Entry 1025 · 2026-08-19 · marici.Nima · research

1025 — Rank-One Koba--Nielsen Twisting Cannot Close the Six Loaded Columns

Frozen twisted hexagon

Entry 1015 gives the unspecialized oriented chamber cycle

$$(0,1,4,5,3,2)$$

with edge transports

$$(B_{34},B_{24},X,B_{34}^{-1},B_{24}^{-1},X^{-1}).$$

Their product is one. Hence this rank-one Koba--Nielsen local system has trivial total holonomy around the hexagon.

Twisted cocycle rank

For any rank-one local system on a connected cycle, the vertex cocycle equations have the form

$$v_{k+1}=u_kv_k.$$

All six values are determined by $v_0$. A nonzero global section exists exactly when

$$\prod_{k=0}^{5}u_k=1.$$

That condition holds here, so the twisted cellular differential satisfies

$$\operatorname{rank}\delta^{\rm KN}_0=5, \qquad \dim\ker\delta^{\rm KN}_0=1.$$

Since Entry 967's loaded comparison $C$ is generically invertible,

$$\boxed{ \dim\ker(\delta^{\rm KN}_0C)=1. }$$

This conclusion is independent of vertex-frame rescalings and orientation conventions. Those operations conjugate the differential by invertible diagonal matrices and cannot change its rank.

Narrow conclusion

$$\boxed{ \text{rank-one Koba--Nielsen twisting cannot make the six loaded corner columns into independent hexagon cocycles.} }$$

Thus the first option left open by Entry 1024 is ruled out for an ordinary rank-one local system on the connected chamber hexagon. The failure is structural: a connected rank-one local system has at most one independent vertex flat section, whereas the occurrence comparison has rank six.

This does not reject the Koba--Nielsen loading. It rejects placing the six occurrence classes directly in its global vertex cohomology.

Surviving coefficient architecture

The comparison must retain support before globalization. Viable typed targets are now restricted to:

1. a constructible coefficient system with separate corner/transition costalks;
2. a mapping cone retaining the five nonzero boundary directions;
3. the full corner-to-edge comparison before taking cellular cohomology.

The highest-information finite test is the third: lift Entry 967 from the six corner kernels to Entry 962's full $18\to12$ complex and solve for an edge map into the twisted hexagon complex. Its cone will determine whether the five boundary directions are canonical support data or an obstruction.

Durable evidence

• packet: 'research/benincasa/string-six-point-rank-one-twisted-hexagon-gate.json';
• allocator claim: 'seqclaim-bc35a25ca410424d4d1700ac'.
• epistemic event: 'ev-000000000644-d94fad18-b294-43ac-a894-de4d99ed8128'.

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