← NS-INRS / DCRP84/X72R67 · Logarithmic Capacity, Matched Atoms, and Scale Escape

NS-INRS · DCRP84/X72R67 Gap Closure 2026-08

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.

This round's status: Gap Closure — Taken from the file's own Executive result / STOP node, meaning preserved, not a full verbatim translation. The original paper itself was authored directly in English; only this page's own commentary (title, status, and summary) is translated here.

Connections

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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