Orbit-Threshold Transfer

Source result: GAT V, Theorem 3.5.
Status: Source-bound theorem in the public preprint.

Catalog summary

For a sharply covariant spectral system, a positive height exclusion holds along every admitted symmetry orbit. It therefore yields an invariant lower bound for the orbit-floor operator, with the same squared-threshold constant.

What the theorem says

A spectral quantity such as energy can change with the observer. To obtain an invariant threshold, the theorem asks for the smallest value that quantity can approach along the full symmetry orbit of a spectral point. Covariance forces every point of that orbit to remain in the spectral support, so a forbidden low-height region excludes the entire orbit whenever it could enter that region.

Hypotheses and scope

Use the orbit-covariant spectral system of GAT V, Definition 3.1. Its space is locally compact, second-countable, and Hausdorff; a second-countable topological group acts continuously by homeomorphisms. On one Hilbert carrier there is a strongly continuous unitary representation and a sharp Borel projection-valued measure obeying

Let be continuous, closed and invariant, and . Require

For , assume some satisfies

This is the closed-form meaning of .

Precise conclusion

Define the orbit floor and its squared spectral operator by

Then

The second inequality is a form inequality on . The orbit infimum need not be attained.

Why it works

The original inequality removes the strict height sublevel below from the spectral support outside . Covariance makes that support invariant. Every point on a retained orbit must therefore stay above the same threshold. Taking the infimum along the orbit gives the invariant statement.

Example

In the relativistic specialization on the forward momentum cone, take and the full proper orthochronous Lorentz action. A future timelike momentum has orbit-floor energy equal to its invariant mass. A nonzero null momentum can be boosted to arbitrarily small positive energy, so its orbit floor is zero. A uniform positive energy exclusion on the full covariant support therefore excludes the nonzero null orbit.

The mass interpretation requires this Lorentz-covariant, positive-energy, same-carrier spectral setting. A lower bound in a truncated or noncovariant description is insufficient.

Claim limits

The result is a lower certificate, not a proof that spectrum occurs at its value. Replacing the symmetry group by a smaller realized subgroup can change the orbit floor and weaken its physical interpretation. Identifying a reconstructed form operator with the physical height generator requires a separate same-carrier theorem and the declared normalization.

Source

GAT V — Invariant Spectral Transfer, Definition 3.1, Lemmas 3.3–3.4, Theorem 3.5, Boundary 3.6, and the relativistic specialization in §6. Public version v1.0.2-webpub1.