Finite games that read the type of a von Neumann algebra

Douglas, Seth; Mghirbi, Nidhal

doi:10.5281/zenodo.23787124

Results · Finite games and operator-algebra type

Finite games that read the type of a von Neumann algebra

Seth Douglas and Nidhal Mghirbi · version 1.0.0 · Zenodo, 30 October 2026 · 36 pages · doi:10.5281/zenodo.23787124 · PDF

In plain wordsA finite game that only infinite quantum systems of the kind quantum field theory uses can win every time; no finite quantum system ever can.

The resultat three depths

In one sentenceThere is a finite game, 243 pairs of questions each answered right or wrong, that two players can win every time with suitable type III algebras, such as those of quantum field theory, and never when either player’s algebra is semifinite, which includes every finite-dimensional system.

As the paper states it, in briefA 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.

For a scientist
One paragraph

The game $G_{III}$ has 45 and 46 questions, at most nine answers and binary scoring. It has a perfect normal strategy on the injective type III_1/2 factor and none whose marginal is normal on a semifinite algebra; its perfect correlation lies in the closure of finite-dimensional correlations. For factors with separable predual, hosting it is equivalent to an invariant normal state on the flow of weights (by Haagerup and Musat’s theorem), so prepared-state hosting cannot see Connes’s ratio invariant. A different quantifier can: every faithful normal state wins the flagged magic-square game exactly when $1 / 2 \in S (M)$ , a nine-question game does the same for $1 / 3$ , and the two together characterize type III₁. Every positive rational is forced as a modular ratio by some finite game. For $G_{III}$ a robust sorted spectral law holds for every normal semifinite strategy, and gives $D_{G_{III}} (ε) = \exp (Θ (ε^{- 1 / 2}))$ . As a conditional consequence, the specified semifinite perturbative gravitational models cannot attain perfect play, while suitable type III quantum-field-theory hosts can; no preparation or energy bound follows.

For an expert
The main theorems, as stated in the paper (Section 1)

Notation from the paper: $R_{λ}$ is the injective type III_$λ$ factor and $R_{\infty}$ the injective type III₁ factor, with separable predual; $v^{worst}$ is the infimum of the value over faithful normal states; $ω_{d}$ and $D_{G}$ refer to spatial POVM strategies of local dimension at most $d$ , mixed states included.

Theorem 1.1 (Finite-game hosting)

The game $G_{III}$ defined below has 45 Alice questions, 46 Bob questions, at most nine answers, and 243 uniformly sampled pairs. It has a perfect normal strategy on $R_{1 / 2}$ . No perfect strategy has a normal marginal on a semifinite ambient algebra of either party. For every factor $M$ with separable predual, the following are equivalent:

$M$ hosts $G_{III}$ ;
$M$ contains an expected copy of $R_{1 / 2}$ ;
its flow of weights has an invariant normal state;
$M$ contains an expected copy of $R_{\infty}$ .

The displayed perfect correlation belongs to $C_{q a}$ , the closure of finite-dimensional spatial correlations, but has no normal semifinite realization.

Theorem 1.2 (Two finite state-response tests)

For every factor $M$ with separable predual,

v_{G_{FMS}}^{worst} (M) = 1 \Leftrightarrow 1 / 2 \in S (M),

v_{G_{3}}^{worst} (M) = 1 \Leftrightarrow 1 / 3 \in S (M) .

Consequently every faithful normal state has value one for both games if and only if $M$ is type III₁.

Theorem 1.3 (Rational ratio forcing)

Every $r \in ℚ_{> 0}$ is forced by some finite binary-scoring game admitting a finite-dimensional perfect strategy: every perfect normal strategy has a nonzero modular $r$ -eigenoperator after support reduction. On the binary magic-square core every rational ray of $ℝ^{4}$ is finitely forceable. The distinguished parameters of rational orthogonal-pair single bridges are exactly ${2 + x^{2} + y^{2} : x, y \in ℚ}$ .

Theorem 1.4 (Robustness and dimension)

For a normal semifinite strategy of $G_{III}$ with density $h$ , let $p (t) = μ_{t} (h)$ be its decreasing singular-value function, normalized by $\int p = 1$ . Put

H (h)^{2} = \int_{0}^{\infty} {(\sqrt{p (t)} - \sqrt{2 p (2 t)})}^{2} d t .

Theorem 7.2 of the companion paper supplies a universal finite constant $C_{F}$ . There is a universal constant $C_{R} = \frac{243}{8} (C_{F}^{2} + 4552)$ , such that $H (h)^{2} \leq C_{R} ℓ_{G_{III}}$ . It holds for either normal semifinite marginal, mixed strategies and POVMs. As $ε ↓ 0$ , the minimum local dimension satisfies

D_{G_{III}} (ε) = \exp (Θ (ε^{- 1 / 2})),

1 - ω_{d} (G_{III}) = Θ ((\ln d)^{- 2}) .

The optimized type-I marginal entropy and normalized finite-factor deficit have exponent $1 / 2$ ; sorted log trace-position and log-density variances have exponent $1$ . The exponents match explicit constructions, not their leading constants.

Theorem 1.5 (Type and dimension cost)

There are finite rational games of minimum host class II with polynomial cost, exponential cost, and a quantum value gap. Every $Δ_{2}^{0}$ exponent in $[1 / 2, 1]$ occurs as an existing log-dimension exponent of a type-III-only game with quantum value one. Every positive finite existing exponent is $Δ_{2}^{0}$ . Type-III-only hosting is d.r.e.-hard; its intersection with quantum value one is $Π_{2}^{0}$ -hard as an unpromised language.

No validated numerical constant is given; the sharp constant is conjectural. The paper proves the game and the exclusion in Section 2 and the expected-subfactor and flow equivalences in Section 3; the analytic equivalence between expected embeddings and invariant flow states is Haagerup and Musat’s theorem, not a new structure theorem.

Questionson the index

Partly answered Which Schmidt spectra can be self-tested by pseudo-telepathy games? posed by others
Answered Is there a finite game that type III algebras can win perfectly and semifinite algebras cannot?
Answered Can finite games characterize type III₁?
Answered Is every positive rational forced as a modular ratio by some finite game?
Open Can any finite game force an irrational modular ratio? left open
Open Can a finite game tell type III_1/2 from type III_2^−1/2 at the worst-state level? left open
Open What is the sharp robustness constant of the type-III-only game? left open
Open How does an algebra’s type constrain the price of perfection? left open

How this was checked

Blind audits: One blind audit with no critical findings
Fix-checks: A fix round and a fix-check.
Verification: Seven block verifications
Lean: None.
Peer review: Not yet.

Each check, what it covers, and what it leaves to the written proof.

Each check, what it covers, and what it leaves to the written proof.
Check	Covers	Does not cover
Finite checks, exact	Fourteen sequential checks on one CPU. Exact rational and quadratic-field arithmetic checks the game predicates, forbidden probabilities, normalization, conference orthogonality, the rational forcing chains, every event in the six further games, nonorthogonal witnesses and graph primitivity; Boolean tables and tail arithmetic for the type/price table.	The imported classification results, universal quantifiers, or any physical state-admissibility premise. Floating-point quadrature and optimization are diagnostics, not proofs.
Independent calculations	The endpoint loss identity, counterexamples to unrestricted profile identities, the sequential-instrument estimate and numerical Airy and Gaussian constants.	The variational and arbitrary-algebra proofs, which are written.
Blind verification and audit	Seven theorem blocks verified separately; one blind audit of the paper with no critical findings; a fix round and a fix-check.	Errors every reader shares. An audit is evidence, not a guarantee.
Lean	Nothing is formalized.
Peer review	Not yet.	A preprint on Zenodo.

The finite checks and their limits are as the paper’s section “How this work was done” describes them. The blind verification and audit record is ours; every audit here is internal. How we check results.

Label checked 29 October 2026.

Novelty and credit

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.

The flagged magic-square gadget and the frustrated-overlap principle are Mančinska’s; the conference geometry is classical. Four-dimensional Kochen–Specker configurations give earlier context (Cabello). Connes’s classification, state-orbit theory and modular expectations supply the analytic tools. Exact Cuntz mechanisms and operational readings of type through embezzlement precede this work (Harris; van Luijk, Stottmeister, Werner and Wilming); the paper’s gravitational corollary is the finite-game counterpart of their task-level observation. The contribution is the particular finite-game equivalences, the rational-ray and ratio-forcing theorems, and the robust gluing law for $G_{III}$ .

Novelty, in the paper’s words: no earlier version of these precise finite-game statements was found in a bounded comparison with the cited literature; this is not an exhaustive priority claim. The base game and its robust sector estimate come from the authors’ companion paper, Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy.

Versionsone, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23787123, which always resolves to the newest.
Version	Date	DOI	Notes
1.0.0	30 Oct 2026	10.5281/zenodo.23787124	PDF hosted here.

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DOI

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github.com/Apsiape/finite-games-read-type

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python paper/anc/verify_hashes.py

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Cite

BibTeX built from the repository’s CITATION.cff.

@misc{douglas2026finitegames,
  author  = {Douglas, Seth and Mghirbi, Nidhal},
  title   = {Finite games that read the type of a von {N}eumann algebra},
  year    = {2026},
  version = {1.0.0},
  doi     = {10.5281/zenodo.23787124},
  url     = {https://doi.org/10.5281/zenodo.23787124},
  note    = {Preprint. Text: CC BY 4.0. Scripts: MIT or Apache 2.0}
}

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