# step 29 — Beyond Finite Fusion: Non-Rational and Continuum Limits via Robust Truncation

## Purpose

This step explains how to extend the RU→string bridge beyond strictly finite RCFT fusion systems.

The idea:

- build finite truncations with explicit perturbation metrics
- use RU robustness and chamber logic to control what persists in the limit

## Why this matters in modern string theory

Recent work develops non-rational Verlinde-type identities and applications (Virasoro TQFT, AdS3/CFT2 ensembles).

Modern minimal-string definitions connect to random matrix models and nonperturbative structure.

Both contexts naturally produce:

- positive kernels or positive spectral densities
- truncation schemes (basis cutoffs, discretizations, orthogonal polynomial truncations)
- questions about “universal” limiting behavior

## RU strategy: certified truncation

Given a positive integral operator K or kernel-like fusion object, pick a truncation scale N and produce a finite template system K_N.

Then:

- treat ||K-K_N|| as a perturbation (in a chosen metric)
- certify universality for K_N with RU, producing ε*(N)
- if truncation error ≤ ε*(N), the conclusion transfers

## Where the difficulty lives

Two hard parts remain:

- choosing a perturbation metric compatible with the physics interpretation
- bounding truncation error sharply enough relative to ε*

Your framework contributes the missing piece:

- once the error is bounded, the universality statement becomes certificate-driven.
