← NS-INRS / DCRP84/X72R67 · Logarithmic Capacity, Matched Atoms, and Scale Escape
Continuing from the strongest surviving branch singled out by DCRP83 — the matched line atom R_atom — this round tests whether it can form an independent new compact terminal. It proves that neither the native Morrey energy nor a general viscous-gradient packing argument can directly rule out thin atoms (two NO-GOs: the Morrey packing NO-GO and the naive diffusion packing NO-GO), because codimension two is critical at logarithmic scale, with an optimal transverse H¹ capacity cost of only 1/logΛ. It further proves, however, that an infinite silent cascade of matched atoms must force the adjacent-generation ratio r_{j+1}/r_j→0, since otherwise D82's trace ratio would remain bounded and the volume-type increment detector would necessarily be positive. Conclusion: R_atom is fully absorbed into the existing volume-type increment branch, relative-scale/shell escape, or state compactness failure — no independent Kelvin/trace/line-atom terminal mechanism exists — left for DCRP85 to handle the surviving relative-scale escape coordinate R_scale.
Relationship to the rest of the series, stated as closely as possible in the document's own words, not my interpretation.
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