Check whether the saved matrix examples represent rotations

The exact feasible-witness diagnostic shows that more linear optimization cannot remove the failing widths. Before another propagation revision, test whether the saved extreme examples preserve a known property of the physical force matrix.

That matrix is M = R_initial R(p) diag(1/(1+g)). Remove the proposed gains to obtain B = M diag(1+g). A true rotation preserves the Gram spectrum: for an exactly orthogonal initial matrix, BᵀB = I. More generally, ||BᵀB-I||_F² = ||R_initialᵀR_initial-I||_F². The latter equality avoids treating a supplied floating initial matrix as exactly orthogonal. Reconstruct M from all saved physical and matrix-error coordinates using rational arithmetic. Compare its Gram defect with that of the original initial matrix.

Local known-answer tests must cover identity, signed/permuted rotations, gain removal, nonorthogonal initial matrices, stretched columns, equal-length but nonorthogonal columns and malformed/nonpositive gains. Distinguish the exact defect from an explicit 1e-8 dimensionless diagnostic allowance for the existing floating matrix conventions. This is a numerical diagnostic threshold, not a measured sensor specification or a replacement scientific gate.

After tests pass, freeze six identities from the completed width diagnostic. Each examines its six saved exactly feasible witnesses and its physical anchor: 36 witness checks and six anchor checks, 42 matrix checks total. An incompatible physical anchor fails qualification and blocks interpretation of that case. Use the original saved generated configurations and source-bound matrix references; verify all input hashes and complete 51-coordinate ordering. Record reconstructed matrices, gains, exact Gram defects, reference defects, allowance and every flagged example. Preserve near-threshold outcomes without replacement or tuning. Four workers use one thread each, with a 60-second cooperative allowance. No LP proposals, native calls/bounds, original aircraft reads, observed fits or protected outcomes. No retries or budget expansion.

A flagged example diagnoses a nonphysical combination admitted by this relaxation. It does not prove all permitted examples are nonphysical, justify discarding the region or establish a narrower valid interval. A clean Gram check is necessary, not sufficient: it does not establish the correct dependence on the proposed orientation, full trajectory feasibility or aircraft uncertainty. Any stronger future constraints need their own derivation and qualification.