{"id":"25fbafe6-50b2-412e-95d0-dd4343cfc451","arxiv_id":"2412.00011","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A formal framework for knowledge in open-ended evolutionary systems yields frame-relative, non-ergodic knowledge and the impossibility of common knowledge.","lead":"A pair of economists builds a formal language for what an observer can know inside a system that keeps generating genuinely new possibilities, such as an economy with ongoing innovation. They argue that such open-ended systems make common knowledge impossible, bind each person's knowledge to a local frame, and make nonlogical reasoning like aesthetics a legitimate source of knowledge.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"OEE2 and OEE3 rest on the unproved Disjointness Conjecture, and Corollary 2 uses that conjecture as a premise rather than deriving it; Corollary 3 adds a further gap by asserting 'agree to disagree' without a formal probabilistic setting.","rationale":"The paper makes a genuine conceptual contribution and its philosophical framing of local knowledge and frame relativity is engaging. It also deserves credit for explicitly labeling the Disjointness claim as a conjecture rather than hiding it. But the formal core does not support the announced results. The only route to OEE2 is Corollary 2, and its proof assumes Ti,t != Tj,t without proving it; Conjecture 1 was supposed to supply that fact, and its sketch appeals to Theorem 1, which concerns a single agent's theory over time, not cross-agent disjointness. A symmetric OEE model in which both agents observe the same new predicates satisfies the paper's working definition of open-endedness, showing that the definitions do not force disjointness. Thus the common-knowledge impossibility result is unsupported. The reader's REJECT verdict is therefore well-founded, and I find no reason to change it. I would add that even a proven OEE2 would not by itself yield OEE3: 'agree to disagree' in the Aumann framework requires a precise probabilistic construction, which the paper never supplies. No machine-checked proof, reproducible implementation, or independent formal verification accompanies the claimed theorems, and the Appendix's references to theorems and lemmas do not fill the gap.","tokens_in":29288,"tokens_out":8635,"duration_ms":102685,"concrete_test":"Construct a minimal two-agent model satisfying the paper's definitions with shared open-ended novelty: set P_i,t = P_j,t = P_0 ∪ {q_0, ..., q_t}, Ti,t = Tj,t, Omega_i,t = Omega_j,t, and Ki,t = Kj,t for all t, with both agents observing the same new predicate each period. If this model is consistent with Def. 7 and the framework's other stated assumptions, then Conjecture 1 is not entailed by the formalism, so Proposition 6 and Corollary 2 lack their needed premise. Separately, re-derive Corollary 3 after formalizing Aumann-style posteriors; if no such model with common knowledge of differing posteriors is produced, OEE3 remains an assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central negative results OEE2 and OEE3 are not established by the paper's formal apparatus. Proposition 6, Corollary 2, and Corollary 3 all require the claim that Ki,t != Kj,t 'in general' under open-ended evolution. That claim is explicitly labeled Conjecture 1 in Section 3.2, and its proof sketch is circular: it argues that if Ki,t = Kj,t then Ti,t = Tj,t, then says that because Theorem 1 rules out general theory convergence, disjointness is 'reasonable.' But Theorem 1 only shows that an individual's theory cannot in general be an inessential extension of its own past theory when new predicates are introduced (Pi,t+1 \\ Pi,t nonempty, Def. 7). It says nothing about whether two different agents' theories or knowledge sets are disjoint from each other. Open-endedness as defined is an individual condition: it is compatible with every agent observing the same sequence of new predicates and sharing the same theories and known-worlds at every time. Corollary 2 then begins with 'Under open-ended evolution, Ti,t != Tj,t != T_Omega in general,' which is exactly the conjecture restated as a fact; so the proof of common-knowledge impossibility has the load-bearing assumption built in. Even granting OEE2, Corollary 3 does not follow: in the Aumann sense, 'agree to disagree' means common knowledge of differing posterior beliefs, and the paper never defines posteriors, common priors, or a probabilistic version of its OEE knowledge sets. The corollary is asserted as a 'direct consequence' rather than demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a syntactic epistemic-logic framework for knowledge in open-ended evolutionary (OEE) systems. It defines per-agent \"known-worlds\" Ω_{i,t}, theories T_{i,t}, predicate sets P_{i,t}, decision procedures δ_{i,t}, contextual knowledge sets K_{i,t}, and local knowledge κ_{i,t}, and it proposes five headline results: OEE1 (believed truths never fully coincide with true truths), OEE2 (common knowledge is generally impossible), OEE3 (agents can agree to disagree), OEE4 (nonlogical search can be as knowledge-generative as logical search), and OEE5 (knowledge is non-ergodic). The formal apparatus is presented in an appendix with definitions, propositions, and corollaries. The central philosophical claim is \"frame relativity\": an observer embedded in an OEE system cannot possess a complete and correct model of the system, and individual knowledge is local and fragmented across agents and time.","tokens_in":29611,"tokens_out":5392,"duration_ms":57581,"significance":"If OEE1–OEE5 were soundly derived, the paper would provide a substantive formal challenge to closed-world epistemic logic and would motivate a research program on institutions, aesthetics, and heuristics as knowledge-generating devices under genuine novelty. The authors are to be credited for making their formalism explicit enough that the gaps are checkable, and for transparently labeling the key cross-agent assumption as a conjecture. However, the significance is currently prospective: the individual-level incompleteness results largely restate the definition of open-endedness, while the cross-agent results OEE2 and OEE3 rest on an unproved conjecture and an undeveloped probabilistic model. The paper supplies no machine-checked proofs and no falsifiable predictions, and several appendix proofs are invalid as written. The central claims are therefore not established by the manuscript in its present form.","major_comments":[{"comment":"The proof of OEE2 uses the unproved Disjointness Conjecture as a premise. Corollary 2 begins \"Under open-ended evolution, T_{i,t} ≠ T_{j,t} ≠ T_Ω in general,\" which is exactly the conjecture restated as a fact rather than a derived conclusion. The proof sketch of Conjecture 1 appeals to Theorem 1, but Theorem 1 concerns a single agent's theory extension across time (P_{i,t+1}\\P_{i,t} ≠ ∅) and says nothing about whether two different agents' theories or knowledge sets are disjoint. Definition 7 is an individual condition; it is compatible with all agents observing the same sequence of new predicates and sharing identical theories and known-worlds at every time. Therefore OEE2 is not established, and Proposition 6's claim that hierarchies are \"in general\" inconsistent inherits the same gap.","section":"§3.2, Conjecture 1; Appendix, Proposition 6 and Corollary 2"},{"comment":"The \"agree to disagree\" conclusion does not follow from Proposition 6 and Corollary 2 as stated. In Aumann's sense, agreeing to disagree means that differing posterior beliefs are common knowledge, which requires a probabilistic apparatus: states with prior probabilities, posterior belief functions, and a definition of common knowledge of those beliefs. The paper never defines posteriors, common priors, or a probabilistic counterpart of K_{i,t}. Corollary 3 is asserted as a \"direct consequence\" without supplying any of the machinery that gives the phrase its standard meaning, so OEE3 is not demonstrated.","section":"Appendix, Corollary 3"},{"comment":"Proposition 8's proof cites \"Theorems 7 and 5,\" but no Theorem 7 exists in the manuscript. More substantively, the argument that an algorithmic process m \"does not halt\" does not establish that there exists a sentence ξ' in Ω for which m is not specified: a non-halting process could still decide every sentence at some finite stage. Corollary 5 then infers from Proposition 8 that T_{i,t} is not recursively enumerable, which is a much stronger claim than the proposition establishes. Thus the frame-relativity result, as formalized, is not proved.","section":"Appendix §6.7, Proposition 8 and Corollary 5"},{"comment":"The finite-τ argument is logically invalid. From the assumption that there are finitely many undecidable disjunctions D_{i,t} with |D_{i,t}| = τ, the proof concludes that T_{i,t+τ} = T_Ω. But open-endedness only guarantees that new predicates arrive at each step; it does not imply that the number of undecidable disjunctions equals the number of steps to the theory of everything, nor that T_Ω is reached after τ steps at all. The further claim that D_{i,t} is uncountably infinite is also unsupported by the preceding argument, which at most suggests an infinite set.","section":"Appendix, Lemma 2"},{"comment":"The individual-level impossibility result OEE1 is true by construction rather than by substantive proof. Definition 7 defines open-endedness as P_{i,t+1}\\P_{i,t} ≠ ∅; Proposition 5 derives the nonemptiness of the adjacent knowledge possible A_{i,t} from this definition; Lemma 1 derives the existence of an undecidable sentence; and Theorem 2 uses Lemma 1 to conclude that knowledge is incomplete. This chain is valid but analytic: it shows that if one defines open-endedness as the arrival of new predicates, then individual knowledge is incomplete. The paper should state this explicitly, because OEE2–OEE5 inherit the definitional character of OEE1 rather than gaining independent empirical or mathematical support.","section":"Definition 7, Proposition 5, Lemma 1, Theorem 2"}],"minor_comments":[{"comment":"The definition text says \"the quotient of the temporally adjacent contextual knowledge possible K_{i,t+1} with the current contextual knowledge possible K_{i,t+1},\" but the displayed formula and the prose clearly require K_{i,t}, not K_{i,t+1}, in the second position.","section":"Definition 13"},{"comment":"There are several notation slips: δ_{i,t}(ξ) + δ_{it},(¬ξ) contains a stray comma in the second term; Proposition 8 refers to \"Theorems 7 and 5\" when only Theorem 5 exists; and the appendix refers to \"SE5\" where Section 2.1 uses \"S5n.\" These should be corrected.","section":"Equations (3.2), (6.3), and Proposition 8"},{"comment":"The statement \"T_{i,t} = P_{i,t}\" equates a theory with a set of nonlogical constants. Since a theory is a set of sentences and P_{i,t} is a set of predicates, this equality is a category error unless an unconventional identification is intended, which should be defined.","section":"Proposition 2(ii)"},{"comment":"The proof contains apparent typographical repetitions (\"T_{i,t} ⊂ T_Ω, T_{i,t} ⊂ T_Ω\") and does not actually prove the claimed generic inconsistency; it merely asserts that equality of theories \"is not true under open-ended evolution,\" which is the conjecture.","section":"Appendix, Proposition 6"},{"comment":"The discussion of the \"ragged partition\" of system-level time is informal and does not engage with the formal definitions of T used elsewhere; it reads as a modeling suggestion rather than a result, and should be labeled as such.","section":"Section 4.6"}],"recommendation":"reject","confidential_remarks":"The central advertised contributions OEE2 and OEE3 are not established: Corollary 2 assumes the Disjointness Conjecture, and Corollary 3 lacks any probabilistic model of beliefs. These are not local presentation issues. A revision would need to prove the conjecture or replace it with a theorem, and would need to build a fully specified probabilistic extension of the framework before 'agree to disagree' can be claimed. Given that the abstract and introduction present these as the paper's primary results, the current manuscript does not meet the bar for publication. The novelty of the framing is not in question, and I do not see evidence of citation or attribution problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe headline: this is a stimulating essay wrapped in a formalism that does not hold together. The authors want to formalize the Hayek/Kauffman picture of radically uncertain, open-ended systems, and they write with real clarity about why that picture matters. But the central theorems OEE1–OEE5 are not established; several proofs use the thing they are out to prove.\n\nWhat is genuinely useful: the distinction between the contextual knowledge possible and the adjacent knowledge possible, and the frame relativity idea, are clean ways to describe why embedded observers cannot have the god's-eye theory of the system. The paper also does a good job situating itself in the OEE literature and in the Aumann–Hintikka tradition. The institutional conclusion—that shared codes emerge as a substitute for impossible common knowledge—is suggestive.\n\nNow the soft spots, and they are not cosmetic. The Disjointness Conjecture (Conjecture 1) is doing all the work, and its proof sketch is circular. It assumes that if two agents share knowledge sets then they share theories, then says theory convergence is impossible by Theorem 1. But Theorem 1 only says an individual's theory cannot in general be an inessential extension of its own past theory when new predicates arrive. It says nothing about whether two agents' theories converge. Open-endedness is defined per individual; it is compatible with every agent seeing the same new predicates and converging to the same theories. So the 'general disjointness' claim is simply unproved. Corollary 2 then starts with 'Ti,t ≠ Tj,t ≠ TΩ in general,' which is the conjecture restated as a premise, and concludes common knowledge is impossible. Corollary 3 asserts agree-to-disagree as a direct consequence, but Aumann's result needs posteriors and common priors; none are defined. Proposition 8 cites a nonexistent 'Theorem 7.' Lemma 2 says that if there were finitely many undecidable disjunctions then Ti,t+τ = TΩ; that does not follow—finitely many may be resolved in τ steps, but the theory may still not be the theory of everything. The non-ergodicity proof also leans on the same nonempty adjacent possible it is trying to establish.\n\nI want to give credit: the paper does not hide these moves, and the prose is honest about the tradition. But the formal machinery is not yet doing the work. If OEE2 and OEE3 were relabeled as conjectures, the paper would be a reasonable essay; as theorems, they fail.\n\nFor peer review: I'd send it out, because the conceptual questions are significant and a good referee can pin down exactly what needs proving. But I would not accept it in anything like this form. The authors need to either prove the Disjointness Conjecture, weaken their claims to conjectures, or drop the Aumann 'agree to disagree' language. The paper deserves a reading group debate, and a serious referee, but the referee should be told to go straight for the conjecture.","headline":"A stimulating essay whose formal theorems are not established: the impossibility results rest on an unproved conjecture and several appendix proofs are invalid.","tokens_in":30145,"tokens_out":3454,"would_cite":false,"duration_ms":36681,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B42","91A26"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that in open-ended evolutionary systems, observers' knowledge is local and frame-relative: common knowledge is generally impossible, and rational agents can agree to disagree.","keywords":["open-ended evolution","epistemic logic","common knowledge","frame relativity","agree to disagree","adjacent possible","non-ergodic knowledge","undecidable disjunctions"],"falsifier":"Construct an open-ended evolutionary system with two observers who, despite continual novelty, maintain identical knowledge sets \\(K_{i,t}=K_{j,t}\\) at every time step, for example by having all new predicates broadcast symmetrically to both; then common knowledge would be possible and OEE2 and OEE3 would fail. Alternatively, exhibit a specific OEE process whose theory revision sequence \\(\\{T_{i,t}\\}_{t}\\) converges to a decidable \\(T_\\$\\Omega$\\), which would falsify OEE1 and frame relativity.","tokens_in":1772,"feed_emoji":"🧠","tokens_out":1933,"duration_ms":49454,"temperature":0.7,"pith_summary":"The paper tries to show that the standard epistemic-logic machinery used in economics, which assumes a complete, shared state space and common knowledge, breaks down when the system generates genuinely novel possibilities. It builds a formal framework for knowledge use and acquisition in open-ended evolutionary (OEE) systems and claims five results: believed truths never fully coincide with true truths, common knowledge is generally impossible, agents can agree to disagree, nonlogical search can be as knowledge-generative as logical search, and knowledge is non-ergodic. A sympathetic reader would care because these results, if correct, would replace the closed-world assumptions of rational-choice theory with a picture in which each observer is bound to a local, evolving frame and coordination must be achieved through institutions, aesthetics, and other nonlogical codes.","feed_headline":"Open-ended systems make common knowledge impossible","feed_subtitle":"A formal model says embedded observers can only know their local frame and can rationally agree to disagree.","key_machinery":"The framework's load-bearing objects are the observer-specific theory \\(T_{i,t}\\), the model \\(M_{i,t}\\), the set of nonlogical constants \\(P_{i,t}\\), and the decision procedure \\(\\delta_{i,t}:\\Omega_{i,t}\\to\\Sigma_{i,t}\\) that assigns truth values to sentences. Local knowledge \\(\\kappa_{i,t}(\\omega)\\) is the set of sentences in state \\(\\omega\\) prefixed with the knowledge operator \\(k_{i,t}\\); the contextual knowledge possible \\(K_{i,t}\\) collects all states decidable by \\(\\delta_{i,t}\\); and the adjacent knowledge possible \\(A_{i,t}=K_{i,t+1}\\setminus K_{i,t}\\) captures the new states that enter between time steps. The Disjointness Conjecture, that \\(K_{i,t}\\neq K_{j,t}\\) in general for different agents, is what carries the impossibility of common knowledge, while the non-convergence of theory revision, grounded in Gödelian undecidability, carries frame relativity.","core_discovery":"The central discovery the paper asserts is frame relativity: a complete and correct theory of an open-ended evolutionary system is possible only for an agent outside the system, while any embedded observer is limited to their own known-world \\(\\Omega_{i,t}\\), theory \\(T_{i,t}\\), and decision procedure \\(\\delta_{i,t}\\). From this, the paper derives five results. OEE1 says an observer's believed-true propositions and the true propositions of the system can never coincide. OEE2 says common knowledge is in general impossible because knowledge hierarchies of different individuals are not mutually consistent. OEE3 says agents can therefore rationally agree to disagree. OEE4 says random, heuristic, or aesthetic search can be as knowledge-generative as logical search because logic alone cannot resolve the undecidable disjunctions an OEE system continually produces. OEE5 says knowledge in such systems is non-ergodic: the time average of local knowledge does not converge to a state-space average because the adjacent possible is always nonempty.","pith_inferences":["Extension: if frame relativity is correct, expert failure and scientific consensus are not merely corrigible by more computation or better data; they are structural limits of any embedded observer, which would extend Hayek's knowledge problem to organized science.","Extension: the framework suggests a testable prediction that faster open-ended change should increase cultural, ideological, and aesthetic fragmentation, because shared nonlogical codes become more valuable and cheaper to maintain in niches.","Extension: the non-ergodicity result gives a formal epistemology for the recent critique of expected utility theory based on ergodicity; one could test it by simulating an artificial-life system where agents with asymmetric observations of novelty repeatedly disagree despite common priors.","Extension: the 'many-selves' reading of reasoning through time implies that preference change is not just a parameter shift but a change in the observer's known-world, which could be operationalized in behavioral experiments on how people reinterpret past choices after radical novelty."],"forward_implications":["No observer embedded in an OEE system can ever reach the theory of everything \\(T_\\Omega\\), and the number of statements true in \\(T_\\Omega\\) but undecidable in \\(T_{i,t}\\) is infinite.","Common knowledge cannot be presumed in open-ended systems, so coordination must be achieved through explicit institutions, shared codes, and other knowledge-sharing mechanisms.","Agents can rationally agree to disagree, overturning the classical result that common priors and common knowledge force agreement.","Nonlogical modes of reasoning such as aesthetics and heuristics are not mere biases; they can be as knowledge-generative as logical deduction when undecidable disjunctions are unavoidable.","Knowledge in OEE systems is non-ergodic, so an individual's time average of knowledge cannot be replaced by an ensemble average over possible states."],"supporting_citations":[{"why":"supplies the syntactic epistemic logic and knowledge-hierarchy formalism that the paper weakens for OEE systems.","marker":"Aumann 1999a"},{"why":"provides the state-as-set-of-sentences construction that the paper adapts for local known-worlds.","marker":"Samet 1990"},{"why":"gives the incompleteness result the paper uses to show undecidable disjunctions and the impossibility of a complete observer theory.","marker":"Gödel 1931"},{"why":"defines open-ended novelty and the theory-altering novelty types that motivate the OEE knowledge framework.","marker":"Banzhaf et al 2016"},{"why":"contributes the concept of the adjacent possible, which the paper formalizes as the nonempty quotient of the knowledge possible.","marker":"Kauffman 1993"},{"why":"supports the claim that no formal logic can eliminate intuition, which the paper relies on for theory non-convergence.","marker":"Turing 1939"},{"why":"supplies the ergodicity framing the paper repurposes to prove that knowledge in OEE systems is non-ergodic.","marker":"Peters 2019"},{"why":"provides the motivating problem of dispersed local knowledge and the paper's titular 'use of knowledge in society.'","marker":"Hayek 1945"}],"fun_headline_variants":["Frame relativity: embedded observers can't know the whole system","In open-ended systems, knowledge is local and non-ergodic","Rational agents in open-ended systems agree to disagree","No common knowledge in open-ended evolutionary systems","Frame relativity limits embedded observers to local theories"],"cache_read_input_tokens":32128,"weakest_assumption_plain":"The proof that common knowledge is impossible and that agents can agree to disagree rests on the unproved Disjointness Conjecture, which asserts that different observers' local knowledge sets are at least partially disjoint across time and agents; the proof sketch appeals to the very non-convergence result the paper is trying to establish.","fun_headline_variants_meta":{"raw":{"variants":["Frame relativity: embedded observers can't know the whole system","In open-ended systems, knowledge is local and non-ergodic","Rational agents in open-ended systems agree to disagree","No common knowledge in open-ended evolutionary systems","Frame relativity limits embedded observers to local theories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000543,"raw_usage":{"total_tokens":2551,"prompt_tokens":847,"completion_tokens":1704,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":1629}},"tokens_in":463,"tokens_out":1704,"duration_ms":10841,"temperature":1.0,"reasoning_tokens":1629,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:05:09.300617+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an open-ended evolutionary system with two observers who, despite continual novelty, maintain identical knowledge sets \\(K_{i,t}=K_{j,t}\\) at every time step, for example by having all new predicates broadcast symmetrically to both; then common knowledge would be possible and OEE2 and OEE3 would fail. Alternatively, exhibit a specific OEE process whose theory revision sequence \\(\\{T_{i,t}\\}_{t}\\) converges to a decidable \\(T_\\$\\Omega$\\), which would falsify OEE1 and frame relativity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the state-as-set-of-sentences construction that the paper adapts for local known-worlds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supports the claim that no formal logic can eliminate intuition, which the paper relies on for theory non-convergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the ergodicity framing the paper repurposes to prove that knowledge in OEE systems is non-ergodic."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the motivating problem of dispersed local knowledge and the paper's titular 'use of knowledge in society.'"}],"review_version":1}