popularity_peak
plain-language theorem explainer
The theorem establishes that style popularity reaches its maximum of 1 exactly at scaled time s=1, where J-cost vanishes. Archaeologists reconstructing Petrie curves and recognition theorists deriving serial succession from J-cost would cite this anchor point. The proof is a direct term reduction that unfolds the popularity definition, discards the non-positive branch by norm_num, and evaluates Jcost(1) to zero.
Claim. At scaled time $s=1$, the popularity function satisfies popularity$(1)=1$, since $J(1)=0$ and thus popularity$(s)=1/(1+J(s))$ evaluates to unity.
background
The module models ceramic style succession as the J-cost trajectory of popularity on a one-dimensional design graph. Popularity is defined for $s>0$ by popularity $s = 1/(1 + $Cost.Jcost$ s)$, with the zero case returning 0; this follows the cost induced by a multiplicative recognizer and the J-cost of any recognition event. The upstream popularity definition states: 'Per-style popularity at scaled time $s = t/τ$: popularity $s = 1 / (1 + J(s))$ for $s > 0$.' The local setting derives the empirical Petrie curve shape from neighboring J-cost minima separated by $J(φ)≈0.118$.
proof idea
The proof is a one-line term wrapper. It unfolds popularity, rewrites the conditional using norm_num to confirm $1>0$, then unfolds Cost.Jcost and applies norm_num to obtain $1/(1+0)=1$.
why it matters
This supplies the peak value required by the PotterySerialCert structure and the pottery_serial_one_statement theorem, which certify boundedness in $[0,1]$ together with the adjacency gap in $(0.11,0.13)$. It fills the canonical peak step in the derivation of serial succession from the Recognition Composition Law and J-uniqueness (T5). The result anchors the forcing chain at the point where J-cost reaches its global minimum.
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