{"id":"3c4dd842-3d99-45a5-81ec-66c1a6ebf30a","arxiv_id":"2608.04315","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Extremal distributions for partially specified robust inference problems satisfy Dinkelbach-type fixed-point conditions, giving a discretization-and-iteration solution framework.","lead":"This paper develops a framework for computing extremal values of ratio-of-expectations quantities when only partial constraints on a probability distribution are known. It characterizes where the extremal distribution places mass through fixed-point conditions, and could benefit robust Bayesian inference, reliability, econometrics, and finance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.8's convergence claim is false for Borel partitions with dense cells; a counterexample with E_1=Q×[0,1] gives ~ϕ_m=0 but ϕ*=1.","rationale":"The reader's weakest assumption identified the treatment of Borel cells as closed in Corollary 2.8's connectedness argument. Our stress test confirms this is not just a proof gap: the corollary is false, and we provide a concrete counterexample with dense Borel level sets. The reader's accompanying concern about R_2.13 containing all bounded Borel simple functions is, however, unfounded: a simple function constant on each Borel level set admits the constant continuous extension to the closure of each piece, so those functions do lie in R_2.13, and the uniform-closure argument in the remark to Theorem 2.15 is basically sound. The central fixed-point characterizations (Theorems 2.9, 2.13, 2.15) do not use Corollary 2.8 in their proofs, and our counterexample does not refute them. Nevertheless, Corollary 2.8 is presented as a core methodological result for piecewise-defined objectives with asymptotically vanishing oscillations, and the paper's abstract claims coverage of piecewise continuous functions without the needed regularity qualifications. The appropriate response is to require the authors to repair or remove Corollary 2.8, and to qualify the piecewise-continuity claims accordingly. This supports the reader's conditional verdict, though with a sharper and more serious defect than the reader described.","tokens_in":78174,"tokens_out":15688,"duration_ms":150162,"concrete_test":"Recompute the dyadic approximating problems in Corollary 2.8 with K=[0,1]^2, E_1=Q×[0,1], E_2=Q^c×[0,1], f=1_{E_2}, g≡1, and verify that ̃ω_f(m)=̃ω_g(m)=0, ~ϕ_m=0 for every m, and ϕ*=1. The contradiction settles that the corollary's conclusion is false as stated; if instead the corollary is restricted to E_ℓ with empty boundary or to piecewise-continuous functions with finite discontinuity sets, the counterexample no longer applies and the connectedness step may be repairable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 2.8 asserts that if the oscillation of f and g on the refined atoms R^(m)_τ = C^(m)_ν ∩ E_ℓ vanishes uniformly, then the discretized infima ~ϕ_m converge to ϕ*. The proof in Appendix F relies on the claim that the family {C^(m)_ν ∩ E_ℓ : ℓ=1,...,L} is finite and closed, and uses connectedness of C^(m)_ν to bound the oscillation over the whole product cell by L times the refined oscillation. This step is invalid because the E_ℓ are only assumed Borel, not closed. In fact, the conclusion of Corollary 2.8 is false. Take K=[0,1]^2, M=2, uniform marginals, E_1 = Q×[0,1], E_2 = Q^c×[0,1], f = 1_{E_2}, g ≡ 1. Then each R^(m)_τ is a constant set for f and g, so ̃ω_f(m) = ̃ω_g(m) = 0 for every m, satisfying the hypothesis. But ϕ* = inf_{P∈D} P(E_2) = 1, since every P∈D has Lebesgue first marginal. For each m, the measure P_m = μ_m × λ_V, with μ_m uniform on one rational point per dyadic interval, belongs to D^(m) and gives P_m(E_2)=0, hence ~ϕ_m = 0 for all m. Thus ~ϕ_m ↛ ϕ*, contradicting Corollary 2.8. This is a concrete failure of the claimed piecewise-continuous extension; it is not merely a missing hypothesis in the proof but a false statement in the methodology section.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a methodological framework for robust inference problems of the form inf_{P∈D} E_P[f(X)]/E_P[g(X)], where D is a class of Borel probability measures on K=[0,1]^d constrained by fixed partition masses, overlapping measurable-set masses, prescribed (density) marginals, or combinations thereof. The principal claims are: (i) a finite-support reduction (Proposition 2.2) showing that the infimum over D equals the infimum over discrete priors with at most n+q support points, with a detailed proof via Richter's theorem and a rank-pruning argument; (ii) fixed-point characterisations of extremal support locations, (E1)-(E2), for analytic objectives (Theorem 2.9), extended by approximation to continuous (Theorem 2.11), bounded piecewise-continuous (Theorem 2.13), and uniform-limit (Theorem 2.15) classes; (iii) a Dinkelbach-type computational pipeline (Algorithms 2.10, 2.14) with convergence proofs; and (iv) several worked applications, including Bernoulli reliability priors (Theorem 3.1), generalised Boole-Fréchet bounds (Theorem 3.2), interval-identified econometric models (Theorem 3.3), VaR aggregation (Theorem 3.4 and Corollary 3.5), and basket options (Theorem 3.6). All proofs are provided in the appendices at the end of the manuscript.","tokens_in":78403,"tokens_out":21644,"duration_ms":184168,"significance":"If the main theorems held as stated, this would be a substantial contribution: the paper unifies a wide family of robust-Bayesian and fixed-marginal optimisation problems under one fixed-point reduction, identifies where extremal mass is placed, and provides a computational route with certified convergence. The strengths are genuine: Proposition 2.2 is proved in full detail with an explicit support bound; Theorem 2.9's analytic characterisation is self-contained and correctly linked to Dinkelbach iteration, whose monotone convergence is established in Appendix H; the applications in Sections 3.3-3.7 exhibit real breadth; and there are no fitted parameters in the central derivations. However, the paper as submitted overclaims the piecewise-continuous case: Corollary 2.8, advertised as the convergence result for piecewise-defined objectives, is false as stated (a counterexample is given below), and the 'exhaustiveness' proof in Appendix K has a gap. The remaining theorems appear to be unaffected, and the errors are localisable, so the claims are likely repairable within the manuscript's scope, but the current text is not correct as it stands.","major_comments":[{"comment":"Corollary 2.8 is false as stated, and its proof is invalid. Take K=[0,1]^2, M=2, uniform marginals (rho_1=rho_2=1), and the fixed Borel partition E_1=Q×[0,1], E_2=Q^c×[0,1], with f=1_{E_2} and g≡1. Letting the B^{(m)}_{j,r} be the dyadic partitions (connected intervals), every refined atom R^{(m)}_τ=C^{(m)}_ν∩E_ℓ is a level set of f, so ~ω_f(m)=~ω_g(m)=0 for all m, satisfying the corollary's hypotheses. But φ* = inf_{P∈D} P(E_2) = 1, since every P∈D has Lebesgue first marginal and λ(Q^c)=1. On the other hand, the measure P_m=μ_m×λ, with μ_m uniform on one rational point in each dyadic interval, lies in D^{(m)} and satisfies P_m(E_2)=0, so ~φ_m=0 for every m; hence ~φ_m does not converge to φ*, contradicting both the convergence claim and the error bound in item (2). The source of the error is in Appendix F, where the family {C^{(m)}_ν∩E_ℓ} is declared 'finite and closed'; the E_ℓ are only Borel, not closed, and the intersection-graph connectedness argument for the connected set C^{(m)}_ν fails exactly for partitions with dense cells. This is not a mere missing hypothesis: the counterexample satisfies every stated assumption. The corollary must be repaired (e.g., by requiring the E_ℓ to be closed, or by replacing the chain argument with a measure-theoretic bound), withdrawn, or explicitly isolated from the rest of the methodology; Section 2.1 currently advertises it as the convergence result for piecewise-defined objectives.","section":"Theorem 2.13 / Appendix K"},{"comment":"The proof of the 'exhaustiveness' claim M_i(φ*)=C_i(φ*) in Appendix K is not valid as written. To show M_i(φ*)⊆C_i(φ*), the proof perturbs the Dinkelbach difference to h(x)+η_m‖x−x*‖² and asserts that analytic approximants to the perturbed objective produce minimisers converging to x*. But C_i(φ*) is defined as the collection of subsequential limits of global minimisers of Dinkelbach differences formed from analytic approximating sequences that converge uniformly to the unperturbed extensions ~f_{i,ℓ}, ~g_{i,ℓ}. Approximants to the pair (f+η_m‖·−x*‖², g) do not converge to the unperturbed extensions; a diagonal argument with η_m→0 would be needed and is not supplied, and the perturbation changes the location of the global minimiser in a way that is not controlled. Since the remark after Theorem 2.13 ('analytic approximation is exhaustive: ... there are no extremal support candidates outside the class of extremal support points obtained from analytic approximations') relies on this equality, the claim needs either a corrected proof or a reformulation that states only what is actually established.","section":"Theorem 2.15 (iii)-(iv), Section 3.5 remark"},{"comment":"Theorem 2.15(iii) asserts that the limiting tuple (x*_1,...,x*_n) 'is the support of some extremal prior P*', but the proof in Appendix L only establishes that P* := Σ_i p_i δ_{x*_i} is the weak limit of the extremal priors P_{m_k} of the approximating problems and that E_{Q_k}[f]/E_{Q_k}[g]→φ* for feasible Q_k⇒P*. For discontinuous objectives in the uniform-limit class, weak convergence does not imply value attainment; the paper itself acknowledges this in the remark following Theorem 3.3 ('a weak limit of these approximating priors may itself fail to attain the robust bound'). Consequently, P* is not shown to be extremal in the value sense, nor is it shown to satisfy the fixed-point conditions (E1)-(E2) for the limiting pair (f,g). The theorem's statement and the corresponding remarks in Section 2.2 should be reformulated to say that P* is a weak-limit point of extremal priors of the approximating problems, with the extremal value obtained in the limit, unless additional regularity is imposed.","section":"Theorem 2.15"}],"minor_comments":[{"comment":"There are several typographical errors: 'whereextremal' in the abstract, missing spaces in the Figure 18 caption ('Attractory∗ and repellery∗∗ fromh(x) =ϕ ∗'), and inconsistent closure notation (K_i vs \\bar{K}_i) in Theorem 2.9 and equation (6). These should be corrected in revision.","section":"Abstract / throughout"},{"comment":"In the statement of Theorem 2.9 and in equation (6), the notation for the closures of the K_i is inconsistent: the text defines \\bar{K}_i but the displays use K_i for both the original sets and their closures. Please standardise, for example by writing \\bar{K}_i for closures throughout.","section":"Theorem 2.9 / equation (6)"},{"comment":"The equality R_{2.13}^{||·||∞} = B_b^+(K) is asserted in a remark rather than proved; the inclusion S_b^+(K) ⊆ R_{2.13}^{||·||∞} is the key step and deserves a one-line justification (constant functions on arbitrary Borel pieces admit continuous extensions to the closures). Because the refined-atom discretisation of Corollary 2.8 fails for arbitrary Borel partitions, the remark should also state explicitly that the level-set approximation route used here is a different construction.","section":"Remark after Theorem 2.15"},{"comment":"Theorem 3.1 and its proof are reproduced verbatim from Salako and Muhammad (2025); the authors should state at the start of Section 3.1 which parts of the paper are new contributions and which are restatements within the new fixed-point framing, so that the novelty of Section 3.1 is unambiguous.","section":"Section 3.1 / Theorem 3.1"},{"comment":"The hypothesis that the B^{(m)}_{j,r} are connected in Corollary 2.8 is present only to support the invalid chain argument; if the corollary is repaired by imposing closedness or additional regularity on the E_ℓ, the proof should state explicitly why connectedness is (or is not) still needed.","section":"Appendix F"},{"comment":"Recommend rephrasing 'is the support of some extremal prior P*' to 'is the support of the weak limit P* of extremal priors of the approximating problems', since value attainment is not proved and is generally false for discontinuous Borel objectives (compare the remark after Theorem 3.3).","section":"Theorem 2.15 (iii)"},{"comment":"In the proof of Proposition 2.2, the case N≤n+q is dispatched with 'there is then nothing more to prove'; since Q_0 may have fewer than n+q support points, the sentence should say explicitly that the support-size bound is then satisfied automatically (zero-weight terms are ignored in the definition of D^{(r)}_{p,m}).","section":"Proposition 2.2"},{"comment":"The supplementary material is referenced as 'Salako and Muhammad (2026)' but consists of the appendices of the present paper; the journal submission should clarify how these appendices will be provided to referees and readers.","section":"Supplementary material / References"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional assessment matches my own reading. The central theorems (Proposition 2.2, Theorems 2.9 and 2.11) appear sound and well-proved; the failure of Corollary 2.8 is a serious but localised error that does not propagate to the proofs of Theorems 2.13 and 2.15 as they stand. I would ask the authors to (i) repair or withdraw Corollary 2.8, (ii) fix the exhaustiveness proof in Appendix K or soften the claim, and (iii) clarify the value-attainment status of 'extremal' priors in Theorem 2.15. I also note that Theorem 3.1 is a verbatim reproduction of a prior preprint, so the novelty disclosure should be sharpened. The fit with the journal's scope is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the bottom line. The fixed-point characterisation for the basic finite-partition problem is a real contribution. Proposition 2.2's finite-support reduction with n+q atoms and pointwise domination is carefully proved, and Theorem 2.9's (E1)-(E2) conditions for analytic f and g are genuinely useful, connecting cleanly to Dinkelbach iteration. The paper is also honest that several applications re-derive known bounds (Laurence-Wang, Rüger, Giacomini-Kitagawa), which is fine.\n\nBut the stress-test note is right, and it is worse than the reader thought: Corollary 2.8 is not merely missing a hypothesis, it is false. Take K=[0,1]^2 with uniform marginals, E1=Q×[0,1], E2=Q^c×[0,1], f=1_{E2}, g≡1. On every refined atom the oscillations vanish, so the corollary's hypothesis holds. Yet phi* = inf_{P in D} P(E2) = 1, while for each m the measure mu_m × lambda, with mu_m uniform on one rational point per dyadic interval, belongs to D^(m) and gives objective 0. So ~phi_m = 0 for all m, and convergence fails. The proof breaks because it treats the Borel cells E_l as closed in the connectedness argument; a finite Borel partition is not a finite closed cover. That invalidates the claimed convergence for piecewise-defined objectives and undermines Theorem 2.15's assertion that the uniform closure contains all bounded Borel functions.\n\nThe analytic/continuous core, and probably the refined piecewise setup of Theorem 2.13 with continuous extensions on closures, still look sound. The paper needs major revision: drop or repair Corollary 2.8, restate the uniform-closure theorem with correct hypotheses, and stop claiming the full bounded-Borel class is covered. The reproduction of Theorem 3.1's proof from a prior paper and the missing application appendices are secondary but also worth cleaning up.\n\nWho is this for? Robust Bayes and model-free bound people will want the finite-partition fixed-point results. But anyone who reads the abstract will overestimate the scope. I would send it to a serious referee, not desk-reject, because the core is valuable and the false claims are identifiable and fixable. A good referee should ask for the convergence claims to be reworked and for the authors to be explicit about what does and does not carry beyond analytic and continuous objectives.","headline":"The finite-partition fixed-point core is solid and worth knowing, but the paper's broadcast claims about piecewise and Borel-uniform classes collapse on Corollary 2.8, which is false as stated.","tokens_in":79023,"tokens_out":3739,"would_cite":false,"duration_ms":38719,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62F35","90C32","62F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fixed-point equations locate the support of extremal priors in robust Bayesian inference, reducing constrained optimisation to a discrete system solvable by iteration.","keywords":["robust Bayesian inference","extremal distributions","fixed-point characterisation","Dinkelbach iteration","fractional programming","partially specified priors","Boole-Fréchet bounds","marginal constraints"],"falsifier":"Construct a pair of bounded Borel functions f, g, one of which is not piecewise continuous with continuous extensions, together with a feasible constraint set D, and exhibit a prior P in D whose ratio is strictly below every value that the claimed fixed-point system can produce; if such a counterexample satisfies the paper's own assumptions, the central characterisation fails.","tokens_in":77861,"feed_emoji":"📌","tokens_out":3390,"duration_ms":29083,"temperature":0.7,"pith_summary":"This paper tries to establish that a large class of robust inference problems — minimising a ratio of expectations over priors that satisfy partial constraints — can be solved by focusing on discrete extremal priors whose support points are pinned down by fixed-point conditions. The central claim is that for analytic objectives the extremal value is attained by a prior placing mass at points satisfying (E1)–(E2), and that continuous, piecewise continuous, and uniform-limit objectives are handled by approximation with convergence guarantees. A sympathetic reader would care because the result turns a potentially infinite-dimensional optimisation over distributions into a finite system that can be solved numerically, and it identifies where extremal priors concentrate mass.","feed_headline":"Fixed-point equations locate worst-case priors in robust inference","feed_subtitle":"A new framework reduces partially specified Bayesian problems to discrete fixed-point systems with explicit support locations.","key_machinery":"The Dinkelbach difference $h_{\\phi,f,g}(x) = f(x) - \\phi g(x)$. Minimising over each partition cell with a fixed $\\phi$ and then updating $\\phi$ as the resulting ratio of expectations gives the fixed-point system, and a Dinkelbach-type iteration converges to the extremal value. The second load-bearing object is the finite-support reduction, showing any feasible prior can be replaced by one with at most $n+q$ atoms without increasing the objective.","core_discovery":"For the generic fractional programme $\\inf_{P \\in D} \\mathbb{E}_P[f]/\\mathbb{E}_P[g]$ under partition-mass, overlapping-set, or marginal constraints, extremal values are attained by discrete priors or weak limits of feasible priors whose support points satisfy the Dinkelbach-type fixed-point conditions (E1) $x_i^* \\in \\arg\\min_{x \\in \\overline{K_i}} (f(x) - \\phi^* g(x))$ and (E2) $\\phi^* = \\sum_i f(x_i^*) p_i / \\sum_i g(x_i^*) p_i$. The same characterisation is extended by approximation to continuous, bounded piecewise continuous, and uniform-limit objective classes, with explicit convergence of extremal values and weak convergence of extremal priors.","pith_inferences":["The paper's main theorems imply a general recipe elsewhere: for any fractional expectation problem with finitely many linear prior constraints, the extremal support lies among the minimisers of a Dinkelbach difference, so one can solve the problem by scanning candidate support locations and fixed-point iteration without designing bespoke arguments per application. This extends beyond the paper's w","The value-quantisation approximation in Theorem 2.15 suggests a testable computational shortcut that the paper only partially exploits: for any bounded Borel f and g, bounding f and g from below and above by quantised simple functions yields certified upper and lower bounds on the extremal value at every approximation level, so convergence can be monitored without knowing the limit in advance.","The attractor/repeller interpretation of fixed points opens a route to sensitivity analysis that the paper leaves implicit: the sign of the Dinkelbach difference close to a fixed point indicates whether small perturbations of the prior constraints will shift extremal mass towards or away from a given support location, which could be used to rank the influence of each constraint.","The exchange fixed-point formulation for fixed-marginal problems suggests a unified test for extremality in any coupling problem: a prior is extremal exactly when every marginal-preserving mass exchange fails to increase the target probability, which could be checked empirically at the level of simulated exchanges in higher-dimensional examples not treated by the paper's closed forms."],"forward_implications":["If the central claim is correct, any partial-prior robust inference problem of the fractional expectation type reduces to solving the finite system (E1)–(E2), and the extremal prior is discrete or a limit of feasible discrete priors.","Overlapping measurable-set constraints and multiple marginal-density constraints are covered by the same reduction, since both can be rewritten as finite partition or product-cell problems.","Continuous and piecewise continuous objectives are solved by analytic approximation, with the approximating extremal values converging to the true infimum and support points converging along subsequences.","The framework gives a route to practical computation: run the Dinkelbach iteration with coordinatewise minimisation over each cell, then read off the extremal prior from the limiting support points.","The results recover and unify known solutions, including the k-out-of-n Boole–Fréchet bounds, robust VaR aggregation with fixed marginals, the Laurence–Wang basket-option lower bound, and interval-identified robust posterior bounds."],"supporting_citations":[{"why":"Supplies the original finite-support reduction for partition-constrained robust Bayes problems that Proposition 2.2 extends to n+q atoms.","marker":"[Moreno and Cano (1991)]"},{"why":"Provides the convexity/atomisation theorem used in the proof of the finite-support reduction.","marker":"[Richter (1957)]"},{"why":"Underlies the moment-set extreme point arguments used for the reduction.","marker":"[Winkler (1988)]"},{"why":"Generalised moments formulation whose Theorem 3 is referenced as the baseline that the paper's discrete formulations extend.","marker":"[Betrò et al. (1994)]"},{"why":"Provides the fixed-point iteration scheme that the paper's Algorithm 2.10 applies to the extremal-prior problem.","marker":"[Dinkelbach (1967)]"},{"why":"Supports the convergence analysis of the Dinkelbach iteration for nonlinear fractional programmes.","marker":"[Schaible (1976)]"},{"why":"Establishes the robust-Bayes posterior-bound interpretation that the paper's fixed-point solutions are designed to compute.","marker":"[Lavine (1991)]"},{"why":"Constitutes the linearisation/bounding baseline whose (E2)-type condition the paper refines into a full fixed-point characterisation.","marker":"[Wasserman et al. (1993)]"},{"why":"Closed-form basket-option lower bound that Theorem 3.6 recovers and re-derives through the fixed-point/anti-diagonal construction.","marker":"[Laurence and Wang (2005)]"},{"why":"Provides the containment-principle robust-posterior bound that Theorem 3.3 re-derives in fixed-point and extremal-prior form.","marker":"[Giacomini and Kitagawa (2021)]"}],"fun_headline_variants":["Fixed-point equations reveal extreme priors in robust inference","Fixed-point conditions pinpoint extremal distributions","Find worst-case priors via fixed-point characterisation","Fixed-point equations solve robust Bayesian priors","Extremal distributions from fixed-point constraints"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain rests on treating each piece of a piecewise-defined function as having a continuous extension to the closure of its piece, and on treating the Borel partition cells as closed when the connectivity argument for the approximation error is applied.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-point equations reveal extreme priors in robust inference","Fixed-point conditions pinpoint extremal distributions","Find worst-case priors via fixed-point characterisation","Fixed-point equations solve robust Bayesian priors","Extremal distributions from fixed-point constraints"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000492,"raw_usage":{"total_tokens":2380,"prompt_tokens":870,"completion_tokens":1510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":1441}},"tokens_in":486,"tokens_out":1510,"duration_ms":11447,"temperature":1.0,"reasoning_tokens":1441,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T19:56:35.042726+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a pair of bounded Borel functions f, g, one of which is not piecewise continuous with continuous extensions, together with a feasible constraint set D, and exhibit a prior P in D whose ratio is strictly below every value that the claimed fixed-point system can produce; if such a counterexample satisfies the paper's own assumptions, the central characterisation fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the convexity/atomisation theorem used in the proof of the finite-support reduction."},{"cited_title":"Sharp Upper and Lower Bounds for Basket Options , journal =","cited_arxiv_id":null,"evidence_quote":"Closed-form basket-option lower bound that Theorem 3.6 recovers and re-derives through the fixed-point/anti-diagonal construction."},{"cited_title":"Econometrica , volume =","cited_arxiv_id":null,"evidence_quote":"Provides the containment-principle robust-posterior bound that Theorem 3.3 re-derives in fixed-point and extremal-prior form."}],"review_version":1}