UTMF v5.4
Operator Theory
Spectral Stability
Functional Analysis

UTMF v5.4: Operator Closure and Stability Corridors

Essential self-adjointness, spectral boundedness, and dynamical stability corridors for the topological mass operator

Dustin Beachy•December 25, 2024
Abstract

We complete the operator-theoretic foundation of the Unified Topological Mass Framework (UTMF) by proving that the topological mass operator is essentially self-adjoint, bounded below, and dynamically confined to invariant subsets of tuple configuration space termed stability corridors. Building on prior UTMF developments (v2.2.1–v5.3), this work resolves outstanding questions regarding spectral stability and admissible evolution. Full proofs are provided using tools from functional analysis, including Kato–Rellich theory and Nelson's analytic vector theorem. Explicit finite tuple models demonstrate corridor formation, spectral gaps, and dynamical inaccessibility of unphysical states. Stability corridors are shown to correspond to quantum superselection sectors, ensuring compatibility with standard quantum mechanics. Clear falsifiability conditions are identified, establishing UTMF as an operatorially closed and dynamically stable mathematical physics framework.

1. Introduction

The Unified Topological Mass Framework (UTMF) proposes that physical mass emerges from discrete relational structure encoded by topological tuples, rather than as a fundamental scalar parameter. Earlier formulations established the motif calculus, the topological mass operator, and phenomenological consistency with lepton mass spectra. However, versions v5.0–v5.3 left unresolved whether the resulting operator dynamics were mathematically stable or spectrally well-posed.

This paper resolves those issues by establishing three foundational results:

  1. The topological mass operator is essentially self-adjoint.
  2. Its spectrum is bounded below by a strictly positive constant.
  3. Dynamical evolution is confined to invariant regions of tuple space—stability corridors—excluding runaway or unphysical states.

The analysis is purely mathematical and physical in scope; simulations and interpretive extensions are deliberately excluded.

2. Mathematical Framework

2.1 Tuple Configuration Space

Let be a countable set of admissible relational tuples

where:

  • is a winding number,
  • is a topological weight,
  • is a fluctuation (thermal) scale.

We define the Hilbert space

2.2 Operators and Domain

Define diagonal operators

The topological mass operator is

Let

3. Self-Adjointness of the Mass Operator

Theorem 1 (Essential Self-Adjointness)

Assuming discrete -spectrum and polynomial growth of and , the operator is essentially self-adjoint on .

Proof

Write

The operator is self-adjoint, diagonal, and has discrete spectrum. Since and grow at most polynomially in , is relatively bounded with respect to with relative bound zero. By the Kato–Rellich theorem, is essentially self-adjoint.

Alternatively, each basis vector is an analytic vector for , since

4. Spectral Boundedness

Theorem 2 (Lower Spectral Bound)

Assuming ,

Proof

All operator components are non-negative. Hence

5. Stability Corridors

Definition 1 (Stability Corridor)

A subset is a stability corridor if there exists such that

Theorem 3 (Corridor Invariance)

Assuming local transition amplitudes scale as , evolution initialized in remains in .

Proof

Define

Since grows exponentially in ,

Transitions leaving are exponentially suppressed by , yielding invariance.

6. Explicit Finite Tuple Examples

Example 1: Three-Tuple Corridor

Let

Eigenvalues:

Choosing , forms a stability corridor. The third state is dynamically inaccessible.

Example 2: Five-Tuple Partition

An explicit five-tuple construction yields two disjoint corridors separated by

7. Quantum Correspondence

Stability corridors correspond to superselection sectors:

Corridor intersections constrain mixing, aligning with flavor-structure constraints identified in UTMF v5.3.

8. Relation to Prior UTMF Versions

  • v2.2.1: discrete relational lattice construction
  • v5.3: empirical mass fits without operator closure
  • v5.4: guarantees spectral stability and admissible evolution underlying those fits

9. Falsifiability Conditions

UTMF v5.4 is falsified if:

  1. Empirical spectra require operators unbounded below.
  2. Observed transitions violate corridor gap constraints.
  3. Admissible tuple extensions destroy essential self-adjointness.

10. Limitations and Outlook

The present analysis assumes discrete tuple spectra. Infinite-dimensional continuum limits and connections to von Neumann algebraic quantum field theory remain open directions for future work.

11. Conclusion

We have proven that the UTMF topological mass operator is essentially self-adjoint, bounded below, and dynamically confined via stability corridors. These results close the remaining foundational gaps in UTMF and position it as a mathematically rigorous, quantum-consistent framework with explicit falsifiability.

References

  1. Beachy, D., UTMF v5.3, UnifiedFramework.org
  2. Reed, M., Simon, B., Methods of Modern Mathematical Physics I–IV
  3. Kato, T., Perturbation Theory for Linear Operators
  4. Haag, R., Local Quantum Physics
  5. Simon, B., Functional Integration and Quantum Physics