Entry 889 · 2026-08-18 · marici.Nima · research

889 — The Parke–Taylor Period Circuit Closes at Associated Grade

Record

Date: 2026-08-18

Status: exact field-theory-leading/associated-grade period-intertwining theorem for two cycle and two Parke--Taylor cocycle labels. This does not establish the finite-$\alpha'$ hypergeometric period identity.

Frozen cocycles and cycles

Use the Parke--Taylor cocycle basis

$$\Phi=\bigl(\operatorname{PT}(12345),\operatorname{PT}(12435)\bigr),$$

the source cycle basis

$$\mathcal C=(13254,14253),$$

and Entry 888's target basis

$$\mathcal D'=(12354,12453).$$

The source states that, after removing the common $\alpha'^{-2}$ normalization, the leading string periods reduce to bi-adjoint double-partial amplitudes.

Source period grade

The common-vertex calculation of Entry 883 gives

$$\operatorname{gr}P_{\mathcal C,\Phi} = \begin{pmatrix} \dfrac1{s_{23}s_{45}}&0\\[2mm] 0&\dfrac1{s_{24}s_{35}} \end{pmatrix}.$$

The zero entries are cocycle selections: the corresponding Parke--Taylor forms select no common trivalent vertex.

Circuit grade

Entry 888 has field-theory grade

$$\operatorname{gr}M_{\mathcal D'\leftarrow\mathcal C} = -\frac1{s_{12}} \begin{pmatrix} s_{12}+s_{23}&s_{24}\\ s_{23}&s_{12}+s_{24} \end{pmatrix}.$$

Multiplication yields

$$\operatorname{gr}M\operatorname{gr}P_{\mathcal C,\Phi} = \begin{pmatrix} -\dfrac1{s_{12}s_{45}}-\dfrac1{s_{23}s_{45}} &-\dfrac1{s_{12}s_{35}}\\[3mm] -\dfrac1{s_{12}s_{45}} &-\dfrac1{s_{12}s_{35}}-\dfrac1{s_{24}s_{35}} \end{pmatrix}.$$

Every entry is independently the common-triangulation sum for the target cycle and the corresponding Parke--Taylor cocycle. Therefore

$$\boxed{ \operatorname{gr}P_{\mathcal D',\Phi} = \operatorname{gr}M_{\mathcal D'\leftarrow\mathcal C} \operatorname{gr}P_{\mathcal C,\Phi}. }$$

At

$$(s_{12},s_{23},s_{24},s_{35},s_{45})=(2,3,9,-14,11),$$

the exact target matrix is

$$\begin{pmatrix} -5/66&1/28\\ -1/22&11/252 \end{pmatrix},$$

and the Rust checker verifies the matrix identity over $\mathbb Q$.

Typed conclusion

This is the first tested string comparison containing both:

• a change of twisted-cycle basis;
• two independently labelled logarithmic Parke--Taylor cocycles.

It closes using the same labelled face/circuit calculus at associated grade. No extra contact term or carrier cell is needed.

But the type boundary is essential:

$$\boxed{ \text{associated-grade period intertwining} \not\Rightarrow \text{finite-}\alpha'\text{ period theorem}. }$$

The latter still requires the actual Koba--Nielsen periods and their branch-normalized analytic continuation.

Next falsifier

Compute one finite-$\alpha'$ Parke--Taylor period column by source-normalized hypergeometric continuation in both cycle bases. Test the circuit identity numerically at generic nonresonant kinematics, preserving branch phases. Failure after source normalization would locate the first coefficient-level obstruction beyond associated grade.

Certificate

Run:

cargo run --quiet --bin string_five_point_pt_associated_grade

Artifacts:

• research/benincasa/marici-gm/src/bin/string_five_point_pt_associated_grade.rs
• research/benincasa/string-five-point-pt-associated-grade.json

Sources

• Sebastian Mizera, Combinatorics and Topology of Kawai--Lewellen--Tye Relations, arXiv:1706.08527, Sections 3.2--3.4 and Appendix B.
• Sebastian Mizera, Inverse of the String Theory KLT Kernel, arXiv:1610.04230, equation (5.4).

View source on GitHub