halfLife
The scaling function assigns to each natural number rung k the value of the golden ratio raised to k. Quantitative linguists modeling lexical persistence in corpora would cite this to derive the observed ratio of high- versus low-frequency word lifetimes. The definition is a direct power expression with no further computation.
claimThe half-life at rung $k$ is given by $phi^k$, where $phi$ is the golden ratio.
background
In the module on lexical decay from the phi-ladder, word longevity scales with the recognition frequency rung k. The scaling function provides the factor $phi^k$, with base lifetime normalized to unity in RS units. This follows the Recognition Science derivation that high-frequency words persist $phi$-times longer per rung on the ladder.
proof idea
The declaration is a direct definition that sets the scaling at rung k to the golden ratio raised to the power k. No lemmas or tactics are applied.
why it matters in Recognition Science
This definition supplies the scaling law that enables the consecutive-rung ratio theorem establishing that half-lives differ by exactly the factor phi and the lexical decay certification structure bounding phi to the fifth between 10 and 12. It realizes the RS derivation of lexical persistence on the phi-ladder, connecting corpus observations to the self-similar fixed point of the J-function and the eight-tick octave. The module states that this scaling matches the 6000-year to 750-year half-life ratio within a factor of 1.4.
scope and limits
- Does not fix a specific base lifetime value.
- Does not incorporate actual corpus frequency statistics.
- Does not derive the numerical half-life values of 6000 or 750 years.
- Does not address deviations from pure exponential scaling.
Lean usage
theorem consecutive_ratio (k : ℕ) : halfLife (k + 1) / halfLife k = phi := by unfold halfLife; rw [pow_succ, div_eq_iff (pow_pos phi_pos k).ne']; ringformal statement (Lean)
27noncomputable def halfLife (k : ℕ) : ℝ := phi ^ k
proof body
Definition body.
28