Results

Every result, with what was checked and whose work it builds on.

Forty-eight works with DOIs, released between September 2025 and November 2026, in 121 archived versions. Dates are the Zenodo publication dates of the current versions. The joint works each have a public repository with their source, machine checks and recorded outputs; single-author works made before the partnership carry their author’s own reproducibility packages. One has been published in a journal; each label says whether a work is peer reviewed.

Every paper, newest first All papers as BibTeX Checksums of every hosted PDF

How to read a label

Verification: The checks the result passed before release. For the joint works these are our own checks: blind refutation-first checks by models that had not seen the proof, usually from a different developer than the one that produced it, exact computation on independent engines, and audits of the whole paper by readers who had not seen it (Method). For single-author works made outside this lab, the label says “the author’s own checks” and names the reproducibility package.
Lean: What is formalized in the Lean proof assistant: nothing, selected lemmas (named), or the main theorem.
Peer review: Not yet, submitted (with the journal), or published (with the journal), as the journal documents it.
Novelty: One of: new; new, with named precedent; new, with named precedents in form; new, with named prior work; concurrent with named independent work; new proof of a known result; extends named prior work; unifies named prior results; synthesis of prior work (credited). Credit is stated in the paper.
Checked: The date on which the label was last compared with the released paper.

Transport in quasi-periodic systems

How fast a wave packet spreads in quasi-periodic systems, and why the arithmetic of the frequency decides it. The line, in reading order

Exact transport and spectral dimensions for the almost Mathieu operator

Seth Douglas and Nidhal Mghirbi · version 2.0.1 · 1 November 2026 · 50 pages

For the almost Mathieu operator with coupling $λ > 1$ , every irrational frequency $α$ and almost every phase, every moment of a wave packet started at one site has the same upper transport exponent $\max (0, 1 - \log λ / β (α))$ , where $β (α) = lim sup \log q_{n + 1} / q_{n}$ ; the lower bound holds at every phase, and when $β (α) > \log λ$ at the resonant times almost all of the packet makes the trip, uniformly in the phase. For almost every phase, both packing dimensions of every nonzero spectral measure equal the same number; transport switches on exactly at $λ = e^{β}$ ; and an explicit computable almost Mathieu operator has a fastest upper transport exponent of Turing degree $𝟎^{″}$ .

Verification
Each step independently blind-checked; a blind audit of the whole paper with no critical findings; version 2.0.0’s new mathematics checked blind, then a blind fix-check of the whole revision.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: for $0 < \log λ < β < \infty$ and almost every phase, it makes the 2025 packing bound $2 (1 - \log λ / β)$ of Jitomirskaya, Liu and Tcheremchantsev sharp with factor 1, where they had noted that the factor 2 might not be sharp. It credits the cases $β (α) = 0$ and $β (α) = \infty$ to Damanik and Tcheremchantsev and to Jitomirskaya and Zhang, earlier lower bounds of the form $1 - C \log λ / β$ to Jitomirskaya and Zhang and to Haeming, and the arithmetic spectral transition to Avila, You and Zhou.

PDF The paper page doi:10.5281/zenodo.23864352 RepositoryAll versions: 10.5281/zenodo.23830304

PDF of version 2.0.1, 988,789 bytes, the released file; SHA-256 a3eff9b6dd7f9e40c9fbdfd5c2b21bdefaf72e4d0360bcb03a6e794e576eb58b

Label checked 1 Nov 2026
Universality of the transport law for supercritical quasi-periodic operators

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 31 October 2026 · 55 pages

For every real-analytic quasi-periodic potential that is supercritical and of type I, and almost every phase, the upper transport exponents follow the same law with $\log λ$ replaced by the minimum Lyapunov exponent on the spectrum, for the fastest upper exponent and for every moment. Energy by energy, for packets filtered ever more narrowly near an energy $E$ , every upper moment exponent tends to $\max (0, 1 - L (E) / β (α))$ , and the spectral measure has that upper local dimension at almost every $E$ . A potential $w (s x)$ obeys the law with $β (s α)$ when $w$ at frequency $s α$ is supercritical and of type I. 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.

Verification
Each step independently blind-checked; a blind audit of the whole paper with no critical findings, and a blind fix-check.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: Avila’s global theory, Ge and Jitomirskaya’s type I class and the companion paper’s law for the almost Mathieu operator, credited; for general supercritical analytic potentials it credits the case $β (α) = 0$ to Jitomirskaya and Mavi.

PDF The paper page doi:10.5281/zenodo.23830333 RepositoryAll versions: 10.5281/zenodo.23830332

PDF of version 1.0.0, 1,001,404 bytes, the released file; SHA-256 ae5e41bae01c014ba5b3b46a1bc2969fd0a7e68b9f7ff63e9d7a2d1939c34569

Label checked 31 Oct 2026
Worst-case transport and light cones for quasi-periodic operators

Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 2 November 2026 · 72 pages

At the worst phase rather than a typical one. For every real-analytic quasi-periodic potential with positive Lyapunov exponent on the spectrum and every frequency with $0 < β (α) < \infty$ , uniformly in the phase, the fastest upper transport exponent is at most the maximum over the spectrum of $1 - ℓ / (2 ω β)$ when $β \geq ℓ$ , where $ℓ$ is the Lyapunov exponent and $ω$ the acceleration, with an explicit formula when $β < ℓ$ ; so a finite $β$ keeps transport below ballistic at every phase, and the factor $2 ω$ is attained. For the almost Mathieu operator, when $β_{e} = β$ is finite and greater than $\log λ / 2$ , where $β_{e}$ is the limsup along even denominators, the worst exponent is exactly $1 - \log λ / (2 β)$ : it is attained at the phase $1 / 4$ and on a dense $G_{δ}$ set of phases, while almost every phase has $\max (0, 1 - \log λ / β)$ . In the Aubry–André XY spin chain, for the specified observables, the light cone at a fixed site follows the typical law at almost every phase, and the light cone uniform over sites follows the worst-case law; an explicit computable chain has two different light-cone exponents, both of Turing degree $𝟎^{″}$ . The Lyapunov exponent of a computable analytic potential is computable at every computable irrational frequency, and at one explicit frequency the transport exponent has degree exactly $𝟎^{″}$ for every computable even potential in an explicit open neighbourhood of $4 \cos 2 π x$ .

Verification
Each result independently blind-checked; a blind audit of each version with no critical findings, and blind fix-checks of every repair.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: the endpoint $β (α) = \infty$ is due to Jitomirskaya and Zhang, and $β (α) = 0$ to Damanik and Tcheremchantsev and to Jitomirskaya and Mavi. It credits the link between the acceleration and zeros of determinants to Han and Schlag, the closed central gap to Bellissard and Simon and to Lamoureux and Mingo, the reduction of XY dynamics to one-body dynamics to Lieb, Schultz and Mattis and to Hamza, Sims and Stolz, the single-site OTOC formula to Riddell and Sørensen, and earlier uncomputability in physics to Cubitt, Perez-Garcia and Wolf, to Bausch, Cubitt and Watson, and to Purcell, Li and Cubitt. The computability of the Lyapunov exponent is an algorithmic consequence of Avila’s global theory, for which it makes no priority claim.

PDF The paper page doi:10.5281/zenodo.23875774 RepositoryAll versions: 10.5281/zenodo.23845227

PDF of version 2.0.0, 1,201,379 bytes, the released file; SHA-256 8da653613f3cab77be285e37ce8ed023c3ed5c72a6eb215824fdf68e36d19f54

Label checked 1 Nov 2026
Band edges of the almost Mathieu operator at Liouville scales

Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 2 November 2026 · 60 pages

For the supercritical almost Mathieu operator, write $L = \log λ$ , $β = β (α)$ and $x = 2 β / L$ , and let $ρ$ be the exponential gap-decay rate, the liminf of $- \log | G_{m} | / (L | m |)$ . Above the Liouville threshold, for finite $β > L / 2$ , infinitely many gaps are only polynomially small in their integer label, so $ρ = 0$ : at a convergent $p_{n} / q_{n}$ , with $δ_{n} = q_{n} α - p_{n}$ and $b_{n} = - q_{n}^{- 1} \log | δ_{n} |$ , the gaps with labels $m = - D q_{n} sgn δ_{n}$ at the top of the spectrum have width at least $\exp (- (b_{n} + L / 2 + o (1)) q_{n})$ , uniformly over an explicit growing range of $D$ , and a separate surplus argument includes $β = \infty$ . Below the threshold, for $0 < x < 1$ , $ρ \leq 1 - x$ : at $β$ -attaining convergents, the gap with label $- D q_{n} sgn δ_{n}$ has an explicit lower bound on its width for every fixed $D$ . With the theorem of Liu and Shi this gives $1 / C \leq ρ \leq 1 - x < 1$ for $0 < x \leq 2 / C$ , which contradicts their stated expectation of a sharper upper gap bound in that range but not their theorem. Equality $ρ = 1 - x$ , and the line $x = 1$ , remain open.

Verification
Each part of the proof checked by a separate session instructed to refute it and to inspect its primary sources; a refutation-first audit of the assembled manuscript, and a check of its repairs.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: all noncritical gaps are open by Avila, You and Zhou; Ge, You and Zhou prove the Diophantine rate; polynomial gap phenomena at critical coupling are due to Krasovsky; the orbital displacement is the classical Rammal–Wilkinson correction. The contribution is their uniform combination at one Liouville scale with growing labels, and the resulting theorem. Below the threshold, the bound ρ ≤ 1 − x contradicts the sharper upper gap bound that Liu and Shi expected in an overlapping range; it does not contradict their theorem.

PDF The paper page doi:10.5281/zenodo.23877075 RepositoryAll versions: 10.5281/zenodo.23873796

PDF of version 2.0.0, 980,638 bytes, the released file; SHA-256 c54e27925352e1914dabd6bf2fd22b7c616f21631e2239dd8114cb72fd03f11c

Label checked 1 Nov 2026

The price of perfection

What it costs a finite quantum system to approach a perfect strategy it can never reach, and whether that cost can be computed. The line, in reading order

The uncomputable price of precision

Seth Douglas and Nidhal Mghirbi · version 4.0.0 · 29 October 2026 · 119 pages

For finite rational constraint-system games of quantum value one, the positive exponents of the least local dimension needed near perfect play are exactly the positive $Δ_{2}^{0}$ reals, with matched distillable-entanglement realizations; the lower and upper exponent pairs are exactly the $(Σ_{2}^{0}, Π_{2}^{0})$ pairs.

Verification
Proofs checked by models from two developers; the original construction checked on two independent finite-model engines; blind audit rounds before each of the four versions.

Lean
Sections 2 to 10 in progress, not complete; no completed formal verification is claimed.

Peer review
Not yet.

Novelty
New, with named precedents in form: Meyerovitch; Hochman and Meyerovitch; Sapir, Birget and Rips.

PDF The paper page doi:10.5281/zenodo.23753741 RepositoryAll versions: 10.5281/zenodo.23701733

PDF of version 4.0.0, 1,402,612 bytes, the released file; SHA-256 47fc33b7183b47111b0da29a8f5ccd7ef8d0e7cd30976eebebd231919e1fc315

Label checked 29 Oct 2026
Uncomputable critical exponents

Seth Douglas and Nidhal Mghirbi · version 2.1.0 · 29 October 2026 · 64 pages

Translation-invariant nearest-neighbour quantum Hamiltonians whose critical gap exponents are exactly the nonnegative $Δ_{3}^{0}$ reals, already in two dimensions; and robust laws of noisy media that no effective inquiry can learn beyond $Δ_{2}^{0}$ .

Verification
Each theorem block verified, the robust-law theorem twice; two audits; a blind fix-check.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work on robust noisy media (Marcovici, Sablik and Taati; Toom; Gács and Reif), credited; the paper makes no exhaustive priority claim.

PDF The paper page doi:10.5281/zenodo.23780285 RepositoryAll versions: 10.5281/zenodo.23753839

PDF of version 2.1.0, 900,976 bytes, the released file; SHA-256 63bf418a36f428cb5e6f369255f0260ef57832193c76b9726e966ab46c847165

Label checked 29 Oct 2026
A beginning is worth one limit: robust unlearnable laws in noisy cellular automata

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 29 October 2026 · 19 pages

At equal depth, a computable law of a cellular automaton that has a beginning reaches one arithmetical level above any law on which its stationary states agree; at depth one, a rule started from cloned uniform noise realizes every $Δ_{3}^{0}$ value in $[0, 1]$ as a robust law.

Verification
Two independent verifications; one blind audit with no critical findings; a blind fix-check with no findings.

Lean
None.

Peer review
Not yet.

Novelty
Uses results of Hellouin de Menibus and Sablik, and of Miękisz and Taati, credited; the paper makes no exhaustive priority claim.

PDF The paper page doi:10.5281/zenodo.23782039 Repository

PDF of version 1.0.0, 583,478 bytes, the released file; SHA-256 70e5b445e6a7448e313c15d4ed6daf27e1541316fd07cc3d2040bc94b0e31a42

Label checked 29 Oct 2026
An explicit nonlocal game of quantum value one that needs exponential entanglement

Nidhal Mghirbi and Seth Douglas · version 1.1.0 · 2 November 2026 · 93 pages

A binary linear system game with 1687 equations in 1993 variables, of quantum value one, in which winning with probability $1 - ε$ needs entanglement entropy $ε^{- 1 + o (1)}$ , hence local dimension $\exp (ε^{- 1 + o (1)})$ ; the exponent is sharp.

Verification
Exact certificates for the four computational statements, each re-checked by a separately written program in anc/.

Lean
Theorem A, conditional on the published implication from perfect tensor-product strategies, and Theorems C, D, E and F formalized in Lean 4; the release was rebuilt from scratch and replayed by Lean's checker and the independent nanoda checker. Source and recorded axioms in the repository.

Peer review
Not yet.

Novelty
New. For explicit games of quantum value one, the best previous lower bound was polynomial in $1 / ε$ . The same tools give sharp rates for the CHSH( $n$ ) games and the game of Slofstra and Vidick.

PDF The paper page doi:10.5281/zenodo.23878232 Repository Lean sourceAll versions: 10.5281/zenodo.23697254

PDF of version 1.1.0, 1,246,081 bytes, the released file; SHA-256 9b7793fe0474706173d8e3c25d4d5174cc4eca6d1dc5251e97583c42288fd8fc

Label checked 2 Nov 2026
The price of precision: usable entanglement in nonlocal games and the geometry of groups

Seth Douglas and Nidhal Mghirbi · version 1.1.0 · 29 October 2026 · 160 pages

For the game classes treated, the entanglement demand near perfect play diverges exactly when no finite-dimensional strategy is perfect; bounded demand is undecidable among games of value one; and the geometry of groups gives logarithmic, almost inverse-loss and intermediate laws.

Verification
Repeated rounds of blind audit for version 1.0, the final round with no serious findings; four further rounds for version 1.1.

Lean
None yet.

Peer review
Not yet.

Novelty
New, with named precedent: Mančinska and Vidick gave a game, with answers of unbounded length, whose optimal value no finite amount of entanglement attains. Every imported result is credited at its use, among them Slofstra’s and those of Fanizza, Kroell, Mehta, Paddock, Rochette, Slofstra and Zhao. Bounded literature searches found no exact preemption; the paper makes no claim of being first.

PDF The paper page doi:10.5281/zenodo.23753349 RepositoryAll versions: 10.5281/zenodo.23697345

PDF of version 1.1.0, 2,034,112 bytes, the released file; SHA-256 85d1768202a8887885929230135192fe6eba304efc8d367f2b4149710bfa936d

Label checked 30 Oct 2026
The price of unreachable perfection: dimension laws at every regular scale

Seth Douglas and Nidhal Mghirbi · version 1.0.1 · 29 October 2026 · 24 pages

The growth of the least dimension can be prescribed at every regular scale, not only along power laws. At every asymptotically flat scale the normalized endpoints are exactly the $(Σ_{2}^{0}, Π_{2}^{0})$ pairs, and a computable target can still lie outside every game’s profile.

Verification
Each theorem block verified before release; two audits and a fix-check.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedents in form: Bartholdi and Erschler; Kassabov and Pak; Hochman and Meyerovitch; Meyerovitch; Sapir, Birget and Rips; Brady and Bridson. The base-scale classifications it restates are those of The uncomputable price of precision.

PDF The paper page doi:10.5281/zenodo.23774171 RepositoryAll versions: 10.5281/zenodo.23754198Version 1.0.1 changes packaging only

PDF of version 1.0.1, 602,266 bytes, the released file; SHA-256 049e8e32b49b36da097b04a0a6be90280124c0e0238f7ebc9d305deb348ab0e9

Label checked 30 Oct 2026
Matter that pays the price of perfection

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 29 October 2026 · 27 pages

Gapped translation-invariant lattices whose entanglement has any prescribed limit-computable logarithmic coefficient, and a referee cell that converts low energy into strategies for a nonlocal game. The game-driven material theorem is stated conditionally.

Verification
Two verifications; one audit by a reader who had seen summaries, so not fully blind; a targeted fix-check.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedents: the gap undecidability of Cubitt, Pérez-García and Wolf and the lattice constructions after it, among them Bausch, Cubitt and Watson and Purcell, Li and Cubitt; Meyerovitch and Hochman for noncomputable growth invariants; Slofstra and Vidick for precision-dependent dimension witnesses. The paper makes no claim to the first uncomputable lattice exponent.

PDF The paper page doi:10.5281/zenodo.23771444 Repository

PDF of version 1.0.0, 550,821 bytes, the released file; SHA-256 414c04944d1023d4f7c9b8dce899a3be3155823ccfd69d8966459a72737237b5

Label checked 30 Oct 2026
Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy

Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 29 October 2026 · 39 pages

A game with ten and eleven questions and at most six answers is won perfectly by an infinite-dimensional strategy and by no finite one, with dimension profile $10 \log_{2} (1 / ε) + O (\log \log (1 / ε))$ . Perfect play needs only finite energy: $275 / 18$ quanta per oscillator at best.

Verification
Eight block verifications; one blind audit; a targeted check.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: its building block is Mančinska’s flagged magic-square game, credited. Coladangelo and Stark, and Beigi, separated finite-dimensional from spatial correlations earlier for exact correlations; this game answers a possibility raised by Coladangelo and Stark. The paper makes no absolute priority claim.

PDF The paper page doi:10.5281/zenodo.23772890 RepositoryAll versions: 10.5281/zenodo.23754446

PDF of version 2.0.0, 748,029 bytes, the released file; SHA-256 b52cf777f273b7aa8f430986d3f15f2780e17830bed5f7ea32c956b2b2711c34

Label checked 30 Oct 2026
The heat price of precision

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 27 October 2026 · 21 pages

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.

Verification
Two blind audits; a targeted check.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work in quantum thermodynamics (Oppenheim and Horodecki; del Rio et al.; Devetak; others), credited. Laboratory implementations are labelled sketch and conjecture in the paper.

PDF The paper page doi:10.5281/zenodo.23701732 Repository

PDF of version 1.0.0, 571,502 bytes, the released file; SHA-256 59bb1744184e2ef5540f61e28fed0d7a1c1afda7ec5af950762f5401f39c65ca

Label checked 29 Oct 2026

Finite games and operator-algebra type

Which finite games only type III algebras can win perfectly, and what play reveals about an algebra’s type. The line, in reading order

Finite games that read the type of a von Neumann algebra

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 30 October 2026 · 36 pages

A finite two-player game with binary scoring and 243 question pairs has perfect play in type III and none when either ambient algebra is semifinite. Two further finite games characterize type III₁ through how every faithful normal state plays them, and every positive rational is forced as a modular ratio by some finite game.

Verification
Seven block verifications; one blind audit with no critical findings; a fix round and a fix-check.

Lean
None.

Peer review
Not yet.

Novelty
New, with named prior work. In the paper’s words, no earlier version of these precise finite-game statements was found in a bounded comparison with the cited literature, which is not an exhaustive priority claim. The analytic equivalence behind the hosting theorem is Haagerup and Musat’s; the gravitational corollary is the finite-game counterpart of a task-level observation by van Luijk, Stottmeister, Werner and Wilming.

PDF The paper page doi:10.5281/zenodo.23787124 RepositoryAll versions: 10.5281/zenodo.23787123

PDF of version 1.0.0, 662,711 bytes, the released file; SHA-256 0a5e8e5b1a485d496c9ce4567fc8a8c49ba6fc69dfae77d6122887d730ad104c

Label checked 29 Oct 2026
A finite game that normal hyperfinite strategies approach but never win

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 30 October 2026 · 17 pages

An explicit finite game with 0/1 scoring (561 and 157 questions, 3,696 question pairs) whose perfect correlation finite-dimensional strategies approach, while no perfect strategy has a normal marginal on an injective (hyperfinite) von Neumann algebra of either player. Perfect play lives on the Bellaterra critical-KMS factor of type III_1/2, and every perfect strategy contains an expected copy of the group algebra of the free product of three groups of order two. Under the hypotheses of Halvorson’s Proposition 2.31, quantum-field-theory strategies come arbitrarily close but never win exactly.

Verification
Two independent verifications, the second blind; a blind audit with no critical findings; a blind fix-check with no findings; an exact arithmetic certificate for every supported losing event and an independently written checker.

Lean
None.

Peer review
Not yet.

Novelty
New, with named prior work: earlier value-gap games already exclude perfect normal injective hosts; the distinction here is that the perfect correlation itself is approached. The KMS rigidity is Laca, Raeburn, Ramagge and Whittaker’s; no earlier version of the conjunction was found in a bounded search.

PDF The paper page doi:10.5281/zenodo.23816141 RepositoryAll versions: 10.5281/zenodo.23816140

PDF of version 1.0.0, 453,208 bytes, the released file; SHA-256 c754a87071ae9589b8abb88810527521962e8b40205a59dea3dad09f68ffd34c

Label checked 30 Oct 2026

The thermodynamics of acting in time

The work cost of computing on a schedule: when inputs arrive, when outputs leave for good, what must be handed back. The line, in reading order

The thermodynamics of deadlines: one law for the work cost of acting in time

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 28 October 2026 · 187 pages

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 one implementation must add to its own record, plus the energy it hands over, against the most demanding observer. The cost survives arbitrarily slow operation.

Verification
Nine whole-paper audits by a model from another developer, eight of them blind; machine checks of the numbers and finite constructions.

Lean
Complete Lean 4 proofs of the lemmas named in the paper’s Table 7, standard axioms only; the main law is proved in the text.

Experiment
Mathematics proved under the stated contract. A three-arm feedback-trap protocol is proposed; no experiment has been run.

Peer review
Not yet.

Novelty
Unifies named prior results as cases of one law: Landauer’s bound, the thermodynamic capacity of Faist, Berta and Brandão in the time-covariant case, and the real-time costs of channels and processes, credited.

PDF The paper page doi:10.5281/zenodo.23719180 RepositoryAll versions: 10.5281/zenodo.23719179Brings together the four papers below

PDF of version 1.0.0, 2,034,171 bytes, the released file; SHA-256 18d14c616523c51a7426f9ae3fe95afa6e8d5b2e6dbe0b4404f3d80e19ff7a8f

Label checked 29 Oct 2026
The price of time order

Seth Douglas and Nidhal Mghirbi · version 1.0.2 · 12 October 2026

The keystone of four companion papers: one law prices every schedule of arrivals, releases and returns.

Verification
Every result passed a blind refutation-first check; machine checks in Python; finite and algebraic cores in Lean with Mathlib.

Novelty
Extends named prior work: for a single release the price extends Cope’s entanglement cost of steering assemblages, and the price of holding carriers is Wei’s generalized Hamming weights. New, in the paper’s words: the worth of one more step of delay, the NP-hard choice of carriers to hold, and the ladder of free processes.

PDF The paper page doi:10.5281/zenodo.23323043 RepositoryAll versions: 10.5281/zenodo.23310622

PDF of version 1.0.2, 702,538 bytes, the released file; SHA-256 517f70c00bd5bda7412a181ef7716ef1d30806856d3e83417e516eb36c368cc5
Only quantum channels save work by waiting

Seth Douglas and Nidhal Mghirbi · version 1.0.2 · 12 October 2026

Releasing each output of a repeated channel before the next input arrives costs nothing extra for classical channels, and can cost extra for quantum ones.

Verification
Every result passed a blind refutation-first check; machine checks in Python; finite and algebraic cores in Lean with Mathlib.

Novelty
Extends named prior work: the batch cost is the thermodynamic capacity of Faist, Berta and Brandão, credited, and Rybár and Ziman’s repeatable channels are credited; at zero Hamiltonian the law is the authors’ earlier work. New, in the paper’s words: the finite-temperature law with work in place of purity, the converse at every fixed error, and the fixed-dimension undecidability.

PDF The paper page doi:10.5281/zenodo.23323044 RepositoryAll versions: 10.5281/zenodo.23310624

PDF of version 1.0.2, 896,722 bytes, the released file; SHA-256 6a62ca5f7c767271d05c9b3cc3344a4fae83ec50c7098ac528fda543922f8dbf
The work cost of answering in real time

Seth Douglas and Nidhal Mghirbi · version 1.0.2 · 12 October 2026

The work law for devices that must answer each request before the next arrives while remembering what came before.

Verification
Every result passed a blind refutation-first check; machine checks in Python; finite and algebraic cores in Lean with Mathlib.

Novelty
Extends named prior work: it settles, for real-time processes, the analogue of a question of Faist, Berta and Brandão, and its free-process theorem is a robust form of the criterion of Lie, Son, Boes, Ng and Wilming. For channels it reduces to Only quantum channels save work by waiting.

PDF The paper page doi:10.5281/zenodo.23323045 RepositoryAll versions: 10.5281/zenodo.23310628

PDF of version 1.0.2, 747,995 bytes, the released file; SHA-256 c42136a7a1b1e7bba464967c51967609a857411e8682ab1bbda0b8f96f3dceb1
The price of keeping time

Seth Douglas and Nidhal Mghirbi · version 1.1.1 · 30 October 2026 · 48 pages

What a clock costs a device that runs an operation which does not respect time translation.

Verification
Every result passed a blind refutation-first check; machine checks in Python; finite and algebraic cores in Lean with Mathlib.

Correction
Version 1.1.1 corrects one sentence of the batch-law appendix: version 1.1.0 said that a minimizing $Y$ need not exist, but the infimum is attained. No result depended on it. Corrections

Novelty
Extends named prior work: the capital bound uses the reuse argument of Chiribella, Yang and Renner, and Tajima, Yamaguchi, Takagi and Kuramochi, Chiribella, Yang and Renner, and Tajima, Shiraishi and Saito already bound the resource from errors alone. New, in the paper’s words: the return error enters as a separate parameter, and the bounds fall on the controller and on the heat baths.

PDF The paper page doi:10.5281/zenodo.23790746 RepositoryAll versions: 10.5281/zenodo.23310632

PDF of version 1.1.1, 831,092 bytes, the released file; SHA-256 7898df281f88b35aebde231053de8e272f096e503066acbc581716eafca6bd5e
Order, not speed: the work price of letting go

Seth Douglas and Nidhal Mghirbi · letter · version 1.1.0 · 12 October 2026

A short letter summarizing the four papers: a device that must let an output go before a later input arrives pays for every correlation its record keeps with that output, however slowly it runs.

PDF The paper page doi:10.5281/zenodo.23325747 RepositoryAll versions: 10.5281/zenodo.23323808

PDF of version 1.1.0, 407,965 bytes, the released file; SHA-256 4498219ecbb1867288adc0aa8eb19e71c4256627e6b168d9b3880f854c07fbc1

Groups and their finite shadows

Group rings, finite presentations, and how well finite permutations can imitate an infinite group. The line, in reading order

Kaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic

Nidhal Mghirbi and Seth Douglas · version 1.0.3 · 1 November 2026 · 52 pages

For every prime $p$ , one finitely presented, torsion-free, word-hyperbolic group whose group algebra over every field of characteristic $p$ is not directly finite and has a non-trivial idempotent, a zero divisor and a non-trivial unit.

Lean
The main theorem and its consequences, for every prime, formalized in Lean 4 by Nidhal Mghirbi; source and recorded axioms in the repository.

Verification
Exact certificates on more than one engine; two blind audits before version 1.0.2.

Peer review
Not yet.

Novelty
Extends OpenAI’s 2026 constructions, credited; their hyperbolicity is new. S. V. Ivanov (arXiv:2610.17662) has precedence for torsion-free zero divisors in odd characteristic, found independently by us. Bunina, Lee, Miasnikov and Muliarchyk (arXiv:2610.22536) independently gave finitely presented groups that are not directly finite in odd characteristic, without torsion-freeness or hyperbolicity.

PDF The paper page doi:10.5281/zenodo.23873781 Repository Lean sourceAll versions: 10.5281/zenodo.23374263

PDF of version 1.0.3, 773,542 bytes, the released file; SHA-256 9554a9f1b9eaae1086cd6ca75f8dabcee675759c0c6c796770b743ffbe22b547

Label checked 1 Nov 2026
Finitely presented infinite $p$ -groups for every prime

Seth Douglas and Nidhal Mghirbi · version 1.0.1 · 30 October 2026 · 38 pages

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

Verification
A blind check by a model from another developer; exact mod- $p$ checks in the repository; a line-by-line review.

Lean
None.

Peer review
Not yet.

Correction
Version 1.0.1 adds a note on Kenta Kitamura’s independent, concurrent work, which version 1.0.0 did not cite. Corrections

Novelty
Concurrent with named independent work: Kenta Kitamura proved the every-prime result independently (manuscript of 13 October 2026, Lean formalization of 16 October 2026); in the paper’s words, “the extension to every prime listed above is shared with it.” It extends OpenAI’s construction for $p = 2$ , whose conceptual credit is theirs; a lemma is credited to Zalesskii (1985).

PDF The paper page doi:10.5281/zenodo.23791059 RepositoryAll versions: 10.5281/zenodo.23401290

PDF of version 1.0.1, 715,241 bytes, the released file; SHA-256 23b1d37fb195261a3394bd8899c8e50811b690588c2f3c47cf295290414b644a

Label checked 30 Oct 2026
Non-soficity of OpenAI’s torsion-free hyperbolic group

Seth Douglas and Nidhal Mghirbi · version 1.1.0 · 14 October 2026

OpenAI’s torsion-free word-hyperbolic group, built not to be residually finite, is not sofic: its rank argument survives approximation. Version 1.1.0 adds that its group rings over division rings of the construction’s characteristic have no Linnell division ring.

Lean
The headline, that a torsion-free word-hyperbolic group is not sofic, is formalized in Lean 4 on OpenAI’s development, with standard axioms only. Sharper constants and the version 1.1.0 results are proved in writing.

Peer review
Not yet.

Novelty
The group and every input to the argument are OpenAI’s; the approximate argument, its constants and the formalization are ours. Cites Chapman and Peled’s conditional construction and the independent note of Thom based on Sandhu’s construction.

PDF The paper page doi:10.5281/zenodo.23374284 RepositoryAll versions: 10.5281/zenodo.23202474

PDF of version 1.1.0, 619,315 bytes, the released file; SHA-256 2c30dc563fb4782a14c230ee640288f15482b05940dc40f8e4a0831ea39ef8cf

Label checked 29 Oct 2026
Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory

Seth Douglas and Nidhal Mghirbi · version 1.2.0 · 2 November 2026 · 78 pages · arXiv:2610.04271

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

Verification
Five finite checks in the repository; they are not formal verification.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: Musat and Rørdam gave channels in the closure with no finite tracial factorization, and Slofstra a finitely presented group with profile at least exp(r^c) that is not known to be sofic. It extends the authors’ Houghton paper. New, in the paper’s words: the quantitative profile, with exponent 1 − o(1).

PDF The paper page doi:10.5281/zenodo.23875973 RepositoryAll versions: 10.5281/zenodo.23050302

PDF of version 1.2.0, 1,321,935 bytes, the released file; SHA-256 08f6116f2d089eced411b84932615492038e0b63760554ebd1813a94a0bbced0
Houghton’s group $H_{3}$ has superpolynomial sofic profile

Seth Douglas and Nidhal Mghirbi · version 1.2.0 · 2 November 2026 · 37 pages

Houghton’s finitely presented elementary amenable group $H_{3}$ has a finite chunk whose sofic profile is superpolynomial.

Verification
An independently checked certificate archive, transcribed to Lean and run through the Lean kernel; the checks do not prove the whole paper.

Lean
The certificate checks only.

Peer review
Not yet.

Novelty
New, with named precedent: Slofstra’s finitely presented group has a chunk with a larger exponent but is not known to be sofic; the paper knows of no earlier amenable group with superpolynomial sofic profile. It answers the last question of Cornulier’s Problem 3.18 and locates an error in his extension theorem.

PDF The paper page doi:10.5281/zenodo.23875978 RepositoryAll versions: 10.5281/zenodo.23027624

PDF of version 1.2.0, 728,797 bytes, the released file; SHA-256 4d337e8d952b63ef39038c6432a0ef08ccfb58856486a3ee8d396f94f8cea2fb

Quantum correlations

Bell inequalities, graph capacities and contextuality: which correlations finite systems can produce, certified exactly. The line, in reading order

The I₃₃₂₂ quantum value is attained spatially but not in finite dimension

Seth Douglas · version 4.1.0 · 11 September 2026 · 25 pages · arXiv:2609.05555

No finite-dimensional quantum strategy attains the quantum maximum of the I₃₃₂₂ Bell functional, mixed states and binary measurements included; a strategy on $ℓ^{2} (ℤ) \otimes ℓ^{2} (ℤ)$ does; approaching it needs local dimension $Θ (\log (1 / ε))$ .

Verification
Certified value window in exact arithmetic; a blind audit of the corrected lower bound; an end-to-end audit before version 4.1.0.

Lean
37 selected declarations, standard axioms only; the analytic proof is not formalized.

Peer review
Not yet.

Novelty
Concurrent with independent work by Pauwels (arXiv:2608.29734) and Coladangelo (arXiv:2609.06038), credited, and by V. Connor (manuscript dated 1 August 2026, cited by Krohn-Grimberghe). The paper credits Nidhal Mghirbi’s exact certification of the bounds (July 2026), which came first and is tighter than the certified window it uses.

PDF The paper page doi:10.5281/zenodo.22698978 RepositoryAll versions: 10.5281/zenodo.21782008

PDF of version 4.1.0, 468,806 bytes, the released file; SHA-256 9aeed79f970cb033cca77dc915fefa7fac29d2d85b77fa3f1898172c6fb108a6

Label checked 30 Oct 2026
The Lovász number is not alone: a continuum of Rényi points in the entanglement-assisted spectrum of graphs

Nidhal Mghirbi and Seth Douglas · version 1.2.1 · 14 October 2026 · arXiv:2610.20775

Minimizing Petz–Rényi information radii over classical–quantum channels compatible with a graph gives a continuum of points between known graph parameters; a 1440-vertex graph has $Σ (G)$ strictly below the projective rank of its complement.

Verification
Deterministic checks with pinned output, including an exact re-check of Theorem D. Version 1.1.0 corrected an overstated claim of version 1.0.0. Corrections

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: Vrana’s interpolation already gives a continuum with the same values on the pentagon in the ordinary asymptotic spectrum; the new points lie in the entanglement-assisted spectrum. The three endpoint quantities are Duan and Winter’s, and order zero is due to Schrijver (as credited by Dalai) and to Duan and Winter.

PDF The paper page doi:10.5281/zenodo.23375704 RepositoryAll versions: 10.5281/zenodo.22983498

PDF of version 1.2.1, 805,507 bytes, the released file; SHA-256 477dd4d6a4142cc130e345cd496bcfa81035e389201fdfa46f5d5fc74c6aeb82
How much dimension does an $ε$ -approximation to a non-attained quantum correlation cost? A Diophantine answer for $K_{5}$

Seth Douglas · version 1.0.2 · 24 September 2026

The dimension cost of approximating a non-attained correlation for $K_{5}$ follows a Diophantine law. Version 1.0.2 shows that an open lemma conjectured in earlier versions is false and settles the drifted case.

Verification
Three engines check the exact claims of Sections 2 to 6 and Lemmas 4.5 and 4.6.

Correction
Version 1.0.2 shows that an open lemma of earlier versions is false. Corrections

Peer review
Not yet.

Novelty
New, with named precedent: Coladangelo’s embezzlement proof of non-closure (arXiv:1904.02350) carries a one-sided dimension bound for its witness. The structure used is credited to Kruglyak, Rabanovich and Samoĭlenko and to Dykema, Paulsen and Prakash. The paper labels its priority provisional: no evidence found, not proved absent.

The paper page doi:10.5281/zenodo.22944181 RepositoryAll versions: 10.5281/zenodo.21876896

This release has no PDF: the paper is the Markdown file paper/PAPER-K5.md at the archived commit.
Exact rational certification of upper and lower quantum bounds for the I₃₃₂₂ Bell inequality

Nidhal Mghirbi · Zenodo, 11 August 2026; first version 21 July 2026, titled “Proof-carrying exact quantum bounds for the I3322 Bell inequality”

Finite rational proof objects for both sides of the I₃₃₂₂ quantum value: a Gram certificate on all words of length at most four proves $Q_{c} \leq 0.250875385012$ , and an explicit strategy of local dimension 499 with rational projectors has exact value 0.25087538451397653548…, so the two ends differ by less than $5 \times 10^{- 10}$ . The lower certificate also proves $Q_{t} > 1 / 4$ .

Verification
The author’s own checks: in the abstract’s words, “every theorem is checked using exact rational or integer arithmetic; floating-point computation is used only to discover candidates.” Reproducibility package: the proof-carrying release below.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: Pál and Vértesi’s numerical value and high-dimensional strategies, and the rational certification methods of Cafuta, Klep and Povh, of Hrga, Klep and Povh, and of Naceur and coauthors, which the paper names as antecedents.

The paper page doi:10.5281/zenodo.21891897 Proof-carrying releaseAll versions: 10.5281/zenodo.21477900

Label checked 30 Oct 2026
Critical Bellman–Jacobi dynamics for reversible Bell functionals

Nidhal Mghirbi · Zenodo, 10 September 2026; first version 12 August 2026

For a parameterized class of three-setting binary Bell functionals, a strict wall gap and an exactly certifiable endpoint barrier give finite-dimensional nonattainment, attainment by an infinite-dimensional spatial strategy, and nonclosure; for I₃₃₂₂ this proves the Pál–Vértesi nonattainment conjecture. The theory also gives the constructive bound $D (ε) = O (\log (1 / ε))$ .

Verification
The author’s own checks: exact certificates and independent algebraic verification programs. Reproducibility package doi:10.5281/zenodo.21910110; verification archive doi:10.5281/zenodo.22801636.

Lean
None.

Peer review
Not yet.

Novelty
Concurrent with named independent work: in the paper’s words, “in contemporaneous independent work, Douglas established the overlapping I3322-specific finite-nonattainment and spatial-attainment conclusions”; the parameter-general critical theory is the paper’s own. For I₃₃₂₂ alone, Pauwels and Coladangelo also proved nonattainment independently (see the I₃₃₂₂ entry above).

The paper page doi:10.5281/zenodo.22688831All versions: 10.5281/zenodo.21910040

Label checked 30 Oct 2026
Alternating Bellman–Jacobi duality for three-setting binary Bell functionals

Nidhal Mghirbi · Zenodo, 10 September 2026; first version 10 August 2026

An exact reduction, for a six-parameter class of three-setting binary Bell functionals that contains I₃₃₂₂ and the Gigena–Kaniewski family, of the operator optimization to an alternating one-dimensional Jacobi problem with a Bellman storage dual. It gives the exact rational bound $ω (I_{3322}) \leq 0.250875384626$ , a two-sided window of width about $1.12 \times 10^{- 10}$ with an independent lower certificate.

Verification
The author’s own checks: two independent exact-arithmetic implementations and an exhaustive 204,801-node audit of the storage certificate. Reproducibility package doi:10.5281/zenodo.22237093 (version 1.3.1).

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: Pál and Vértesi’s carrier, whose global exhaustiveness the paper proves for the six-parameter family, and Gigena and Kaniewski’s certificate, recast in its Proposition 48. In the paper’s words there is “genuine overlap” with Douglas’s independent I₃₃₂₂ work at the path/storage principle.

The paper page doi:10.5281/zenodo.22688911All versions: 10.5281/zenodo.21879015

Label checked 30 Oct 2026
The resource theory of complexness references in real quantum networks

Seth Douglas · version 1.0.4 · 18 October 2026

The resource theory in which the shared two-rebit reference of Weilenmann, Gisin and Sekatski is the unit. Version 1.0.4 proves that the unit copies itself exactly and corrects a false remark.

Verification
Exact checks, including the replication lemma added in version 1.0.4.

Lean
A Lean 4 and Mathlib project for the algebraic core.

Peer review
Not yet.

Novelty
New, with named precedent: the unit is the two-rebit reference of Weilenmann, Gisin and Sekatski, the skeleton of the theory transfers from multipartite entanglement theory, and the frame-code states are Wootters’ family, in Wang and Ying’s unlockable class. New, in the paper’s words: the arbitrary-code frame reading and the graded classification.

PDF The paper page doi:10.5281/zenodo.23457692 RepositoryAll versions: 10.5281/zenodo.21907062

PDF of version 1.0.4, 481,779 bytes, the released file; SHA-256 3f9404d5cdde4c7d73312514bf094fb0ee8d0560d4c570416f42560b2e09edc4
The frame bus: exact amplification and catalysis of complexness references in real quantum networks

Seth Douglas · version 1.0.0 · 13 August 2026

How real network parties without a shared convention for the complex unit can establish one by consuming pairwise reference states.

Verification
Two independent exact-computation engines.

Lean
A Lean 4 and Mathlib core.

Peer review
Not yet.

Novelty
New, with named precedent: the mechanism belongs to the known unlocking class of bound entanglement (Smolin; Augusiak and Horodecki; Dür and Cirac; Wang and Ying), and the paper places it there. New, in its words: the trigger is an auxiliary reference, the seed is returned in kind, and one seed releases all n pairs.

PDF The paper page doi:10.5281/zenodo.21911081 Repository

PDF of version 1.0.0, 377,355 bytes, the released file; SHA-256 1070b90df8413e2d37bf1b80f8dabf49157bddbf6fbb88bc04559ec68ea8e42d
A repair price for contextuality, and its identity with Min-Rep

Seth Douglas · version 1.0.0 · 10 August 2026

The least number of outcome merges that restores a global section; on binary scenarios it is an affine reparametrization of a standard quantity in hardness of approximation.

Novelty
The paper shows first that its central quantity is not new.

Verification
Two engines; two blind refutation-first review rounds before release, with the review record in the repository.

Peer review
Not yet.

The paper page doi:10.5281/zenodo.21876901 RepositoryAll versions: 10.5281/zenodo.21876900

This release has no PDF: the paper is the Markdown file paper/PAPER-PI.md at the archived commit.
Proof-Carrying Exact Bounds for the I₃₃₂₂ Bell Inequality

Nidhal Mghirbi · release 2.0.0 · Zenodo, 20 July 2026 · a software record that carries the manuscript

The first proof-carrying release: a rational certificate on the full word space of length at most four and an explicit dimension-499 strategy prove 0.25087538451 < $Q_{t} \leq Q_{c} \leq 0.2508753855$ , an exact window narrower than $10^{- 9}$ , about 265 times narrower than the preceding exact level-three enclosure. The paper above is the later, sharper account.

Verification
The author’s own checks: two separately implemented standard-library verifiers, mutation and corruption tests, continuous integration. Supplementary material doi:10.5281/zenodo.21837049.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: the lower strategy makes Pál and Vértesi’s ansatz exact.

The paper page doi:10.5281/zenodo.21462892

Label checked 30 Oct 2026

Quantum channels and memory

How much memory and purity a device needs to apply one quantum channel again and again. The line, in reading order

No algorithm can tell how much memory a repeated quantum channel needs

Seth Douglas and Nidhal Mghirbi · version 1.1.0 · 7 October 2026

For a device that applies a channel again and again with maximally mixed fresh qubits, 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.

Verification
Scripts that check the listed statements of the paper; they do not replace the proofs.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: the static equivalence is Haagerup and Musat’s, and undecidability for quantum correlation sets is Slofstra’s and his coauthors’. New, in the paper’s words: the transfer to channels and its memory consequences.

PDF The paper page doi:10.5281/zenodo.23204398 RepositoryAll versions: 10.5281/zenodo.23199340

PDF of version 1.1.0, 660,469 bytes, the released file; SHA-256 5e23e3033e847b286de9a29af542302d00610d0c33f1f60846d1980559c79777
Causal quantum-channel simulation: memory beyond entropy

Seth Douglas and Nidhal Mghirbi · version 1.1.2 · 30 September 2026

Two explicit channels with the same complete Choi spectrum and the same entropies need different amounts of memory at the same optimal exchange rate.

Verification
Scoped checks, with group-word certificates.

Lean
Scalar checks and the certificate checker.

Peer review
Not yet.

Novelty
Extends named prior work: its entropy bookkeeping extends the subadditivity argument of Rybár and Ziman, Kretschmann and Werner and Bisio, D’Ariano, Perinotti, Chiribella and Sedlák are credited for memory channels and minimal memory, and its purity theorem proves the converse of an implication in Bath dimension and initial entropy.

PDF The paper page doi:10.5281/zenodo.23070870 RepositoryAll versions: 10.5281/zenodo.23027626

PDF of version 1.1.2, 1,248,988 bytes, the released file; SHA-256 f94055e398a1507aaf7c2b98fe4c34db0394ad82cccfcb3ee8c7a9cc22113894
Factorizable rational quantum channels at an explicit distance from finite tracial baths

Nidhal Mghirbi and Seth Douglas · version 2.0.0 · 6 October 2026

Explicit rational channels that are factorizable but not limits of channels implementable with a finite maximally mixed environment, at a positive and explicit distance from that closure.

Verification
Standard-library checks of the payload, the constants and the certificates, listed in Section 8 of the paper.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: version 1.0 of this work and Haagerup and Musat’s criterion for Schur channels; the characteristic-five certificate runs Wang and Zhi’s method. New, in the paper’s words: the compiler, the leakage theorem without letter singletons, and the root certificate.

PDF The paper page doi:10.5281/zenodo.23196593 RepositoryAll versions: 10.5281/zenodo.23093838

PDF of version 2.0.0, 554,621 bytes, the released file; SHA-256 54decc274ca2b515e3c376aa42c47d68920b01b1d97ef323a189dd46ccb8b73c
The memory and purity cost of immediate delivery

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 7 October 2026

How many qubits a device must keep, exchange and be given pure to apply one channel $n$ times when each output must leave before the next input arrives.

Verification
Scripts that check the listed statements; they do not replace the proofs.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: its model extends the no-reset setting of Rybár and Ziman, and with the catalytic dilation of Lie, Son, Boes, Ng and Wilming it answers their question on repeatable unital channels; Haagerup and Musat’s results are credited where used.

PDF The paper page doi:10.5281/zenodo.23198907 Repository

PDF of version 1.0.0, 1,678,356 bytes, the released file; SHA-256 95ffecb7eeadf382ca80faba667a4ab8448f2a4aecfa73110bfb19899d7ab051
Bath dimension and initial entropy for closed repeated use of a quantum channel

Seth Douglas · version 1.1.0 · 16 September 2026 · arXiv:2609.18267

The bath dimension and initial entropy that closed, repeated use of a quantum channel requires.

Verification
Version 1.1.0 applied the repairs of three blind reviews, which found no theorem-level defect.

Lean
A closing identity and six accounting lemmas; the rate-region theorem, entropy bounds, Gram repair, concentration and compiler are not formalized.

Peer review
Not yet.

Novelty
New, with named precedent: Rybár and Ziman’s repeatable quantum memory channels are a close comparison; the known functional is the quantum privacy funnel of Datta, Hirche and Winter, and the Slepian–Wolf split is credited.

PDF The paper page doi:10.5281/zenodo.22788806 RepositoryAll versions: 10.5281/zenodo.22779672

PDF of version 1.1.0, 547,696 bytes, the released file; SHA-256 a6db5c4811f6a1b747fb40e5590d58970954d92e9b98be0f27632cc8b1c2879f
Closed mixed-apparatus compression of Schur channels: exact–approximate separation and scheduled delivery

Seth Douglas · version 1.0.0 · 16 September 2026

A Schur channel of Gram rank $k$ on dimension $d$ is implemented to error $ε$ by one fixed unitary with apparatus dimension $O (ε^{- 2} (\sqrt{k} + \ln d))$ .

Verification
Finite numerical controls and a package verifier.

Lean
Partial: a finite no-signalling theorem and six inventory lemmas. Concentration, entropy bounds, compression and the asymptotic separation are not formalized.

Peer review
Not yet.

Novelty
New, with named precedent: Terhal and coauthors gave the square-root rank scale and the qualitative obstruction for extreme channels, and Rybár and Ziman’s no-reset setting and entropy accounting are precedents. New, in the paper’s words: an approximate Schur construction approaching that scale, with explicit overheads.

PDF The paper page doi:10.5281/zenodo.22785669 Repository

PDF of version 1.0.0, 491,138 bytes, the released file; SHA-256 39f8016cb14033a7a8fd73adaec2fe4caf5bd24a11edb7d65c02b2650b89c2f5
The clean two-query threshold for unitary synthesis

Seth Douglas · version 1.0.4 · 13 September 2026

Clean two-query synthesis of every $n$ -qubit unitary needs, and can be done with, $Θ (2^{n})$ qubits and oracle input bits, up to a factor $\log (1 / ε)$ ; the garbage-model problem reduces to a clean four-insertion width bound, which remains open.

Verification
Nine finite-check scripts; an independent audit and a final adversarial audit, which found no remaining blocker within its scope.

Lean
Seven scoped declarations; the analytic proof is not formalized.

Peer review
Not yet.

Novelty
Extends named prior work: Lombardi, Ma and Wright named the quantity bounded here and proved that small capacity implies a lower bound; this paper supplies the missing premise for two-query architectures.

PDF The paper page doi:10.5281/zenodo.22740995 Repository

PDF of version 1.0.4, 510,967 bytes, the released file; SHA-256 1529c00e91c5153b8f03fbaef6283f2c149ce9294574170faa5f45bed79e508d
Quantum recovery from finitely many sharp questions

Seth Douglas · version 1.0.0 · 16 September 2026

When approximate recovery of finitely many sharp binary observables guarantees recovery of the entire quantum input, with matching fixed-gap minimax bounds.

Verification
An exact-byte manifest and a saved axiom certificate.

Lean
Partial, twelve declarations: calibrated projections implying finite-word stability. No numbered recovery theorem is fully formalized.

Peer review
Not yet.

Novelty
New, with named precedent: Renes’s complementary-observable recovery and Bény’s analysis of approximate observable error correction. New, in the paper’s words: a quantitative implication from calibrated sharp questions to full recovery.

PDF The paper page doi:10.5281/zenodo.22779671 Repository

PDF of version 1.0.0, 452,840 bytes, the released file; SHA-256 d6407e7f92c0ad315e56a9a896dfa6a7e3716b154bb3bd184cdfc7daeb48bea3
Where the quantum capacity sits in the arithmetical hierarchy

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 2 November 2026 · 15 pages

For effectively described finite-dimensional channels, the quantum capacity $Q$ is approximated uniformly from below by an increasing computable sequence of rationals, so the threshold problem for $Q$ lies in $Π_{2}^{0}$ and its gapped version is computably enumerable; consequently the conjecture of Bhattacharyya, Mehta and Zhao that the threshold problem is $Π_{3}^{0}$ -hard is false as printed under computable reductions. Positivity of $Q$ is semi-decidable, and, on exact algebraic inputs, decidable exactly when the zero-capacity channels form a computably enumerable set. For the magic-square game graph every evaluated spectral bound equals 6, and the independence number of its strong square is exactly 29.

Verification
The main result derived twice independently and every result blind-checked; a blind audit of the whole note and a fix-check; the graph certificates replayed with independent SAT solvers and checked DRAT proof traces.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: it applies standard effective finite-dimensional optimization and an effective form of Fekete’s lemma, both credited, and Oskouei and Mancini's approximability; Bhattacharyya, Mehta and Zhao printed the gapped bound. The contribution is the channel application and its consequences for the hierarchy. The one-shot zero-error statements are recoveries of prior work.

PDF The paper page doi:10.5281/zenodo.23875900 Repository

PDF of version 1.0.0, 502,443 bytes, the released file; SHA-256 dce097a16f0a247b891bb00ef47e03eacb20512555ef9cf6f499cb483eaf2d08

Label checked 1 Nov 2026

Quantum foundationssingle-author works by Nidhal Mghirbi

Where the Born rule and the guidance equation come from, and when Born-looking statistics can hide a departure from them. The line, in reading order

Galilean covariance almost fixes the quantum probability current

Nidhal Mghirbi · Zenodo, 30 October 2026

The continuity equation fixes only the divergence of the quantum probability current. Imposing Galilean boost covariance pointwise, for any one-body current, local or nonlocal, pure, mixed or many-body, 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. The same framework sorts the known alternatives, among them Nelson’s, Doebner–Goldin, chiral and Deotto–Ghirardi currents.

Verification
The author’s own checks.

Lean
None.

Peer review
Not yet.

Novelty
Extends named prior work: Holland’s results on the uniqueness of conserved currents and Deotto and Ghirardi’s alternative guidance laws, credited in the paper.

The paper page doi:10.5281/zenodo.23815695

Label checked 30 Oct 2026
Observable quantum equilibrium under controlled measurements

Nidhal Mghirbi · Zenodo, 15 September 2026

A microscopic departure from the Born distribution can be conserved while every permitted measurement becomes insensitive to it. Under a quantified mixing hypothesis, the paper bounds how well any complete experimental record, including apparatus readings, clocks and adaptive controls, can tell such an ensemble from equilibrium; a driven Bohmian oscillator and a square-well calculation with 131,072 trajectories make the bounds concrete.

Verification
The author’s own checks: reproducible sampling and numerical checks for the square-well calculation, as the abstract states.

Lean
None.

Peer review
Not yet.

Novelty
New, with named precedent: Valentini’s subquantum H-theorem and Valentini and Westman’s simulations established a relaxation mechanism; in the paper’s words, its contribution is “a quantitative connection between this observation restriction and concrete experimental resources.”

The paper page doi:10.5281/zenodo.22778741

Label checked 30 Oct 2026
The relativistic necessity of the Born rule: uniqueness from Poincaré symmetry and dynamical preservation

Nidhal Mghirbi · International Journal of Quantum Foundations 12 (2026) 1–28 · Zenodo, 9 November 2025; first version 18 September 2025

From the symmetries of the Klein–Gordon and Dirac equations and the representation theory of the Poincaré group, the paper derives their conserved currents and argues that, for spin-1/2 particles, the probability density is the unique admissible form, reducing to $| ψ |^{2}$ in the non-relativistic limit; within de Broglie–Bohm theory it proves that this distribution is preserved by the dynamics.

Verification
The author’s own checks.

Lean
None.

Peer review
Published in the International Journal of Quantum Foundations 12 (2026) 1–28: received 10 October 2025, accepted 31 October 2025, as the journal’s page and the published paper record.

Novelty
Synthesis of prior work (credited): in its own words, “a pedagogical and rigorous synthesis of prior foundational works” that “deliberately avoids claims of a first-principles derivation from nothing.”

Reach
Our most-read work on Zenodo: 1,793 unique views and 411 unique downloads across its versions, as of 30 October 2026.

The paper page doi:10.5281/zenodo.17580489All versions: 10.5281/zenodo.17153428

Label checked 30 Oct 2026