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Entry 1036 · 2026-08-19 · marici.Nima · research

1036 — The Six Loaded Walls Are Not the Facets of One Associahedron

Primary regularization type

The generalized Pochhammer construction regularizes one loaded associahedron using its actual face poset. A codimension-(k) face contributes the product of the (k) inverse boundary factors of the facets meeting there. Thus its edge and corner terms require one coherent facet local system, not merely six available factors with the right determinant.

This is the construction in Section 4.1 of Mizera, Combinatorics and Topology of Kawai–Lewellen–Tye Relations, arXiv:1706.08527.

Two distinct six-tuples

The native chamber hexagon of Entries 979 and 1015 has half-monodromy transports

(B34,B24,X,B34−1,B24−1,X−1).(B_{34},B_{24},X,B_{34}^{-1},B_{24}^{-1},X^{-1}).

Hence its total holonomy is

1,1,

and its three opposite pairs are inverse.

The loaded boundary complex instead has occurrence monodromies

(M1,M2,M2,M3,M4,M4),(M_1,M_2,M_2,M_3,M_4,M_4),

where

M1=(ZA2)2,M2=(ZA2B24)2,M3=(A3/Z)2,M4=(A3B34/Z)2.\begin{aligned} M_1&=(ZA_2)^2,& M_2&=(ZA_2B_{24})^2,\\ M_3&=(A_3/Z)^2,& M_4&=(A_3B_{34}/Z)^2. \end{aligned}

Their generic product is

M1M22M3M42≠1.\boxed{ M_1M_2^2M_3M_4^2\neq1. }

Moreover, the loaded multiset contains no inverse pair. No relabelling can turn it into the facet monodromies of the native hexagon.

Narrow result

the six loaded factors are not the six facets of one loaded associahedron.\boxed{ \text{the six loaded factors are not the six facets of one loaded associahedron.} }

They are six occurrence-labelled tubular boundaries drawn from four source walls, with two walls appearing twice. Entry 1032's diagonal Pochhammer cancellation is therefore valid locally at each occurrence, but it cannot be promoted by simply declaring those six occurrences to be one hexagon's facets.

This independently explains why Entry 1025 found that a rank-one Koba–Nielsen local system on the hexagon cannot close all six loaded columns.

Revised global object

The correct target is an occurrence-resolved Cousin/Čech diagram:

⨁iReg⁡Di⟹⨁i<jReg⁡Di∩Dj⟹⋯ ,\bigoplus_i \operatorname{Reg}_{D_i} \Longrightarrow \bigoplus_{i<j}\operatorname{Reg}_{D_i\cap D_j} \Longrightarrow\cdots,

where the repeated (M_2) and (M_4) occurrences remain distinct and corner terms exist only for source-derived wall intersections.

The native chamber hexagon and this wall-occurrence diagram may later be compared, but they must not be identified cell by cell.

Next falsifier

Freeze the actual incidence graph among the four loaded source walls from the source character blocks. Construct only those codimension-two Pochhammer terms whose wall intersections are present. Test whether their Cousin differential closes. A surviving class is a genuine global comparison extension; exactness closes the static loaded regularization without fitting a hexagon.

Durable verification

View source on GitHub