# Verification report

On (0,1) with Dirichlet boundary conditions, V_t(x)=t x has M_2(V_t)-M_3(V_t)>t^2/20000 for every 0<t<=1. Every real bounded V with ||V-x||_infinity<=10^-5 has M_2(V)-M_3(V)>1/40000. Thus x+10^-5 x^2 is an explicit smooth strictly convex counterexample to Lavine's potential-mean monotonicity question.

# Originating-system mathematical audit

1. The domain(0,1) is bounded and convex. V=x is convex, and V=x+10^-5 x^2 is smooth, positive in the open interval and strictly convex. The source imposes no symmetry; an affine counterexample is already admissible.
2. For bounded real potentials the one-dimensional Dirichlet spectrum is simple, and eigenfunctions are normalized in L2. Expectations are of the same V defining the eigenfunctions, not unperturbed eigenfunctions or eigenvalue derivatives with a missing factor t.
3. Centering W=x-1/2 gives norm1/2. At n2 andn3 the free spectral gap is at least3pi^2. The fixed Riesz contour of radius3pi^2/2 is in the resolvent for all complex |z|<3pi^2. Rankone is preserved throughout this disk. Thus there is one analytic eigenvalue, not a choice of potentially colliding branches.
4. Normal resolvent spectral inclusion and isolation of the contour imply |lambda_n(z)-E_n|<=|z|/2. It is not an unjustified eigenvalue Lipschitz estimate for general nonnormal operators. Reflection gives exact evenness. Cauchy on radius27 is valid on a larger analytic disk.
5. The displayed v solves the differentiated ODE, has both zero endpoint values for a=n pi, and is orthogonal to phi. Symbolic integration verifies the coefficient for symbolic positive integer n; it is (a^2-15)/(48a^4), consistent with the sum calculation in the credited literature. It is a Taylor coefficient, not lambda'' itself.
6. Hellmann-Feynman gives M_n(tx)=t/2+t lambda'_n(t). Therefore the second-minus-third leading term is C t^2. Summing both derivative tails introduces factor54. The exact rational tail1457/7154784 is for all orders, not a truncated computation. For0<t<=1 its contribution is at most R t^4.
7. The bound49/5<pi^2<10 gives C>263/1036800. Subtraction yields429503/8585740800>1/20000. All reported inequalities are checked exactly using fractions; optional numerical results are corroborative only.
8. For V0=x, min-max gives |mu_j-E_j|<=1, so the gap from n2 or3 to every other eigenvalue exceeds25. Bounded perturbation moves each ordered eigenvalue by at mostdelta. On the orthogonal complement of the unperturbed eigenfunction, the resolvent bound is1/(25-delta). Projecting the perturbed eigenvalue equation gives the claimed eigenvector estimate with aligned signs.
9. The potential itself also changes, contributingdelta to each expectation. The remaining term is at most2||V0|| ||w-u||. The two expectation errors total at most2delta+8delta/(25-delta), not merely an eigenfunction error. At delta10^-5 this is2899999/124999950000<1/40000. The strictly convex polynomial lies in this ball in essential supremum norm.
10. The theorem refutes the all-index sequence, not the n1-to-n2 comparison. It does not challenge the first total-energy gap theorem. There is one exact source-subquestion closure, not several solved bundled questions. Known Stark coefficients and standard tools are credited. Novelty uncertainty is separate from proof validity.

No gap found in the complete written argument. This is a self-audit, not independent peer review or machine formalization.

# Bounded literature review,25October2026

Searches: Lavine potential means; Lavine potential increasing sequence; Lavine potential counterexample eigenvalues; potential means Schrodinger; potential energy expectation convex eigenstates monotonic; potential energy expectation monotonicity eigenvalues; loweigenvalues Lavine monotonicity; infinite square well Stark monotonic; Quantum mechanical sum rules for two model systems; The eigenvalue gap for one-dimensional convex potentials.

Primary sources read: MFO2009/06 report, AIM Low Eigenvalues problem list, Belloni-Robinett2008 arXiv0802.2217 and its DOI landing metadata. Lavine1994 author's accessible text concerns the first two total energies and comments on their potential expectations; it is not used to claim a higher-index monotonicity theorem. Fundamental-gap literature is context, not a proof of the target claim. Secondary search hits and their crawl dates are not mathematical authority. The classical Stark coefficient is a known ingredient and no discovery credit is claimed for it. The small-coupling sign already follows from that coefficient and elementary perturbation theory. The additional contribution here is the explicit t=1 certificate, a uniform range0<t<=1 and a quantitative perturbation neighborhood giving a fixed strictly convex polynomial.

No direct published answer to the precise Lavine question or this quantitative neighborhood was located in the bounded search. This is not an exhaustive novelty certificate and does not rule out an unindexed or unpublished observation. The manuscript makes no absolute-priority claim.

All four final pages were visually inspected. No overflow, missing symbols or unresolved citations were found. The built-in compiler and Tectonic succeeded. Author name is prominent; affiliation/contact is a footnote; AI assistance appears in prose.