IndisputableMonolith.Information.PolarCodeGapFromPhi
The PolarCodeGapFromPhi module defines referenceGap as the polar code gap at rung 0 for minimal block length together with gapAt and its ratio and positivity properties. Researchers working on information bounds inside Recognition Science would cite these objects when placing code performance on the phi-ladder. The module consists entirely of definitions and elementary lemmas that import only the Constants module.
claimLet $ ext{referenceGap}$ be the reference polar code gap at rung 0. For integer rung $r \ge 0$ the gap function $ ext{gapAt}(r)$ satisfies the recurrence and ratio relations induced by the phi fixed point, with $ ext{gapAt_pos}$ asserting positivity and $ ext{gapAt_succ_ratio}$, $ ext{gapAt_adjacent_ratio}$ giving the successive and adjacent ratios.
background
The module sits in the Information domain and imports only Constants, whose sole documented content is the RS-native time quantum $ au_0 = 1$ tick. All sibling declarations (referenceGap, gapAt, gapAt_pos, gapAt_succ_ratio, gapAt_adjacent_ratio, PolarCodeCert, polarCodeCert) are therefore expressed directly in terms of this unit and the phi-ladder already fixed by the upstream forcing chain.
No additional notation or external constants are introduced; the module therefore supplies the base layer for any later polar-code constructions that remain inside RS-native units.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The declarations supply the concrete gap values required by any downstream information-theoretic result that invokes polar codes inside the Recognition framework. They sit immediately after the phi self-similar fixed point (T6) and the eight-tick octave (T7) and therefore constitute the first explicit information-domain objects built on those landmarks. No used_by edges are recorded yet.
scope and limits
- Does not derive the gap expressions from the J-uniqueness equation.
- Does not evaluate numerical gap values or compare them with classical polar-code bounds.
- Does not connect the gap to mass formulas, the alpha band, or spatial dimension D=3.