Entry 900 · 2026-08-18 · marici.Nima · research

900 — The Trivalent Six-Point Vertex Has No Residual Braid Associator

Frozen source vertex

Use the source intersection

$$m_{\alpha'}(123456\mid126435) = \frac1{ \sin\pi s_{12} \sin\pi s_{34} \sin\pi s_{345}}.$$

The corresponding common vertex has the compatible normal set

$$\mathcal N=(12,34,345).$$

All subamplitudes are trivalent. No contact term or nontrivial lower-point coefficient block is inserted.

Six ordered histories

There are six orders in which the three normal residues can be taken. For a permutation $\sigma\in S_3$, tubular orientation transforms by

$$d\arg H_{\sigma(1)}\wedge d\arg H_{\sigma(2)}\wedge d\arg H_{\sigma(3)} = \operatorname{sgn}(\sigma) d\arg H_1\wedge d\arg H_2\wedge d\arg H_3.$$

The ordered triple residue transforms by the same sign:

$$\operatorname{Res}_{\sigma(3)} \operatorname{Res}_{\sigma(2)} \operatorname{Res}_{\sigma(1)} = \operatorname{sgn}(\sigma) \operatorname{Res}_{3} \operatorname{Res}_{2} \operatorname{Res}_{1}.$$

Hence every ordered history has total transition

$$\operatorname{sgn}(\sigma)^2=1.$$

Braid test

The two reduced words for reversing three normals are

$$s_1s_2s_1, \qquad s_2s_1s_2.$$

Both reach the same order and each gives orientation sign $-1$ and residue sign $-1$. Their total maps are identical:

$$(-1)(-1)=+1.$$

Thus the braid relation closes without a residual scalar, sign, or rank-one associator.

Source coefficient reconstruction

Each normal contributes the source half-monodromy cell

$$2i\frac{e^{\pi i s_H}}{e^{2\pi i s_H}-1} = \frac1{\sin\pi s_H}.$$

Their product reproduces the complete source formula. At a generic nonresonant point the numerical error is

$$4.56\times10^{-16}.$$

The durable packet is at

research/benincasa/string-six-point-vertex-braid.json.

Narrow result

For this source codimension-three trivalent vertex,

$$\boxed{ \text{all six histories} \simeq \text{one oriented triple residue}, \qquad \text{braid associator}=0. }$$

The existing flagged-normal and Koszul orientation calculus is sufficient. No higher string-specific coherence primitive is required at this vertex.

Scope boundary

This conclusion is restricted to a normally crossing vertex with a rank-one Koba–Nielsen local system and trivalent subamplitudes. It does not control a six-point block in which different vertices and an internal four-point self-intersection mix.

Next falsifier

Use the source $2\times2$ six-point KLT block containing both trivalent and four-point self-intersection contributions. Reconstruct every entry from Pochhammer cells, invert the block, and test whether the mixed sine sums arise without a new determinant divisor or coherence term.

View source on GitHub