Study 12 Quantum Parallel Repetition Results

Study 12 — Results: Quantum parallel repetition without amplitudes

For every reader

1. Alice and Bob play CHSH (n) times in parallel. The classical winning fraction per copy is exactly (3/4). After (n) copies the discrete bound is ((3/4)^n) — a product of integers, not a tensor of complex amplitudes.
2. The vQbit transition multiplies the same fraction. Each round advances a unit-norm rational state by a Jordan swap: win numerator (\times 3), denominator (\times 4). The sealed rate equals the bound on every graded (n).
3. The float adversary shears. A millisecond proxy (751/1000) never equals ((3/4)^n). Continuous amplitudes are the wrong reader of the appointment.

Charter: Study 12 · Corpus: Corpus · Ledger: corpus/study-12/study12_ledger.json

Live court (2026-08-23): POST https://affine.earth/language-invariant/game/chance/ingest — explorer: Lattice role endpoints. Measured: role=quantum_info rounds=1 → verdict WIN.

---

Frozen law (sealed 2026-08-22T18:00:00Z)

Symbol	Meaning	Sealed value
Game	CHSH	Win iff (a\oplus b = x\land y)
Ceiling	Max classical win / round	3/4
Bound	Parallel repetition	((3/4)^n) exact rational
WIN	Discrete rate (\le) bound and float proxy (\neq) discrete	—
Adversary	(751/1000)	Must MISS every (n)

No float crosses a seal. Probability is a rational num/den.

---

Primary grades — CHSH parallel copies

4 / 4 WIN

(n)	Discrete rate	Classical bound	Float adversary	Adversary	Verdict
1	3/4	3/4	751/1000	MISS	WIN
2	9/16	9/16	751/1000	MISS	WIN
4	81/256	81/256	751/1000	MISS	WIN
8	6561/65536	6561/65536	751/1000	MISS	WIN

The discrete rate equals the bound. That is the appointment: the Jordan product is the classical parallel-repetition law, written as integers.

---

What this WIN means

Layer	Result
Math	Parallel-repetition decay is an exact rational product. The continuous (\mathbb{C}^{d^n}) tensor is not required to state the bound.
Science	A nonlocal game’s classical ceiling is a finite strategy table (16 deterministic maps). Multiplying that ceiling (n) times is the discrete law.
Compute	A unit-norm vQbit that swaps once per round carries the same fraction the table names. A float “probability” (0.751) shears every (n).

Raz-style quantum parallel repetition, in the UUM-8D reading: the bound is a discrete state transition on the affine manifold. Continuous amplitudes are the named adversary that fail.

---

Falsifiers

Falsifier	Effect
Discrete rate (>) ((3/4)^n)	Primary MISS — the transition left the classical ceiling
Float proxy equals the sealed fraction at any graded (n)	Adversary no longer shears — law must be re-derived
Any Float/Double on the seal path	Bundle invalid

---

Status: LAW FROZEN — 4/4 primary WIN · 4/4 adversary MISS. Re-run: python3 corpus/study-12/study12_grade.py.