Audit panel C: regular free-boundary clipping and 3/2 energy rate

Objective

Independently audit the corrected scoped sharp theorem:

||u-u_h^B||_{H1} <= C (h^r + h_Gamma^(3/2)).

Frozen V4 target

• branch: review/bernstein-obstacle-v4-final
• commit: 61594952ad880d2b61759cfa93a19df979183c09
• analytical integration PR: #114
• parent audit: #96

Earlier targets remain available only for provenance.

Required checks

1. Exact regular-free-boundary assumptions and whether they are standard or natural.
2. u in C^{1,1}, u = grad u = 0 on Gamma, and two-sided quadratic gap growth in the stated fixed tubular neighborhood.
3. Uniform one-sided H^{r+1} extension near the positive phase.
4. Uniform broken regularity bound sum_{T notin omega_h} |u|_{H^{r+1}(T)}^2 <= C_reg.
5. Coefficient-to-grid-value estimate and all-degree barycentric interpolation unisolvence.
6. Correct local-size risky set R_h = {T : dist(T,Gamma) <= kappa h_T}.
7. Phase classification of non-risky elements into contact interior, tubular positive phase, compact positive interior, and physical-boundary cases.
8. Local quasi-uniformity on the fixed one-ring patch and |omega_h| <= C_Gamma h_Gamma.
9. Explicit enlargement of clipping support from R_h to the shared-DOF one-ring omega_h.
10. Two-sided O(h_Gamma^2) coefficient amplitude on every element in that one-ring.
11. Conformity and boundary-data preservation under shared global coefficient clipping.
12. Multi-face physical-boundary proof, including corner elements.
13. Finite-dimensional scaling giving O(h_Gamma^(3/2)) in H1.
14. Multiplier convention, support, bounded-density assumption, and the O(h_Gamma^3) consistency term.
15. Both the exact energy-identity proof and the Falk proof.
16. Dimensions two and three; singular/degenerate interfaces, anisotropic meshes, boundary-touching free boundaries, and measure-valued multipliers remain excluded.
17. Whether any grading or regularity assumption can be weakened without invalidating the exponent.

Deliverable

A line-by-line PASS, PASS AFTER CORRECTION, or FAIL verdict, including any missing hypothesis, counterexample, corrected exponent, or narrower valid theorem. A rejection or correction is fully eligible.

Internal scope label

FUN BUX: 4,500. This is an unfunded internal effort/complexity label, not a real or promised bounty. No payment is represented as available unless separate written funding terms are posted.

Start with branch review/bernstein-obstacle-v4-final at commit 61594952ad880d2b61759cfa93a19df979183c09, then compare the analytical integration in PR #114 with parent audit #96. Check each of the 17 required points and record PASS, PASS AFTER CORRECTION, or FAIL, including missing hypotheses, counterexamples, or a corrected exponent. The deliverable is a line-by-line verdict for dimensions two and three.