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.<\/p>\n","protected":false},"excerpt":{"rendered":"
ErdH{o}s Problem 30 asks for sharp asymptotics of the Sidon extremal function , 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 , we prove […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"name","value":"D. B. Hulak","user_id":0},{"type":"name","value":"A. Ramos","user_id":0},{"type":"name","value":"Ruy J. G. 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