{"id":"16ea56cd-8627-49d9-9eec-7c85863cdc77","arxiv_id":"2608.02482","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For piecewise constant fan subsolutions of the isentropic Euler equations, the entropy-rate and action-rate admissibility criteria reduce to two coefficients, and for the Krupa–Szekelyhidi contact-discontinuity example both criteria prefer the convex-integration solutions over the classical solution","lead":"This paper derives exact formulas for the energy and action rates of convex-integration solutions of the 2D isentropic Euler equations relative to self-similar reference solutions, showing they depend only on the underlying fan. Applying these to the Krupa–Szekelyhidi planar contact discontinuity, it proves with certified exact arithmetic that both Dafermos's entropy-rate criterion and the action-rate criterion prefer the wild convex-integration solutions over the classical c","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Pairwise-preference claim hinges on exact pointwise identities and an unexecuted exact-arithmetic certificate; both merit independent verification.","rationale":"The structural derivation in Sections 3-5 is elementary and correct; the algebra D = 1/2 K + I, E = -1/2 K + I is exact, and the KS application uses only the fan data. The exact-arithmetic certificate is a strong piece of evidence, and the reported margins (~10^10) make it unlikely that a small perturbation changes the signs. My concern is therefore not a demonstrated flaw but a verification gap: the pointwise identities and the Newton-Kantorovich certificate are load-bearing and are not independently established within the paper. Since the ancillary code is provided and reproducible, and the reader's residual uncertainty about the certificate is reasonable, I do not see a need to change the ACCEPT verdict. The same caveat about unexecuted scripts applies, hence the concrete test.","tokens_in":15378,"tokens_out":30024,"duration_ms":290945,"concrete_test":"Run KS_coefficients.mpl and certify.py in a clean environment (or independently re-implement the exact-rational checks) and confirm the inequalities (6.4), (7.1)-(7.2) and the bounds delta_0, L_D, L_E. In parallel, consult [11] or [13] for the statement guaranteeing rho = bar-rho and |v|^2 = C_i a.e.; if the construction only provides these up to a tolerance epsilon, recompute D_KS and E_KS over the worst-case deviations and check whether the strict signs D_KS > 0, E_KS < 0 persist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that every convex-integration realization generated by the KS fan is preferred to the planar contact by both criteria depends on two conditions that are asserted rather than proved here. (1) Section 2 states that convex integration yields solutions with rho = bar-rho and |v|^2 = C_i almost everywhere in each fan region; this is cited from [11]/[13] but not re-derived. If the construction only gives approximate constraints, Proposition 2.8 fails and D_KS, E_KS become realization-dependent, so the 'every realization' conclusion is unsupported. (2) Theorem 6.5 transfers the sign inequalities from the rational parameter vector to the exact fan via Newton-Kantorovich bounds produced by certify.py (delta_0 <= 2.45e-13, L_D <= 53890, L_E <= 49688). The certificates are not reproduced in the manuscript; a bug in the LDL^T or interval-arithmetic routines would invalidate the transfer. The margins are enormous (orders 10^10), so these are verification gaps, not demonstrated errors.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies admissibility criteria for weak solutions of the two-dimensional isentropic Euler equations obtained by convex integration from piecewise constant fan subsolutions. The main structural result is that, relative to any piecewise constant self-similar reference solution, the action-rate and entropy-rate coefficients of every convex integration realization generated by a given fan depend only on the fan, not on the particular realization. These coefficients are shown to decompose as D = K/2 + I and E = -K/2 + I in terms of a weighted kinetic-energy mismatch K and internal-energy mismatch I, yielding a complete two-sector characterization of when the action and entropy rate criteria agree. The theory is applied to the Krupa–Szekelyhidi fan for Riemann data whose classical solution is a planar contact discontinuity. Using exact rational arithmetic and certified Newton–Kantorovich bounds, the paper establishes D_KS > 0 and E_KS < 0 for the exact fan, so that both criteria strictly prefer every convex integration solution associated with that fan to the classical contact solution.","tokens_in":15681,"tokens_out":18351,"duration_ms":183250,"significance":"If the result holds, it is a valuable and concrete contribution to the ongoing discussion of selection criteria for nonunique weak solutions of the compressible Euler equations. The paper provides a transparent, essentially algebraic reduction of the two criteria to a pair of coefficients, and it demonstrates on a nontrivial example that both criteria can select the same 'wild' solutions over the classical self-similar solution. The strengths of the paper are its explicit, parameter-free derivation of the D/E decomposition; the clean separation of the finite-dimensional coefficient evaluation from the existence of the exact fan; and the use of exact rational arithmetic and certified bounds with enormous margins rather than floating-point heuristics. The two verification concerns raised in the stress-test — the exact pointwise identities for convex integration realizations and the unexecuted certification scripts — are real but, on reading the manuscript, they do not land as demonstrated errors: the identities are stated as explicit hypotheses imported from the cited literature, and the certificate is described in enough algorithmic detail with code made available. I did not indepen","major_comments":[],"minor_comments":[{"comment":"The pointwise identities ρ = ρ̄ and |v|^2 = C_i a.e. in each fan region are load-bearing for Proposition 2.8 and hence for the entire paper. Since they are imported from the convex integration literature, please give the specific theorem or proposition number in [11] or [13] that guarantees them, rather than only the summary statement, so that a reader does not have to reconstruct the argument.","section":"Section 2, items (i)-(iii)"},{"comment":"The proof of Proposition 6.3 and 6.4 assumes e(ρ_3^*) = ê_3, which the exact normalization of [11] does not satisfy; the corrected chain rule appears only in Remark 7.4. Please integrate the affine-e computation into the main proof of the Lipschitz bounds and state Theorem 7.3 under the true normalization, so that the formal proof does not rest on a provisional assumption.","section":"Sections 6.2 and 7 / Remark 7.4"},{"comment":"The Newton–Kantorovich certificates for δ_KS, L_D, L_E, and the persistence of the inequality constraints are delegated to certify.py and the Maple worksheet. For archival and reproducibility, include the scripts or a detailed verification log as supplementary material, and state explicitly in the main text that the output has been generated by running them to completion.","section":"Section 7 and Theorem 7.2"},{"comment":"The statement that the margin is 'greater than 10^10' is informal. Please replace it with the formal certified inequalities, e.g. L_D δ_0 ≤ 1.4×10^{-8} < 6609 and L_E δ_0 ≤ 1.3×10^{-8} < 1407, which are the actual quantitative bounds.","section":"Theorem 6.5"}],"recommendation":"minor_revision","confidential_remarks":"The paper's conclusion depends crucially on the correctness of the ancillary scripts, which I did not execute. I recommend that the editor request a reproducibility archive, ideally with a versioned DOI, and a short transcript of the key certified inequalities. The mathematical structure is sound and the presentation is generally clear; the requested changes are local and do not affect the central derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Pairwise, this paper earns its claims. The new content is the finite-dimensional reduction of the two admissibility criteria for piecewise constant fan subsolutions—the formulas D = 1/2 K + I and E = -1/2 K + I, the phase diagram, and the explicit certified evaluation showing that for the Krupa–Székelyhidi fan both the entropy-rate and least-action criteria prefer the convex-integration family to the planar contact discontinuity. That is a concrete, reusable result, and it is not fitted: the fan parameters come from [11], and the sign inequalities are strict with enormous margins.\n\nThe paper does well in being precise about what it does not prove. Section 6 explicitly separates the conditional transfer to the exact fan from the existence argument in [11], and Section 7 describes the exact-arithmetic certification in enough detail that an independent reader can rerun it. The Maple worksheet and certify.py scripts are readily available. That is the right way to handle computer-assisted parts.\n\nSoft spots, in proportion. The strongest claim—that every convex-integration realization generated by the KS fan is preferred to the classical solution—rests on the pointwise identities ρ = ρ̄ and |v|² = C_i a.e., which are imported from [11] and [13] and not re-derived here. That is a legitimate citation, not a circular step, but it makes the 'every realization' wording conditional on the cited construction being exactly what the author assumes. I would prefer the author to flag that dependency more prominently. The Newton–Kantorovich transfer itself is the other soft spot: I did not execute certify.py, so I am relying on the description plus the large margins. The margins are so large that a bug capable of flipping the sign is implausible, but independent execution is the natural check. The specially constructed pressure law is artificial, which the author acknowledges in the introduction; it is standard for this nonuniqueness program.\n\nOn balance, the central argument holds up. The decomposition theorem is elementary and correct; the application is new and convincing. This paper is for people working on selection criteria and nonuniqueness for the multidimensional Euler equations, and it deserves a serious referee. I would send it to peer review with minor revisions.","headline":"A clean, honest computation: the K/I decomposition is new and the certified KS-fan evaluation is convincing, with the main caveats being the cited convex-integration identities and the unexecuted exact-arithmetic certificate.","tokens_in":16116,"tokens_out":3028,"would_cite":true,"duration_ms":366371,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","35L65","76N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that for piecewise constant fan subsolutions, action-rate and entropy-rate admissibility coefficients both reduce to a single kinetic/internal-energy decomposition, and that for a specific contact-discontinuity Riemann fan","keywords":["isentropic Euler equations","convex integration","fan subsolution","entropy rate criterion","least action principle","contact discontinuity","admissibility criteria","Riemann problem"],"falsifier":"Evaluate D and E at the exact fan parameters with an independent exact-arithmetic implementation: if either D is not positive or E is not negative, the central conclusion fails. Because the certified margins are large (D > 6609, E < -1407, correction bound 2.45e-13), a counterexample would have to come from a certified error in the auxiliary verification or from a violation of the |v|^2 = C_i identity on a set of positive measure.","tokens_in":15289,"feed_emoji":"🌀","tokens_out":5795,"duration_ms":51880,"temperature":0.7,"pith_summary":"The paper's central claim is that two rival selection criteria for weak solutions of the isentropic Euler equations—the entropy rate criterion and the action rate criterion—are governed by the same pair of numbers whenever the solutions come from a piecewise constant fan subsolution. For such fans the paper derives explicit formulas and proves the reduction D = K/2 + I and E = -K/2 + I, where K is a weighted kinetic-energy mismatch and I a weighted internal-energy mismatch; the criteria therefore agree exactly when |I| < |K|/2. The paper then applies this to a fan constructed numerically for Riemann data whose classical solution is a planar contact discontinuity. Certified exact rational computations show D > 0 and E < 0 at the exact fan, so both criteria prefer every convex integration realization generated by the fan to the classical contact discontinuity. This matters because it says the standard selection criteria do not restore uniqueness for this Riemann problem; they point instead toward the nonunique multidimensional solutions.","feed_headline":"Both criteria rank convex-integration fans above the contact wave","feed_subtitle":"Exact rational arithmetic shows both action-rate and entropy-rate tests favor the nonunique fan solutions.","key_machinery":"The central object is a piecewise constant fan subsolution: a self-similar partition of spacetime into conical regions on which the coarse density, velocity, and relaxed stress are constant and satisfy a strict convex-integration subsolution condition; each convex integration realization has density equal to the fan density and |v|^2 equal to a prescribed constant in each intermediate region. The load-bearing identities are D = K/2 + I and E = -K/2 + I, where K represents the weighted kinetic-energy mismatch and I the weighted internal-energy mismatch. These reduce the comparison of the two admissibility criteria to the location of the pair (K, I) relative to the lines I = ±K/2. The paper al","core_discovery":"The core discovery is that all convex integration realizations belonging to the same admissible piecewise constant fan subsolution are identical from the standpoint of both criteria: their action difference and energy-rate difference relative to any common piecewise constant self-similar reference solution depend only on the fan, not on the internal oscillations inserted by convex integration. The action and entropy coefficients are computed in closed form as sums over the overlap of fan and reference regions, and are decomposed into kinetic and internal mismatches. For the specific planar-contact-discontinuity fan, exact arithmetic shows the action coefficient is positive and the entropy co","pith_inferences":["If the K/I reduction extends beyond piecewise constant fans to more general subsolutions, the (K, I) plane could become a general diagnostic for when dissipation-based selection criteria agree; this extension is not proved in the paper.","The result suggests that scalar dissipation or action rates may be structurally blind to the difference between a one-dimensional contact and genuinely multidimensional oscillations, since both carry the same macroscopic energy profile; that reading goes beyond the paper.","The certification pipeline—exact rational coefficient evaluation plus interval/Newton–Kantorovich transfer—could be applied to other numerically constructed fans, for example the two-shock fans where the paper predicts the criteria disagree, to test the characterization."],"forward_implications":["Any two convex integration realizations generated by the same fan are indistinguishable to both criteria: selection, if it happens, is determined by the fan, not by the realization.","The two criteria agree precisely when the weighted kinetic mismatch dominates the internal mismatch; outside that sector they select opposite solutions.","For the contact-discontinuity Riemann data, neither the entropy rate criterion nor the action criterion selects the classical solution—both select the convex integration family.","The explicit finite-dimensional formulas turn the admissibility comparison into a checkable algebraic condition, so the same computation can be rerun for any other fan subsolution."],"fun_headline_variants":["Entropy and action rates depend only on the fan","Entropy and action both favor fan solutions over contact wave","Fan subsolutions: entropy and action rates are fan-only","Exact arithmetic shows both criteria prefer fan solutions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The sign transfer from rational approximations to the exact fan assumes the Newton–Kantorovich certificates are correct and that every convex integration realization of the fan satisfies density equal to the fan density and |v|^2 equal to the prescribed constant almost everywhere in each fan region; if the constructed solutions only approximate these identities, the signs of the coefficients could change.","fun_headline_variants_meta":{"raw":{"variants":["Entropy and action rates depend only on the fan","Entropy and action both favor fan solutions over contact wave","Fan subsolutions: entropy and action rates are fan-only","Exact arithmetic shows both criteria prefer fan solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1401,"prompt_tokens":620,"completion_tokens":781,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":364,"completion_tokens_details":{"reasoning_tokens":717}},"tokens_in":364,"tokens_out":781,"duration_ms":7010,"temperature":1.0,"reasoning_tokens":717,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:25:10.159717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate D and E at the exact fan parameters with an independent exact-arithmetic implementation: if either D is not positive or E is not negative, the central conclusion fails. Because the certified margins are large (D > 6609, E < -1407, correction bound 2.45e-13), a counterexample would have to come from a certified error in the auxiliary verification or from a violation of the |v|^2 = C_i identity on a set of positive measure.","supporting_citations":[],"review_version":1}