{"id":"0c5963ff-3fd0-4221-b2f5-79af4aff7e0d","arxiv_id":"1908.05128","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper confirms the Medvedoff-Morrison conjecture for all k>n, for (ℓ^e, ℓ^f) with e not dividing f and k≠4, and for k=2^e≥4 with n not a power of 2, showing the shuffle group is the full alternating or symmetric group.","lead":"This paper studies what happens when a deck of cards is cut into k piles and shuffled with all possible pile permutations. It proves an open conjecture about when these shuffles can generate every possible ordering of the deck, for three large families of pile counts and pile sizes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Undocumented Magma checks are the load-bearing point: Lemma 5.2, Theorem 5.10, and the exceptional cases of Theorem 1.9 and Corollary 1.10 all delegate essential finite verifications to Magma without code or logs; the three headline families rest on those checks.","rationale":"The reader identified the same weakest assumption I would single out: the repeated, load-bearing use of undocumented Magma computations. I read the rest of the paper in good faith and found no clear mathematical inconsistency in the analytic proofs. The product, affine, primitivity, 2-transitivity, and cascading arguments are structured carefully, and the reliance on CFSG-based classifications and standard results from the literature is appropriate. However, the paper's central claims for the three families depend on finite checks that are reported without any way to audit them. Lemma 5.2 is needed to pass from a Bochert bound to a complete result for all k > n; Theorem 5.10 is needed to rule out every remaining almost simple 2-transitive candidate; and Theorem 1.9's exceptional cases are needed before Corollary 1.10 can cover k = 4 and k = 8. Supplying scripts, logs, or an independent reproduction would settle the concern. Since this is a reproducibility gap rather than a demonstrated error, and the reader's conditional verdict already encodes exactly this worry, I would not change the verdict: conditional acceptance with a request for auditability is the right call. I also noted a minor proof-label mismatch in the proof of Theorem 1.4, where part (1) is said to follow from a special case of Proposition 3.3, but that theorem is not used in the headline conjecture families and does not alter my assessment.","tokens_in":36216,"tokens_out":15493,"duration_ms":159819,"concrete_test":"Reproduce the finite checks in a fresh Magma or GAP session using only the generators in Definition 2.1. Specifically: (1) for every 2 ≤ n < k ≤ 14 compute Sh(Sym(k), n) and Sh(Alt(k), n) and verify the conclusion of Lemma 5.2; (2) for each row of Table 3, reconstruct a 2-transitive affine group P of degree k and verify that Sh(P, n) contains Alt(kn), contradicting the assumption that its socle is T; (3) compute Sh(Sym(4), 3), Sh(Sym(4), 6), and Sh(Sym(8), 3) and verify they are Sym(12), Alt(24), and Sym(24) respectively, with the parity depending on n as stated. If all computations reproduce exactly, the concern is resolved; if any case fails, the corresponding theorem needs correction or its hypothesis restricted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The analytic framework of the paper is coherent, but the proof that the three families confirm Conjecture 1.3 passes through several finite computations that are asserted rather than documented. Lemma 5.2 needs Magma for every pair 2 ≤ n < k ≤ 14 with P = Alt(k) or Sym(k); Theorem 5.10 needs Magma to eliminate each of the fifteen candidates in Table 3; and Theorem 1.9 plus Corollary 1.10 need Magma for the exceptional cases (4,3), (4,6), and (8,3). No scripts, input files, logs, or machine-readable outputs are supplied, so a reader cannot check whether the computations used the same generators from Definition 2.1, whether any cases were missed, or whether the reported inclusions and equalities are correct. These checks are not peripheral: removing them removes the exclusion of the almost simple candidates in Theorem 5.10, the k ≤ 14 case of Theorem 1.8(1), and the exceptional cascading cases in Corollary 1.10. A wrong finite computation here could leave one of the headline families unsupported. This is a reproducibility and audibility gap, not a discovered mathematical contradiction; it is the weakest point in an otherwise carefully structured argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generalised shuffle group Sh(P,n) ≤ Sym(kn) generated by the standard k-pile out-shuffle and the pile permutations from P, following Medvedoff-Morrison and Diaconis-Graham-Kantor. The main result is confirmation of Medvedoff-Morrison's Conjecture 1.3 for three doubly infinite families: (i) k>n (Theorem 1.8(1)), (ii) k=ℓ^e, n=ℓ^f with e∤f and k≠4 (Corollary 1.5), and (iii) k=2^e with n not a power of 2 (Corollary 1.10). Along the way the paper proves structure theorems for Sh(P,n) in the 'power case' (Theorem 1.4), primitivity (Theorem 1.6), and 2-transitivity results (Theorem 1.8).","tokens_in":36477,"tokens_out":11063,"duration_ms":95335,"significance":"The results are substantial, confirming a long-standing conjecture in broad families and providing a general framework for shuffle groups with arbitrary pile permutation groups. The analytic development is coherent: the power-case identification (Section 3), the primitivity proof using orbital digraphs (Theorem 1.6), and the classification analysis via Burnside and the classification of 2-transitive groups (Section 5) are all carefully argued. The paper also introduces the cascading shuffle group technique (Section 6), which is elegant. However, the completeness and verifiability of the main theorems rests on several asserted finite computations in Magma, for which no scripts or logs are supplied.","major_comments":[{"comment":"The proof of Theorem 1.8(1) for k>n is completed by Lemma 5.2, where the remaining cases 2≤n n (with the one excluded (4,2) case), (k,n) = (ℓ^e, ℓ^f) with e not dividing f and k ≠ 4, and k = 2^e with n not a power of 2. They also introduce generalised shuffle groups Sh(P,n) for arbitrary P and the cascading construction, which is a new tool and gives the cleanest route to the k = 2^e result. The paper is well organized and mostly self-contained given standard CFSG-based classifications; the product-action analysis in Section 3, the primitivity theorem, and the 2-transitive analysis all read as careful, honest mathematics. The citation pattern is appropriate: they build directly on Diaconis–Graham–Kantor and Medvedoff–Morrison and are explicit about what is new. I found no circularity or fitted parameters.\n\nThe soft spot is exactly where the reader and stress-test put it: the Magma computations. Lemma 5.2 (all pairs 2 ≤ n < k ≤ 14), Theorem 5.10 (eliminating the fifteen almost simple candidates in Table 3), and the exceptional cases (4,3), (4,6), (8,3) in Theorem 1.9/Corollary 1.10 all say \"we used Magma\" without supplying scripts, input files, or logs. These checks are not peripheral; they are load-bearing for the three headline families. If any one of those computations is wrong, a family could collapse. I want to be clear: nothing in the paper suggests the authors are hiding a flaw, and the computations look plausible—but for a paper whose main results rest on finite checks, the absence of reproducible code is a real reproducibility gap. It does not undermine the analytic framework, but it makes an independent audit impossible without redoing the work.\n\nWho is this for? Group theorists and combinatorists working on shuffle groups, permutation groups, and the Medvedoff–Morrison conjecture. A serious referee should engage with it, and an editor should send it to review rather than desk reject. But the referee report should ask for the Magma code or detailed verification data, and ideally for a second independent check of the finite cases. My own read is that the mathematics is sound and the conclusion is likely correct, but the paper as submitted should be conditional on making those computations auditable.","headline":"Solid group theory paper that confirms Medvedoff-Morrison for three infinite families and introduces generalized shuffle groups; the main weakness is the repeated reliance on undocumented Magma checks for load-bearing finite cases.","tokens_in":737,"tokens_out":1321,"would_cite":true,"duration_ms":26120,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B25","05E18"],"pacs":[],"model":"deepseek-v4-flash","headline":"Generalized perfect shuffles are proved to generate the full symmetric or alternating group in three infinite families.","keywords":["card shuffling","perfect shuffles","permutation groups","primitive groups","shuffle groups","product action","affine groups","cascading shuffle groups"],"falsifier":"Independently recompute $\\operatorname{Sh}(\\operatorname{Sym}(k), n)$ for each pair $2\\le nn$, all pairs $(k,n)=(\\ell^e,\\ell^f)$ with $e\\nmid f$ and $k\\neq 4$, and all $k=2^e\\ge 4$ with $n$ not a power of $2$. It also sets up a broader theory of shuffle groups $\\operatorname{Sh}(P,n)$ for arbitrary pile-permutation groups $P$, proving primitivity and 2-transitivity results that may apply well beyond the conjecture.","feed_headline":"Many-handed perfect shuffles generate all deck permutations","feed_subtitle":"Conjecture from 1987 confirmed for k>n, prime-power pile counts, and all 2-power pile numbers.","key_machinery":"The load-bearing construction is the shuffle $\\sigma$ itself: on the $kn-1$ nonzero card positions it acts as multiplication by $k$ modulo $kn-1$, fixing the top and bottom cards (Lemma 2.2). In the power case $k=\\ell^e$, $n=\\ell^f$, writing cards in base $\\ell$ identifies the deck with the set $[\\ell]^{e+f}$ of $(e+f)$-tuples, under which $\\sigma$ becomes the cyclic shift of coordinates by $e$ places and the pile-permutation group becomes a wreath product in product action — the group that permutes the coordinates of each tuple independently and then permutes the coordinates themselves. This yields $\\operatorname{Sh}(P,\\ell^f) = P \\wr C_{1+f/e}$ when $e\\mid f$, and the full symmetric or affine group when $e\\nmid f$ (Theorem 1.4), the affine group being the transformations $x\\mapsto xA+b$ of a vector space over the prime field. For $k>n$, the key tools are 2-transitivity of $\\operatorname{Sh}(P,n)$ for 2-transitive $P$, minimal-degree bounds (Bochert's bound and the minimal degree of a wreath product in product action), the dichotomy that a finite 2-transitive group is either affine or almost simple, and Zsigmondy's theorem on primitive prime divisors to eliminate classical candidates. For $k=2^e$, the cascade groups $G_t=\\operatorname{Sh}(V_t,2^{e-t}n)$ with $V_t$ the elementary abelian group of order $2^t$ — the translations of a $t$-dimensional vector space over the two-element field — form a nested chain; the base $G_1$ is known from earlier work, and the equalities among the $G_t$ force the overgroup $\\operatorname{Sh}(\\operatorname{Sym}(2^e), n)$ to contain the alternating group.","core_discovery":"The central discovery is that the generalized shuffle group $\\operatorname{Sh}(\\operatorname{Sym}(k), n)$ equals $\\operatorname{Alt}(kn)$ or $\\operatorname{Sym}(kn)$ in three doubly infinite families: whenever $k>n\\ge 2$ (with the single exception $(k,n)=(4,2)$, where it is $\\operatorname{AGL}(3,2)$, the affine group on a 3-dimensional vector space over the field of 2 elements); whenever $k=\\ell^e$ and $n=\\ell^f$ with $f$ not a multiple of $e$ and $k\\neq 4$; and whenever $k=2^e\\ge 4$ and $n$ is not a power of $2$. The alternatives are decided by parity: the group lies in $\\operatorname{Alt}(kn)$ exactly when $n\\equiv 0 \\pmod 4$, or $n\\equiv 2 \\pmod 4$ and $k\\equiv 0$ or $1 \\pmod 4$, or $n$ is odd and the pile group lies in $\\operatorname{Alt}(k)$; otherwise it is $\\operatorname{Sym}(kn)$. The proof passes through a structural theorem for 'power case' shuffle groups (where $k$ and $n$ are powers of the same integer), a general primitivity theorem for primitive pile groups, a full analysis of the $k>n$ 2-transitive case using the classification of finite 2-transitive groups, and a cascade argument showing that shuffle groups with elementary abelian pile groups often coincide.","pith_inferences":["The cascade mechanism suggests an analogue for odd primes: if the structure of $\\operatorname{Sh}(\\operatorname{Sym}(3), m)$ were known for all $m$, the same nesting argument could confirm the conjecture for $k=3^e$ and $n$ not a power of $3$ — the paper explicitly notes this is the missing ingredient.","The computational evidence that even the cyclic pile group $C_k$ generates $\\operatorname{Alt}(kn)$ for $k\\le 13$ and $n\\le 1000$ hints that the 'as large as possible' phenomenon may hold far beyond $\\operatorname{Sym}(k)$; if so, the full symmetric group is not needed to mix the deck completely.","The theorem that $\\operatorname{Sh}(P,n)$ is primitive for primitive non-regular $P$ gives a route to classify all pairs $(P,M)$ where $\\operatorname{Sh}(P,n)$ is trapped in a maximal subgroup $M$ — the question the paper poses as Question 1.11 — because the only maximal overgroups to worry about are the primitive ones.","If the conjecture is true for all remaining pairs, then the minimal degree of $\\operatorname{Sh}(\\operatorname{Sym}(k),n)$ should be $2n$ (from the transposition of two piles), a quantity one could test computationally for small $k,n$ to detect a counterexample before constructing the whole group."],"forward_implications":["For $k>n$, the shuffle group is the full symmetric or alternating group, so the dealer can achieve every permutation of the deck (or every even permutation).","The structure of $\\operatorname{Sh}(\\operatorname{Sym}(2^e), n)$ is now completely known for every $n$: combine Theorem 1.2 for $n=2^f$, the affine result for $k=4$ and $n=2^f$, and Corollary 1.10 otherwise.","The conjecture is confirmed for all prime-power pile counts $k=\\ell^e$ when the deck count $n$ is also a power of $\\ell$ with exponent not dividing $e$; in the prime case this yields the affine group $\\operatorname{AGL}(e+f,\\ell)$.","Any counterexample to the conjecture must have $n\\ge k$, so future work can focus on the regime where piles are at least as large as their number.","If $P$ is a 2-transitive almost simple group (a group between a nonabelian finite simple group and its automorphism group) and $k>n$, then $\\operatorname{Sh}(P,n)$ is almost simple; if $P$ is affine and $n$ is not a prime power of the same prime, $\\operatorname{Sh}(P,n)$ contains $\\operatorname{Alt}(kn)$."],"supporting_citations":[{"why":"Supplies the base case $k=2$ shuffle group and its complete structure, which the paper generalises.","marker":"[8]"},{"why":"Introduces the group $\\operatorname{Sh}(\\operatorname{Sym}(k), n)$, proves the power-of-$k$ case, and poses the original small-$k$ conjectures.","marker":"[19]"},{"why":"States Conjecture 1.3, proves the affine case $\\operatorname{Sh}(\\operatorname{Sym}(4), 2^f)$, and handles the pair $(9,3)$.","marker":"[5]"},{"why":"Classifies maximal subgroups of symmetric and alternating groups, used in Corollary 1.5 to force $\\operatorname{Sh}(\\operatorname{Sym}(k),n)$ to be maximal.","marker":"[16]"},{"why":"Classifies subgroups of prime-power index in finite simple groups, used in Lemma 5.3 to rule out almost-simple $P$ when $\\operatorname{Sh}(P,n)$ is affine.","marker":"[13]"},{"why":"Provides coprime-representation and classical-subgroup bounds, used in Lemmas 5.3 and 5.9 to eliminate classical 2-transitive groups.","marker":"[15]"},{"why":"Supplies the theorem on imprimitive groups containing certain cycles, used in Corollary 1.10 to conclude that primitivity forces the alternating group.","marker":"[24]"},{"why":"Zsigmondy's theorem on primitive prime divisors, used throughout Section 5 to constrain prime-power degrees $k=p^e$.","marker":"[25]"},{"why":"The computer algebra system used for the finite verifications in Lemmas 5.2, 5.4, 5.9, Theorem 5.10 and Theorem 1.9.","marker":"[3]"}],"fun_headline_variants":["Many-handed shuffles confirm 1987 conjecture","Shuffle groups: three families prove all permutations","Perfect shuffles with many hands: conjecture proven","Medvedoff-Morrison conjecture confirmed for three families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on finite group computations performed with the Magma computer algebra system and reported without the underlying scripts or output logs: the verification for all pairs $2\\le n