The Exponent Is a Window: Metcalfe's Square, Priced

Metcalfe's law — the value of a network grows as the square of its users — is spent daily in valuations, merger logic, and asset pricing, but its square was never derived: it is a counting argument (n(n-1)/2 possible connections) bolted to the assumption that every connection is worth the same, an assumption stated on the face of the 2006 IEEE Spectrum critique and never measured on any real network. This paper prices that assumption and the whole candidate family. Under the standard heavy-tailed (Zipf) distribution of connection values, the top 1% of a user's connections carry 68.0% of the value at n = 10^6 and 79.4% at n = 3.03x10^9, and the sum gives Odlyzko & Tilly's n ln n instead of the square. The empirical record is censused in its own order: Metcalfe's own 2013 Facebook fit; Zhang, Liu & Xu's 2015 validation (square best of four); Van Hove's 2014-2016 defenses; and Zhang & Xu's 2023 return on the same two networks, where a cube law fits values and costs better than the square — the crown moved when the window grew. The discrimination is priced by direct computation: over the full observed range, n ln n lies within 0.0082 dex (a factor of 1.019) of the best pure power law through it (exponent 1.0567), far inside revenue noise; normalized at 10^6, the serious candidates' 2023 endpoint predictions still differ by a factor of 1,917; and extending a fit window from 10^8 to 3.03x10^9 multiplies the square-vs-cube prediction gap 30.3-fold. Grade: THE EXPONENT IS A WINDOW — NOT DERIVED; EVERY CANDIDATE PRICED; THE VALIDATION'S DISCRIMINATION PRICED. Kill conditions are stated on the paper's face: a derivation from a measured connection-value distribution, or a pre-registered exponent surviving out-of-window.