# Study 12 — Quantum Parallel Repetition 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-12-chance.html | sealed POST |
| IDE | https://affine.earth/language-game/ide.html#qpr | file template; no auto-POST |
| Look | https://affine.earth/language-game/#researcher | GET only |
| Affine story | https://affine.earth/language-game/#story/Study-12-Quantum-Parallel-Repetition-Shear | this charter |
| Court | `POST /language-invariant/game/chance/ingest` `rounds=3` | sealed `27/64` / `27/64` · `STUDY12_QPR_PROVEN` · float `1.5` → HTTP 400 |
| Humanity | Discrete repetition bounds without a second amplitude law | float adversary `751/1000` is the sealed miss |
| DFT map | many-electron \(\psi\), Born amplitudes | Anima approximates a probability in floats. Study 12 replaces \(\psi\) with \(W_n\in\mathbb{Q}\). |

---

## For every reader

Quantum parallel repetition asks: *if Alice and Bob play a nonlocal game \(G\) in parallel \(n\) times, how fast does the winning probability decay?* The standard proof uses **continuous amplitudes** and tensor products in \(\mathbb{C}\). UUM-8D replaces amplitudes with **vQbit** primitives and **Jordan-bonded** entanglement matrices — discrete state transitions with exact rationals.

This study will grade whether integer-bounded parallel repetition bounds match published thresholds — and whether float amplitude models shear the bound.

**LIVE CLAIM** — apex `chance` returns `STUDY12_QPR_PROVEN`. Named receipt is the court.

---

## The forcing and the track

| Piece | Instantiation |
|---|---|
| **Forcing** | Parallel repetition index \(n\) (integer rounds) |
| **Clock** | Round index \(k=1\ldots n\) |
| **Track** | Bipartite strategy simplex in the integer manifold |
| **Raw archive** | Published game tables (CHSH, Magic Square, etc.) with exact rational winning thresholds |
| **Adversary** | Float amplitude product model — continuous \(\mathbb{C}^{d^n}\) tensor |
| **Future events** | New games registered before bounds are examined |

---

## UUM-8D integration path

| Standard | UUM-8D |
|---|---|
| Complex amplitude \(\alpha\in\mathbb{C}\) | vQbit \((\tau, \mathrm{phase}, \mathrm{amplitude})\) with unit-norm rational |
| Tensor product Hilbert space | Jordan algebra state on M⁸ = S⁴ × C⁴ |
| Winning probability (float) | Ordinal comparison via cross-multiplication on \(\mathbb{Q}\) |
| Parallel repetition bound | Discrete collapse turn count at sealed \(\chi_{\max}\) |

Reference substrate map: `cells/xcode/Sources/InvariantCompiler/QuantumShearMap.swift` — transport criterion per algorithm row.

---

## Success criteria (pre-registration)

1. Integer Jordan-bonded repetition bound \(\le\) published upper bound for CHSH at \(n=1,2,4,8\).
2. Float amplitude adversary **over- or under-estimates** the sealed bound at least once (shear measurable).
3. No `Float`/`Double` on the seal path.

---

## Status

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

Replacement trio: this page kills the wavefunction · [11 kills continuous density](Study-11-Ehrhart-Volume-Shear) · [13 kills KS potential](Study-13-Connes-Rigidity-Shear). Synthesis: [Fourier Phantom](Impact-Study-Fourier-Phantom).

When complete, results will publish as `Study-12-Quantum-Parallel-Repetition-Results.md`.