# RU to String Theory Object Dictionary

## Purpose

This document specifies a reduction template that takes a string-theoretic “positive propagation with constraints” situation and converts it into the RU pipeline inputs.

If the template cannot be instantiated, that is not a failure. It is a certified boundary: the situation is not in the scope of the RU universality engine.

## Reduction template at a glance

| RU pipeline object | RU meaning | String-theory analogue that often matches | What must be true for the reduction to be valid |
|---|---|---|---|
| Reduced directed graph \(G=(V,E)\) | finite state graph for allowed transitions | fusion/channel graph, quiver, sector adjacency, allowed OPE channel transitions | the state set is finite after the chosen truncation or sector restriction |
| Edge cost \(c(e)\) | additive objective; extremal mean \(\lambda_*\) sets the “minimal growth” regime | conformal weight shifts, action increments, energy penalties, selection costs | costs must be well-defined on edges and comparable across paths |
| Templates \(T_e\in\mathbb{R}_+^{d\times d}\) | nonnegative propagation blocks | fusion matrices, nonnegative transfer operators, incidence matrices of allowed moves | templates must be entrywise nonnegative and composable along paths |
| Potential / calibrated subaction \(u\) | normalizes costs into reduced costs \(c_u\ge 0\) | gauge-like reweighting that makes “locally optimal” edges explicit | a potential exists for the chosen cost objective on the reduced graph |
| Tight edges \(E^{(0)}=\{c_u=0\}\) | locally optimal transitions | dominant channels or optimal sector moves | the reduced costs separate tight vs non-tight with a margin \(\gamma>0\) if robustness is desired |
| Tight SCC(s) | recurrent core candidates | irreducible subsectors of the fusion/channel system | at least one SCC exists; aperiodicity determines whether mixing is possible |
| Active support sets \(S(v)\) | descriptor state supports; active face tracking | superselection supports, allowed label subsets, boundary condition supports | supports must be consistent with propagation under tight transitions |
| Doeblin word \(w\) with \(\delta>0\) | a finite block that spreads mass across the active face | a short sequence of fusions/transitions that reaches all supports with uniform positivity | there exists a finite block with strictly positive lower bounds on the active face |
| Recurrence predicate RC1 | Doeblin block is unavoidable on the induced subshift | mixing is forced by finite-state structure, not by external ergodicity assumptions | avoidance graph has no directed cycle (no “word-avoidance loop”) |
| Hilbert-metric contraction | projective mixing to a unique direction | convergence to a universal dominant “shape” (PF direction) under normalized propagation | Doeblin plus recurrence yields explicit contraction and an explicit exponential rate |

## How to read “string theory analogue” correctly

This dictionary is intentionally conservative.

- It does not claim that every string-theory construction fits.
- It claims that several important ones do fit, once you identify the correct finite sector and the correct positive propagation operator.

The key scope condition is positivity on a cone plus finite template control.

## Failure modes and what they mean

| RU failure mode | What it means in RU | String-theory interpretation (typical) |
|---|---|---|
| No aperiodic tight SCC (OG1-style) | the tight core is periodic or decomposes into purely cyclic behavior | dominant sector cycles without mixing; multiple disjoint periodic components |
| Active face inconsistency (OG2-style) | supports do not propagate consistently along tight edges | superselection/boundary constraints prevent a uniform active sector description |
| No positive Doeblin block (OG3-style) | propagation never achieves uniform positivity on the active face | reducibility or lack of primitivity; no finite “mixing word” exists in the chosen sector |
| Avoidable Doeblin word (OG4-style) | a Doeblin word exists but can be avoided forever along tight paths | symbolic constraints allow an infinite avoidance subshift; mixing is not forced |
| Tight gap \(\gamma\) not positive | tight set unstable under perturbation | small deformations reorder which channels are dominant; no stable core |
| No uniform bounds on templates | explicit constants cannot be made global | parameter region too large; need chamberwise analysis or further truncation |

## What the bridge produces when the reduction succeeds

When a reduction succeeds, the bridge produces two things:

- a standard RU **robust certificate** (see `sources/ru/robust_certificate_schema.json`)
- optionally, a **global universality report** certifying a region in parameter space (see `sources/ru/global_universality_report_schema.json`)

Those objects are the “proof carriers” that make universality claims auditable.
