# Source statement and exact accepted scope

Frozen record OWR-14740-003, record_id 30003169, ulamai/UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e. The unchanged statement hash is bfe4fbc5d06866ff73cecd0b12ed5fc40d4e84ae4af175ffb9bd8cc5f7fe32f6.

The record asks for a bandwidth-independent B such that h^-2 I2 tends to B after dropping nh^(d+4) -> 0 in a decomposition MDM=I1+I2.

The canonical source is J. E. Chacón, joint work with W. Polonik, “Asymptotics and optimal bandwidth selection for level set estimation,” Oberwolfach Report 32/2016, pp. 1832–1833, DOI 10.4171/OWR/2016/32. The actual conjecture is on p.1833, visually inspected in sources/owr-conjecture-page.png. It supplies the additional variance premise sqrt(nh^d) I1 -> A(f,K,c), and allows Lebesgue weight g=1 and probability weight g=f.

The accepted theorem is a complete counterexample to simultaneous bandwidth-independent A and B, retaining that variance premise. It uses d=2, a smooth Gaussian density, a nonnegative compactly supported radial C2 kernel, and three monotone critical bandwidth sequences t=1,2,3. For every decomposition satisfying the stated variance premise on all three sequences, the claimed universal B is impossible. A second complete result computes unequal limits for an exact, explicitly defined centered comparison.

The report does not give a formula for I1 and I2. Our centered comparison is not attributed to Cadre or silently substituted for that undefined notation. Although the joint proposal in the report is disproved, we conservatively withhold a whole frozen-record closure and a claim identifying the literal I2 in the author's intended decomposition. This is a complete scoped counterexample, not an unresolved mathematical proof.

The exact-source closure increment, auxiliary-question closure increment and frozen-record closure increment are all zero. The complete scoped-result and manuscript increments are one. Broad level-set estimation, all weights, all bandwidth regimes and all possible bias-term definitions are not claimed solved.

# Self audit of the complete scoped counterexample

Accepted scope: papers/OWR-14740-003/main.tex, Theorem 1 and Corollaries 2–3. Original source identification and novelty are distinct from mathematical correctness.

1. Kernel validity. K(x)=4/pi (1-|x|^2)^3 on the unit disk and zero outside is nonnegative, integrates to one, is radial nonincreasing, and its first two derivatives match zero at the boundary. Its coordinate moment is 1/10 and squared L2 norm is 16/(7*pi). It meets the compactness and smoothness restrictions motivating the source. The density is smooth with bounded derivatives and regular circle boundary.
2. Dimension. The final construction is two-dimensional. The earlier proposed one-dimensional Gaussian example was not used, avoiding Cadre's distinct d=1 bandwidth restrictions and a noncompact kernel.
3. Meaning of the claim. We do not assume a formula for the source's undefined I2. If both limits with universal A,B existed for any exact decomposition, the risk limit would be affine in t. The explicit risk limit is strictly convex. This proves the joint obstruction for every such decomposition. The stronger claim that a particular unwritten I2 has been identified is withheld.
4. Bias uniformity. Gaussian fourth derivatives are bounded. Compact support and vanishing odd kernel moments give a uniform Taylor remainder O(h^4). The boundary bias is b h^2, b=c/10; the moving radial mean is (b-2cu)h^2.
5. Variance and normal limit. The variance formula includes the negative f_h(x)^2/n term; after division by h^4 it vanishes. Each centered summand of the h^-2 normalized estimator is bounded by 2||K||/(n h^4)=O(h^2). Total variance tends to b^2 t^2>0. Lindeberg follows without assuming a uniform central limit theorem.
6. Noncompact domain. Outside radii [1,3], the level gap is uniformly positive. The integrated variance is bounded by R_K/(n h^2)=O(h^4). Thus the total far-region contribution is o(h^2), including for Lebesgue measure on an infinite-area domain. A pointwise constant bound alone would not have justified this step.
7. Domination at the boundary. On [1,3] the radial derivative is bounded away from zero. Beyond a fixed |u|, bias cannot change the true side of the level. Chebyshev gives C/u^2. The annular integrand extended by zero has an integrable envelope; dominated convergence is legitimate.
8. Sign conventions. Radius r<2 is inside L and uses P(fhat<c); r>2 is outside and uses P(fhat>=c). The normal law is continuous. The integrated limiting probability is E|W|, not E W, with W=(1+tZ)/20.
9. Coefficients. Polar Jacobian at r=2 is 4*pi; multiplying by E|W| yields pi*w*/5 G(t). Lambda=3200 e^2/7 is c R_K / b^2. All normalization exponents agree.
10. Convexity. G''(t)=2 phi(1/t)/t^3>0 for every t>0. Three distinct arithmetic-progression parameters defeat every affine A t/sqrt(Lambda)+B, irrespective of signs of A,B. Numerical evaluations are illustrative only.
11. Centered comparison. The centered field need not be a density, which is explicitly stated. Its level-set risk is well-defined; the same integrated variance bounds show finite expectation. Its mean is exactly f. Removing b from the normal mean leaves C_w t E|Z|. The residual derivative is 2 C_w(phi(1/t)-phi(0))<0.
12. Quantitative gap. The second difference is the integral of G'' against a unit-mass triangular kernel on [1,3]. G''>=2 phi(1)/27. The coefficient sum 1+2+1=4 bounds the maximum of the three affine errors from below by C_w phi(1)/54.
13. Existing theory. Doss–Weng Theorem 2.1 and Qiao Theorem 3.2/Corollary 3.1 already encompass the nonlinear folded-normal phenomenon. They are credited. This note contributes an explicit source-connected obstruction and an elementary direct proof; it does not claim a new general risk expansion.

All proof obligations above have explicit arguments in the manuscript. This is originating-system self-audit, not independent review or proof-assistant verification. The checker tests symbolic constants/identities and three illustrative numerical integrals; it does not establish asymptotic probability claims.

# Literature and contribution audit on 27 October 2026

The source report is Chacón (joint with Polonik), OWR32/2016, pp.1832–1833, DOI https://doi.org/10.4171/OWR/2016/32. The publisher PDF was downloaded over verified TLS, hashed, extracted, and the conjecture page visually read.

Cadre, “Kernel estimation of density level sets,” Journal of Multivariate Analysis 97 (2006), 999–1023, DOI https://doi.org/10.1016/j.jmva.2005.05.004, is the cited source of undersmoothing asymptotics. The public author-submitted preprint https://arxiv.org/abs/math/0501221 was obtained and read. It does not exhibit the report's literal two-term formula. The publisher page returned 403; the author's later Rennes PDF failed TLS verification and was not fetched with verification disabled. These versions must not be conflated.

Doss and Weng, “Bandwidth selection for kernel density estimators of multivariate level sets and highest density regions,” Electronic Journal of Statistics 12 (2018), 4313–4376, DOI https://doi.org/10.1214/18-EJS1501, https://arxiv.org/abs/1806.00731v2: Theorem 2.1 gives a folded-normal risk law for probability-weighted loss, compactly supported kernels and regular critical bandwidths in dimensions at least two. The same shape is already in this prior work. Its theorem page and assumptions were inspected. The NSF PDF transport timed out; the author-uploaded arXiv version was obtained successfully over verified TLS.

Qiao, “Asymptotics and Optimal Bandwidth for Nonparametric Estimation of Density Level Sets,” https://arxiv.org/abs/1707.09697v3 (2019): Theorem 3.2, Corollary 3.1 and Remark 3.4 give broad weighted risk expressions and distinguish them from a sum of separate upper bounds. Theorem 3.1 has stronger bandwidth requirements and is not used as a substitute for the expectation theorem. The relevant theorem page was visually inspected. Product-kernel assumptions are not transferred silently to our radial polynomial kernel; our proof is direct and does not invoke that theorem.

Bounded public searches combined the exact source title, Chacón/Polonik, conjecture, bias, I2, bandwidth and level-set phrases. Saved search records include unsuccessful and irrelevant hits, with no claims based on those hits. No identical explicit three-bandwidth obstruction tied to OWR32/2016 was located. This is not an exhaustive literature or priority certification.

Contribution assessment: the nonlinear Gaussian risk law and general bandwidth selection are prior theory. The present complete result is the explicit smooth compact-kernel counterexample to simultaneous universal variance and bias constants, its quantitative affine-approximation obstruction, and the exact centered residual calculation. The general phenomenon is not counted as a discovery. Whole frozen-record closure is withheld because the report does not define its I2.