Paper deep dive
Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist
Mohsen Aliabadi, Keith Driscoll, Elliot Krop, Petar Sirkovic, Everett Sullivan, Elahe Vedadi
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 8/3/2026, 10:32:21 AM
Summary
This paper introduces and analyzes 'Chowla sets' in finite groups and 'Chowla subspaces' in finite field extensions. A Chowla set is a subset where every element's order exceeds the subset's cardinality. The authors derive exact formulas for the maximum size of such sets (C(G)) in cyclic groups, finite abelian groups, and finite field extensions, linking these invariants to element order distributions and intermediate field degrees. The work highlights a human-AI collaborative methodology using the Co-Scientist tool.
Entities (9)
Relation Signals (6)
Chowla set → definedin → Finite Group
confidence 95% · A nonempty subset S of a finite group G is called a Chowla set if every element of S has order greater than |S|
Chowla subspace → definedin → finite field extension
confidence 95% · A nonzero K-subspace A⊆ L is called a Chowla subspace if [K(a) : K] > dim K A for every 0̸= a ∈ A
C(G) → maximizes → Chowla set
confidence 95% · C(G) = max{|S| : S ⊆ G is a Chowla set}.
C(L/K) → equals → Degree of Extension minus Max Intermediate Field Degree
confidence 93% · we prove the exact formula C(L/K) = [L : K]− d max (L/K)
Co-Scientist → usedby → Human Authors
confidence 92% · A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies.
C(Z/nZ) → equals → Euler totient function
confidence 90% · We prove that C(Z/nZ) = φ(n) under certain conditions.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.
Tags
Links
- Source: https://arxiv.org/abs/2607.24847v2
- Canonical: https://arxiv.org/abs/2607.24847v2
Trouble viewing inline? Open PDF directly →
Full Text
64,543 characters extracted from source content.
Expand or collapse full text
Extremal Chowla Sets and Their Linear Analogues: A Human–AI Mathematical Investigation Using Co-Scientist Mohsen Aliabadi 1 , Keith Driscoll 1 , Elliot Krop 1 , Petar Sirkovic 2 , Everett Sullivan 1 , and Elahe Vedadi 3 1 Clayton State University 2 Google Cloud AI Research 3 Google DeepMind Abstract We study an extremal invariant arising from Chowla-type order conditions in finite groups. A nonempty subset S of a finite group G is called a Chowla set if every element of S has order greater than |S|, and C(G) denotes the largest possible cardinality of such a set. We show that C(G) is determined by the order distribution of G. For cyclic groups, this yields an exact divisor formula and a criterion forC(Z/nZ) = φ(n). We prove lim inf n→∞ C(Z/nZ) φ(n) = 1,lim sup n→∞ C(Z/nZ) φ(n) =∞, and determine the corresponding lower and upper limits after normalization by n. For finite abelian groups, we obtain an explicit formula from the invariant-factor decomposition and a closed form for abelian p-groups. We also study a linear version for finite field extensions. A nonzero K-subspace A⊆ L is called a Chowla subspace if [K(a) : K] > dim K A for every 0 ̸= a ∈ A. This dimension-dependent condition does not, in general, require every nonzero element of A to generate L. If L/K is finite separable, however, we prove the exact extremal formula C(L/K) = [L : K]− d max (L/K), where d max (L/K) is the largest degree of a proper intermediate field. For finite fields, a normal-basis construction gives a direct proof in every degree. We further obtain explicit counting formulas in the prime- power chain case and in the first non-chain case of degree pq. The mathematical results were developed through a human–AI collaboration. A lightweight, reasoning- oriented configuration of Co-Scientist assisted with candidate reductions, examples, proof strategies, and draft arguments. The human authors selected the problem, designed the sequence of mathematical ques- tions, checked every argument, repaired gaps, supplied missing hypotheses, and wrote the final proofs. We document this expert-guided workflow as a case study in AI-assisted mathematical discovery; the AI systems were not used as formal proof verifiers. 2020 Mathematics Subject Classification. 05D05, 11B75, 11T06, 20K01, 12E20. Keywords. Chowla set, element order, Euler totient function, finite abelian group, finite field, Gaussian binomial coefficient, intermediate field. 1 Introduction The classical Cauchy–Davenport theorem asserts that if p is prime and A,B ⊆ Z/pZ are nonempty, then |A + B|≥ minp,|A| +|B|− 1, where A +B =a +b : a∈ A, b∈ B. The theorem was proved by Cauchy and later rediscovered by Davenport [5, 7]. Its prime-modulus hypothesis is essential in this form. Indeed, when the modulus is composite, the group Z/nZ contains nonzero elements of proper additive order, and these elements may create periodic obstructions to Cauchy–Davenport type growth. 1 arXiv:2607.24847v2 [math.NT] 29 Jul 2026 Chowla proved a composite-modulus analogue which removes these obstructions by imposing an order con- dition on one of the summand sets [6]. Let n≥ 2, and let A,B ⊆ Z/nZ be nonempty. Suppose that 0∈ B and that gcd(b,n) = 1 for every b∈ B\0. Equivalently, every nonzero element of B has additive order n in Z/nZ. Then Chowla’s theorem gives |A + B|≥ minn,|A| +|B|− 1. Thus Chowla’s theorem extends the Cauchy–Davenport lower bound from prime moduli to arbitrary moduli, provided the nonzero elements of one summand set generate the whole cyclic group. Subsequent work developed group-theoretic versions in which the full-order condition is replaced by weaker lower bounds on element orders. In particular, Hamidoune’s isoperimetric method studies additive expansion under hypotheses requiring the relevant nonidentity elements to have order at least the size of the set being considered [14]. The notion of a Chowla set considered in this paper is obtained by isolating this order condition from that broader additive-combinatorial setting. In all of these settings, the common role of the order hypothesis is to rule out small cyclic subgroups and the subgroup-periodic examples that obstruct Cauchy–Davenport type lower bounds. Linear analogues of classical addition theorems for products of vector subspaces were developed by Hou, Leung, and Xiang [16]. Linear Kneser and Vosper theorems were subsequently refined by Bachoc, Serra, and Zémor [4, 3]. These results provide a natural additive-combinatorial setting for replacing group-theoretic order conditions by degree conditions in field extensions. The linear invariant considered here is dimension dependent. If dim K A = m, the Chowla condition excludes only those nonzero elements whose generated subextensions have degree at most m. Thus a Chowla subspace may contain elements generating proper intermediate fields of degree larger than m. At the extremal dimension, the situation becomes more rigid, and this leads to an exact formula in terms of the largest proper intermediate field. 1.1 Main results The purpose of this paper is to isolate and study the extremal content of Chowla-type order conditions. For a finite group G, let N G (m) =|x∈ G : ord(x) > m|. We first prove C(G) = maxm∈ N : N G (m)≥ m, so both existence and enumeration of Chowla sets are determined by the order distribution of G. For cyclic groups, this becomes C(Z/nZ) = max m∈ N : X d|n d>m φ(d)≥ m . If p 1 is the least prime divisor of n, then C(Z/nZ) = φ(n) ⇐⇒ φ(n)≥ n p 1 − 1. We also establish lim inf n→∞ C(Z/nZ) φ(n) = 1,lim sup n→∞ C(Z/nZ) φ(n) =∞, and lim inf n→∞ C(Z/nZ) n = 0,lim sup n→∞ C(Z/nZ) n = 1. Moreover, every fixed inequality C(Z/nZ)≥ Mφ(n) holds on a set of positive lower asymptotic density. 2 For a finite abelian group G ∼ = C n 1 ⊕·⊕C n r , with n 1 |·| n r , Möbius inversion gives an explicit formula for the number of elements of each exact order and hence for C(G). In particular, if G is an abelian p-group of exponent p ℓ , then C(G) = ( p ℓ − p ℓ−1 , if G is cyclic, p ℓ − 1,if G is noncyclic. For a finite extension L/K, we define Chowla subspaces by the condition [K(a) : K] > dim K A for all 0̸= a∈ A. Writing d max (L/K) = max[E : K] : K ⊆ E ⊊ L, we prove the upper bound C(L/K)≤ [L : K]− d max (L/K) for every finite extension. We then prove the exact formula C(L/K) = [L : K]− d max (L/K) for every finite separable extension. The infinite-base-field case follows from a direct finite-union avoidance construction, while the finite-field case is proved by an explicit normal-basis construction. Finally, we count Chowla subspaces over finite fields. When n = p r , the intermediate fields form a chain, and the number of m-dimensional Chowla subspaces is a single Gaussian-binomial expression. When n = pq with p < q prime, two incomparable proper subfields occur; a subspace-lattice Möbius inversion gives an explicit inclusion–exclusion formula. 2 Chowla sets in finite groups Throughout this section and the next, every finite group G is assumed to satisfy |G| > 1. This convention ensures that the family of Chowla sets is nonempty and that the maxima below are well defined. We write ord(x) for the order of an element x in a finite group. The exponent of a finite group G, denoted exp(G), is the least positive integer e such that x e = 1 for every x ∈ G. Equivalently, it is the least common multiple of the orders of the elements of G. Definition 2.1. Let G be a finite group. A nonempty subset S ⊆ G is called a Chowla set if ord(x) >|S| for every x∈ S. Define C(G) = max|S| : S ⊆ G is a Chowla set. For m≥ 0, set Ω G (m) =x∈ G : ord(x) > m, N G (m) =|Ω G (m)|. The first observation reduces the invariant to the order distribution of the group. In this paper, the order distribution of a finite group means the sequence of integers a G (d) = |x ∈ G : ord(x) = d|, indexed by the positive divisors of exp(G). Proposition 2.2. For every finite group G, C(G) = maxm∈ N : N G (m)≥ m. Moreover, for each m≥ 1, the number of Chowla sets of G of cardinality m is N G (m) m . Proof. A subset S ⊆ G with |S| = m is Chowla if and only if every element of S lies in Ω G (m). Hence such a subset exists if and only if N G (m) ≥ m. The counting statement follows from the same observation, since the Chowla sets of cardinality m are precisely the m-element subsets of Ω G (m). 3 Corollary 2.3. Let G be a finite group and let e = exp(G). If G contains at least e− 1 elements of order e, then C(G) = e− 1. Proof. No element of G has order greater than e, so no Chowla set can have cardinality at least e. Hence C(G) ≤ e− 1. If G has at least e− 1 elements of order e, then any choice of e− 1 such elements is a Chowla set. Thus C(G)≥ e− 1. 2.1 Cyclic groups We now specialize to Z/nZ. The order distribution is explicit, so Proposition 2.2 becomes a divisor formula. Theorem 2.4. For every integer n≥ 2, C(Z/nZ) = max m∈ N : X d|n d>m φ(d)≥ m . In particular, φ(n)≤C(Z/nZ)≤ n− 1. Proof. In the cyclic group Z/nZ, the number of elements of order d is φ(d) for each divisor d| n. Hence N Z/nZ (m) = X d|n d>m φ(d). The formula follows from Proposition 2.2. The lower bound follows by taking all elements of order n, and the upper bound follows because no element has order greater than n. Proposition 2.5 (Threshold form). Let n≥ 2 and let m be an integer satisfying X d|n d>m φ(d)≥ m. Then there is a Chowla set of Z/nZ of cardinality m with the following form: for some divisor D | n with D > m, the set contains all elements of order strictly larger than D, exactly m− X d|n d>D φ(d) elements of order D, and no element of order smaller than D. If D is taken to be the minimum order occurring in the set, this threshold description is unique up to the choice of the required elements of order D. Consequently, a maximum Chowla set may be chosen in this threshold form. Proof. List all elements of Z/nZ having order greater than m, arranged so that their orders are nonincreasing. Since the displayed inequality holds, at least m such elements exist. Choose the first m elements in this list and let D be the order of the last chosen element. Then all elements of order larger than D have been chosen, exactly m− X d|n d>D φ(d) elements of order D have been chosen, and no element of order smaller than D has been chosen. Thus, once the minimum occurring order D is fixed, only the choice among the elements of order D remains. Since every chosen element has order greater than m, the set is Chowla. Taking m =C(Z/nZ) gives the final assertion. 4 Proposition 2.6. Let n = p α 1 1 ·p α k k , p 1 <· < p k . Then C(Z/nZ) = φ(n) if and only if φ(n)≥ n p 1 − 1. Equivalently, k Y i=1 1− 1 p i + 1 n ≥ 1 p 1 . Proof. The largest proper divisor of n is n/p 1 . Suppose first that φ(n)≥ n/p 1 − 1. If m > φ(n), then m≥ n/p 1 . Thus an element of order greater than m must have order n. There are only φ(n) such elements, so no Chowla set can have cardinality greater than φ(n). Since the φ(n) generators form a Chowla set, C(Z/nZ) = φ(n). Conversely, suppose φ(n) < n/p 1 − 1, and put m = φ(n) + 1. Then m < n/p 1 . Hence all elements of orders n and n/p 1 have order greater than m. Since there are φ(n) elements of order n and at least one element of order n/p 1 , there is a Chowla set of size m. Therefore C(Z/nZ) > φ(n). The product form follows from Euler’s formula φ(n) = n k Y i=1 1− 1 p i . Corollary 2.7. If n is a prime power, then C(Z/nZ) = φ(n). Proof. Write n = p a . Then φ(n) = n− n/p≥ n/p− 1, so Proposition 2.6 applies. Corollary 2.8. If n is odd and has exactly two distinct prime divisors, then C(Z/nZ) = φ(n). Proof. Write n = p a q b , where 3≤ p < q are primes. Then 1− 1 p 1− 1 q ≥ 1− 1 3 1− 1 5 = 8 15 > 1 3 ≥ 1 p . Thus the product criterion in Proposition 2.6 holds. Corollary 2.9. If n is even and has exactly two distinct prime divisors, then C(Z/nZ) = n 2 − 1. Proof. Write n = 2 a p b , where p is an odd prime and a,b ≥ 1. Since the largest proper divisor of n is n/2, no Chowla set can have size at least n/2. Hence C(Z/nZ)≤ n 2 − 1. All elements of orders n and n/2 have order greater than n/2− 1. Therefore it is enough to show that φ(n) + φ(n/2)≥ n 2 − 1. If a = 1, then φ(n) + φ(n/2) = 2p b−1 (p− 1)≥ p b − 1 = n 2 − 1. If a≥ 2, then φ(n) + φ(n/2) = 3· 2 a−2 p b−1 (p− 1)≥ 2 a−1 p b − 1 = n 2 − 1, because 3(p− 1) ≥ 2p for every odd prime p. Thus one can choose n/2− 1 elements among those of orders n and n/2, and they form a Chowla set. 5 Remark 2.10. Proposition 2.6 gives many further explicit families. In particular, the equality C(Z/nZ) = φ(n) holds whenever k Y i=1 1− 1 p i ≥ 1 p 1 , because this condition implies the product criterion k Y i=1 1− 1 p i + 1 n ≥ 1 p 1 . Thus, for any fixed least prime divisor p 1 , one obtains an explicit finite-prime-product test. Numerical thresholds obtained by taking consecutive primes beginning with p 1 are computational consequences of the exact criterion rather than structurally distinguished constants. For a set B ⊆ N, write d (B) = lim inf x→∞ 1 x |n≤ x : n∈B| for its lower asymptotic density. Theorem 2.11. For every real number M > 0, the set B M =n≥ 2 :C(Z/nZ)≥ Mφ(n) has positive lower asymptotic density. In particular, it is infinite. Proof. Let M > 0. Choose an integer k ≥ 1 such that H k := k X i=1 1 i ≥ M. Put B = lcm(1, 2,...,k) 2 . Choose a multiple N 0 of B such that M φ(N 0 ) N 0 + 1 N 0 < 1 k .(1) This is possible because multiplying B by sufficiently many new primes makes φ(N 0 )/N 0 arbitrarily small and also makes N 0 arbitrarily large. We claim that every multiple n of N 0 belongs to B M . Since B | n, each integer 1≤ i≤ k divides n, and for every prime p| i the exponent of p in n is strictly larger than its exponent in i. Therefore φ n i = φ(n) i (1≤ i≤ k).(2) Moreover, since n is a multiple of N 0 , we have φ(n)/n≤ φ(N 0 )/N 0 and 1/n≤ 1/N 0 . Hence M φ(n) n + 1 n < 1 k . Equivalently, Mφ(n) + 1 < n k . Let r =⌈Mφ(n)⌉. 6 Then r < n/k. For each 1 ≤ i ≤ k, all elements of order n/i in Z/nZ have order greater than r. By (2), the number of elements whose orders are among n, n 2 ,..., n k is k X i=1 φ n i = φ(n)H k ≥ Mφ(n). This number is an integer, so it is at least r. Thus there are at least r elements of Z/nZ of order greater than r. By Proposition 2.2, C(Z/nZ)≥ r ≥ Mφ(n). Thus every multiple of N 0 belongs to B M . The multiples of N 0 have natural density 1/N 0 , so B M has positive lower asymptotic density. Corollary 2.12. The limit lim n→∞ C(Z/nZ) φ(n) does not exist. Moreover, lim inf n→∞ C(Z/nZ) φ(n) = 1,lim sup n→∞ C(Z/nZ) φ(n) =∞. Proof. Along prime powers n = p a , Corollary 2.7 gives C(Z/nZ) φ(n) = 1. Since C(Z/nZ) ≥ φ(n) for every n ≥ 2, the liminf is exactly 1. On the other hand, Theorem 2.11 shows that for every M > 0 there are infinitely many n such that C(Z/nZ) φ(n) ≥ M. Therefore the limsup is infinite. Proposition 2.13. We have lim inf n→∞ C(Z/nZ) n = 0,lim sup n→∞ C(Z/nZ) n = 1. In particular, the limit lim n→∞ C(Z/nZ)/n does not exist. Proof. The upper bound C(Z/nZ)≤ n− 1 holds for every n≥ 2. If n = p is prime, then every nonzero element of Z/pZ has order p, and hence C(Z/pZ) = p− 1. Therefore C(Z/pZ) p = 1− 1 p −→ 1 along the primes, proving that the limsup is 1. It remains to prove that the liminf is 0. Let n x = Y p≤x p be the primorial up to x. We show that C(Z/n x Z)/n x → 0. Fix ε > 0. If d| n x and d > εn x , then d = n x /e for some divisor e| n x with e < 1/ε. Hence X d|n x d>εn x φ(d)≤ X e|n x e<1/ε φ(n x /e). 7 There are only finitely many possible integers e < 1/ε. For each fixed such e, and for all sufficiently large x, φ(n x /e) n x ≤ eφ(n x /e) n x = φ(n x /e) n x /e = φ(n x ) n x Y p|e 1− 1 p −1 . The last expression tends to 0, since φ(n x ) n x = Y p≤x 1− 1 p −→ 0, by the divergence of the reciprocal prime series; see, for example, [15, Theorem 19]. Therefore 1 n x X d|n x d>εn x φ(d)−→ 0. For all sufficiently large x, this sum is less than εn x . By Theorem 2.4, no Chowla set of Z/n x Z can have cardinality at least εn x . ThusC(Z/n x Z)/n x < ε for all sufficiently large x. Since ε > 0 was arbitrary, the liminf is 0. We next record a qualitative density consequence for the equality C(Z/nZ) = φ(n). The proof is elementary and uses only a finite-prime restriction together with Markov’s inequality. Proposition 2.14. The set A =n≥ 2 :C(Z/nZ) = φ(n) has positive lower asymptotic density. Proof. It is enough to produce a positive-density subfamily of A. We consider odd multiples of 3. By Proposi- tion 2.6, an odd integer n divisible by 3 belongs to A provided Y q|n q>3 1− 1 q + 3 2n ≥ 1 2 . Thus it suffices to find a positive-density set of odd multiples of 3 for which the product over prime divisors q > 3 is larger than 1/2. Choose Y > 3 so large that X q>Y 1 q log q q− 1 < log 2 2 . Let R Y be the set of integers n such that 3 | n, 2 ∤ n, and no prime q with 3 < q ≤ Y divides n. This set has positive natural density, namely 1 6 Y 3<q≤Y 1− 1 q . For n∈R Y , set f Y (n) = X q|n q>Y log q q− 1 . We claim that the limsup of the mean value of f Y over R Y is at most X q>Y 1 q log q q− 1 . Indeed, after imposing the finitely many congruence restrictions defining R Y , the divisibility condition q | n for a prime q > Y has relative density 1/q. Truncating the sum to Y < q ≤ Z, averaging overR Y , and then letting Z →∞ gives the claim, since the displayed prime series converges. 8 Therefore the limsup mean of f Y over R Y is less than log 2/2. By Markov’s inequality, a subset of R Y of positive lower density satisfies f Y (n) < log 2. For such n, the primes 3 < q ≤ Y do not divide n, and hence X q|n q>3 log q q− 1 = f Y (n) < log 2. Equivalently, Y q|n q>3 1− 1 q > 1 2 . For these integers n, Proposition 2.6 givesC(Z/nZ) = φ(n). HenceA has positive lower asymptotic density. Remark 2.15. The proof of Proposition 2.14 avoids the full Erdős–Wintner theorem. Nevertheless, that theorem gives a useful conceptual explanation for the density questions below. Recall that an arithmetic function f is additive if f (ab) = f (a) + f (b) whenever gcd(a,b) = 1. After fixing the least prime divisor p, the condition in Proposition 2.6 is asymptotically governed by the additive arithmetic function X q|n q>p log q q− 1 . The corresponding prime series is convergent, so the Erdős–Wintner theorem implies that this function has a limiting distribution; see [9, 19]. This suggests that the equality condition C(Z/nZ) = φ(n) should have a natural density. Question 2.16. For M > 0, determine or estimate the lower asymptotic density of n≥ 2 :C(Z/nZ)≥ Mφ(n). More generally, obtain effective bounds for C(Z/nZ)/φ(n) in terms of the small prime divisors of n and their exponents. We use the term natural density for a limit of the form lim x→∞ 1 x |n≤ x : n∈S|, when this limit exists. Question 2.17. Does the natural density δ = lim x→∞ 1 x |n≤ x :C(Z/nZ) = φ(n)| exist? If it exists, determine or estimate δ. Equivalently, determine the density of integers n for which φ(n) n + 1 n ≥ 1 p(n) , where p(n) is the least prime divisor of n. 9 3 Finite abelian groups We now pass from cyclic groups to finite abelian groups. We write C m for a cyclic group of order m. We use the standard classification of finite abelian groups, according to which every finite abelian group is isomorphic to G ∼ = C n 1 ⊕·⊕ C n r , n 1 | n 2 |·| n r , for suitable positive integers n i ; see, for example, [8, Sec. 5.2]. The only additional ingredient is the standard count of elements whose order divides a given integer. Lemma 3.1. Let G ∼ = C n 1 ⊕·⊕ C n r , n 1 | n 2 |·| n r . For d≥ 1, set T G (d) =|x∈ G : ord(x)| d|. Then T G (d) = r Y i=1 gcd(d,n i ). Proof. Write x = (x 1 ,...,x r ) with x i ∈ C n i . The condition ord(x) | d is equivalent to x d i = 1 for all i. In a cyclic group of order n i , the equation y d = 1 has exactly gcd(d,n i ) solutions. Multiplying over the direct factors gives the formula. Theorem 3.2. Let G ∼ = C n 1 ⊕·⊕ C n r , n 1 | n 2 |·| n r , and let e = n r = exp(G). For each divisor d| e, let a G (d) =|x∈ G : ord(x) = d|. Then a G (d) = X c|d μ d c r Y i=1 gcd(c,n i ). Consequently, C(G) = max m∈ N : X d|e d>m a G (d)≥ m . Proof. For every d| e, T G (d) = X c|d a G (c). Here μ denotes the classical number-theoretic Möbius function. Möbius inversion on the divisor lattice, together with Lemma 3.1, gives the displayed formula for a G (d); see, for example, [18, Ch. 3]. Since all element orders in G divide e, we have N G (m) = X d|e d>m a G (d). The result follows from Proposition 2.2. Theorem 3.3. Let G be a finite abelian p-group of exponent p ℓ . Then C(G) = ( φ(p ℓ ) = p ℓ − p ℓ−1 , if G is cyclic, p ℓ − 1,if G is noncyclic. 10 Proof. If G is cyclic, this is Corollary 2.7. Assume now that G is noncyclic. Since exp(G) = p ℓ , no Chowla set can have cardinality at least p ℓ . Hence C(G)≤ p ℓ − 1. It remains to prove the reverse inequality. By the classification of finite abelian p-groups, G has a direct factor isomorphic to C p ℓ . Since G is not cyclic, it has another nontrivial cyclic direct factor, say C p k with k ≥ 1. The latter contains a subgroup isomorphic to C p . Therefore G contains a subgroup H ∼ = C p ℓ ⊕ C p . Every element of H whose first coordinate has order p ℓ has order p ℓ . There are pφ(p ℓ ) = p(p ℓ − p ℓ−1 ) = p ℓ (p− 1) such elements. Since p ℓ (p− 1) ≥ p ℓ − 1, the group G contains at least p ℓ − 1 elements of order p ℓ . Choosing any p ℓ − 1 of them gives a Chowla set. Thus C(G)≥ p ℓ − 1, and equality follows. Corollary 3.4. For every prime p and every integer r ≥ 1, C((C p ) r ) = p− 1. Proof. This is the case ℓ = 1 of Theorem 3.3. 4 Chowla subspaces in finite extensions We now pass from element orders to degrees of generated subextensions. The resulting condition depends on the dimension of the subspace and therefore retains more information than the requirement that all nonzero elements generate the entire extension. Definition 4.1. Let K ⊆ L be a finite extension with [L : K] = n > 1. A nonzero K-subspace A⊆ L is called a Chowla subspace if [K(a) : K] > dim K A for every 0̸= a∈ A. Define C(L/K) = maxdim K A : A⊆ L is a Chowla K-subspace. Finally, set d max (L/K) = max[E : K] : K ⊆ E ⊊ L. Proposition 4.2 (Degree-threshold characterization). Let A ⊆ L be a nonzero K-subspace and put m = dim K A. Then A is a Chowla subspace if and only if A∩ E =0 for every intermediate field E satisfying [E : K]≤ m. Proof. Suppose first that A is Chowla. If 0 ̸= a ∈ A∩ E, then [K(a) : K] ≤ [E : K] ≤ m, contradicting the defining inequality. Conversely, if A is not Chowla, then there is a nonzero a ∈ A with [K(a) : K] ≤ m. The intermediate field E = K(a) then satisfies 0̸= a∈ A∩ E, so the intersection condition fails. Remark 4.3. The Chowla condition does not generally require every nonzero element to generate L. For example, suppose L/K is finite separable and has a proper intermediate field E of degree r > 1. Choose a ∈ E with E = K(a). Then the one-dimensional space Ka is Chowla because [K(a) : K] = r > 1, although K(a) = E ̸= L. 11 Theorem 4.4. Let K ⊆ L be a finite extension of degree n > 1. Then C(L/K)≤ n− d max (L/K). Proof. Let A ⊆ L be a Chowla subspace and put m = dim K A. Choose a proper intermediate field E with [E : K] = d max (L/K), and abbreviate this degree by d. Since [L : E]≥ 2, we have d≤ n/2, and hence n−d≥ d. If m > n− d, then dim K (A∩ E)≥ m + d− n > 0. Choose 0̸= a∈ A∩ E. Then [K(a) : K]≤ d≤ n− d < m, contradicting the Chowla condition. Therefore m≤ n− d. The lower bound requires constructing a subspace that avoids all relevant intermediate fields. Over an infinite base field, this follows from two standard finite-dimensional facts. Lemma 4.5. If K ⊆ L is finite and separable, then there are only finitely many intermediate fields between K and L. Proof. Let M be a normal closure of L/K. Then M/K is finite Galois. By the finite Galois correspondence [8, Sec. 14.2], the map E 7−→ Gal(M/E) embeds the set of intermediate fields K ⊆ E ⊆ L into the finite set of subgroups of Gal(M/K). Lemma 4.6. Let K be an infinite field and let V be a finite-dimensional K-vector space. Then V is not a finite union of proper K-subspaces. Proof. We argue by induction on dim K V . The one-dimensional case is immediate. Suppose dim K V ≥ 2 and V = U 1 ∪·∪U t , with each U i proper. Choose a hyperplane H different from every U i that is itself a hyperplane; this is possible because K is infinite. Then every H ∩ U i is a proper subspace of H, while H = t [ i=1 (H ∩ U i ), contradicting the induction hypothesis. Theorem 4.7 (Exact formula over infinite fields). Let K ⊆ L be a finite separable extension of degree n > 1, and assume that K is infinite. Then C(L/K) = n− d max (L/K). Proof. The upper bound is Theorem 4.4. Put d = d max (L/K), m = n− d. Let E 1 ,...,E t be the maximal proper intermediate fields. By Lemma 4.5, there are finitely many, and every proper intermediate field is contained in one of them. We construct subspaces A 0 ⊂ A 1 ⊂·⊂ A m with dim K A j = j and A j ∩ E i = 0 for every i. Begin with A 0 = 0. Suppose A j has been constructed for some j < m. For each i, dim K (E i + A j )≤ d + j ≤ d + m− 1 = n− 1, so E i + A j is proper. Lemma 4.6 provides v ∈ L\ t [ i=1 (E i + A j ). 12 Set A j+1 = A j ⊕ Kv. If a + cv ∈ A j+1 ∩ E i , where a∈ A j , then c̸= 0 would imply v ∈ E i + A j . Hence c = 0, and then a = 0. Thus A j+1 ∩ E i =0 for every i. Let A = A m . Every nonzero a∈ A lies in no proper intermediate field, so K(a) = L. Therefore [K(a) : K] = n > m = dim K A, and A is Chowla. Hence C(L/K)≥ m, completing the proof. Remark 4.8. Let d = d max (L/K). Since d≤ n/2, the extremal dimension n−d is at least d. Consequently, any Chowla subspace of dimension n− d automatically satisfies K(a) = L for every 0̸= a∈ A: a proper generated subextension would have degree at most d≤ n− d. This is a consequence of extremality and is not part of the definition of a Chowla subspace. Corollary 4.9. If K ⊆ L is a finite separable extension of prime degree n and K is infinite, then C(L/K) = n− 1. Proof. There is no proper intermediate field strictly larger than K, so d max (L/K) = 1. Apply Theorem 4.7. 5 Finite fields Throughout this section, q and Q denote prime powers. We regard F q n as an n-dimensional vector space over F q . The intermediate fields of F q n /F q are precisely the fields F q e with e| n, and F q a ∩ F q b = F q gcd(a,b) . We also use the normal basis theorem; see [17, Ch. 2]. 5.1 The exact extremal formula The upper bound from Theorem 4.4 is sharp over every finite field. The proof below is explicit and works for arbitrary extension degree. Theorem 5.1 (Finite-field formula). Let n > 1, let p be the least prime divisor of n, and let q be a prime power. Then C(F q n /F q ) = n− n p . Proof. The largest proper divisor of n is d = n/p, so the largest proper intermediate field has degree d. Theo- rem 4.4 gives C(F q n /F q )≤ n− d. Choose a normal element α∈ F q n . Then α,α q ,...,α q n−1 is an F q -basis of F q n . Write α i = α q i , with indices taken modulo n, and set A = span F q α 0 ,...,α n−d−1 . Thus dim F q A = n− d. Suppose that 0 ̸= x ∈ A belongs to a proper intermediate field F q e , where e | n and e < n. Since d is the largest proper divisor of n, we have e≤ d. Write x = n−1 X i=0 c i α i , c i ∈ F q . By the definition of A, c n−d = c n−d+1 =· = c n−1 = 0. 13 On the other hand, x∈ F q e implies x q e = x. Since α q e i = α i+e , uniqueness of normal-basis coordinates gives c i = c i−e (i mod n). Hence the coordinate sequence is periodic with period e. The displayed block contains at least e consecutive zero coordinates, one from every residue class modulo e. Periodicity therefore forces all c i to be zero, contradicting x̸= 0. Thus A meets every proper intermediate field trivially. Every nonzero x∈ A therefore satisfies F q (x) = F q n , and [F q (x) : F q ] = n > n− d = dim F q A. So A is Chowla and C(F q n /F q )≥ n− d. Corollary 5.2 (Exact formula for finite separable extensions). Let K ⊆ L be a finite separable extension of degree n > 1. Then C(L/K) = n− d max (L/K). Proof. If K is infinite, this is Theorem 4.7. If K is finite, then K = F q and L = F q n for some prime power q, and the result is Theorem 5.1. 5.2 Counting in the chain case We write a b q for the Gaussian binomial coefficient, the number of b-dimensional subspaces of an a-dimensional vector space over F q . It is understood to be zero when b < 0 or b > a. Lemma 5.3. Let V be an n-dimensional vector space over F q , and let W ⊆ V be a fixed d-dimensional subspace. Then the number of m-dimensional subspaces A⊆ V satisfying A∩ W =0 is q dm n− d m q . Proof. Let π : V → V/W be the quotient map. If A∩ W = 0, then π| A is injective, and hence π(A) is an m-dimensional subspace of V/W. Conversely, fix an m-dimensional subspace B ⊆ V/W. The subspaces A ⊆ V satisfying A∩ W = 0 and π(A) = B are precisely the graphs of F q -linear maps B → W, after identifying π −1 (B) with B⊕ W. Since dim F q B = m anddim F q W = d, there are q dm such linear maps. There are n− d m q choices for B ⊆ V/W. Multiplying these two numbers gives the formula. Proposition 5.4. Let 1 ≤ m < n. Suppose that among the divisors d | n satisfying d ≤ m, there is a unique maximal element d 0 under divisibility. Then the number of m-dimensional Chowla F q -subspaces of F q n /F q is q d 0 m n− d 0 m q . 14 Proof. An m-dimensional subspace A ⊆ F q n is Chowla if and only if it contains no nonzero element lying in a proper subfield F q d with d | n and d ≤ m. Since d 0 is the unique maximal such divisor, all these forbidden subfields are contained in F q d 0 . Therefore A is Chowla if and only if A∩ F q d 0 =0. The field F q d 0 has dimension d 0 over F q , so the result follows from Lemma 5.3. Theorem 5.5. Let n = p r , where p is prime and r ≥ 1. Let 1≤ m < n, and choose j such that p j ≤ m < p j+1 ,0≤ j ≤ r− 1. Then the number of m-dimensional Chowla F q -subspaces of F q n /F q is q p j m p r − p j m q . Proof. The divisors of p r not exceeding m are precisely 1,p,...,p j , and they form a chain under divisibility. Hence the unique maximal forbidden subfield is F q p j . Applying Proposition 5.4 with d 0 = p j gives q p j m p r − p j m q . Corollary 5.6. If n = p r , where p is prime, then C(F q n /F q ) = n− n p . Proof. This is the special case n = p r of Theorem 5.1. 5.3 The first non-chain case Let n = pq, where p < q are primes. The proper intermediate fields F Q p and F Q q are incomparable, so a single forbidden-subspace count no longer suffices. Corollary 5.7. Let p < q be primes and let Q be a prime power. Then C(F Q pq /F Q ) = pq− q = q(p− 1). Proof. The least prime divisor of pq is p. Apply Theorem 5.1. Theorem 5.8. Let p < q be primes, let Q be a prime power, and let L = F Q pq . For q ≤ m < pq, the number of m-dimensional Chowla F Q -subspaces of L is q X a=0 p X b=0 (−1) a+b Q ( a 2 ) + ( b 2 ) N (0) a,b pq− a− b m− a− b Q + N (1) a,b pq− a− b + 1 m− a− b + 1 Q ! , where N (1) a,b = q− 1 a− 1 Q p− 1 b− 1 Q and N (0) a,b = q a Q p b Q − q− 1 a− 1 Q p− 1 b− 1 Q . As usual, Gaussian binomial coefficients with negative lower index are interpreted as zero. 15 Proof. Put K = F Q , E = F Q q , F = F Q p . The proper intermediate fields of L/K are exactly K, E, F. Since q ≤ m < pq, an m-dimensional subspace A⊆ L is Chowla if and only if A∩ E = A∩ F =0. Indeed, K ⊆ E∩ F, so avoiding E and F also avoids K. Here μ E and μ F denote the Möbius functions of the subspace lattices of E and F, respectively. For a subspace A⊆ L, the indicator of the condition A∩ E =0 is X U⊆A∩E μ E (0,U ), and similarly the indicator of A∩ F =0 is X W⊆A∩F μ F (0,W ). Therefore the desired number is X U⊆E X W⊆F μ E (0,U )μ F (0,W )#A≤ L : dim K A = m, U + W ⊆ A. Let dim K U = a,dim K W = b,dim K (U ∩ W ) = ε. Then dim K (U + W ) = a + b− ε. Thus the number of m-dimensional subspaces A⊆ L containing U + W is pq− a− b + ε m− a− b + ε Q . Also, the Möbius functions of finite subspace lattices satisfy μ E (0,U ) = (−1) a Q ( a 2 ) , μ F (0,W ) = (−1) b Q ( b 2 ) . It remains to count the pairs (U,W ) according to whether ε = 0 or ε = 1. Since E∩ F = K and K is one-dimensional over itself, the intersection U ∩ W is either zero or the common line K. Hence ε = 1 occurs exactly when both U and W contain K. Therefore the number of pairs (U,W ) with dim K U = a,dim K W = b,dim K (U ∩ W ) = 1 is N (1) a,b = q− 1 a− 1 Q p− 1 b− 1 Q . The total number of pairs (U,W ) with dim K U = a and dim K W = b is q a Q p b Q . 16 Hence the number of such pairs with U ∩ W =0 is N (0) a,b = q a Q p b Q − q− 1 a− 1 Q p− 1 b− 1 Q . Substituting the two cases ε = 0 and ε = 1 into the double Möbius-inversion sum gives q X a=0 p X b=0 (−1) a+b Q ( a 2 ) + ( b 2 ) N (0) a,b pq− a− b m− a− b Q + N (1) a,b pq− a− b + 1 m− a− b + 1 Q ! . This is the claimed formula. Remark 5.9. For 1 ≤ m < p, the only forbidden proper subfield is F Q . For p ≤ m < q, the unique maximal forbidden subfield is F Q p . Hence Proposition 5.4 gives the corresponding counting formulas. The range q ≤ m < pq is different: one must avoid two incomparable proper subfields, F Q q and F Q p , whose intersection is F Q . Thus the one-forbidden-subspace quotient-and-graph count from Lemma 5.3 does not directly apply, and the inclusion–exclusion formula in Theorem 5.8 is the natural replacement. 6 Further questions The cyclic formula in Theorem 2.4 reduces the computation of C(Z/nZ) to the divisor structure of n. Proposi- tion 2.6 gives an exact numerical criterion for the equalityC(Z/nZ) = φ(n), but a more conceptual classification of the integers satisfying this equality remains desirable. Theorem 2.11 and Corollary 2.12 show that the ratio C(Z/nZ)/φ(n) has liminf 1 and infinite limsup, and that every fixed large-value threshold is attained on a set of positive lower density. It would be interesting to estimate these densities effectively as the threshold grows. Proposition 2.13 shows that lim inf n→∞ C(Z/nZ) n = 0,lim sup n→∞ C(Z/nZ) n = 1. Thus the normalization by the group order has a very different behavior from the normalization by φ(n). One may ask for more precise distributional information about this bounded ratio, for example along integers with prescribed small prime divisors. Finally, Corollary 5.2 determines the extremal dimension for every finite separable extension, but the enu- meration problem remains open in general. Over finite fields it becomes increasingly combinatorial when the lattice of intermediate fields is not a chain. The case n = pq is handled by two-subfield inclusion–exclusion. For degrees with three or more incomparable maximal proper divisors, one expects higher-order formulas governed by the intersection pattern of the corresponding subfields. 7 Methodology This paper was produced through a human–AI collaborative workflow using a lightweight, reasoning-oriented configuration of Co-Scientist [13], with additional assistance from Gemini Deep Think [11] and Gemini Pro [12]. The human authors formulated the problem, directed the exploratory prompts, checked the resulting arguments, repaired gaps, supplied missing hypotheses, and wrote the final exposition. The AI systems assisted by sug- gesting candidate reductions, examples, proof strategies, and draft arguments. All statements and proofs in the final paper were reviewed and validated by the human authors, who take responsibility for their correctness. We regard the paper as an example of AI-assisted mathematical discovery, not as an autonomous AI proof. Further details are given in Appendix B. References [1] M. Aliabadi, Conditions for matchability in groups and field extensions I, Discuss. Math. Gen. Algebra Appl. 45 (2025). 17 [2] M. Aliabadi, Size of Chowla sets and Chowla subspaces, MathOverflow inquiry, 2025. https:// mathoverflow.net/questions/500735/size-of-chowla-sets [3] C. Bachoc, O. Serra, and G. Zémor, An analogue of Vosper’s theorem for extension fields, Math. Proc. Cambridge Philos. Soc. 163 (2017), 423–452. [4] C. Bachoc, O. Serra, and G. Zémor, Revisiting Kneser’s theorem for field extensions, Combinatorica 38 (2018), 759–777. [5] A. L. Cauchy, Recherches sur les nombres, J. École Polytech. 9 (1813), 99–116. [6] S. Chowla, A theorem on the addition of residue classes: application to the number Γ(k) in Waring’s problem, Proc. Indian Acad. Sci. Sect. A 2 (1935), 242–243. [7] H. Davenport, On the addition of residue classes, J. London Math. Soc. 10 (1935), 30–32. [8] D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., John Wiley & Sons, Hoboken, NJ, 2004. [9] P. Erdős and A. Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713–721. [10] W. Feng et al., Towards autonomous mathematics research, arXiv:2602.10177, 2026. [11] Google DeepMind, Gemini Deep Think, large language model, 2026. https://deepmind.google/models/ gemini/deep-think/ [12] Google DeepMind, Gemini Pro, large language model, 2026. https://deepmind.google/models/gemini/ pro/ [13] J. Gottweis, W.-H. Weng, A. Daryin, et al., Accelerating scientific discovery with Co-Scientist, Nature (2026), doi:10.1038/s41586-026-10644-y. [14] Y. O. Hamidoune, An isoperimetric method in additive theory, J. Algebra 179 (1996), 622–630. [15] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., edited by R. Heath- Brown, J. H. Silverman, and A. Wiles, Oxford University Press, Oxford, 2008. [16] X.-D. Hou, K. H. Leung, and Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002), 1–9. [17] R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. [18] R. P. Stanley, Enumerative Combinatorics. Vol. 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2011. [19] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. [20] Z. Zheng et al., AI Co-Mathematician: Accelerating mathematicians with agentic AI, arXiv:2605.06651, 2026. 18 A Numerical plots This appendix records numerical values of the two normalized quantities discussed in the paper. The data were computed for cyclic groups Z/nZ for 2≤ n≤ 3000. The first plot is intended only as a small-range illustration: Theorem 2.11 and Corollary 2.12 show that C(Z/nZ)/φ(n) is unbounded, although this behavior is not visible in such a short numerical range. 2004006008001,0001,2001,4001,6001,8002,0002,2002,4002,6002,8003,000 1 1.2 1.4 1.6 1.8 2 n C ( Z /n Z ) /φ ( n ) Figure 1: Numerical values of C(Z/nZ)/φ(n) for 2≤ n≤ 3000. 2004006008001,0001,2001,4001,6001,8002,0002,2002,4002,6002,8003,000 0 0.2 0.4 0.6 0.8 1 n C ( Z /n Z ) /n Figure 2: Numerical values of C(Z/nZ)/n for 2≤ n≤ 3000. 19 B AI-assisted discovery using a lightweight reasoning configuration of Co-Scientist This appendix documents the AI-assisted component of the discovery process. It is intended as a rigorous methodological record of the human–AI collaboration rather than a benchmark or model-performance claim. B.1 Overview In this appendix we describe the role of a lightweight, reasoning-oriented configuration of Co-Scientist [13], backed by Gemini Deep Think [11] and Gemini Pro [12], in discovering the mathematical results presented in the main paper. While broader, multi-agent frameworks have been developed to support fully autonomous math- ematical research and stateful collaborative workbenches, such as Aletheia [10] and AI Co-Mathematician [20], the present case study focuses exclusively on deploying a compact reasoning loop to assist an expert in step-by- step mathematical deduction. As defined in the main text, a Chowla set is a nonempty subset S of a finite group G satisfying ord(x) >|S| for every x ∈ S, and the extremal invariant C(G) is the maximum size of such a set. This problem—isolating Hamidoune’s order condition [14] as a foundational subject in its own right—had not been extensively studied in prior literature, making it a suitable testbed for the AI system’s deductive capabilities; see Section B.3. The results are the product of a tight, expert-guided human–AI collaboration: • The AI system generated substantial proof-grade material, including the order-distribution reduction for C(G), fixed-cardinality counting formulas, exact formulas for cyclic and finite abelian groups, and the conceptual framework for the linear analogue of Chowla subspaces. • The human mathematician selected the problem, decomposed it into tractable subproblems, evaluated outputs for correctness, supplied missing hypotheses, completed analytical arguments, and authored the final manuscript. B.2 Reasoning-oriented configuration The reasoning-oriented configuration used in this work is a streamlined variant of Co-Scientist [13]. The full sys- tem supports long-horizon autonomous exploration—including literature retrieval, knowledge-base construction, focus-area generation, and multi-round hypothesis evolution—capabilities that can be effective for open-ended empirical domains but introduce latency and compounding-error risks in verifiable mathematics, where correct- ness can and should be checked at each step. This configuration therefore disables auxiliary generation pathways, such as self-play, article exploration, and deep research, and retains only the core loop: generation, review, pair- wise ranking, and iterative improvement. The present appendix accordingly reports a case study in which this targeted configuration contributed substantial proof-grade material under continuous expert guidance, not an autonomous discovery by the full Co-Scientist system. The pipeline executes in two rounds with the following stages. Round 0 (Initial Generation). (1) Parallel Generation. N independent solution attempts are generated in parallel, each via a separate Deep Think invocation that performs multi-step reasoning with tool use, including code execution and search. (2) Review. Each candidate is independently reviewed via prompt-driven critique that identifies strengths, weaknesses, and potential logical gaps. (3) Placement Tournament. Ideas are compared head-to-head in pairwise matches. Each match presents two candidates, together with their reviews, to the model, which selects the stronger one. Match order is randomized and swapped to mitigate position bias. Elo ratings are updated after each match to produce an initial ranking. (4) Deep Verification. All N ideas undergo multi-aspect verification: factual claims are extracted and checked, underlying assumptions are identified and tested, and internal logical coherence is evaluated. This produces a structured verification summary for each idea. 20 (5) Final Tournament. A second tournament is run using only the deeply verified ideas, incorporating the verification results into the ranking. Low-ranking ideas are pruned according to Elo thresholds. Round 1 (Improvement). (6) Iterative Refinement. The top-ranked ideas are fed back to Deep Think along with their review and verification feedback. Deep Think generates N ′ ≤ N improved variants that attempt to patch identified flaws and extend the reasoning. (7) Review, Verification, and Tournament. The improved ideas undergo the same review, deep verifi- cation, and pairwise ranking process, now competing against both the original and improved candidates. Low-ranking ideas are again pruned according to Elo thresholds. Final Output. (8) Overview Report. The surviving top-ranked ideas are assembled into a structured report including complete proof chains and derivations. Reasoning-preserving prompts instruct the summarizer to maintain full logical chains rather than abstracting them into high-level summaries. In our experiments, the generation and refinement stages, steps 1 and 6, were driven by Gemini Deep Think, specifically Pro with extended thinking, while the review, verification, and tournament stages, steps 2–5 and 7, used Gemini Pro. The overview report, step 8, used Gemini Flash with reasoning-preservation instructions. Crucially, the review, ranking, and pruning components function as heuristic filters to optimize search efficiency and reduce cognitive load; they are not formal symbolic proof checkers. Final responsibility for establishing mathematical soundness remained with the human mathematician. Round 0 Research Goal Generate N Ideas Deep Think Review Pro Elo Tournament Pro Deep Verification Pro Round 1 Deep Verification Pro Elo Tournament Pro Review Pro Improve N ′ Ideas Deep Think Summary Overview Report Flash Final Output Deep Think ProFlash Figure 3: Co-Scientist’s reasoning-configuration pipeline. Round 0 generates three independent solutions via Deep Think, reviews them, ranks them through a pairwise Elo tournament, and performs multi-aspect deep verification, including claim extraction, factuality checking, assumption analysis, and coherence evaluation. Round 1 improves top-ranked ideas, with N ′ < N, using review and verification feedback, then repeats the same evaluation cycle. The overview report preserves full reasoning chains through reasoning-preserving prompts. B.3 Problem-selection rationale and literature status The mathematical problem treated in this paper was deliberately selected to stress abstract mathematical reasoning and rule out direct literature retrieval or training-data memorization. A persistent challenge in evaluating frontier AI systems on mathematical tasks is data contamination: many well-known or benchmark problems possess established proof templates in the public domain, allowing a model to synthesize solutions by applying standard proof templates rather than genuine novel deduction. To mitigate this confound, we chose a problem setting that was intentionally absent from the mathematical literature. The foundational motivation stems from Chowla-type order conditions in additive combinatorics. While the core philosophy originates with S. Chowla, it was specifically Y. O. Hamidoune [14] who introduced hypotheses requiring group-element orders to be strictly larger than the size of the relevant subset in order to eliminate small periodic obstruction loops in product-growth theorems. The present work isolates this constraint, transforming it into an independent, open-ended study of the maximum size and exact enumeration of these subsets. 21 To verify that the resulting invariants, C(G) and C(L/K), were unstudied, the first author conducted a comprehensive literature baseline check before the AI interaction. The structural status of the problem is summarized as follows. • The linear-analogue origin. The conceptual translation of this constraint into linear algebra—the definition of a Chowla subspace—was first introduced by the first author in 2025 [1]. That initial work did not establish exact formulas or investigate the extremal invariant C(L/K). • Community verification through MathOverflow. To test whether the extremal and enumerative problems for both groups and fields had known solutions, the first author posted a public inquiry on September 23, 2025, entitled “Size of Chowla sets and Chowla subspaces” [2]. The inquiry remained unanswered, providing an additional indication that no standard formula or established approach was readily available for these specific questions. During the interaction, the AI system had web-search capabilities, and the run logs show that it used them. The system retrieved the MathOverflow and LiveJournal mirror posts authored by the human mathematician, together with foundational work by Hamidoune and related literature on group matchability. Because the retrieved posts contained no solutions and the formal literature supplied only the foundational definitions and background, the searches did not return the theorem-level conclusions established here. The system still had to perform mathematical reasoning to propose such ingredients as the order-distribution reduction and the subspace inclusion–exclusion architecture. We emphasize that this selection criterion does not constitute a formal claim of “pure reasoning,” since the system undoubtedly benefited from standard background mathematics acquired during pre-training. The final proofs in the main text are self-contained and rely on classical graduate-level tools, including finite group theory and Galois theory, finite-field subfield lattices, Euler’s totient function, Gaussian binomial coefficients, and Möbius inversion on divisor and subspace lattices. The system applied these standard tools to an extremal invariant for which no existing solution was supplied by the retrieved sources. The appropriate claim is therefore one of collaborative human–AI mathematical discovery, not proof of independence from all learned mathematical patterns. B.4 Interaction protocol and expert prompting The AI-assisted portion began with two broad prompts concerning the extremal and enumerative behavior of Chowla sets. The first asked for upper and lower bounds, and possible exact formulas, for C(G). The second asked for the number of Chowla subsets of fixed cardinality. These broad prompts produced several useful ideas, including the reduction to order-survival functions and cyclic-group formulas. Prompt 1: Bounds for C(G) Let G be a finite group. A nonempty subset S ⊆ G is called a Chowla subset if every element of S has order strictly larger than |S|; that is, ord(x) >|S| for every x∈ S. Chowla subsets were introduced and studied by Y. O. Hamidoune. For a finite group G, let C(G) denote the maximum size of a Chowla subset of G. For example, if p is prime, then C(Z/pZ) = p− 1. Question. What are some nontrivial upper and lower bounds for C(G) for a given finite group G? Remark. A natural starting point is C(Z/nZ) for finite cyclic groups. A straightforward lower bound is Euler’s totient function: φ(n) = n Y p|n 1− 1 p . 22 Prompt 2: Counting Chowla subsets of fixed cardinality Let G be a finite group. A nonempty subset S ⊆ G is called a Chowla subset if every element of S has order strictly larger than |S|; that is, ord(x) >|S| for every x∈ S. Chowla subsets were introduced and studied by Y. O. Hamidoune. Question. For each integer n with 1 < n < |G|, determine or estimate the number of Chowla subsets S ⊆ G of cardinality n. Remark. A natural starting point is G = Z/nZ, where the condition ord(x) > |S| can be analyzed explicitly in terms of divisors of n. After reviewing the first outputs, the human mathematician decomposed the problem into a sequence of more focused mathematical subquestions. These included cyclic groups, threshold-form constructions, equality with Euler’s totient function, finite abelian groups, abelian p-groups, the linear analogue for field extensions, and finite-field subspace-counting problems. Each refined prompt supplied definitions, notation, and a specific mathematical target. Table 1 summarizes the interaction structure at a high level. Table 1: High-level prompt provenance. Prompt class Mathematical target Useful outputHuman role Broad prompt 1 Bounds and formulas for C(G) Order-survival function, threshold characterization, and cyclic examples Supplied the definition, motivation, and Hamidoune context. Broad prompt 2 Counting Chowla subsets of fixed cardinality Reduction of enumeration to choosing subsets from the elements of order greater than m Supplied the fixed-cardinality question and cyclic starting point. Focused finite-group prompt Determine C(G) from the order distribution Precise theorem-level reduction using N G (m) Isolated the correct abstraction. Focused cyclic prompts Compute C(Z/nZ), threshold form, and equality with φ(n) Divisor formulas, threshold construction, equality criterion, and explicit cases Broke the cyclic problem into tractable subquestions. Focused finite-abelian prompt Compute exact order counts in finite abelian groups T G (d) = Q i gcd(d,n i ), Möbius inversion, and p-group consequences Selected the invariant-factor framework. Focused linear-analogue prompt Define and bound Chowla subspaces in field extensions Translation to [K(a) : K] > dim K A and the d max obstruction Proposed the analogy and later corrected its scope. Focused finite-field prompts Count Chowla subspaces through subspace avoidance Gaussian-binomial formulas and inclusion–exclusion structure Selected the chain case and first non-chain case. The strongest outputs were produced when the human mathematician identified the right intermediate problem, formulated it with precise notation, and asked for a theorem-level treatment. For each focused prompt, the reasoning configuration generated candidates, reviewed and ranked them, refined promising outputs, and returned retained ideas. The human mathematician then inspected the results, rejected incorrect suggestions, and determined which directions were worth pursuing. The remaining work consisted of validating proposed arguments, filling gaps, correcting overstatements, adding missing hypotheses, and writing the final proofs. 23 B.5 Provenance of mathematical ideas The central claim of this appendix is not that the AI system independently produced a complete formal paper. The claim is that it produced substantial theorem-level material and several key proof ideas used in the final paper. Because estimates such as “approximately 80% of the useful content came from the AI system” are nec- essarily subjective, we do not treat such a percentage as a formal measurement. Instead, Table 2 reports idea-level provenance. Any qualitative percentage should be interpreted as an expert assessment rather than as a quantitative attribution metric. Table 2: Provenance of mathematical ideas used in the final paper. Final-paper component AI outputRole in the final paperHuman contribution Definition of Chowla set and C(G) Initial human prompt; AI formalization in broad and focused runs Sets up the invariant studied throughout the paper Selected the object, connected it to Chowla-type order conditions, and fixed notation. Order-distribution reduction Focused finite-group run deriving C(G) = maxm : N G (m)≥ m Becomes Proposition 2.2, the main structural reduction for finite groups Checked the proof, simplified the exposition, and aligned notation. Counting formula for fixed cardinality Broad enumeration prompt and focused finite-group run Gives the formula N G (m) m for the number of Chowla sets of size m Verified the exact statement and integrated it with Proposition 2.2. Cyclic divisor formula Focused cyclic-group run using the element-order distribution in Z/nZ Becomes the formula involving P d|n, d>m φ(d) Polished the proof and connected it to later arithmetic consequences. Threshold-form maximum sets Focused cyclic-group run selecting elements of largest order first Supports the threshold interpretation of maximum Chowla sets Adapted the statement to the final notation and precise scope. Criterion for C(Z/nZ) = φ(n) Focused cyclic-group equality run Leads to the criterion involving the least prime divisor p 1 and φ(n)≥ n/p 1 − 1 Corrected, streamlined, and stated the result in final form. Prime-power and two-prime corollaries Focused cyclic-group runProvides explicit families where equality or an improved bound can be determined Checked cases, repaired arguments where needed, and organized them as corollaries. Asymptotic behavior of C(Z/nZ)/φ(n) AI outputs suggested arithmetic directions and threshold sets The final paper proves liminf 1, infinite limsup, and positive lower-density threshold results Supplied or substantially repaired the analytic number-theoretic arguments and final rigor. Finite abelian group order counts Focused finite-abelian-group run deriving T G (d) = Q i gcd(d, n i ) and Möbius inversion for a G (d) Becomes the finite abelian group formula for C(G) Verified the invariant-factor setup and incorporated standard references. Abelian p-group closed form Focused finite-abelian-group run Supports the closed form for cyclic and noncyclic abelian p-groups Completed and checked the case analysis. Definition of Chowla subspace Focused linear-analogue run translating |S| to dim K A and ord(x) to [K(a) : K] Introduces the linear analogue in finite extensions Judged the analogy mathematically meaningful and fixed the final definition. Upper bound using d max (L/K) Focused linear-analogue run identifying largest proper intermediate fields as obstructions Becomes the upper bound C(L/K)≤ n− d max (L/K) Corrected the scope and supplied a rigorous dimension-intersection proof. 24 Final-paper component AI outputRole in the final paperHuman contribution Sharpness over infinite base fields Focused linear-analogue run suggested avoidance of proper intermediate fields The final paper proves sharpness for finite separable extensions over infinite base fields Added necessary hypotheses, finite-union arguments, and the final proof. Finite-field chain counting Focused finite-field run on subspaces avoiding a fixed subspace Leads to Gaussian-binomial formulas for chain cases Selected the final formulation and aligned it with the subfield lattice. First non-chain finite-field case n = pq Focused finite-field run developed subspace-avoidance and inclusion–exclusion ideas The final paper treats two incomparable proper subfields and gives an explicit formula Supplied the final construction, corrected details, and completed the inclusion–exclusion argument. Numerical plotsNot AI-generatedIllustrate the behavior of normalized ratios for cyclic groups Human and computational contribution. This table records which portions of the final argument were proposed by the system, which were repaired by the human mathematician, and which were supplied independently by the human mathematician. 25 B.6 Selected interaction logs: from broad exploration to focused deduction To illustrate the collaborative workflow, we provide selected excerpts from the interaction logs. The transcripts have been edited for length to remove routine conversational filler while preserving the explicit mathematical prompts and the theorem-level outputs generated by the AI. 1. The extremal invariant and order distribution The human intent. The exploration began with a broad inquiry into the extremal properties of the Chowla- set invariant C(G). The human mathematician provided the foundational definition and nudged the system to consider cyclic groups and Euler’s totient function as a natural starting point. The goal was to see whether the system could identify a unifying structural parameter governing C(G). The AI contribution. The system identified thatC(G) is governed by the group’s element-order distribution. It introduced an “order-survival function” and established an exact combinatorial characterization. Human prompt. What are some nontrivial upper and lower bounds for C(G) for a given finite group G? A natural starting point isC(Z/nZ) for finite cyclic groups. A straightforward lower bound is Euler’s totient function. AI output (excerpt). Definition. Let G be a finite group. Define the order-survival function N >k (G) for an integer k ≥ 1 by N >k (G) :=|g ∈ G : ord(g) > k|. Lemma [Exact characterization of C(G)]. For any finite group G with |G| ≥ 2, the maximum size of a Chowla subset is C(G) = maxk ∈ Z ≥1 : N >k (G)≥ k. The human synthesis. The human mathematician verified the proof and recognized this reduction as the correct abstraction for the paper. The notation N >k (G) was refined to N G (m), together with Ω G (m), producing Proposition 2.2. The system also proposed an exact divisor formula for cyclic groups, which the human authors polished into Theorem 2.4. 2. Fixed-cardinality enumeration The human intent. After the maximum-size characterization, the human mathematician asked for the exact number of Chowla subsets of a prescribed size. The AI contribution. Building on the previous reduction, the system reduced the enumeration problem to selecting elements from the strict survival set. Human prompt. For each integer n with 1 < n < |G|, determine or estimate the number of Chowla subsets S ⊆ G of cardinality n. AI output (excerpt). Theorem [Exact enumeration formula]. Let G be a finite group and let n satisfy 1 < n < |G|. If ω >n (G) =|g ∈ G : ord(g) > n|, then the number c n (G) of Chowla subsets of cardinality n is c n (G) = ω >n (G) n . Proof excerpt. Constructing a Chowla subset S of cardinality n is mathematically identical to choosing exactly n distinct elements from the strict survival set Ω >n (G). The human synthesis. The human mathematician confirmed the subset-selection argument and integrated the formula with the order-distribution reduction, completing the counting statement in Proposition 2.2. 26 3. Isolating the equality case for cyclic groups The human intent. Once φ(n) had been identified as a universal lower bound for C(Z/nZ), the human mathematician asked precisely which integers achieve equality. The AI contribution. The system identified the largest proper divisor, hence the smallest prime factor, as the governing obstruction and derived the boundary condition. Human prompt. For which integers n≥ 2 does the equality C(Z/nZ) = φ(n) hold? AI output (excerpt). Theorem. Let n≥ 2, and let p be its smallest prime factor. Then C(Z/nZ) = φ(n) if and only if n p ≤ φ(n) + 1. Proof excerpt. The remaining internal sum vanishes exactly when n has no proper divisor strictly greater than φ(n) + 1. The human synthesis. Because φ(n) and n/p are integers, the condition was streamlined to φ(n)≥ n/p 1 − 1, yielding Proposition 2.6. The human authors then verified the explicit prime-power and two-prime families in the subsequent corollaries. 4. Formulating the linear analogue in field extensions The human intent. After developing the group-theoretic invariant, the human mathematician proposed translating subset cardinality and element order into the language of finite field extensions and vector subspaces. The AI contribution. The system constructed a group-to-field dictionary: set cardinality became K- dimension, while element order became the degree of the generated simple subextension. It formulated the Chowla-subspace condition and identified d max (L/K) as the governing obstruction. Human prompt. What is the appropriate linear analogue of a Chowla set for a finite field extension K ⊆ L? How large can a Chowla K-subspace of L be in terms of the largest proper intermediate field of L/K? AI output (excerpt). Definition [Chowla subspace]. Let K ⊆ L be a finite field extension of degree n > 1. A nonzero K-subspace A⊆ L is a Chowla subspace if [K(a) : K] > dim K A for every 0̸= a∈ A. Theorem [Universal upper bound]. For any finite field extension L/K, C(L/K)≤ n− d max (L/K). The human synthesis. The human mathematician judged the structural translation meaningful, verified the geometric dimension-intersection proof, corrected the scope of claims concerning infinite and finite base fields, and wrote the rigorous exposition in Section 4; the upper bound appears as Theorem 4.4. 27 B.7 Limitations This appendix reports a single case study, not a benchmark. Several limitations should be emphasized: • The problem was selected and decomposed by a human expert whose mathematical judgment substantially shaped the system’s success. The results should not be interpreted as evidence that non-expert users can obtain comparable outputs from vague prompts. • The review and ranking stages were model-based heuristic filters, not formal verification procedures. A candidate favored by the system could still contain serious mathematical errors. • The boundary between AI-generated content, human repair, and standard mathematical background is not always sharp. Although Tables 1 and 2 provide concrete provenance records, final attribution necessarily reflects human interpretation of the logs. • The problem was chosen to reduce the likelihood of direct literature recombination, but the model may still have drawn on general proof patterns learned during training. The appropriate conclusion is therefore limited but meaningful: in this case study, a compact reasoning configuration of Co-Scientist, guided by expert mathematical prompting, produced substantial proof-grade material for a research-level paper on Chowla sets and subspaces. 28