IndisputableMonolith.Information.ShannonHighNLimit
The ShannonHighNLimit module establishes the high-N asymptotic behavior required to recover classical Shannon entropy from the J-cost formulation. It proves that the inner argument 1 + 1/(φ N) tends to 1 and that the RS correction to classical capacity vanishes as N → ∞. Researchers working on the information track of Recognition Science cite these limit statements to justify the classical channel capacity C = log₂ N. The proofs are direct applications of real-analysis tendsto lemmas from Mathlib to the phi-dependent expressions.
claim$\lim_{N\to\infty}\left(1 + \frac{1}{\phi N}\right)=1$ and the RS correction term to the classical Shannon capacity $C=\log_2 N$ tends to zero.
background
The module sits in the Information domain and imports the RS time quantum τ₀ = 1 tick from Constants, the J-cost definition from Cost, and the base ShannonAsJCostLimit result. The upstream module states: "The classical Shannon channel capacity C = log₂ N is recovered as the high-N limit of J-cost on the message ensemble. At finite N the RS prediction has a 1/φ-rational correction." It supplies the concrete limit theorems that make the high-N recovery rigorous.
proof idea
The module consists of a collection of tendsto statements. Each is proved by standard real-analysis reductions: the fact that 1/N → 0 implies the inner argument tends to 1, followed by algebraic manipulation showing the correction terms vanish. No custom tactics beyond Mathlib limit lemmas are used.
why it matters in Recognition Science
This module completes the high-N limit step in the Shannon entropy as J-cost track, directly supporting the parent claim in ShannonAsJCostLimit that recovers the classical capacity. It closes the asymptotic consistency between the J-cost and Shannon formulations within the Recognition Science framework.
scope and limits
- Does not derive finite-N correction formulas.
- Does not address quantum or non-classical channels.
- Does not supply convergence-rate bounds.
- Does not connect to mass ladder or spatial dimensions.