On sums of $\mathscr{P}$-free forms under mis\`ere play

Alfie Davies; Rebecca Milley; Sarah Miller

REVIEW 3 major objections 4 minor 15 references

On sums of $\mathscr{P}$-free forms under mis\`ere play

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Every P-free blocking game is invertible in misère play.

desk verdict The framework is promising, but the proof of Theorem 3.6 is invalid and the main result is not established as written. read the letter →

arxiv 2506.05257 v1 pith:XYYCNQJQ submitted 2025-06-05 math.CO

Alfie Davies , Sarah Miller , Rebecca Milley This is my paper

classification math.CO MSC 91A46

keywords misèreplayP-freegamesblockinguniverseinvertibleelementstippingpointsdead-endingoutcome-stablemonoids

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Misère play is hostile to invertibility: in the full game universe no nonzero game has an inverse, and in restricted universes an inverse can differ from the conjugate. This paper establishes a structural condition that guarantees invertibility: in any monoid of games that is outcome-stable, hereditary, integer-invertible, and has a technical 'Property X', the P-free games (those with no subposition of outcome P) are closed under addition. The blocking universe, which contains all dead-ending games and many more, satisfies these hypotheses. Consequently every P-free blocking game is invertible modulo the blocking universe, extending a characterisation previously known only for dead-ending games. This gives a route to the invertible subgroups of many misère monoids.

What carries the argument

The engine is a generalised tipping-point theory. For a game G and an integer k, the L-, N-, and R-tipping points are the smallest nonnegative integers $\ell(G)$, $n(G)$, $r(G)$ at which the outcome of $G+k$ is respectively L, N, or R. Theorem 3.6 shows that for a P-free game in an outcome-stable monoid the outcome sequence splits into three contiguous blocks—L, then N, then R—and the subsequent lemmas convert this ordering into inequalities between tipping points that decide the outcome of $G+H$. The capstone is Theorem 3.19: outcome-stable plus hereditary plus integer-invertible plus Property X implies pf(A) is a monoid. Property X handles the one case, two N-outcome summands where one is an end with $r(G) = \ell(H) = 1$, that the tipping-point inequalities do not decide on their own.

What would settle it

Find a P-free blocking game G for which $G + \overline{G}$ has outcome P; Lemma 4.10 and Theorem 4.12 would fail. A direct way to look is to compute the outcome of $G + \overline{G}$ for all P-free blocking games born by day 4 or 5; the first violation would be a counterexample, while exhaustiveness would support the theorem.

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Extended reading notes

Core claim

The central claim, Theorem 4.12, is that if G is a P-free blocking game then G is B-invertible: the P-free blocking games form a subgroup of the invertible subgroup of the blocking monoid. The proof does not proceed case-by-case on blocking games. It first proves a general theorem: whenever a hereditary monoid A is outcome-stable, integer-invertible, and satisfies Property X, the set pf(A) of strictly P-free games in A is closed under addition. The blocking universe B is shown to meet these conditions, and for every P-free G in B the symmetric sum $G + \overline{G}$ is Left B-strong, which is exactly what the comparison theorem requires for invertibility. The result is a direct generalisation of the earlier dead-ending characterisation: instead of using the special fact that the only N-outcome dead-ending end is zero, the proof isolates the structural properties that make the tipping-point argument work.

Load-bearing premise

The load-bearing premise is an external test for Left B-strong games, cited from a submitted companion paper, whose proof is not reproduced here; the proof that the blocking universe is integer-invertible depends on it.

Editorial extensions

If this is right

The set of P-free blocking games is closed under addition, so pf(B) is itself a monoid.
Every P-free blocking game is B-invertible, so pf(B) is a subgroup of the invertible subgroup of the blocking monoid.
For any conjugate-closed submonoid A of a universe U with pf(U) contained in the invertible subgroup of U, either pf(A) is empty or it is a subgroup of the invertible subgroup of A.
Rulesets that are blocking but not dead-ending, such as maze and cricket pitch, now inherit the P-free subgroup conclusion, whereas the dead-ending theorem could not reach them.
The general theorem applies to every universe between the smallest universe containing the integer 1 and the blocking universe, so the P-free subgroup is nontrivial for all of them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Property X is the least stable part of the theorem: the paper's counterexamples to dropping it also fail integer-invertibility, so a directed search for an outcome-stable, hereditary, integer-invertible monoid that violates Property X would settle whether the condition is genuinely needed.
The same machinery suggests a concrete computation: enumerate the P-free elements born by day n in the smallest universe containing the integer 1; a day-by-day enumeration would give the first explicit nontrivial example of a P-free subgroup.
If the companion equality pf(B) equals the full invertible subgroup of B is combined with the main theorem, invertibility in the blocking universe becomes exactly the absence of a P subposition; I would expect this equality, rather than the mere inclusion, to be what transfers to other universes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper generalizes Milley–Renault's tipping-point theory to a monoid-level framework and claims that P-free blocking games are invertible. The framework itself—outcome-stable, integer-invertible, Property X—is a useful way to organize the proof strategy, and the paper is clearly written and honest about its limitations. If the main theorem were true, it would partially resolve a genuine open problem and give a new class of universes beyond dead-ending where P-free games coincide with invertible games. That part is real and worth pursuing.

The problem is the technical core. The proof of Theorem 3.6, which underpins everything downstream, uses outcome-composition rules that are not part of Definition 3.4 and are false in general misère play. For k > l(G), the proof labels the summand k−l(G) as L, but positive integers are R in misère. Later it uses N+R = N, which is also not a valid composition rule—0 has outcome N and 1 has outcome R, yet 0+1 is R. The stress-test counterexample with G = -1 in the dead-ending universe is correct: the proof would give o(-1+1)=o(0)=L, but the true outcome is N. So Theorem 3.6 is not proven, and since Theorems 3.13–3.19 and the blocking application all build on it, the central claim pf_B(B) ≤ B^× is not supported by the text.

There is also an internal inconsistency in the definition of tipping points. Definition 3.1 says l(G) is the smallest non-negative integer with o(G+l(G))=L, but Theorem 3.6 uses -l(G) in the bounds, and for G=0 no non-negative integer gives outcome L. The existence proof in Theorem 3.2 appears to rely on negative integers. This suggests the definition is simply mistaken about the range, but as written it is a contradiction.

The reliance on Theorem 4.3 from a companion paper by two of the authors is a secondary concern; the internal flaw is enough to invalidate the current version. That said, the approach is likely repairable—the intended composition rules might hold in integer-invertible monoids with extra assumptions, and the blocking-universe result may still be true. But the preprint as submitted does not make the case.

Who is this for? Researchers in misère combinatorial game theory who are interested in invertibility and monoid structure. It deserves a serious referee, not a desk reject, because the question is important and the framework is valuable. The referee should focus on the tipping-point proofs before looking at the applications. I would not cite this version in my own work, but I would read a corrected version.

Referee Report

3 major / 4 minor

Summary. The paper proposes a general framework, based on Milley and Renault's tipping-point arguments, for showing that sets of P-free forms are closed under addition in restricted misère monoids. The authors define outcome-stable and integer-invertible monoids, introduce a technical 'Property X', and prove (Theorem 3.19) that under these hypotheses pf(A) is a monoid. They then apply the framework to the recently studied blocking universe B, proving that pf(B) is additively closed and that every P-free blocking game is B-invertible (Theorem 4.12). Consequences for conjugate-closed submonoids of B are also given.

Significance. If the results were correct, this would be a meaningful advance in misère combinatorial game theory: it would extend Milley and Renault's characterization of invertibility in the dead-ending universe to the strictly larger blocking universe and would give a reusable set of sufficient conditions for P-free forms to form a monoid. The paper is well organized, contains useful examples and clear open problems, and states crisp, falsifiable claims such as Theorem 4.12. However, the central tipping-point proof contains load-bearing errors, so the significance of the paper cannot be assessed from the text as it stands.

major comments (3)

[§3.1, Definition 3.1 and Theorem 3.2] The L-tipping point is not well-defined as stated. For the zero game G=0, o(0)=N and o(0+n)=R for every n≥1, so no non-negative integer l satisfies o(0+l)=L. The proof's assertion that o(G+\bar{b}(G)+1)=L by symmetry is also false for G=0, since \bar{0}+1 is the game 1, which has outcome R in misère play. The intended convention appears to be that l(G) satisfies o(G+\overline{l(G)})=L, but that is not what Definition 3.1 says. Since Theorem 3.6 and all later tipping-point arguments rely on the existence and meaning of l(G), this is a foundational issue, not a typo alone.
[§3.1, proof of Theorem 3.6] The proof uses two outcome-composition steps that are not consequences of Definition 3.4 and are false in the paper's own setting. For k>l(G), the summand k-l(G) is a positive integer, which has outcome R in misère play, yet the proof labels it L. For n(G)<k<r(G), the proof uses the rule N+R=N; this rule is not part of outcome-stability, and it fails in the dead-ending universe E, which the paper itself identifies as outcome-stable: 0 has outcome N, 1 has outcome R, but 0+1 has outcome R. Therefore the proof of Theorem 3.6 is invalid. Since Theorem 3.6 underpins Lemmas 3.7–3.11, Theorem 3.17, and Theorem 3.19, the monoid closure theorem and Theorem 4.12 are not established by the current text.
[§4, Lemma 4.5] The proof that B is integer-invertible is not self-contained. The maintenance part is deferred to induction, and the proviso is obtained from Theorem 4.3, cited as [4, Theorem 3.1] from a submitted companion paper by Davies and Milley. No proof or sketch of Theorem 4.3 is given in the manuscript. Because Theorem 4.9 and Theorem 4.12 depend on Lemma 4.5, the main application to the blocking universe is conditional on a result the reader cannot verify from the submitted material. Even if Theorem 3.6 were repaired, this dependency would need to be addressed, either by including the proof of Theorem 4.3 or by establishing integer-invertibility directly.

minor comments (4)

[§3.1, Definition 3.1] The definition of the N-tipping point contains a duplicated clause: it reads 'o(G+n(G))=N, or o(G+n(G))=N'.
[§3.1, Theorem 3.2] The line 'o(G+\tilde{b}(G)+1)=L by symmetry' appears to be a typo; as written it is false for G=0. It should presumably read something like o(G+\overline{\tilde{b}(G)+1})=L.
[Throughout §3] The sign convention for l(G) is used inconsistently. Definition 3.1 says o(G+l(G))=L, but later proofs, for instance Lemma 3.13, write H+\overline{l(H)} to obtain outcome L. The authors should clarify whether l(G) is defined by adding a positive integer or by adding its conjugate, and adjust the surrounding text and displayed equations accordingly.
[§3, Figure 1] Figure 1 is helpful, but the text should state explicitly how the tipping points l(G), r(G), and n(G) are positioned with respect to adding n versus adding -n; the figure and equations are only consistent under the reading that o(G-l(G))=L, not o(G+l(G))=L.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Section 3 tipping-point machinery is derived from stated axioms, and the main monoid theorem does not reduce to its inputs; the only flagged item is a load-bearing self-citation in the blocking-universe application.

full rationale

The paper's derivation chain is not circular in the definitional or fit-to-input sense. Section 3 builds tipping points from the explicit hypotheses of outcome-stability (Definition 3.4), integer-invertibility (Definition 3.12), heredity, and Property X; Theorem 3.6, Lemmas 3.7–3.15, Theorem 3.17, and Theorem 3.19 are proved by induction and case analysis from these definitions, not by assuming that pf(A) is additively closed. Section 4 supplies internal proofs that the blocking universe B is outcome-stable (Lemma 4.2), has Property X (Lemma 4.8), and is hereditary. The main external dependency is Theorem 4.3, cited from the same authors' submitted companion paper [4] and used in Lemma 4.4 to prove Lemma 4.5 (B integer-invertible). This is a genuine self-citation and an omitted proof in the current preprint, so it is worth flagging as a verifiability concern; however, it is not circular: Theorem 4.3 is a parameter-free characterization of Left B-strong games whose stated assumptions do not include the target result pf_B(B) ≤ B^×, and the paper does not rename or presuppose that characterization. The proof of Theorem 3.6 also contains a questionable composition assertion when it labels the positive integer k−l(G) as an L-position and effectively uses L+R=L; that is a correctness risk, not a circularity, because it does not make the conclusion equivalent to the hypotheses. Overall, no circular reduction is exhibited, so the score stays low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No numerical parameters are introduced. The new postulate is Property X, which has independent support only through its verification for B and E. The paper depends on several external results, including the unpublished companion theorem [4], and its own proof of Theorem 3.6 uses an unstated outcome-composition rule. See the red flags for locations.

assumptions (5)

standard math The comparison theorem for universes (Larsson-Nowakowski-Santos, Theorem 2.1) is valid.
Invoked in Section 2 as Theorem 2.1 to compare games modulo a universe; it underlies the maintenance and proviso conditions used throughout.
domain assumption Universes satisfy the conjugate property and contain integer inverses when they contain 1 (Larsson et al. [6]; Davies-Yadav [2]).
Used to make sense of integer-invertible monoids and to identify U^× with U^× for universes; see Section 2 and Theorem 4.11.
domain assumption Theorem 4.3 of [4], a test for Left B-strong games, is correct and applies to the augmented forms used in Lemma 4.5.
Lemma 4.5's proof of B integer-invertibility says 'the proviso is given by Theorem 4.4', which itself relies on Theorem 4.3 from the submitted companion paper [4]. Not verified in this preprint.
ad hoc to paper The L-, N-, and R-tipping points as defined in Definition 3.1 exist for all games (the paper's Theorem 3.2 is not valid as written).
Definition 3.1 says non-negative integers, but the existence proof (Theorem 3.2) adds conjugate (negative) integers; for 0 no non-negative l reaches outcome L. The theory as written is inconsistent, yet the later theorems assume these tipping points exist for all integers.
standard math Induction on formal birthday is a valid proof principle for game forms.
Used in Lemma 3.17 and in Section 4 (e.g., Lemma 4.1, Lemma 4.7) to prove results for all games by induction on game tree height.

invented entities (1)

Property X independent evidence
purpose: A technical condition on pairs of N-outcome P-free games (Definition 3.16) needed in the hypothesis of Theorem 3.19 to prove closure of pf(A) under addition; its necessity is left as Open problem 3.22.
It is verified for the blocking universe B (Theorem 4.8) and trivially satisfies for the dead-ending universe E, giving it grounding in two independent, previously studied universes.

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Cite this review

Pith. "Pith review of On sums of $\mathscr{P}$-free forms under mis\`ere play." pith.science (2026). https://pith.science/paper/XYYCNQJQ

@misc{pith2026250605257,
  author       = {Pith},
  title        = {Pith review of: On sums of $\mathscrP$-free forms under mis\`ere play},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYYCNQJQ}},
  note         = {Machine review of arXiv:2506.05257}
}

abstract

Milley and Renault proved an interesting characterisation of invertible elements in the dead-ending universe: they are the games with no subpositions of outcome $\mathscr{P}$ (the '$\mathscr{P}$-free' games). We generalise their approach to obtain a stronger result and show in particular that the set of $\mathscr{P}$-free blocking games is closed under addition, which yields that every $\mathscr{P}$-free blocking game is invertible modulo the blocking universe. This has consequences for the invertible subgroups of various other mis\`ere monoids.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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work page

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Alfie Davies and Vishal Yadav. Invertibility in the misère multiverse,

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Davies, Neil A

Alfie M. Davies, Neil A. McKay, Rebecca Milley, Richard J. Nowakowski, and Carlos P. Santos. Pocancellation theorems in Combinatorial Game Theory. Submitted,2024

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