{"id":"b1218a78-10b0-4d6d-badc-541ce851db2c","arxiv_id":"2607.21024","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bounded entire viscosity solutions of Hessian inclusion equations are constant precisely when the admissible set is Liouville admissible.","lead":"This paper proves that certain degenerate Hessian equations have constant bounded entire solutions exactly when their defining admissible set satisfies a recursive geometric condition. The result gives a unified framework that recovers known Liouville theorems for k-Hessian and LYZ-type equations and produces new anisotropic examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 2.2's supersolution clause says A^c, but (P1), (P6), and (P9) all operate with Q\\int A; under the literal reading the converse's smooth pullback in Prop. 7.3 is not a supersolution, so the iff statement is ill-posed.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing issue: the supersolution clause in Definition 2.2 is inconsistently written as A^c while the proofs use Q\\int A. I agree with that assessment rather than the more drastic alternatives. The mathematical architecture—recursive quotient reduction, induction on dimension, mollification and comparison arguments—appears coherent and the main idea does not seem to depend on unpublished black boxes in a way that affects the theorem's core. The definitional ambiguity is not a mere cosmetic typo, because the converse direction constructs a smooth solution whose Hessian lies on ∂A, and whether that construction is a supersolution depends precisely on whether boundary Hessians are allowed. Under the literal A^c reading, the smooth pullback is not a supersolution and Proposition 7.3 collapses. Under the intended Q\\int A reading, the construction works because a smooth function with Hessian in ∂A satisfies the supersolution test: any lower test function with Hessian in int A would force the function's Hessian into int A by adding a positive semidefinite term, contradicting ∂A. Since the correction is straightforward and the rest of the proof is consistent with it, the appropriate verdict remains conditional: the paper should be accepted only after Definition 2.2 is amended and the affected properties are re-verified. This does not change the reader's CONDITIONAL verdict, hence UNCHANGED.","tokens_in":26446,"tokens_out":18080,"duration_ms":200750,"concrete_test":"Adopt the corrected supersolution definition Hess φ ∈ Q\\int A and re-run the main proof chain (P1), (P6), (P9), Proposition 7.3. For a concrete check, take A=Q+(R), ∂A={0}, and verify that the constant function u≡0 is a viscosity supersolution under the corrected definition but fails under the literal A^c definition (φ≡0 gives Hess φ=0∈A). Then, for a non-Liouville set from Example 2.4, check explicitly that the smooth pullback constructed in Proposition 7.3 satisfies the supersolution condition pointwise and is therefore a genuine solution. If either check fails, the theorem needs a revised formulation; if both pass, only the wording of Definition 2.2 needs correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is an iff between Liouville admissibility and constancy of every bounded C^{0,α} viscosity solution of Hess u ∈ ∂A. That theorem is only meaningful if 'viscosity solution' is unambiguously defined. Definition 2.2 says supersolutions require test functions touching from below to have Hessian in A^c. But (P1) proves a C² function with Hessian in Σ=∂A is a supersolution by showing Hess φ ∉ int A; this is the condition Q\\int A, not Q\\A. Likewise, the comparison argument in (P6) and the restriction argument in (P9) rely on the same Q\\int A condition: (P9) concludes that a pullback with Hessian in ∂(A/N) is a supersolution of the quotient. The key converse, Proposition 7.3, produces a smooth bounded nonconstant function q*_N[u]_ε whose Hessian lies in ∂A. Under the literal A^c reading, this function is not a supersolution: taking φ=q*_N[u]_ε itself, Hess φ ∈ ∂A⊂A, so Hess φ ∉ A^c. Thus the constructed function would not be a viscosity solution and the converse fails as written. The intended definition is evidently Q\\int A, which is also the closed, stable condition needed for (P2) and for the comparison principle. But unless the text is corrected and the proofs re-verified with this definition, the main theorem's statement is ambiguous at a point on which both directions depend.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies entire viscosity solutions of translation-invariant Hessian inclusions Hess_F u ∈ ∂A, where A is a closed convex elliptic subset of the space of symmetric/Hermitian forms. It introduces quotient reductions A/N and defines a set A to be Liouville admissible if every quotient is either boundary-compatible or contained in a terminal set. Theorem 2.9 asserts that A is Liouville admissible if and only if every bounded globally C^{0,α} entire viscosity solution of Hess_F u ∈ ∂A is constant. The forward direction is proved by dimension induction (Prop. 7.1, Thm. 7.2); the converse is proved by constructing a nonconstant smooth pullback from a nonterminal quotient (Prop. 7.3). Spectral admissible sets are shown to be Liouville admissible (Thm. 3.12), and polarizations of univariate Gårding polynomials satisfying the monotone root sequence condition are used to produce classes of examples, including k-Hessian cones and Fu–Yau–Zhang's σ_{n-1}+σ_n equation. Appendices discuss maximal spectral terminal cones and anisotropic examples.","tokens_in":26784,"tokens_out":17798,"duration_ms":186073,"significance":"The proposed geometric characterization is attractive and, if the technical inconsistencies are repaired, would provide a genuine unification of known Liouville theorems (Dinew–Kołodziej, Székelyhidi, Fu–Yau–Zhang) and a systematic source of new anisotropic examples. The paper is transparent about dependencies: the Gårding polarization input is quoted from the authors' preprints [13,14], but the main equivalence is proved from viscosity theory and does not fit the examples to the theorem; there are no fitted parameters. The proof structure is coherent and substantial. However, the manuscript's central definition of viscosity solution is internally inconsistent, and the Gårding cone used for the examples is left open/closed ambiguous; these issues must be fixed before the main theorem is acceptable. No circularity was detected.","major_comments":[{"comment":"Definition 2.2's supersolution clause (A^c) conflicts with every proof in the paper. (P1) proves a C² function w with Hess w∈Σ is a supersolution by showing Hess φ∉int A, i.e. the condition Q(V)\\int A; literally, taking φ=w gives Hess φ∈∂A⊂A, not A^c. (P6) and (P9) also use Q(V)\\A. The converse (Prop. 7.3) constructs q_N^*[u]_ε, a smooth bounded function with Hessian in Σ; under the literal A^c definition this is not a supersolution, so the 'only if' direction fails. The intended condition is evidently Q(V)\\int A; Definition 2.2 and the comparison/restriction proofs must be corrected and re-verified with that definition. This ambiguity affects both directions of Theorem 2.9.","section":"Definition 2.2; (P1), (P6), (P9), Prop. 7.3"},{"comment":"The Gårding component C_g from Theorem 4.3 is the unique connected component of {g>0}, hence open, but Theorem 4.4 asserts A_g={λ:λ∈C_g} is closed, as required by Definition 2.1, and the proof uses 'C_g is closed' and '0∈∂C_g'. The inequalities in Remark 4.5 are strict. Example 4.6 silently switches to the closed k-Hessian cone. The authors should define A_g as the spectral lift of the closure of C_g, prove that the closure is admissible, and state the boundary equation with the corresponding non-strict inequalities; otherwise Theorem 1.3 is not formulated for admissible sets.","section":"Section 4, Theorem 4.4 and Remark 4.5"}],"minor_comments":[{"comment":"The notation A^c should be defined explicitly as the complement in Q(V), and the intended supersolution condition (Q(V)\\int A vs. Q(V)\\A) should be stated unambiguously.","section":"Definition 2.2"},{"comment":"The example is only piecewise C²; the text asserts u_{11}≡0 and concludes D²u∈Σ without verifying the viscosity conditions at |x'|=1. A short viscosity verification or a smooth approximation would make the example rigorous.","section":"Example 2.4"},{"comment":"The sentence 'because w is subharmonic, [w]_ε ≥ w' should specify that subharmonicity is with respect to the metric g and follows from (A2) and (A4).","section":"Proposition 6.1, (P5)"},{"comment":"The relationship between the metric g used for spectrality and the metric h in (A4) deserves a clearer statement; the proof appears to use h=g implicitly.","section":"Theorem 3.12 and Section 3.4"},{"comment":"The same symbol Γ^+_k is used for both the open set {σ_1>0,...,σ_k>0} and the closed cone {σ_1≥0,...,σ_k≥0}; please distinguish these and make clear which is the admissible set.","section":"Section 4, Example 4.6"}],"recommendation":"major_revision","confidential_remarks":"The definitional inconsistency is the main obstacle; it is repairable without changing the architecture of the proof. The paper relies on two unpublished preprints [13,14] for the Gårding polarization; an editor may wish to have those checked. I found no circularity or fitted parameters. If the definitional and open/closed-cone issues are fixed, the paper would be a solid contribution to the Liouville-theory literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the stress-test note is correct. Definition 2.2 defines supersolutions using A^c, but (P1), (P6), and (P9) all operate with Q \\ int A, and (P6) even writes Q \\ A. Taken literally, a smooth function with Hessian in ∂A is not a supersolution—the test function can be the function itself. That breaks (P1) and, more importantly, the converse construction in Proposition 7.3. The intended definition is plainly Q \\ int A, and most of the proofs already use that, so this is a repairable definitional bug, not a dead end. But until it is fixed and the proofs rechecked under that definition, the statement of Theorem 2.9 is ambiguous at the exact point both directions rely on.\n\nWhat is genuinely good: Liouville admissibility—the recursive quotient condition—is a new and natural packaging of rigidity for translation-invariant Hessian inclusions. The forward direction via dimension induction and the converse via quotient pullback form a coherent strategy. Theorem 3.12, showing spectral admissible sets are automatically Liouville admissible, is clean and strong. The Gårding polynomial construction gives real new examples, including mixed elementary-symmetric sets and anisotropic constructions, and recovers Dinew–Kołodziej and Fu–Yau–Zhang as special cases. The reliance on the authors' own preprints [13,14] is a dependency, not circularity, though a referee will need to see those preprints.\n\nSofter spots: the open/closed issue for the Gårding cone C_g is under-addressed—Theorem 4.4 treats it as closed after defining it via strict inequalities. That is a minor notation/closure fix. The proof also leans on several black-box results from unpublished work, which is acceptable if the preprints check out.\n\nBottom line: this is worth a serious referee. The correction to Definition 2.2 should be required, and the proofs re-verified with Q \\ int A. The main idea is sound and the contribution is substantial; the current version just needs to be made internally consistent.","headline":"The central iff theorem is new and the overall strategy is sound, but Definition 2.2's supersolution condition is inconsistent with the proofs and must be fixed before the main statement is trustworthy.","tokens_in":27294,"tokens_out":4828,"would_cite":true,"duration_ms":55608,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B53","35J60","35D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Liouville property for Hessian inclusions is characterized by a recursive geometric condition on the admissible set.","keywords":["Liouville theorem","Hessian equations","viscosity solutions","admissible sets","quotient reduction","terminal sets","Gårding polynomials","monotone root sequence"],"falsifier":"Directly check the function $u$ in Example 2.4. At a point on the boundary $\\{|x'|=1\\}$ where $u = -1$, take a test function $\\phi$ touching $u$ from below with $\\text{Hess} \\phi$ having $11$-entry $0$ and $(22+33+44)$-entry $0$, so $\\text{Hess} \\phi \\in \\partial A$. Under the literal Definition 2.2 such $\\phi$ is forbidden because $\\text{Hess} \\phi \\notin A^c$, so $u$ is not a supersolution and the example collapses; under the $Q \\setminus \\mathrm{int} A$ reading, $\\text{Hess} \\phi \\notin \\mathrm{int} A$ holds and $u$ qualifies. This single check decides whether the intended viscosity notion — and hence the main theorem — is the weak one used in the proofs.","tokens_in":26284,"feed_emoji":"📐","tokens_out":7594,"duration_ms":74499,"temperature":0.7,"texified_at":"2026-08-05T21:37:46.213455+00:00","pith_summary":"The paper asserts that, for translation-invariant real and complex Hessian equations, whether every bounded Hölder-continuous entire viscosity solution of $Hess_F u \\in \\partial A$ must be constant is fully determined by the geometry of the admissible set $A$, not by the algebraic form of the boundary equation. It introduces Liouville admissible sets: after quotienting by any proper linear subspace, the reduced boundary set must either reproduce the boundary of the reduced admissible set or collapse into a terminal set, where bounded-above subsolutions are already constant. The main theorem proves this geometric condition is equivalent to the analytic Liouville property. As a consequence, spectral (rotation-invariant) admissible sets are automatically Liouville admissible, and a large family of examples arises by polarizing univariate polynomials with monotone root sequences, recovering standard k-Hessian equations and mixed equations such as $\\sigma_{n-1}+\\sigma_n=0$. The upshot is that a classic rigidity question in PDEs becomes a checkable condition on the defining set.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8147,"prompt_tokens":897,"completion_tokens":7250,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":897,"completion_tokens_details":{"reasoning_tokens":6337}},"feed_headline":"Hessian rigidity reduced to quotient geometry of admissible sets","feed_subtitle":"Bounded Hölder entire solutions of Hess_F u ∈ ∂A are constant exactly when A is Liouville admissible.","key_machinery":"Quotient reduction of an admissible set: for a linear subspace $N$, $A/N$ is the set of forms $B$ on $V/N$ whose pullback $q_N^*B$ belongs to $A$; this transports the inclusion problem down to a lower-dimensional space. The boundary dichotomy (Proposition 3.8) says that for any proper quotient, either $\\partial(A/N)=\\Sigma/N$ (the boundary structure persists) or $\\Sigma/N=A/N$, in which case the quotient ceases to be a genuine boundary problem. A terminal set is any admissible set for which every continuous bounded-above viscosity subsolution of $Hess_F u \\in T$ is constant, with the positive semidefinite cone as the basic example and the real logarithmic cone as a larger real example. Liouville admissibility is the recursive r","core_discovery":"The central claim (Theorem 2.9) is the equivalence: an admissible set $A \\subset Q(V)$ is Liouville admissible if and only if every bounded, globally $C^{0,\\alpha}$ entire viscosity solution of the boundary inclusion $Hess_F u \\in \\partial A$ is constant, for any $0 < \\alpha \\leq 1$. Here 'admissible' means a closed convex subset of the space of real symmetric (or Hermitian) forms that contains 0, is closed under adding positive semidefinite forms, and lies in a half-space of positive trace. A set is Liouville admissible if for every nonzero proper linear subspace $N$, the quotient set $A/N$ either has boundary coinciding with the reduced boundary ($\\partial(A/N) = \\Sigma/N$) or is contained in a terminal set — a set for which every bounded-abov","pith_inferences":["The same recursive quotient-dichotomy may apply to other translation-invariant subequation inclusions beyond Hessian ones, as long as the equation class admits a restriction/quotient calculus and a terminal-set classification; the paper's argument uses only convexity, ellipticity, and mollification, not the special form of the Hessian.","The viscosity-solution wording in Definition 2.2 appears inconsistent with the proofs: supersolutions are actually used with the condition Hess φ ∉ int A (i.e., Q \\ int A), not the literal Hess φ ∈ A^c = Q \\ A. Under the literal complement, a smooth function with Hessian exactly on ∂A would not be a supersolution, and the converse construction in Proposition 7.3 would not produce a viscosity solut","The identification of maximal spectral terminal sets (positive semidefinite cone in the complex case, the real logarithmic cone in the real case) suggests that the class of terminal sets — and therefore of Liouville admissible sets — may be larger than the paper's examples; any sharp characterization of terminal sets in the non-spectral case would tighten the condition.","For non-spectral admissible sets built as finite intersections of pullbacks of spectral sets (as in Appendix B), the theorem predicts exact rigidity; one could attempt to construct explicit nonconstant bounded solutions when one of the terminal-containment hypotheses is dropped, which would delineate the boundary of the condition."],"forward_implications":["If the main theorem is correct, then verifying a Liouville theorem for any translation-invariant Hessian inclusion reduces to checking a recursive, purely linear-algebraic condition on quotient sets.","Every spectral (orthogonally or unitarily invariant) admissible set — including the cones for the Laplace, k-Hessian, and Monge–Ampère equations — is automatically Liouville admissible, so rigidity holds for the associated boundary equation.","The framework recovers earlier Liouville theorems for complex k-Hessian equations and for the critical LYZ-type equation σ_{n−1}+σ_n=0 without requiring gradient bounds beyond bounded C^{0,α} regularity.","Polarization of univariate Gårding polynomials satisfying the monotone root sequence condition yields a supply of new Liouville-rigid equations, including non-stable examples not obtainable from real-stable polynomials.","Liouville admissibility is closed under finite intersections and linear pullbacks, so anisotropic (non-spectral) rigid equations can be produced by combining multiple admissible sets."],"fun_headline_variants":["Constant solutions pinned to quotient geometry of admissible sets","Liouville property: a geometric condition on admissible sets","Bounded Hessian solutions are constant when set is Liouville admissible","Admissible set geometry decides if entire solutions are constant","Rigidity for Hessian equations: a recursion on quotient sets"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that a viscosity supersolution of the inclusion $Hess_F u \\in \\partial A$ is defined by requiring every test function's Hessian to lie outside the interior of $A$ ($Q \\setminus \\mathrm{int} A$); if one instead enforces the literal complement $A^c$ as written in Definition 2.2, then solutions touching the boundary cease to be supersolutions and both the example and the converse direction unravel.","fun_headline_variants_meta":{"raw":{"variants":["Constant solutions pinned to quotient geometry of admissible sets","Liouville property: a geometric condition on admissible sets","Bounded Hessian solutions are constant when set is Liouville admissible","Admissible set geometry decides if entire solutions are constant","Rigidity for Hessian equations: a recursion on quotient sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1095,"prompt_tokens":722,"completion_tokens":373,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":293}},"tokens_in":466,"tokens_out":373,"duration_ms":4559,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:44:55.596548+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly check the function $u$ in Example 2.4. At a point on the boundary $\\{|x'|=1\\}$ where $u = -1$, take a test function $\\phi$ touching $u$ from below with $\\text{Hess} \\phi$ having $11$-entry $0$ and $(22+33+44)$-entry $0$, so $\\text{Hess} \\phi \\in \\partial A$. Under the literal Definition 2.2 such $\\phi$ is forbidden because $\\text{Hess} \\phi \\notin A^c$, so $u$ is not a supersolution and the example collapses; under the $Q \\setminus \\mathrm{int} A$ reading, $\\text{Hess} \\phi \\notin \\mathrm{int} A$ holds and $u$ qualifies. This single check decides whether the intended viscosity notion — and hence the main theorem — is the weak one used in the proofs.","supporting_citations":[],"review_version":1}