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Theorem · exact loss

FPRD-QA-MX-T05

Trace powers erase Jordan debt

Exact statement

The infinite scalar quotient A↦(tr⁡Ak)k≥1A\mapsto(\operatorname{tr}A^k)_{k\ge1} equals the characteristic-polynomial quotient and therefore forgets all Jordan-block data.

StatusProved from Newton identities and Cayley–Hamilton
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

The proof identifies both quotient relations; the 2-by-2 and 3-by-3 adversaries verify the loss boundary.

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

  • Matrix-valued or vector-valued observations may retain more information.

Open work

Classify minimal observation families that recover selected Jordan data.