correction
plain-language theorem explainer
The φ-ladder supplies a finite-N correction factor for quantum channel capacity equal to 1/(φ N) at positive integer input-symbol count N. Information theorists deriving entanglement-assisted capacity bounds within the Recognition Science framework cite this factor when comparing classical and quantum limits. The definition is a direct algebraic assignment with no additional lemmas or tactics.
Claim. The finite-N correction factor for quantum channel capacity is given by $c(N) = 1/ (φ N)$ for positive integers $N$, where $φ$ is the golden ratio.
background
In the Recognition Science framework the classical Shannon capacity is $C = log_2(1 + S/N)$. The quantum analog incorporates an RS finite-N correction that scales as $log_2(1 + 1/(φ N))$ per input symbol, the same φ-suppressed term already present in the classical Shannon bound. The module states that the entanglement-assisted-to-classical capacity ratio for an N-symbol block channel is therefore 1 + 1/(φ N), with adjacent-N ratios differing by (N+1)/N · 1/φ to leading order and the correction vanishing as 1/N rather than 1/N².
proof idea
The definition is a direct one-line algebraic expression that multiplies the golden ratio phi by the natural number N and takes the reciprocal.
why it matters
This definition supplies the correction term used in higher-order constants such as the fine-structure constant seed and curvature space derivations. It appears in downstream results including water bond angle predictions and alpha higher-order terms. The factor connects to the eight-tick octave and D=3 spatial dimensions through the phi-ladder, filling the finite-N gap in the quantum channel capacity analysis. It touches the open question of distinguishing RS predictions from classical-only models via the 1/N versus 1/N² decay.
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