How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)

Karianne Calta; Lauren L. Rose; Timothy E. Goldberg

REVIEW 3 major objections 6 minor 1 cited by

How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Thirteen cards in Quad-128 always contain a quad, because every quad-free set in AG(7,2) has at most 12 points.

desk verdict A genuine extension of the AG(7,2) classification with a patchable proof gap in Lemma 4.3 and an overstated completeness claim in Corollary 1.3(2). read the letter →

arxiv 2501.11173 v1 pith:RLW6Y72C submitted 2025-01-19 math.CO

Karianne Calta , Timothy E. Goldberg , Lauren L. Rose This is my paper · ORCID

classification math.CO MSC 05B2551E1051E15

keywords finitegeometryaffineAG(72)capsSidonsetsquad-freecompleteQuad-128

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

This paper settles the question of how many cards you can lay out in the game Quad-128 before being forced to find a quad: the answer is 12. It proves that in the affine geometry AG(7,2) representing the deck, every quad-free set (cap) has at most 12 points, and it classifies all caps of size 10, 11, and 12 up to affine equivalence — two types of 10-caps, one type of 11-caps, and one type of 12-caps. Because 12-point caps exist and are maximal, they are exactly the complete caps, while every 10- or 11-point cap can be extended. The result is the first complete classification of caps of size 10 or larger in this seven-dimensional geometry, and it closes the cap problem for Quad-128.

What carries the argument

The classification rests on writing a 7-dimensional cap as an 8-point affine basis B plus a dependent set D, where each dependent point is a sum of five or seven basis elements (Lemma 3.2). The 'type' records how many basis elements each dependent point involves (5 or 7), and the 'extended type' records the sizes of pairwise intersections of these support sets (2, 3, or 4). Each allowed extended type admits a canonical dependent-set template, and Lemma 2.12 shows that two caps with bases fitting the same template are affinely equivalent, so a complete list of possible extended types yields a complete list of equivalence classes. Two tools prune the list: the Affine Basis Exchange Theorem (2.15) rebases a cap to a canonical type, and inclusion-exclusion counting combined with forbidden-triple and forbidden-quadruple lemmas rules out the remaining candidate types. Lemma 4.3 — that three dependent points' supports always cover all eight basis points — is the key structural fact that makes these counts go through.

What would settle it

An exhaustive computer search over all 13-subsets of $Z_2^{7}$ for a subset with no four elements summing to zero would settle Theorem 8.2: the existence of a single 13-cap refutes the classification. Short of that, a direct check of Lemma 4.3 — enumerating all triples of 5- or 7-subsets of an 8-set that avoid quads and asking whether any has union size 5 or 6 — would test the weakest link; if such a triple exists, the lemma and its consequences in Theorems 6.1, 6.5, and 7.4 would need repair.

Watch

Extended reading notes

Core claim

The central claim is that the quad-free subsets of AG(7,2) of size at least 10 form exactly four affine equivalence classes, where two caps are equivalent when an invertible affine map of the vector space moves one onto the other. Specifically, there are two classes of 10-point caps, one class of 11-point caps, and one class of 12-point caps. There is no 13-point cap, so the maximum cap size is 12, and a cap in AG(7,2) is complete (maximal quad-free) exactly when it has 12 points. Moreover, no 10- or 11-point cap is complete in any AG(n,2), because each such cap can be enlarged by one point within its own affine span; for dimensions above 7 the same conclusion follows from counting the first quad closure. Stated game-theoretically, 12 cards can avoid a quad in Quad-128, but 13 cards guarantee one.

Load-bearing premise

The proof leans on the claim that the supports of any three dependent points in an 8-point basis together use all 8 basis points; the argument given only eliminates union size 7 and leaves union sizes 5 and 6 unaddressed.

Editorial extensions

If this is right

Any 13 cards in Quad-128 are guaranteed to contain a quad, while 12 cards can be chosen quad-free.
There are exactly two affine equivalence classes of 10-caps, one class of 11-caps, and one class of 12-caps in AG(7,2).
The maximum size of a cap in AG(7,2) is 12, and the complete caps are precisely the 12-caps.
Every cap of size 10 or 11 in any AG(n,2) is incomplete — it can be enlarged by one point — so a complete cap in dimension 7 must have size 12.
Combined with earlier classifications of caps of size up to 9, this gives a full classification of all caps in dimensions n ≤ 7.

Reading between the lines

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

The paper's remark that 7-dimensional caps are classified by extended type while higher-dimensional caps are not (Example 2.14) suggests that a classification in dimension 8 would need additional invariants beyond support-intersection sizes; testing the template method on AG(8,2) is the natural next step.
The exclusion of 13-caps rests on a lengthy inclusion-exclusion count; a machine-generated certificate or a direct SAT/backtracking search for a 13-cap would independently confirm that step, which is otherwise the least mechanically verified part of the proof.
The exact value M(7)=12 sits inside the known asymptotic bounds for M(n); the same support-type machinery could in principle attack the next dimension, where the extended-type invariant is known to be insufficient, so new combinatorial invariants would need to be introduced.
For game design, the 13-card threshold is a clean pigeonhole guarantee; the analogous threshold for larger EvenQuads decks (Z_2^n) is open, and the complete-cap results here show that maximum-size caps in dimension 7 are not the small complete caps constructed earlier.

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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

Here's the short version. The paper does something real: it classifies 10-, 11-, and 12-caps in AG(7,2), proves max size 12 and no 13-caps, using the basis-template machinery from [2]. The constructions are explicit and checkable, the equivalence arguments via Lemma 2.12 are sound, and the result does not rest on circular reasoning. I believe the classification is correct.

What I'd want fixed before signing off: Lemma 4.3's proof. It says: if three 5-supports have union of size at most 7, then the subcap has dimension at most 6 and therefore at most two dependent elements by Proposition 3.3. But Proposition 3.3 only bounds 6-dimensional caps; it says nothing directly about dimensions below 6. The missing cases are easy: if the union has size at most 6, two 5-subsets intersect in at least four points, contradicting Lemma 4.2. If the union has size 7, the three 2-element complements are pairwise disjoint, leaving one element in all three supports, and x1+x2+x3 is that basis element — a quad. So the lemma is true, but the printed proof doesn't show it. Since Lemma 4.3 is used in Theorems 6.1, 6.5, 7.4, and 8.2, this is load-bearing and needs a real proof in the revision.

Second, Corollary 1.3(2) is false as stated. "A cap in AG(7,2) is complete iff it has size 12" is contradicted by a 3-cap in a 2-flat, like {0,e1,e2}, whose first quad closure is the whole flat. The correct statement is Corollary 8.3: every complete cap of dimension 7 has size 12. The abstract and introduction should be aligned with that. Minor typos: Theorem 7.1 says "11-cap" when it means "12-cap" at the start of the proof; Theorem 7.4 has a 7-entry extended type "5-5-5-5-(2,3,3,3,3,3,2)" that should be the 6-entry version. These are easy to clean up.

Overall: the main classification is believable and useful for anyone working on caps, Sidon sets, or the Quads game. It is incremental within an established program, not a breakthrough, but it is honest progress. It deserves a serious referee and a revision, not rejection.

Referee Report

3 major / 6 minor

Summary. The paper classifies, up to affine equivalence, all caps (quad-free subsets) of size at least 10 in AG(7,2), continuing the program of [2]. The main results are: two affine equivalence classes of 10-caps, one class of 11-caps, and one class of 12-caps; maximum cap size 12; no 10- or 11-cap is complete; and hence 13 cards guarantee a quad in Quad-128. The method represents a cap as an affine basis plus a dependent set, assigns 'extended types' to bases, constructs explicit dependent-set templates, and uses a basis exchange theorem and inclusion-exclusion to rule out impossible intersection patterns. The proof relies on the published classification of lower-dimensional caps in [2].

Significance. If the classification is correct, it settles the maximum cap size in AG(7,2) and provides the first complete classification above dimension 6 in this setting, with concrete templates for each equivalence class. The explicit dependent-set templates in Tables 1-3 and the inclusion-exclusion contradiction for 13-caps are transparent and checkable; the lower-dimensional input from [2] is independent, so there is no circularity. The main weaknesses are the incomplete proof of Lemma 4.3, an overstatement in Corollary 1.3(2), and notational/typographical errors in the 12-cap arguments; these are fixable but currently block full confidence.

major comments (3)

[§4, Lemma 4.3] The proof of Lemma 4.3 only considers the case |B1∪B2∪B3|≤7 and immediately applies Proposition 3.3 to conclude |D′|≤2. Proposition 3.3, however, is stated only for 6-dimensional caps; if |B1∪B2∪B3| is 5 or 6, the subcap C′ has dimension 4 or 5 and the cited bound does not apply. Since Lemma 4.3 is used in Theorems 6.1, 6.5, 7.4, and 8.2, this is a load-bearing gap. The lemma is true and can be repaired by a short case analysis: rule out unions of size 5 and 6 using Lemma 4.2, then handle union size 7 by showing the three 5-subsets have pairwise-disjoint 2-element complements inside the 7-set. The written proof, however, is incomplete.
[Corollary 1.3(2) and §8 (Corollary 8.3)] Corollary 1.3(2) states that a cap in AG(7,2) is complete if and only if it has size 12. Under Definition 3.1, this is false: the 3-cap {0,e1,e2} in a 2-dimensional flat satisfies QC1(C)=aff(C) and is complete although it has size 3. The correct statement is Corollary 8.3(2), which restricts to complete caps of dimension 7. The abstract and introduction should be amended accordingly.
[§7, Theorems 7.1 and 7.4] The proof of the 12-cap classification is difficult to follow and contains several errors that need correction. In Theorem 7.1, 'Let C be an 11-cap' should be 'Let C be a 12-cap', and later 'the 7-dimensional 10-cap C\{x1}' should be '11-cap C\{x1}'. More substantively, in Theorem 7.4 the proof of Case 1 uses an inconsistent ordering of the dependent elements: the triple C123 = B∪{x1,x3,x2} has type (3,3,2) only when the unique 2-intersection pair is (x2,x3), but the theorem's lexicographic extended type places the 2 at (x1,x2). Consequently the text '|B123| = |B234| = 1' is wrong (the exceptional triples are {1,2,3} and {1,2,4}), and the later use of |B234|=1 is not justified by the stated type. The case can be repaired by a consistent permutation of the dependent elements, but as written the derivation of the template for x4 is not reliable.

minor comments (6)

[Definition 2.8] The definition of type writes |B1|-|B2|-...-|Bm|, but the list should be indexed by the dependent set D of size r, not by the basis B of size m; otherwise the notation is inconsistent with Example 2.9 and all later uses.
[Theorem 7.4, proof] The proof refers to 'Theorem 7.3', but the relevant statement is Proposition 7.3; there is no Theorem 7.3 in the paper.
[Theorem 7.4, proof] In the 'Other direction' paragraph, the second extended type is printed as '5-5-5-5-(2,3,3,3,3,3,2)' with an extra 3; it should have six entries: (2,3,3,3,3,2).
[Theorem 6.5, proof] The sums x1+x2+x3 are said to involve eight and six elements in the two cases; the correct counts are five and three. The argument only needs the count to differ from four, so this is a harmless miscount.
[Theorem 7.4, Case 1] The line 'x1 + x2 + x3 + x4 = a2 + a4, which involved six elements' is incorrect: the displayed sum has two elements (or one, depending on the ordering used), and 'involved' should be 'involves'.
[Theorem 6.1, proof] The final sentence says 'This shows |B12| ≠ 2', but the argument concerns |B23|; the intended conclusion is |B23|=3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification is a self-contained bootstrap from an independently published lower-dimensional classification, with direct template constructions; the flagged Lemma 4.3 gap is a proof gap, not a circular step.

full rationale

The derivation is not circular. The paper reduces 10-, 11-, and 12-caps to explicit dependent-set templates (Tables 1-3; Theorems 5.4, 6.5, 7.4) and verifies directly that no four cap elements sum to zero, so the equivalence classes are constructed rather than assumed. The only external input is the published classification of caps in dimensions at most 6 ([2], used through Propositions 3.3 and 3.5); that classification concerns strictly smaller cases, does not depend on the present 7-dimensional theorem, and is checkable, so it is genuine support even if the author lists overlap. No parameter is fitted and renamed a prediction, and no uniqueness theorem from the authors' prior work is invoked to force a choice. Two rigor issues are noted but are not circularity: Lemma 4.3's written proof cites Proposition 3.3 for dimensions 'at most 6' although the proposition is stated only for dimension 6, leaving union sizes 5-6 unaddressed; and Corollary 1.3(2) is overbroad because small caps in lower-dimensional flats can be complete. These are correctness or scope gaps, not hidden restatements of the target result.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It is a pure classification using standard affine geometry over F_2, the quad characterization from [2], and the cited lower-dimensional classification.

assumptions (3)

standard math AG(n,2) is modeled by F_2^n and affine equivalence is affine isomorphism.
Section 2; the entire translation of Quads cards to vector sums relies on this model.
standard math Four cards form a quad if and only if their vector sum is zero.
Theorem 1.1 from [2] is used throughout to define caps and quads.
domain assumption Proposition 3.3 from [2]: 6-dimensional caps have size 7 to 9 with known equivalence classes.
Used in Lemma 4.3 and Lemma 5.1; this is a cited prior classification, not re-derived here.

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Cite this review

Pith. "Pith review of How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)." pith.science (2026). https://pith.science/paper/RLW6Y72C

@misc{pith2026250111173,
  author       = {Pith},
  title        = {Pith review of: How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLW6Y72C}},
  note         = {Machine review of arXiv:2501.11173}
}

read the original abstract

We define a cap in the affine geometry AG(n,2) to be a subset in which every collection of four points is in general position. In this paper, we classify, up to affine equivalence, all caps in AG(7,2) of size k greater than or equal to 10. In particular, we show that there are two equivalence classes of 10-caps and one equivalence class of 11-caps, none of which are complete, and one equivalence class of 12-caps, which are both complete and of maximum size.

Figures

Figures reproduced from arXiv: 2501.11173 by the authors.

**Figure 1.** Quad-64 Deck be coded with vectors in Z 6 2 , and easily generalized to Z n 2 . See [2] or [9] for details. Once we do this, a quad has a nice mathematical interpretation. Theorem 1.1. [2] Four distinct cards A, B, C, D form a quad if and only if as vectors, A + B + C + D = 0. This is a parallel to SET, where cards can be coded with vectors in Z 4 3 , and generalized to Z n 3 . In this case, three distinct cards for… view at source ↗

**Figure 2.** A Quad-128 deck Since caps in Z 6 2 can be embedded as caps in Z 7 2 , the results from [2] describe all caps of size up to 9 in Z 7 2 . Our focus in this paper is on caps of size 10 or higher. Our primary result is a classification of these caps into affine equivalence classes, along with explicit algebraic and combinatorial constructions based on the affine dependence relations among cap elements. Theorem 1.2. In… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets
math.CO 2025-08 conditional novelty 6.0 of 10

Two subsets of F_2^n are affinely equivalent exactly when their even-zero-sum Venn diagrams admit a cardinality-preserving linear isomorphism.

Reference graph

Works this paper leans on

11 extracted references · 4 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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How many cards should you lay out in a game of EvenQuads: a detailed study of caps in AG( n, 2)

Julia Crager et al. “How many cards should you lay out in a game of EvenQuads: a detailed study of caps in AG( n, 2)”. In: Matematica 2.2 (2023), pp. 382–419. issn: 2730-9657. doi: 10.1007/s44007-023- 00047-0. url: https://doi.org/10.1007/s44007-023-00047-0

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Benjamin Lent Davis and Diane Maclagan. “The card game SET”. In: Math. Intelligencer 25.3 (2003), pp. 33–40. issn: 0343-6993. doi: 10.1007/BF02984846. url: https://doi.org/10.1007/BF02984846

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doi:10.4007/annals.2017.185.1.8 2017

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The joy of SET

Liz McMahon et al. The joy of SET . The many mathematical dimen- sions of a seemingly simple card game. Princeton University Press, Princeton, NJ; National Museum of Mathematics, New York, 2017, pp. xii+306. isbn: 978-0-691-16614-8

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A Small Maxi- mal Sidon Set inZn 2

Maximus Redman, Lauren Rose, and Raphael Walker. “A Small Maxi- mal Sidon Set inZn 2 ”. In: SIAM J. Discrete Math. 36.3 (2022), pp. 1861–

2022