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The exact fourth-vector feasibility wall of the canonical MUB triple in dimension six

Noah Erlwein

Published Jun 12, 2026• 0 citations
Publisher

Abstract

For the canonical product MUB triple {Z₂ ⊗ Z₃, X₂ ⊗ Q₀, Y₂ ⊗ Q₁} in dimension six, known to admit no vector unbiased to all three bases, we prove an exact two-sided theorem for the projector-space feasibility functional f, where f(v) = 1 would characterize a candidate fourth unbiased vector. We show that the global maximum of f is W = (88 + 3√6) / 100 ≈ 0.9534846923. The bound is attained by an explicit algebraic unit vector, verified entirely in exact arithmetic over the field Q(ζ₂₄, √5). Its symmetry orbit contains 72 distinct maximizing rays. The matching global upper bound f ≤ W is established via an exact positive-semidefinite certificate at the level-2 Veronese lift. The certificate consists of a symmetric matrix over Q(√2, √3), whose polynomial identity and positivity are machine-verified in exact arithmetic and closed deterministically under the triple's 432-element symmetry group. No randomized computation or floating-point arithmetic appears anywhere in the proof chain. We further prove, via exact separating functionals, that degree-4 and degree-6 sum-of-squares certificates do not exist. In this precise sense, the lift level employed is minimal and necessary. Companion results include: • an exact product-vector ceiling for aligned product triples; • the exact wall value for the second known triple; • a complete Weyl-board census in dimension six, together with a Lean 4 kernel-verified combinatorial layer. These results completely characterize the pinned canonical triple. They do not resolve the MUB(6) problem, which remains open. The deposit contains two files: (1) the manuscript (PDF); (2) a complete verification bundle (ZIP; 91 files) containing all computational modules, tests, evidence receipts, execution logs, a SHA-256 manifest, and a minimal independent verifier (verify_global_certificate.py). The verifier is implemented in standard-library-only Python, shares no code with the certificate-construction pipeline, loads the stored exact certificate matrix, and independently re-establishes both directions of the global bound in approximately 150 seconds. Reproduction is available via: make verify-global Code is released under the Apache License 2.0 (with SPDX headers throughout the bundle). The manuscript text is released under CC BY 4.0. This research made extensive use of AI systems for conjecture generation, computation, proof drafting, and adversarial review. However, no mathematical claim depends on the correctness of any AI system. Every reported result is verified by exact rational arithmetic, exact interval arithmetic, or the Lean 4 kernel, and all verification artifacts are included in the deposit.

DOI
10.5281/zenodo.20670933

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An Independent Reproduction and Four Extensions of the MUB(6) Fourth-Vector Feasibility Wall

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Title. An Independent Reproduction and Four Extensions of the MUB(6) Fourth-Vector Feasibility Wall Abstract. We present a companion note that cites N. Erlwein’s exact fourth-vector feasibility wall for a canonical mutually unbiased product triple in dimension six (Zenodo 20682233; theorem W = (88 + 3√6)/100). We do not redistribute the upstream software bundle. Independently, we (A) maintain a pure-Rust exact verification surface; (B) raise a d=10 closed-form hunt to a 460-decimal-place stationarity anchor with a widened PSLQ menu (no relation found); (C) run a full Cl(2)×Cl(5) single-qudit Clifford census at d=10, finding a single antiunitary Z/2 real structure; and (D) issue exact rational finite-cover exclusion certificates on a specified cell of the gauge-fixed candidate 5-torus. MUB(6)=3 and MUB(10) remain open. Local certificates mint no authority. Why a researcher should care. The note strengthens a rare exact, machine-checkable obstruction artifact in d=6 quantum information with a second language (Rust), higher-precision d=10 evidence, and independently checkable finite covers—while keeping claim boundaries explicit. What this does not claim. Not MUB(6)=3; not MUB(10); not a closed form at d=10; not joint authorship of Erlwein’s wall; not a global moduli theorem. Reproduction. cd crate && cargo test -j 1cargo run --release --bin mub_cover -- --region quadrant DOI. Pending human mint. Upstream wall: 10.5281/zenodo.20682233.

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Mutually Unbiased Bases in Dimension Twelve: Tensor Saturation, Strong Unextendibility, and the Flat-Sum Trichotomy

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In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.

•Oct 3, 2026•0 citations

No Szöllősi matrix lies in a quadruplet of mutually unbiased bases in dimension six

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•Sep 30, 2026•0 citations

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Tensor Saturation and the Nonexistence of a Fourth Mutually Unbiased Basis in Dimension Six

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AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.

•Dec 16, 2025•0 citations
For the canonical product MUB triple {Z₂ ⊗ Z₃, X₂ ⊗ Q₀, Y₂ ⊗ Q₁} in dimension six, known to admit no vector unbiased to all three bases, we prove an exact two-sided theorem for the projector-space feasibility functional f, where f(v) = 1 would characterize a candidate fourth unbiased vector. We show that the global maximum of f is W = (88 + 3√6) / 100 ≈ 0.9534846923. The bound is attained by an explicit algebraic unit vector, verified entirely in exact arithmetic over the field Q(ζ₂₄, √5). Its symmetry orbit contains 72 distinct maximizing rays. The matching global upper bound f ≤ W is established via an exact positive-semidefinite certificate at the level-2 Veronese lift. The certificate consists of a symmetric matrix over Q(√2, √3), whose polynomial identity and positivity are machine-verified in exact arithmetic and closed deterministically under the triple's 432-element symmetry group. No randomized computation or floating-point arithmetic appears anywhere in the proof chain. We further prove, via exact separating functionals, that degree-4 and degree-6 sum-of-squares certificates do not exist. In this precise sense, the lift level employed is minimal and necessary. Companion results include: • an exact product-vector ceiling for aligned product triples; • the exact wall value for the second known triple; • a complete Weyl-board census in dimension six, together with a Lean 4 kernel-verified combinatorial layer. These results completely characterize the pinned canonical triple. They do not resolve the MUB(6) problem, which remains open. The deposit contains two files: (1) the manuscript (PDF); (2) a complete verification bundle (ZIP; 91 files) containing all computational modules, tests, evidence receipts, execution logs, a SHA-256 manifest, and a minimal independent verifier (verify_global_certificate.py). The verifier is implemented in standard-library-only Python, shares no code with the certificate-construction pipeline, loads the stored exact certificate matrix, and independently re-establishes both directions of the global bound in approximately 150 seconds. Reproduction is available via: make verify-global Code is released under the Apache License 2.0 (with SPDX headers throughout the bundle). The manuscript text is released under CC BY 4.0. This research made extensive use of AI systems for conjecture generation, computation, proof drafting, and adversarial review. However, no mathematical claim depends on the correctness of any AI system. Every reported result is verified by exact rational arithmetic, exact interval arithmetic, or the Lean 4 kernel, and all verification artifacts are included in the deposit.
Title. An Independent Reproduction and Four Extensions of the MUB(6) Fourth-Vector Feasibility Wall Abstract. We present a companion note that cites N. Erlwein’s exact fourth-vector feasibility wall for a canonical mutually unbiased product triple in dimension six (Zenodo 20682233; theorem W = (88 + 3√6)/100). We do not redistribute the upstream software bundle. Independently, we (A) maintain a pure-Rust exact verification surface; (B) raise a d=10 closed-form hunt to a 460-decimal-place stationarity anchor with a widened PSLQ menu (no relation found); (C) run a full Cl(2)×Cl(5) single-qudit Clifford census at d=10, finding a single antiunitary Z/2 real structure; and (D) issue exact rational finite-cover exclusion certificates on a specified cell of the gauge-fixed candidate 5-torus. MUB(6)=3 and MUB(10) remain open. Local certificates mint no authority. Why a researcher should care. The note strengthens a rare exact, machine-checkable obstruction artifact in d=6 quantum information with a second language (Rust), higher-precision d=10 evidence, and independently checkable finite covers—while keeping claim boundaries explicit. What this does not claim. Not MUB(6)=3; not MUB(10); not a closed form at d=10; not joint authorship of Erlwein’s wall; not a global moduli theorem. Reproduction. cd crate && cargo test -j 1cargo run --release --bin mub_cover -- --region quadrant DOI. Pending human mint. Upstream wall: 10.5281/zenodo.20682233.
Abstract Setting. The maximal number N(12) of mutually unbiased bases (MUBs) of C12 is unknown; no composite non-prime-power dimension has a known exact value. The quantum MacNeish (tensor) bound gives N(12) ≥ min{N(3),N(4)} = 4, and within Array Cosmology (AC) the integer 4 = k is the ququart rank of the rendering algebra R12 = M3(C)⊗ M4(C). This paper maps the d = 12 MUB landscape and states the tensor saturation conjecture N(12) = 4. Canonical sector. (i) Strong unextendibility [A]: no vector of C12 is unbiased to all four bases of the canonical product quadruple; the proof closes through an entanglement-forcing theorem, a flatness–circle-structure reduction, a monomial-rigidity step, and an atomic lemma whose ninebranches collapse to f = U+UV+V withexactminimum5/4 = (k+1)/k > 1. (ii)Localrigidity[A]: the canonical quadruple is rigid modulo its 191-dimensional gauge orbit; the Jacobian has exact rank 385 and exact nullity 191, certified by an explicit integer basis of the kernel verified in exact arithmetic, by modular arithmetic at p = 1000033, and by an independent gauge-dimension proof
In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.
Unrefereed candidate excluding the specified closed two-circulant Szollosi family from MUB quadruplets in complex dimension six. The Fourier-family result re-proves prior work. The unrestricted dimension-six maximum remains conditional on MMVW Conjecture 2. The package separates an untrusted hull generator from an interval and graph checker; canonical half-open ownership repairs boundary coverage, and input-bound clean replay prevents stale cache acceptance. No end-to-end formal verification, unaffiliated reproduction or external human peer review is claimed. See the replay receipt for exact measured scope. Original prose and data CC0; original code MIT; third-party and review material retain their terms.
We study the canonical USp(4)-covariant quantum compression problem in which the fifteen-dimensional carrier W = V5 ⊕ V10 is encoded through a four-dimensional covariant memory M = V4 and decoded, and we ask for the optimal entanglement fidelity F*(p) of the composite channel on a trace-one, sector-weighted reference state ρ_p. This problem is the operational face of a structural fact: the same pair of sectors 5 ⊕ 10 that makes USp(4) on C^4 state-homogeneous but not observable-uniform is exactly the modal content a minimal covariant memory is asked to carry. Our contribution is not a resolution of the underlying capacity question but a demonstration that this problem admits an exact reduction and a machine-checkable global bound: the reduction and the final lower- and upper-bound certificates are exact, while only the intermediate relaxation-solving step is numerical. We give an exact representation-theoretic reduction — Wedderburn splitting of the covariant commutant over Q(i), an induced-involution positivity bridge, and an exact trace-preservation reduction to fifteen free rational coordinates — turning F*(p) into a bilinear polynomial optimization over a product of two explicit semialgebraic bodies, each the affine preimage of the product cone (R+ × SOC4 × SOC4). Exploiting correlative sparsity we build an order-two sum-of-squares relaxation, solve it numerically for discovery, reconstruct the resulting SOS upper bounds in exact rational arithmetic, and separately certify exact rational feasible lower bounds from explicit witnesses. At p = 1/2 and p = 7/8: 0.0749 ≤ F*(1/2) ≤ 0.21, and 0.1546 ≤ F*(7/8) ≤ 0.643 (exact rationals in the paper). The upper bounds are Positivstellensatz certificates whose polynomial identity and positive semidefiniteness are verified with no optimizer in the loop by a standalone verifier we provide. Throughout we keep three visibly separate registers — exact theorems, numerical diagnostics, and open or conditional interpretations — and we are explicit that the exact optimum, the exactness of the relaxation, and the width of the certified interval remain open. Package: this record includes the manuscript, a single 30-page combined archival PDF (cover + manuscript + auditable reduction-supplement listing) inside the package, and a verification package with two independent machine-checkable obligations (the exact reduction chain and the certified two-sided enclosure). Manuscript licensed CC BY 4.0; code licensed MIT.
AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.