HUNT packet: the hidden braid-arrangement filtration in subleading Yang–Mills collinear limits

Mode: HUNT
Selected 7 September 2026
Status: active research target; no theorem is claimed

1. Exact question

Put (m=n-3). Dong and Stieberger associate to the symmetric strict collinear limit of (n)-point Yang–Mills amplitudes a canonical increasing filtration of row spaces

$$0\subset F_n^1\subset F_n^2\subset\cdots\subset F_n^m \subset K^{S_m}, \qquad F_n^k=\operatorname{rowspan}(W_n^{(1)},\ldots,W_n^{(k)}),$$

over the function field (K) of generic kinematics. Computations through (n=9) give

$$\dim(F_n^k/F_n^{k-1})= \left[{m\atop k}\right], \qquad F_n^m=K^{S_m}.$$

Their paper conjectures these statements for every (n), and explicitly notes that the equal dimensions with braid-arrangement cohomology are not yet explained by an intertwining map.

The selected question is deliberately stronger and more structural:

Does the open–closed monodromy construction identify the filtration (F_n^), up to a canonical kinematic change of chamber basis, with the opposite dual of the Heaviside filtration on the Varchenko–Gelfand algebra of the type-(A_{m-1}) braid arrangement?

An affirmative identification would prove the all-multiplicity rank formula. If its compatible pairing is the arrangement contravariant/Varchenko form, it should also explain the conjectural determinant factorization.

2. Why this is not another presentation census

Both sides are already functions on one intrinsic finite set. The real braid arrangement has (m!) chambers, canonically indexed by permutations in (S_m). The BCJ-reduced collinear coefficient vector also has (m!) orderings. No choice of residual configurations is being added.

The Varchenko–Gelfand algebra

$$VG_m=\operatorname{Fun}(S_m,K)$$

has its canonical Heaviside filtration (H_{q}), generated by the half-space indicators (h_{ij}()=1_{(i)<(j)}). Classical arrangement theory gives

$$\dim H_{\le q}/H_{\le q-1} =\left[{m\atop m-q}\right].$$

Thus the amplitude increments occur in the reverse order. With the chamber trace pairing

$$\langle f,g\rangle_{\rm ch}=\sum_{\sigma\in S_m}f(\sigma)g(\sigma),$$

define the opposite dual filtration

$$T_k=(H_{\le m-k-1})^\perp, \qquad k=1,\ldots,m,$$

where (H_{}=0). Then

$$\dim(T_k/T_{k-1})=\left[{m\atop k}\right]$$

for every (m), before any amplitude calculation. The rank coincidence is therefore the shadow of a specific candidate filtration, not just a matching integer sequence.

The target is to construct or refute one map between (F_n^•) and (T_•). If this fails at the first equivariance or circuit-relation gate, the route stops. We will not replace the map by higher-multiplicity rank enumeration.

3. Known frontier

Dong and Stieberger prove the open–closed collinear hierarchy and verify its full-rank Stirling decomposition through (n=9). Their (k)-th new block has rank (left[{n-3k}]). Even (k) give homogeneous BCJ-like relations; odd (k) give Einstein–Yang–Mills amplitudes and their higher-derivative corrections. The theorem for arbitrary (n) remains conjectural.

They also conjecture a factorization of the determinant of the square reconstruction matrix. Its factors are indexed by unordered bipartitions (I|I^c) of the (m+1=n-2) hard labels, with multiplicity

$$(|I|-1)!(|I^c|-1)!.$$

The factorization is proved at (n=7), while the displayed all-(n) form is conjectural from (n=8) onward.

The mathematical ingredients already exist separately:

• Varchenko and Gel’fand identify the filtered algebra of chamber functions; its associated graded has the braid-arrangement Poincare series.
• Brauner identifies the graded pieces equivariantly with Eulerian representations and Whitney homology.
• Orlik–Solomon and Lehrer–Solomon describe the complex arrangement cohomology and its cycle decomposition.
• Mizera derives KLT relations from twisted period relations.
• Lam identifies arrangement intersection pairings with Varchenko and Schechtman–Varchenko forms and extends scattering constructions to matroids.
• Mazloumi and Stieberger realize ordinary EYM amplitudes as intersections of twisted forms.

What is missing is precisely their composition in the strict subleading open–closed projection.

4. Exact calibration

The attached dependency-free script checks, for (2m):

1. the Stirling blocks sum to (m!);
2. the even and odd blocks each sum to (m!/2);
3. the amplitude order (k=1,ldots,m) is the reverse of the graded Heaviside order (q=0,ldots,m-1);
4. the weighted degree $$\sum_k k\left[{m\atop k}\right]=m!H_m$$ equals the total degree contributed by all conjectural unordered bipartition factors.

Item 4 is especially suggestive. The same flats that govern arrangement pairing determinants carry exactly the factorial multiplicities appearing in the amplitude determinant. This is evidence for the proposed pairing target, not a proof that the two matrices coincide.

5. First ATTACK

The first attack has three gates, in this order.

Gate A: remove the noncanonical row bases

Work only with the cumulative spaces (F_n^k), which are independent of the choice of maximal row basis inside (W_n^{(k)}). Reconstruct the published (n=5,6) matrices from the open–closed kernel and express their rows as functions on the chambers (S_m).

Gate B: test the canonical dual filtration

Construct (H_{q}) directly from the Heaviside functions of the braid arrangement. Determine whether there is a single diagonal kinematic chamber weight (D_n), or the canonical Varchenko pairing matrix (V_n), for which

$$F_n^kD_n=T_k \quad\text{or}\quad F_n^kV_n=T_k \qquad (k=1,\ldots,m)$$

at (n=5,6). This is a simultaneous equality of filtrations, not a rank comparison.

Gate C: circuit relations

If Gate B succeeds, prove that the degree-leading open–closed kernels kill the Arnold three-term circuit relations and respect deletion–contraction. That would identify the associated graded map for every (m); the classical Hilbert series would then prove the rank conjecture.

If Gate B fails for every canonical pairing and the failure is not repaired by the unique sign/orientation character, stop. Searching arbitrary changes of basis would recreate the presentation problem this target was selected to avoid.

6. What would count as success

The desired result is an all-multiplicity structural theorem: a natural filtered isomorphism, or a natural associated-graded surjection that becomes an isomorphism by the known dimensions. A new verification at (n=10), a larger rank table, or a fitted determinant is not success.

A negative result is also valuable if it identifies the first classical arrangement relation violated by the physical row filtration. It would show that the Stirling pattern reflects a different universal object and would close this route without a census.

7. Candidate comparison

The hunt also considered:

• equality of homological and naive support in big tensor-triangulated categories;
• stable triviality of three-dimensional quantum cellular automata;
• a Carroll–Cotton boundary characterization of asymptotically flat self-dual gravity; and
• the membrane/Gromov–Witten correspondence for Calabi–Yau fivefolds.

Each is substantial. The support and QCA problems naturally branch by ambient category or lattice model. The Carroll–Cotton route inherits counterterm and phase-space choices close to the obstruction that stopped the JT program. The membrane route currently defaults to localization graphs. The collinear-amplitude problem uniquely offers a canonical common carrier, an explicit missing intertwiner, and a low-multiplicity falsification gate.

8. Sources and credit

The target and physical conjectures are entirely due to Jin Dong and Stephan Stieberger, Subleading Collinear Limits of Yang–Mills Amplitudes from Gravity, arXiv:2609.00153v1, submitted 31 August 2026: https://arxiv.org/html/2609.00153v1.

The Varchenko–Gelfand calibration is influenced by Galen Dorpalen-Barry’s modern treatment of the chamber-function filtration and Sarah Brauner’s equivariant description. The proposed pairing route is influenced by Sebastian Mizera’s twisted-period formulation and Thomas Lam’s matroid intersection-pairing framework. These are intellectual influences; none proves the selected identification.

The complete source ledger is included beside this packet.