Internal proof review: convex-domain ground states

Henry Zweiman. September 16, 2026.

This is an internal audit by the same AI-assisted preparation process, not an independent referee report. No subagents or external reviewers were used for this revision.

Exact claim reviewed

For every nonempty bounded open convex domain in every dimension, the positive Dirichlet logarithmic Schrodinger ground state is unique. Every positive log-concave solution is this ground state. The sharp energy inequality holds on all of H^1_0 with equality precisely for its real multiples. Positive solutions of greater energy are not excluded.

Adversarial checks and their resolutions

The Hessian minimum need not be interior on a rough domain. The proof does not minimize on the original domain. It subtracts a positive level t and minimizes the logarithmic Hessian of U-t on {U>t}. These level domains have smooth uniformly convex boundaries compactly inside the original domain. Their smallest Hessian eigenvalue tends to infinity at the boundary.
Ordinary Hopf arguments can fail for a non-Lipschitz reaction at zero. No such Hopf theorem is used in the new section. The normal derivative on {U=t} is nonzero because it is a regular positive level of the smooth strictly log-concave U. The earlier ball proof retains its separate boundary argument.
The shifted reaction has an extra Hessian term. Both B_t' and B_t'' are retained in (8.6). Below U=1, both have the favorable sign. Above U=1, the minimizing point lies in a fixed compact set, where the original positive Hessian supplies a fixed lower bound delta and the error is O(t). Dividing by an uncontrolled lambda_t is avoided.
Strong minimum principles require the correct scalar inequality. Positive-semidefinite matrix order gives |He|^2 <= tr(H) e^THe. The scalar inequality has the supersolution sign, with a locally bounded nonnegative zeroth-order coefficient after enlarging it. The same argument applies to H-I after obtaining nonnegativity. A zero then forces an affine, respectively quadratic, restriction to a finite maximal line segment, contradicting boundary blow-up.
The curvature bound is only pointwise strict. No uniform global gap above I is claimed or used. Transport gives a strictly positive remainder separately on each compact image of a source ball.
Transport regularity cannot be imported from smooth domains with densities bounded away from zero at the boundary. Contraction first gives a Lipschitz map. The bounded target density and positive source density give det(DT)>=c on each source ball. With eigenvalues <=1 this yields local uniform ellipticity, an injective open map with image in the interior, and then interior Evans-Krylov and Schauder regularity. Only a full-measure image is needed.
The transport theorem has an erratum. The original Theorem 11 was read, the erratum's existence and DOI verified, and the precise contraction statement checked in the later primary paper of De Philippis and Figalli, Theorem 1.1, which explicitly cites both. The publisher did not provide the erratum's full text through the accessible page; this is recorded, rather than claiming to have read it. The proof uses the later verified formulation, not the original uncorrected proof alone.
Strict contraction must follow without a uniform curvature gap. If a smooth transport Hessian has eigenvalue one at any source point, the corresponding fixed directional second derivative attains a global maximum. Twice differentiating log det gives a nonpositive left side and a strictly positive right side because the target Hessian exceeds 2I at that image point. This is a local contradiction.
The logarithmic Sobolev constant is normalization-sensitive. The Gaussian is pi^(-N/2) exp(-|x|^2), whose coefficient for Ent(h^2) is one. The Ornstein-Uhlenbeck gradient decays as exp(-2s); integration of exp(-4s) yields 1/4 for positive F and coefficient one for h^2.
The ratio of two solutions need not have finite weighted energy. The ground-state identity and entropy difference are used only for compactly supported competitors. Their combination is an ordinary H^1-continuous energy functional, and the retained right-hand side is local. Strong H^1 approximation extends the inequality and its local remainders without splitting possibly divergent global quotient integrals.
Equality may be lost in approximation. It is not. Each fixed compact remainder passes to the limit. Zero total deficit forces a zero weak gradient of v/U on their full-measure union. Since v/U is H^1 locally, connectedness forces one constant across the whole domain.
Ground state means Nehari minimization in the source. The proof compares every nonzero Nehari competitor, not just critical solutions. There the action is half its squared L^2 norm. The same equality case identifies the two real minimizers and the unique positive one.
The result must not become full positive-solution uniqueness. The manuscript explicitly leaves positive higher-energy solutions open. Neither the new proof nor the prior shape branch supplies their exclusion.

Proof dependency chain

GMS log-concave existence -> positive interior Hessian -> positive-level regularization -> Gaussian-strength Hessian -> Caffarelli contraction -> local ellipticity and strict contraction -> local strict logarithmic Sobolev remainder -> compactly supported ground-state identity -> strong H^1 approximation -> unique Nehari minimizer.

Each arrow has an explicit argument or a named classical input in Section 8. No numerical experiment is used as proof. Source validation and PDF checks do not constitute a theorem verifier.

Internal assessment

No unresolved mathematical gap was identified in these checks. The broad ground-state question in GMS Remark 1.6 is answered by the manuscript's hand proof, subject to normal external mathematical scrutiny. This is a substantial extension from balls and nearby domains to all bounded convex domains and all dimensions; it is not a new numbered paper. Priority searches found no matching earlier bounded-convex ground-state uniqueness theorem, but they do not certify absolute novelty. The preprint remains unreviewed.

Historical review of revisions 1.0 and 1.1

Internal review of the convex-deformation theorem

Henry Zweiman. September 16, 2026. Originating-assistant review, not independent expert verification.

Exact question. The published P46 Section 7 asks whether the normalized square-root-log conclusion survives controlled deformations. The theorem constructs arbitrarily small smooth uniformly convex deformations where it fails. It does not rename the weaker small-rescaling concavity property as the question being disproved.
Cubic obstruction. Along a line, sqrt(w)=|t|sqrt(a+b t+O(t^2)). Its second derivative has opposite one-sided limits b/sqrt(a) and -b/sqrt(a). Both must be nonnegative for convexity. C4 regularity permits the differentiated remainder.
Tensor conclusion. Vanishing of every diagonal cubic value implies vanishing of the symmetric tensor by polarization. A nonzero cubic yields failure arbitrarily close to the maximum.
Ball input. The already proved estimate -D2 log U >= A I with A>1 is used, not inferred merely from strict log-concavity. The positive boundary slope and C2,gamma regularity come from the previous boundary argument, which addresses the non-Lipschitz nonlinearity.
Ground-state eigenvalue. L U=-2U exactly. The ground-state form identity makes this the simple first eigenvalue, rather than only a negative test quotient.
Potential domain. U is comparable to distance. The logarithmic potential is infinitesimally form-bounded by Hardy's inequality, so the form domain remains H01 and the resolvent is compact.
Poincare constant. The density is U squared, hence its potential Hessian is at least 2A. The constant is 1/(2A), giving L >= 2A-2 on the perpendicular complement.
Vanishing boundary density. The proof first uses smooth weighted Neumann problems on smaller concentric balls. The convex-boundary Bochner term has the correct nonnegative sign. Radius exhaustion and form approximation justify the desired weighted inequality.
Kernel exclusion. Testing a zero mode against U makes it perpendicular to U; positivity on that complement gives zero. No eigenvalue-count guess is used.
Holder-space invertibility. Division by a defining function loses one derivative. Multiplication by rho log rho is Holder of every exponent below one. The potential term is compact from C2,gamma Dirichlet functions to C0,gamma, so Fredholm index zero and kernel exclusion give an isomorphism.
Boundary nonlinearity. The implicit-function theorem is not applied to the scalar function at zero. Writing u=rho b with b uniformly positive makes 2rho b log rho + 2rho b log b an analytic map in the stated Banach spaces.
Genuine domains. The explicit radial graph is smooth, converges in every Ck norm, and has positive principal curvatures for small parameter. The radial cutoff gives a diffeomorphism identical near the old maximum.
Positive branch. The branch stays in the positive-Hopf cone. It solves the actual equation on the deformed domain, not a formal first-order residual equation.
Shape derivative. Its Eulerian boundary value is -Uprime(R)Y3, with the negative outward derivative from the ball. The material and Eulerian derivatives differ by DU dot V; V vanishes near zero.
Boundary regularity of the derivative. A nonzero boundary trace times the logarithmic potential can prevent C2 regularity at the boundary. The proof asserts only C1,gamma there and interior smoothness.
Angular equation. Degree three gives eigenvalue 3(N+1) on the unit sphere. Harmonic projection commutes with the radial linear operator. Homogeneous Dirichlet uniqueness removes every other mode.
Nonzero cubic coefficient. The regular radial equation after factoring r cubed is an integral equation in dimension N+6. Zero initial coefficient forces the zero solution and contradicts the nonzero boundary data. A negative-part test in the angular subspace gives its positive sign.
Actual maximum. The strict value gap away from the old center and a negative-definite Hessian nearby yield a unique global maximum. Its first derivative with respect to the shape parameter is zero because the degree-three shape derivative has zero gradient at zero.
Moving-center correction. The maximum moves by O(epsilon squared). Therefore the base fourth derivative does not cancel the order-epsilon cubic tensor. The calculation is performed at the moved maximum.
Logarithmic jet. At a critical point the third derivative of -log u is exactly -D3u/u. The e1 diagonal value has nonzero leading coefficient -6c/U(0).
Ordinary log-concavity. On an interior compact set it follows from the original strict bound and C2 convergence. A common boundary collar follows from uniform Hopf slope, positive curvature and C2,gamma bounds; the Schur complement handles the mixed terms.
Scope of uniqueness. Only the branch near the ball solution is classified by the implicit-function theorem. Global uniqueness of all positive solutions on arbitrary deformed domains is not asserted.
Priority distinction. GMS's final page-44 question is about balls and was answered by the previous revision. The present theorem answers P46's subsequent deformation question. Ishige-Salani-Takatsu's small-rescaling definition is explicitly distinguished. The weighted Poincare input is classical and credited.
Evidence boundary. The argument is a hand proof with no numerical PDE or finite parameter search as a dependency. MathJax, compilation, rendered pages and public byte checks establish artifact consistency, not independent correctness or novelty.

Internal conclusion: The proof supports the stated counterexample near every ball in every dimension at least two. Publish as a revision of P46 after final artifact checks. Do not claim an extraordinarily verified broad conjecture solution, a new paper count, or completion of the larger goal.

Review of the retained ball results

Internal proof and artifact review

Henry Zweiman. September 15, 2026.

This is the originating assistant's fresh audit of its own proof, not an independent referee report or human mathematical review. No subagents were used in this stage. Artifact results are recorded separately in artifact-check.json.

Proof audit

Normalization and center. Differentiating the radial equation gives z''+((N+1)/r-2rz)z'=2z(z-1), with z(0)=2 log U(0)/N. The center boundary term in the integrating factor vanishes because z'=O(r). The Volterra equation gives uniqueness near any positive center and legitimizes the needed even expansions without assuming the global series in advance.
Amplitude and signs. A central value between zero and the Gaussian threshold would force a positive decreasing z bounded above by its center value, contradicting blowup of the logarithm at a finite zero. Equality is the exact Gaussian. Above the threshold the same identity makes z strictly increasing. Radial symmetry is credited to the published non-Lipschitz result; it is not deduced from the concavity being proved.
Recurrence. The symmetric convolution factor is n+1, and the denominator is (n+1)(2n+N+2). Separating the two endpoint products for n>=1 proves coefficient positivity; n=0 is handled separately. Dropping the negative term yields c_n<=A^(n+1), giving a genuinely convergent local solution. The integrated series satisfies the original equation because its residual has zero derivative and zero central limit.
Global convergence. A positive series with a finite convergence radius is singular at the positive real endpoint. The manuscript proves this with monotone convergence and the Taylor series at the hypothetical regular endpoint. A smaller radius than R^2 would contradict analytic ordinary-ODE continuation of the actual positive solution. A larger radius would contradict its logarithmic blowup. Division by 2j preserves the second series' radius.
Convexity. Positive even coefficients express sqrt(w) as a norm of nonnegative convex monomials. The probability identity in (4.3) was recomputed: its right-hand side is 2 variance + mean*(mean-1). All moments are finite within the convergence radius. The center is a norm-type cusp, not a C2 point. The exponent threshold follows from the radial leading power r^(2 beta), so necessity is local and applies to every solution.
Boundary. Bounded continuous forcing gives finite U'(R) and U''(R). Assuming zero slope, energy dissipation and monotonicity of -U' near the boundary give v<=C U sqrt(log(1/U)). The resulting infinite travel-time integral contradicts finite R. This replaces an unqualified Hopf lemma with a complete proof valid when f'(0)=-infinity, including N=1.
Positive powers. The Hessian condition is alpha<=w''/w'^2. The ratio is positive, tends to infinity at the center and one at the boundary, and is strictly below one near the boundary. Its positive minimum is attained in the interior. Center and boundary extension are explicitly checked.
Parameter and product consequences. For n>=1 the c_n are nonnegative polynomials in delta=A-1 without constant term. Their scaled comparison yields the normalized-profile inequalities. The simple-zero asymptotic excludes equal first zeros for different centers. This uniqueness conclusion is credited prior work. The product formula solves the PDE by separation and its transformed concavity follows from the finite-dimensional norm argument.

Dependency and novelty audit

The externally used PDE results are existence and radial symmetry from Gallo--Mosconi--Squassina. The central IVP and analytic continuation facts are standard ODE results; the potentially delicate real-axis series singularity is proved in full. Neither finite computation nor symbolic tests are used as a substitute for any universal argument. No numerical experiment is needed by this manuscript.

The early note 0178 tentatively raised uniqueness as potentially new. The subsequent primary-source check found Ben Chrouda 2022 and corrected that interpretation before publication. The current June 2026 Liu--Sun--Zou v2 and current author version of the parabolic concavity paper were checked. The precise extent of the source inspection and its limits are recorded in ASSESSMENT.md and source-audit.json.

Original revision limits (historical)

The original revision was for balls, with the stated product corollary. It does not settle stronger concavity on arbitrary convex domains or compute the optimal positive-power exponent. The proof and assessment remain open to expert correction. PDF compilation, MathJax parsing and public hash equality certify artifact consistency, not mathematics or originality.