Dirichlet Passage Pencil

Meaning

The Dirichlet passage pencil is the finite positive matrix problem that measures the least ordinary-gluing energy carried by a selected charged passage after every allowed interior relaxation and every common Dirichlet zero mode have been removed.

Write the collar stiffness matrix in interior and boundary variables. Its Friedrichs Schur complement is

This is not a convenient approximation to the bulk energy. It is the exact least energy among all interior extensions with the prescribed boundary value.

On the selected source space , with inclusion , Klauder defect , and ordinary boundary realization , the passage matrix is

Relative to the Gram matrix , its least generalized eigenvalue is the local passage floor.

Example (illustrative)

A selected source vector can have a nonzero incompatibility and still fail to cost energy if its realized boundary value lies entirely in . The quotient boundary map removes those zero modes. The exact nonescape test is

Scope and limits

A floating-point eigenvalue is discovery evidence, not a rigorous certificate. Website copy must retain the need for exact or certified-rank arithmetic and a rigorous interval lower endpoint for every normalized block.

Source

Phenomenological Implications I, §§6.1–6.3, equations (23)–(27).

Read the source manuscript.