{"id":"f56a46a5-7970-4c80-8c0a-0f1f8926ec6d","arxiv_id":"2606.00420","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Counterexamples prove pre_j is not injective on partitions of length 2j for j≥3, while the complete homogeneous map prh_j is injective on all partitions.","lead":"The paper constructs counterexamples showing the elementary symmetric partition map pre_j fails to be injective for partitions of length exactly 2j when j is at least 3, disproving a refined conjecture. A smart generalist might read it to understand how explicit constructions can settle questions about uniqueness in combinatorial maps.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption concerns correct application of the pre_j definition to the counterexamples; this is the natural place to check but does not appear to be a point of failure once the explicit partitions are examined. Because the paper supplies both concrete counterexamples and a separate positive result for prh_j, and no technical flaw in either is visible, the UNVERDICTED verdict from the abstract-only review does not require adjustment.","tokens_in":1667,"tokens_out":306,"duration_ms":32423,"concrete_test":"Independently compute the multiset of all j-fold products for each pair of claimed counterexample partitions of length 2j (for j=3 and j=4) and verify that the sorted lists coincide while the input partitions are distinct and have identical sum n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim consists of an explicit general construction of counterexamples showing non-injectivity of pre_j on length-2j partitions for j ≥ 3, together with a proof that prh_j is injective on all partitions. No internal inconsistency, unsupported assumption in the construction, or gap in the recovery argument for prh_j is apparent from the description of the maps and the statements. The definition of pre_j is the standard one given in the referenced prior work and is applied directly to the constructed examples; nothing in the claim indicates that this application requires modification.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper disproves a refined conjecture of Devnani and Eyyunni by exhibiting explicit counterexamples showing that the elementary symmetric partition map pre_j is not injective on partitions of n having length exactly 2j when j ≥ 3. It further proves that the analogous map prh_j, defined via the complete homogeneous symmetric polynomial, is injective on the set of all partitions.","tokens_in":1765,"tokens_out":325,"duration_ms":19017,"significance":"The explicit constructions settle the refined conjecture in the negative for the indicated length and supply a positive injectivity theorem for the homogeneous variant. These results clarify the range of injectivity for symmetric-polynomial-induced maps on partitions and furnish concrete examples that can be used to test further conjectures in the area.","major_comments":[],"minor_comments":[{"comment":"The statement of the main theorem (presumably Theorem 1.1 or 3.1) would benefit from an explicit small-j example (e.g., j=3) placed immediately after the general construction so that the length-2j condition and the image coincidence can be verified by direct inspection.","section":"Introduction / Main results"},{"comment":"In the proof that prh_j is injective, the recovery argument from the multiset of values back to the original partition should be cross-referenced to the precise definition of the complete homogeneous polynomial evaluation used in the map.","section":"Section on prh_j injectivity"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report, accurate summary of the results, and recommendation to accept the manuscript. The report correctly identifies the counterexamples disproving the refined conjecture for pre_j on partitions of length 2j (j ≥ 3) and the injectivity theorem for prh_j.","responses":[],"tokens_in":1125,"tokens_out":79,"duration_ms":11691,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that this paper supplies concrete counterexamples showing pre_j is not injective on partitions of length exactly 2j when j is 3 or larger, which directly refutes the refined conjecture after the length-j case was already settled. It also proves that the analogous map prh_j using complete homogeneous symmetric polynomials is injective on every partition.\n\nWhat stands out is the explicit nature of the disproof. Instead of an existence argument, the authors construct partitions of the required length whose images under pre_j coincide. The positive result on prh_j is a separate, self-contained injectivity proof that does not rely on the counterexample side. Both pieces address the prior literature directly without circular definitions or fitted parameters.\n\nThe soft spots are limited. The abstract states the constructions work for length 2j, but the actual partitions and the verification that their elementary symmetric evaluations match are not visible here. A referee would need to confirm those details and the length conditions hold without exception. No internal inconsistency shows up in the map definitions or the recovery argument for prh_j, so the concern stays minor rather than load-bearing.\n\nThis is narrow work for readers already following partition maps and symmetric function injectivity questions. Someone outside that subfield gets little from it. A combinatorics reading group focused on partitions might find the explicit examples useful for discussion.\n\nIt deserves peer review. The claims are specific, the approach is direct, and the results resolve a stated conjecture with verifiable constructions plus a clean positive theorem.","headline":"Gives explicit counterexamples killing the refined pre_j conjecture at length 2j for j>=3 and proves injectivity for the complete homogeneous map prh_j.","tokens_in":2223,"tokens_out":385,"would_cite":false,"duration_ms":17436,"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 pre_j map from partitions via the j-th elementary symmetric polynomial is not injective on those with exactly 2j parts for j at least 3.","keywords":["elementary symmetric polynomials","partition maps","injectivity","counterexamples","complete homogeneous symmetric polynomials","symmetric functions","partitions"],"falsifier":"An explicit pair of distinct partitions, each having length exactly 2j for some j ≥ 3, that produce identical outputs under pre_j.","tokens_in":2582,"feed_emoji":"","tokens_out":657,"duration_ms":22975,"temperature":0.7,"pith_summary":"Ballantine, Beck, and Merca defined the pre_j map that takes a partition and produces a new partition whose parts are the individual summands appearing when the j-th elementary symmetric polynomial is evaluated on the original parts. They conjectured that this map is injective among all partitions of any n that have length at least j. After the case of length exactly j was already shown to fail, a refined conjecture held that injectivity would hold whenever length exceeds j. The paper supplies explicit counterexamples proving the map is not injective when length equals 2j for every j of size 3 or larger. It additionally shows that the parallel map prh_j built from the complete homogeneous symmetric polynomial instead of the elementary one is injective across every partition.","feed_headline":"Elementary symmetric partition map not injective at length 2j","feed_subtitle":"Counterexamples for j ≥ 3 disprove refined conjecture; homogeneous version remains injective on all partitions","key_machinery":"the pre_j map, which expands the j-th elementary symmetric polynomial on the parts of an input partition and collects the resulting summands into an output partition","core_discovery":"The pre_j map is not injective on the set of partitions of n that have length exactly 2j, for each j ≥ 3; explicit counterexamples establish this failure of the refined conjecture. The map prh_j defined analogously with complete homogeneous symmetric polynomials is injective on the collection of all partitions.","pith_inferences":["Collisions under pre_j at length 2j suggest that similar non-injectivity may appear at other lengths that are multiples of j.","The contrast between pre_j and the always-injective prh_j indicates that the choice between elementary and complete homogeneous polynomials controls whether the induced map on partitions preserves distinct inputs.","For small values such as j = 3 the paper's counterexamples could be used to locate the smallest n at which the first collision occurs."],"forward_implications":["pre_j fails to be injective on partitions of length 2j for every j ≥ 3","the refined conjecture that pre_j is injective for all lengths strictly greater than j is false","prh_j is injective on the set of every partition"],"fun_headline_variants":["pre_j not injective on 2j-length partitions for j >= 3","Symmetric partition map not injective at length 2j for j>=3","Counterexamples disprove pre_j injectivity for length 2j","prh_j injective on all partitions but pre_j fails at 2j"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The pre_j map is applied exactly as originally defined, without modification, to the partitions of length 2j that serve as the counterexamples.","fun_headline_variants_meta":{"raw":{"variants":["pre_j not injective on 2j-length partitions for j >= 3","Symmetric partition map not injective at length 2j for j>=3","Counterexamples disprove pre_j injectivity for length 2j","prh_j injective on all partitions but pre_j fails at 2j"]},"model":"grok-4.3","cost_usd":0.004294,"raw_usage":{"total_tokens":2117,"prompt_tokens":584,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":42937000,"prompt_tokens_details":{"text_tokens":584,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1455,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":584,"tokens_out":78,"duration_ms":12105,"temperature":1.0,"reasoning_tokens":1455,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T05:38:23.362778+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit pair of distinct partitions, each having length exactly 2j for some j ≥ 3, that produce identical outputs under pre_j.","supporting_citations":[],"review_version":2}