this framework and thereby satisfy the FEI conjecture.","tokens_in":1890,"tokens_out":386,"duration_ms":20206,"significance":"If the claimed verifications of the key inequality hold, the work supplies new, explicitly described classes of functions satisfying the FEI conjecture beyond the previously known symmetric functions and read-k decision trees. The stopping-binary-tree reduction is a non-circular structural device that isolates the inequality as the only additional condition needed; this may prove useful for further targeted verifications. The paper also records auxiliary results on these classes that could be of independent interest to experts.","major_comments":[{"comment":"The central claim that the three listed classes satisfy the FEI conjecture rests entirely on the assertion that the key inequality holds at the branching nodes of the chosen stopping trees. The manuscript states that the classes 'fit this framework' and 'demonstrate that they satisfy' the conjecture, but does not exhibit the explicit calculations or case analysis establishing the inequality for δ-tribes or the separation properties; this verification is load-bearing for the result.","section":"Abstract / framework description"}],"minor_comments":[{"comment":"Abstract contains the typographical repetition 'and, and,' in the sentence listing the three classes.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their thoughtful summary and for highlighting the potential utility of the stopping-binary-tree framework. We address the single major comment below.","responses":[{"response":"We agree that the explicit verification of the key inequality at branching nodes is load-bearing and that the current manuscript does not contain the full case analysis. In the revised version we will insert detailed calculations establishing the inequality for each of the three classes (δ-tribes, monotone functions with the tribe-separation property, and functions with the semi-separation property), including the choice of branching variable m at each node and the resulting entropy-influence bounds.","revision_made":"yes","referee_comment":"[Abstract / framework description] The central claim that the three listed classes satisfy the FEI conjecture rests entirely on the assertion that the key inequality holds at the branching nodes of the chosen stopping trees. The manuscript states that the classes 'fit this framework' and 'demonstrate that they satisfy' the conjecture, but does not exhibit the explicit calculations or case analysis establishing the inequality for δ-tribes or the separation properties; this verification is load-bearing for the result."}],"tokens_in":1384,"tokens_out":254,"duration_ms":13417,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This note adds three new classes of Boolean functions to the list where the Fourier Entropy-Influence conjecture holds: δ-tribes functions, monotone functions with the tribe separation property, and functions with the semi-separation property. It organizes the checks around a stopping binary tree where the conjecture follows if a local inequality on entropy versus m-influence holds at the branches and the base case holds at the stopping nodes.\n\nThe reduction is logically direct and extends earlier verification techniques without circularity or free parameters. The framework itself looks reusable for other restricted classes.\n\nThe limitation is that the result stands or falls on whether the key inequality actually holds for these specific families at the chosen nodes. The abstract asserts that it does but supplies no calculations or edge-case checks, so the strength of the support cannot be judged from the given material. This remains incremental verification rather than a general advance or new applications.\n\nThe work is for specialists tracking the FEI conjecture in Boolean Fourier analysis. Readers already following the literature on tribes and decision trees may find the tree method convenient for testing further examples. The internal logic is consistent and the citation pattern is standard.\n\nI would send it to peer review for a referee to inspect the inequality verifications.","headline":"The paper adds three more families to the verified cases for the FEI conjecture using a stopping binary tree reduction.","tokens_in":2354,"tokens_out":315,"would_cite":false,"duration_ms":16886,"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":"The Fourier Entropy-Influence conjecture holds for δ-tribes functions, monotone functions with the tribe separation property, and functions with the semi-separation property.","keywords":["Fourier Entropy-Influence conjecture","Boolean functions","tribes functions","monotone functions","Fourier analysis","influence","entropy","separation property"],"falsifier":"A function belonging to one of the three classes in which Fourier entropy exceeds total influence, or in which the key inequality fails to hold at any branching node of its stopping binary tree.","tokens_in":2662,"feed_emoji":"","tokens_out":674,"duration_ms":18298,"temperature":0.7,"pith_summary":"The Fourier Entropy-Influence conjecture states that the Fourier entropy of any Boolean function is at most its total influence. The paper verifies the conjecture for several new families by introducing a stopping binary tree and a key inequality that relates entropy differences of a function and its subfunctions to the m-influence. Functions in the identified classes satisfy the inequality at every branching node of the tree and the conjecture itself at the stopping nodes. This recursive structure yields the result for the whole function. The approach shows how the conjecture can be established class by class without proving the general case.","feed_headline":"δ-tribes and separated Boolean functions satisfy entropy-influence bound","feed_subtitle":"The Fourier Entropy-Influence conjecture holds for these classes via a recursive argument on stopping binary trees and a key inequality at b","key_machinery":"The stopping binary tree together with the key inequality that bounds the difference between the entropy of f and the average entropy of the subfunctions f± by the m-influence of f.","core_discovery":"The authors establish that δ-tribes functions, monotone Boolean functions with the tribe separation property, and Boolean functions with the semi-separation property all satisfy the Fourier Entropy-Influence conjecture. They do so by defining a stopping binary tree such that any function obeying the key inequality at its branching nodes and the conjecture at its stopping nodes obeys the conjecture overall. These three classes are shown to meet both requirements.","pith_inferences":["The separation properties may indicate a structural condition on how influences are distributed across coordinates that makes the key inequality easier to verify.","The same tree-based reduction could be tested on additional families such as read-k functions or symmetric functions to see whether the inequality continues to hold.","If the key inequality turns out to fail for some functions, those counterexamples would isolate the precise obstacle to a general proof."],"forward_implications":["The Fourier Entropy-Influence conjecture holds for every δ-tribes function.","The Fourier Entropy-Influence conjecture holds for every monotone Boolean function with the tribe separation property.","The Fourier Entropy-Influence conjecture holds for every Boolean function with the semi-separation property.","If the key inequality holds at every branching node for every Boolean function, then the Fourier Entropy-Influence conjecture holds for all Boolean functions by induction on the stopping binary tree."],"fun_headline_variants":["δ-tribes satisfy Fourier entropy-influence conjecture","Tribe separation verifies entropy-influence bound","Semi-separation property meets FEI conjecture","Stopping trees confirm FEI for separated functions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The key inequality holds at the branching nodes of the stopping binary tree for the identified function classes.","fun_headline_variants_meta":{"raw":{"variants":["δ-tribes satisfy Fourier entropy-influence conjecture","Tribe separation verifies entropy-influence bound","Semi-separation property meets FEI conjecture","Stopping trees confirm FEI for separated functions"]},"model":"grok-4.3","cost_usd":0.009177,"raw_usage":{"total_tokens":4046,"prompt_tokens":697,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":91765500,"prompt_tokens_details":{"text_tokens":697,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3295,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":697,"tokens_out":54,"duration_ms":28094,"temperature":1.0,"reasoning_tokens":3295,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T21:50:17.893691+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A function belonging to one of the three classes in which Fourier entropy exceeds total influence, or in which the key inequality fails to hold at any branching node of its stopping binary tree.","supporting_citations":[],"review_version":1}