Markov-Compatible Seam Extension

Source result: GAT VIII, Proposition 14.2.
Status: Source-bound proposition in the public preprint.

Catalog summary

A seam form can be added to the electric–magnetic Dirichlet form while preserving the Dirichlet property if it is gauge invariant, satisfies the normal-contraction inequality, has the required common domain/core, and yields a closed form sum. Nonnegativity alone is insufficient.

What the proposition says

A seam contribution may be needed when physical pieces are joined. Adding a nonnegative energy term does not automatically preserve the contraction property that makes an evolution Markovian. This proposition states the additional conditions under which the seam term is compatible with that analytic structure.

Hypotheses and scope

Start with the electric–magnetic Dirichlet form of Theorem 14.1. On the same configuration-space carrier, let be a closed nonnegative gauge-invariant quadratic form whose domain contains a common core for the electric–magnetic form. Require stability of its real form domain under every normal contraction , with

Assume the form sum is closed on its common domain. These are actual requirements on the seam realization, not consequences of calling the seam energy positive.

Precise conclusion

The form , restricted to the gauge-invariant carrier, is again a symmetric Dirichlet form.

Why it works

Both summands decrease under normal contractions, so their sum does as well. Gauge invariance is retained termwise. The closedness and common-domain assumptions make that contraction property a statement about a valid closed form.

Example and nonexample

In a two-point illustrative model, the seam energy , with , is contraction-compatible because .

In contrast, is a nonnegative closed quadratic form, but it fails the contraction test. For , . Clipping each coordinate to gives , for which . Thus positivity alone does not ensure Markov compatibility. These finite-dimensional models isolate the analytic distinction; they are not physical seam certificates.

Claim limits

The proposition does not establish the contraction property for every physical seam operator, remove the need to control form domains, or prove conservativity. It also does not make a subsequent sharp detector compression Markovian.

Source

GAT VIII, Proposition 14.2, equation (14.5), and the following qualifications. Retain “Proposition” as the source designation.