# step 25 — Worked Bridge Example C: Bootstrap Positivity as a Finite RU Propagation Problem

## Purpose

This step gives a conservative, scope-labeled bridge from modern conformal bootstrap practice into your RU framework.

It is intentionally phrased as:

- a finite truncation bridge
- an algorithmic positivity bridge
- not a claim that the full infinite-dimensional bootstrap is already an RU instance

Numerical bootstrap rests on crossing symmetry and positivity constraints implied by unitarity / reflection positivity.

## Bootstrap data and positivity

Bootstrap constraints restrict:

- operator dimensions
- OPE coefficients (with positivity/reality constraints in unitary cases)

Reflection positivity is one route by which positivity enters the bootstrap machinery.

## A conservative RU reduction template

### Step 1: choose a finite truncation

Pick:

- a finite set of operators (or a finite basis expansion)
- a finite set of constraints (derivative functionals, crossing equations sampled at points, etc.)

This yields a finite vector of unknown nonnegative coefficients x ∈ R_+^d.

### Step 2: define a positive propagation operator

Many bootstrap algorithms can be organized as repeated application of:

- constraint-imposing updates
- feasibility propagation
- cone-preserving transforms

A conservative approach is to define templates T_e that map a feasible coefficient vector to a new feasible candidate vector while preserving nonnegativity (e.g., via reweighting and projection onto an inner approximation cone).

The RU graph G encodes which update steps are allowed in which order, yielding a constrained template product system.

### Step 3: define an objective cost

Define a cost that measures “tightness to feasibility” or “distance to a target extremal bound” so that a tight set corresponds to extremal propagation steps.

RU then extracts a tight core representing the dominant constraint-satisfying dynamics under the chosen update scheme.

## What RU would deliver if the reduction is valid

If the templates are truly cone-preserving and the tight system admits a Doeblin + recurrence certificate, RU yields:

- uniqueness and stability of a projective “shape” of extremal feasible vectors
- explicit mixing/contraction rates for the normalized propagation

If it fails, RU yields obstruction witnesses that identify why the propagation does not force universality (periodicity, reducibility, avoidable mixing blocks).

## Why this matters as a bridge to string theory

Bootstrap constraints are a major modern tool for constraining 2D and higher-dimensional CFTs, including those that can serve as string worldsheet theories.

Your RU lens offers a new way to talk about universality inside bootstrap computations:

- not “we solved it numerically,” but “the finite propagation has a certified mixing core, hence a stable universal shape in the extremal regime.”

## Scope and honesty flags

This bridge is valid only if:

- the chosen bootstrap update scheme can be expressed as a finite cone-preserving template system
- truncation errors are controlled (or treated as perturbations within an RU robustness budget ε*)

step 26 turns those words into a chamber/robustness framework.
