{"id":"66e344fc-c96d-48f2-916c-741c6e775d71","arxiv_id":"1908.07055","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A Paley type partial difference set exists in an abelian group of odd order v exactly when v is a prime power congruent to 1 mod 4 or v = n^4 or 9n^4 for odd n > 1.","lead":"This paper proves exactly which group sizes can host a Paley type partial difference set, a symmetric subset used to build strongly regular graphs. It closes a 25-year-old existence question by showing the only non-prime-power sizes are fourth powers or nine times fourth powers of odd integers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof's only external dependency is the quoted Theorem 2.1, and its application to the Hall subgroup H checks out cleanly.","rationale":"The paper's central claim is an existence classification. The necessary direction is the only nontrivial part; it depends on Ma's structure theorem and on Theorem 2.1. I stress-tested the latter where it is least secure: the sub-PDS induced on the Hall subgroup H of order u^2. The parameters align: beta = -1, pi = u, theta = 0, Delta_1 = |H|, k_1 = (u^2 - 1)/2, and the nonempty and not-H\\{e} conditions are met. Thus the contradiction for odd r is exactly sound. The only residual risk is a misquotation of a published theorem, which is not a defect of this paper's argument unless the quoted form is wrong; I have no evidence that it is. For the converse, the dependence on Polhill (2010) is explicit and published. Consequently the ACCEPT verdict stands unchanged; if one wanted extra caution, a source-level check of Theorem 2.1 would be the single useful verification.","tokens_in":3716,"tokens_out":11367,"duration_ms":115507,"concrete_test":"Verify the quotation of Theorem 2.1 against the original Arasu-Jungnickel-Ma-Pott paper (or the Ma 1994 sources), in particular the exact hypotheses of the Moreover congruence. If the theorem is correctly quoted, recheck the two-line contradiction in Theorem 2.2; if any hidden condition is missing, the necessary direction of Theorem 1.2 would need an independent proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the argument at its least protected point, the invocation of Theorem 2.1 in Theorem 2.2. For a Paley-type PDS, beta = -1, Delta = p^(2r) u^2 and delta = p^r u. With H of order u^2, the coprimality and odd-index hypotheses hold, pi = gcd(u^2, p^r u) = u > 1, and the defining interval forces theta = 0. Theorem 2.1 then gives beta_1 = -1 and Delta_1 = u^2, hence k_1 = (u^2 - 1)/2, so D_1 is nonempty and not H \\ {e}. If r were odd, the Moreover clause would require theta = (p-1)/2 mod (p-1), contradicting theta = 0. This is exactly how r even is forced. I find no internal gap: the only way the proof could fail is if Theorem 2.1 is misquoted or carries an unstated hypothesis, but that is a published theorem quoted in the form given by Arasu et al., and no such defect is apparent. The non-prime-power case then follows by applying Theorem 2.2 to each prime p_i >= 5, giving |G| = n^4 or 9n^4, and Polhill's constructions supply the converse. I therefore have no load-bearing objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Paley type partial difference sets in abelian groups, i.e. regular PDSs with parameters (v, (v-1)/2, (v-5)/4, (v-1)/4). It first recalls a theorem of Ma stating that if the discriminant Delta is not a square, then v must be a prime power congruent to 1 modulo 4. The main work concerns square v. Using a sub-PDS theorem of Arasu, Jungnickel, Ma, and Pott (Theorem 2.1), the paper proves Theorem 2.2: no Paley type PDS can exist in an abelian group of order p^{2r}u^2 with p >= 5, gcd(p,u)=1, and u>1 when r is odd. Applying this to each prime divisor p_i >= 5 of a square non-prime-power order forces every such exponent to be a multiple of 4, so the order is n^4 or 9n^4 for odd n>1 (Corollary 2.3). Combined with Polhill's 2010 constructions and the classical Paley construction over finite fields, this yields the complete characterization in Theorem 1.2: for odd v>1, a Paley type PDS exists in some abelian group of order v if and only if v is a prime power congruent to 1 modulo 4, or v = n^4 or 9n^4 for odd n>1.","tokens_in":4014,"tokens_out":10090,"duration_ms":102331,"significance":"If correct, this result completely answers the motivating existence question for odd orders, the natural completion of a line of work by Ma, Arasu-Jungnickel-Ma-Pott, Polhill, and the author. The proof is short and transparent: the main technical input is the quoted Theorem 2.1, and the application to a subgroup of order u^2 is clean and checks out. The paper gives explicit credit to prior work and, in the acknowledgement, appropriately notes the related Lemma 2.4 of Leung and Ma. The necessary condition derived here is sharp because matching constructions were already known, making this a meaningful and publishable contribution.","major_comments":[],"minor_comments":[{"comment":"In the proof of Corollary 2.3, the quantity u should be defined as u = sqrt(|G|/p_i^{2t_i}), not as |G|/p_i^{2t_i}; the group order is p_i^{2t_i} u^2 in Theorem 2.2, and |G|/p_i^{2t_i} is a square because all other exponents are even.","section":"§2, Corollary 2.3"},{"comment":"In the proof of Theorem 2.2, the sentence \"Also D \\neq H \\setminus {e}\" should read \"Also D_1 \\neq H \\setminus {e}\", since D_1 is the intersection of D with H and the condition in Theorem 2.1 concerns D_1.","section":"§2, Theorem 2.2 proof"},{"comment":"There are several typographical errors that should be corrected: \"Athough\" for \"Although\", \"Thorem\" for \"Theorem\", and \"relative prime\" for \"relatively prime\".","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is very short and relies on a substantial quoted theorem (Theorem 2.1 from Arasu et al.) without proof. This is acceptable if the journal's policy permits use of established results; the application of the theorem is correct. The overlap with Leung and Ma's Lemma 2.4 is acknowledged, and the proof given here appears to be an independent derivation of that lemma. No concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a clean short proof that completes the classification of orders for Paley-type PDSs in abelian groups. The main theorem is exactly the if-and-only-if statement: prime power v ≡ 1 mod 4, or v = n^4 or 9n^4 for odd n > 1. The necessary side was the open part; the paper proves Theorem 2.2, that for p ≥ 5 the exponent r in p^{2r}u^2 must be even, and derives the order restriction.\n\nWhat I like: the proof is short and transparent. It uses the Arasu–Jungnickel–Ma–Pott theorem (2.1) to pull the sub-PDS D1 back to a subgroup of order u^2, computes β1 = −1 and k1 = (u^2−1)/2, and then uses the congruence clause to force r even. I checked the calculation: π = gcd(u^2, p^r u) = u, the interval for θ contains −1 only for θ = 0, and the congruence for odd r gives θ ≡ (p−1)/2 mod p−1, which contradicts θ = 0. So the load-bearing step holds. The reduction from the corollary to the main theorem is fine, and Polhill’s constructions are cited for the converse. The paper also openly acknowledges that Leung and Ma had listed a similar lemma without proof; that’s the right thing to do, and it keeps the novelty claim honest.\n\nSoft spots, all minor: Theorem 2.1 is quoted without proof. That is fine given it is published, but the “Moreover” clause is doing real work, so a reader might want the statement checked against the original. There is a small typo in the proof of Theorem 2.2: it says “D ≠ H \\ {e}” where it should say “D1 ≠ H \\ {e}”. The claim is true with D1, and the printed statement is meaningless in context. Also the paper could have noted explicitly that primes p = 3 cause no problem in Corollary 2.3; the conclusion still follows because the constraint is only on p ≥ 5, and an odd exponent on 3 just produces the 9n^4 factor. Not a gap, but a line of explanation would help.\n\nBottom line: the result is correct, the prose is clear, and the history is handled honestly. It is a solid subfield contribution, not a revolution. Anyone working on partial difference sets or strongly regular Cayley graphs will want this for the classification statement. I would send it to a serious combinatorics journal like JCTB or Designs, Codes and Cryptography. It deserves peer review.","headline":"Clean, correct proof that completes the 25-year-old existence classification for Paley-type PDSs in abelian groups; a bit narrow, very solid.","tokens_in":4519,"tokens_out":4086,"would_cite":true,"duration_ms":36020,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B10","05E30","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a Paley type partial difference set exists in some abelian group of odd order $v>1$ exactly when $v$ is a prime power congruent to 1 modulo 4, or $v=n^4$ or $9n^4$ for odd $n>1$.","keywords":["partial difference sets","Paley type","abelian groups","Cayley graphs","strongly regular graphs","sub-partial difference sets","prime powers"],"falsifier":"Run a complete character-theoretic search for a $(1225, 612, 305, 306)$-partial difference set in each of the four abelian groups of order $5^2 \\cdot 7^2$; finding one would refute Theorem 1.2, which predicts none because 1225 is neither a prime power congruent to 1 modulo 4 nor of the form $n^4$ or $9n^4$.","tokens_in":3532,"feed_emoji":"🔢","tokens_out":17093,"duration_ms":152363,"temperature":0.7,"pith_summary":"This paper closes a long-open existence question: for which odd orders $v>1$ does some abelian group contain a Paley type partial difference set? The author proves that when $v$ is not a prime power, such a set can exist only if $v=n^4$ or $v=9n^4$ for an odd integer $n>1$. Since the classical finite-field construction gives all prime powers congruent to 1 modulo 4, and an earlier construction gives the orders $n^4$ and $9n^4$, the characterization is complete. The upshot is a clean division: either the order is a prime power congruent to 1 modulo 4, or it is a fourth power, possibly multiplied by 9.","feed_headline":"Paley difference set orders: n^4, 9n^4, prime powers ≡1 mod4","feed_subtitle":"Non-prime-power orders must be fourth powers or 9 times fourth powers, and known constructions cover exactly those cases.","key_machinery":"The load-bearing mechanism is the sub-partial-difference-set theorem (Theorem 2.1, quoted from [1]): when a nontrivial regular PDS with square discriminant $\\Delta=\\delta^2$ is restricted to a subgroup $H$ whose order is coprime to its index and whose index is odd, the intersection $D\\cap H$ is again a regular PDS with parameters controlled by $\\pi=\\gcd(|H|,\\delta)$ and an integer $\\theta$ chosen so that $(2\\theta-1)\\pi \\le \\beta < (2\\theta+1)\\pi$. The crucial clause says that if $\\delta=p^r\\pi$ with $p\\ge 5$ and $\\gcd(p,\\pi)=1$, then odd $r$ forces $\\theta\\equiv (p-1)/2 \\pmod{p-1}$. For a Paley type PDS one has $\\beta=-1$, so in the chosen subgroup $\\theta=0$; the congruence then contradicts odd $r$. This single mechanism extracts the square structure of the group order from the PDS parameters.","core_discovery":"The paper's central claim is Theorem 1.2: for an odd integer $v>1$, a Paley type partial difference set (PDS) exists in some abelian group of order $v$ if and only if $v$ is a prime power with $v\\equiv 1 \\pmod{4}$, or $v=n^4$ or $9n^4$ for odd $n>1$. The new ingredient is the necessity direction for orders that are not prime powers. Given a Paley type PDS in a group of order $p^{2r}u^2$ with $p\\ge 5$ and $\\gcd(p,u)=1$, the author takes a subgroup $H$ of order $u^2$ and applies a theorem on sub-partial-difference sets to force the auxiliary integer $\\theta$ to be $0$; the theorem's congruence condition then rules out odd $r$. Hence every prime $p\\ge 5$ appears with exponent divisible by 4, leaving only fourth powers possibly multiplied by 9. Sufficiency comes from the existing finite-field and group constructions.","pith_inferences":["The classification also settles which abelian groups can carry strongly regular Cayley graphs with $\\lambda-\\mu=-1$, since a Paley type PDS is exactly the Cayley subset that produces such a graph; the paper states the result only in PDS language.","The necessity proof uses only the order of the group, not its abelian structure, so the finer question of which particular abelian groups of order $n^4$ or $9n^4$ admit such a set remains open; the known construction provides at least one group for each admissible order.","The exceptional factor 9 points to the prime 3 as the only prime whose square can appear outside the fourth-power part; working out the $p=3$ case of the congruence machinery might show whether this exception is forced or merely an artifact of the construction."],"forward_implications":["The existence question for Paley type partial difference sets in abelian groups of odd order is fully answered, with no order left undecided.","Any order of the form $p^2 q^2$ with distinct primes $p,q$ at least 5 cannot support a Paley type PDS; only fourth-power exponents, with the single possible factor 9, survive.","All admissible non-prime-power orders already have explicit constructions, so the necessary condition is sharp.","For prime-power orders, the classical finite-field construction is the only source needed."],"supporting_citations":[{"why":"It supplies Theorem 2.1, the sub-partial-difference-set parameter and congruence result that drives the necessity proof.","marker":"[1]"},{"why":"It gives the theorem that a non-square discriminant forces Paley type parameters and order $p^{2s+1}$, reducing the problem to square orders.","marker":"[4]"},{"why":"It provides the classical finite-field construction of Paley type PDS for every prime power congruent to 1 modulo 4, covering the prime-power case.","marker":"[7]"},{"why":"It provides constructions of Paley type PDS in groups of orders $n^4$ and $9n^4$ for odd $n>1$, exactly the non-prime-power orders allowed by the theorem.","marker":"[9]"}],"fun_headline_variants":["Non-prime-power Paley PDS orders must be n^4 or 9n^4","Paley-type difference sets: only prime powers ≡1 mod4, n^4, 9n^4","Complete classification of Paley-type PDS in abelian groups","For abelian groups, Paley PDS orders are exactly prime powers ≡1 mod4, n^4, 9n^4","New result settles Paley-type difference set existence in abelian groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on a previously established theorem about how a partial difference set restricts to a subgroup, including a precise congruence condition on an auxiliary integer; if that theorem is wrong or does not apply to the chosen subgroup, the proof that rules out non-prime-power orders collapses.","fun_headline_variants_meta":{"raw":{"variants":["Non-prime-power Paley PDS orders must be n^4 or 9n^4","Paley-type difference sets: only prime powers ≡1 mod4, n^4, 9n^4","Complete classification of Paley-type PDS in abelian groups","For abelian groups, Paley PDS orders are exactly prime powers ≡1 mod4, n^4, 9n^4","New result settles Paley-type difference set existence in abelian groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001111,"raw_usage":{"total_tokens":4631,"prompt_tokens":948,"completion_tokens":3683,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":3561}},"tokens_in":564,"tokens_out":3683,"duration_ms":26141,"temperature":1.0,"reasoning_tokens":3561,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:27:34.942917+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a complete character-theoretic search for a $(1225, 612, 305, 306)$-partial difference set in each of the four abelian groups of order $5^2 \\cdot 7^2$; finding one would refute Theorem 1.2, which predicts none because 1225 is neither a prime power congruent to 1 modulo 4 nor of the form $n^4$ or $9n^4$.","supporting_citations":[{"cited_title":"Arasu, D","cited_arxiv_id":null,"evidence_quote":"It supplies Theorem 2.1, the sub-partial-difference-set parameter and congruence result that drives the necessity proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the theorem that a non-square discriminant forces Paley type parameters and order $p^{2s+1}$, reducing the problem to square orders."},{"cited_title":"Paley, On orthogonal matrices, J","cited_arxiv_id":null,"evidence_quote":"It provides the classical finite-field construction of Paley type PDS for every prime power congruent to 1 modulo 4, covering the prime-power case."},{"cited_title":"Polhill, Paley partial difference sets in groups of orde r n4 and 9n4 for any odd n > 1, Journal of Combinatorial Theory , Series A 117, 1027–1036 (2010)","cited_arxiv_id":null,"evidence_quote":"It provides constructions of Paley type PDS in groups of orders $n^4$ and $9n^4$ for odd $n>1$, exactly the non-prime-power orders allowed by the theorem."}],"review_version":1}