The Measure Chosen by Its Requirements: Shannon's Entropy and the Price of the Grouping Property

In 1948 Claude Shannon founded information theory on a measure: the uncertainty of a source is H = -\sum p_i \log p_i, reached by stating three requirements on any such measure — continuity in the probabilities, monotonic growth for equally likely choices, and the grouping property, that a choice broken into successive choices weighs its parts by their probabilities — and showing the logarithmic form is the only one satisfying them. The requirements are stated in the paper as what a measure should do; they are not derived from anything beneath them. This paper verifies the Shannon form against its own requirements on declared distributions (the grouping identity closes with residual +0.000000 on both declared cases), then weakens the load-bearing requirement — grouping — to additivity for independent systems only, and computes the consequence: the one-parameter Rényi family walks in, and on the declared skewed source (0.7, 0.2, 0.1) its members spread across 1.070389 bits around a Shannon value of 1.156780 bits, while failing the grouping identity itself by -0.081460 bits at \alpha = 2. The entropy, and the coding theorems built on it, are the 1948 paper's derivations and stand entire. What is counted here is the derivation status of the selecting step: 1 postulate set, 0 free numbers, 0 derivations in 1948 of the requirements — the measure of information is exactly as forced as its requirements are. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.