Exact Quantum Values and Permutation-Blind Maximizers in Cyclic Bell Inequalities

Abstract

Perito, D’Avino, Jung, Mironowicz, Acín, and Augusiak introduced two cyclic Bell families for efficient certification of maximal global randomness. For the first reduced operator they proved the general bound \(d\sqrt2\), supplied a strategy of value \(2\csc(\pi/(2d))\), and conjectured that the latter value is exact. We prove that formula in every dimension and in the commuting-operator model. Every attained finite-dimensional tensor-product exact maximizer of the first augmented family has all equality roots with equal multiplicity on Alice’s support. Yet conditional permutations of those roots remain invisible to the first-harmonic score. For every \(d\ge4\), a final-two swap maximizes both augmented families while producing a nonuniform designated joint-output distribution. Thus the intended scalar-value implication of the first-family maximal-violation conjecture fails in these dimensions. This does not challenge certification conditioned on the complete canonical behavior, identify the worst-case guessing probability, or show that the canonical strategy lacks maximal randomness.

Main results

The corresponding first augmented family has value \(2\csc(\pi/(2d))+1\). The second reduced and augmented families have values \(d\) and \(d+1\), respectively, with the sum-of-squares identity of Perito et al., Eqs. (22)–(23), whose prefactor is \(1/(2d)\), also admitting the commuting-operator reading.

Relation to the originating work

The originating authors defined both cyclic Bell families, supplied the canonical maximally entangled strategies and matching lower bounds, proved the second-family sum-of-squares bound, and developed the full-behavior numerical randomness framework. This manuscript adopts those definitions and credits those contributions directly.

A localized normalization discrepancy in the source is treated as a technical convention issue. The displayed Hermitian operator definition, its subsequent equations, and stated small-dimensional values determine the convention used here; the discrepancy is not used rhetorically.

Exact scope of the randomness conclusion

A full-behavior SDP can return \(G=1/d^2\) for the fixed canonical distribution while another distribution attains the same scalar Bell maximum and has \(G>1/d^2\). The root-swapped behavior differs in higher Fourier correlators and therefore does not satisfy the canonical full-behavior constraints.

The construction does not dispute a calculation that fixes the complete canonical behavior, does not show that the canonical strategy lacks maximal randomness, and does not determine the globally worst guessing probability. Passing from a fixed-behavior conclusion to a value-only conclusion would require uniqueness, suitable self-testing, or additional observed correlators.

In particular, the maximum does not determine a unique behavior even modulo local output relabelings: a relabeling can only permute a target table and cannot turn the canonical-order uniform table into the final-two-swap nonuniform table.

Proof architecture

1. A sharp roots-of-unity extremum bounds the scalar function \(\sum_y|1+\omega^yz|\) and identifies its equality phases.
2. A kernel-safe polar positive-factor identity and continuous functional calculus lift that scalar inequality to arbitrary commuting Alice–Bob unitaries.
3. The canonical finite strategy supplies the matching lower bound and proves equality of \(q\), \(qa\), and \(qc\).
4. For every attained finite-dimensional tensor-product exact maximizer of the first augmented family, state-support cancellation and a reflection-product rank bound force every equality root to have the same supported multiplicity.
5. A weighted-cycle theorem gives exact product conditions under which permutations of the equality phases remain order-\(d\) projective maximizers.
6. Fourier analysis of the cycle weights shows that a final-two swap preserves all Bell-visible correlators but makes the designated joint table nonuniform.
7. An exact discrete Fourier calculation transports the same Bob observables into the second family and annihilates every factor of the sum-of-squares certificate of Perito et al., Eqs. (22)–(23), whose prefactor is \(1/(2d)\).

A standard one-input locality theorem supplies the universal setting baseline. A self-contained binary benchmark recovers the prior-art value \(3\sqrt3\) and two private bits, while a private-MUB lemma gives a sufficient operator criterion for higher-dimensional designs. More specialized observations about the standard two-input qudit strategy and a separately bounded third-setting repair are confined to an appendix and do not imply a general \(2\times2\) or \(2\times3\) impossibility theorem.

Verification and review artifacts

From the repository root, the unified replay command is:

cd cyclic_bell_exact_values_and_randomness
./reproduce.sh

The scientific source and verification scripts for this revision are frozen at 609f8c6ffc083b665804890dd82fc739d414ea9d.

The replay covers scalar and polar identities, canonical/source observables, support-rigidity ranks and phases, deliberately inadmissible phase-permutation controls, all-dimensional formulas, two independent \(d=4\) exact certificates, the second-family certificate, the binary and private-MUB checks, retained setting checks, the manuscript build, and website metadata and links.

• verification/README.md — prerequisites, commands, and verification boundary.
• verification_report.txt — captured unified replay result.
• CLAIMS_LEDGER.md — every retained theorem and its support.
• PROOF_DEPENDENCY_MAP.md — theorem-to-proof dependencies.
• ADVERSARIAL_REVIEW.md — fresh skeptical reconstruction and adversarial regression tests.
• REVIEW_RESPONSE.md — item-by-item disposition of the author-ready revision review.
• THEOREM_CROSSWALK.md — every substantive predecessor result and its canonical disposition.
• CLAIMS_NOT_MADE.md — concise boundaries the revision deliberately does not cross.
• manifest.sha256 — canonical package integrity hashes.

Current priority statement

The family definitions, canonical strategies, first-family \(d\sqrt2\) bound and matching lower value, second-family sum-of-squares identity, and binary \(d=2\) benchmark are prior work. The candidate contributions here are the all-dimensional sharp first-family upper bound, its commuting-operator strengthening, the phase-permutation mechanism, the explicit nonuniform maximizers, and their second-family extension. The support-rigidity and private-MUB statements are retained for completeness and are not promoted as lead novelty claims.

The accompanying audit compares these claims against current primary sources. The manuscript uses qualified language—“to our knowledge”—and makes no claim of absolute or definitive priority. Concurrent, unindexed, or unpublished work may exist.

Read the fresh priority audit and source-comparison table.

Focused source-author review packet

A compact packet is provided for Perito and collaborators to inspect the theorem boundary without navigating the full repository. It contains the normalization convention, theorem comparison, Bell-value versus full-behavior distinction, three load-bearing proof points, collaborative open problems, and the shortest exact replay commands.

Historical standalone versions

The three prior papers remain available as timestamped historical artifacts. Their landing routes now point here, while their original PDF URLs, source snapshots, manifests, hashes, and Git history remain unchanged and directly accessible.

Earlier deployed bytes also remain directly accessible through Git history: the initial exact-value PDF has SHA-256 947b601903de18ea6ffbd8e49ba2bfe261c32342c4d1a71dd96ed9f283ec6c94, and the initial counterexample PDF has SHA-256 73c2e2ab39de79357d7b441a2a4c30f901a13e99e15bf086540783041f7d6051.

The initial releases and strengthened revisions are recorded in the repository’s publication history. Git history has not been rewritten, and no historical DOI has been reassigned to this synthesis.

Citation and availability

Alec Kriebel, “Exact Quantum Values and Permutation-Blind Maximizers in Cyclic Bell Inequalities: Sharp operator bounds, equality structure, and limits of Bell-value randomness certification,” Version 1.1.0, unrefereed manuscript, 9 August 2026.

AI-assistance disclosure

Generative-AI systems were used extensively in exploration, proof development, software generation, verification design, literature organization, and drafting. Alec Kriebel directed the work, selected the released claims, and is responsible for the decision to publish these materials. Model output was not treated as evidence by itself; claims were retained only with written proofs and replayable checks.