--- title: "One continuous route to four quantum bases is ruled out" date: 2026-09-30 version: "3.0-candidate" doi: 10.5281/zenodo.23051062 pdf: https://github.com/ipitchford/szollosi-mub-exclusion/releases/download/v3.0-candidate/paper.pdf repository: https://github.com/ipitchford/szollosi-mub-exclusion archive: https://zenodo.org/records/23051062 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # One continuous route to four quantum bases is ruled out ## Summary Quantum measurements can be complementary in a precise sense: knowing the outcome in one measurement basis leaves every outcome in another equally likely. Such bases are called mutually unbiased. In six dimensions, three mutually unbiased bases are known, but whether a fourth can exist remains open. This candidate rules out one whole route to a fourth basis: the specified two-parameter Szöllősi family of complex Hadamard matrices. It includes the family's boundary and degenerate cases, not just sampled parameter values. It does **not** settle the full six-dimensional problem. The method separates finding possible solutions from checking them. A numerical program proposes small regions containing candidate vectors. A separately implemented checker must prove that those regions cover every relevant vector and that they cannot form the two additional bases required for a quartet. ## Summary for specialists The target is the closed two-circulant family $S=\begin{pmatrix}A&B\\B^*&-A^*\end{pmatrix}$, with the specified unimodular circulant blocks and Hadamard constraint. The claimed theorem excludes an MUB quadruplet containing the standard basis and the columns of $S/\sqrt6$ for every parameter in that family, including repeated-root boundaries. This is Conjecture 3 in the programme of Matolcsi, Matszangosz, Varga and Weiner. Their separate algebraic Conjecture 2 remains necessary for the proposed route to the unrestricted three-basis maximum. A Fourier-family replay in the package re-establishes the known Jaming–Matolcsi–Móra–Szöllősi–Weiner exclusion; it is not presented as a new theorem. ## Technical account After phase normalization, a vector unbiased to the standard basis has five free phases. For each parameter cell, the generator proposes phase-space boxes, called hulls, covering possible vectors unbiased to the Hadamard basis. The checker uses outward-rounded interval arithmetic, modulus and mean-value tests, and a parametric Krawczyk test to verify coverage. A quartet would require twelve vectors forming two orthonormal six-element bases. Verified separation prevents two of those vectors sharing a hull. Possible orthogonality and unbiasedness relations therefore define a necessary graph configuration: two six-cliques with all required cross-relations. Its absence excludes the quartet. Extra possible edges make this test harder, not easier. A hull need not contain exactly one isolated root. The parameter cover distinguishes half-open ownership cells from their closed interval enclosures. A boundary point has one canonical owner; the proof follows that owner through subdivision. Merely showing that a discarded closed cell touches the boundary would not establish a gap in coverage. ## Evidence, assurance and limitations The certificate inputs comprise 14,658 Szöllősi and 1,935 Fourier parameter certificates. These are certificate counts, not counts of distinct roots. The revised driver binds replay receipts to the actual hulls, family, table, checker and runtime, and rejects stale caches. The package also retains the review findings, negative controls and the precise trusted-computing boundary. Fresh local replay passed both complete covers. Of the Szöllősi boxes, 14,583 passed directly and 75 required fresh parameter subdivision: 38 to level 10, 29 to level 11 and 8 to level 12. No required box remained unchecked. The versioned replay receipt binds the exact source, data, binary and environment; historical success records were not imported into this run. This is an unrefereed candidate. Implementation separation from the generator does not establish unaffiliated reproduction or human peer review. The argument is not formalized end to end in a proof assistant. Its larger-family extensions remain research directions, not checked consequences. ## Relationship to earlier work The broad discretisation strategy descends from the 2009 Fourier exclusion. The contribution claimed here is the uniform exclusion for the specified Szöllősi family and an inspectable hull-and-checker implementation. A recent correction to an earlier exclusion argument makes it important to distinguish a claimed theorem from a valid proof; it does not show that the theorem itself is false. The source audit records that history without asserting exhaustive novelty or priority clearance. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Mathematical quantum-information researchers | Inspect an explicit exclusion target in the dimension-six programme. | The unrestricted MUB problem remains open. | | Computer-assisted-proof researchers | Reuse the separation of hull generation, interval coverage and graph obstruction. | Audit the family-to-checker semantic bridge. | | Researchers studying degenerate solution sets | Examine a method that need not isolate every root. | Larger algebraic families require new encodings and branch coverage. | ## Why the problem matters Dimension six is the first dimension that is not a prime power. It is a central test case for understanding how arithmetic constrains complementary quantum measurements. Excluding one continuous family reduces the available routes; it does not yet deliver a new device or an experimentally validated advantage. ## How to inspect or reproduce the recorded checks Start with `AI_INDEX.md` and `CERTIFICATE_INTERFACE.md`. Build the kernel, run the normal and optimized regression tests, then execute `chk/full_replay.py` with a new output directory. Preserve both complete-cover summaries and their bound environment records. Graph checks alone are insufficient: phase-space coverage is the computationally expensive obligation. ## What would improve assurance next An unaffiliated full replay and a separate audit of the parameter-ownership, interval-arithmetic and graph arguments would strengthen assurance. A proof-assistant implementation is a further possibility, not something this release supplies. Extending the method to larger families requires new verified encodings and coverage of algebraic branches; it is not merely a data swap. ## What is in the evidence package The manuscript and editable source; the hull proposals and unit-circle table; the interval/graph checker; clean-replay orchestration; exact symmetry scripts; boundary, stale-cache and mixed-term regressions; source comparisons; review dispositions; provenance and licence boundaries; and an AI index for reuse. ## Verification status Unrefereed candidate with complete fresh local replay. Not end-to-end formally verified or externally peer reviewed. The full dimension-six MUB problem remains open. ## References 1. Matolcsi, Matszangosz, Varga and Weiner (2026), Triplets of mutually unbiased bases: Conjecture 3 and the conditional reduction in Proposition 4.2. 2. Jaming et al. (2009), A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6: the prior Fourier exclusion. 3. Szöllősi (2010), A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids. 4. McNulty and Weigert (2025), correction identifying a failed earlier argument; not a disproof of the target theorem.