{"id":"b30fea6d-ff0b-4d7b-b01a-56f4a681f61b","arxiv_id":"2607.05558","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Interpretive agreement between two signal models is ordered by cone inclusion of jointly acceptable surplus vectors, which holds exactly when each model is a convex combination of the other pair, with cosine similarity the unique rotation-invariant completion.","lead":"The paper defines a decision-theoretic order on pairs of subjective signal models: one pair is more agreeable when it admits a larger set of jointly acceptable signal-contingent plans relative to a shared reservation payoff. The order is prior-free, equals a convex-hull relation on the models, and has cosine similarity as its unique rotation-invariant scalar completion.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's identification of the reservation-spanning assumption as the weakest modeling choice is accurate, but it is not a load-bearing threat to the theorems as stated. The paper works throughout under the fixed-scalar reservation (Definition 1 and the geometric setup of §2.2), for which the spanning property holds and the necessity direction of Proposition 1 goes through. The authors themselves flag the prior-optimal alternative in footnote 6 and develop the restricted-cone comparison in Online Appendix B.2, so the caveat is already transparent. The uniqueness argument for cosine (Proposition 5) relies only on rotation invariance of the Euclidean structure on surplus space and the elementary fact that cone aperture equals the angle between normals; that geometry is independent of how the reservation is generated. Applications (speculative-trade nesting, normalized Pareto expansion, rationalizing-model enlargement) inherit the same clean dual-cone logic. No algebraic gap, circularity, or unstated domain restriction undermines the strongest claim. The contribution is therefore solid as a theoretical advance, and the reader's ACCEPT / high-confidence assessment needs no adjustment.","tokens_in":32513,"tokens_out":531,"duration_ms":5035,"concrete_test":"Independently re-derive the dual-cone step of Proposition 1: verify that C(p̂m,p̂m') ⊆ C(pm,pm') if and only if cone(pm,pm') ⊆ cone(p̂m,p̂m'), then confirm that probability normalization forces the coefficients to sum to one, yielding the convex-combination statement. If that equivalence fails for any full-support p, the characterization collapses; otherwise the claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Proposition 1's convex-hull characterization of the inclusion preorder, and Proposition 5's uniqueness of the rotation-invariant cosine completion) rest on transparent dual-cone geometry and standard separating-hyperplane arguments. The reservation-spanning assumption that the reader flags is already treated carefully by the authors: under the fixed-scalar reservation, finite decision problems do span R^{Ω×S} (so full-space cone inclusion is necessary and sufficient), while Online Appendix B.2 correctly records that the prior-optimal-action variant makes full-space inclusion only sufficient. That is a modeling choice, not a hidden gap in the proofs. The dual-cone reduction, the prior-cancellation argument, and the aperture-based uniqueness argument for cosine all check out; no circularity or missing step appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies joint decision-making by two agents who share prior, state space, actions, payoffs, reservation payoff, and Bayesian updating, but may disagree about signal likelihoods. It defines an inclusion preorder over pairs of subjective models: (m, m') is more agreeable than (m̂, m̂') if the cone of jointly acceptable surplus vectors under the former contains that under the latter, uniformly across decision problems. Proposition 1 shows this is equivalent to each model in the more agreeable pair being a convex combination of the two models in the less agreeable pair, and that the comparison is prior-independent. Proposition 5 shows that the unique rotation-invariant strict completion of the (incomplete) inclusion preorder is ranking by cosine similarity of the induced joint distributions. Applications establish that greater agreement shrinks speculative-trade transfer intervals (Proposition 2), expands a suitably normalized ex-ante Pareto frontier (Proposition 3), and enlarges the set of single-model rationalizations (Proposition 4). The order is independent of Blackwell dominance and selects quadratic over KL-type Bregman divergences among rotation-invariant comparisons.","tokens_in":32713,"tokens_out":689,"duration_ms":5224,"significance":"If the characterizations hold, the paper supplies a clean, decision-theoretic, prior-free partial order on interpretive disagreement that is grounded in joint participation constraints rather than ad-hoc statistical distance. The convex-hull representation (Proposition 1) and the uniqueness of the cosine completion under rotation invariance (Proposition 5) are transparent and rest on standard dual-cone and separating-hyperplane arguments that are fully written out in Appendix A. The applications give the order immediate economic content for speculative trade, Pareto frontiers, and external rationalizability. The careful treatment of the reservation-payoff spanning assumption (footnote 6 and Online Appendix B.2) and the explicit comparison with Blackwell and Bregman divergences further strengthen the contribution. The results are machine-checkable from the written proofs and require no free parameters.","major_comments":[],"minor_comments":[{"comment":"In the abstract and introduction the phrase \"selects quadratic over KL-type Bregman divergences\" is slightly loose; Proposition 9 shows that the only rotation-invariant Bregman divergence on the sphere is a multiple of squared Euclidean distance (hence cosine). A one-sentence clarification would prevent misreading.","section":null},{"comment":"Figure 1 and Figure 2 are helpful, but the captions could more explicitly state the coordinates (θ11, θ22) and the meaning of the dashed segments so that a reader skimming the figures alone can recover the geometry.","section":null},{"comment":"A few minor typos appear (e.g., \"suprlus\" near the end of §4.2, \"won\" for \"down\" in the same paragraph). A careful proof-reading pass would catch them.","section":null},{"comment":"The online appendix material on introspection-proofness (B.1) and the prior-optimal reservation variant (B.2) is useful; a brief pointer in the main-text conclusion would help readers locate it.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is unusually clean for a theory submission: the central geometry is standard, the proofs are complete, and the authors themselves flag the only modeling caveat (reservation spanning). I see no reason to delay acceptance for further rounds."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real contribution is a prior-free preorder on pairs of signal structures: one pair is more agreeable than another exactly when each model is a convex combination of the other pair (Prop 1). The unique rotation-invariant completion of that incomplete order is cosine similarity of the induced joints (Prop 5). Both results are new relative to the common-prior, misspecification, and Blackwell literatures they cite, and the dual-cone / separating-hyperplane arguments check out cleanly in the appendix.\n\nWhat works well is the geometry. Beliefs become normals in Arrow-Debreu surplus space; joint acceptability is the intersection of half-spaces; larger cones mean more agreement. The applications then follow without extra machinery: the same order shrinks speculative-trade intervals, expands a normalized Pareto frontier, and enlarges the set of single-model rationalizations. The independence from Blackwell and the selection of quadratic over KL-type Bregman divergences are cleanly shown and useful for positioning.\n\nThe main modeling choice is the fixed scalar reservation payoff that lets finite decision problems span the whole surplus space. The authors flag this themselves and correctly record in Online Appendix B.2 that under a prior-optimal reservation action the full-space inclusion becomes only sufficient. That is a deliberate modeling decision, not a hole in the proofs. The prior-uniform cosine order is incomplete (as they prove), so the scalar completion is fixed-prior; that is a limitation of the object, not a flaw.\n\nThis is for people who work on information, committees, trade under heterogeneous models, or AI-assisted diagnosis. The math is standard and fully written; citations are appropriate. I would send it to referees without hesitation and would cite the convex-hull characterization and the cosine uniqueness result myself.","headline":"Clean geometric order on interpretive disagreement with a sharp convex-hull characterization and unique cosine completion; solid theory paper.","tokens_in":33286,"tokens_out":444,"would_cite":true,"duration_ms":5084,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Interpretive agreement is convex combination of signal models, completed by cosine similarity.","keywords":["interpretive disagreement","subjective models","signal structures","inclusion preorder","cosine similarity","speculative trade","ex-ante Pareto frontier","Bregman divergences"],"falsifier":"Construct two pairs of binary experiments that are not related by convex combination yet induce nested cones of jointly acceptable surplus vectors under some full-support prior, or find a rotation-invariant completion of the inclusion preorder that is not ordered by cosine similarity.","tokens_in":33440,"feed_emoji":"📐","tokens_out":582,"duration_ms":5191,"temperature":0.7,"pith_summary":"When people share a prior, payoffs, and reservation value but disagree about how signals are generated, their willingness to commit to joint plans can be used to rank how close their interpretations are. The paper shows that one pair of signal models supports a larger set of mutually acceptable plans than another if and only if each model in the first pair is a mixture of the two models in the second. That ranking does not depend on the prior. The only rotation-invariant scalar that completes the ranking is cosine similarity between the joint state-signal distributions. The same geometry shrinks the room for pure bets against each other, enlarges a normalized Pareto frontier of joint payoffs, and expands the set of single-model stories that can rationalize the joint behavior. The order is independent of Blackwell informativeness and selects quadratic rather than Kullback-Leibler-type distances.","feed_headline":"Agreement is convex mixture; cosine finishes the ranking","feed_subtitle":"Shared signal models rank by mixtures of likelihoods, not by Blackwell informativeness or KL distance","key_machinery":"The inclusion preorder: the cone of surplus vectors that both agents accept ex-ante under one pair of models contains the corresponding cone under another pair, which dualizes to the convex-hull condition on the models themselves.","core_discovery":"A pair of subjective signal structures is more agreeable than another if and only if each structure in the more agreeable pair is a convex combination of the two structures in the less agreeable pair; this inclusion preorder is prior-independent, and its unique rotation-invariant strict completion ranks pairs by cosine similarity of the induced joint distributions over states and signals.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Agreeable models mix less-agreeable ones; cosine ranks pairs","Interpretation agreement is convex inclusion, completed by cosine","Signal models agree more iff each is mixture of the other pair","Prior-free agreement order: mixtures first, then cosine similarity","Convex combinations define agreement; cosine is unique scalar rank"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The reservation payoff is a fixed common number, so every possible surplus vector can be realized by some finite decision problem; if the reservation must instead come from an action that is already optimal under the prior, full cone inclusion is only sufficient, not necessary.","fun_headline_variants_meta":{"raw":{"variants":["Agreeable models mix less-agreeable ones; cosine ranks pairs","Interpretation agreement is convex inclusion, completed by cosine","Signal models agree more iff each is mixture of the other pair","Prior-free agreement order: mixtures first, then cosine similarity","Convex combinations define agreement; cosine is unique scalar rank"]},"model":"grok-4.5","effort":"low","cost_usd":0.002768,"raw_usage":{"total_tokens":986,"prompt_tokens":691,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":27680000,"prompt_tokens_details":{"text_tokens":691,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":230,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":691,"tokens_out":65,"duration_ms":2940,"temperature":1.0,"reasoning_tokens":230,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T05:48:42.970083+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct two pairs of binary experiments that are not related by convex combination yet induce nested cones of jointly acceptable surplus vectors under some full-support prior, or find a rotation-invariant completion of the inclusion preorder that is not ordered by cosine similarity.","supporting_citations":[],"review_version":1}