Frobenius floor and conditioning debt

Exact statement

The conjugacy-orbit Frobenius infimum is $\sum_i|\lambda_i|^2$ , attained exactly for semisimple matrices; for $0<t\le1$ , the displayed diagonal scaling of $J_m(\lambda)$ has excess $(m-1)t^2$ and condition number $t^{-(m-1)}$ .

StatusSelf-contained proof and exact checks

External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

The matrix calibration separates the invariant quotient, its closed semisimple core, the Jordan receipt lost inside a fibre, and the quotient that is universal for a declared family of future observations.

Proof or evidence

Schur form proves the floor and diagonal Jordan scaling proves nonattainment and the displayed resource tradeoff.

Self-contained matrix calibration and public audit →

Verification notes

The archived proof and failure boundaries were reconstructed; the public dependency-free verifier checks 44 partitions, 233 dominance comparisons, six scaling rows, the smallest collision, and sharp observation chains through dimension seven.

Limitations

The exhibited condition number is a witness, not a global optimality theorem.

Open work

Determine optimal conditioning for prescribed distance above the orbit floor.