# Worked Example: Rational CFT Fusion Matrices as RU Templates

## Purpose

## example audited artifact (concrete)

This engine now includes a fully instantiated, mechanically auditable bridge artifact:

- `docs/Example_Fibonacci_Minimal.md`

It ships a complete host instance plus an RU robust certificate (`sources/ru/robust_certificate_fibonacci_fusion.json`) and a bridge wrapper (`sources/ru_bridge/bridge_certificate_fibonacci_fusion.json`).


This example shows a fully concrete reduction: a rational conformal field theory (RCFT) fusion system is converted into a constrained positive propagation instance.

The point is not to claim novelty about RCFT itself.
The novelty is that your RU pipeline gives a **finite, certificate-driven** way to decide and quantify:

- whether the fusion-driven propagation mixes projectively (a Doeblin-style condition)
- whether that mixing is forced on all tight trajectories (derived recurrence, OG4 exclusion)
- how stable the conclusion is under controlled perturbations of the input data

## The RCFT data used

An RCFT provides:

- a finite label set \(\mathcal{I}\) of primary fields (or simple objects)
- nonnegative integer fusion coefficients \(N_{ij}^{\;k}\) describing how sectors compose

For each fixed label \(i\), define the fusion matrix \(N_i\) by
- \((N_i)_j^{\;k}=N_{ij}^{\;k}\)

These matrices have nonnegative entries and encode a positive propagation on the cone \(\mathbb{R}_+^{|\mathcal{I}|}\).

## RU reduction

Choose a constrained propagation setting:

- states \(V\) represent allowed “macro-sectors” or constraints on which labels may appear next
- edges \(e:u\to v\) encode an allowed use of a generator label \(i(e)\in\mathcal{I}\)
- templates \(T_e:=N_{i(e)}\)

If additional constraints exist (selection rules, boundary conditions, defect lines), they are encoded by restricting the allowed edges, or by working with block templates that incorporate boundary projections.

### Costs and tight core

To apply the RU tight-core machinery, you choose an additive cost on edges.

Common choices that remain within the RU formalism:

- uniform cost \(c(e)=0\) to study maximal growth structure
- label penalties \(c(e)=\kappa(i(e))\) to bias toward a restricted set of generators
- energetic costs derived from a physical objective you care about (if the mapping is justified)

Given \(c\), RU constructs a calibrated potential \(u\) and reduced costs \(c_u\ge 0\).
The tight edges are exactly those with \(c_u(e)=0\).
The induced symbolic system on the tight edges is the candidate “dominant fusion channel system.”

## What the RU conclusions mean here

### Doeblin block and primitivity

A Doeblin word in RU corresponds to a finite product
- \(W = T_{e_L}\cdots T_{e_1}\)

that is strictly positive on the relevant active face.

In the fusion-matrix setting, a sufficient condition is that some constrained product yields a matrix with strictly positive entries on the chosen sector supports.

Interpretation:

- the constrained fusion dynamics is **primitive** on the active subsector
- repeated use of allowed generators cannot avoid spreading support

### Hilbert-metric contraction and “universal direction”

When Doeblin plus recurrence holds, RU gives:

- projective contraction of normalized propagation
- an explicit exponential contraction rate

In this setting, that means:

- repeated constrained fusion pushes any positive initial vector toward a universal projective shape on the active face
- the convergence has an explicit rate once bounds and a Doeblin margin \(\delta\) are certified

This is the certificate-driven version of the informal statement:
- “the dominant fusion sector has a unique asymptotic profile”

## Obstruction taxonomy in RCFT language

| RU obstruction class | RU meaning | RCFT/fusion interpretation |
|---|---|---|
| OG1 | no aperiodic tight SCC | the constrained fusion graph is periodic or decomposes into periodic components |
| OG2 | active support inconsistent | superselection or constraint map prevents a stable active subsector |
| OG3 | no Doeblin block exists | constrained fusion never becomes primitive on the active subsector |
| OG4 | Doeblin word is avoidable | there exists an infinite constrained fusion path that avoids all mixing blocks |

This gives a clean classification of why “dominant sector universality” fails, in finite terms.

## How to engine the result as an auditable certificate

Once the reduction is fixed, you emit a robust certificate with fields:

- tight-core data: tight edges, tight gap \(\gamma\), SCC periods
- Doeblin data: word \(w\), length \(L\), margin \(\delta\), uniform bounds \(M,U\)
- recurrence data: RC1 holds or OG4 witness
- rate pack: explicit contraction constants and rate

The exact schema is in:

- `sources/ru/robust_certificate_schema.json`

This makes the statement “fusion propagation mixes universally” not a slogan but a checkable claim.

## What is genuinely new in this bridge

Even if RCFT fusion theory is classical, the bridge contributes a new kind of object:

- a portable **universality certificate** that can be produced for a given constrained fusion system
- explicit margins and rates that are stable under perturbations
- a finite obstruction witness when it fails

That is precisely the shape of a -style “global universality” demonstration.
