{"id":"118c9f8d-53bf-42ad-ad23-a866f3238d0e","arxiv_id":"2606.25250","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends eventual stability of quadratic iterates f = x² + 1/c to Q(i) by proving the case c ≡ 2 mod 4 is stable, supporting a prior conjecture with an auxiliary check due to lack of ordering.","lead":"The paper extends results on eventual stability of iterates of f = x² + 1/c from the rationals to the Gaussian rationals Q(i). It proves stability holds for the case c ≡ 2 mod 4, giving evidence toward a conjecture on irreducibility of all iterates.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's note that the review was abstract-only already flags the absence of verifiable detail; without the actual arguments no further technical concern can be raised or refuted.","tokens_in":1727,"tokens_out":187,"duration_ms":17657,"concrete_test":"Retrieve the full manuscript and examine the section that defines the auxiliary function and proves stability for c ≡ 2 mod 4; confirm that the function is shown to imply bounded factorization for all iterates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The provided context supplies only the abstract and a reader summary performed on the abstract alone. No full proofs, explicit definitions of the auxiliary function, or detailed arguments for the c ≡ 2 mod 4 case are available to scrutinize. Consequently no load-bearing internal inconsistency or unsupported assumption in the central stability claim can be located.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends the study of eventual stability for quadratic maps f(x) = x² + 1/c from Q to the Gaussian field Q(i). It proves that the previously elusive case c ≡ 2 mod 4 (as a Z-class) yields eventually stable iterates over Q(i), and discusses the extension of the irreducibility conjecture 'if f² is irreducible then all fⁿ are irreducible' to Q(i), where an auxiliary function is required to check a specific n because Q(i) lacks a linear order. Stability for the c ≡ 1 mod 2 class (now viewed in Z[i]) receives less complete treatment than in the rational case.","tokens_in":1783,"tokens_out":397,"duration_ms":12584,"significance":"If the central stability result for c ≡ 2 mod 4 holds, the manuscript supplies concrete supporting evidence for Conjecture 1 of the cited prior work by resolving an open congruence class over a new base field. The introduction of an auxiliary function to handle the absence of ordering is a technically natural adaptation, though the incomplete treatment of the odd-c case limits the overall scope.","major_comments":[],"minor_comments":[{"comment":"The abstract states that an auxiliary function is used to check a specific n, but the manuscript should include an explicit definition or reference to its construction in the main text (e.g., near the statement of the extended conjecture).","section":null},{"comment":"The claim that stability for c ≡ 1 mod 2 'is not as fully handled' should be made precise by indicating which subcases remain open and whether they are expected to follow the same argument as over Z.","section":null},{"comment":"Notation for the equivalence classes (Z versus Z[i]) should be clarified at first use to avoid confusion when moving between the rational and Gaussian settings.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment and recommendation of minor revision. The report correctly identifies our main contribution (the stability result for c ≡ 2 mod 4) and the technical adaptation needed for the irreducibility conjecture. We address the noted limitation on the odd-c case below.","responses":[{"response":"We agree this is a genuine limitation of the present manuscript. The primary goal was to settle the previously open congruence class c ≡ 2 mod 4 over Q(i), which supplies new evidence for Conjecture 1 of the cited work. The odd-c case over Z[i] would require additional tools (e.g., a fuller analysis of primes above 2 and possible ramification in the dynatomic extensions) that go beyond the scope of this short note. We will add a clarifying paragraph in the introduction stating the restricted scope and indicating why a complete treatment is deferred.","revision_made":"yes","referee_comment":"Stability for the c ≡ 1 mod 2 class (now viewed in Z[i]) receives less complete treatment than in the rational case."},{"response":"We appreciate the referee’s recognition that the auxiliary function is a natural device. It is introduced precisely to replace the ordering argument used over Q when verifying the base case of the irreducibility conjecture for a specific iterate. We will incorporate the referee’s phrasing into a revised introduction to emphasize both the adaptation and the acknowledged limitation on scope.","revision_made":"partial","referee_comment":"The introduction of an auxiliary function to handle the absence of ordering is a technically natural adaptation, though the incomplete treatment of the odd-c case limits the overall scope."}],"tokens_in":1286,"tokens_out":369,"duration_ms":23440,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"McDermott has worked out the stability for the c ≡ 2 mod 4 case over Q(i) that was left open in the Q paper. That's the main new piece.\n\nThe paper adapts the auxiliary function method to handle the lack of ordering on Q(i) and checks the irreducibility for that specific congruence class. It gives concrete evidence that the conjecture holds in this new setting for the elusive case. The approach stays close to the cited work, which keeps the argument grounded.\n\nOn the positive side, the extension is direct and the focus on the Z equivalence class that was missing is useful. No sign of circular reasoning or reliance on unverified assumptions beyond what's in the prior paper.\n\nThe weaker part is the admission that stability for c ≡ 1 mod 2 over Z[i] is not as complete. That leaves the overall picture for Q(i) still partial. Also, the abstract does not show the detailed steps for the new proof, so one would need to verify the auxiliary function calculations in the full text to be sure there are no hidden gaps in the factorization arguments.\n\nThis is aimed at researchers tracking the eventual stability conjecture in arithmetic dynamics. Someone already reading the Q version will get value from seeing how the method transfers, but it is not a broad advance.\n\nI would bring it to a reading group for the specific case it resolves. I would not cite it in my own work soon because it is a narrow extension. It deserves peer review because the new case is a real gap that the field cares about filling, even if the paper needs work on the second class.","headline":"McDermott fills the c ≡ 2 mod 4 stability gap over Q(i) but leaves the odd case incomplete.","tokens_in":2228,"tokens_out":397,"would_cite":false,"duration_ms":21507,"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":"Quadratic polynomials x² + 1/c are eventually stable over Q(i) when c ≡ 2 mod 4.","keywords":["quadratic polynomials","eventual stability","iterates","irreducibility","Gaussian rationals","number fields","congruence classes"],"falsifier":"An explicit c ≡ 2 mod 4 together with a concrete large n such that f^n factors into more than the predicted number of irreducibles over Q(i).","tokens_in":2626,"feed_emoji":"","tokens_out":655,"duration_ms":19924,"temperature":0.7,"pith_summary":"The paper extends the study of eventually stable quadratic polynomials from the rationals to the Gaussian rationals Q(i). It proves that f = x² + 1/c is eventually stable, meaning the number of irreducible factors of its iterates stays bounded, precisely when c falls into the congruence class 2 mod 4. This supplies concrete evidence for the conjecture that irreducibility of the second iterate implies irreducibility of all higher iterates. The proof adapts the rational case but introduces an auxiliary function to handle the missing linear order on Q(i).","feed_headline":"Quadratics stabilize over Q(i) for c ≡ 2 mod 4","feed_subtitle":"The case that resisted proof over the rationals now holds in the Gaussian rationals, supporting the iterate-irreducibility conjecture.","key_machinery":"Eventual stability of polynomial iterates, i.e., the property that the number of irreducible factors of the n-fold composition f^n stays bounded by a constant independent of n.","core_discovery":"When c belongs to the Z-equivalence class 2 mod 4, the quadratic f = x² + 1/c satisfies eventual stability over Q(i): the number of irreducible factors of f^n remains bounded independently of n. The argument proceeds by showing that if f² is irreducible then all higher iterates remain irreducible, with an auxiliary function supplying the necessary check for a specific n because Q(i) lacks a total order.","pith_inferences":["The same auxiliary-function technique might resolve the remaining odd-c cases in Z[i] with additional computation.","Analogous stability statements could be tested over other quadratic fields once suitable auxiliary functions are identified.","The bounded-factor property implies that the dynamical system generated by f has only finitely many distinct prime divisors in its orbit factorizations."],"forward_implications":["If f² is irreducible over Q(i) then every higher iterate f^n remains irreducible.","The previously unresolved congruence class 2 mod 4 now satisfies the bounded-factor condition.","The conjecture linking irreducibility of f² to irreducibility of all iterates holds for this class in Q(i).","Stability results carry over from Q to Q(i) at least for the even-c case."],"fun_headline_variants":["Stability for quadratics over Q(i) with c ≡ 2 mod 4","c ≡ 2 mod 4 case stable for iterates over Q(i)","Quadratic iterates stable over Q(i) for c ≡ 2 mod 4","Eventual stability in Q(i) for c ≡ 2 mod 4 quadratics"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That an auxiliary function suffices to verify the irreducibility condition at the needed finite stage even without a linear order on Q(i).","fun_headline_variants_meta":{"raw":{"variants":["Stability for quadratics over Q(i) with c ≡ 2 mod 4","c ≡ 2 mod 4 case stable for iterates over Q(i)","Quadratic iterates stable over Q(i) for c ≡ 2 mod 4","Eventual stability in Q(i) for c ≡ 2 mod 4 quadratics"]},"model":"grok-4.3","cost_usd":0.005936,"raw_usage":{"total_tokens":2826,"prompt_tokens":688,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":59362000,"prompt_tokens_details":{"text_tokens":688,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2054,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":688,"tokens_out":84,"duration_ms":16595,"temperature":1.0,"reasoning_tokens":2054,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T21:36:30.170786+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit c ≡ 2 mod 4 together with a concrete large n such that f^n factors into more than the predicted number of irreducibles over Q(i).","supporting_citations":[],"review_version":2}