Entry 901 · 2026-08-18 · marici.Nima · research

901 — The Mixed Six-Point KLT Block Introduces No New Determinant Divisor

Frozen source block

Take the first source $2\times2$ six-point inverse-KLT block

$$M= \begin{pmatrix} \dfrac1{S_{12}S_{34}S_{345}} & -\dfrac{C_{34}+C_{35}}{S_{12}S_{345}} \\[3mm] -\dfrac{C_{34}+C_{45}}{S_{12}S_{345}} & \dfrac1{S_{12}S_{34}S_{345}} \end{pmatrix},$$

where

$$S_I=\sin(\pi s_I), \qquad C_I=\cot(\pi s_I),$$

and source kinematics gives

$$s_{345}=s_{34}+s_{35}+s_{45}.$$

The diagonal entries are trivalent common-vertex contributions. The off-diagonal entries mix two four-point self-intersection histories. This is the first tested block where multiple vertices and an internal coefficient sum coexist.

Pochhammer reconstruction

Every entry is built before inversion from the source cells

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

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

No target inverse entry or determinant factor is used to choose the block representative.

Determinant factorization

The only nontrivial identity required is

$$\sin(\pi s_{35})\sin(\pi s_{45}) - \sin\pi(s_{34}+s_{35}) \sin\pi(s_{34}+s_{45}) = -\sin(\pi s_{34})\sin(\pi s_{345}).$$

It yields

$$\boxed{ \det M = -\frac1{ S_{12}^{,2}S_{34}S_{35}S_{45}S_{345}}. }$$

Thus block inversion introduces no divisor involving the mixed sums

$$s_{34}+s_{35}, \qquad s_{34}+s_{45}.$$

Those sums occur only in numerator entries of the inverse.

Published inverse recovered

Direct inversion gives

$$M^{-1} = \begin{pmatrix} -S_{12}S_{35}S_{45} & -S_{12}S_{45}\sin\pi(s_{34}+s_{35}) \\[2mm] -S_{12}S_{35}\sin\pi(s_{34}+s_{45}) & -S_{12}S_{35}S_{45} \end{pmatrix},$$

exactly matching the frozen source formula.

At a generic nonresonant point, the maximum inverse-entry error is

$$1.11\times10^{-16},$$

and the matrix-product error is

$$2.22\times10^{-16}.$$

The durable packet is at

research/benincasa/string-six-point-klt-block.json.

Narrow result

The first mixed six-point block closes under the existing coefficient calculus:

$$\boxed{ \text{Pochhammer local cells} + \text{four-point self-intersection sums} + \text{source kinematic relation} \Longrightarrow \text{published block inverse}. }$$

Neither a new carrier divisor nor a higher coherence correction appears. The mixed sine sums are numerator transport data, not new support.

Scope boundary

This proves one $2\times2$ block only. The complete six-point KLT matrix has three relabelled blocks and a nontrivial basis organization. Compatibility of the blockwise construction with all relabelling transitions is not yet established.

Next falsifier

Assemble the complete source six-point block-diagonal kernel, derive the relabelling maps between its three $2\times2$ blocks, and test cyclic/parity covariance. Any mismatch must be classified as orientation transport, coefficient monodromy, or genuinely missing coherence.

View source on GitHub