Audit panel E: nonlinear Bernstein grand theorem and codimension law

Objective

Independently audit the proposed Bernstein certified-inner-approximation grand theorem on branch agent/bernstein-grand-theorem.

The theorem extends the corrected obstacle package from symmetric quadratic energies to Lipschitz strongly monotone, possibly nonlinear and nonsymmetric, variational inequalities. It also proposes a unified geometric law for scalar interior obstacles and planar frictionless Signorini contact.

Exact claims to test

E1. Nonlinear certified Falk estimate

For K_h ⊂ K and strongly monotone Lipschitz F, verify

||u-u_h||_V^2
≤ (L^2/alpha^2) ||u-v_h||_V^2
  + (2/alpha) <F(u), v_h-u>

for every v_h ∈ K_h, without symmetry or a potential.

E2. Perturbed-operator estimate

Audit equation (5.3), including all signs and dependence on the a posteriori consistency defect.

E3. Codimension repair law

For a constraint manifold of ambient codimension c, audit

||d_h||_V = O(h_Sigma^((c+3)/2))

under the stated stable-lifting and patch-count assumptions.

E4. Universal multiplier law

Audit the claim that quadratic gap amplitude and an O(h_Sigma) active transition strip give

<F(u), v_h-u> = O(h_Sigma^3)

for both volume obstacles and boundary contact, yielding a common final h_Sigma^(3/2) term.

E5. Planar Signorini lifting

Audit the global normal-control update

U_i_tilde = U_i + min(c_i,0) n,
c_i = g_i - U_i·n,

including conformity, essential-boundary compatibility, support, and physical H^1 scaling.

E6. Conservative inexact data

Check the one-sided directions:

• psi_h^+ >= psi for v >= psi;
• g_h^- <= g for g-v·n >= 0.

Audit the additive consistency terms in equation (14.1).

E7. Prior-art collision

Search nonlinear VI approximation, high-order bounds-constrained FEM, Bernstein/Bézier contact, hp/mortar/BEM contact, Mosco stability, and active-transition error analysis. Identify which exact combined claims, if any, remain new.

Required verdict

Return PASS, PASS AFTER STATED CORRECTION, or FAIL for E1–E7. A counterexample, prior-art collision, or narrower corrected theorem counts as a successful audit outcome.

The report must satisfy math/bernstein_obstacle/EXTERNAL_REVIEW_PROTOCOL.md and must not treat internal AI work or repository-owner review as independent endorsement.

Start with math/bernstein_obstacle/EXTERNAL_REVIEW_PROTOCOL.md and inspect branch agent/bernstein-grand-theorem alongside the stated E1–E7 claims. Done means an independent report following the protocol, giving PASS, PASS AFTER STATED CORRECTION, or FAIL for each claim and recording counterexamples, corrections, or prior-art collisions without treating internal review as endorsement.