No-Leakage Principle

Status: Exact canonical theorem binding still required. Related published statements are identified below.

Catalog summary

The reconstruction must preserve the declared ground and spectral structure through its limiting comparisons. Published comparison–Mosco results explain preservation of a coercive lower bound and the exact thin ground; the registry’s broader no-leakage title still needs its precise theorem and source binding.

What this entry concerns

A sequence of finite or positive-thickness systems can have a spectral separation at every stage and still lose that separation in a limit. A rigorous limiting argument must control the forms, the maps between carriers, the complete-ground projections, and the lower bound uniformly at the required scope.

The archive currently uses “No-Leakage Principle” as a registry label without a canonical theorem number or statement. That label should not collapse several different results: identifying the limiting closed form, identifying its complete ground, controlling isolated spectral projections, and establishing physical no leakage on the final represented branch.

A precise related result available in the public sources

GAT III, Theorem 7.1 and Appendix B, states a thin complete-ground identification theorem under its source-locked comparison–Mosco hypotheses M1–M6:

1. Controlled comparison maps and form domains on the moving carriers.
2. A vanishing relative form defect in the stated comparison norm.
3. Comparison-weak compactness at the certified scope.
4. Transport of the complete-ground projections.
5. One uniform positive lower bound on the ground complements along a cofinal tail.
6. Inclusion-compatible recovery and monotone exhaustion.

Under that package, the limiting closed nonnegative form satisfies

The associated resolvents and heat semigroups converge strongly in the declared comparison sense. The stronger norm-resolvent conclusion requires the stronger relative-form estimate specified by the source.

This establishes that the transported complete-ground space is exactly the zero-form space. It does not assume that the ground is one-dimensional.

Where Mosco and Riesz enter

Comparison–Mosco convergence concerns closed forms and their limiting operators on the declared carrier geometry. Riesz projection arguments concern an isolated spectral component and require the corresponding resolvent and spectral-separation hypotheses.

GAT III’s Theorem 7.1 expressly obtains its thin complete-ground identity from its comparison–Mosco and ground-transport package without requiring a separate Riesz argument there. GAT V, after Theorem 10.5, separately preserves the role of Riesz stability in identifying an isolated limiting zero projection in the broader reconstruction route. These statements have different hypotheses and jobs.

Example: why pointwise positivity is insufficient

On , let

Every has exactly the same one-dimensional ground space and a positive off-ground eigenvalue. Nevertheless , so the limiting kernel is all of . The positive eigenvalue has collapsed to zero. Even finite-dimensional form convergence does not prevent this: the missing condition is a uniform positive lower bound.

This elementary example explains one necessary control. It is not a counterexample to a theorem whose uniform-coercivity hypotheses are satisfied.

What must be fixed before this becomes a fully bound theorem page

Specify which no-leakage result this record denotes, its manuscript version and theorem number, its carrier, its projection and form hypotheses, and whether its conclusion is analytic or the stronger physical branch statement.

GAT V’s public source crosswalk points to frozen Paper V, Hypothesis 7.2 and Theorems 7.5, 7.9, and 7.13 for the comparison–Mosco/Riesz package, and Theorem 13.18 for final physical no leakage on declared branches. These are cross-references reported by GAT V, not primary statements independently inspected for this editorial pass. They do not authorize assigning one of those numbers to this registry entry by inference.

Claim limits

A retained form lower bound does not by itself identify invariant relativistic mass, prove occupancy at the lower-bound value, or identify the physical lineage of an arbitrarily chosen carrier. The exact source binding remains pending; the underlying mathematics is not thereby declared an open problem.

Sources

GAT III, Theorem 7.1 and Appendix B, especially Theorem B.1; GAT V, Hypothesis 10.4, Theorem 10.5 and its following qualification, and Appendix A.2. These clarify related results while the exact source for THM-NOLEAK-001 remains unassigned.