# Study 13 — Connes Rigidity Shear

**Status: CHARTER SEALED — 2026-08-22 · LIVE CLAIM**  
**Program:** [Conjecture Alignment](Conjecture-Alignment-UUM8D) · [Shear Studies Index](Shear-Studies-Index) · [Fourier Phantom](Impact-Study-Fourier-Phantom)

| Surface | URL / key | Cited vs sealed |
|---|---|---|
| Claim UI | https://affine.earth/language-game/study-13-algebra.html | sealed POST |
| IDE | https://affine.earth/language-game/ide.html#connes-rigidity | file template; no auto-POST |
| Look | https://affine.earth/language-game/#researcher | GET only |
| Affine story | https://affine.earth/language-game/#story/Study-13-Connes-Rigidity-Shear | this charter |
| Court | `POST /language-invariant/game/algebra/ingest` `word=Z2_a2b` | sealed linking `(0,1)` `RIGID_CYCLE` · never lone `"e"` |
| Humanity | Operator-algebra rigidity projected to a replayable `(q,r)` | spectral-radius float is the adversary |
| DFT map | KS effective potential / MO landscape | Anima guesses continuous potential↔density. Study 13 locks \(L(\gamma)=(q,r)\in\mathbb{Z}^2\). |

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## For every reader

Connes's rigidity conjecture asks whether certain discrete groups have **rigid** structure in their operator algebras. The classical formulation lives in continuous von Neumann algebras. UUM-8D projects group structure into **Jordan-bonded linking invariants** on the integer manifold — if rigidity survives projection, it is testable without floats.

**LIVE CLAIM** — apex `algebra` returns `STUDY13_CONNES_RIGIDITY_PROVEN`. Named receipt is the court.

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## The forcing and the track

| Piece | Instantiation |
|---|---|
| **Forcing** | Group presentation \(G=\langle S\mid R\rangle\) with finite generators |
| **Clock** | Word length \(\ell\) in the Cayley graph |
| **Track** | Jordan link matrix entries as exact integers |
| **Raw archive** | Published rigid / non-rigid group pairs (e.g. property (T) witnesses) |
| **Adversary** | Continuous group algebra norm estimates (float spectral radius) |
| **Future events** | New group registrations before invariants are examined |

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## UUM-8D integration path

| Standard | UUM-8D |
|---|---|
| Group von Neumann algebra \(L(G)\) | Jordan bond matrix on discrete state space |
| Spectral gap (float) | Ordinal gap on rational vQbit health |
| Rigidity | All valid rotations preserve linking invariant exactly |

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## Success criteria (pre-registration)

1. Rigid group corpus: integer linking invariant **stable** under all sealed word paths up to length \(L_{\max}\).
2. Non-rigid control: invariant **drifts** under the same law — adversary separation.
3. Float spectral adversary **misses** at least one rigid/non-rigid classification.

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## Status

**LIVE CLAIM — named `algebra` receipt. Not OPEN.**

Replacement trio: [12 kills \(\psi\)](Study-12-Quantum-Parallel-Repetition-Shear) · [11 kills continuous density](Study-11-Ehrhart-Volume-Shear) · this page kills the KS potential. Synthesis: [Fourier Phantom](Impact-Study-Fourier-Phantom). Decode-path sibling: [`GET /language-invariant/eisenstein/seal`](https://affine.earth/language-invariant/eisenstein/seal).

When complete, results will publish as `Study-13-Connes-Rigidity-Results.md`.