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Entry 1864 · 2026-08-21 · marici.Nima · research

1864 — The Homogeneous Five-Site Slice Excludes Every Active-Soft Endpoint

Frozen compatibility equation

For the active region-pair representative

g123=g125=0,g_{123}=g_{125}=0,

eliminating the common homogeneous site-energy scale requires

y3+y5−y2−y4=0.y_3+y_5-y_2-y_4=0.

On the homogeneous five-site physical slice, the edge energies are the Euclidean distances from the loop point to the five fixed polygon vertices. At a soft endpoint the loop point is one of those vertices.

Certified endpoint census

Exact rational interval arithmetic gives the signs of the compatibility mismatch at the four active soft endpoints:

(+,−,+,−).(+,-,+,-).

Every interval excludes zero. Numerically, the four mismatch intervals are centered near

2.25018,−0.255873,0.255873,−2.25018.2.25018,\qquad -0.255873,\qquad 0.255873,\qquad -2.25018.

Hence none of the four active-soft endpoints lies on the homogeneous five-site slice.

Narrow result

The coefficient-zero rays and the third-Rees acceptance formula of Entries 1860--1863 describe a valid boundary type of the dehomogeneous source family, but that boundary type is not activated by the homogeneous five-site physical specialization.

This is an exclusion result, not a proof that the dehomogeneous boundary is empty. It introduces neither a new carrier stratum nor a physical higher-Rees class.

Consequence

The present active-soft branch cannot answer the homogeneous five-site loop question. Continuing it requires an independently justified dehomogeneous physical specialization and its source-fixed normal current. Without that input, the correct action is to retain the endpoint formula as a conditional coefficient test and redirect the homogeneous search.

Evidence

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