Entry 904 · 2026-08-18 · marici.Nima · research

904 — The Dense Six-Point Momentum-Kernel Determinant Closes on Channel Sines

Frozen source object

Use the string momentum kernel of Bjerrum-Bohr, Damgaard, Sondergaard, and Vanhove, arXiv:1010.3933, equation (17):

$$\mathcal S_{\alpha'}[i_1,\ldots,i_k\mid j_1,\ldots,j_k]_p = (\pi\alpha'/2)^{-k} \prod_{t=1}^k \sin\!\left(\pi\alpha'\left( p\cdot k_{i_t} +\sum_{q>t}\theta(i_t,i_q)k_{i_t}\cdot k_{i_q} \right)\right),$$

where $\theta=1$ precisely when the pair has opposite order in the two words.

At six points freeze the dense basis

$$S_3=(234,243,324,342,423,432)$$

on both variances and take $p=k_1$. No block-atlas basis change is used.

Exact Laurent encoding

Write

$$x_i=e^{i\pi\alpha' k_1\cdot k_i}, \qquad y_{ij}=e^{i\pi\alpha' k_i\cdot k_j},$$

and represent each sine, up to its common invertible scalar, by

$$\sin(\arg X)\longmapsto X-X^{-1}.$$

This converts the complete dense $6\times6$ determinant into an exact Laurent calculation.

Determinant certificate

The channel factors and multiplicities are

$$\begin{array}{c|c} \text{Laurent monomial} & \text{multiplicity}\\ \hline x_2,x_3,x_4 & 2,2,2\\ y_{23},y_{24},y_{34} & 2,2,2\\ y_{23}y_{24}y_{34} & 1\\ x_2x_3y_{23},x_2x_4y_{24},x_3x_4y_{34} & 1,1,1\\ x_2x_3x_4y_{23}y_{24}y_{34} & 2. \end{array}$$

Equivalently, these are the one-particle, internal two-particle, internal three-particle, pivoted two-particle, and full pivoted channel sines.

For every factor, the exact truncated-series determinant has the displayed valuation on both branches

$$X=+1, \qquad X=-1.$$

The multiplicities sum to

$$6+6+1+3+2=18,$$

which saturates the determinant degree bound. At six additional generic points the quotient by the displayed product is the constant $-1$. The complete calculation replicates over the independent primes

$$2305843009213693951, \qquad 2305843009213693921.$$

The durable checker is

research/benincasa/marici-gm/src/bin/string_six_point_dense_momentum_kernel.rs,

and its convention packet is

research/benincasa/string-six-point-dense-momentum-kernel.json.

Narrow result

The standard dense six-point momentum kernel introduces no determinant divisor beyond source-derived channel sines in the exact two-prime certificate:

$$\boxed{ \det\mathcal S_{\alpha'}^{(6)} \sim -\prod_A \sin(\pi\alpha' s_A)^{m_A}, }$$

with the eleven labelled factors and multiplicities above.

In particular, the internal three-particle factor

$$\sin\!\bigl(\pi\alpha'(k_2+k_3+k_4)^2/2\bigr)$$

is required. Omitting it leaves one unsaturated determinant degree and can be mistaken for a new mixed divisor.

Implication for the sector comparison

Entry 903's block diagonalization did not hide a dense-basis obstruction. The same incidence channels reappear in a different coefficient presentation:

$$\text{associahedral channel carrier} + \text{rank-one Koba--Nielsen local system} + \text{momentum-kernel basis transport}.$$

This strengthens the string-sector analogue of H2: changing from the Pochhammer block atlas to a canonical dense source basis changes the coefficient matrix, not the carrier divisor set.

Scope boundary and next falsifier

This is a determinant/support theorem at six points. It does not construct a chain-level equivalence between the dense momentum-kernel basis and Entry 903's block atlas.

The next test is therefore coherence rather than another determinant scan: derive the explicit twisted-cycle transition matrix between the two source bases and verify that it intertwines their residue factorizations on every channel wall, including codimension-two intersections.

View source on GitHub