# step 24 — Worked Bridge Example B: Fusion, Quantum Dimensions, and Forced Universality

## Purpose

This worked example shows how a standard string-theory-adjacent structure already contains a natural “RU shape”:

- fusion rule matrices act as nonnegative templates
- quantum dimensions form Perron–Frobenius eigenvectors
- normalized fusion propagation often exhibits “universal direction” behavior

Your RU pipeline upgrades this from “PF folklore” to a certificate + obstruction story:

- either forced mixing with explicit rates
- or a finite witness for why universality fails

## Starting point: fusion matrices and Perron–Frobenius eigenvectors

In unitary fusion settings, the vector of quantum dimensions is the Perron–Frobenius eigenvector of each fusion rule matrix (standard in TQFT/anyon literature).

That is already a universality statement, but it is incomplete in two ways:

- it is single-step (one matrix at a time), while constrained dynamics uses sequences
- it does not classify why universality fails under constraints (periodicity, avoidance, reducibility)

RU supplies the missing layers.

## RU reduction

### State space

Let the labels (simple objects / primaries) be a finite set I.

Let a constraint system select allowed sequences of generators i_1,i_2, (boundary conditions, defect insertion constraints, or a chosen generating set).

### Templates

For each generator i, define the nonnegative fusion matrix N_i with entries (N_i)_j^k = N_{ij}^k.

Set RU templates on edges by T_e := N_{i(e)}.

### Costs and tight core

Choose a cost model:

- the “unbiased” case c(e)=0 studies maximal growth structure
- penalty costs c(e)=κ(i(e)) bias which generators count as tight

RU calibrates and extracts the tight edges E^(0), giving an induced constrained symbolic system on dominant generators/channels.

## What RU certifies here

### Doeblin word

A Doeblin word is a finite constrained product W that is strictly positive on the active face.

In fusion language, that is a primitivity witness for the constrained fusion dynamics.

### Recurrence vs avoidability

Even if a Doeblin block exists, the constrained system might avoid it forever.

RU detects this by producing either:

- RC1 (unavoidable Doeblin blocks) → forced universality
- or an OG4 avoidance witness → universality not forced globally

### Quantitative contraction

When Doeblin + recurrence holds, RU outputs:

- explicit Hilbert-metric contraction constants
- an explicit exponential contraction rate

This turns “fusion dynamics converges to a universal profile” into a numerical certificate.

## Why this is a natural string-theory connection

Fusion categories and RCFT data sit inside many string constructions:

- worldsheet CFT sectors on orbifolds and compactifications
- boundary CFT for D-branes
- topological strings and 3D TFT structures

The bridge claim is:

- wherever the string-theory side produces finite fusion-like propagation under constraints, RU can certify whether universality is forced and quantify it.

## Deliverable expectation

A fully instantiated version of this worked example would include:

- the chosen finite label set I
- the constraint graph G=(V,E)
- the selected templates T_e
- an RU robust certificate object referencing the schemas in `sources/ru/`

This document is the blueprint for producing that certificate in any concrete model.
