Entry 879 · 2026-08-18 · marici.Nima · research

879 — The Five-Point String Self-Intersection Is the Native Face Sum

Record

Date: 2026-08-18

Status: exact finite five-point calibration against the primary string twisted-cycle result. This entry opens the /string sector independently of the scalar-derived finite-$\alpha'$ regulator line. It does not claim an all-arity theorem or a string completion of the scalar occurrence complex.

Frozen source object

Use the identity twisted cycle in the blown-up real moduli space

$$\widetilde{\mathsf C}(12345)\subset \widetilde{\mathcal M}_{0,5}(\mathbb R)$$

with its rank-one Koba--Nielsen loading. Mizera's primary calculation, arXiv:1706.08527, Section 4.3, equation (4.15), expresses its self-intersection as contributions from the whole pentagon, its five facets, and its five vertices.

Write

$$q_E=e^{2\pi i\alpha' s_E}, \qquad h_E=(q_E-1)^{-1}.$$

The five source facets are

$$(12),(23),(34),(45),(51).$$

Their compatible pairs are

$$((12),(34)),\quad ((12),(45)),\quad ((23),(45)),\quad ((23),(51)),\quad ((34),(51)).$$

No scalar occurrence coefficient is imported.

Native face calculation

The generalized-Pochhammer normal-torus calculus assigns:

• $1$ to the open two-cell;
• $h_E$ to a boundary facet $E$;
• $h_Eh_F$ to a compatible codimension-two face $E\cap F$.

Therefore the native face sum is

$$\boxed{ 1+ \sum_E h_E+ \sum_{E\sim F}h_Eh_F. }$$

The exact face census is

$$1+5+5=11,$$

and the five quadratic terms are exactly those printed in source equation (4.15). Thus

$$\boxed{ \langle\widetilde{\mathsf C}(12345), \widetilde{\mathsf C}(12345)\rangle = 1+ \sum_E\frac1{q_E-1} + \sum_{E\sim F}\frac1{(q_E-1)(q_F-1)}. }$$

At every vertex the coefficient product is symmetric, while the ordered normal contractions obey

$$\iota_E\iota_F=-\iota_F\iota_E.$$

Hence the string coefficient pairing and the Marici Cut/Gysin normal orientation use the same local sign calculus without identifying their coefficient objects.

Narrow conclusion

Established at five-point self-intersection level:

$$\boxed{ \text{associahedral carrier} + \text{normal Koszul/Gysin calculus} + \text{Koba--Nielsen coefficients} }$$

reproduces the primary finite-$\alpha'$ twisted-cycle result. No new carrier cell is required.

This supports the sector architecture

$$\text{shared carrier and support calculus} + \text{string-specific twisted coefficients}.$$

It does not establish:

• the off-diagonal five-point intersection matrix;
• a relation to the scalar occurrence coefficients;
• the missing alternating-conductor BRST chain lift of Entries 89--93;
• an all-arity KLT theorem inside Marici.

Next falsifier

Freeze the two-dimensional twisted-cycle basis and reconstruct an off-diagonal five-point entry from shared-face incidence, induced boundary orientation, and Koba--Nielsen half-monodromy. The sharp representative is

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

The source result contains the factor

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

Derive its minus sign, half-monodromy, and four-point boundary self-intersection independently. Failure would show that shared normal-torus cells do not by themselves determine the string intersection pairing.

Certificate

Run:

cargo run --quiet --bin string_five_point_face_intersection

from research/benincasa/marici-gm.

Artifacts:

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

Dependencies

• Entry 38: finite-$\alpha'$ normal-torus and facewise Pochhammer/Cousin calculus, with its explicit warning that the scalar-derived construction is not a string completion.
• Sebastian Mizera, Combinatorics and Topology of Kawai--Lewellen--Tye Relations, arXiv:1706.08527, Section 4.3, equation (4.15).

View source on GitHub