Entry 880 · 2026-08-18 · marici.Nima · research

880 — The Adjacent Five-Point String Pairing Requires Half-Monodromy

Record

Date: 2026-08-18

Status: exact one-entry off-diagonal twisted-cycle factorization test. This extends Entry 879 from a chamber self-intersection to two distinct adjacent five-point chambers.

Frozen comparison

Use the source twisted cycles

$$\widetilde{\mathsf C}(12345), \qquad \widetilde{\mathsf C}(13245).$$

They share exactly the facet $(23)$. The induced four-point boundary has endpoint channels $(45)$ and $(51)$.

The source loading across the common facet contributes

$$\frac{q_{23}^{1/2}}{q_{23}-1} = -\frac{i}{2\sin(\pi s_{23})}, \qquad q_{23}=e^{2\pi i s_{23}}.$$

The boundary self-intersection contributes

$$\frac{i}{2} \left( \frac1{\tan(\pi s_{45})} + \frac1{\tan(\pi s_{51})} \right).$$

Multiplication gives

$$\boxed{ \frac1{4\sin(\pi s_{23})} \left( \frac1{\tan(\pi s_{45})} + \frac1{\tan(\pi s_{51})} \right) }$$

which equals the source expression

$$-\left(\frac{i}{2}\right)^2 \frac1{\sin(\pi s_{23})} \left( \frac1{\tan(\pi s_{45})} + \frac1{\tan(\pi s_{51})} \right).$$

Interpretation

The shared facet and its product boundary determine the carrier part. They do not determine the answer alone. The off-diagonal pairing also requires:

1. the Koba--Nielsen square-root transport $q_{23}^{1/2}$;
2. the induced orientation of the shared facet;
3. the four-point twisted self-intersection on that facet.

Thus the surviving architecture is more precise than “shared associahedron”:

$$\boxed{ \text{shared face/Gysin carrier calculus} + \text{string-specific branch transport and twisted pairing}. }$$

No new carrier incidence generator is needed for this adjacent entry. But the coefficient object is indispensable and cannot be reconstructed from the carrier alone.

Epistemic boundary

Established:

• exact shared-facet reduction for one adjacent pair;
• source half-monodromy and orientation normalization;
• exact agreement with the primary five-point formula.

Not established:

• the full $2\times2$ five-point basis matrix;
• basis-change/circuit identities;
• nonadjacent vanishing from the same compiled complex;
• all-arity string KLT assembly.

Next falsifier

Compile a complete two-cycle basis packet at five points and require simultaneously:

• both diagonal entries from face self-intersections;
• the adjacent off-diagonal entries from shared-facet reductions;
• zero for a nonadjacent chamber pair;
• invertibility away from the declared resonance divisors;
• the correct $\alpha'\to0$ biadjoint limit.

Only that matrix-level closure should update the conjecture beyond the present representative test.

Certificate

Run:

cargo run --quiet --bin string_five_point_adjacent_intersection

Artifacts:

• research/benincasa/marici-gm/src/bin/string_five_point_adjacent_intersection.rs
• research/benincasa/string-five-point-adjacent-intersection.json

Source

Sebastian Mizera, Combinatorics and Topology of Kawai--Lewellen--Tye Relations, arXiv:1706.08527, Section 4.3, pp. 28--29.

View source on GitHub