What question did this study set out to answer?

To establish exact algebraic foundations, modular lifts, and machine-verified bounds for the cyclotomic values governing Ramanujan's third-order mock theta function.

August 16, 2026Open Access

CRT-Theta Lifts and Toric Resolvents for Cyclotomic Mock-Theta Values: Exact Foundations for a Hecke-Projected Arithmetic Detector

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JIJoesph D. Burke IIIDallas Independent School District

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Implication

Theoretical analysis reveals exact algebraic structures of cyclotomic mock-theta values with Lean 4 proofs, establishing unconditional bounds for quadratic Galois resolvents.

Key Points

To establish exact algebraic foundations, modular lifts, and machine-verified bounds for the cyclotomic values governing Ramanujan's third-order mock theta function.
Reformulated cyclotomic values as centred Chinese Remainder Theorem-weighted unary-theta packets with explicit chamber rules for winding numbers.
Constructed level-39 theta packets, Shimura lifts, local gates, Hecke systems, and an analytic detector via simultaneous all-cusp truncation.
Formalized and machine-checked 59 mathematical statements in Lean 4 against Mathlib alongside multi-script exact arithmetic verification.
Derived exact formulas for power-basis coefficients, histograms, constant terms, and traces of cyclotomic mock-theta values.
Obtained the first unconditional bound on the integer invariant Mp by identifying its character sum as a Kloosterman sum with dominant Dirichlet character size √p.
Formally verified all 59 core statements without unproven axioms or native decision procedures in Lean 4.

Cite This Study

Joesph D. Burke III (2026) studied this question.

synapsesocial.com/papers/6a817a11f2fb91fc834ad990 https://doi.org/10.5281/zenodo.21933676

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Implication

Theoretical analysis reveals exact algebraic structures of cyclotomic mock-theta values with Lean 4 proofs, establishing unconditional bounds for quadratic Galois resolvents.

CRT-Theta Lifts and Toric Resolvents for Cyclotomic Mock-Theta Values: Exact Foundations for a Hecke-Projected Arithmetic Detector

Discussion

Member takes

Implication

Key Points

Cite This Study

CRT-Theta Lifts and Toric Resolvents for Cyclotomic Mock-Theta Values: Exact Foundations for a Hecke-Projected Arithmetic Detector

Discussion

Member takes

Implication

Key Points

Cite This Study

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