Six-point stopping result

The matching dimensions do not come from the proposed pairing

The five-point positive control works: the natural diagonal rescaling sends its single EYM row to the unique sum-zero chamber function. The obstruction first appears for the nontrivial three-level flag at six points.

At the exact massless specialization$$(s_{12},s_{13},s_{14},s_{23},s_{24},s_{34}) =(2,3,5,7,11,-28),$$every factor in Dong and Stieberger's six-point reconstruction determinant is nonzero. The cumulative degree-at-most-two physical row space has a one-dimensional right kernel, generated by

$$d=\left(\frac{10}{13},\frac{20}{39},-\frac{10}{7}, \frac67,-\frac{10}{9},1\right)^t.$$

Any diagonal map taking the second physical level to the opposite Heaviside level is therefore forced to use this diagonal. But the first EYM block then violates the first Heaviside orthogonality condition:

$$G_6\operatorname{diag}(d)h_{23}=(20,-20)^t\ne0.$$

The stronger Varchenko test also fails. Giving the three A2 hyperplanes completely arbitrary weights$(x,y,z)$, the four equations requiring just one ordinary EYM row to lie in the opposite first Heaviside piece have unit Gröbner ideal. An elementary elimination reduces the only possible values of$z$ to

$$(z-1)(z+1)(189z^2-608z+189)=0,$$

and the required product relation fails at every root. The same exact test rejects all left and right sign-character twists.

Exact target

Replace an observed rank pattern by an intrinsic map

Dong and Stieberger organize the symmetric strict subleading collinear data of an$n$-point Yang–Mills amplitude into cumulative row spaces$F_n^1\subset\cdots\subset F_n^{n-3}$. Their computations through nine particles give

$$\dim(F_n^k/F_n^{k-1})=\left[{n-3\atop k}\right].$$

The paper conjectures this at every multiplicity and explicitly says that no intertwining map to the matching mathematical modules is known. The program asks whether the open–closed monodromy construction is the opposite dual of the canonical chamber-function filtration of the type-A braid arrangement.

Intrinsic object

Functions on chambers already have the right filtration

Put $m=n-3$. The chambers of the real braid arrangement are canonically the permutations$S_m$, exactly the column labels of the BCJ-reduced amplitude system. The Varchenko–Gelfand algebra$VG_m=\operatorname{Fun}(S_m,K)$is filtered by the Heaviside functions recording which side of each collision hyperplane a chamber occupies.

$$\dim \operatorname{gr}^{q}VG_m =\left[{m\atop m-q}\right].$$

Taking the orthogonal complements of the low-degree pieces under the chamber trace pairing reverses this order and gives increments $\left[{m\atop k}\right]$without an amplitude calculation. Brauner's equivariant theory further identifies these pieces with Eulerian representations and Whitney homology.

Second structural clue

The determinant points to the arrangement pairing

Dong and Stieberger's conjectural determinant factors over unordered bipartitions of the hard labels with multiplicity$(|I|-1)!(|I^c|-1)!$. Exact arithmetic confirms that the total factor degree equals

$$\sum_{k=1}^{m}k\left[{m\atop k}\right]=m!H_m,$$

the degree forced by the Stirling blocks. Lam's arrangement-intersection framework identifies canonical scattering pairings with Varchenko forms, while Mizera's twisted-period formulationderives open/closed string relations from the same kind of pairing. This makes the pairing hypothesis precise, but does not prove it for the strict collinear projection.

Stopping boundary

The route ends before an arbitrary basis search

Any two flags with these dimensions are related by some element of $GL_6(K)$, so allowing an unrestricted change of basis would make the comparison empty. The failed diagonal, Varchenko, and orientation tests exhaust the canonical maps declared in the hunt packet. The Stirling pattern may still reflect a kinematic local system or another universal object, but pursuing that requires a new target rather than a larger rank census.

Scope and credit

A conjecture from physics, with classical mathematics as the bridge

The amplitude hierarchy, rank conjecture, and determinant conjecture are due to Dong and Stieberger. The selected route is influenced by Varchenko–Gelfand chamber filtrations, the Orlik–Solomon/Whitney-homology description of their graded pieces, and twisted intersection formulations of scattering amplitudes. None of these sources already supplies the missing strict-collinear intertwiner.