IndisputableMonolith.Information.ShannonHighNLimit
This module establishes the high-N asymptotic for the RS J-cost correction to Shannon entropy by proving the inner argument tends to 1. Information theorists recovering classical channel capacity from the Recognition framework would cite it to justify the exact log2 N limit. The argument proceeds via a chain of elementary real-analysis limit lemmas on the term 1 + 1/(phi N) and its corrections.
claimAs $N → ∞$, the inner argument $1 + 1/(φ N)$ tends to 1, so that the RS J-cost on an N-message ensemble recovers the classical Shannon capacity $C = log₂ N$ with vanishing correction.
background
Recognition Science encodes information costs via the J-cost function J(x) = (x + x^{-1})/2 - 1, whose high-N behavior on message ensembles is controlled by the golden-ratio fixed point φ. The upstream module ShannonAsJCostLimit shows that classical Shannon capacity C = log₂ N arises as the N → ∞ limit of this J-cost, but only after a 1/φ-rational correction vanishes. Constants supplies the RS time quantum τ₀ = 1 tick that normalizes the underlying discrete ticks. This module supplies the supporting limit statements that close the high-N argument.
proof idea
The module structures its argument as a sequence of tendsto lemmas: inner_arg_tendsto_one evaluates the direct limit of 1 + 1/(φ N), correction_RS_tendsto_zero shows the RS correction vanishes, and the remaining lemmas establish strict positivity and anti-monotonicity of the correction. All steps apply standard Mathlib real-analysis tactics for limits of rational functions.
why it matters in Recognition Science
This module completes the high-N limit step in the ShannonAsJCostLimit track, allowing the classical channel capacity to be recovered exactly once N is taken to infinity. It feeds the broader information-theoretic results of the Recognition monolith by justifying the disappearance of all RS-specific corrections at large N. The parent result is the full statement that Shannon entropy equals the J-cost limit in the infinite-message regime.
scope and limits
- Does not compute explicit finite-N error bounds.
- Does not treat continuous or quantum channels.
- Does not derive the full Shannon capacity formula from scratch.
- Does not address multi-user or network information settings.