3. Lexical Half-Life

Data & measurement

Continuity metrics emphasize intergenerational transmission, corpus completeness, and structural stability. Keep cohort definitions fixed across measurement waves. Symbols: t (Half-life time (in years or generations)); c (Decay constant (annual retention rate, 0-1)); ln (Natural logarithm). Log instrument versions, sample frames, and cleaning rules whenever estimates are refreshed so longitudinal comparisons stay valid.

Solution & proof

Conceptual summary: How long it takes for half of a set of native Kifuliiru words to fall out of active use. This predicts when vocabulary might "expire" if current trends continue. Treat this as a measurement recipe: map each symbol to an empirical quantity, substitute estimates, and simplify with ordinary algebra (including logarithms, min/max caps, or piecewise branches where shown). Where limits or integrals appear, approximate with discrete sums on cohorts or time steps when closed forms are impractical. Interpret the result against thresholds in the cited source and report uncertainty on inputs.

Examples

1. 1. Word problem — field log

   Word problem

   A measurement round produced the table below. Each symbol matches the Variables section for “Lexical Half-Life”. Find the computed value using the formula above.

   Illustrative inputs—replace with your field numbers

   Quantity	Symbol	Value
   Half-life time (in years or generations)	t	2
   Decay constant (annual retention rate, 0-1)	c	0.63
   Natural logarithm	ln	—

   Solution

   Step 1 — From the table, assign t = 2; c = 0.63.

   Step 2 — Substitute into the definition of “Lexical Half-Life” at the top of the page (same structure as the source formula).

   Step 3 — Finish any remaining algebra (sums, caps, logs, or limits) by hand or in a CAS when the expression still contains Σ, integrals, or non-numeric parameters. Tie final numbers back to your corpus or community records.

   $$\text{Given: }t=2,c=0.63$$

   $$t=\frac{\ln (0.5)}{\ln (c)}$$

2. 2. Second scenario — adjusted inputs

   Word problem

   The same measure applies with corrected inputs for the next report (see table). Recompute using the same formula.

   Revised values for the next report (illustrative)

   Quantity	Symbol	Value
   Half-life time (in years or generations)	t	2
   Decay constant (annual retention rate, 0-1)	c	0.63
   Natural logarithm	ln	—

   Solution

   Step 1 — From the table, assign t = 2; c = 0.63.

   Step 2 — Substitute into the definition of “Lexical Half-Life” at the top of the page (same structure as the source formula).

   Step 3 — Finish any remaining algebra (sums, caps, logs, or limits) by hand or in a CAS when the expression still contains Σ, integrals, or non-numeric parameters. Tie final numbers back to your corpus or community records.

   $$\text{Given: }t=2,c=0.63$$

   $$t=\frac{\ln (0.5)}{\ln (c)}$$

Variables

Source

ALL_FORMULAS.md — Continuity

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