{"id":"2a644a32-d6db-4bf4-9ef8-0743f04b1956","arxiv_id":"2509.07169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two new partition identities are proved: three-colored partitions with certain forbidden differences have generating functions equal to a distinct-parts factor times the first or second Rogers-Ramanujan product.","lead":"A mathematician proves two new identities that connect three-colored number partitions to products tied to the famous Rogers-Ramanujan series. The proof combines hand-written functional equations with computer algebra checks, but the Maple code files are referenced and not included.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Atomic relation crel1 for T is misprinted: its last term should be x q^{a+2} T_{a+4,b+2,c+2}, not x^2 q^{a+2}; as printed the relation is false.","rationale":"Reading the paper in good faith, the central claim is the pair of partition identities, and the proof architecture — deriving functional equations and using atomic relations — is coherent. The hand-derived parts (q-Pascal relations, functional equations for Q1, Q2, Q3, and the combinatorial derivations of (3.2) and (3.6)) check out at low orders and are not the source of concern. The decisive step is the Maple verification that certain S- and T-expressions are linear combinations of the atomic relations. Because the Maple files are not included, the reader's conditional verdict is reasonable. My stress-test uncovered a further, concrete problem: the printed crel1 for the T family is internally inconsistent. The text explicitly states that factors of x^2 are changed to x for the T relations, yet crel1 keeps x^2 in its final term. A direct derivation from (2.9) and an explicit coefficient check show the correct factor is x, not x^2. Since the proof of Theorem 2 relies on a typed linear combination that includes crel1, the printed verification cannot be correct as written. This does not demonstrate that the identities are false — the error is likely a typo — but it materially strengthens the need for the missing computational evidence and for a corrected statement of crel1. I therefore keep the reader's CONDITIONAL verdict, with the explicit conditions being the correction of crel1 and the release of the Maple verification files.","tokens_in":13873,"tokens_out":29323,"duration_ms":205933,"concrete_test":"Evaluate the coefficient of x in crel1_{0,0,0} as a q-series: with the printed x^2 q^2 T_{4,2,2} the relation fails at the first order, while with x q^2 it holds. Then, using the corrected relation (x q^{a+2} T_{a+4,b+2,c+2} instead of x^2 q^{a+2}), re-run the linear-combination check for (3.7) — the target T_{2,0,1} - (1+xq+xq^2)T_{3,1,2} + x^2 q^4 T_{4,2,3} - xq^3(1-xq^2)T_{5,3,4} — and confirm whether the typed coefficients in Section 3.2 still reproduce the target. If the coefficients change, the proof of Theorem 2 needs substantive revision, not just a typo fix.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The printed atomic relation crel1_{a,b,c} for T_{a,b,c} (Section 2.1) contains a factor x^2 q^{a+2} in its final term. This contradicts the sentence immediately above it, which says the T relations are obtained from the S relations by changing factors of x^2 to x, and it contradicts a direct derivation from the shift equation (2.9): shifting i to i+1 increases the x-exponent of T_{a,b,c}(x) by exactly 1, so the two terms produced by the (1+q^{i+j+k+1}) split are x q^{a+1} T_{a+3,b+1,c+1} and x q^{a+2} T_{a+4,b+2,c+2}. Testing crel1_{0,0,0} at the coefficient of x gives a nonzero q-series with the printed x^2; with x instead, the coefficient vanishes. This is load-bearing because the proof of Theorem 2 (equation (3.7)) provides a typed linear combination using crel1_{2,0,1} and crel1_{2,0,3}; if crel1 is false as printed, that combination is false. The referenced Maple files MC6 proof A-E are not included, so one cannot tell whether the computer used a corrected relation. The decisive algebraic verification is therefore not merely unverifiable: one printed piece of it is demonstrably wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves two mod-10 partition identities. Theorem 1 states that the generating function for three-colored partitions in Gamma with no part 1_R equals 1/((q;q^2)_infinity (q,q^4;q^5)_infinity), and Theorem 2 states the analogous identity with no parts 1_R, 2_R, 1_B, with product 1/((q;q^2)_infinity (q^2,q^3;q^5)_infinity). The proofs refine both sides by a variable x marking the number of parts, derive functional equations for the refined generating functions Q_1,Q_2,Q_3 of the colored partitions and for two auxiliary pair-of-partitions generating functions, and then use the atomic-relations method to show that the q-trinomial multisums S_{a,b,c} and T_{a,b,c} satisfy the same equations. The algebraic verifications are delegated to Maple files that are referenced but not included; one conjecture refining Theorem 1 is stated in Section 4.","tokens_in":14118,"tokens_out":14841,"duration_ms":120080,"significance":"The identities, if correct, are attractive new examples of colored partition identities with modulus 10, connected to the Rogers-Ramanujan products and to level-3 standard modules of A_1^(1); the atomic-relations method is a promising proof paradigm, and the paper usefully exhibits small cases and prints one full linear combination. The manuscript is honest about its reliance on computer verification and gives enough combinatorial structure that the main functional equations are plausible. However, the current text is not self-contained: a printed atomic relation is false, and the decisive symbolic checks are not reproducible from the submission, so the significance is conditional on a revision that fixes these gaps.","major_comments":[{"comment":"The printed atomic relation crel1 for T_{a,b,c} is false. Its final term is x^2 q^{a+2} T_{a+4,b+2,c+2}, but the shift equation (2.9) and the sentence introducing the T-relations (which says x^2 factors are changed to x) both require x q^{a+2} T_{a+4,b+2,c+2}. For example, with the printed relation, the coefficient of x in crel1_{0,0,0} is q^2 rather than 0. This is load-bearing: the displayed linear combination proving (3.7) in Section 3.2 uses crel1_{2,0,1} and crel1_{2,0,3}, so that combination is incorrect as printed. Please correct the typo and re-verify all identities involving crel1.","section":"Section 2.1"},{"comment":"The crucial symbolic verifications are not contained in the paper. After display (3.1), (3.3), (3.4), (3.5), and (3.7), the proof asserts that the substituted expression is a linear combination of atomic relations and refers to files 'MC6 proof A.txt' through 'MC6 proof E.txt', but none of these files is included. From the text alone, the central algebraic step is an assertion. Please include the Maple code and output (or an equivalent machine-readable certificate), or display the explicit linear combinations in the paper, so that the proofs are checkable.","section":"Sections 3.1 and 3.2"}],"minor_comments":[{"comment":"In the justification of (2.12), the bullet for the case where the partition contains 2_R says the rest is counted by Q_1(xq^2), but the displayed term is x q^2 Q_3(xq^2), and the listed forbidden parts support Q_3. Please correct the wording.","section":"Section 2.2"},{"comment":"The substitutions of T-expressions into (2.15) and (2.16) are said to use (2.8), but the relevant shift equation for T_{a,b,c} is (2.9). Please update these cross-references.","section":"Sections 3.1 and 3.2"},{"comment":"After substituting (1+xq)S_{3,0,1}(x) into (3.2), the displayed equation divides by (1+xq); since this factor has constant term 1 as a formal power series, the division is legitimate, but a brief remark would avoid confusion.","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The misprinted crel1 suggests that the omitted Maple files may have been generated with the corrected relation; please ask the author to confirm and to submit the files. The novelty of the identities relative to the atomic-relations papers [6,21] is reasonably clear, but the proof's reproducibility is the main issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two new mod-10 three-colored partition identities, proved via atomic relations. The sum sides are new; the product sides are known (Stembridge; level-3 A1^(1) characters). The combinatorial derivations of the functional equations (2.12)–(2.14) are careful, and the bijective arguments for the product-side equations (3.1) and (3.5) check out by hand. The identities are a genuine extension of the author's prior program, and there is no circularity or fitting involved: both sides are compared to independent benchmarks. This is a solid contribution to an active subfield.\n\nThe main problems are about verifiability, not correctness. The decisive steps—showing that the substituted multisums satisfy (3.2), (2.15), and (2.16) as linear combinations of atomic relations—are delegated to Maple files (MC6 proof A-E) that are not included in the preprint. That alone makes the proof conditional. The stress-test note is right: the printed crel1 for T is misprinted. Its last term should be x q^{a+2} T_{a+4,b+2,c+2}, not x^2 q^{a+2}; as printed the relation is false. The sentence above says the T relations come from the S relations by changing factors of x^2 to x, so a reader can guess the correction, but the displayed equation is wrong and the typed linear combination in Section 3.2 uses crel1, so the proof as written has a concrete gap. This looks like a typo, not a conceptual error, but it needs fixing.\n\nThe paper is for q-series and partition identity specialists, especially those who use automated verification. The result is plausible and well-motivated, but the proof is not fully checkable from the text. It deserves a serious referee: request the code or explicit linear combinations and get the typo corrected before publication.","headline":"Solid new identities, but the proof's computational crux is unverifiable from the preprint and one printed atomic relation is misprinted; worth refereeing with revisions.","tokens_in":14643,"tokens_out":3240,"would_cite":true,"duration_ms":26863,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A17","11P84","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves two identities equating the generating functions of restricted three-colored partitions with the products $1/((q;q^2)_\\infty(q,q^4;q^5)_\\infty)$ and $1/((q;q^2)_\\infty(q^2,q^3;q^5)_\\infty)$.","keywords":["three-colored partitions","partition identities","Rogers-Ramanujan identities","q-trinomial coefficients","atomic relations","functional equations","generating functions"],"falsifier":"Independently rerun the five Maple verifications (or expand both sides of the substituted functional equations by hand or computer and check they reduce to zero); any failure of a reduction would invalidate the proof. Separately, enumerate all three-colored partitions in $\\Gamma$ with the stated exclusions up to $n=30$ and compare coefficients with the series expansions of the two product sides; a first mismatch would falsify the corresponding theorem.","tokens_in":13631,"feed_emoji":"🧩","tokens_out":10561,"duration_ms":83970,"temperature":0.7,"pith_summary":"The paper establishes a pair of exact product formulas for the generating functions of three-colored partitions into distinct parts, drawn from a set $\\Gamma$ with color-dependent difference constraints. One identity forbids the red part 1; the other forbids red 1, red 2, and blue 1. The products are $1/((q;q^2)_\\infty(q,q^4;q^5)_\\infty)$ and $1/((q;q^2)_\\infty(q^2,q^3;q^5)_\\infty)$, namely the odd-distinct partition generating function times a Rogers-Ramanujan product. These identities give closed forms for the restricted partition counts, and because the same products arise as principally specialized characters of level-3 standard modules of $A_1^{(1)}$, they tie the colored partitions to affine Lie algebra representation theory. The proof is a template that introduces $q$-trinomial multisums, derives functional equations for bivariate refinements, and uses Maple to check linear atomic-relation combinations.","feed_headline":"Three-colored partition counts equal Rogers-Ramanujan products mod 10","feed_subtitle":"These identities connect restricted three-colored partitions to the Rogers-Ramanujan family of q-series.","key_machinery":"The machinery has three parts. First, the $q$-trinomial coefficient $\\binom{i+j+k}{i,j,k}_q = (q;q)_{i+j+k}/((q;q)_i(q;q)_j(q;q)_k)$ and the multisums $S_{a,b,c}(x)$ and $T_{a,b,c}(x)$ built from it; these carry the refined counting. Second, the atomic relations rel1–rel9 (and crel1–crel9 for $T$), which are linear relations among shifted multisums proved from $q$-Pascal identities and let the computer rewrite any substituted functional equation as zero. Third, functional equations for the colored-partition generating functions $Q_1,Q_2,Q_3$, untangled by the Murray-Miller algorithm into the single equations (2.15) and (2.16). The paper shows that particular multisum combinations satisfy those same functional equations, then checks initial conditions, forcing equality of generating functions.","core_discovery":"The central claim is that the two restricted generating functions factor completely: $\\sum_{n\\ge 0} A(n)q^n = 1/((q;q^2)_\\infty(q,q^4;q^5)_\\infty)$ (Theorem 1) and $\\sum_{n\\ge 0} A^*(n)q^n = 1/((q;q^2)_\\infty(q^2,q^3;q^5)_\\infty)$ (Theorem 2), where $A(n)$ counts three-colored partitions in $\\Gamma$ with no $1_R$ and $A^*(n)$ counts those with no $1_R, 2_R, 1_B$. The proof establishes stronger bivariate statements first: the part-count generating functions $Q_1(x)$ and $Q_3(x)$ are identified with explicit combinations of the $T$-multisums, $T_{1,0,1}(x)+xqT_{3,1,2}(x)=Q_1(x)$ and $T_{2,0,1}(x)=Q_3(x)$, and the $S$-multisums are identified with the sum sides of the first and second Rogers-Ramanujan identities via equations (3.1) and (3.5). Setting $x=1$ and applying Euler's odd-distinct identity and the Rogers-Ramanujan identities yields the product forms.","pith_inferences":["Editorial inference: because the product sides are principally specialized characters of level-3 standard modules of $A_1^{(1)}$, the restricted three-colored partitions may provide a direct combinatorial model for those characters, potentially supporting a crystal-theoretic bijection.","Editorial inference: the five cited Maple verifications are essential to the proof, and since the files are not included, an independent computer algebra check of the linear combinations would settle whether the proof's central step is sound.","Editorial inference: the method suggests a recipe for proving other conjectured Nahm-sum identities: insert auxiliary variables, derive a bivariate functional equation from a combinatorial interpretation of the product side, and verify the multisum side via atomic relations."],"forward_implications":["The two restricted three-colored partition families have explicit product generating functions, so their coefficients can be computed from the products rather than by enumerating $\\Gamma$.","The stronger bivariate identities provide information about the number of parts in these partitions, not just their total size.","The identities join the Rogers-Ramanujan family, giving colored-partition interpretations of products that also appear as level-3 characters of $A_1^{(1)}$ standard modules.","The proof strategy—derive functional equations for bivariate refinements, then use atomic relations to verify the equations—extends to other conjectured Rogers-Ramanujan-type multisum identities, including the refinement proposed in Section 4."],"supporting_citations":[{"why":"Supplies the two Rogers-Ramanujan identities used at the end of the proofs to turn the multisum sides into the product forms.","marker":"[28]"},{"why":"Provides the Murray-Miller algorithm used to untangle the system of functional equations into the single equations (2.15) and (2.16).","marker":"[26]"},{"why":"Introduces the method of atomic relations that the paper adapts to prove the multisum identities.","marker":"[21]"},{"why":"Earlier paper with the same atomic-relations paradigm, referenced as the source of the identity-proving template.","marker":"[6]"},{"why":"The computerized search through which the identities were discovered.","marker":"[20]"}],"fun_headline_variants":["Three-colored partitions meet Rogers-Ramanujan mod 10","Mod 10 identity: three colors tie into Rogers-Ramanujan","Three-colored partition sums factor as Rogers-Ramanujan mod 10","Mod 10: three-colored partitions yield Rogers-Ramanujan products"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the Maple files MC6 proof A.txt through E.txt confirming that substituted multisum expressions are linear combinations of atomic relations, and those files are not included in the preprint, so the central computational step cannot be checked from the text alone.","fun_headline_variants_meta":{"raw":{"variants":["Three-colored partitions meet Rogers-Ramanujan mod 10","Mod 10 identity: three colors tie into Rogers-Ramanujan","Three-colored partition sums factor as Rogers-Ramanujan mod 10","Mod 10: three-colored partitions yield Rogers-Ramanujan products"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":2972,"prompt_tokens":863,"completion_tokens":2109,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2045}},"tokens_in":479,"tokens_out":2109,"duration_ms":14275,"temperature":1.0,"reasoning_tokens":2045,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:12:08.752834+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently rerun the five Maple verifications (or expand both sides of the substituted functional equations by hand or computer and check they reduce to zero); any failure of a reduction would invalidate the proof. Separately, enumerate all three-colored partitions in $\\Gamma$ with the stated exclusions up to $n=30$ and compare coefficients with the series expansions of the two product sides; a first mismatch would falsify the corresponding theorem.","supporting_citations":[{"cited_title":"Murray and Kenneth S","cited_arxiv_id":null,"evidence_quote":"Provides the Murray-Miller algorithm used to untangle the system of functional equations into the single equations (2.15) and (2.16)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the method of atomic relations that the paper adapts to prove the multisum identities."},{"cited_title":"Russell, and Christopher Sadowski","cited_arxiv_id":null,"evidence_quote":"Earlier paper with the same atomic-relations paradigm, referenced as the source of the identity-proving template."},{"cited_title":"Russell.IdentityFinderand some new identities of Rogers-Ramanujan type.Exp","cited_arxiv_id":null,"evidence_quote":"The computerized search through which the identities were discovered."}],"review_version":2}