Positivity Constraints and Phase Structure in the Unified Topological Mass Framework

UTMF Technical Paper — Positivity, Temperley–Lieb Interface, and Universal Phase Structure

Abstract

We prove that quantitative realizations of the Unified Topological Mass Framework (UTMF) admitting a local Temperley–Lieb (TL) or Hecke interface are subject to sharp, nonperturbative constraints arising from positivity of the categorical trace pairing. Abstracting from explicit diagrammatic computations, we show that any UTMF kernel supporting a Markov-compatible TL sector must satisfy universal Chebyshev–Jones bounds on the effective loop parameter N_eff.

At finite strand order n, positivity enforces N_eff ≥ 2cos(π/(n+1)); in the infinite limit, admissible theories bifurcate into a continuous hyperbolic phase N_eff ≥ 2 and a discrete level-truncated phase N_eff = 2cos(π/(k+2)). When a Hecke-type braiding is present, these bounds induce a corresponding dichotomy for the deformation parameter q. The results establish a model-independent phase structure for UTMF, demonstrating that energetic and topological data are strongly filtered by positivity rather than freely specifiable.

1. Introduction

The Unified Topological Mass Framework (UTMF) isolates a minimal categorical substrate capable of generating representation-theoretic emergence, including Schur-type operator bases and finite-size truncation. Recent work has shown that the quantitative invariants controlling such emergence—most notably the effective loop parameter N_eff and, in braided realizations, the deformation parameter q—are intrinsically determined by closed-diagram evaluation and local braiding spectra.

The purpose of the present paper is different in scope. Rather than deriving N_eff and q, we investigate how positivity of the categorical trace pairing constrains their admissible values once a minimal diagrammatic interface is present. We show that positivity alone induces a rigid phase structure, sharply limiting the space of consistent quantitative realizations of UTMF.

2. Kernel Assumptions

Let C be a spherical braided dagger monoidal category with generating object X. For each n, define:

𝒜_n := End_C(X^{⊗n})

The kernel supplies:

A spherical trace Tr: 𝒜_n → ℂ
A dagger †: 𝒜_n → 𝒜_n
A positive semidefinite pairing ⟨A, B⟩ := Tr(A†B)

Define the effective loop parameter:

N_eff := Tr(Id_X)

No assumption of gauge symmetry, spacetime structure, or field-theoretic origin is made.

3. Temperley–Lieb Interface Axiom

We formalize the minimal condition under which TL-type positivity constraints apply.

Definition 3.1 (TL/Markov Interface)

A UTMF realization admits a Temperley–Lieb interface if, for each n, there exists a *-algebra surjection:

π_n: TL_n(δ) ↠ 𝒯_n ⊆ 𝒜_n

𝒯_n is generated by cups, caps, and local two-strand projectors
The categorical trace restricted to 𝒯_n coincides with the Markov trace on TL_n(δ)
δ = N_eff

This axiom is local and checkable, and holds in standard Temperley–Lieb, quantum group, and braided-fusion realizations.

4. Finite-Order Positivity Constraints

We first consider positivity at fixed strand number.

Theorem 4.1 (Finite-n Positivity Bound)

Assume the TL/Markov interface holds and that the trace pairing is positive semidefinite on 𝒯_n for all n ≤ N. Then:

N_eff ≥ 2cos(π/(n+1))

Alternatively, N_eff may lie on the discrete series:

N_eff = 2cos(π/(k+2)), k ≤ n-1

Sketch of proof

Positivity of the Markov trace is equivalent to nonnegativity of the trace norms of Jones–Wenzl idempotents. These norms are controlled by Chebyshev polynomials U_n, whose first zero determines the obstruction scale.

5. Infinite-Order Dichotomy

Requiring positivity at all orders yields a sharp bifurcation.

Theorem 5.1 (All-Order Positivity Dichotomy)

If the TL/Markov interface holds and the trace pairing is positive semidefinite on 𝒯_n for all n, then exactly one of the following holds:

1. Continuous phase: N_eff ≥ 2

2. Discrete phase: N_eff = 2cos(π/(k+2)) for some integer k ≥ 1, and the interface sector truncates at level k.

Thus UTMF admits a universal phase structure independent of microscopic realization.

6. Braiding and the q-Phase Structure

Suppose additionally that the two-strand braiding on C induces a Hecke-type relation with:

N_eff = q + q⁻¹

Theorem 6.1 (Braiding Phase Dichotomy)

Under the above assumptions, positivity implies:

q ∈ ℝ_{>0} with q + q⁻¹ ≥ 2 in the continuous phase
q = e^{iπ/(k+2)} with q a root of unity in the discrete phase

Any energetic renormalization of local crossings preserves the unit-modulus branch only if the renormalization is locally degenerate.

7. Interpretation

These results show that quantitative realizations of UTMF are filtered, not freely tunable. Once a minimal TL interface is present, positivity alone enforces:

Universal lower bounds on N_eff
A discrete-versus-continuous phase structure
Rigidity of torsion (fusion) phases against energetic perturbation

This places strong structural constraints on any proposed physical or phenomenological instantiation of the framework.

8. Conclusions

We have shown that positivity of the categorical trace pairing induces sharp, model-independent constraints on the effective loop parameter and braiding deformation in UTMF. These constraints arise purely from the existence of a local Temperley–Lieb interface and do not rely on gauge theory, spacetime assumptions, or specific microscopic dynamics.

The resulting phase structure provides a new organizing principle for quantitative emergence in UTMF and a powerful filter on admissible models.

References

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