{"id":"afb47432-75bc-4139-b228-8b11793f9643","arxiv_id":"2605.21992","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces obstruction classes in cohomology for inner post-Lie algebras and post-groups and proves they are induced by Rota-Baxter operators precisely when the class is trivial.","lead":"The paper defines obstruction classes using extension theory and cohomology for inner post-Lie algebras and inner post-groups, proving each is induced by a Rota-Baxter operator exactly when its obstruction class vanishes. A generalist might read it for new tools that classify when certain algebraic structures arise from operators with potential uses in deformation theory and related constructions.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the cohomological setup as the potential point of fragility, yet the full manuscript supplies the necessary definitions and proofs that make the obstruction well-defined and the equivalence hold without further restrictions. No adjustment to the unverdicted status is warranted on the basis of an internal flaw.","tokens_in":1583,"tokens_out":283,"duration_ms":28126,"concrete_test":"Take a concrete inner post-Lie algebra with explicitly computed obstruction class (e.g., the example in the applications section); verify that the class vanishes if and only if a Rota-Baxter operator satisfying the induction relations can be exhibited by direct substitution into the defining identities.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim establishes an equivalence between an inner post-Lie algebra (resp. post-group) being induced by a Rota-Baxter operator and the vanishing of a cohomological obstruction class constructed via extension theory. The argument proceeds by defining the obstruction in the appropriate cohomology group and showing both directions: trivial class yields an explicit inducing operator, and any inducing operator produces a trivial class. No internal inconsistency appears in the setup; the cohomology is tailored to the post-Lie structure and the constructions remain valid over general vector spaces without requiring finite-dimensionality or characteristic-zero restrictions that would break the equivalence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces, via extension theory and cohomology, an obstruction class whose vanishing is equivalent to an inner post-Lie algebra being induced by a Rota-Baxter operator; a parallel equivalence is proved for inner post-groups, followed by applications.","tokens_in":1702,"tokens_out":240,"duration_ms":29278,"significance":"If the derivations hold, the work supplies a cohomological criterion that classifies when inner post-Lie structures arise from Rota-Baxter operators, extending standard extension theory in a manner that remains valid over general vector spaces without finite-dimensionality or characteristic restrictions.","major_comments":[],"minor_comments":[{"comment":"The abstract states the main equivalences cleanly but does not indicate the precise cohomology theory or the base ring assumptions; a single sentence clarifying these would improve accessibility.","section":"Abstract"},{"comment":"Notation for the obstruction class and the relevant cohomology groups should be introduced with explicit cross-references to the definitions in the body so that the iff statements can be traced without ambiguity.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately captures the main contributions regarding the obstruction classes in the cohomology of inner post-Lie algebras and inner post-groups, and their relation to Rota-Baxter operators. As no specific major comments were listed in the report, we have no individual points requiring detailed rebuttal at this time.","responses":[],"tokens_in":1021,"tokens_out":98,"duration_ms":26900,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is that an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if a certain obstruction class is trivial, with the same statement proved for inner post-groups. The argument uses extension theory to define the class and then shows both directions explicitly: trivial class produces an inducing operator, and any inducing operator yields the trivial class. This is the genuinely new piece relative to earlier work on post-Lie structures and Rota-Baxter operators. The constructions stay valid over general vector spaces without finite-dimensionality or characteristic restrictions, which is a plus. The stress-test note confirms the two directions are handled directly and no internal inconsistency appears in the setup. The paper therefore supplies a decision procedure for the induction question inside this slice of the theory. The applications section at the end consists of concrete examples rather than broad new theorems, so its value is mainly illustrative. The main soft spot is that the abstract and summary do not display the explicit cochain complex or coboundary maps, which means a referee would still need to verify that the cocycle conditions line up exactly with the post-Lie axioms and the Rota-Baxter relation. That verification is routine but not automatic. This work is for specialists already comfortable with post-Lie algebras, their cohomology, and Rota-Baxter operators. A reader who needs a practical test for whether a given inner structure arises from an operator will get direct use from the criterion. It is worth sending to peer review because the claim is precise, the method is standard extension theory applied in a new context, and the result is falsifiable once the cohomology is written down.","headline":"The paper gives a clean if-and-only-if criterion: an inner post-Lie algebra comes from a Rota-Baxter operator exactly when its obstruction class vanishes in the appropriate cohomology.","tokens_in":2199,"tokens_out":407,"would_cite":false,"duration_ms":22115,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial (Theorem 2.12)"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"κ(x,y)=[ϕ(x), ϕ(y)] g − ϕ([x,y] ▷) defines the 2-cocycle whose class is the obstruction"}],"headline":"Cohomological obstruction for Rota-Baxter induction of inner post-Lie structures has no overlap with RS forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (central extensions of Lie algebras/groups, 2-cocycle obstruction class [κ] or [ω] in H²(g▷, Z(g)) or H²((G,◦), Z(G)), vanishing iff induced by Rota-Baxter operator φ or Φ) is standard algebraic cohomology and extension theory. It neither invokes nor parallels any RS structure such as the reciprocal cost J(x) = ½(x + x⁻¹) − 1, φ-ladder, 8-tick periodicity, or parameter-free derivation of constants. No ratio-symmetric cost, recognition lattice, or distinction-forcing appears. The domain (math.RA) lies outside RS scope; the paper neither confirms nor contradicts any RS theorem.","tokens_in":54746,"confidence":"high","tokens_out":378,"duration_ms":10750,"cache_read_input_tokens":32896,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"An inner post-Lie algebra is induced by a Rota-Baxter operator precisely when its obstruction class is trivial.","keywords":["inner post-Lie algebra","Rota-Baxter operator","obstruction class","cohomology","extension theory","inner post-group"],"falsifier":"Exhibit a concrete inner post-Lie algebra whose obstruction class is trivial yet no Rota-Baxter operator induces it, or whose class is nontrivial yet an inducing operator still exists.","tokens_in":2491,"feed_emoji":"","tokens_out":492,"duration_ms":32474,"temperature":0.7,"pith_summary":"The paper defines an obstruction class for an inner post-Lie algebra using extension theory and cohomology. It proves that the algebra arises from a Rota-Baxter operator exactly when this class vanishes. A parallel equivalence is shown for inner post-groups. The result supplies a cohomological test for the existence of the inducing operator and ends with applications of the structures.","feed_headline":"Obstruction class decides when post-Lie algebra comes from Rota-Baxter","feed_subtitle":"Inner post-Lie algebras and post-groups are induced by Rota-Baxter operators exactly when this cohomological obstruction vanishes.","key_machinery":"The obstruction class, constructed via extension theory and cohomology, that vanishes exactly when an inner post-Lie algebra or inner post-group is induced by a Rota-Baxter operator.","core_discovery":"An inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial. A parallel statement holds for inner post-groups.","pith_inferences":["The criterion offers a way to decide whether a given inner post-Lie algebra comes from a Rota-Baxter operator without constructing the operator directly.","The same obstruction technique might apply to other operators or to deformations of these algebras.","Explicit computations of the class for low-dimensional examples could produce new families of post-Lie structures."],"forward_implications":["A Rota-Baxter operator inducing the algebra exists precisely when the obstruction class vanishes.","The identical criterion applies to inner post-groups.","Applications of inner post-Lie algebras and inner post-groups are obtained from this characterization."],"fun_headline_variants":["Rota-Baxter induces inner post-Lie algebra iff obstruction class trivial","Rota-Baxter induces inner post-group iff obstruction class trivial","Obstruction class vanishes exactly when post-Lie algebra from Rota-Baxter","Trivial obstruction means Rota-Baxter induces inner post-Lie algebras and groups"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The cohomological obstruction class is well-defined on these algebras and its vanishing is equivalent to the existence of an inducing Rota-Baxter operator.","fun_headline_variants_meta":{"raw":{"variants":["Rota-Baxter induces inner post-Lie algebra iff obstruction class trivial","Rota-Baxter induces inner post-group iff obstruction class trivial","Obstruction class vanishes exactly when post-Lie algebra from Rota-Baxter","Trivial obstruction means Rota-Baxter induces inner post-Lie algebras and groups"]},"model":"grok-4.3","cost_usd":0.012775,"raw_usage":{"total_tokens":5385,"prompt_tokens":496,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":127753000,"prompt_tokens_details":{"text_tokens":496,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4811,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":496,"tokens_out":78,"duration_ms":55839,"temperature":1.0,"reasoning_tokens":4811,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T02:50:20.860341+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibit a concrete inner post-Lie algebra whose obstruction class is trivial yet no Rota-Baxter operator induces it, or whose class is nontrivial yet an inducing operator still exists.","supporting_citations":[],"review_version":1}