HZ Henry Zweiman
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Internal proof review

Henry Zweiman. September 15, 2026.

This review was performed by the originating assistant, not by a separate agent or an outside mathematician. It records the proof obligations checked and the limits of the result. Formatting checks and source searches are separate from proof validation.

Dependency and sign checks

  1. Analytic branch. The Robin form has common H1 domain. The derivative of the normalized eigenpair equation is invertible because the first eigenvalue is simple: pairing with u fixes the scalar derivative, normalization fixes the kernel component, and Fredholm inversion fixes the orthogonal component. Positive normalization patches the local branches over the whole real line. The argument needs only local analytic expansions, not one convergent power series on all of R.
  2. Gauge. With nu_i dot gamma=-1, w=exp(-alpha gamma dot x)u has zero normal derivative on both rays. Direct differentiation gives Delta w=-2 alpha gamma dot grad w-(lambda+alpha^2|gamma|^2)w. Both signs were checked.
  3. Published regularity input. Dauge's author-posted Theorem 8.1 applies to compactly supported homogeneous-Neumann weak solutions on R times a sector. At k=0 its criterion is 0<2-2/p<pi/omega. Multiplying the planar solution by a compactly supported smooth function of an extra coordinate proves the planar form used in Lemma 2.1. This reduction avoids a mixed condition at an artificial outer arc.
  4. Analytic H2 upgrade. The cutoff and its Laplacian are analytic in H1 and L2 respectively. The weak Neumann graph space is closed, and its inclusion into H2 is bounded by the closed graph theorem and Dauge's regularity result. Analytic dependence therefore transfers through a fixed bounded map. Analyticity in C2 or C1,1 is neither assumed nor asserted.
  5. Different p ranges. A p just above two, below 2/(2-beta) at an obtuse corner, gives C1 regularity. Separately, H2 gives grad w in every finite Lp in dimension two, and hence the forcing F in any finite Lp. The coefficient formula uses p above 2/(2-beta) only for F. Applying W2,p regularity to w at that larger p would be wrong; the manuscript explicitly avoids it.
  6. Corner positivity. For alpha>0, k=lambda+alpha^2|gamma|^2 is positive. The weighted equation is -div(rho grad w)=k rho w. Positivity on the outer closed arc follows from interior positivity and the ordinary Hopf lemma on the open edges. Testing with (m-w)+ makes the left side negative and the right side nonnegative. The resulting zero gradient and zero arc trace show w>=m at the vertex. No smooth-boundary Hopf lemma is applied at a corner.

Coefficient and obstruction checks

  1. Angular projection. The cosine mode has derivative zero at both ray angles, so angular integration gives a''+a'/r-beta^2 a/r^2=b with no missing boundary term. Its homogeneous solutions are r^beta and r^(-beta); the latter is excluded by H1. The particular solution in (3.6) has the displayed positive/negative order of its two integrals.
  2. Direct formula check. For b(r)=r^(kappa-2), kappa>beta, the particular solution is r^kappa/(kappa^2-beta^2). This agrees with direct application of the radial operator. For a=r^beta and b=0, (3.3) returns coefficient one. For an angular constant both projections vanish. These are hand checks of the formula, not numerical evidence for the theorem.
  3. Continuity. With area measure s ds dtheta, the singular weight is s^(-beta). It is in L^{p'} exactly when beta p'<2. Thus p>2/(2-beta) makes the coefficient functional continuous. The residual is O(r^(2-2/p)), strictly higher order than r^beta. The fixed arc trace is bounded on H2 on a slightly larger sector.
  4. Semiconcavity. In the interior, semiconcavity bounds the two Hessian eigenvalues above. The PDE bounds their sum below and above because the relevant function and gradient are bounded near the positive corner. Each eigenvalue is therefore bounded below as well. Convexity of the sector lets integration along segments extend the gradient Lipschitz bound to the vertex. Multiplication by the gauge preserves C1,1. Its zero gradient at the vertex makes the first cosine projection O(r^2), forcing c=0.
  5. Nonzero seed. Differentiating the normalization gives integral v=0, and testing the differentiated equation with one gives mu=perimeter/area. At a vertex, v+mu|x|^2/4-gamma dot x is harmonic Neumann. Only the first mode at an obtuse corner has exponent in (1,2). If all such coefficients vanished, the Fourier expansions and smooth open-edge regularity would give C2 on the closure, contradicting ACH Corollary 8.3. The differentiated forcing is constant, so c'(0) is exactly the nonzero first-mode coefficient.
  6. Quantifiers. A nontrivial real-analytic function on R has finitely many zeros in any compact interval. The proof gives a fixed vertex, a fixed polygon, and arbitrarily large bad parameters. It does not just give a polygon that changes with alpha. It does not show that all positive parameters are bad, or that only finitely many exceptional parameters exist on the unbounded half-line.
  7. Examples. The displayed parallelogram has side lengths 2 and sqrt(2), so the tangential-quadrilateral necessary condition fails; its angles are pi/4 and 3pi/4. Products with intervals preserve the Robin parameter on every face. The positive product is the ground state, and slice restriction transfers failure of log-concavity. These examples are part of the same theorem and paper.

Outcome

No remaining logical gap was identified in the stated theorem during this internal audit. The proof uses the established ACH first-variation rigidity and Dauge Neumann regularity theorem; it does not independently reprove the entirety of either source. The source audit names the exact statements inspected. No finite-element experiment, symbolic verifier, or artifact check is used as a substitute for a universal argument. Historical priority and the mathematical conclusion remain subject to expert review and correction.

September 15 revision: interval lifting

The Section 6 proof was checked directly. Failure of semiconcavity for a C2 function on a convex open set is exactly absence of a global upper Hessian bound. Bounded gradient and a fixed nonzero interval logarithmic derivative keep the correcting tangent component bounded. Its Hessian cost is finite for each fixed positive parameter, while the polygonal positive Hessian is unbounded. Taylor expansion is taken at one fixed interior point after the positive curvature has been obtained; no uniform Taylor remainder near the corner is assumed. Both endpoints strictly exceed their midpoint, proving nonconvexity of a superlevel set. The interval derivative is explicitly nonzero. The product Robin parameter is identical on every face. Higher-dimensional slicing retains the same midpoint witness. All analytic and regularity inputs are proved in Sections 2-4; no new PDE regularity theorem is assumed.

This is an originating-assistant audit, not an independent review. The full ACH Conjecture 2 remains unresolved.

Revision 1.2: weighted cone proof audit

  1. A radial cutoff preserves Neumann data and has L2 Laplacian; convex-domain H2 regularity therefore gives h = partial_e psi in local H1, including the vertex.
  2. Transverse concavity transfers by positive homogeneity only because all cone points have positive e height.
  3. Face differentiation gives H nu = a nu. Eliminating the mixed component yields (7.6) exactly. Its sign depends on the strict k < 1/sqrt(2) hypothesis.
  4. Codimension-two logarithmic cutoffs have vanishing H1 energy. They remove the skeleton from nonnegative tests, then pass the facet flux inequality to all compact tests by the H1 continuity of the form. Thus no boundary Hessian trace at an edge is assumed.
  5. The exponential test is integrable by the separately stated strict-dual condition. Homogeneous annular norms give vanishing cutoff errors; q is nonnegative, so the resulting integral identity forces it to vanish.
  6. A negative-semidefinite transverse block with zero trace is zero. Transverse affinity, homogeneity and harmonicity then force psi = 0 for degree between one and two.
  7. The Robin corollary assumes a nonzero leading mode at an eligible cone. Its tangent correction uses a nonzero limiting derivative along gamma and is of order rho^(beta-1); the fixed midpoint witness then survives the Neumann perturbation.
  8. The triangular-cone example proves nonempty degree range variationally. Positive spanning gives the strict-dual property; bounded convergence gives a Rayleigh quotient tending to two; ACH's equality classification gives the strict lower bound on each pointed cone. No explicit epsilon or numerical eigenvalue is asserted.
  9. The exact matrix (7.16) shows the sign step fails in the remaining angle range. It is not a counterexample to the full conjecture.

This is an originating-assistant proof audit. The full conjecture, independent human review, and broad priority certification remain outside the established result.

Revision 2.0: full polyhedral quasiconcavity proof

The following is an originating-assistant hand-proof audit, not independent review. Earlier revision sections are historical and their full-conjecture limitations are superseded by Section 8.

No finite computations are used as proof. At revision 2.0, the unrestricted local cone statement and the sharp Robin gap remained open in this manuscript; the gap limitation is superseded by revision 3.0. Artifact validation is recorded separately in artifact-check.json.

Revision 3.0: positive gap degeneration and monotonicity

This is a separate audit by the originating assistant. The Section 9 argument does not depend on Sections 2--8.

  1. Geometry: The domain is the intersection of finitely many strict linear inequalities after expanding absolute values, so it is open, convex, bounded and full dimensional. Its slice is exactly epsilon K_(1-|t|). Slice integration and each facet graph factor reproduce V, S_Y and S_Z in (9.5). There is no missing facet at |X|=1: the Y width there is zero. The extra edges at s=0 or the kink at t=0 have zero surface measure.
  2. Barrier: Differentiation gives r'(s)=(sqrt26/3-sqrt2)/(2/3+s)^2, strictly between zero and one. The exact middle barrier is (3/14)(sqrt26-3sqrt2)>1/7. All rational comparisons are justified by squaring positive quantities.
  3. Poincare: The double-integral variance identity, segment Cauchy--Schwarz and the substitution z=(1-t)x+ty give the coefficient (3/2)L^2 in dimension three. Splitting at t=1/2 is essential; changing variables near t=1 would give a divergent bound. The estimate is proved directly, so no sharp Poincare theorem or unverified proof in an external source is required.
  4. Trace: The common inball gives x dot nu >=1/6, and the outer ball gives |x|<=3. The divergence formula yields 1440g^2 <=1600g^2 for mean-zero w. Surface area <=16 then gives |integral_boundary w|<=160g. These estimates are uniform even where the facet combinatorics change at s=0.
  5. Transverse bound: The cross term is at worst -320 beta |m|g. Completing the square with half the gradient energy costs 51200 beta^2 m^2. The remaining half controls beta r ||w||^2 for beta<=1/4096. Volume >=1/2 converts the remainder to 102400 beta^2 ||f||^2, safely below 110000. The scaled error is 110000 alpha^2, not an error of order 1/epsilon.
  6. Slicing and sign: Lateral measure is dt times transverse measure multiplied by one or sqrt(1+epsilon^2/2). These factors may be dropped only because alpha>0. Caps are dropped in the lower bound but retained in the upper bound. The derivative in t is ambient, so no coordinate-transformation metric or moving-average derivative is omitted.
  7. Upper trial: The layer width epsilon^(1/3) balances gradient cost ell^(-2) against ratio variation ell/epsilon. The derivative bound is 15/(2ell^2); the sloping correction is <=alpha epsilon; the single nonzero cap costs <=3alpha/ell. These yield A_alpha=15/2+5alpha.
  8. Localization: Subtracting alpha r0/epsilon from the lower and upper energy bounds controls integral(r-r0)u0^2. Its restriction to |t|<=1/2 has the positive barrier delta, giving exactly (9.20). No pointwise eigenfunction approximation is assumed.
  9. Gap: Simplicity makes the positive ground state even. The bounded odd cutoff gives exact orthogonality. Testing with u0 f^2 is legitimate in H1 and cancels the complete Robin term. The denominator is at least 1-M; the numerator is at most 4M. Subtracting two independently coarse eigenvalue estimates would not suffice, and is not used.
  10. Fixed diameter and dimension: The explicit constants are bounded on compact positive parameter intervals. Scaling changes alpha to a_epsilon alpha, with a_epsilon tending to D/2. The product construction has an eigenvalue obtained by exciting only the four-dimensional factor, so its fundamental gap is no larger. Diameters are then normalized exactly.
  11. Concrete example: At epsilon=10^(-30), alpha=1, 7(110000+25/2)10^(-10)<1/1000. Hence the domain gap is less than 4/999<1/100. Its diameter is in [2,21/10]. The interval first excited eigenvalue is >=pi^2/D^2>100/49 and its ground eigenvalue is <=2/D<=1. The comparator gap is therefore >51/49>1. These are hand estimates, not a numerical spectral experiment.
  12. Monotonicity: The same segment calculation in dimension four has coefficient 7D^2/3, giving a Neumann gap >1/20 on the explicit domain. Finite-dimensional Neumann trial spaces and the nonnegative Robin term prove continuity at zero by min--max. A small positive parameter therefore has gap >1/20, whereas parameter one has gap <1/100. This proves failure on the positive parameter axis itself.
  13. Quantifiers and priority: The universal all-dimensional conjecture is refuted by dimension four; the dimension-two and dimension-three versions are left open. Negative-Robin degeneration, known volume/surface geometry, rectangular-box results and the interval theorem are credited. The Smits attribution is explicitly via Laugesen because the original full text was unavailable.

No remaining logical gap was identified in this written argument. Independent expert review and an exhaustive priority determination are not claimed. The new proof is a universal variational argument, with an optional explicit instance; its validity does not rely on any finite computation or artifact check.

Revision 3.1: nested localization and spectral ratios

  1. Central barrier: On |t|<=3/4, s=1-|t|>=1/4. The exact excess is delta_*=(3/22)(sqrt(26)-3sqrt(2))>15/154>1/16. Subtracting the upper ground eigenvalue estimate from the transverse lower form bound yields (10.5). All smallness conditions are uniform on compact positive parameter intervals.
  2. Cutoff identity: The bounded Lipschitz cutoff eta and u_0 eta^2 are valid H1 multipliers/test functions. The complete Robin boundary term cancels in the ground-state identity. Applying the supported-function barrier to u_0 eta proves M(a)<=epsilon M(b)/(kappa (b-a)^2); no interface boundary condition or moving-fiber derivative is introduced.
  3. Exponential iteration: N intervals between 1/4 and 3/4 have width 1/(2N). Taking N=floor(sqrt(kappa)/(4sqrt(epsilon))) bounds each factor by 1/4. The floor costs a factor of four. The odd function of slope four gives gap <=16M/(1-M). The resulting constant 256/3 is below the stated 128. This is an upper bound, not a sharp asymptotic or matching lower bound.
  4. Explicit example: At alpha=1, epsilon=10^(-30), (25/2)epsilon^(1/3)+110000epsilon<1/32. Two cutoffs of width 1/4 yield M(1/4)<=(512epsilon)^2<1/2 and gap<=2^23 epsilon^2. Since E0>=1/epsilon, the relative gap is <=2^23 epsilon^3.
  5. Prior Dirichlet input: Andrews--Clutterbuck Corollary 1.4 gives the gap lower bound for convex domains, including the nonsmooth case by its stated approximation. The box of longitudinal length one and three transverse lengths epsilon/6 lies strictly within the domain: |x|+5|y|<epsilon/2 and |x|+|z|+epsilon|t|<2epsilon/3. Its first Dirichlet eigenvalue is pi^2(1+108/epsilon^2). Division by this upper bound and D^2<=5 gives relative gap >=3epsilon^2/545>epsilon^2/200. All inequality directions were checked.
  6. Fixed-domain limit: Monotone Robin forms give finite limiting eigenvalues bounded by the Dirichlet values. Compact H1-to-L2 embedding and trace continuity yield orthonormal zero-trace limits of the first j+1 eigenfunctions. Lower semicontinuity on each linear combination and Dirichlet min--max give the reverse eigenvalue inequality. The domain is fixed throughout this limit; no uniform rate in epsilon is assumed.
  7. Products and scaling: Added interval ground energies are positive and the product gap is at most the original gap. The inscribed product box bounds the Dirichlet ground energy by (105+d)pi^2/epsilon^2. Its diameter tends to two. Robin scaling changes alpha to a_epsilon alpha, in a compact positive interval, while the Dirichlet ratio is scale invariant. Thus the quantifiers for every D and starting alpha are justified.
  8. Distinct conjectures: The conclusion refutes Laugesen B, separately from A. It does not refute the theorem that alpha times the ratio increases, the ratio result on balls, or prove failure of concavity for the second eigenvalue itself. Dimensions two and three remain open here.

No remaining gap was identified in Section 10's written proof. This review is by the originating assistant; it is not an independent mathematical certification.

Revision 4.0: asymmetric end regions and second-eigenvalue concavity

  1. Convexity: A_h is affine, and the last transverse constraint imposes both affine branches of the concave tent L. The domain is a finite intersection of half-spaces, with nonempty interior. The peak t=1/3 gives L=2/3, so all unscaled fibers lie in the already audited K_s family after dilation by A in [1,2].
  2. Endpoint potentials: At the left end A-1 is O(h^2 tau); at the right end A-1=bh+O(h^2 tau). Dividing R_h-r0 by h gives k a_L tau and k a_R tau-b r0. The lower bound m a_q tau/4-b r0 holds on the whole rescaled half, preventing mass escape. The tilt b is fixed independently of alpha.
  3. Transverse control: beta<=1/8192 lets one retain one quarter of the transverse gradient energy. Dilation preserves this coefficient and the quadratic remainder. Physical rescaling gives transverse coefficient h^(-4), potential alpha(R-r0)/h, and remainder 110000 alpha^2 h^2. The common leading subtraction is affine in alpha.
  4. Facets and caps: Lateral graph factors are >=1 and 1+O(epsilon^2), with normals explicitly read from the domain inequalities. Lower bounds discard nonnegative exterior caps; upper trials retain their alpha h endpoint terms. Artificial cuts carry Neumann or Dirichlet conditions, without a Robin penalty. The correct limiting endpoint condition is Neumann.
  5. Fiber map: The explicit gauges are uniformly positive on the sphere and jointly Lipschitz. The radial map and inverse have uniform Lipschitz bounds; its tau derivative is O(h). The a.e. determinant is f^3 by homogeneity. The derivative matrix need not converge to identity; only bounded inverse derivatives and local uniform convergence of the Jacobian are used. Chain rules hold for bi-Lipschitz Sobolev changes of variables despite switching facets.
  6. Longitudinal derivative: The transformed derivative is partial_tau v-B dot grad_Y v, with B=O(h). The loss Ch|grad_Y v|^2 is absorbed by retained transverse energy. The longitudinal coefficient tends to one. No derivative of a moving-fiber mean is used.
  7. Compactness: Coercivity controls longitudinal derivatives, h^(-4) transverse derivatives, and the tau moment. Local Rellich compactness makes limits fiber constant, and the tail bound preserves global norms and orthogonality. No H1 extension by zero across a Neumann cut is assumed. A fixed potential shift makes the integrands nonnegative for lower-limit restriction; the shift is subtracted using norm convergence.
  8. Ordered spectrum: Upper trials are constant on actual fibers and compactly supported in the endpoint coordinate, hence admissible for either cut. Density and finite-dimensional polarization give upper bounds at every fixed index. Simultaneous compactness of the first j+1 eigenvectors and the limit min--max principle give lower bounds, including multiplicities. Cut bracketing then gives the original domain's ordered union limit.
  9. Strict witness: The limiting first two eigenvalues are the two ground branches on a neighborhood of alpha=1 because both first excited states are separated there. Their difference is C(alpha^(2/3)-alpha). The midpoint defect is C eta/6+O(eta^2)>0. Convergence at three fixed parameters transfers the strict inequality; differentiating an asymptotic expansion is unnecessary. The crossing is in the decoupled limit, not asserted for the connected domain's simple ground state.
  10. Products and diameter: The interval estimate 2alpha/epsilon-8alpha^2<=E0<=2alpha/epsilon follows from a direct one-dimensional mean/trace calculation. Its excited gap has order epsilon^(-2), above the O(h^(-2)) four-dimensional gap. Added ground terms have only a vanishing centered nonlinear remainder. Scaling one fixed domain transfers the midpoint witness with all three parameters divided by the same length factor.
  11. Scope: The result addresses constant positive Robin parameter and zero potential on full-dimensional convex polyhedra in d>=4. It leaves dimensions two and three and a quantitative h threshold open. It does not assert an actual crossing, a smooth-domain example, or failure of first-eigenvalue concavity. The qualitative convergence is proved here; no external Airy or thin-domain asymptotic is assumed.

No remaining gap was identified in the written Section 11 proof by the originating assistant. Independent expert review has not occurred.

Revision 4.1: Section 12 internal proof audit

The originating assistant found no remaining gap in this written section. This is an internal AI-assisted audit, not independent verification or expert review.

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