bibkey: huang2025signpatterns
authors: "Zeyu Huang, Timothy Huber, James McLaughlin, Pengjun Wang, Yan Xu and Dongxi Ye"
year: 2025
title: "Sign-patterns of Certain Infinite Products"
doi: 10.48550/arXiv.2507.16644
url: https://arxiv.org/abs/2507.16644v1
claim: "Classical cubic theta definitions, Euler product identities, and the three-core lattice exponent used in the study of infinite-product signs."
strata_touched:
  - D5/S3/Arith/ModularForms/CubicThetaLattice
  - D5/S3/Arith/ModularForms/CubicThetaShifted
  - D5/S3/Arith/ModularForms/CubicThetaJacobi
  - D5/S3/Arith/ModularForms/CubicThetaRootProduct
  - D5/S3/Arith/ModularForms/CubicThetaJacobiCube
  - D5/S3/Arith/ModularForms/CubicThetaJacobiColumns
  - D5/S3/Arith/ModularForms/CubicThetaCubeDissection
  - D5/S3/Arith/ModularForms/CubicThetaResidueSums
  - D5/S3/Arith/ModularForms/CubicThetaRootEvaluation
  - D5/S3/Arith/ModularForms/CubicThetaCoreProduct
  - D5/S3/Arith/ModularForms/CubicThetaProductTwist
  - D5/S3/Arith/ModularForms/CubicThetaWeightedNorm
  - D5/S3/Arith/ModularForms/CubicThetaTwistedColumns
  - D5/S3/Arith/ModularForms/CubicThetaWeightedProduct
  - D5/S3/Arith/ModularForms/CubicThetaIdentities
  - D5/S3/Arith/ModularForms/EtaSignK6
license: citation-only
triage: anchor

Cubic theta lattices and their product interpretation

Verified locator

DOI: 10.48550/arXiv.2507.16644

URL: https://arxiv.org/abs/2507.16644v1

The inspected v1 source is signpatternsaoc.tex, extracted text SHA256 e870441977cf9615a6e7a32daba63a7843bf668e7196c42087e4355a4cd780b6. The subsection titled “The Borwein cubic theta functions” defines the norm theta series with exponent x^2+x*y+y^2 and records the classical identities in the display labelled abc1. The proof of the lemma labelled lem9 quotes Cooper, Ramanujan’s Theta Functions (2017), Theorem 3.15, for the three-core product and the sum over integer triples of sum zero. The substitution (m1,m2,m3)=(x,y,-x-y) gives the exponent 3*(x^2+x*y+y^2)-2*x-y. Cooper’s book was not independently inspected for this note; its theorem is attributed as quoted by the inspected source.

Formal anchors and scope

D5/S3/Arith/ModularForms/CubicThetaLattice.norm_theta_dissection proves the all-index lattice identity A(X)=A(X^3)+6*X*T_lattice(X^3). The proof constructs a norm-tripling equivalence on the residue-zero fibres and six disjoint rotated copies of the shifted lattice on the residue-one fibres. Residue-two fibres are empty. The construction and the finite-fibre bounds are repository proofs of classical lattice mathematics, with no mathematical novelty claim.

D5/S3/Arith/ModularForms/CubicThetaLattice.hasSum_lattice_theta identifies the coefficient-cardinality series with the full integer lattice sums. The positivity and constant terms refer to these lattice series.

D5/S3/Arith/ModularForms/CubicThetaShifted.shiftedCoreEquiv gives three disjoint exhaustive affine copies of each core fibre in the shifted norm fibre Q(x,y)-x-y=n. Its series identity is shifted_theta_eq_three_core.

D5/S3/Arith/ModularForms/CubicThetaJacobi reuses the repository’s InversionSeq207TripleProduct.formal_triple_product and InversionSeq207Euler.euler_coefficient_construction. Its tendsto_finite_jacobi proves that the actual finite Jacobi products converge coefficientwise to the normalized supplier, while jacobiSeries_column identifies every integer Laurent column. The supplier, including its Durfee normalization, is attributed repository mathematics, not a new cubic identity.

D5/S3/Arith/ModularForms/CubicThetaRootProduct.jacobi_cubic_root_product proves the specialization J(w;X)=(1-w)E(X^3) under w^2+w+1=0, by exact finite factorization followed by a coefficientwise limit. It does not by itself identify the Euler quotient with the core lattice.

D5/S3/Arith/ModularForms/CubicThetaJacobiCube.hasSum_jacobi_cube proves local finiteness and the full integer-triple sum for the cube of the actual Jacobi product. The complete pair sums for each fixed Laurent index are recorded by D5/S3/Arith/ModularForms/CubicThetaJacobiColumns.hasSum_cube_column. The anchors cube_column_zero and cube_column_one identify its first two columns with A and -3*T_lattice, respectively. The three anchors in D5/S3/Arith/ModularForms/CubicThetaCubeDissection, cube_column_three_mul, cube_column_three_mul_add_one, and cube_column_three_mul_add_two, evaluate every integer Laurent column, including all negative indices, by full affine lattice reindexing.

D5/S3/Arith/ModularForms/CubicThetaResidueSums.hasSum_residue_zero uses a complete sign-reversing involution. Its hasSum_residue_one and hasSum_residue_two are the two forms of Euler’s pentagonal series. D5/S3/Arith/ModularForms/CubicThetaRootEvaluation.evaluated_jacobi_cube combines all columns, with finite Laurent support justifying evaluation.

D5/S3/Arith/ModularForms/CubicThetaCoreProduct.core_product_eq_theta proves the actual identity E(X^3)^3/E(X)=T_lattice using unit inverses. Its coeff_core_product counts the full finite fibre 3*(x^2+x*y+y^2)-2*x-y=n for every natural index, including zero. The proof compares the complete cube evaluation with the actual Jacobi finite-product limit in the rational quadratic algebra w^2+w+1=0, then cancels a proven unit. This is a formal proof of the attributed classical three-core identity, not an assumed theta identity or a finite coefficient test.

D5/S3/Arith/ModularForms/CubicThetaProductTwist.three_jacobi_product proves the product of three root-twisted Jacobi factors by finite factorization and coefficientwise limits, without division. D5/S3/Arith/ModularForms/CubicThetaWeightedNorm.weighted_norm_dissection counts the full phase-weighted norm fibres using the same six core cosets. The complete Laurent constant column is identified by D5/S3/Arith/ModularForms/CubicThetaTwistedColumns.twisted_column_zero. Together these prove D5/S3/Arith/ModularForms/CubicThetaWeightedProduct.norm_dissection_mul_euler.

D5/S3/Arith/ModularForms/CubicThetaIdentities.B_dissection proves B=A(X^3)-3*X*T(X^3) for the actual quotient B=E(X)^3/E(X^3). Its cubic_identity proves A^3=B^3+27*X*T^3. The cubic defect satisfies a homogeneous recurrence; the generic coefficient-induction theorem D5/S3/Combinatorics/PowerSeries/ExpandingRecurrence.eq_zero_of_mul_eq_X_expand closes every degree. Thus all three classical identities are derived from product and lattice mathematics, with no theta identity assumed.

D5/S3/Arith/ModularForms/EtaSignK6.first_dissection and D5/S3/Arith/ModularForms/EtaSignK6.second_dissection prove the two finite polynomial expansions for the actual normalized k6 quotient. The all-index strict signs are in D5/S3/Arith/ModularForms/EtaSignK6.all_index_signs. The nonzero-modulo-three argument and general reduction are reused mathematics from lem9; the second dissection completes the selected k6 table entry. No claim of literature priority, worldwide absence, unique status, independent formal acceptance, CI or publication follows from these formal anchors.

The printed progressions in the proof of lem9 omit a factor -9 in residue one and retain an undilated denominator after extraction. They are not used as stated. The formal first dissection retains the correct factors -9 and 27 and uses the rescaled reciprocal 1/E(X)^3.

Citation and implementation scope only; no source text or PDF is vendored.