# Verification and scope

For every fixed n, alpha>0 and volume M, positive Robin eigenvalues satisfy rho_k>=a*k^(2/(2n-1)) on every bounded open convex domain and rho_k>=b*k^(1/n) on every bounded Lipschitz open set, with explicit positive constants depending only on n,alpha,M. Both exponents are sharp over the respective classes. The one-dimensional case is included. The general-domain half is explicitly credited as a consequence of van den Berg--Bucur (2014).

# Canonical source and scope

Frozen UnsolvedMath v1.6.0 record OWR-14297736-026, record 30005615,
commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e.
The immutable corpus and original source record are unchanged.
The full source is Geometric Spectral Theory, OWR2023/36,
doi:10.4171/owr/2023/36, printed pp2017--2106.

The record has merged two questions. Pedro Freitas's p2093 paragraph
asks broadly about analogous Polya inequalities for families of manifolds.
The next page separately states a conjecture for the positive Robin
Laplacian on Euclidean domains: fixed alpha,M>0 admit a,b>0 depending
only on alpha,M,n with rho_k>=a*k^(2/(2n-1)) for convex domains and
rho_k>=b*k^(1/n) for general domains, all k>=1, |Omega|=M.
Printed p2094 / PDF78 was visually inspected and agrees with this wording.

The accepted result proves this entire two-part Robin conjecture in the
standard bounded Lipschitz form setting (bounded open convex domains
automatically qualify). General sets may be disconnected. In dimension
one bounded Lipschitz sets are finite unions of intervals. No arbitrary
crack-trace convention or unbounded-domain extension is claimed.
The separate manifold question remains unaddressed, so this is one exact
source-question closure, but ZERO whole frozen-record closures.

# Detailed self-audit

1. Quantifiers and form: alpha>0 is fixed, M>0 is fixed, and constants
depend only on n,alpha,M. Robin means outward du/dnu+alpha*u=0 and
positive boundary energy on H1. Domains in the general statement are
bounded Lipschitz and may be disconnected. Convex domains are Lipschitz.
The separate manifold question is not marked solved.

2. Interval inequality: |v(x)|^2<=2|v(0)|^2+2x*int|v'|^2.
Integration gives 2*l*|v(0)|^2+l^2*int|v'|^2. Average the two endpoint
versions, obtaining l*(|v(0)|^2+|v(l)|^2)+l^2*int|v'|^2. In slicing,
endpoint integration is boundary integration with |nu dot e|<=1.
Thus ||u||^2<=max(w^2,w/alpha)*q, yielding MIN(w^-2,alpha/w), not max.
This proof extends by H1 density and the trace theorem.

3. Counting: cubes have side pi/(2 sqrt(n) sqrt(lambda)); intersections
are convex and have diameter<=pi/(2sqrt(lambda)). If the low-energy
space has dimension above the number of cubes, its mean on every cube
can be set to zero by linear constraints. Poincare then forces energy
>=4lambda times mass, contradicting the spectral bound <=lambda.
Multiplicity and <= at an eigenvalue cause no issue. Boundary null
grid faces do not remove volume. No Robin domain monotonicity is used.

4. John geometry: the non-symmetric dilation is n, not sqrt(n).
The axes are ordinary Euclidean semiaxes after translation and rotation.
The product lies between R^n/n^n and R^n. Only a1<=R and an>=R/n follow;
we never assume all ai>=R/n. The slab width 2n*a1 is an upper bound on
the true width, which is the correct direction for the decreasing lower
bound. No affine pullback of the differential operator is used.

5. Inversion: lambda>=min(w^-2,alpha/w) implies
w>=min(lambda^-1/2,alpha/lambda). It does not imply the corresponding
maximum. This controls each reciprocal axis by 2n*D*lambda for
lambda>=lambda0. The largest axis is separately bounded by n/R, so
one counting factor costs sqrt(lambda) and n-1 cost lambda.
The product exponent is n-1/2, hence the eigenvalue exponent 2/(2n-1).
lambda0>0 uniformly also handles k=1; no hidden large-k threshold occurs.

6. General domains: zero extension of |u|^2 is BV for Lipschitz sets.
Holder supplies M^-1/n*||u||^2, while the boundary integral is at most
q/alpha. Therefore s<=2sqrt(t)+t/alpha. Its positive root gives gamma.
N_alpha<=N0 uses rho1>=gamma in the correct direction. The heat trace
constant is (192n)^n; choosing time n/lambda cancels n^n and leaves
(192eN0)^n*M*lambda^n. This argument is an application of the credited
2014 result, not a new heat-kernel theorem.

7. Sharpness: the box trial space has dimension k and functions vanish
on the two end faces. Each of the other 2(n-1) faces contributes alpha/t
times the volume norm. Optimizing the explicit bound gives
t^(2n-1)=alpha*M^2/(pi^2*k^2) and the coefficient (2n-1). No assumption
about the exact ordering of product Robin modes is needed. On k balls,
constants on components give a k-dimensional space of common quotient
n*alpha/r. These are variable-domain, fixed-volume upper bounds, not
fixed-domain Weyl asymptotics or extremizer assertions.

8. Dimension one: the Neumann k-th value is pi^2*(k-1)^2/M^2.
For k>=2, (k-1)^2>=k^2/4. Finite unions are counted by active interval
lengths, then the same positive spectral floor removes sqrt(lambda).
The k=1 case is explicitly covered.

9. Diagnostics: 13 exact positive LDL pivots verify the unit-interval
trace form on polynomials through degree12. Independent Robin phase
roots and heap enumeration check 540 lower bounds and72 box upper
bounds;60 rescalings check units. Numerical roots are NOT interval
arithmetic certificates. The written inequalities supply the proof.

10. Acceptance is an originating-system self-audit; no independent
referee, formal proof checker, or absolute novelty certification is
claimed. Source citations and theoretical dependencies are explicit.

# Bounded primary-source comparison

Freitas--Kennedy, IMRN2021, establishes the planar rectangle and finite
union-of-rectangles versions, with optimal exponents 2/3 and 1/2.
The OWR2023/36 source explicitly asks for convex and general Euclidean
extensions. Its arbitrary manifold question is a separate paragraph.

Van den Berg--Bucur, JFA266(2014), Proposition7 and eqs7/15, already
give the heat trace <=(192n)^n N_alpha^n |Omega| t^(-n), where
N_alpha=c_n(alpha^(-1)+alpha/rho_1), c_n=1/(n omega_n^(1/n)).
Their BV inequality plus Holder gives a uniform positive rho_1 lower
bound at fixed alpha and volume. The general-domain k^(1/n) conclusion
is therefore credited to that existing theorem, not presented as novel.

The convex proof uses John's classical E subset K subset nE, a direct
one-dimensional trace estimate and the Payne--Weinberger Poincare
inequality. The partition method is already used by Gittins--Lena in
Appendix A; their curvature-based bound is not imported or extended by
assertion. Our full moment-constraint counting proof is written out.
An anisotropic change of the Laplacian or Robin domain monotonicity is
never used. The single long axis is essential to the counting exponent.

Antunes--Freitas--Kennedy, ESAIM COCV19(2013)438--459,
doi10.1051/cocv/2012016, already uses equal balls for the general upper
growth order. The box trial subspace gives the matching convex upper
order directly. No extremizer or optimal constant is identified.

On26October2026 bounded searches combined Robin/positive Robin,
convex domains, lower eigenvalue bounds, k^(2/(2n-1)), k^(2/(2d-1)),
Polya, John ellipsoid, heat trace, and Freitas--Kennedy. The original
report and the above primary papers were compared. No identical
general convex proof was located. This is not exhaustive priority
certification. Bucur--Cito's geometric-control paper concerns negative
parameters and is not evidence for the present positive result.
Frank--Laptev--Suragan's 2026 generalized-inradius paper addresses the
lowest eigenvalue, not this uniform all-k conclusion. Third-party PDFs
remain local and are excluded from all upload archives.

## Supplementary diagnostics

```json
{
  "box_lower_eigenvalue_checks": 540,
  "box_sharpness_upper_checks": 72,
  "continuum_proof_required": true,
  "exact_interval_polynomial_degree": 12,
  "exact_positive_ldl_pivots": 13,
  "first_pivots": [
    "1",
    "17/12",
    "49/180",
    "2321/47600",
    "12209/2160900"
  ],
  "formal_verification": false,
  "largest_sharpness_upper_ratio": 0.9962739040063832,
  "minimum_lower_ratio": 37.71910779980836,
  "minimum_slab_ground_ratio": 1.9276396742174733,
  "numerical_checks_are_not_rigorous_spectral_certificates": true,
  "scaling_checks": 60,
  "status": "PASS"
}
```