What we asked, and what is still open.

1. Answered

   Does any finite-dimensional strategy reach the quantum maximum of the I₃₃₂₂ Bell inequality?

   No. An infinite-dimensional spatial strategy reaches it, and coming within $\epsilon$ needs local dimension of order $\log (1/\epsilon )$.

   Posed byPál and Vértesi, 2010: they conjectured that no finite dimension suffices and posed “as a challenge to provide an analytical proof of our conjecture”, as the paper quotes them.
   Our paperSeth Douglas, The I₃₃₂₂ quantum value is attained spatially but not in finite dimension; and, for a parameter family that contains I₃₃₂₂, Nidhal Mghirbi, Critical Bellman–Jacobi dynamics for reversible Bell functionals, which calls the first contemporaneous and independent.
   CreditConcurrent and independent: Pauwels (arXiv:2608.29734), Coladangelo (arXiv:2609.06038) and V. Connor (manuscript of 1 August 2026, as cited by Krohn-Grimberghe). Exact certification of the bounds came first, from Nidhal Mghirbi, in July 2026.

   As printed

   “Taking as a conjecture that our construction is optimal (in the sense that no finite dimension suffices to achieve the true quantum maximum for I₃₃₂₂), […] We pose it as a challenge to provide an analytical proof of our conjecture.”

2. Extended

   Do Kaplansky’s group-ring conjectures hold for every torsion-free group?

   No. Each conjecture was first disproved by others: the unit conjecture by Gardam (2021, over ${𝔽}_2$), extended by Murray to every positive characteristic and by Gardam to the complex numbers; the zero-divisor conjecture by OpenAI (2026, over ${𝔽}_2$), extended by S. V. Ivanov to every odd characteristic; the direct-finiteness conjecture by OpenAI (2026, over ${𝔽}_2$, first for a group with torsion), and the idempotent conjecture by OpenAI’s torsion-free group whose group algebra over ${𝔽}_2$ is not directly finite (2026). The paper extends these answers: for every prime $p$, one finitely presented, torsion-free, word-hyperbolic group fails all four conjectures over every field of characteristic $p$; and OpenAI’s characteristic-two random groups are word-hyperbolic with probability $1-o(1)$. In characteristic zero the zero-divisor conjecture remains open.

   Posed byKaplansky, 1957: Problem 6 of his list asks the zero-divisor question for every group without elements of finite order (quoted below). Higman’s 1940 thesis contains the unit and zero-divisor questions, Fox asked the zero-divisor question for integral group rings in 1953, and Kaplansky returned to the problems in 1970 and 1972, as the paper cites them. The restriction to hyperbolic groups is the paper’s own question, question 22.
   Our paperNidhal Mghirbi and Seth Douglas, Kaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic.
   CreditGardam (units over ${𝔽}_2$, 2021, and over the complex numbers), Murray (units in every positive characteristic), OpenAI (zero divisors and direct finiteness, 2026) and S. V. Ivanov (zero divisors in odd characteristic, arXiv:2610.17662) answered first. The paper extends OpenAI’s constructions; Bunina, Lee, Miasnikov and Muliarchyk (arXiv:2610.22536) independently gave finitely presented groups that are not directly finite in every odd characteristic, without torsion-freeness or hyperbolicity.

   As printed

   “Let F be a field, G a group. The group algebra A(G) of G is the set of all finite linear combinations of group elements with coefficients in F, with the natural operations of addition and multiplication. Problem 6. If G has no elements of finite order, does A(G) have no divisors of 0?”

3. Extended

   Can an infinite periodic group have an ordinary finite presentation?

   Yes. OpenAI answered it first, with an infinite, finitely presented, residually finite 2-group; our paper gives one for every prime $p$, from one presentation over the integers.

   Posed byRecorded as an open question by Ol’shanskii and Sapir (2003, p. 45) and again by Haettel and Osajda (2025), after Burnside’s question of 1902, as the paper cites them.
   Our paperSeth Douglas and Nidhal Mghirbi, Finitely presented infinite p-groups for every prime.
   CreditThe answer to the question is OpenAI’s. Concurrent and independent: Kenta Kitamura proved the every-prime result (manuscript of 13 October 2026); in the paper’s words, “the extension to every prime listed above is shared with it.” A lemma is due to Zalesskii (1985).

   As printed

   “This question is one of a series of similar questions about finding finitely presented “monsters”, i.e. groups with unusual properties. Probably the most famous problem in that series is the problem about finding a finitely presented infinite torsion group.”

4. Answered

   Can a sofic group have a sofic profile that is unbounded and not linear?

   Yes. Houghton’s group $H_3$, amenable and so sofic, has a chunk with superpolynomial profile; a finitely presented elementary amenable group has a chunk with profile $\exp (r^{1+o(1)})$.

   Posed byCornulier, 2013, the last question of his Problem 3.18, as the papers cite it.
   Our paperSeth Douglas and Nidhal Mghirbi, Houghton’s group H₃ has superpolynomial sofic profile; the quantitative form in Amenable groups with nearly exponential sofic profile.
   CreditSlofstra’s finitely presented group has a chunk with a larger exponent, which answers Cornulier’s second question, but it is not known to be sofic. The Houghton paper also locates an error in Cornulier’s extension theorem and proves a corrected bound.

   As printed

   “Problem 3.18. Develop methods to compute lower bounds for the sofic profile of explicit groups. Is there any group for which the sofic profile is unbounded and not $\simeq r$? Can such a group be sofic?”

5. Answered

   Can a game with a binary score witness the separation between finite-dimensional and spatial quantum correlations?

   Yes. A finite game with a 0/1 winning predicate, ten and eleven questions and at most six answers, is won perfectly only outside the finite-dimensional set.

   Posed byColadangelo and Stark raised the possibility that a binary or integer-valued score might not, in the second-page remark of their 2017 preprint (arXiv:1708.06522), as the paper cites it.
   Our paperSeth Douglas and Nidhal Mghirbi, Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy.
   CreditColadangelo and Stark (2018) and Beigi separated finite-dimensional from spatial correlations earlier, for exact correlations. The building block is Mančinska’s flagged magic-square game. The paper makes no absolute priority claim.

   As printed

   “It is plausible that $C_q\ne C_{qs}$, but the separation is witnessed only by correlations which do not arise from non-local games with a binary (or integer-valued) scoring function.”

6. Partly answered

   Which Schmidt spectra can be self-tested by pseudo-telepathy games?

   Every positive rational is forced as a modular ratio by some binary-scoring finite game. Partial: in the paper’s words, “prescribing all rational spectra remains a different problem.”

   Posed byMančinska, Open Question 2 of The right entanglement for perfect nonlocality and zero-error communication (arXiv:2610.06512, 2026), in her words, as the paper cites it.
   Our paperSeth Douglas and Nidhal Mghirbi, Finite games that read the type of a von Neumann algebra, Theorem 1.3.
   CreditMančinska’s own example already has squared ratio 25/4, and her counting obstruction limits perfect-game self-test spectra to a countable set.

   As printed

   “Open question 2. Which Schmidt spectra can be self-tested by pseudo-telepathy games?”

7. Partly answered

   Can every $n$-qubit unitary be implemented by a polynomial-size circuit making few queries to a classical oracle?

   For two sequential queries in the clean model: it needs, and can be done with, $\mathrm{\Theta }(2^n)$ qubits and oracle input bits, up to a factor $\log (1/\epsilon )$. Partial: the garbage model reduces to a width bound that remains open, and more queries are not treated.

   Posed byAaronson and Kuperberg, 2007, as the paper cites them.
   Our paperSeth Douglas, The clean two-query threshold for unitary synthesis.
   CreditAt one query the answer was known to be no. Lombardi, Ma and Wright named the quantity bounded here and proved that small capacity implies the lower bound; the paper supplies the missing premise.

   As printed

   “(4) Is there a polynomial-time quantum oracle algorithm $Q$, such that for every $n$-qubit unitary transformation $U$, there exists a classical oracle $A$ such that $Q^A$ approximately implements $U$? Alternatively, would any such algorithm require more than poly($n$) queries to $A$?”

8. Answered

   Which unital channels are repeatable by one device with a finite memory and no resets?

   Some unital channels that are not random-unitary are repeatable, and some unital channels are not.

   Posed byRybár and Ziman, 2008, who showed that random-unitary channels are repeatable and non-unital ones are not, and left the other unital channels open (their Section IV), as the paper cites them.
   Our paperSeth Douglas and Nidhal Mghirbi, The memory and purity cost of immediate delivery; restated in The thermodynamics of deadlines.
   CreditThe positive half rests on known results: the catalytic dilation of Lie, Son, Boes, Ng and Wilming, and an example of Haagerup and Musat.

   As printed

   “For general systems we leave the question of repetable implementation of unital, but not random unitary channels open. […] It is an open problem whether there are some unital but not random unitary channels that are implementable in the described repeatable way.”

9. Partly answered

   Do thermal operations reach the thermodynamic capacity of a channel that is not time-covariant?

   For a clock that must return: the batch cost of every channel is a single-letter quantity, equal to the capacity whenever the environment can carry the channel’s energy change, and for a Gibbs-preserving channel only then; even unlimited returned coherence does not reach the capacity of some channels. Partial: a reference that is not returned is outside this answer.

   Posed byFaist, Berta and Brandão, 2019 and 2021 (their Section 9), who expected that a large amount of coherence would be needed, as the paper cites them.
   Our paperSeth Douglas and Nidhal Mghirbi, The price of keeping time; the analogous question for real-time implementation in The work cost of answering in real time.
   CreditThe capital bound uses the reuse argument of Chiribella, Yang and Renner; Tajima, Yamaguchi, Takagi and Kuramochi, Chiribella, Yang and Renner, and Tajima, Shiraishi and Saito bound the resource from errors alone.

   As printed

   “Whether or not it is possible to universally implement any i.i.d. channel that is not time-covariant using thermal operations is still an open question. We expect that such a protocol might in general need a very large amount of coherence, much like the requirement of large embezzling states for the reverse Shannon theorem [15, 16].”

10. Partly answered

    Can thermal operations with a time reference achieve what Gibbs-preserving maps achieve?

    A channel version, for a reference that must return: on a qubit they always can, including their own coherence-creating example; on a qutrit they can fail. Partial: the channel version, at the level of work per use in batch.

    Posed byFaist, Oppenheim and Renner, 2015, as the paper cites them.
    Our paperSeth Douglas and Nidhal Mghirbi, The price of keeping time.
    CreditFaist, Oppenheim and Renner also warned that a reference returned only approximately can embezzle work; the paper shows that such a reference can lower the battery’s share of the work, but not the total once the controller’s free energy is counted.

    As printed

    “It is still an open question whether transformations achievable by Gibbs-Preserving Maps coincide with Thermal Operations combined with some form of time reference.”

11. Answered

    Do the middle links of the Duan–Winter chain of zero-error quantities ever separate, and what are they on the pentagon?

    The pentagon conjecture holds. For every graph $S_{0,\mathrm{NS}}(G)=\log \mathrm{\Sigma }(G)$, so one of the two links is always an equality, and some graph has $C_{\min }(G)<S_{0,\mathrm{NS}}(G)$, so the other can be strict.

    Posed byDuan and Winter, 2016: they conjectured that on the pentagon every term except the first equals $\log \frac{5}{2}$, listed an additivity as unknown, and wrote that whether the other two inequalities can be strict “remains an open question”, as the paper quotes them.
    Our paperNidhal Mghirbi and Seth Douglas, The Lovász number is not alone, the corollaries “the pentagon conjecture” and “the middle links”.
    CreditWang and Duan answered a different question of Duan and Winter, strict submultiplicativity for fixed non-commutative graphs, with explicit examples.

    As printed

    “We think that also the leftmost inequality in (53) can be strict. But although the pentagon $G=C_5$ seems to be a candidate, for which we conjecture (based on some ad hoc calculations) that $C_{\min }(G)=S_{0,\mathrm{N}\mathrm{S}}(G)$ $=\log \Sigma (G)$ $=\log {\alpha }^{\ast }(G)$ $=\log \frac{5}{2}$, whereas $\log \vartheta (G)=\frac{1}{2}\log 5$, a rigorous proof of this has so far eluded us. Whether the other two inequalities can be strict remains an open question.”

12. Answered

    Is the factor 2 in the packing-dimension bound near the arithmetic transition sharp?

    No. For $0<\log \lambda <\beta <\infty$ and almost every phase, both packing dimensions of every nonzero spectral measure equal $\max (0,1-\log \lambda /\beta )$ exactly, so the sharp factor is 1.

    Posed byJitomirskaya, Liu and Tcheremchantsev, 2025: Theorem 1.1 bounds the upper packing dimension by $2(1-\log \lambda /\beta )$, and their Remark 1.1 says the factor 2 may not be sharp.
    Our paperSeth Douglas and Nidhal Mghirbi, Exact transport and spectral dimensions for the almost Mathieu operator, Theorem C.
    CreditJLT’s phase set (the α-Diophantine phases) is contained in the paper’s, with proof in version 2.0.1, so the sharp factor 1 holds at every phase their theorem covers.

    As printed

    “also suggestive that the factor of 2 may be not sharp.”

13. Partly answered

    Does the dimension bound of Jitomirskaya, Liu and Tcheremchantsev extend, with $\log \lambda$ replaced by the minimal Lyapunov exponent, to the other cases where a sharp arithmetic transition is known?

    In sharp form, for supercritical type I analytic potentials with $0<\beta <\infty$: the upper local dimension is $\max \{0,1-L(E)/\beta \}$ almost everywhere, and at phases in a full-measure set every nonzero spectral measure has upper packing dimension at most $\max \{0,1-{\ell }_{\ast }/\beta \}$, with ${\ell }_{\ast }$ the least Lyapunov exponent on the spectrum: their formulation with factor 1 in place of 2. Caveats: the phase set is not theirs, and only analytic potentials of type I are covered.

    Posed byJitomirskaya, Liu and Tcheremchantsev, 2025, in their introduction; the surrounding text names operators of type I among those cases.
    Our paperSeth Douglas and Nidhal Mghirbi, Universality of the transport law for supercritical quasi-periodic operators, Theorem B(3) and Corollary 6.13.
    CreditTheir bound carries the factor 2; Levi (2025) extended it, with the same factor, to monotone potentials, which the paper does not cover.

    As printed

    “we believe that the same formulation with $\ln \lambda$ replaced by $\min L(E)$ should work for the other cases where sharp arithmetic transition has been established”

None of these answers has been through journal review. Partly answered means the paper settles a stated case and leaves the rest open; Extended means someone else answered the question first and our paper extends the answer. Our own questions, answered and open, follow below by line.

1. 47Gap closed

   How fast does a wave packet spread in the supercritical Aubry–André model at Liouville frequencies?

   Exactly, for the upper exponents: for coupling $\lambda >1$, every irrational frequency $\alpha$ and almost every phase, every moment of a packet started at one site has upper transport exponent $\max (0,1-\log \lambda /\beta (\alpha ))$, where $\beta (\alpha )=\mathrm{lim\; sup}\log q_{n+1}/q_n$. The lower bound holds at every phase, and when $\beta (\alpha )>\log \lambda$ at the resonant times almost all of the packet makes the trip. Transport switches on exactly at $\lambda =e^{\beta }$.

   Gap closed inExact transport and spectral dimensions for the almost Mathieu operator, Theorems A and B.
   Prior workthe arithmetic spectral transition of Avila, You and Zhou; the cases $\beta (\alpha )=0$ (Damanik and Tcheremchantsev) and $\beta (\alpha )=\infty$ (Jitomirskaya and Zhang).

2. 48Answered

   Is the factor 2 in the packing-dimension bound near the arithmetic transition sharp?

   No. For $0<\log \lambda <\beta <\infty$ and almost every phase, both packing dimensions of every nonzero spectral measure equal $\max (0,1-\log \lambda /\beta )$ exactly, so the sharp factor is 1.

   Answered inExact transport and spectral dimensions for the almost Mathieu operator, Theorem C.
   Prior workthe 2025 bound of Jitomirskaya, Liu and Tcheremchantsev, who noted that the factor might not be sharp.

   As printed

   “also suggestive that the factor of 2 may be not sharp.”

3. 49Gap partly closed

   Is this transport law universal?

   Yes for every real-analytic quasi-periodic potential that is supercritical and of type I: for almost every phase the upper transport exponents follow the law with $\log \lambda$ replaced by the minimum Lyapunov exponent on the spectrum, for every moment, and energy by energy in the limit of narrow energy filters; a potential $w(sx)$ obeys it with $\beta (s\alpha )$ when $w$ at frequency $s\alpha$ is supercritical and of type I. Partial: for every real-analytic potential with positive Lyapunov exponent, the law for the fastest upper exponent is proved conditionally on one explicit weak-separation hypothesis, open beyond type I.

   Gap partly closed inUniversality of the transport law for supercritical quasi-periodic operators, Theorems A to D.
   Prior workAvila’s global theory; Ge and Jitomirskaya’s type I class; the case $\beta (\alpha )=0$ for general supercritical analytic potentials, observed by Jitomirskaya and Mavi.

4. 50Open

   Does the weak-separation hypothesis hold beyond type I?

   It is needed only for primitive real-analytic potentials with acceleration at least two. If it holds for them, the law for the fastest upper transport exponent holds, for almost every phase, for every real-analytic quasi-periodic potential with positive Lyapunov exponent on its spectrum.

   Posed inUniversality of the transport law for supercritical quasi-periodic operators, Section 8.

5. 51Partly proved

   How fast can a wave packet spread at the worst phase, rather than a typical one?

   For every real-analytic potential with positive Lyapunov exponent on the spectrum and every frequency with $0<\beta (\alpha )<\infty$, uniformly in the phase, the fastest upper transport exponent is at most the maximum over the spectrum of $1-\ell /(2\omega \beta )$ when $\beta \ge \ell$, with an explicit formula when $\beta <\ell$; so transport is never ballistic at a finite Liouville strength, and the factor $2\omega$ is attained in this class. For the almost Mathieu operator, when ${\beta }_e=\beta$ is finite and greater than $\log \lambda /2$, the worst exponent is exactly $1-\log \lambda /(2\beta )$. Partial: the exact worst-case law for primitive potentials, and for the almost Mathieu operator without the parity condition, remain open (questions 55 and 56).

   Partly proved inWorst-case transport and light cones for quasi-periodic operators, Theorems A and B.
   Prior workthe endpoints $\beta (\alpha )=0$ (Damanik and Tcheremchantsev; Jitomirskaya and Mavi) and $\beta (\alpha )=\infty$ (Jitomirskaya and Zhang); the closed central gap of Bellissard and Simon and of Lamoureux and Mingo.

6. 52Proved

   Is the worst phase rare?

   Not in the sense of category. For the almost Mathieu operator, when ${\beta }_e=\beta$ is finite and greater than $\log \lambda /2$, the exponent is $1-\log \lambda /(2\beta )$ on a dense $G_{\delta }$ set of phases, while it is $\max (0,1-\log \lambda /\beta )$ at almost every phase: measure and category give different answers.

   Proved inWorst-case transport and light cones for quasi-periodic operators, Theorem C.
   Prior workthe Baire argument is standard; the almost-every-phase law is the companion paper’s.

7. 53Proved

   How fast do signals travel in the quasi-periodic XY spin chain?

   For the Aubry–André chain and the specified even sources and receivers: at almost every phase, the light cone at a fixed site follows the typical law $\max (0,1-\log \lambda /\beta )$; when ${\beta }_e=\beta$ is finite and greater than $\log \lambda /2$, the light cone uniform over sites follows the worst-case law $1-\log \lambda /(2\beta )$. For every supercritical analytic field, the uniform light cone obeys the all-phase upper bound. An explicit computable chain has two different light-cone exponents, both of Turing degree ${\mathrm{𝟎}}^{″}$.

   Proved inWorst-case transport and light cones for quasi-periodic operators, Theorem D.
   Prior workthe reduction to one-body dynamics of Lieb, Schultz and Mattis, used by Hamza, Sims and Stolz; the Fibonacci chain of Damanik, Lemm, Lukic and Yessen; the OTOC formula of Riddell and Sørensen.

8. 54Proved

   Where does the uncomputability of transport come from?

   From the frequency alone. The Lyapunov exponent, the density of states and the spectrum of a computable analytic potential are computable at every computable irrational frequency; and at one explicit computable frequency and the phase $1/4$, the transport exponent has Turing degree exactly ${\mathrm{𝟎}}^{″}$ for every computable even potential in an explicit open neighbourhood of $4\cos 2\pi x$.

   Proved inWorst-case transport and light cones for quasi-periodic operators, Theorem E.
   Prior workthe computability of the Lyapunov exponent is an algorithmic consequence of Avila’s global theory; earlier robust and continuous-parameter uncomputability is due to Cubitt, Perez-Garcia and Wolf, to Bausch, Cubitt and Watson, and to Purcell, Li and Cubitt.

9. 55Open

   Is the worst-phase factor two, rather than twice the acceleration, for primitive potentials?

   Conjectured: if the potential is $w(sx)$ with $w$ primitive and positive Lyapunov exponent on the spectrum, and the Liouville strength of $s\alpha$ is finite and at least the largest Lyapunov exponent on the spectrum, then the worst-phase exponent is exactly $1-{\ell }_{\ast }/(2\beta (s\alpha ))$: the worst phase halves the localization strength. Division by twice the acceleration is attained by potentials with a hidden period; whether a primitive potential can do worse than halving is open.

   Posed inWorst-case transport and light cones for quasi-periodic operators, Conjecture 11.1.

10. 56Open

    Does the exact worst-case law of the almost Mathieu operator hold without the parity condition?

    It is proved when the Liouville strength is attained along even denominators. It would follow for every irrational frequency with $\log \lambda \le \beta <\infty$ from a monotonicity conjecture on the extreme gaps of the rational approximants, which has been checked only on finite numerical grids.

    Posed inWorst-case transport and light cones for quasi-periodic operators, Conjecture 11.5 and Proposition 11.6.

1. 1Answered

   Can the rate at which the least dimension grows near perfect play be computed from a finite game’s rules?

   Not in general. For finite rational Boolean constraint-system games of quantum value one, the positive dimension exponents are exactly the positive ${\mathrm{\Delta }}_2^0$ reals, the limits of computable sequences of rationals; each is matched by distillable entanglement on the original mixed state.

   Answered inThe uncomputable price of precision, Theorem 1.1.
   Prior workMousavi, Nezhadi and Yuen (value one is ${\mathrm{\Pi }}_2^0$-complete); Slofstra and Vidick (the hyperlinear profile); precedents in form by Meyerovitch, by Hochman and Meyerovitch, and by Sapir, Birget and Rips.

2. 2Answered

   Which lower and upper exponents can one fixed game have?

   Exactly the pairs $0<L\le U$ with $L\in {\mathrm{\Sigma }}_2^0$ and $U\in {\mathrm{\Pi }}_2^0$, each realized with the same usable-entanglement pair. Some computable whole profiles are realized by no game.

   Answered inThe uncomputable price of precision, front theorem “Exponent pairs and whole profiles”; The price of unreachable perfection for every regular scale.

3. 3Open

   Can a finite rational value-one game with no finite-dimensional perfect strategy have dimension exponent zero?

   Finite-dimensional perfection forces exponent zero; the converse is open. Every construction so far pays for compulsory clocks, so a zero example needs a cheaper verifier or a different mechanism.

   Posed inThe uncomputable price of precision, Problem “Zero without perfection”.

4. 4Open

   Is every positive usable-entanglement coefficient of a finite rational game limit-computable?

   Every positive ${\mathrm{\Delta }}_2^0$ real is realized; the reverse inclusion is open, because finite-dimensional feasibility does not compute a regularized distillation rate. A weak pure rounding conjecture would settle it.

   Posed inThe uncomputable price of precision and The price of precision.
   Prior workPaddock’s maximally entangled rounding, the closest precedent.

5. 5Open

   Which finite power exponents above one occur for a fixed finite rational game?

   Fixed games with infinite upper power exponents exist; existing finite limits above one are open.

   Posed inThe uncomputable price of precision, Problem “Power-law usable demand”.

6. 6Answered

   When does near-perfect play need unbounded entanglement, and can one tell?

   For synchronous games with tested diagonals and for constraint-variable consistency games, the demand diverges exactly when no finite-dimensional strategy is perfect. Among games of value one, bounded demand is undecidable.

   Answered inThe price of precision: usable entanglement in nonlocal games and the geometry of groups.

7. 7Answered

   How much entanglement can an explicit game of quantum value one force?

   Entanglement entropy ${\epsilon }^{-1+o(1)}$ at loss $\epsilon$, hence local dimension $\exp ({\epsilon }^{-1+o(1)})$, for a binary linear system game with 1687 equations; the exponent is sharp. The same tools give the game of Slofstra and Vidick local dimension $\mathrm{\Theta }({\epsilon }^{-1/2})$.

   Answered inAn explicit nonlocal game of quantum value one that needs exponential entanglement.
   Prior workfor explicit two-player games of value one, the best previous lower bound was polynomial in $1/\epsilon$, as the paper records.

8. 8Answered

   Can perfect play in a finite game require infinite dimension yet only finite energy?

   Yes. A binary-scoring game with ten and eleven questions is won perfectly on ${\ell }^2\otimes {\ell }^2$, never in finite dimension, and has a perfect state of the massive free scalar field with every polynomial energy moment finite.

   Answered inPerfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy.
   Prior workMančinska’s flagged magic-square game, the building block.

9. 9Answered

   Are the critical exponents of lattice matter computable from its Hamiltonian?

   Not in general. Translation-invariant nearest-neighbour Hamiltonian families have existing critical gap exponents that are exactly the nonnegative ${\mathrm{\Delta }}_3^0$ reals, already in two dimensions.

   Answered inUncomputable critical exponents.

10. 10Answered

    Can the robust laws of a noisy cellular automaton be unlearnable?

    Yes. A noisy rule started from cloned uniform noise realizes every ${\mathrm{\Delta }}_3^0$ value in $[0,1]$ as a robust law; at computable kernels no effective inquiry converges with positive probability to a fixed such law outside ${\mathrm{\Delta }}_2^0$. At equal depth, the laws of an automaton with a beginning reach one arithmetical level higher than any law on which its stationary states agree.

    Answered inA beginning is worth one limit and Uncomputable critical exponents.

11. 11Open

    Can cloned preparation be removed from the realization of robust unlearnable laws?

    Posed inA beginning is worth one limit.

12. 12Answered

    Can gapped lattice matter carry any limit-computable amount of entanglement?

    Gapped translation-invariant lattices have entanglement with any prescribed limit-computable logarithmic coefficient. The paper’s further, game-driven material theorem is stated conditionally.

    Answered inMatter that pays the price of perfection.

13. 13Answered

    Does resetting a device before its parts communicate cost unbounded heat?

    For games whose usable-entanglement demand diverges, resetting locally before communicating forces divergent work, while bringing the systems back together before reset keeps the operating work bounded.

    Answered inThe heat price of precision.

14. 14Open

    Which positive local-reset heat coefficients occur?

    The spectrum lies between the positive ${\mathrm{\Delta }}_2^0$ reals and the positive reals left-c.e. in the halting set; equality is not proved.

    Posed inThe uncomputable price of precision, Problem “The full local-heat spectrum”.

1. 15Answered

   Is there a finite game that type III algebras can win perfectly and semifinite algebras cannot?

   Yes. A game with 243 question pairs has a perfect normal strategy on a type III factor and none whose marginal is normal on a semifinite algebra. For factors with separable predual, hosting it is equivalent to an invariant normal state on the flow of weights.

   Answered inFinite games that read the type of a von Neumann algebra, Theorem 1.1.
   Prior workHaagerup and Musat (the analytic equivalence); Mančinska (the flagged magic-square gadget); van Luijk, Stottmeister, Werner and Wilming (an operational reading of type through embezzlement).

2. 16Answered

   Can finite games characterize type III₁?

   Yes. Every faithful normal state has value one for two fixed finite games exactly when the factor is type III₁. One prepared state in dimension 25 wins both, so the universal-state premise cannot be dropped.

   Answered inFinite games that read the type of a von Neumann algebra, Theorem 1.2.
   Prior workConnes’s classification and invariant.

3. 17Answered

   Is every positive rational forced as a modular ratio by some finite game?

   Yes, by a binary-scoring game with a finite-dimensional perfect strategy.

   Answered inFinite games that read the type of a von Neumann algebra, Theorem 1.3.

4. 18Open

   Can any finite game force an irrational modular ratio?

   Posed inFinite games that read the type of a von Neumann algebra, Section 8.

5. 19Open

   Can a finite game tell type III_1/2 from type III_2^−1/2 at the worst-state level?

   Their Connes invariants meet the positive rationals in the same set, so rational membership tests agree.

   Posed inFinite games that read the type of a von Neumann algebra, Section 8.

6. 20Open

   What is the sharp robustness constant of the type-III-only game?

   The robust law is proved; 121.5 is the natural candidate for the constant, and no validated numerical constant is given.

   Posed inFinite games that read the type of a von Neumann algebra, Section 8.

7. 21Open

   How does an algebra’s type constrain the price of perfection?

   Eight cells of the paper’s type-against-price table are open, among them polynomial-price type-III-only games and exponentially expensive games of minimum type I_∞.

   Posed inFinite games that read the type of a von Neumann algebra, Sections 7 and 8.

1. 22Partly answered

   Do Kaplansky’s group-ring conjectures hold for torsion-free hyperbolic groups?

   Not in positive characteristic: for every prime $p$ one finitely presented, torsion-free, word-hyperbolic group fails all four over every field of characteristic $p$. In characteristic zero the unit and zero-divisor conjectures remain open for these groups; the idempotent conjecture and direct finiteness hold there.

   Partly answered inKaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic.
   Prior workOpenAI’s 2026 constructions, which it extends; S. V. Ivanov (arXiv:2610.17662) has precedence for torsion-free zero divisors in odd characteristic; Bunina, Lee, Miasnikov and Muliarchyk (arXiv:2610.22536), independently, for direct finiteness without torsion-freeness or hyperbolicity. For torsion-free groups in general, others disproved each conjecture first; see the printed question.

2. 23Answered

   Is there a finitely presented infinite $p$-group for every prime?

   Yes: an infinite, residually finite group with an ordinary finite presentation in which every element has finite $p$-power order. One presentation over the integers serves every prime.

   Answered inFinitely presented infinite p-groups for every prime.
   Prior workOpenAI’s construction for $p=2$, whose conceptual credit is theirs; a lemma of Zalesskii (1985). Concurrent and independent: Kenta Kitamura also proved the every-prime result (manuscript of 13 October 2026), as version 1.0.1 of the paper notes.

3. 24Answered

   Is OpenAI’s torsion-free hyperbolic group sofic?

   No: the rank argument that shows it is not residually finite survives approximation.

   Answered inNon-soficity of OpenAI’s torsion-free hyperbolic group.
   Prior workthe group and every input to the argument are OpenAI’s; Chapman and Peled’s conditional construction; Thom’s independent note based on Sandhu’s construction.

4. 25Answered

   Can a finitely presented amenable group have nearly exponential sofic profile?

   Yes: a finitely presented elementary amenable group with a finite piece whose sofic profile is $\exp (r^{1+o(1)})$.

   Answered inAmenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory.

5. 26Answered

   Is the sofic profile of Houghton’s group $H_3$ polynomial?

   No: $H_3$ has a finite chunk whose sofic profile is superpolynomial.

   Answered inHoughton’s group H₃ has superpolynomial sofic profile.

1. 27Answered

   Can a finite quantum system reach the largest violation of the I₃₃₂₂ Bell inequality?

   No. A spatial strategy on ${\ell }^2(\mathbb{Z})\otimes {\ell }^2(\mathbb{Z})$ attains the quantum maximum; no finite-dimensional strategy does, mixed states and binary measurements included. Coming within $\epsilon$ needs local dimension $\mathrm{\Theta }(\log (1/\epsilon ))$.

   Answered inThe I₃₃₂₂ quantum value is attained spatially but not in finite dimension.
   Prior workconjectured by Pál and Vértesi (2010); concurrent and independent work by Pauwels (arXiv:2608.29734) and by Coladangelo (arXiv:2609.06038).

2. 28Claimed answered elsewhere

   What is the exact quantum value of I₃₃₂₂?

   Claimed answered elsewhere (Krohn-Grimberghe 2026, computer-assisted; not independently verified by us): the value is claimed to be the unique intersection of two analytically characterized curves, isolated in a rational interval of width ${10}^{-55}$; no algebraic equation for it is claimed, and the work is not peer reviewed. Our papers use certified windows: about $3.6\times {10}^{-9}$ in the I₃₃₂₂ paper, and exact rational enclosures down to about $1.1\times {10}^{-10}$ in Nidhal Mghirbi’s (q42).

   Left open inThe I₃₃₂₂ quantum value …
   Claimed inKrohn-Grimberghe, arXiv:2609.29693 (31 August 2026).

3. 29Answered

   Is the Lovász number the only Rényi point in the entanglement-assisted spectrum of graphs?

   No. Petz–Rényi information radii over classical–quantum channels compatible with a graph give a continuum of points between known graph parameters.

   Answered inThe Lovász number is not alone.
   Prior workDuan and Winter; Li and Zuiddam.

4. 30Open

   Is the projective rank of a graph’s complement in its entanglement-assisted spectrum?

   Li and Zuiddam leave it open. A 1440-vertex graph has $\mathrm{\Sigma }(G)$ strictly below that projective rank, which closes one route to equality without deciding the question.

   Discussed inThe Lovász number is not alone, Theorem D.

5. 31Answered

   What does drift in the marginals cost when approximating the non-attained $K_5$ correlation?

   At the distinguished parameter $t_{\ast }=1/\sqrt{5}$, the cost against the fixed target is $\mathrm{\Theta }(\min ({\epsilon }^{-1/4},{\delta }^{-1/4}))$: the same fourth root, not the ${\delta }^{-1/2}$ an earlier version conjectured.

   Answered inA Diophantine answer for K₅, version 1.0.2, Theorem 5.3.
   Prior workthe non-closure witness of Dykema, Paulsen and Prakash.

6. 32Answered

   Is the least number of outcome merges that restores a global section a new measure of contextuality?

   No. On binary scenarios it is an affine reparametrization of Min-Rep, a standard quantity in hardness of approximation; the paper itself shows this.

   Answered inA repair price for contextuality, and its identity with Min-Rep.

7. 41Open

   What is the sharp constant of the I₃₃₂₂ dimension rate?

   The least local dimension within $\epsilon$ of the value is $\mathrm{\Theta }(\log (1/\epsilon ))$, with the explicit upper constant 23.9010650 (natural logarithms). The lower constants are existential; no explicit onset or sharp constant is claimed.

   Left open inThe I₃₃₂₂ quantum value …, Theorem (Rate). Split from question 28 on 30 October 2026.

8. 42Answered

   How tightly can exact rational arithmetic enclose the I₃₃₂₂ quantum value?

   A Gram certificate on all words of length at most four and a rational strategy of local dimension 499 enclose it in a window narrower than $5\times {10}^{-10}$; a Bellman storage certificate lowers the upper end to 0.250875384626, a window of about $1.12\times {10}^{-10}$.

   Answered inExact rational certification of upper and lower quantum bounds for the I₃₃₂₂ Bell inequality and Alternating Bellman–Jacobi duality, both by Nidhal Mghirbi.
   Prior workPál and Vértesi’s numerical value; rational certification methods of Cafuta, Klep and Povh, of Hrga, Klep and Povh, and of Naceur and coauthors.

9. 43Answered

   Which three-setting binary Bell functionals reach their quantum maximum only in infinite dimension?

   A parameterized class of reversible functionals that contains I₃₃₂₂: for each, the supremum is approached in finite dimension and attained only by an infinite-dimensional spatial strategy, and the set of finite-dimensional correlations is not closed.

   Answered inCritical Bellman–Jacobi dynamics for reversible Bell functionals, by Nidhal Mghirbi.
   Prior workPál and Vértesi’s conjecture; for I₃₃₂₂ itself, Douglas’s paper, which the paper calls contemporaneous and independent, and Pauwels and Coladangelo.

1. 33Answered

   Can an algorithm tell how much memory a repeated quantum channel needs?

   No. The answer turns on one set of channels: inside it, channels with a flat probe can be served with memory that grows more slowly than the number of uses; outside it, a device with sublinear memory fails with probability tending to one.

   Answered inNo algorithm can tell how much memory a repeated quantum channel needs.

2. 34Answered

   Do a channel’s spectrum and entropies determine the memory needed to simulate it causally?

   No. Two explicit channels with the same complete Choi spectrum and the same entropies need different amounts of memory at the same optimal exchange rate.

   Answered inCausal quantum-channel simulation: memory beyond entropy.

3. 35Answered

   Is every factorizable rational channel a limit of channels with a finite maximally mixed environment?

   No. Explicit rational channels are factorizable yet at a positive, explicit distance from that closure.

   Answered inFactorizable rational quantum channels at an explicit distance from finite tracial baths.

4. 36Answered

   How many qubits does clean two-query synthesis of every $n$-qubit unitary need?

   $\mathrm{\Theta }(2^n)$ qubits and oracle input bits, up to a factor $\log (1/\epsilon )$: a Gaussian-mean-width lower bound and a permutation-transport construction.

   Answered inThe clean two-query threshold for unitary synthesis.

5. 37Open

   Does the clean four-insertion width bound hold?

   The garbage-model two-query problem reduces to it.

   Posed inThe clean two-query threshold for unitary synthesis.

6. 38Answered

   When does recovering finitely many sharp binary observables recover the whole quantum input?

   The paper gives the condition, with matching fixed-gap minimax bounds.

   Answered inQuantum recovery from finitely many sharp questions.

1. 39Answered

   What does it cost, in work, to act on a schedule?

   For every finite schedule in which each carrier returns with its Hamiltonian and the controller comes back, the optimal work per use is the least entropy an implementation must add to its own record, plus the energy it hands over, against the most demanding observer. The cost survives arbitrarily slow operation.

   Answered inThe thermodynamics of deadlines, with its four companion papers.
   Prior workLandauer’s bound and the thermodynamic capacity of Faist, Berta and Brandão (in the time-covariant case) are cases of the law.

2. 40Open

   Does the predicted price of early release appear in a laboratory?

   A two-qubit process that is free when its output is held costs every early-release device at least $0.87k_{\mathrm{B}}T\ln 2$ per episode at one percent error. A three-arm feedback-trap protocol for a classical version is proposed; it has not been run.

   Proposed inThe thermodynamics of deadlines.

1. 44Answered

   Does Galilean symmetry fix the quantum probability current, and with it the guidance equation?

   Almost. Imposing Galilean boost covariance pointwise on any one-body current leaves exactly the Schrödinger current plus a linear functional of the density. Rotational covariance with conservation removes that term in three or more dimensions, and conservation with the momentum–translation relation removes it in every dimension.

   Answered inGalilean covariance almost fixes the quantum probability current, by Nidhal Mghirbi.
   Prior workHolland’s results on the uniqueness of conserved currents; Deotto and Ghirardi’s alternative guidance laws.

2. 45Answered

   When can restricted measurements hide a conserved departure from quantum equilibrium?

   Under a quantified mixing hypothesis, the distinguishability of every permitted complete record is bounded by the product of dynamical decay and the sensitivity of the record family, and a Fisher-information bound expresses that sensitivity through calibrated device quantities.

   Answered inObservable quantum equilibrium under controlled measurements, by Nidhal Mghirbi.
   Prior workValentini’s subquantum H-theorem; Valentini and Westman’s simulations.

3. 46Open

   Can distributions out of quantum equilibrium be physically accessed?

   In the paper’s words, “the physical accessibility of non-equilibrium distributions remains an open question.”

   Posed inThe relativistic necessity of the Born rule, by Nidhal Mghirbi.