Audit the Bernstein grand order-interval and Banach-scale clipping theorems

Audit target

Please independently audit the new analytical extension on:

• branch: research/bernstein-grand-order-interval
• commit: b6fa26c1d45d5ad0c633cd6fb9455b126a4782c1
• theorem note: math/bernstein_obstacle/GRAND_ORDER_INTERVAL_THEOREM.md

This is a research target, not a replacement for the immutable corrected v2 theorem.

Principal claims

Theorem A — bilateral order-interval Mosco convergence

For fixed-degree conforming simplicial Bernstein finite elements in W_0^{1,p}, conservative sampled obstacle envelopes

psi_h^+ = B_h psi + E_psi h^2,

phi_h^- = B_h phi - E_phi h^2

and coefficient inequalities produce an exactly feasible discrete set K_h^B subset K_{psi,phi} that Mosco-converges to the continuous bilateral obstacle interval.

Theorem B — nonlinear uniformly convex transfer

Minimizers of coercive uniformly convex p-growth energies over K_h^B converge strongly in W_0^{1,p}. This includes weighted p-Dirichlet energies.

Theorem C — codimension–growth clipping law

If coefficient corrections have amplitude O(h^beta) on a patch of measure O(h^kappa), then

||d_h||_{W^{1,q}} <= C h^(beta - 1 + kappa/q).

Quadratic contact on a codimension-one patch gives h^(1+1/q), whose q=2 case is the corrected h^(3/2) theorem.

Required mathematical checks

1. Uniform L^infinity Bernstein sampling error for fixed degree on shape-regular simplices.
2. Global conformity from shared physical barycentric lattice values.
3. Conservative envelope signs and exact pointwise inclusion K_h^B subset K.
4. Existence of a strict collar-supported feasible function.
5. Strong W^{1,p} density of strict W^{2,infinity} feasible functions using interiorization, variable-interval projection, zero extension and mollification.
6. Diagonal Mosco recovery with the obstacle-envelope h^2 shifts.
7. Weak closedness of the bilateral order interval.
8. Uniform-convexity argument upgrading weak/energy convergence to strong convergence.
9. Applicability to weighted p-Dirichlet energies for every 1<p<infinity.
10. Reference-element norm equivalence and patch summation in Theorem C.

Required collision search

Compare against:

• Kirby–Shapero high-order bounds-satisfying variational inequalities;
• Allen–Kirby bounds-constrained Bernstein approximation;
• higher-order p-Laplacian obstacle FEM;
• the 2026 hp/SEM obstacle theorem using transformed GLL constraints and Bernstein positivity;
• Mosco convergence for moving unilateral and bilateral obstacle sets;
• bounds-preserving Bernstein/Bezier FEM and limiter literature.

The report must distinguish novelty of individual ingredients from novelty of their combination.

Nonlinear-rate boundary

The note does not claim a completed sharp p-Laplacian free-boundary rate. The predicted exponent requires a separate operator-specific Bregman/Falk estimate and a verified contact-growth law. Treat any unconditional nonlinear sharp-rate statement as an error.

Deliverable

A numbered PASS, PASS AFTER CORRECTION, or FAIL report with precise derivations, counterexamples, references, and the strongest valid theorem after any corrections.

Read math/bernstein_obstacle/GRAND_ORDER_INTERVAL_THEOREM.md at commit b6fa26c1d45d5ad0c633cd6fb9455b126a4782c1 on research/bernstein-grand-order-interval, then work through the ten mathematical checks and the collision-search references. Done means a numbered PASS, PASS AFTER CORRECTION, or FAIL report with derivations, counterexamples, references, and the strongest valid corrected theorem.