Entry 1231 · 2026-08-20 · marici.Nima · research

Entry 1231 — Five-Site Triple Collisions Close by the $A_2$ Čech Relation

Correctly supported census

Entry 1230 requires both root equality and the corresponding signed-energy wall. For each three-edge set $\{i,j,k\}$, retain the three pair supports

$$(F_i=F_j,X_A=0), \qquad (F_i=F_k,X_B=0), \qquad (F_j=F_k,X_C=0).$$

Each edge pair has two complementary source occurrences. Therefore the complete labelled census has

$$\binom53 2^3=80$$

triple overlaps. They form 16 free $C_5$-orbits.

Energy-wall ranks

Exact ranks of the three source indicator vectors give

$$\boxed{ 30\text{ overlaps of energy rank }2, \qquad 50\text{ overlaps of energy rank }3. }$$

For all 50 rank-three overlaps, the total-energy vector lies in the span. Hence those strata force

$$E_T=0.$$

The other 30 do not. This separates ordinary higher marked incidence from incidence already lying on the scattering/nearby-cycle boundary.

Root-normal complex

On a labelled Kummer chart, the three root-equality normals form the standard $A_2$ arrangement:

$$\varepsilon_{ij}=0, \qquad \varepsilon_{jk}=0, \qquad \varepsilon_{ik} = \varepsilon_{ij}+\varepsilon_{jk}=0.$$

The three pair-residue lines map to the triple costalk by the oriented row

$$\boxed{(1,-1,1)}.$$

This map has rank one and kernel rank two, exactly matching the top Orlik–Solomon rank of the (A_2) arrangement.

Therefore every labelled triple overlap is generated by the three existing pair cells with their single circuit relation:

$$\boxed{ H_{ m top}^{A_2} = \ker(1,-1,1), \qquad \dim H_{ m top}^{A_2}=2. }$$

No higher Čech excess occurs.

Classification

The rank-two triples are ordinary marked relative-incidence coefficient objects. The rank-three triples are their existing total-energy nearby-cycle restrictions. Neither class requires a new carrier stratum.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_triple_collision_cech.rs
• research/benincasa/results/five-site-triple-collision-cech.json

Next falsifier

Compute the iterated nearby-cycle compatibility for the 50 total-energy-forced overlaps. In particular, test whether the (A_2) kernel commutes with specialization at $E_T=0$ and whether its second normal/Rees grade contains coefficient data not generated by the existing Kummer and total-energy maps.

View source on GitHub