IndisputableMonolith.NumberTheory.Primes.Resonance
The Resonance module in NumberTheory.Primes introduces the Cicada Lemma on prime synchronization. Researchers working on periodic encounters or resonance phenomena in number theory would cite it for its direct LCM identity. The module imports the Basic primes layer and supplies the lemma as a building block with no internal proof obligations.
claimIf $p$ is prime and $p$ does not divide $k$, then $lcm(p,k)=p k$.
background
The module imports Mathlib and the Basic primes module, whose design goals are reuse of Nat.Prime, axiom-free and sorry-free code, and small sanity theorems that confirm correct wiring before analytic extensions. It introduces the Cicada Lemma, which states that a prime maximizes the interval to first synchronization with any non-multiple integer k. The local setting is algebraic number theory restricted to primes, positioned for use in larger Recognition Science constructions.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module is imported by the Primes aggregator, which downstream code prefers over individual files. It supplies the Cicada Lemma to the prime foundations, filling the synchronization-maximization step that supports resonance arguments in the Recognition framework.
scope and limits
- Does not treat composite numbers or their LCM behavior.
- Does not include numerical examples or computations.
- Does not link to physical constants, phi-ladder, or forcing chain.
- Does not address prime density or infinitude results.