IndisputableMonolith.Verification.CPT.RankCertification
Rank-certification layer for CPT window identifiability: distinct exponential nodes make the associated Hankel matrix full rank via a Vandermonde factorization. Analysts cite it to justify that finite window measurements separate distinct node sets. The argument is classical linear algebra (Vandermonde determinant nonzero, Hankel as a product, hence det nonzero) plus a nonempty identifiability locus.
claimIf the node set is pairwise distinct, the Hankel matrix built from exponential-sum window data factors through a Vandermonde matrix and therefore has nonzero determinant (full column rank). Consequently the identifiability locus of distinct-node configurations is nonempty.
background
CPT verification sits on reusable Core interfaces (decision tags, resolved sets, class-restricted domination) and on Window Identifiability, which equates injective reconstruction from finite window measurements with trivial kernel of the measurement map and with a full-column-rank predicate.
This module supplies the concrete rank engine behind that predicate. Nodes are treated as distinct exponential parameters. Window data are packaged as exponential-sum samples; the associated Hankel matrix is the natural moment matrix of those samples. Classical Vandermonde theory then controls its determinant.
The local goal is purely algebraic: certify that distinctness of nodes forces the Hankel determinant away from zero, so the measurement map is injective on that locus.
proof idea
Structure is standard Hankel–Vandermonde algebra, not a single deep lemma. Distinct-node hypotheses feed a nonzero Vandermonde determinant. The Hankel matrix of the exponential-sum data is identified with a Vandermonde product factorization. Nonvanishing of the factors yields nonzero Hankel determinant, hence full column rank. A final existence statement records that the distinct-node identifiability locus is nonempty. Supporting definitions package the exponential-sum data and the Hankel construction used throughout.
why it matters in Recognition Science
Window identifiability (paper Thms. 4.5 / 6.5, exported as WINDOW_*) needs a concrete full-rank certificate; without it the measurement map could collapse distinct node configurations. This module closes that linear-algebra gap and is re-exported through the CPT Export surface for the P→B→A pipeline composition.
In the broader Recognition verification stack it is the place where “distinct nodes” becomes a checkable rank condition rather than an informal modeling assumption. Downstream citation should go through the export aliases once the window theorems invoke these rank facts.
scope and limits
- Does not prove the full CPT window theorems; only the rank/Hankel ingredients.
- Does not handle repeated (non-distinct) nodes or Jordan-block generalized eigenspaces.
- Does not address noise, numerical stability, or floating-point rank tests.
- Does not introduce new physics constants or forcing-chain steps (T0–T8).
- Does not by itself discharge the P→B→A pipeline composition.