{"id":"58cb500e-adca-412b-8cdc-b23b10274dd8","arxiv_id":"1908.07741","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact generating functions for p_nu(50n+8), p_omega(40n+12), spt_omega(10n+3), spt_omega(10n+5), spt_omega(50n+23), spt_omega(50n+25) and new congruences modulo powers of 5.","lead":"This paper derives new exact generating functions and congruence identities for partition functions that count partitions related to Ramanujan's mock theta functions gamma(q) and nu(q). It offers elementary proofs for several previously known results and new congruences modulo powers of 5 for a smallest parts function.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of identity (1.36) is the load-bearing weakness: Corollary 1.9's new mod 5^ℓ congruence families depend on it.","rationale":"The paper's two fully proved theorems (1.1 and 1.4) are substantial and internally consistent, and my spot check of Lemma 2.2 against its small-q expansions shows no inconsistency; those lemmas are from the authors' earlier paper [8] and can be cited. The genuine soft spot is Eq. (1.36), which is not proved in this paper; the authors say so explicitly. Corollary 1.9, which contains the new high-power congruence families, is derived from (1.35)/(1.36) and then by further dissections that are also abbreviated. This makes the central claim conditional: the exact identity and its congruence consequences are unverified. The reader's weakest_assumption pointed to Lemmas 2.2–2.3; that is related but not identical, so I mark partial agreement. A direct q-series check of (1.36) would settle the concern; if it passes, the conditional verdict can stand with confidence; if it fails, the paper should be rejected. Since I have not shown a falsehood, I do not move the verdict; I would keep it conditional.","tokens_in":19123,"tokens_out":22086,"duration_ms":190404,"concrete_test":"Compute the left side of (1.36) numerically from the known generating function (1.23): for N=50, let a(n) be the coefficient of q^{25n+12} in E_2^9/E_1^6, so a(n)=sptω(50n+25). Compare a(n) with the coefficient of q^n in the 17-term eta quotient on the right-hand side of (1.36), for 0≤n≤50. If all coefficients agree exactly (or even modulo 5^6), the identity is strongly supported; if any coefficient differs, Theorem 1.8 and the Corollary 1.9 congruences derived from (1.36) are false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is not the quoted Lemmas 2.2–2.3, which are published results from [8], but the paper's own admitted omission of the proof of Theorem 1.8's identity (1.36): 'The identity (1.36) can be proved in a similar way, therefore we omit the proof' (Section 6). Identity (1.36) is a 17-term eta-quotient identity with coefficients up to 1024×10^18, and it is the input for every congruence in Corollary 1.9, including the new families sptω(5^{2k+ℓ−1}(10n+5)) ≡ sptω(5^{ℓ−1}(10n+5)) mod 5^ℓ for ℓ=1,...,6 and the mod 25, 625, 15625 statements (1.38)–(1.40). Because those congruences are obtained by iterated 5-dissections of (1.36), any coefficient or exponent error in (1.36) can change the residues used and invalidate the congruence claims. Theorem 1.7's (1.34) and the higher-ℓ cases of (1.37) are also stated with 'omitting details' or 'similar fashion', but (1.36) is the sharpest case: no proof is given at all. Until (1.36) is independently verified, the paper's headline congruence families rest on an unproved assertion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives exact q-series generating functions for the partition functions p_nu, p_omega and the smallest-parts functions spt_omega, spt_omega at certain arithmetic progressions, using 5-dissections together with identities from the authors' earlier paper [8]. The main results are Theorem 1.1 for p_nu(50n+8), Theorem 1.4 for p_omega(40n+12), Theorem 1.7 for spt_omega(10n+3) and spt_omega(50n+23), and Theorem 1.8 for spt_omega(10n+5) and spt_omega(50n+25). From these generating functions the authors deduce new congruences modulo powers of 5, including Corollary 1.2, Corollary 1.5, and the families in Corollary 1.9. The proofs are elementary but computation-heavy; Theorems 1.1 and 1.4 are written out in detail, while parts of Theorems 1.7 and 1.8 and Corollary 1.9 are stated with omitted details.","tokens_in":19403,"tokens_out":5252,"duration_ms":55942,"significance":"If the identities and congruences are correct, they are explicit, nontrivial additions to the literature on partition functions associated with mock theta functions and on smallest-parts functions. The paper gives concrete and falsifiable statements, and the detailed proofs of Theorems 1.1 and 1.4 are checkable by the methods of the field. The substantial new congruences in Corollary 1.9 depend essentially on identity (1.36), whose proof is omitted; as written, the manuscript therefore does not fully establish its headline claims. The paper is nonetheless well organized and the computational work is impressive, provided the missing derivations can be supplied.","major_comments":[{"comment":"The proof of Theorem 1.8 states that (1.36) 'can be proved in a similar way, therefore we omit the proof.' This is a 17-term eta-quotient identity with coefficients as large as 1024 x 10^18, and it is the sole input for every congruence in Corollary 1.9, including the new families spt_omega(5^{2k+l-1}(10n+5)) congruent to spt_omega(5^{l-1}(10n+5)) modulo 5^l. A single coefficient or exponent error in (1.36) could change the residues obtained by subsequent 5-dissections and invalidate the congruence claims. Please provide a complete proof or a verifiable derivation of (1.36) before acceptance.","section":"Section 6, Eq. (1.36)"},{"comment":"The derivation of (1.34) relies on four displayed dissections introduced by the phrase 'omitting details.' In particular, the third and fourth displayed dissections have many large coefficients, and these computations are exactly what produce (1.34) and the subsequent congruences (1.31) and (1.32). The reader cannot check these steps from the manuscript as it stands. Please supply the intermediate dissections or an accompanying reproducible computation.","section":"Section 5, Theorem 1.7"},{"comment":"The cases l = 4, 5, 6 of (1.37) are not proved; the text says 'The remaining cases of (1.37) can be proved in a similar fashion' and then records only successive generating functions. Since (1.39) and (1.40) are new congruence statements, they require either a proof or an explicit statement that they are computer-assisted and verified to a specified order. Please provide the missing details or clearly separate the proved cases from the conjectural ones.","section":"Section 6, Corollary 1.9"}],"minor_comments":[{"comment":"The name 'Wladherr' following Eq. (1.5) should be 'Waldherr'.","section":"Section 1"},{"comment":"The notation (q15,q35,q50;q50)_infty is used without definition; please define the multiple-parameter product notation or replace it with an equivalent product expression.","section":"Section 2, Eq. (2.5)"},{"comment":"There are several typographical slips, such as 'spt_omega (1250n + 625))qn' with a double parenthesis, which should be corrected.","section":"Section 6, displayed equations near (6.7)"},{"comment":"The transition 'from (1.36) and (2.3)' in (6.4) is terse; it would help the reader if the congruence modulus being reduced at each stage were stated explicitly.","section":"Section 6, proof of Corollary 1.9"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the computations appear plausible, but the omitted proof of (1.36) is load-bearing for the main congruence families. I would ask for a full derivation of (1.36) and for the missing details in Theorem 1.7 and Corollary 1.9 before acceptance. The reliance on [8] is acceptable since those lemmas are published, though the present paper should make explicit which identities are being quoted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful content here is real. Theorems 1.1 and 1.4 give new exact generating functions for p_nu(50n+8) and p_omega(40n+12), and the proofs are written out in enough detail that I can follow the 5-dissection machinery and check the algebra. Theorem 1.7's generating function for spt_omega(10n+3) and (1.34) for spt_omega(50n+23) are also plausible and the proof is sketched honestly, though with some 'omitting details' in the extraction step. The paper is careful about credit: it notes that (1.31) and (1.32) were already proved by Wang and Yang, and Corollary 1.5 is attributed to Xia. The reliance on Lemmas 2.2–2.3 from the authors' earlier paper is legitimate; those are published, traceable identities, and self-citation here is not a red flag.\n\nThe soft spot is exactly where the stress test points. Theorem 1.8's identity (1.36) is not proved at all—the text says it 'can be proved in a similar way' and omits the proof. That identity is a large eta-quotient with enormous coefficients, and every congruence in Corollary 1.9, including the new mod 5^ℓ families, is derived from it by iterated 5-dissection. If any coefficient or exponent in (1.36) is wrong, those congruence families could collapse. This is not a manufactured flaw; the authors flag the omission themselves. It is also not fatal to the whole paper, because Theorems 1.1 and 1.4 are proved in full and stand independently, and (1.35) in Theorem 1.8 is proved. But as it stands, the paper's headline high-power congruence claims rest on an unverified assertion.\n\nI also want to note that the derivation of the mod 5^4, 5^5, and 5^6 congruences in Corollary 1.9 is largely recorded as a list of successive generating functions with 'can be proved in a similar fashion.' That is the kind of thing that is routine for the authors but not independently checkable by the reader without substantial work. A referee should insist on at least one fully worked iteration, or a verification of (1.36) via a computer algebra system.\n\nWho is this for? Specialists in partition congruences and mock theta functions. They will get value from the explicit generating functions and the congruence families, and they will be able to fill in the omitted algebra. The paper deserves a serious referee and probably acceptance after the missing proof of (1.36) is supplied, not a desk rejection. I would take it for review, but I would send it back for a revision that includes that proof or a computer-checked verification.","headline":"A solid, within-subfield paper with two fully proved generating-function theorems and a real soft spot: the unproved identity (1.36) that carries the new high-power congruences.","tokens_in":20039,"tokens_out":1255,"would_cite":true,"duration_ms":14103,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P83","05A15","05A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves new exact q-series generating functions for partition functions tied to the third-order mock theta functions $\\omega(q)$ and $\\nu(q)$, and derives new divisibility families modulo powers of 5.","keywords":["partition congruence","smallest parts function","mock theta function","generating function","5-dissection","Rogers-Ramanujan continued fraction","eta-quotient","totally symmetric plane partitions"],"falsifier":"For $n=0$, compute $p_\\nu(8)$ directly from the definition (partitions of 8 with distinct parts and each odd part less than twice the smallest part) and compare it with the constant coefficient of the right-hand side of Theorem 1.1; then repeat for $n=1,2$ against the coefficients of $q$ and $q^2$, which would test $p_\\nu(58)$ and $p_\\nu(108)$. A mismatch at any of these first three values would refute the identity, and the congruence $p_\\nu(6250n+5208)\\equiv 0 \\pmod{125}$ can be tested independently by enumerating the relevant partitions for $n=0,1,2$.","tokens_in":18894,"feed_emoji":"🔢","tokens_out":7285,"duration_ms":59308,"temperature":0.7,"pith_summary":"The paper establishes new exact generating functions for four partition functions attached to the third-order mock $\\theta$ functions $\\omega(q)$ and $\\nu(q)$: $p_\\nu(50n+8)$, $p_\\omega(40n+12)$, $\\mathrm{spt}_\\omega(10n+3)$, $\\mathrm{spt}_\\omega(50n+23)$, $\\mathrm{spt}_\\omega(10n+5)$, and $\\mathrm{spt}_\\omega(50n+25)$. Each is written as a finite sum of explicit eta-quotients with integer coefficients. From these formulas the authors derive new congruences modulo powers of 5, the strongest being $p_\\nu(6250n+5208)\\equiv 0 \\pmod{125}$ and the family $\\mathrm{spt}_\\omega(5^{2k+\\ell-1}(10n+5))\\equiv \\mathrm{spt}_\\omega(5^{\\ell-1}(10n+5))\\pmod{5^\\ell}$ for $\\ell=1,\\dots,6$. A sympathetic reader would care because these are exact finite descriptions of counting functions whose divisibility behavior had previously been known only through scattered congruences, and the method reduces each congruence to an algebraic extraction using a fixed set of 5-dissection identities.","feed_headline":"Exact formulas unlock 5-power congruences for mock-theta partitions","feed_subtitle":"New q-series identities give p_nu(6250n+5208) ≡ 0 mod 125 and spt_omega congruence families.","key_machinery":"The working tool is a 5-dissection calculus built on the Rogers-Ramanujan continued fraction: writing $T(q)=q^{1/5}/R(q)$, the identities in Lemma 2.1 express $E_1$, $1/E_1$, $\\phi(-q)$, and a related eta-quotient in powers of $T(q)$, letting one extract the coefficient of $q^{5n+r}$ in a generating function. Two further identities from the authors' earlier work, Lemmas 2.2 and 2.3, link $T(q)$, $T(q^2)$, and the eta quotient $K=E_2 E_5^5/(E_1 E_{10}^5)$; these are applied repeatedly to rearrange and reduce the extracted series into the compact eta-quotient forms of the theorems.","core_discovery":"On the paper's own terms: the sequence $p_\\nu(50n+8)$ can be written exactly as $5\\bigl(E_2^5 E_4^5/(E_1^6 E_{10}^2) + 160 q E_2^{11} E_4^5/E_1^{14} + 2000 q^2 E_2^{11} E_4^{10}/E_1^{20}\\bigr)$ (Theorem 1.1), the sequence $p_\\omega(40n+12)$ has the nine-term eta-quotient expansion of Theorem 1.4, and the two smallest-parts functions satisfy the explicit generating functions of Theorems 1.7 and 1.8. These are not asymptotic or congruence-only statements: they are exact $q$-series equalities, and each corollary divisibility claim follows by reducing the right-hand side modulo a power of 5 and extracting a specific residue class. The same identities yield the previously known congruences (1.31) and (1.32), the new Corollary 1.9, and Theorem 1.3, which translates the $p_\\nu$ result into a congruence for 1-shell totally symmetric plane partitions. The proofs are self-contained in the sense that every new step is an application of 5-dissections and eta-quotient manipulations, starting from known generating functions for the base residue classes.","pith_inferences":["The same dissection-and-extract routine that proves Theorem 1.1 and Theorem 1.4 should, with more labor, produce explicit generating functions for $p_\\nu(250n+208)$ and $p_\\omega(200n+92)$, giving a ladder of congruences modulo 625 and higher.","The pattern in Corollary 1.9 suggests a general theorem: $\\mathrm{spt}_\\omega(5^{2k+\\ell-1}(10n+5))\\equiv \\mathrm{spt}_\\omega(5^{\\ell-1}(10n+5))\\pmod{5^\\ell}$ for all $\\ell$; the paper leaves this as Conjecture 1.10, so a proof would follow if the same reduction pattern persists at every level.","Because $p_\\nu(2n)=f(6n+1)$, the new $p_\\nu$ formulas translate directly into enumerative information about 1-shell totally symmetric plane partitions, connecting the mock-theta congruences to plane-partition geometry.","A direct computation of the first few coefficients from (1.13) and (1.35), independent of the lemmas, would confirm the extraction chain and could be done by generating all partitions for $n$ up to a few hundred."],"forward_implications":["If Theorem 1.1 is correct, the congruence ladder for $p_\\nu(50n+8)$ extends uniformly: reducing (1.13) modulo 5 and 25 recovers (1.12), and the same formula yields $p_\\nu(6250n+5208)\\equiv 0 \\pmod{125}$.","Theorem 1.7 gives back the known congruences (1.31) and (1.32), previously proved with modular forms, by an elementary generating-function route.","Theorem 1.8 and Corollary 1.9 produce infinite families of congruences for $\\mathrm{spt}_\\omega$ along arithmetic progressions, including iterations of $\\mathrm{spt}_\\omega(250n+125)\\equiv \\mathrm{spt}_\\omega(10n+5)\\pmod{5}$.","Since $p_\\nu(2n)=f(6n+1)$, Theorem 1.1 yields an exact generating function for $f(150n+25)$ and the new congruence $f(18750n+15625)\\equiv 0 \\pmod{125}$."],"supporting_citations":[{"why":"Supplies Lemmas 2.2 and 2.3, the identities in $x=T(q)$, $y=T(q^2)$, and $K=E_2E_5^5/(E_1E_{10}^5)$ used in every theorem's reduction.","marker":"[8]"},{"why":"Introduces the partition functions $p_\\omega(n)$ and $p_\\nu(n)$, proves their generating functions (1.5) and (1.6), and establishes the baseline congruences that the paper refines.","marker":"[5]"},{"why":"Gives the exact generating functions (1.8), (1.9), (1.22), and (1.23) that serve as starting points for the new progressions, plus the conjecture whose initial cases are reproved here.","marker":"[28]"},{"why":"Provides the identity for $p_\\nu(10n+8)$ used as the starting point for Theorem 1.1 and for the mod-25 congruence (1.12).","marker":"[30]"},{"why":"Proves the congruence stated in Corollary 1.5, which the paper recovers and whose iteration argument it follows.","marker":"[31]"},{"why":"Establishes the $p_\\nu(2n)$ generating function that links $p_\\nu$ to 1-shell totally symmetric plane partitions and underlies Theorem 1.3.","marker":"[20]"}],"fun_headline_variants":["Exact generating functions yield 5-power partition congruences","Mock theta partitions: explicit q-series prove mod 125 congruences","New eta-quotient identities give 5-adic partition divisibility","Exact formulas for mock theta functions lead to mod-125 congruences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quoted identities in Lemmas 2.2 and 2.3 are taken from the authors' earlier paper without proof in this one, and if either is incorrect, the new generating functions and the congruences built on them do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Exact generating functions yield 5-power partition congruences","Mock theta partitions: explicit q-series prove mod 125 congruences","New eta-quotient identities give 5-adic partition divisibility","Exact formulas for mock theta functions lead to mod-125 congruences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1524,"prompt_tokens":895,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":554}},"tokens_in":511,"tokens_out":629,"duration_ms":98375,"temperature":1.0,"reasoning_tokens":554,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:56:42.342727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $n=0$, compute $p_\\nu(8)$ directly from the definition (partitions of 8 with distinct parts and each odd part less than twice the smallest part) and compare it with the constant coefficient of the right-hand side of Theorem 1.1; then repeat for $n=1,2$ against the coefficients of $q$ and $q^2$, which would test $p_\\nu(58)$ and $p_\\nu(108)$. A mismatch at any of these first three values would refute the identity, and the congruence $p_\\nu(6250n+5208)\\equiv 0 \\pmod{125}$ can be tested independently by enumerating the relevant partitions for $n=0,1,2$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemmas 2.2 and 2.3, the identities in $x=T(q)$, $y=T(q^2)$, and $K=E_2E_5^5/(E_1E_{10}^5)$ used in every theorem's reduction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the partition functions $p_\\omega(n)$ and $p_\\nu(n)$, proves their generating functions (1.5) and (1.6), and establishes the baseline congruences that the paper refines."},{"cited_title":"Wang, New congruences for partitions related to mock thet a functions, J","cited_arxiv_id":null,"evidence_quote":"Gives the exact generating functions (1.8), (1.9), (1.22), and (1.23) that serve as starting points for the new progressions, plus the conjecture whose initial cases are reproved here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the identity for $p_\\nu(10n+8)$ used as the starting point for Theorem 1.1 and for the mod-25 congruence (1.12)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the congruence stated in Corollary 1.5, which the paper recovers and whose iteration argument it follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the $p_\\nu(2n)$ generating function that links $p_\\nu$ to 1-shell totally symmetric plane partitions and underlies Theorem 1.3."}],"review_version":1}