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Entry 881 · 2026-08-18 · marici.Nima · research

881 — The Five-Point String Basis Matrix Is Generically Nondegenerate

Record

Date: 2026-08-18

Status: exact matrix-structure and field-theory-leading nondegeneracy theorem for the two-cycle five-point basis. The result combines Entries 879--880 and adds physical kinematic closure plus a nonadjacent control. It does not give a closed-form determinant at arbitrary α′\alpha'.

Frozen basis and kinematics

Use

B=(C~(12345),C~(13245)).\mathcal B= \bigl( \widetilde{\mathsf C}(12345), \widetilde{\mathsf C}(13245) \bigr).

The five planar invariants are

(s12,s23,s34,s45,s51).(s_{12},s_{23},s_{34},s_{45},s_{51}).

The second cyclic order introduces no independent kinematics:

s13=s45−s12−s23,s24=s51−s23−s34.\boxed{ s_{13}=s_{45}-s_{12}-s_{23}, \qquad s_{24}=s_{51}-s_{23}-s_{34}. }

Matrix structure

The diagonal entries are the native face sums of Entry 879 for cyclic orders

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

and

(13,23,24,45,51).(13,23,24,45,51).

The off-diagonal entries are the shared-(23)(23)-facet reduction of Entry 880. Thus every matrix entry is derived from:

face incidence+normal orientation+Koba–Nielsen branch transport.\text{face incidence} + \text{normal orientation} + \text{Koba--Nielsen branch transport}.

For the leading α′−2\alpha'^{-2} coefficient, remove the common nonzero normalization and write

D(x0,…,x4)=∑i∈Z/51xixi+2,D(x_0,\ldots,x_4) = \sum_{i\in\mathbb Z/5} \frac1{x_i x_{i+2}},

O=1s23(1s45+1s51).O= \frac1{s_{23}} \left(\frac1{s_{45}}+\frac1{s_{51}}\right).

Then

Mmlead∼(D(s12,s23,s34,s45,s51)OOD(s13,s23,s24,s45,s51)).M_{ m lead} \sim \begin{pmatrix} D(s_{12},s_{23},s_{34},s_{45},s_{51})&O\\ O&D(s_{13},s_{23},s_{24},s_{45},s_{51}) \end{pmatrix}.

At the exact physical sample

(s12,s23,s34,s45,s51)=(2,3,5,11,17),(s_{12},s_{23},s_{34},s_{45},s_{51})=(2,3,5,11,17),

one has

(s13,s24)=(6,9)(s_{13},s_{24})=(6,9)

and

det⁡Mmlead=24515147≠0\boxed{ \det M_{ m lead}=\frac{245}{15147}\neq0 }

after the stated common normalization. Therefore the leading determinant is not the zero rational function. The finite-α′\alpha' twisted intersection matrix is generically nondegenerate near α′=0\alpha'=0, away from its declared resonance divisors.

Nonadjacent control

The chambers

12345,1352412345,\qquad13524

share no face in M~0,5(R)\widetilde{\mathcal M}_{0,5}(\mathbb R). Their twisted-cycle intersection therefore vanishes, matching the primary source.

Narrow update

The complete five-point evidence now supports:

shared associahedral incidence/Gysin calculus+string-specific Koba–Nielsen pairing\boxed{ \text{shared associahedral incidence/Gysin calculus} + \text{string-specific Koba--Nielsen pairing} }

for diagonal, adjacent, and nonadjacent intersection types. Carrier incidence determines which terms may occur; the coefficient local system determines their trigonometric weights and half-monodromies.

This is evidence for the shared-carrier/sector-specific-coefficients hypothesis. It is not evidence that string and scattering coefficient objects are identical.

Next falsifier

Move from the intersection matrix to its circuit identity. Freeze the source two-cycle and two-cocycle bases and test whether the Marici support calculus reconstructs the five-point twisted period/KLT relation, including basis changes and normalization. This is the first test involving both homology and cohomology rather than the cycle pairing alone.

Certificate

Run:

cargo run --quiet --bin string_five_point_basis_matrix

Artifacts:

Source

Sebastian Mizera, Combinatorics and Topology of Kawai--Lewellen--Tye Relations, arXiv:1706.08527, Sections 3.4 and 4.3.

View source on GitHub