# step 26 — Global Universality Across Moduli Boxes: Chamber Structure and Robustness

## Purpose

This step connects the “global” part of your RU global-universality program to string theory in the most concrete way:

- string theory often has families of data depending on parameters (moduli, couplings, boundary data)
- your RU global report schema is designed to certify universality uniformly across a region

The core idea is chamber stability:

- if a tight gap γ>0 separates the dominant core, then the core structure is stable under small perturbations
- this stability induces chambers in parameter space where the universality type is constant

## The global universality report object

The shipped schema `sources/ru/global_universality_report_schema.json` is the proof carrier for region-wide claims:

- it records the region R (a parameter box or domain)
- a finite set of sample points
- local robust certificates at those points
- a verified coverage rule yielding a union of certified balls covering R

## Chamber decomposition (conceptual)

Define a type label τ consisting of:

- tight SCC structure + periods
- existence/nonexistence of Doeblin word
- recurrence vs avoidability
- explicit contraction rate (when present)

Then τ is locally constant wherever γ>0 and Doeblin margins remain positive.

Boundaries occur where γ=0 (core changes), δ=0 (mixing collapses), or periods change.

This gives a mathematically precise “universality phase diagram.”

## How to use this in practice (algorithm)



###  audited artifacts (concrete)

- `docs/Example_Fibonacci_Minimal.md` (minimal audited bridge)
- `docs/Example_ConstraintGraph_Mixing.md` (nontrivial constraint graph, certified mixing)
- `docs/Example_OG4_AvoidableDoeblin.md` (explicit OG4 boundary witness)
- `docs/Global_Report_Fibonacci_Alpha.md` (chamberwise “moduli box” report)




- Choose a parameter region R (a “moduli box” where truncations remain valid).
- At each sampled point p in R:
  - build the RU instance via the bridge dictionary
  - compute a robust certificate with ε*(p)
- Use the global report logic to cover R by balls of radius ε* (in the chosen perturbation metric)
- Output:
  - a region-wide universality label τ
  - or a list of boundary zones where margins collapse

## Why this matches string reality

String constructions often feature:

- stability within chambers
- sharp boundaries where a new sector becomes dominant

Your RU global framework is a direct way to express those phenomena in a certificate language.
