Why this page exists — twice in eight days we
committed compute to a question that turned out to be answered, both times because we
named an object in our own vocabulary instead of the community's. The cure is to stop
generating targets and start reading the ones the field states in print. Each card below
carries three separate things and they must not be run together: the problem
statement, quoted verbatim from a published review; a plain-terms
gloss, which is our paraphrase and not the authors'; and our
position, which is ours alone and is the part we can be wrong about.
Ordered by how much we are committing, not by problem number.
Below: what our instrument can decide, then the problems in full — and, at the foot,
what we host and what we do not.
What our instrument decides
CAN
Exact certified exclusion over a zero-dimensional algebraic ideal: msolve (F4/FGLM) proves the companion variety finite, then rational interval arithmetic certifies each candidate pair as NOT mutually unbiased. The goal is a certificate a referee re-checks with no solver — NOT YET DELIVERED: the certificate published so far lists witnesses by index without the companion vectors, so it can be checked for internal consistency but not recomputed from the file alone. Until the self-contained version exists, treat solver-free re-checking as the design target, not a capability we have shipped.
CANNOT
It decides ONE matrix, or a finite set of them. It cannot decide a CONTINUOUS family: K6(3) is a three-parameter family, so it contains uncountably many matrices and no finite ideal describes all of them at once. Any claim we make about a family is a claim about the specific parameter values we ran.
Problem 10.6 is written as 'for certain parameter values' precisely because the parameter-point case is the tractable one; the review separately calls a general analytic proof 'a major step forward'. Reading 10.6 as within reach and the general case as within reach is the same extrapolation error that produced sealed predictions 1, 3 and 5. See LESSONS V7.
What we are engaging
4
Our instrument fits, or would if something upstream cleared. These are the four that can cost core-hours.
Show that a given pair of MU bases in C6 does not extend to a triple (or quadruple) of MU bases.
Our gloss —
The review's own fallback when the bigger problems are too hard: complete Table 2 by replacing numerical evidence of unextendibility with rigorous proof.
Our position
The review states the blocker in its own words, and it is a HARDWARE blocker: "For the symmetric, Hermitian and Szöllősi non-affine families, i.e. M6(1), B6(1) and X6(2), the available computational memory was insufficient to produce the relevant Gröbner basis, making certain approximations necessary. Thus, no rigorous conclusion regarding the existence of a fourth MU basis containing these families could be drawn." That is precisely the task our instrument performs — produce the basis, then replace the approximation with an exact certificate — and M6(1) is one of the three families Problem 10.6 names. A question abandoned for want of memory is an opportunity; a question abandoned because the mathematics resists is a wall. This one is on the record as the first kind.
Risk — Two risks, stated precisely. (1) The April 2025 invalidation of the family-level Chen–Yu chain touched M6(1) only — NOT the Karlsson families K6(2) or K6(3), which were never closed by that chain and are open for older reasons. An earlier version of this page said 'these families were reopened', which overstated it for two of the three. (2) The memory wall is a decade-old note in a review published four months ago; hardware has moved, and we will not be the only people who can read that sentence. If someone has already cleared it, the desk gate below should find them before we spend a core-hour.
Find all pairs of 6 x 6 complex Hadamard matrices H1 and H2 such that their product H1-dagger H2 is another Hadamard matrix.
Our gloss —
Equivalent to identifying all MU triples in d=6. Conjecture 8.1 proposes the Fourier family F6(2), its transpose, and Szollosi's X6(2) as the only candidates.
Our position
Conjecture 8.1 names a FINITE candidate list, and checking named candidates is what our certificates do. Needs a desk gate first: the conjecture may already have been verified at the parameter points that matter.
For d = p1^n1 ... pr^nr not a prime power, is a set of min_i (p_i^n_i + 1) nice MU bases strongly unextendible?
Our gloss —
Related to Conjecture 6.2, confirmed for d <= 15.
Our position
A conjecture verified to d <= 15 is a bounded finite check with an obvious next value. The classic compute-overhang profile -- verified this far because that is where someone stopped, not where the mathematics stopped (LESSONS R5). Cheap to gate.
Show that no single vector is mutually unbiased to any triple of three MU bases when d = 6.
Our gloss —
Strong unextendibility of triples. The review notes this would probably need an exhaustive classification of all MU triples.
Our position
The vector-search half is exactly what our companion enumeration computes, and we would run it directly. The blocker is upstream: the exhaustive triple classification does not exist, and building it is Problem 10.5 plus Problem 10.4. We cannot start at the end of a chain whose first link is open.
Show that no set of four MU bases exists when d = 6.
Our gloss —
Zauner's conjecture in its d=6 form. Sufficient to settle the complete-set question.
Our position
Our certified k=8 run is one slice of this: it shows no fourth MU basis accompanies any BH(6,8) class. That slice is Brierley-Weigert 2009 -- a reproduction, not a discovery. The full problem is far out of reach.
Do inequivalent complete sets of MU bases exist in prime dimensions?
Our gloss —
Known to exist for d = p^n with n > 1; no example found for d prime.
Our position
Existence question over prime dimensions -- a YES needs one object. Small primes are finite-ish. Worth a desk pass, but far from our instrument's strength.
Can one improve the lower bound provided in Thm. 6.1, to find larger sets of MU bases in composite dimensions?
Our gloss —
For d = 2p the known constructions yield only three MU bases no matter how large d gets. A constructive question -- a YES needs one object.
Our position
Witness-shaped, which fits our bias toward chasing a YES. But the search space is a continuous manifold, not a finite grid, so our SAT machinery does not apply and our algebraic machinery has no finite ideal to chew.
Is the set of (mu + 2) MU bases from Thm. 6.11 unextendible?
Our gloss —
Unextendibility of the Latin-square construction in square dimensions.
Our position
Latin-square adjacency is our native vocabulary from the A6 work. Square dimensions means d=9,16,25... -- d=36 is the interesting one and is enormous.
Classify all complex Hadamard matrices of order six.
Our gloss —
Conjecture 7.1: the known list (isolated S6, three-parameter K6(3), four-parameter G6(4)) is expected to be exhaustive, but no proof exists.
Our position
A classification theorem over continuous families. Our instrument decides points, not families. No amount of compute closes this.
For a quantum particle with one degree of freedom, is the Heisenberg-Weyl type triple of MU bases with basis-independent overlaps strongly unextendible?
We do not mirror papers here — not even the
openly licensed ones. Every record links out to the publisher's copy. Problem statements
appear verbatim and attributed because a reworded open problem is a different open
problem; nothing else of theirs is reproduced.
Licences are per item, not per source.
The review cited on every card above is CC-BY 4.0 and could lawfully be republished in full;
the 2009 paper it
cites is under publisher copyright and could not. That difference is why this is curated
by hand rather than by a crawler.
Status assessments are ours alone and
carry no endorsement from the cited authors. If you are one of them and an assessment here
is wrong — or you are already working on one of these and would rather we did not — say so
and we will change it. The constitution
records how corrections are handled.