{"id":"6a82fe71-bd18-4728-8fd8-afb2a0187895","arxiv_id":"2607.09793","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"full","parameter_count":0,"one_line_summary":"Cohen's Conjecture 66 fails: C_σ(3959)=697 > 696 = C_σ(31)+C_σ(3928) at m=31, n=3928.","lead":"A short note gives an explicit counterexample to Cohen's 2025 conjecture that the counting function for Sophie Germain cyclic numbers is subadditive. The witness is elementary, lies inside the million range the conjecture's author searched, and is checked by the Lean 4 kernel.","discovery_kind":"incremental","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper’s sole load-bearing step is the verification that eleven concrete integers satisfy the Sophie-Germain-cyclic predicate. That step is elementary, fully enumerated in the text, cross-checked computationally by the author, and re-checked by the Lean kernel without native decision procedures. The reader already flags precisely this list as the weakest assumption and correctly judges the residual risk low. No further soft spot (definitional ambiguity, range error, or incomplete formalization) appears. Consequently the ACCEPT / HIGH-confidence verdict stands; the concrete test above is merely a belt-and-suspenders recomputation that is expected to succeed.","tokens_in":4096,"tokens_out":483,"duration_ms":5200,"concrete_test":"Independently recompute, for each of the eleven k in {3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959}, both gcd(k,φ(k)) and gcd(2k+1,φ(2k+1)) with a second library (e.g., Sage or PARI); confirm all eleven pairs equal 1 and that the initial segment [1,31] still yields exactly ten.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a finite, fully explicit arithmetic counterexample: the eleven listed integers in (3928,3959] are each Sophie Germain cyclic while [1,31] contains only ten, so C_σ(3959)=C_σ(3928)+11 > C_σ(31)+C_σ(3928). The paper supplies the lists, an independent double-check of φ, a consistency check against Cohen’s tabulated C_σ(598)=120, and a Lean 4 kernel proof that depends only on propext/Classical.choice/Quot.sound (no native_decide). The reader correctly isolates the eleven memberships as the sole arithmetic content; those memberships are machine-checked, so residual doubt is negligible. No hidden analytic assumption, asymptotic extrapolation, or circular appeal is present.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper disproves Cohen's Conjecture 66 asserting subadditivity of the counting function C_σ of Sophie Germain cyclic numbers (n such that both n and 2n+1 satisfy gcd(k,φ(k))=1). It exhibits the explicit pair m=31, n=3928 for which C_σ(3959)=697>696=C_σ(31)+C_σ(3928). The short proof lists the ten Sophie Germain cyclic integers in [1,31] and the eleven in the window (3928,3959], verifies the defining conditions (with an illustrative calculation for 3929), and concludes that the denser window of length 31 violates the claimed inequality. The identical statement is formalized in Lean 4 over mathlib; the kernel accepts the proof of the negation of the conjecture using only the three standard axioms propext, Classical.choice and Quot.sound.","tokens_in":4263,"tokens_out":757,"duration_ms":47160,"significance":"The note settles a conjecture that Cohen left open after an unsuccessful search to 10^6 (later attributed by him to a coding error). The counterexample is elementary, fully explicit and lies well inside the searched range, so the result is definitive rather than asymptotic. Explicit strengths include the hand-checkable lists, independent double computation of the relevant C_σ values (cross-checked against Cohen's tabulated C_σ(598)=120), and especially the machine-checked Lean formalization that never relies on native_decide. While the mathematical novelty is that of a single finite counterexample to an analog of a believed-false prime conjecture, the combination of transparency and kernel-level verification makes the note a clean, high-rigor contribution to the literature on cyclic numbers.","major_comments":[],"minor_comments":[{"comment":"The absolute values C_σ(3959)=697 and C_σ(3928)=686 are stated in the abstract and Theorem 1, yet the proof (and the Lean argument) only ever uses the difference of 11 versus 10. A single sentence noting the independently computed value of C_σ(3928) would make the numerical claims fully transparent without lengthening the argument.","section":"Theorem 1 and §2"},{"comment":"Beyond the worked example for 3929, the remaining ten window members are simply declared Sophie Germain cyclic. A compact one-line table (or footnote) recording φ(n), φ(2n+1) and the two gcds for each would render the note completely self-contained for a reader who does not consult the Lean source.","section":"Proof of Theorem 1"},{"comment":"The statement that #print axioms returns only the three standard axioms is valuable; a brief pointer to the precise location of the supplementary Lean file (or an inline excerpt of the key Nat.count lemmas) would make the formal claim easier to inspect.","section":"§3"}],"recommendation":"accept","confidential_remarks":"Correct, self-contained short note; the Lean kernel verification is a genuine plus. Suitable for immediate acceptance in a venue that publishes brief contributions on integer sequences or elementary counterexamples (e.g., J. Integer Sequences). No novelty, priority or scope concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a short, correct note that settles one open conjecture of Cohen by exhibiting a concrete pair: m=31, n=3928. The length-31 window after 3928 contains eleven Sophie Germain cyclic numbers while [1,31] contains only ten, so C_σ(3959) exceeds C_σ(31)+C_σ(3928). That is the whole argument.\n\nWhat is new is the explicit witness and the density comparison. Cohen had searched to 10^6 and found nothing (later attributed to a coding error); the paper supplies the lists, double-checks φ, matches Cohen's tabulated C_σ(598)=120, and ships a Lean 4 formalization over mathlib that the kernel accepts with only the three standard axioms. No native_decide, no black-box computation. For a finite arithmetic claim that is as solid as it gets.\n\nSoft spots are minor and do not touch the result. Significance is local: it closes one analog of the Hardy–Littlewood subadditivity conjecture inside the cyclic-number program. The paper does not claim more. The eleven memberships are the only arithmetic content that needs checking; they are listed and machine-verified, so residual doubt is negligible. Citation pattern is tight and appropriate.\n\nAnyone following Cohen's cyclic-number analogs, or interested in formal verification of elementary number-theoretic counterexamples, will get value from it. It is short enough for a reading group if people want a clean example of a density-window disproof plus Lean. I would send it to peer review without hesitation; a referee can confirm the lists and the Lean file in an afternoon. Worth citing if you work in this niche; otherwise it is a solid local correction.","headline":"Clean, machine-checked counterexample that kills Cohen's Conjecture 66 with an explicit short window denser than [1,31].","tokens_in":4820,"tokens_out":446,"would_cite":false,"duration_ms":4257,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A25","11N25","11Y55"],"pacs":[],"model":"grok-4.5","headline":"A short length-31 window holds more Sophie Germain cyclic numbers than the initial segment [1,31], breaking Cohen's subadditivity conjecture.","keywords":["Sophie Germain cyclic numbers","cyclic numbers","subadditivity","counterexample","Euler totient","counting function","Lean formalization"],"falsifier":"Recompute gcd(k,φ(k)) and gcd(2k+1,φ(2k+1)) for each of the eleven explicit values k in {3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959}; any single failure falsifies the claimed inequality.","tokens_in":5008,"feed_emoji":"🔢","tokens_out":678,"duration_ms":5242,"temperature":0.7,"pith_summary":"Cohen conjectured that the counting function for Sophie Germain cyclic numbers is subadditive: the count up to m+n never exceeds the sum of the separate counts up to m and up to n. The paper shows this fails already for the small pair m=31, n=3928. The reason is elementary density comparison: the open interval (3928,3959] contains eleven Sophie Germain cyclic integers while [1,31] contains only ten, so the cumulative count jumps by eleven rather than by at most ten. Because the counterexample sits well inside the range Cohen reported searching, the note also records that the earlier exhaustive search contained a coding error. The arithmetic verification is short enough to be machine-checked by the Lean 4 kernel, making the refutation both human-readable and formally certified.","feed_headline":"Length-31 window breaks Sophie Germain cyclic subadditivity","feed_subtitle":"Eleven such numbers sit between 3928 and 3959, one more than the ten up to 31","key_machinery":"Sophie Germain cyclic numbers: positive integers n for which both n and 2n+1 satisfy gcd(k,φ(k))=1. Their counting function C_σ supplies the inequality that is shown to fail by a direct window comparison of length 31.","core_discovery":"At m=31 and n=3928 one has C_σ(3959)=697 > 696 = C_σ(31)+C_σ(3928). Equivalently, the eleven integers 3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959 are all Sophie Germain cyclic, one more than the ten such integers that lie in [1,31]. This single density excess falsifies Conjecture 66.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["m=31 n=3928 disproves Cohen subadditivity for Sophie Germain cyclics","Eleven Sophie Germain cyclics in [3929,3959] beat ten up to 31","C_σ(3959)=697 exceeds C_σ(31)+C_σ(3928), falsifying Conjecture 66","Short Lean-checked counterexample to Sophie Germain cyclic subadditivity","Length-31 window yields one extra Sophie Germain cyclic past 3928"],"cache_read_input_tokens":128,"weakest_assumption_plain":"That each of the eleven listed integers in the window (3928,3959] really is Sophie Germain cyclic; if any one fails the gcd condition the excess count disappears and the counterexample collapses.","fun_headline_variants_meta":{"raw":{"variants":["m=31 n=3928 disproves Cohen subadditivity for Sophie Germain cyclics","Eleven Sophie Germain cyclics in [3929,3959] beat ten up to 31","C_σ(3959)=697 exceeds C_σ(31)+C_σ(3928), falsifying Conjecture 66","Short Lean-checked counterexample to Sophie Germain cyclic subadditivity","Length-31 window yields one extra Sophie Germain cyclic past 3928"]},"model":"grok-4.5","effort":"low","cost_usd":0.004322,"raw_usage":{"total_tokens":1301,"prompt_tokens":775,"num_sources_used":0,"completion_tokens":126,"cost_in_usd_ticks":43220000,"prompt_tokens_details":{"text_tokens":775,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":400,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":775,"tokens_out":126,"duration_ms":4847,"temperature":1.0,"reasoning_tokens":400,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T15:38:54.927620+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Recompute gcd(k,φ(k)) and gcd(2k+1,φ(2k+1)) for each of the eleven explicit values k in {3929,3931,3935,3941,3943,3945,3947,3949,3953,3957,3959}; any single failure falsifies the claimed inequality.","supporting_citations":[],"review_version":1}