# step 28 — Referee Map: The  Claim, Precisely

## Purpose

This step provides a referee-facing summary of what your engine contributes when viewed through the string theory bridge.

## One-sentence claim

Your RU gap-free global universality framework provides a finite, certificate-driven method to decide and quantify when constrained positive propagation exhibits forced projective universality, and it supplies explicit obstruction witnesses when it does not—making it directly plug-and-play for string-theory-adjacent positive structures (fusion/channel systems, transfer operators, truncated positivity propagations).

## Core contributions (what is new)

- A robust certificate object with margins:
  - tight gap γ, Doeblin margin δ, recurrence predicate, and explicit contraction rates
- A global region report:
  - a finite coverage procedure certifying a whole parameter region (moduli box) with uniform conclusions
- A reduction dictionary from string-theory-adjacent data to RU instances
- A worked example suite plus an obstruction atlas

## What makes it -level

It transforms informal arguments:

- constraints force structure
- dominant channels govern asymptotics
- mixing implies universality

into a finite verification pipeline:

- compute tight core
- attempt Doeblin witness
- certify or produce finite obstruction
- compute explicit rates
- export a machine-auditable certificate

## What you are not claiming

- not claiming string theory is experimentally verified
- not claiming every string construction fits RU

You are claiming:

- when a construction fits the finite positivity template, universality can be decided and quantified with certificates.
