rateAtRung
plain-language theorem explainer
Replacement rate at rung k on the phi-ladder equals referenceRate times phi to the power k. Linguists modeling Swadesh list decay would cite this to quantify how replacement accelerates across word categories. It is defined directly as an algebraic product without further computation.
Claim. The replacement rate at rung $k$ on the phi-ladder is $r_0 phi^k$, where $r_0$ is the reference core-vocabulary replacement rate.
background
The module places Swadesh-list vocabulary on a phi-ladder, with core terms at rung 0 and peripheral terms at higher rungs. The upstream referenceRate is defined as 1 in RS-native units to serve as the baseline for the most stable words. This follows the Recognition Science prediction that adjacent rung ratios equal phi, matching empirical factors of 5-10 for two-rung steps observed by Pagel et al.
proof idea
The declaration is a direct definition multiplying the reference rate by the k-th power of phi. It relies only on the definition of referenceRate and the exponentiation operation in the reals.
why it matters
This definition supplies the rate function used by downstream results including rateAtRung_pos, rateAtRung_succ_ratio, rate_adjacent_ratio, and the SwadeshDecayCert structure. It implements the module's structural prediction that rung steps multiply the rate by phi, aligning with T6 where phi is the self-similar fixed point.
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