# step 27 — Obstruction Atlas: Translating OG Witnesses into String-Theory Language

## Purpose

To make a universality claim credible at maximal level, you must explain the boundaries.

This step builds an obstruction atlas:

- each RU obstruction type is translated into string-theory-adjacent meanings
- each comes with an explicit finite witness style

## The four core obstruction types (bridge interpretation)

### OG1 — Periodic tight core (no aperiodic SCC)

- RU meaning: the dominant constrained system is periodic; projective mixing is not forced.
- String-language meaning: the dominant channel dynamics cycles; no genuine mixing mechanism in the chosen sector.

Witness style:
- SCC decomposition + gcd-of-cycle-lengths certificate of periodicity.

### OG2 — Active face inconsistency

- RU meaning: no consistent active support set is preserved by tight propagation.
- String-language meaning: superselection/boundary constraints prevent a stable “active subsector” description.

Witness style:
- a finite contradiction: a support that must appear after some tight move but cannot be supported elsewhere.

### OG3 — No Doeblin block exists (no primitive mixing product)

- RU meaning: even with all allowed tight moves, no finite block spreads positivity uniformly.
- String-language meaning: constrained fusion/channel propagation is reducible; no finite word produces full mixing on the active sector.

Witness style:
- product-automaton reachability certificate that a full-support word does not exist.

### OG4 — Doeblin exists but is avoidable

- RU meaning: mixing blocks exist but can be avoided forever along tight paths.
- String-language meaning: constraints admit an infinite avoidance subtheory; universality is not forced globally.

Witness style:
- avoidance automaton cycle witness.

## Why this atlas matters



See also: `docs/Example_OG1_PeriodicCore.md` for a fully audited OG1 witness.





See also: `docs/Example_OG4_AvoidableDoeblin.md` for a fully audited OG4 witness.



In string theory and CFT, it is common to say:

- “the dominant sector should mix”
- “constraints should force universality”

Your atlas turns these into a finite classification:

- you either certify universality
- or you produce a concrete witness of what structure prevents it


See also: `docs/Example_OG2_ActiveFaceInconsistency.md` for a fully audited OG2 witness.


See also: `docs/Example_OG3_NoDoeblin.md` for a fully audited OG3 witness.


Real-data OG3 witness: `docs/OG3_Ising_NoDoeblin.md`.


Real-host OG1/OG4 boundaries (Fibonacci):
- `docs/OG1_Fibonacci_PeriodicCore.md`
- `docs/OG4_Fibonacci_AvoidableDoeblin.md`


Real-host-derived OG2 witness (Ising + projection constraint): `docs/OG2_Ising_Projection_Inconsistency.md`.


Real-host OG3 witness (Toric code fusion): `docs/OG3_ToricCode_NoDoeblin.md`.
