Formalize Fourier bounds for Shunia's integer-root conjecture

[rbajaj5]

Summary

• add the arithmetic data and reduced coefficient encoding for Shunia's integer-root formula
• prove the quotient-ring modular evaluation identity
• formalize the discrete Fourier transform and inverse-transform formulas on ZMod n
• prove a uniform chordal separation bound for nontrivial roots of unity
• derive spectral and exponential decay estimates, including Taylor-series bounds for 6 ≤ n

Why

This is a collaboration checkpoint toward a formal proof of Conjecture 6.1 from Joseph M. Shunia, Polynomial quotient rings and Kronecker substitution for deriving combinatorial identities.

Current status

This is intentionally a draft. The new file compiles and contains no sorry, admit, or added axioms, but it does not yet state or prove the final public theorem. Remaining work includes the small cases n = 2, 3, 4, 5, coefficient-ratio estimates, the no-wrap Kronecker argument, and the final Nat.nthRoot endgame. The module is not yet imported by Mathlib.lean, so it has no downstream user impact.

Validation

lake env lean Mathlib/NumberTheory/Shunia.lean

The command succeeds against the current mathlib toolchain.

[fqlx]

sweet!