# Adversarial review: tolerance asymmetry

Task: break the approximation-price argument in `lab/2026-09-16-approximation-price/RESULT.md`, `price.py`, and `detect.py`.

Bottom line: the CHSH communication floor is real but rediscovered, the linear CHSH knob is real inside its narrow model, and the Sycamore facts are real. The bridge tying them together is not real enough. The argument currently moves from "published high-n verification is weak for one RCS benchmark" to "adequate rendering policies are wide" by comparing unlike statistics and by treating epistemic verification slack as physical simulation slack.

## (a) Commensurability step

Grade: FATAL.

What breaks: the central four-orders-of-magnitude comparison.

The proposed meta-quantity, "fraction of the ideal quantum prediction the experiment pins down," is not invariant across the two cases. In CHSH, the meaningful nonlocal signal is not the whole Tsirelson value `S = 2.828...`; it is the excess above the local bound, `S - 2 = 0.828...`, because the communication lower bound is `p >= (S - 2)/2`. In XEB, the baseline is not a local polytope bound but the uniform distribution, with ideal fidelity `F_XEB = 1` and fully depolarized output near `F_XEB = 0`. These baselines are doing different mathematical work.

Changing the normalization already changes the headline. The report divides the CHSH 5-sigma tolerance by something like the ideal statistic; dividing by the nonlocal excess instead gives a different fraction. For `N = 1e9`, `5 sqrt(8/N) = 4.47e-4`; relative to `2 sqrt(2)` this is `1.58e-4`, relative to the nonlocal excess it is `5.40e-4`, and relative to the communication floor it is `5.40e-4` in saved bits. All are defensible normalizations. That arbitrariness is the warning sign.

More importantly, CHSH under-delivery by `delta` has a proved operational meaning in average communication cost because Pironio's CHSH result maps Bell violation to communication. XEB under-delivery by `1 - F` does not map to the same resource. It is a single benchmark statistic for a noisy circuit family. Calling both "fractional tolerance" erases the distinct denominators, null hypotheses, convex sets, and cost models.

The author is entitled to say: CHSH and RCS records certify different quantities, and the RCS fidelity benchmark leaves much less of the ideal distribution present than a high-statistics CHSH test leaves of the ideal CHSH value. The author is not entitled to say this is a commensurate four-order widening of the same tolerance knob.

Sources: S. Pironio, "Violations of Bell inequalities as lower bounds on the communication cost of nonlocal correlations," Phys. Rev. A 68, 062102 (2003), arXiv:quant-ph/0304176; F. Arute et al., "Quantum supremacy using a programmable superconducting processor," Nature 574, 505-510 (2019), doi:10.1038/s41586-019-1666-5.

## (b) Is `p >= (S - 2)/2` right?

Grade: SERIOUS for the in-file derivation; MINOR for the corrected theorem.

What breaks: the homemade mixture derivation is too narrow, but the numerical lower bound survives under the right formalism.

The simple derivation in `price.py` assumes

```text
P = p P_comm + (1 - p) P_local
```

with a single setting-independent `p`, a genuinely local no-message branch, and an arbitrary communication branch bounded only by `S <= 4`. Under those assumptions, the algebra is fine.

But this is not a general model of "a rendering policy." If the probability of sending a message depends on settings, hidden variables, or a global scheduler, the conditional no-message subensemble can become setting-biased. Then the no-message data need not be described by one fixed local behavior across all four CHSH settings. This is the same structural danger as postselection/detection-loophole reasoning: conditioning can manufacture apparent nonlocality unless the conditioning event is itself local and handled in the protocol model.

Pironio's 2003 result fixes this by not relying on that mixture story. He decomposes correlations into deterministic strategies grouped by their minimal average communication cost. For CHSH, he proves

```text
C_bar(p) >= B(p)/2 - 1
```

and, for no-signaling correlations violating CHSH, also proves sufficiency: `C_bar(p) = B(p)/2 - 1`. For maximal CHSH violation this is `sqrt(2) - 1 = 0.4142` bits. That is stronger and cleaner than the report's derivation.

Remaining caveats still matter. Pironio's `C_bar` averages over inputs and shared randomness in the Bell-scenario communication model. If a renderer is allowed centralized nonlocal access to future settings, superdeterministic setting correlations, hidden global state updates, or nonlocal precomputed outputs, the theorem may simply not apply. If the distribution of settings is nonuniform, average communication per experimental round can differ from the uniform CHSH cost while preserving the conditional CHSH table. The report should explicitly bind the model to standard Bell-simulation assumptions.

Sources: Pironio 2003, Sections II-III; especially Eq. (boundchsh), `C_bar(p) >= B(p)/2 - 1`, and Proposition 4, `C_bar(p) = B(p)/2 - 1` for no-signaling CHSH correlations.

## (c) Is the bound trivial / known?

Grade: FATAL to novelty; MINOR to validity.

What breaks: any claim that Section 1 is a new derived floor.

This is not just "probably prior art." Pironio's abstract states the exact headline result:

> "to produce using classical resources the correlations that maximally violate the CHSH inequality, sqrt(2)-1 ~= 0.4142 bits of communication are necessary and sufficient."

Inside the paper, the CHSH section derives `C_bar(p) >= B(p)/2 - 1`, then proves saturation for no-signaling correlations. That is exactly the report's `p >= (S - 2)/2` at Tsirelson, and it is stronger because it establishes optimality rather than only a lower bound from a two-branch mixture.

Earlier surrounding work includes Brassard, Cleve & Tapp, "Cost of exactly simulating quantum entanglement with classical communication," Phys. Rev. Lett. 83, 1874-1877 (1999), arXiv:quant-ph/9901035, which gives constant communication for one Bell pair and Omega(2^n) for n Bell states; and Toner-Bacon-style one-bit simulation work. But the exact average-communication CHSH value `0.4142` belongs squarely to Pironio 2003.

The report should mark this as established prior art, not as tonight's derivation except pedagogically.

Sources: Pironio 2003; G. Brassard, R. Cleve, A. Tapp, Phys. Rev. Lett. 83, 1874-1877 (1999), arXiv:quant-ph/9901035.

## (d) Sycamore / XEB as "the tolerance a renderer gets"

Grade: FATAL to the "99.8% wrong" framing; SERIOUS to the cost conclusion.

What breaks: interpreting `F_XEB ~= 0.002` as a free error budget analogous to CHSH `delta`.

The Sycamore noise model is approximately

```text
P_device = F P_ideal + (1 - F) U
```

where `U` is close to uniform. That does mean the measured output distribution is mostly depolarized. But it does not mean the renderer can be "99.8% wrong" about an ideal quantum circuit in the same sense that a CHSH renderer under-delivers `S` by `delta`. It means the experimental target itself is a noisy distribution with a small ideal component.

If the task is to reproduce the noisy device distribution, a black-box strategy can output uniform samples most of the time and pay for ideal-distribution information only in the `F`-weighted component. That suggests a linear factor of roughly `1/F` or `F` in sample-generation cost, not automatic destruction of the exponential dependence of producing the ideal component. At `F = 0.00224`, that is a factor about 450, large but not an exponential rescue by itself.

The Pan-Chen-Zhang result matters because their tensor-network method exploited this finite-fidelity target and the specific circuit geometry/depth well enough to beat the 2019 experiment's own XEB fidelity. But that is an algorithmic fact about a benchmark, not a theorem that low XEB fidelity always makes high-n quantum behavior cheap.

There is also a verification-target confusion. Arute et al.'s 5-sigma statement `F > 0.1%` is not the complete acceptance criterion for any future renderer; it was a confidence statement for a particular experiment and estimator. A simulator trying to sustain an observer-rich universe would face adaptive, cross-checking, non-XEB measurements, not only one scalar XEB threshold.

Sources: Arute et al. 2019; Y. Pan, Y. Chen, P. Zhang, "Solving the Sampling Problem of the Sycamore Quantum Circuits," Phys. Rev. Lett. 129, 090502 (2022), arXiv:2111.03011.

## (e) Pan-Chen-Zhang generalization

Grade: SERIOUS, bordering FATAL if the conclusion says "large-n nature is cheap."

What breaks: generalizing from one RCS family to classical rendering of nature.

Pan, Chen & Zhang solved a particular sampling problem: Google's 53-qubit, 20-cycle Sycamore random circuits at target fidelity around `0.0037`, producing one million uncorrelated bitstrings in about 15 hours on 512 GPUs. Their method is tensor-network contraction tailored to the circuit graph, depth, slicing choices, and target fidelity. It is strong evidence against the original 2019 hardness estimate. It is not evidence that arbitrary large-n quantum processes are cheap to render.

Known limits point the other way. Google's later RCS work, "Phase transitions in random circuit sampling" (Nature, 2024, doi:10.1038/s41586-024-07998-6), explicitly says classical algorithms advanced and raised the threshold, then reports 67-qubit, 32-cycle RCS with estimated fidelity `1.5e-3`, 70 million samples for a single circuit, and estimates approximate simulation on Frontier at around `1e4` years under RAM constraints, or 12 years only under an unrealistic secondary-storage assumption with negligible bandwidth. The same paper says Pan-style tensor contraction sampled the 2019 RCS experiment in 15 hours, so this is not an ignored counterpoint; it is the moved goalpost.

There is also asymptotic nuance. Aharonov, Gao, Landau, Liu & Vazirani give a polynomial-time classical algorithm for noisy RCS at constant noise in the anti-concentration regime, but their abstract states the algorithm is not practical and does not address finite-size RCS supremacy experiments. That cuts against sweeping claims in both directions.

Correct claim: the 2019 Sycamore benchmark at its published fidelity was classically matched/exceeded by later tensor-network simulation. Incorrect claim: therefore high-n classical rendering is generally cheap.

Sources: Pan et al. 2022; Google Quantum AI and Collaborators, "Phase transitions in random circuit sampling," Nature (2024), doi:10.1038/s41586-024-07998-6; D. Aharonov, X. Gao, Z. Landau, Y. Liu, U. Vazirani, "A polynomial-time classical algorithm for noisy random circuit sampling," STOC 2023, arXiv:2211.03999.

## (f) Selection bias in "the record"

Grade: FATAL to the claimed n-trend.

What breaks: the comparison `n=1 tight` versus `n=53 loose` as a general law of measurement records.

The author compares experiments optimized for different objectives. CHSH tests are optimized to estimate one low-dimensional Bell parameter with high statistical power. RCS experiments are optimized for computational hardness and only weakly/indirectly verified because full verification is classically hard. That is not an n-trend; it is a task-selection effect.

A fairer high-n comparison gives a mixed picture, not a monotone collapse to 99.8% tolerance:

- Scalable Bell/graph-state tests exist at high n. Yang et al., "Testing Scalable Bell Inequalities for Quantum Graph States on IBM Quantum Devices," IEEE JETCAS 12(3), 638-647 (2022), arXiv:2101.10307, report violations for path graph states up to 57 qubits on a 65-qubit IBM device, full 65-qubit connection-graph violations on most devices tested, and 8192 shots per circuit. These are not loophole-free cosmic Bell tests, and they use readout mitigation, but they directly refute "large n only has XEB at 0.2% fidelity."
- High-n entanglement verification can pin down fidelities at percent-to-few-percent precision. "Generation of genuine entanglement up to 51 superconducting qubits," Nature (2023), doi:10.1038/s41586-023-06195-1, reports 51-qubit one-dimensional and 30-qubit two-dimensional cluster states with fidelities `0.637 +/- 0.030` and `0.671 +/- 0.006`. That is not CHSH-tight, but it is nowhere near "0.998 fractional tolerance."
- Quantum error-correction experiments at comparable qubit counts verify operational logical error rates, not global state fidelity. Google Quantum AI, "Suppressing quantum errors by scaling a surface code logical qubit," Nature, doi:10.1038/s41586-022-05434-1, reports a 72-qubit device, a 49-qubit distance-5 surface code, and logical error per cycle `epsilon_5 = (2.914 +/- 0.016)%` versus `epsilon_3 = (3.028 +/- 0.023)%` for distance-3 subsets. Again, high-n record strength depends on what is measured.

The n=1 side is also cherry-picked. `detect.py` uses `N = 1e9` as if that is the operative actual Bell record. Some loophole-free Bell tests have far fewer trials: Hensen et al. used 245 trials and found `S = 2.42 +/- 0.20`; Storz et al., "Loophole-free Bell inequality violation with superconducting circuits," Nature 617, 265-270 (2023), report per-test trial counts around `80659` in sets of 13, i.e. about `1e6`, not `1e9`. There may be high-rate non-loophole-free optical Bell datasets, but the report needs to specify which record it means. "The field passed N ~ 1e9 long ago" is not a clean statement about the loophole-free record.

The real trend is not "tolerance widens with n." It is "verification objective determines tolerance." RCS deliberately chooses circuits where exact verification is hard; stabilizer/graph/QEC experiments choose observables that remain verifiable at high n.

Sources: Yang et al. 2022; Nature doi:10.1038/s41586-023-06195-1; Google Quantum AI, Nature doi:10.1038/s41586-022-05434-1; B. Hensen et al., Nature 526, 682-686 (2015); S. Storz et al., Nature 617, 265-270 (2023).

## (g) Conclusion

Grade: FATAL.

What breaks: the move from weak measurement record to wide adequate-policy space.

Even granting every numerical premise, the conclusion does not follow. "The measurement record is weak at large n" is an epistemic claim about human certification. "The space of rendering policies adequate for observers is wide" is an ontological/computational claim about what a simulator would have to generate for a universe containing observers, laboratories, memories, counterfactual experimental choices, and future adaptive tests.

A renderer adequate only to past published records could be extremely sloppy. A renderer adequate to embedded observers who may choose new measurements tomorrow has to preserve a much larger family of conditional distributions. The report slides between those notions of adequacy. The broader the observers' possible actions, the closer the renderer must come to a generative model of the physical world, not merely a table of already-published benchmark statistics.

The strongest supported conclusion is much narrower: current high-n random-circuit sampling certification is weak compared with high-statistics low-n Bell certification, and the 2019 Sycamore XEB target was later matched by a classical tensor-network sampler. That is a useful fact about experimental verification and benchmark design. It is not evidence that classical rendering policies adequate for a universe are generally wide.

## Additional implementation notes

`price.py` is acceptable as an explanatory toy but should cite Pironio directly and stop calling the bound its own rigorous derivation over rendering policies. `detect.py` computes the CHSH estimator variance correctly for balanced settings at the Tsirelson correlators, but the inference "the measurement record sets delta" should be marked as a modeling choice, not a theorem. The code's `N = 1e9` examples are hypothetical unless tied to a named Bell dataset.

## Verdict

Gate: FAIL / revise before ledger movement. The author is entitled to claim that the 2019 Sycamore XEB record was a weak, noisy, classically matched high-n benchmark; they are not entitled to claim a commensurate four-order widening of a general approximation knob or that adequate rendering policies for observers are therefore wide.