The Motif Calculus Kernel of UTMF isolates the minimal algebraic structure required to produce representation-theoretic emergence, including Schur and restricted-Schur operator bases and finite-size truncation. While the kernel establishes these phenomena qualitatively, quantitative emergence depends on two scalars: an effective loop value and, in the braided case, a deformation parameter .

In standard gauge-theoretic or topological quantum field theoretic constructions, these quantities are specified as external inputs. The purpose of this paper is to show that within UTMF they arederived quantities, determined by intrinsic evaluations of closed motifs and local braiding data.

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

The kernel provides:

1. A dagger
2. A spherical trace
3. A braiding
4. A positive semidefinite pairing

Define the loop parameter:

To extract numerical invariants, we introduce a minimal quantitative layer.

3.1 Closed Diagram Evaluation

Let be a closed diagram in . Assume:

• An energy functional , additive under disjoint union
• A topological amplitude , multiplicative under disjoint union
• A scale

Define:

3.2 Trace by Closure

For , define

where denotes the spherical closure. Cyclicity follows from sphericality.

Let denote the closed loop labeled by .

Thus the loop parameter factorizes into a topological dimension and an energetic weight.

5.1 Local Semisimplicity Assumption

For the purpose of extracting a Hecke deformation parameter, we assume:

with two simple channels.

5.2 Braiding Eigenvalues

Naturality implies:

where are the channel projectors.

Define:

Let be the minimal polynomial of . Trace cyclicity yields:

In the Hecke normalization , one has , and is obtained by identifying the roots of .

Assume a local skein relation for :

If is local, then:

provides the effective skein coefficient in . Thus is ultimately fixed by relative energetic costs of local motifs.

In the Temperley–Lieb category , the loop evaluates to . Hence:

The two-strand algebra yields eigenvalues with:

Trace-moment reconstruction recovers exactly.

The derived values and control:

• Finite-size truncation via norm factors
• Additional level truncation at roots of unity
• Stability of representation sectors under non-planar mixing

We have derived the effective loop parameter and braiding deformation directly from the Motif Calculus Kernel with a minimal energetic extension. These quantities are not external inputs but intrinsic invariants determined by closed-motif evaluation and local braiding spectra. This completes the quantitative foundation underlying representation-theoretic emergence in UTMF.