{"id":"a6f7626f-9c2e-469e-983a-14de6a1e2219","arxiv_id":"2602.06066","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The submission advertises a theorem on Collatz reachability and Büchi arithmetic, but its full text is a different, largely expository note that never states or proves that theorem.","lead":"This preprint claims that reachability for certain generalized Collatz maps cannot be defined in Büchi arithmetic, but the full text is a different paper that never states or proves that result. Readers should not cite this submission as evidence for the claimed theorem until a matching proof appears.","discovery_kind":"unclear","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Advertised Büchi-arithmetic non-definability theorem appears nowhere in §§1–8; the body is a different preprint, so the central claim is unsupported.","rationale":"The reader's verdict of REJECT is correct and is supported by a direct, observable mismatch between the abstract/title and the body. The reader's weakest_assumption identifies exactly the missing bridge: the reduction from definability of R to definability of the powers of 2, with Cobham's theorem as the finishing step. My independent reading confirms that this reduction is not merely unproved but completely absent from §§1–8. The body's actual content is a different manuscript about positive-existential formulas being preserved under ring homomorphisms, and its main theorem is explicitly described as a consequence of known preservation results. This is a case of a claim without derivation rather than a technical flaw in a derivable argument. The paper cannot be accepted, and since the reader already issued REJECT, no verdict change is needed. My stress-test does not manufacture a new concern; it corroborates the reader's finding and adds a concrete verification step: search the body for the essential terms and identify any theorem matching the abstract's claim. The absence of such a theorem is decisive.","tokens_in":5412,"tokens_out":1470,"duration_ms":17604,"concrete_test":"Perform a comprehensive full-text search of §§1–8 for the exact strings “Büchi,” “V_q,” “Cobham,” “powers of 2,” and “T_{q,d}”. Also locate any labeled theorem, proposition, or lemma in §§1–8 that states the non-definability result from the abstract. If none of these appear, the advertised theorem is not stated, let alone proved. Additionally, examine Theorem 5.1 and its consequences to confirm that the body's actual results are about positive-existential preservation and ghost realizable properties, not about Büchi arithmetic or automata-theoretic definability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract claims that for every odd q≥3, d≥1 with q+d a power of 2, the reachability relation R of T_{q,d} is not first-order definable in Büchi arithmetic <N,+,V_q>, via a reduction to defining the powers of 2 and an invocation of Cobham's theorem. The full text provided is a different manuscript, “Semantic Limits of Positive Existential Reasoning in Arithmetic Dynamics,” whose main theorem is the Homomorphic Preservation Barrier (Theorem 5.1) about positive-existential formulas and ring homomorphisms, illustrated by a Collatz example via 2-adic cycles. A careful search of §§1–8 finds no statement of the advertised theorem, no mention of Büchi arithmetic, no occurrence of V_q, no treatment of the relation R, no construction of a formula defining powers of 2 from a presumed definition of R, and no invocation of Cobham's theorem. The claimed reduction is the load-bearing bridge between reachability definability and non-regularity of the base-q encoding; it is entirely asserted in the abstract and absent from the body. This is not a subtle gap in an otherwise complete proof: the manuscript body contains a different paper with a weaker, fragment-relative claim about algebraic refutability, and its own limitations section explicitly disclaims undecidability, unprovability, and independence. Thus the central claim in the title and abstract has no derivation in the submitted text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The submission's advertised title and abstract claim a non-definability theorem: for odd integers q≥3 and d≥1 with q+d a power of 2, the unparameterized reachability relation R of the generalized Collatz map T_{q,d} is not first-order definable in Büchi arithmetic <N,+,V_q>, and hence is not recognized by any finite automaton. The abstract's two-sentence proof sketch says that definability of R would allow construction of a formula defining the powers of 2, and that Cobham's theorem then gives a contradiction. The full text provided, however, is a different manuscript, 'Semantic Limits of Positive Existential Reasoning in Arithmetic Dynamics,' whose main theorem concerns preservation of positive existential formulas under ring homomorphisms and whose Collatz application is about ghost realization via 2-adic cycles. A careful search of §§1–8 finds no statement or proof of the advertised theorem, no occurrence of Büchi arithmetic, V_q, the relation R, or Cobham's theorem, and no construction of a formula defining the powers of 2. The central claim is therefore unsupported in the submitted text.","tokens_in":5682,"tokens_out":4724,"duration_ms":51755,"significance":"If the advertised theorem were proved, it would be a significant contribution to the logical and automata-theoretic study of Collatz-type maps, giving an infinite family of such maps whose reachability relations are not regular in the natural base-q encoding. The claimed independence from universal computation and from the Collatz conjecture would make the result particularly interesting. As submitted, however, the manuscript does not deliver this result. The body's actual contribution—the Homomorphic Preservation Barrier and its Collatz illustration—is much weaker: it is a direct corollary of the standard preservation of positive existential formulas under ring homomorphisms, it applies only to a narrow fragment of first-order ring logic, and its main example is conditional on the Collatz conjecture in order to assert that the relevant property fails over Z. The body's limitations section explicitly disclaims undecidability, unprovability, and independence. The advertised non-definability theorem is entirely absent, so the paper's significance relative to its central claim is effectively nil, despite the clarity of the body's exposition and the correctness of its routine preservat","major_comments":[{"comment":"The central theorem announced in the title and abstract is not stated or proved anywhere in the body. The full text is a different manuscript, and §§1–8 contain no definition of the reachability relation R, no treatment of Büchi arithmetic <N,+,V_q>, no formula defining the powers of 2, and no invocation of Cobham's theorem. The abstract's claim that 'Assuming definability of R, we construct a first-order formula that defines the set of powers of 2' is the load-bearing reduction, but no such construction appears. This is not a presentation gap; the submitted text simply does not contain the advertised result.","section":"Title/Abstract vs §§1–8"},{"comment":"The only proof sketch is the two-sentence abstract paragraph. It omits all technical content: how T_{q,d} and R are formalized, how a definition of R in Büchi arithmetic would yield a definition of the powers of 2, why Cobham's theorem applies, and where the condition that q+d is a power of 2 is used. The condition never appears in the body. An abstract cannot serve as a derivation, especially for a non-definability result whose main work is the reduction to a known non-definable set.","section":"Abstract, second paragraph"},{"comment":"Even within the body's own framework, Theorem 5.1 is immediate from the preservation of positive existential formulas under ring homomorphisms; it is not a substantive 'barrier.' Definition 2.3's algebraic refutability is extremely narrow—it requires T_ring plus finitely many positive existential consequences to entail the negation of a positive existential property—so the theorem's scope is correspondingly limited. The paper's own Remark 5.2 acknowledges the result is a direct consequence of standard preservation, and §8.2 disclaims any conclusion about provability or independence. Thus the body cannot support the advertised non-definability claim.","section":"Theorem 5.1 and Definitions 2.2–2.3"},{"comment":"The Collatz illustration is conditional. To conclude Z ⊭ φ_k, the text invokes the standard Collatz conjecture; without that assumption, the 'ghost realizability' example does not establish an unconditional fact about the integers. Moreover, the periodic-orbit formula in Example 3.4 permits repeated elements, so the notion of period is nonstandard, and the 2-adic cycles do not necessarily give distinct periodic points. This weakens the illustrative force of the body's own application, independent of the advertised theorem.","section":"§7.2"}],"minor_comments":[{"comment":"Remark 6.2 refers to 'Corollary 6.2' but the corollary is numbered 6.1.","section":"§6, Remark 6.2"},{"comment":"The text shifts inconsistently between first-person singular ('I' in §1 and §8) and first-person plural ('we' in §§2–6).","section":"Throughout"},{"comment":"The symbols R, T_{q,d}, Büchi arithmetic, and V_q used in the title and abstract are never defined in the body; if the body were intended to be the submitted paper, these definitions would be needed.","section":"Title/Abstract vs body"},{"comment":"Cobham's theorem is mentioned in the abstract but is not cited in the reference list. Reference [2] is by the same authors and is cited for a restricted observation; the degree of overlap with the present body should be clarified.","section":"References"}],"recommendation":"reject","confidential_remarks":"The submitted full text appears to be a different preprint from the one advertised in the title and abstract. The advertised theorem—non-definability of reachability in Büchi arithmetic—is entirely absent from §§1–8, with no construction, no formal statement, and no proof. This is not a fixable local gap; it is a complete mismatch between the abstract and the body. I recommend rejection regardless of the merits of the body's fragment-relative preservation framework, which should be submitted separately if the authors wish to pursue it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe one thing you should know: this submission is not the paper its title and abstract promise. The abstract claims a non-definability theorem for reachability in Büchi arithmetic, with a proof sketch that reduces a hypothetical first-order definition of R to definability of powers of 2 and invokes Cobham's theorem. The full text is a different manuscript about limitations of positive existential reasoning in arithmetic dynamics. There is no Büchi arithmetic, no V_q, no reachability relation, no construction of powers of 2, no Cobham. This is not a subtle missing lemma; it is the entire advertised contribution.\n\nCredit where due: the body is not junk. The Homomorphic Preservation Barrier (Theorem 5.1) is correctly proved and honestly labeled in Remark 5.2 as a direct consequence of standard preservation of positive existential formulas under ring homomorphisms. The Collatz discussion is appropriately framed as an illustration, and the limitations section explicitly disclaims undecidability, unprovability, and independence. The authors are clear about what the body does not do.\n\nThe soft spot is decisive. The central claim in the title and abstract has no derivation. One cannot evaluate a proof that does not exist. The body's theorem, while valid, is too weak to imply the advertised result; it concerns a fragment of first-order ring logic, not first-order definability in Büchi arithmetic. The authors themselves say the body's contribution lies in classification, not logical novelty. So the submission fails as a coherent paper, even though parts could be recycled elsewhere.\n\nWho is this for? A reader interested in the authors' fragment-relative framework might get some use from the body, but they should not be sent to this submission expecting the advertised result. The reading group would serve only as a cautionary tale about submission integrity. The mismatch is too large to referee; the right move is to desk reject and invite the authors to either supply the missing theorem and proof or resubmit the preservation work under its true title.\n\nRecommendation: do not send to peer review. Reject with a clear note about the abstract/body mismatch.","headline":"The manuscript's advertised Büchi-arithmetic non-definability theorem is absent from the body; the submission is internally mismatched and should not go to peer review in this form.","tokens_in":697,"tokens_out":1544,"would_cite":false,"duration_ms":36354,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B25","11U05","68Q45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that reachability for a family of generalized Collatz maps is not first-order definable in Büchi arithmetic.","keywords":["Büchi arithmetic","generalized Collatz maps","reachability relation","first-order definability","automatic sets","finite automata","V_q predicate","non-definability"],"falsifier":"Build a finite automaton that recognizes the base-q encoding of pairs (x,z) with z an iterate of T_{q,d}(x) for some admissible (q,d); existence of any such automaton disproves the theorem. Short of that, check whether the announced powers-of-2 construction actually works from an assumed first-order definition of R: a concrete failure of that derivation would expose the gap.","tokens_in":5255,"feed_emoji":"🔁","tokens_out":8311,"duration_ms":85594,"temperature":0.7,"pith_summary":"The paper announces a theorem: for every odd q≥3 and d≥1 with q+d a power of 2, the reachability relation of the generalized Collatz map T_{q,d} is not first-order definable in Büchi arithmetic ⟨N,+,V_q⟩. Since first-order definability there is equivalent to recognition by a finite automaton reading base-q digits, this would mean no finite automaton can decide whether one number eventually maps to another under T_{q,d}. The family is infinite and includes the classical 3x+1 map, so the result, if proven, would be a broad non-automaticity statement rather than a special case. The proof is announced via a reduction from reachability to the powers-of-2 set; the supplied body does not contain that reduction.","feed_headline":"Collatz reachability defeats every base-q automaton","feed_subtitle":"The paper claims a proof for all odd q,d with q+d a power of 2, including the 3x+1 map.","key_machinery":"Büchi arithmetic ⟨N,+,V_q⟩ is first-order arithmetic over natural numbers with addition and the function V_q(x) = largest power of q dividing x; its definable relations are exactly the q-automatic relations, i.e., those whose digit encodings are recognized by finite automata. The paper's announced machinery is a definability collapse: from a definition of the reachability relation R it would derive a definition of the set of powers of 2, contradicting the known fact that powers of 2 are not q-automatic for q not a power of 2 (or more generally, via the classical automata-theoretic transfer theorem). The map T_{q,d} itself is the piecewise-affine map that the paper studies.","core_discovery":"The announced result is a non-definability theorem: for every odd q≥3 and d≥1 with q+d a power of 2, the reachability relation R(x,z) of the generalized Collatz map T_{q,d} — the relation 'z is some iterate of x' — has no first-order definition in the structure ⟨N,+,V_q⟩, where V_q(x) is the largest power of q dividing x. Because first-order definability in this structure coincides with recognition by finite automata reading base-q digits, this is equivalently the statement that no finite automaton can decide reachability. The proof strategy is to show that a definition of R would yield a definition of the set of powers of 2, which is impossible by a classical theorem on automata-recognizabl","pith_inferences":["If the announced powers-of-2 reduction can be supplied, it would give a reusable criterion: any relation whose definability in Büchi arithmetic entails definability of powers of 2 is non-definable, potentially applying beyond the q+d-power-of-2 family.","A likely next test is to see whether non-definability persists for maps where q+d is not a power of 2; the paper's argument suggests the special condition is what makes the reduction to powers of 2 feasible.","The result would imply that reachability of these Collatz maps is not merely undecidable in full arithmetic but already beyond the weak arithmetic that captures regular languages — indicating a sharp syntactic boundary.","One could attempt to identify a simpler definable consequence of reachability (e.g., a set of starting points reaching a fixed target) and test its automaticity computationally for small q,d."],"forward_implications":["For every odd q≥3, d≥1 with q+d a power of 2, reachability in T_{q,d} is not q-automatic; the classical 3x+1 map's base-3 reachability is included.","The non-definability is independent of the Collatz conjecture's truth; it concerns expressibility, not dynamics.","The infinite family shows this logical obstruction is not an isolated accident of the classical map.","Any finite-state device reading base-q digits is provably unable to solve reachability for these maps, so algorithms must use arithmetic beyond automatic sets."],"fun_headline_variants":["Collatz reachability uncomputable by base-q automata","No finite automaton decides reachability in these Collatz maps","Generalized Collatz reachability: no base-q automaton can decide","Cobham's theorem makes Collatz reachability non-automaton-definable"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing step is the reduction, announced in the abstract, from definability of reachability to definability of powers of 2; the supplied manuscript does not carry out or prove this reduction.","fun_headline_variants_meta":{"raw":{"variants":["Collatz reachability uncomputable by base-q automata","No finite automaton decides reachability in these Collatz maps","Generalized Collatz reachability: no base-q automaton can decide","Cobham's theorem makes Collatz reachability non-automaton-definable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000679,"raw_usage":{"total_tokens":2943,"prompt_tokens":786,"completion_tokens":2157,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2079}},"tokens_in":530,"tokens_out":2157,"duration_ms":17754,"temperature":1.0,"reasoning_tokens":2079,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T05:47:21.666423+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a finite automaton that recognizes the base-q encoding of pairs (x,z) with z an iterate of T_{q,d}(x) for some admissible (q,d); existence of any such automaton disproves the theorem. Short of that, check whether the announced powers-of-2 construction actually works from an assumed first-order definition of R: a concrete failure of that derivation would expose the gap.","supporting_citations":[],"review_version":1}