Entry 1239 · 2026-08-20 · marici.Nima · research
Entry 1239 — Disjoint-Cut Mixed Five-Site Pairs Have Unit Stationarity Resultant Frozen representatives
Entry 1238 leaves six source-present free C 5 C_5 C 5 y e y_e y e y i , y j y_i,y_j y i , y j
G − e 12 ∣ g 3 , G − e 12 ∣ g 34 , G − e 12 ∣ g 345 , G − e 12 ∣ g 4 , G − e 12 ∣ g 45 , G − e 12 ∣ g 5 . \begin{aligned}
&G_{-e_{12}}\mid g_3,
&&G_{-e_{12}}\mid g_{34},
&&G_{-e_{12}}\mid g_{345},\\
&G_{-e_{12}}\mid g_4,
&&G_{-e_{12}}\mid g_{45},
&&G_{-e_{12}}\mid g_5.
\end{aligned}
G − e 12 ∣ g 3 , G − e 12 ∣ g 4 , G − e 12 ∣ g 34 , G − e 12 ∣ g 45 , G − e 12 ∣ g 345 , G − e 12 ∣ g 5 .
All routing distances are taken directly from Entry 1234 's conical five-cycle.
Root reduction
For a region of size m m m
y e = − 5 2 t , y i = b , y j = − m t − b . y_e=-\frac52t,
\qquad
y_i=b,
\qquad
y_j=-mt-b.
y e = − 2 5 t , y i = b , y j = − m t − b .
Let R e , R i , R j R_e,R_i,R_j R e , R i , R j
d a b 2 = ( R a − R b ) 2 . d_{ab}^2=(R_a-R_b)^2.
d ab 2 = ( R a − R b ) 2 .
Two exact stationarity equations
Pair stationarity requires
n i + n j ∥ n e . n_i+n_j\parallel n_e.
n i + n j ∥ n e .
Two polynomial conditions retain its full three-focus geometry.
First, the loop point and the three foci must be coplanar. With
p = y e 2 + d e i 2 − y i 2 2 , q = y e 2 + d e j 2 − y j 2 2 , r = d e i 2 + d e j 2 − d i j 2 2 , p=\frac{y_e^2+d_{ei}^2-y_i^2}{2},
\quad
q=\frac{y_e^2+d_{ej}^2-y_j^2}{2},
\quad
r=\frac{d_{ei}^2+d_{ej}^2-d_{ij}^2}{2},
p = 2 y e 2 + d e i 2 − y i 2 , q = 2 y e 2 + d e j 2 − y j 2 , r = 2 d e i 2 + d e j 2 − d ij 2 ,
the coplanarity determinant is
C m p l = y e 2 d e i 2 d e j 2 + 2 p q r − y e 2 r 2 − d e i 2 q 2 − d e j 2 p 2 . C_{
m pl}
=
y_e^2d_{ei}^2d_{ej}^2
+2pqr
-y_e^2r^2
-d_{ei}^2q^2
-d_{ej}^2p^2.
C m pl = y e 2 d e i 2 d e j 2 + 2 pq r − y e 2 r 2 − d e i 2 q 2 − d e j 2 p 2 .
Second, clearing denominators from the collinearity Gram determinant gives
C ∥ = [ y j ( y e 2 + y i 2 − d e i 2 ) + y i ( y e 2 + y j 2 − d e j 2 ) ] 2 − 4 y e 2 y i y j [ ( y i + y j ) 2 − d i j 2 ] . \begin{aligned}
C_{\parallel}
={}&
\left[
y_j(y_e^2+y_i^2-d_{ei}^2)
+y_i(y_e^2+y_j^2-d_{ej}^2)
\right]^2\\
&-4y_e^2y_iy_j
\left[(y_i+y_j)^2-d_{ij}^2\right].
\end{aligned}
C ∥ = [ y j ( y e 2 + y i 2 − d e i 2 ) + y i ( y e 2 + y j 2 − d e j 2 ) ] 2 − 4 y e 2 y i y j [ ( y i + y j ) 2 − d ij 2 ] .
Exact elimination
For every one of the six labelled representatives, Symbolica computes
Res b ( C p l , C ∥ ) = 1 \boxed{
\operatorname{Res}_b(C_{\rm pl},C_{\parallel})=1
}
Res b ( C pl , C ∥ ) = 1
over Q ( z , t ) \mathbb Q(z,t) Q ( z , t ) z 2 = 5 z^2=5 z 2 = 5 Q ( 5 , t ) \mathbb Q(\sqrt5,t) Q ( 5 , t )
Therefore the full coplanarity-plus-collinearity system has no common algebraic solution for any of the six source-present routing configurations.
Result
the six disjoint-cut M 1 + A m pair orbits have no stationary Landau solution. \boxed{
\text{the six disjoint-cut }M1+A_m\text{ pair orbits have no stationary Landau solution.}
}
the six disjoint-cut M 1 + A m pair orbits have no stationary Landau solution.
Combined with Entries 1237–1238, 21 of the 49 compatible pair orbits are now classified:
seven are confined to t = 0 t=0 t = 0
eight project only to existing one-wall thresholds;
six have unit stationarity ideal.
No new carrier or coefficient divisor has appeared in these classes.
Artifacts
research/benincasa/marici-gm/src/bin/five_site_mixed_pair_landau_disjoint.rsresearch/benincasa/results/five-site-mixed-pair-landau-disjoint.json
Next falsifier
Proceed to the 28 remaining connected-region pair orbits. Begin with pairs whose two cut supports intersect in two occurrences, then one, then zero. Preserve exact labelled routing distances; do not infer elimination from the coarse arc-size profile.
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