IndisputableMonolith.Cost.FrequencyLadder
The module Cost.FrequencyLadder defines the J-cost applied to frequency ratios r = f2/f1. Researchers building the phi-ladder and harmonic structures in RS-native units cite these definitions when computing costs for self-similar ratios. It supplies frequencyRatioCost together with unit and nonnegativity properties as direct specializations of the core J function.
claimFor frequency ratio $r = f_2/f_1$, define frequencyRatioCost$(r) := J(r)$ where $J(x) = (x + x^{-1})/2 - 1$. The cost satisfies frequencyRatioCost$(r) = 0$ if and only if $r = 1$, and frequencyRatioCost$(r) = J(r) = J(1/r)$.
background
Recognition Science derives all physics from the single functional equation encoded in the J-cost. Constants fixes the fundamental time quantum as τ₀ = 1 tick in RS-native units. JcostCore supplies the core J function obeying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). FrequencyLadder specializes this cost to ratios of frequencies to construct the frequency ladder that feeds the self-similar fixed point.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
FrequencyLadder supplies the frequency cost measure that feeds phi_is_self_similar, phi_unique_self_similar, phi_cost_fixed_point and PhiHarmonicForced. These results close the T6 step of the forcing chain by establishing phi as the unique self-similar fixed point of the J-cost on ratios.
scope and limits
- Does not compute explicit numerical costs for concrete frequency values.
- Does not extend the ratio cost to amplitude or phase variables.
- Does not derive the spatial dimension D = 3 or the alpha band.
- Does not address time-dependent or non-harmonic frequency shifts.