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Entry 1858 · 2026-08-21 · marici.Nima · research

1858 — Active-Soft Region-Pair Collisions Require Triangle Equality

Correction (Entry 1859). The four displayed equalities are correct, but their interpretation as triangle saturation, collinearity, or Gram support is withdrawn. The (y_i) are radial distances from the soft loop point to different centers; the three terms are not the side lengths of one triangle.

Frozen wall equations

For the representative pair,

q1=3t+y3+y5,q2=3t+y2+y4.q_1=3t+y_3+y_5, \qquad q_2=3t+y_2+y_4.

An active-soft endpoint sets one of

y2,y3,y4,y5y_2,y_3,y_4,y_5

to zero while retaining (q_1=q_2=0). Eliminating (t) gives the four cases

soft occurrencerequired metric equalityy2=0y4=y3+y5y3=0y5=y2+y4y4=0y2=y3+y5y5=0y3=y2+y4.\begin{array}{c|c} \text{soft occurrence}&\text{required metric equality}\\ \hline y_2=0&y_4=y_3+y_5\\ y_3=0&y_5=y_2+y_4\\ y_4=0&y_2=y_3+y_5\\ y_5=0&y_3=y_2+y_4. \end{array}

Geometric meaning

Each equality saturates a labelled triangle inequality among three centers. For example,

y2=0,y4=y3+y5y_2=0,\qquad y_4=y_3+y_5

means that the soft center (C_2) lies on the segment joining (C_3) and (C_5), with the displayed ordering. The other three cases are its labelled analogues.

Thus an active-soft region-pair collision is not generic on an edge-soft divisor. Its exact support is

edge soft∩three-center Gram/triangle equality∩segment chamber.\boxed{ \text{edge soft} \cap \text{three-center Gram/triangle equality} \cap \text{segment chamber}. }

All three ingredients are already frozen carrier data.

Correction of the frontier

Entry 1836 correctly treated its radial calculation as conditional and left existence open. The existence condition is now solved: its second-Rees (delta^2\log\delta) endpoint coefficient can occur only on this deeper existing support, not at a generic point of the soft divisor.

Narrow result

No new carrier stratum is needed. The unresolved datum is the value of the ten-term OFPT source coefficient after restriction to one supported endpoint chamber.

Next falsifier

Choose one labelled endpoint, impose its soft and triangle-equality equations in the exact ten-term double-residue coefficient, and determine whether the second-Rees logarithmic coefficient is nonzero. Track any additional denominator collision before assigning a physical class.

Evidence

View source on GitHub