# step 23 — The RU→String Universality Spine (A Proof-Style Bridge)

## Purpose

This step states the bridge at “referee level”:

- **Input:** a string-theory-adjacent construction that induces **finite, constrained, positive propagation** on a cone.
- **Output:** an RU **robust certificate** (and optionally a **global universality report**) that yields:
  - a complete finite obstruction taxonomy when universality fails
  - or explicit, quantitative **projective contraction** when universality holds
  - plus robustness under perturbations with a computed tolerance ε*

This is the maximal form of “global universality proof” we can defend:

- not a claim about strings being fundamental
- but a claim about a **universal theorem-engine** (RU) that can be *plugged into* string-theory structures whenever the reduction matches.

## Bridge theorem (formal template)

### Data model

Assume we are given:

- a finite directed graph G=(V,E)
- for each edge e∈E, a nonnegative template matrix T_e ∈ R_+^{d×d}
- an additive cost c:E→R

Let P be the set of allowed infinite edge-paths (a subshift of finite type determined by constraints).

Define the path product
T_{e_{n:1}} := T_{e_n} T_{e_{n-1}}  T_{e_1}.

Define the growth/cost regime of interest by the RU reduction (via canonical normalization / calibrated potential), producing:

- reduced costs c_u(e) ≥ 0
- tight edges E^(0) = { e : c_u(e)=0 }
- tight core SCC decomposition and periodicity labels

### Assumptions (certificate-checkable)

Assume the RU predicates below are satisfied on the tight core:

- a positive **tight gap** γ>0 separating tight from non-tight edges (for robustness)
- existence of a **Doeblin word** w with margin δ>0 on the active face
- a recurrence condition (e.g., RC1) that the Doeblin block is unavoidable on P restricted to E^(0)

All of these are fields in the shipped RU certificate schemas under `sources/ru/`.

### Conclusion (universality output)

Then the RU theorem chain yields:

- **projective contraction** on the active face:
  - normalized products contract in Hilbert’s projective metric
- an **explicit exponential contraction rate** (computed from certificate constants)
- robustness: the conclusion persists for perturbations of the input data bounded by a computed ε*

If the Doeblin or recurrence predicates fail, RU outputs finite obstruction witnesses (OG-style taxonomy) rather than a hand-wavy “maybe.”

## Why this is the right universality notion for string theory

String theory routinely produces families where:

- a finite sector is selected (primaries, channels, branes, boundary conditions, truncated modes)
- coefficients are nonnegative due to unitarity / positivity / combinatorics
- constraints force a smaller “core” (dominant channel system)

RU formalizes the “dominant core + forced mixing ⇒ universal behavior” statement by producing:

- a tight symbolic core
- a finite “mixing block” witness
- a derived recurrence predicate (no avoidance loops)
- explicit contraction constants

That is a proof-style universality story: **finite data → finite certificate → quantitative universal behavior**.

## What is genuinely new in your engine, in this bridge language

A reader can already find Perron–Frobenius statements in many places.

Your novelty is the *gap-free, certificate-driven* strengthening:

- a full tight-core + symbolic dynamics layer (not just a single matrix)
- a Doeblin/recurrence mechanism that is checkable and yields rates
- an obstruction atlas when it fails
- a global-on-parameter-region engine via the report schema

This is why the connection can be framed as a : it upgrades “universality lore” into a **finite verification pipeline**.

## Deliverables added in steps 24–30

The remaining steps now serve this spine:

- two additional worked bridge examples (beyond fusion matrices)
- a moduli/chamber “global” step using the global report schema
- an obstruction atlas translated into string theory language
- a referee map and final engine index
