Oriented Seam Stokes Law

Status

Proved under stated corridor/Green hypotheses.

Statement

Across a source-free corridor, the center label and signed conormal response propagate with orientation, while the positive quadratic passage costs remain positive and add over disjoint collars.

Under the theorem’s hypotheses:

1. the relative passage datum is orientation odd;
2. the Dirichlet passage energy is orientation even;
3. internal signed conormal terms cancel;
4. the center sector is constant across homologous separators.

The compact version is:

Oriented passage cancels internally; positive passage cost does not.

Why it works

There are two Stokes statements. The cohomological identity transports the discrete label. The Green identity transports the signed conormal response. They share orientation behavior but are not the same theorem. On adjacent cells, outward normals reverse and signed internal boundary terms cancel. The quadratic form assigned to each independently supported ordinary-gluing defect does not acquire that sign and therefore does not cancel.

Example (illustrative)

Two adjacent collar cells contribute equal and opposite conormal flux at their common internal face. Their signed boundary terms disappear from the corridor sum. If each cell contains a charged ordinary-gluing defect with positive Dirichlet energy, those two nonnegative energies remain in the sum.

Scope and limits

The topological Stokes statement does not prove a Hilbert-space frame inequality, and the Green identity does not create the center class. Neutral side boundary is an actual hypothesis: charge may otherwise leave through the corridor wall.

Source

Phenomenological Implications I, §§7.2–7.3, equations (30)–(31), Theorem 7.1, and Remark 7.2.

Read the source manuscript.