practical_two
plain-language theorem explainer
The inequality σ_1(2) ≥ 2 holds, with the divisor sum equaling 3, confirming a base verification for the number 2 under the practical criterion. Number theorists working on divisor sums and practical integers would cite this as an initial case. The proof reduces to direct numerical evaluation via a native decision tactic.
Claim. Let σ_1(n) be the sum of positive divisors of n. Then σ_1(2) ≥ 2.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function and extending to the sum-of-divisors function. The sigma abbreviation maps each k to ArithmeticFunction.sigma k, so sigma 1 denotes σ_1. The local setting keeps statements basic to stabilize interfaces before Dirichlet algebra layers. Upstream results include the sigma definition as the sum-of-divisors function.
proof idea
The proof is a one-line wrapper that invokes native_decide to evaluate the concrete inequality by computation, confirming the divisor sum of 2 equals 3.
why it matters
This supplies a verified base case inside the arithmetic functions module for divisor-sum properties. It supports the Recognition Science monolith through foundational number theory without engaging the forcing chain T0-T8, RCL, or phi-ladder directly. No downstream theorems are listed.
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