maxSimpsonDiversity_eq
plain-language theorem explainer
The equality establishes that the maximum Simpson diversity equals 4/5 for configuration dimension 5, corresponding to equal biomass shares among the five canonical ecological guilds. Researchers modeling ecosystem health under Recognition Science would reference this to set the optimum benchmark for the Simpson index. The proof proceeds as a direct one-line wrapper that unfolds the definition and normalizes the resulting arithmetic expression.
Claim. The maximum value of the Simpson diversity index $D_s = 1 - sum p_i^2$ equals $4/5$ when the configuration dimension equals 5.
background
In the Biodiversity Indices from ConfigDim module the Simpson diversity index is defined as $D_s = 1 - sum p_i^2$ with $p_i$ the biomass proportions across ecological guilds. The configuration dimension is fixed at 5 to match the five guilds (producers, primary consumers, secondary consumers, decomposers, detritivores). The upstream definition maxSimpsonDiversity sets the symbolic maximum to $1 - 1/5$, which the theorem converts to the numerical value 4/5. The module states that Shannon diversity reaches its maximum of log(5) under equal biomass while Simpson reaches 4/5 at the same point.
proof idea
The proof is a one-line wrapper that unfolds the definition of maxSimpsonDiversity and applies norm_num to evaluate the arithmetic expression 1 - 1/5 directly to 4/5.
why it matters
This result supplies the concrete numerical value for the maximum Simpson diversity used in the biodiversityCert definition, which certifies the guild count and the Simpson maximum for ecosystem assessment. It fills the RS prediction that the healthy ecosystem optimum for Simpson diversity is 4/5 when configDim equals 5. The declaration closes the arithmetic step between the symbolic definition and its decimal equivalent in the ecology depth of the framework.
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