Finite Passage Certificate

Status

Criterion proved; physical block certificate open.

Open finite certificate — DP2/P06. The criterion is proved, but a completed numerical pure confinement certificate still requires

and a rigorous positive lower endpoint for the least generalized eigenvalue of every normalized passage block, together with the named imported completion certificates.

Statement

A normalized selected passage block has a strictly positive Dirichlet passage eigenvalue exactly when its selected ordinary-boundary realization remains injective after quotienting the common Dirichlet zero modes.

Let , and let be the quotient boundary map. Then

When the factorized comparison constant is available,

The generalized spectrum is unchanged by the selected associator and agrees in the conjugate sectors.

Why it works

This theorem locates the exact finite gate. A selected source direction may be nontrivial and ordinarily nonadmissible while its boundary realization still lands in a zero mode of the Dirichlet form. No topological or refinement argument is permitted to hide that possibility.

Example (illustrative)

In a finite block, assemble , , , and the passage matrix. Exact or certified-rank arithmetic must first show full column rank. Rigorous interval arithmetic must then enclose the least generalized eigenvalue strictly above zero. A floating-point value near zero cannot decide either question.

Scope and limits

The theorem gives the acceptance test; it does not report that every physical normalized passage block has passed it. The website must display:

OPEN FINITE CERTIFICATE: construct the physical block matrices, prove , and certify a positive lower eigenvalue endpoint for every normalized block.

Source

Phenomenological Implications I, §6, Proposition 6.1, Theorem 6.2, §12.1, and proof obligation P06.

Read the source manuscript.