Entry 898 · 2026-08-18 · marici.Nima · research

898 — The First Six-Point Boundary-Order Ambiguity Is Koszul-Coherent

Frozen source face

Use Mizera's six-point intersection

$$\left\langle \mathsf C(123456), \mathsf C(124365) \right\rangle.$$

The two associahedra share the codimension-two edge

$$F=(34)\cap(1234),$$

where momentum conservation identifies the second normal letter with

$$s_{1234}=s_{56}.$$

Its source value is

$$\boxed{ \left(\frac i2\right)^3 \frac1{\sin\pi s_{34}} \frac1{\sin\pi s_{56}} \left( \frac1{\tan\pi s_{12}} + \frac1{\tan\pi s_{234}} \right). }$$

The edge has two compatible normal histories:

$$(34)\to(1234), \qquad (1234)\to(34).$$

Typed order comparison

The source orientation convention is

$$d\arg H_{34}\wedge d\arg H_{1234}>0.$$

Swapping the two normals contributes

$$d\arg H_{1234}\wedge d\arg H_{34} = -d\arg H_{34}\wedge d\arg H_{1234}.$$

The ordered double residue changes by the same Koszul sign:

$$\operatorname{Res}_{1234}\operatorname{Res}_{34} = -\operatorname{Res}_{34}\operatorname{Res}_{1234}.$$

Therefore the total transition is

$$(-1)_{\rm orientation}(-1)_{\rm residue}=+1.$$

The two iterated regularization histories define the same geometric coefficient class.

Pochhammer reconstruction

Every trigonometric factor is independently compiled from Entry 895's exponential cells:

$$\csc(\pi s)= 2i\frac{e^{\pi i s}}{e^{2\pi i s}-1},$$

$$\cot(\pi s)= 2i\left(\frac1{e^{2\pi i s}-1}+\frac12\right).$$

At a generic nonresonant point, this assembly reproduces the source six-point formula with error

$$2.87\times10^{-17}.$$

After applying both order-transition signs, the two histories agree exactly in the checker. The evidence packet is at

research/benincasa/string-six-point-koszul-coherence.json.

Narrow result

The first six-point boundary-order ambiguity closes under ordinary oriented residue calculus:

$$\boxed{ \text{compatible normal histories} + \text{Koszul residue sign} + \text{tubular-orientation sign} \Longrightarrow \text{one class}. }$$

No new string-specific coherence generator is needed for this source codimension-two edge.

This is stronger than mere commutativity of scalar sine factors. Literal commutativity would discard the variance and orientation data; coherence holds because two independently forced antisymmetries cancel.

Scope boundary

One edge does not prove all six-point iterated regularizations are strictly Koszul-coherent. In particular, this test has a rank-one Koba–Nielsen coefficient and normally crossing compatible normals.

Next falsifier

Test a six-point codimension-three vertex with three compatible normals. Compare all six regularization orders and verify the full permutation sign law and braid coherence. A surviving associator there would be the first justified higher coefficient-coherence datum.

View source on GitHub