Formalizing the Prime-Field Singer Construction and Sidon Set Infrastructure in Lean 4

ErdH{o}s Problem 30 asks for sharp asymptotics of the Sidon extremal function \(h(N)\), and Singer’s construction is the classical source of lower-bound examples matching the main term. We present a Lean 4 formalization of Singer’s Sidon set construction for prime fields, together with reusable Sidon-set infrastructure for additive combinatorics. For every prime \(p\), we prove the existence of a Sidon set modulo \(p^2+p+1\) of cardinality \(p+1\). The proof proceeds through a non-trivial algebraic chain: construction of the Galois field \(mathrm{GF}(p^3)\), analysis of the trace kernel as a 2-dimensional subspace, a geometric argument via subspace intersections establishing the multiplicative Sidon property in the quotient group, and a combinatorial bridge transferring this to modular integer arithmetic. Around this central result, we develop a reusable Sidon set library for additive combinatorics. It comprises interval Sidon sets, modular Sidon sets, the extremal function \(h(N)\), Lindstrom’s cross-difference inequality, a Johnson-route shift-incidence upper bound of the form \(h(N) leq sqrt{N} + N^{1/4} + O(1)\), exact representation-function identities, and unconditional two-sided \(h(N)=Theta(sqrt{N})\) bounds with exact floor-rounded finite statements for \(N geq 5\). We further formalize a conditional reduction: subpolynomial prime gaps together with a full subpolynomial upper-error hypothesis for \(h(N)\) imply the ErdH{o}s Problem 30 estimate \(h(N)=sqrt{N}+O_varepsilon(N^varepsilon)\) for every \(varepsilon>0\). The core Singer/Sidon and transfer development comprises 6,382 lines of Lean 4 with zero active uses of sorry. We describe the mathematical lessons learned, focusing on how formalization clarifies the precise scope of classical arguments and forces explicit treatment of the algebraic-combinatorial interface.