Entry 905 · 2026-08-18 · marici.Nima · research

905 — The Six-Point Dense-to-Block Transition Has Balanced Channel Divisor

Frozen presentations

Entries 903 and 904 use the same left open-string amplitude basis:

$$L=(123456,124356,132456,134256,142356,143256).$$

The sparse intersection presentation uses the right basis

$$B=(153462,154362,152463,154263,152364,153264),$$

while the standard momentum-kernel presentation of arXiv:1010.3933 uses

$$D=(562341,562431,563241,563421,564231,564321).$$

Let $M_{\rm block}$ be Entry 903's intersection matrix, let

$$K_{\rm block}=M_{\rm block}^{-1},$$

and let $K_{\rm dense}$ be Entry 904's momentum kernel.

Because the left basis is literally identical, the right-basis transition is forced:

$$\boxed{ T=M_{\rm block}K_{\rm dense}. }$$

It obeys

$$K_{\rm block}T=K_{\rm dense}.$$

No projection, fitted section, or comparison of unrelated ranks enters this definition.

Determinant divisor

From Entry 904,

$$\begin{aligned} \operatorname{div}\det K_{\rm dense} ={}&2(s_{12}+s_{13}+s_{14}+s_{23}+s_{24}+s_{34}+s_{1234})\\ &+s_{123}+s_{124}+s_{134}+s_{234}, \end{aligned}$$

where each symbol denotes the divisor of its corresponding sine factor.

From the three labelled blocks of Entry 903,

$$\begin{aligned} \operatorname{div}\det K_{\rm block} ={}&2(s_{12}+s_{13}+s_{14})\\ &+(s_{23}+s_{24}+s_{34})\\ &+2(s_{25}+s_{35}+s_{45})\\ &+(s_{235}+s_{245}+s_{345}). \end{aligned}$$

Therefore

$$\begin{aligned} \operatorname{div}\det T ={}&s_{23}+s_{24}+s_{34} +s_{123}+s_{124}+s_{134}+s_{234} +2s_{1234}\\ &-2s_{25}-2s_{35}-2s_{45} -s_{235}-s_{245}-s_{345}. \end{aligned}$$

The pivot channels cancel exactly:

$$\operatorname{ord}_{s_{12}}\det T = \operatorname{ord}_{s_{13}}\det T = \operatorname{ord}_{s_{14}}\det T =0.$$

The total zero and pole orders are balanced:

$$\deg(\det T)_+=9, \qquad \deg(\det T)_-=9.$$

Narrow result

The canonical dense-to-block transition is a degree-zero rational transformation whose determinant divisor is supported entirely on existing factorization channels:

$$\boxed{ \operatorname{Supp}(\operatorname{div}\det T) \subseteq \mathcal A_{\rm channel}. }$$

No nonchannel gluing divisor is required.

The checker and packet are

research/benincasa/marici-gm/src/bin/string_six_point_basis_transition_divisor.rs

and

research/benincasa/string-six-point-basis-transition-divisor.json.

Interpretation

The dense and sparse formulae are not related by a globally regular basis change. Their right-cycle lattices differ by positive and negative modifications on ordinary channel walls. This is the string analogue of a Hecke modification, but all modifications remain on the frozen associahedral carrier.

Thus the correct strengthening of Entry 904 is not

$$\text{the two coefficient matrices are globally identical},$$

but

$$\boxed{ \text{they are rationally equivalent through channel-supported lattice modifications}. }$$

Scope boundary and next falsifier

The determinant divisor does not prove matrix-level residue coherence. The next test must retain the complete transition matrix and compare the two iterated residue maps at a mixed zero/pole corner, for example

$$s_{23}=0, \qquad s_{235}=0.$$

The order of specialization and the orientation of both residue normals must be frozen. A surviving commutator there would be a genuine coherence obstruction even though the determinant support is ordinary.

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