FRe-encoded probabilities and Bell
Two papers by Czachor report Bell-inequality violations from a classical local model by changing the arithmetic used on probabilities. Read as mathematics, the models are consistent. Read as probability that an experimenter can count, they break rules every lab relies on.
Any re-encoding of probabilities that respects mixing two sources, or merging two outcomes, is the identity. The models apply a nonlinear g to every probability, so the CHSH values above 2 come with those rules switched off.
Drag the paper point, or use the sliders. Red: the encoded distribution.
- encode, then merge 2 and 3
- 0.333333
- merge 2 and 3, then encode
- 0.5
- difference
- -0.1667
The Bell numbers, read in ordinary arithmetic
- CHSH at level 1
- 2.828427
- g^1(t p + (1 - t) q)
- 0.172746
- t g^1(p) + (1 - t) g^1(q)
- 0.1875
- mixing defect
- -0.01475
A re-encoding that keeps ordinary mixing and merging is the identity. The Bell values above 2 come from a re-encoding that keeps neither: only level 0 has zero mixing defect at every t, and level 0 gives exactly the classical 2.
The precise statement
If phi: [0, 1] -> [0, 1] fixes 0 and 1 and phi(t p + (1 - t) q) = t phi(p) + (1 - t) phi(q), then phi(t) = t (take p = 1, q = 0; no continuity needed). On the simplex, a map that fixes the deterministic outcomes and commutes with binary mixing is the identity.
For phi applied coordinatewise and renormalized, merging outcomes after encoding equals merging before exactly when phi is additive. Square-and-renormalize sends (1/2, 1/4, 1/4) to (0.6667, 0.1667, 0.1667): merging afterwards gives 0.333333, merging first gives 0.5.
Where it breaks
The Czachor model is the classical uniform-arc model with every probability pushed through g. Normalization survives only because every classical marginal is exactly 1/2. Additivity does not: an arc of classical measure 1/8 gets P' = 0.0732233, two of them sum to 0.146447, yet their union gets 0.25.
In ordinary arithmetic the hierarchy reaches CHSH = 2.82843 at level 1 and 4 by level 7. Read in the transported arithmetic, level 1 gives the classical 2. The authors say as much: the violation appears when probabilities of one level are combined by the rules of another.
Prior art and sources
Classical results it reduces to
- Kolmogorov additivity
- Fine 1982 (joint distributions and Bell inequalities)
- Non-additive measures (Pap; Grabisch, Marichal, Mesiar, Pap)
- Cauchy functional equation
NNC anchors
- Arithmetic loophole in Bell's theorem (czachor_2020_arithmetic_loophole_bell)
- Faking Quantum Probabilities: Beyond Bell's Theorem and Tsirelson Bounds (czachor_2021_faking_quantum_probabilities)
OpenAI families named (context only)
- Threshold repetition for entangled games (family 277)
- Entanglement without distillable secret key (family 272)
Model-written and unreviewed. No result on this page depends on one.
Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X3/FINDINGS.md. Every number on this page is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.