{"id":"415c4437-75dc-4ee8-b961-cf1f67e226fb","arxiv_id":"1908.03937","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New congruence families for fractional partition functions modulo arbitrarily high prime powers, obtained by lifting Chan-Wang's results through lacunary Dedekind eta powers and Hecke eigenforms.","lead":"Fractional partition numbers count partitions weighted by a rational power of the generating function. This paper proves that for many choices of that power, the numbers vanish modulo high powers of a prime, extending earlier congruences from modulus ℓ to modulus ℓ^k.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 for d=14 and d=26 rests on unverified Serre eigenform decompositions in §2.2; if these are wrong, the lifting proof breaks.","rationale":"I agree with the reader that the weakest assumption is the Serre decomposition into Hecke eigenforms, especially for d=14 and d=26. In the worked cases the internal steps are coherent: the support arguments give a(ℓ)=0, and the subsequent extraction step forces the right-hand side to vanish once multiplicativity and the vanishing of the ℓ-th coefficients are available. The gaps I also noticed (the unstated v≥2 condition in the second application of Lemma 3, the oldform issue for d=6 and d=8, and the vacuity of Theorem 4 for ℓ=2,3) are real but repairable and do not affect the main construction for v≥2. The central claim is likely correct, but the proof for d=14 and d=26 would be significantly strengthened by an explicit verification or derivation of the asserted eigenform decompositions and their ℓ-integrality. Since the reader's verdict is already CONDITIONAL and my concern matches the reader's weakest assumption, no change to the verdict is needed.","tokens_in":9434,"tokens_out":25186,"duration_ms":250634,"concrete_test":"Use a computer algebra system (e.g., Sage) to compute the q-expansions of both sides of Eqs. (2) and (3) to order q^200 and confirm the identities; verify that each bracketed form is an eigenvector for the Hecke operators T_p for the first several primes p at the relevant level; and check that for every d-satisfactory prime ℓ with ℓ<1000, the ℓ-th coefficient of each bracketed eigenform is 0. If any identity, eigenform property, or vanishing check fails, Theorem 2 for d=14 or d=26 is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central lifting argument in Theorem 2 depends on the assertion in §2.2, Eqs. (1)–(3), that the relevant eta powers are linear combinations of normalized cuspidal Hecke eigenforms whose ℓ-th Fourier coefficient vanishes for d-satisfactory ℓ and whose coefficients are multiplicative. For d=10, the support argument (exponents 1 or 5 mod 12) makes this transparent for ℓ≡3 mod 4. For d=14 and d=26, however, the proof only says 'similar arguments,' and the eigenform property of the forms in Eqs. (2)–(3), their normalization, and vanishing of a±(ℓ) are not demonstrated. The denominators 1/(720√−3) and 1/32617728 also require checking ℓ-integrality for all d-satisfactory primes beyond the explicit exclusions of 5 and 11. If any of these decompositions is not exactly as stated, the extraction step that forces the right-hand side to vanish would fail, and Theorem 2 for those d would be unsupported. This is the load-bearing input: it is cited rather than derived, and the paper does not supply the level, Nebentypus, or a verification of the eigenform relations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies congruences for fractional partition functions p_alpha(n), the coefficients of (q;q)_infty^alpha. Building on Chan-Wang's theorem giving congruences p_alpha(ell n + c) ≡ 0 (mod ell) when ell | a - d b for d in {4,6,8,10,14,26}, the paper uses Serre's explicit decompositions of lacunary eta powers into Hecke eigenforms to lift the modulus to higher powers of ell. Theorem 2 claims p_alpha(ell^2 n + r) ≡ 0 (mod ell^{ord_ell(alpha-d)}) under suitable hypotheses; Theorem 3 gives an analogous result for d = 2 with exponent ord_ell(alpha-2)-1; Theorem 4 drops the 2-satisfactory condition at the cost of a finite choice of w. The paper includes examples intended to show sharpness, including p_{-1/8}(7^2 n+5) ≡ 0 (mod 7^2).","tokens_in":9617,"tokens_out":41793,"duration_ms":406773,"significance":"If the proof is completed, the results are a clean and useful strengthening of the Chan-Wang congruences, and the d = 2 cases are new. The main method is transparent: extract ell-divisible terms from the generating function and use the vanishing of ell-th coefficients of lacunary eta powers together with Hecke multiplicativity. The explicit sharpness examples are valuable and appear to be correctly computed for Theorems 2 and 3. The strengths include the detailed d = 4 and d = 10 arguments, the explicit verification of sharpness, and the honest reliance on the external Chan-Wang and Serre results rather than on fitted constants. The main weakness is that several load-bearing steps for d = 6, 8, 14, 26 and for the v = 1 case are either not written out or are asserted by reference to Serre without the necessary level, Nebentypus, and integrality data.","major_comments":[{"comment":"The two-step reduction using Lemma 3 is only valid when v = ord_ell(alpha-d) is at least 2; for v = 1, Equation (9) contains the exponent ell^{v-2} = ell^{-1}, which is not defined. The theorem as stated includes v = 1, so the proof needs a separate argument for this case, for example a direct appeal to Theorem 1 with m = ell n, or a one-step extraction from Equation (6) using the vanishing of a_4(ell m) for ell ∤ m. As written, the displayed proof does not cover all cases claimed in Theorem 2.","section":"§3, Proof of Theorem 2 (d = 4 case)"},{"comment":"For d = 14 and d = 26, the paper asserts that the displayed linear combinations are normalized cuspidal Hecke eigenforms and that their ell-th coefficients vanish for d-satisfactory ell, but it supplies no level, Nebentypus, normalization check, or a precise citation to the relevant result in Serre. This is load-bearing: the extraction step in the proof of Theorem 2 requires a_d(ell n) = 0 for all n coprime to ell, and that conclusion is obtained exactly from the eigenform decomposition and multiplicativity. Please provide the missing data or a detailed verification of Eqs. (2) and (3), and check the ell-adic integrality of the constants 720√-3 and 32617728 for every d-satisfactory prime, not only the excluded 5 and 11.","section":"§2.2, Eqs. (2)-(3) and §3, d = 14 and d = 26 cases"},{"comment":"The proof says that 'similar conclusions' hold for d = 6 and 8, citing Martin for the statement that eta(tau)^d are Hecke eigenforms for d in {2,4,6,8,12}. However, the forms actually used in Theorem 2 are eta(4 tau)^6 and eta(3 tau)^8, which are scaled (and twisted) versions of eta(tau)^6 and eta(tau)^8. Martin's theorem as stated does not directly apply to these scaled forms, and the multiplicative property of their coefficients, which is needed to conclude a_6(ell n) = a_8(ell n) = 0 for n coprime to ell, must be justified explicitly.","section":"§2.2 and §3, d = 6 and d = 8 cases"},{"comment":"The recurrence in Eq. (14) is written as a_2(ell^{i+1}) = a_2(ell^i) a_2(ell) - a_2(ell^{i-1}), which assumes that the Nebentypus character of eta(12 tau)^2 evaluated at ell is 1. The text says 'Because chi(2) = 1 from Lemma 2', which appears to be a typo for chi(ell) = 1. More importantly, the level and Nebentypus of eta(12 tau)^2 are not identified, so Lemma 1 cannot be applied without additional information. The recurrence is only needed for non-2-satisfactory primes ell ≡ 1 (mod 12), where the character value is indeed 1, but the proof should state this explicitly and justify the eigenform property of eta(12 tau)^2. Since Lemma 4 is used in both Theorem 3 and Theorem 4, this is a load-bearing gap.","section":"§3, Lemma 4 and Eq. (14)"}],"minor_comments":[{"comment":"The chosen value r = (11 · 13^{12} - 1)/12 is not an integer, because 11 · 13^{12} - 1 ≡ 10 (mod 12); hence the example does not actually produce an arithmetic progression. The example would be valid if 11 were replaced by 1, i.e. r = (13^{12} - 1)/12.","section":"§1, Example after Theorem 4"},{"comment":"The displayed formula for a_2(13^k) appears to have a sign error: with a_2(13) = -2, the recurrence in Eq. (14) gives a_2(13^k) = (-1)^k (k+1), not (-1)^{k+1}(k+1). The divisibility conclusion a_2(13^{12}) ≡ 0 (mod 13) is unaffected, but the formula should be corrected.","section":"§1, Example after Theorem 4"},{"comment":"The notation 'k ∈ Z(ell)' is imprecise; it should be specified as an integer k coprime to ell, or written as k ∈ Z_{(ell)} with the definition given.","section":"Throughout, notation Z(ell)"},{"comment":"The statements of Theorems 2-4 do not explicitly mention the standing assumption gcd(ell, b) = 1 from Theorem 5. While ord_ell(alpha-d) ≥ 0 forces this in the nontrivial cases, stating the assumption explicitly would prevent ambiguity, especially for readers checking the well-definedness of congruences modulo powers of ell.","section":"Theorems 2-4"}],"recommendation":"major_revision","confidential_remarks":"The central theorems are likely correct and the paper is a reasonable contribution, but the proof as written is not self-contained for d = 14 and d = 26, and the v = 1 case of Theorem 2 is not handled by the displayed argument. The erroneous Example after Theorem 4 (non-integral r) should be fixed even though it does not affect the theorem. If the editors are willing to accept an expanded citation to Serre in place of a full verification, the revision could be minor, but as it stands the load-bearing gaps justify a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine extension of Chan and Wang, and the core idea—using lacunarity of eta powers plus Hecke multiplicativity to lift mod ℓ congruences to mod ℓ^k—is sound as far as it goes. I'd send it to a referee.\n\nWhat's actually new: Theorem 2 lifts Chan–Wang's modulo ℓ congruences to modulus ℓ^{ord_ℓ(α−d)} for d in {4,6,8,10,14,26}, giving arbitrarily high prime powers for suitable α. Theorem 3 covers d=2, which Chan–Wang didn't have, and Theorem 4 drops the satisfactory-prime condition for d=2. The examples compute p_{−1/8}(7^2 n+5) ≡ 0 mod 7^2 and show sharpness; the d=10 extraction is transparent because the support of the eigenforms is exponents 1 or 5 mod 12.\n\nWhere it's soft: the d=14 and d=26 cases rest on Serre's decompositions in §2.2, Eqs (2) and (3), and the paper just says 'similar arguments.' The constants 1/(720√−3) and 1/32617728 raise ℓ-integrality questions that aren't addressed beyond excluding 5 and 11. If those decompositions are wrong, the extraction step breaks. I don't think they are—Serre's paper is reliable—but the proof should cite the exact level, Nebentypus, and eigenform relations, or include a verification. Also the proof of Theorem 2 writes out only d=4 and d=10; d=6,8,14,26 are 'similar' but not shown. The v=1 case needs an unstated appeal to Theorem 1 to justify the first division by ℓ; it's a real gap, but minor and fixable. Theorem 4's statement is a bit unusual but correct as far as I can tell; the example with ℓ=13 and w=12 checks out.\n\nThe citation pattern is fine—Chan–Wang, Serre, Martin are the right sources, and the paper doesn't fit constants or hide circular dependencies. The reader's stress-test about the eigenform decompositions is the right concern, but it's a completeness issue, not evidence of error.\n\nWho's this for: anyone working on partition congruences or eta-quotient lacunarity. It deserves a serious referee; I'd accept it for review and ask for the omitted cases to be filled in or referenced precisely.","headline":"Solid extension of Chan–Wang to higher prime-power moduli; the d=14/26 cases lean on unverified Serre eigenform decompositions, but the worked cases and examples carry the paper.","tokens_in":10141,"tokens_out":1793,"would_cite":true,"duration_ms":16841,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P83","11F11","11F33"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that fractional partition congruences modulo a prime ℓ lift to modulo ℓ^{ord_ℓ(α−d)}, a power that can be made arbitrarily large by choosing α close to d.","keywords":["fractional partition function","partition congruences","eta function","lacunarity","Hecke eigenforms","q-Pochhammer symbol","prime power congruences","modular forms"],"falsifier":"For d=14, ℓ=11, take r=4 (so 12·4+7=55 has 11-adic valuation 1) and α=135 (so α−14=121=11²). Compute the coefficient $p_{135}(121n+4)$ for n=0,1,2,…; if any value is not divisible by 11², Theorem 2 is false, and if the first nonzero value is not divisible by 11³, the bound $\\operatorname{ord}_{\\ell}(\\alpha-d)$ is sharp in that case.","tokens_in":9187,"feed_emoji":"🔢","tokens_out":10093,"duration_ms":98451,"temperature":0.7,"pith_summary":"This paper establishes that congruences for fractional partition functions—coefficients p_α(n) of $(q;q)^\\alpha_\\infty$ for rational α—hold modulo arbitrarily high powers of a prime, not just modulo the prime itself. The main theorem says that for each d in {4,6,8,10,14,26} and each d-satisfactory prime ℓ, if α is chosen so that ℓ divides α−d exactly k times, then $p_\\alpha(\\ell^2 n+r)$ is divisible by $\\ell^k$ for every n, provided r satisfies a specified 24-adic normalization condition. Because $k=\\operatorname{ord}_\\ell(\\alpha-d)$ can be made as large as one likes by taking α close to d, this yields an infinite supply of prime-power congruences. The argument uses the fact that certain even powers of the eta function are lacunary and can be written as sums of Hecke eigenforms with vanishing coefficients at ℓ. The paper also proves slightly weaker d=2 analogues, including a version that drops the congruence condition on ℓ at the cost of a larger arithmetic progression difference.","feed_headline":"Partition congruences reach arbitrary prime powers","feed_subtitle":"Choosing α close to d makes p_α(ℓ²n+r) divisible by ℓ^k for every k.","key_machinery":"The load-bearing object is the eta function $\\eta(\\tau)=q^{1/24}(q;q)_\\infty$ and its even powers. For d in {2,4,6,8,10,14,26}, a classical result classifies these powers as lacunary: almost all of their Fourier coefficients vanish. More importantly, each $\\eta(\\frac{24}{\\gcd(d,24)}\\tau)^d$ is written explicitly as a linear combination of normalized cuspidal Hecke eigenforms; for the primes ℓ allowed in the theorems, the ℓ-th Fourier coefficient of each eigenform is zero, and multiplicativity of eigenform coefficients then forces $a_d(\\ell m)=0$ for every m prime to ℓ. This vanishing is what makes the coefficient extraction work: after applying the Frobenius congruence $(q;q)_{\\infty}^{\\ell^r \\alpha}\\equiv (q^{\\ell};q^{\\ell})_{\\infty}^{\\ell^{r-1}\\alpha}\\pmod{\\ell^r}$, the unwanted terms drop out and only the $p_\\alpha(\\ell^2 n+r)$ terms remain. The d=2 case uses the same machinery with the periodicity of the sequence $a_2(\\ell^i)$ in place of strict vanishing.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2: for d in {4,6,8,10,14,26}, ℓ a d-satisfactory prime, and r satisfying $\\operatorname{ord}_{\\ell}(\\frac{24}{\\gcd(d,24)}r+\\frac{d}{\\gcd(d,24)})=1$, the congruence $$p_\\$\\alpha$(\\$ell^{2}$ n+r)\\equiv 0\\pmod{\\$ell^{{\\operatorname{ord}}$_\\ell(\\$\\alpha$-d)}}$$ holds for all n. Since $\\operatorname{ord}_\\ell(\\alpha-d)$ can be arbitrarily large, this is a strengthening of the earlier mod-ℓ congruences. The d=2 case is handled separately: Theorem 3 gives the same shape with the exponent reduced by one, and Theorem 4 removes the restriction $\\ell\\not\\equiv 1\\pmod{12}$ by allowing the progression difference to be $\\ell^{w+1}$ instead of $\\ell^2$. The paper demonstrates sharpness in two worked examples, showing that the modulus cannot in general be raised by an additional power of ℓ.","pith_inferences":["The same coefficient-extraction template should apply to other q-series that factor as an eta power times an ℓ-adically small factor, not just to $(q;q)_\\infty^\\alpha$; any such factorization would yield analogous prime-power congruences.","The d=2 periodicity argument gives an explicit bound $w<\\ell^{2v}$ on the progression shift in Theorem 4, so the existence statement is constructive and could be converted into an algorithm for producing the congruences.","Because α can be chosen as $d+\\ell^K$ times any rational with denominator prime to ℓ, Theorem 2 implies that for a fixed satisfactory ℓ there are infinitely many distinct rational exponents α, accumulating ℓ-adically at d, each with a congruence modulo $\\ell^K$."],"forward_implications":["For each listed d and each satisfactory ℓ, choosing α with $\\operatorname{ord}_\\ell(\\alpha-d)=K$ yields $p_\\alpha(\\ell^2 n+r)\\equiv 0\\pmod{\\ell^K}$ for all n, so the modulus can be any prescribed prime power.","The d=2 results extend the reach to a case absent from the earlier mod-ℓ theorem: for example, $p_{1/13}(25n+7)\\equiv 0\\pmod{5}$ follows from Theorem 3.","Theorem 4 shows that even when ℓ fails the d=2 congruence condition, suitable congruences exist after enlarging the arithmetic progression difference from $\\ell^2$ to $\\ell^{w+1}$.","The sharpness examples imply the exponent $\\operatorname{ord}_\\ell(\\alpha-d)$ cannot generally be increased: $p_{-1/8}(5)$ is not divisible by $7^3$, so the modulus in that case is exactly $7^2$.","Together these results turn the earlier mod-ℓ congruences into a prime-power congruence theory for fractional partition functions."],"supporting_citations":[{"why":"Supplies the base mod-ℓ congruences and the Frobenius congruence lemma used to move exponents through q-Pochhammer symbols.","marker":"[4]"},{"why":"Supplies the classification of lacunary even powers of the eta function and the explicit decompositions into Hecke eigenforms.","marker":"[7]"},{"why":"Provides the Nebentypus character of each eta power, used in the Hecke eigenvalue relation that gives coefficient multiplicativity.","marker":"[3]"},{"why":"Shows which eta powers are themselves Hecke eigenforms, used for d=2,4,6,8.","marker":"[5]"}],"fun_headline_variants":["Fractional partition congruences reach arbitrary ℓ^k","Arbitrarily deep congruences for fractional partition functions","From mod ℓ to mod ℓ^k: fractional partition congruences","New ℓ^k congruences for fractional partition functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that each eta power in the list really can be decomposed into modular building blocks whose ℓ-th coefficients vanish at the relevant primes; if any of these decompositions were missing or failed, the coefficient extraction that eliminates the right-hand side would break.","fun_headline_variants_meta":{"raw":{"variants":["Fractional partition congruences reach arbitrary ℓ^k","Arbitrarily deep congruences for fractional partition functions","From mod ℓ to mod ℓ^k: fractional partition congruences","New ℓ^k congruences for fractional partition functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000565,"raw_usage":{"total_tokens":2670,"prompt_tokens":925,"completion_tokens":1745,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":1677}},"tokens_in":541,"tokens_out":1745,"duration_ms":15742,"temperature":1.0,"reasoning_tokens":1677,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:59:30.055383+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For d=14, ℓ=11, take r=4 (so 12·4+7=55 has 11-adic valuation 1) and α=135 (so α−14=121=11²). Compute the coefficient $p_{135}(121n+4)$ for n=0,1,2,…; if any value is not divisible by 11², Theorem 2 is false, and if the first nonzero value is not divisible by 11³, the bound $\\operatorname{ord}_{\\ell}(\\alpha-d)$ is sharp in that case.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the base mod-ℓ congruences and the Frobenius congruence lemma used to move exponents through q-Pochhammer symbols."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classification of lacunary even powers of the eta function and the explicit decompositions into Hecke eigenforms."},{"cited_title":"Carney, A","cited_arxiv_id":null,"evidence_quote":"Provides the Nebentypus character of each eta power, used in the Hecke eigenvalue relation that gives coefficient multiplicativity."},{"cited_title":"Martin, Multiplicative η-quotients, Trans","cited_arxiv_id":null,"evidence_quote":"Shows which eta powers are themselves Hecke eigenforms, used for d=2,4,6,8."}],"review_version":1}