IndisputableMonolith.Information.PolarCodeGapFromPhi
The module PolarCodeGapFromPhi defines reference polar code gaps derived from the phi self-similar fixed point at rung 0 for minimal block length. It supplies gapAt together with positivity and ratio lemmas for rung-dependent gaps. Information theorists applying Recognition Science to coding cite these definitions when computing phi-ladder gaps. The module is purely definitional and builds directly on the Constants import.
claimreferenceGap denotes the polar code gap at rung 0; gapAt : ℕ → ℝ computes the gap at rung r, satisfying gapAt_pos (gapAt r > 0) and gapAt_succ_ratio (gapAt (r+1) / gapAt r = constant derived from phi).
background
Recognition Science places information theory on the phi-ladder with gaps measured in RS-native units. The module imports Constants, where τ₀ = 1 tick is the fundamental time quantum. It introduces referenceGap as the base gap at rung 0 (minimal block length) and gapAt for successive rungs, together with adjacent-ratio and successor-ratio properties.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the gap primitives required by PolarCodeCert and polarCodeCert in the Information domain. It fills the rung-0 reference needed for phi-derived coding bounds and connects to the broader RS forcing chain via the phi fixed point.
scope and limits
- Does not prove optimality of the resulting polar codes.
- Does not compute explicit block lengths or error rates.
- Does not extend beyond rung-dependent gap ratios.
- Does not incorporate physical constants other than phi.