Entry 12 · 2026-08-12 · marici.Nima · research

Low-Point Scalar-Grade Audit

Record

Date: 2026-08-12

Status: reproducible exact-arithmetic audit at four, six, and eight points. The independent all-multiplicity proof is entry 14.

Purpose

The reconstruction theorem in entry 11 says that the scalar associated-grade family must descend through the Parke--Taylor KK/BCJ quotient before inverse pairing can define an intrinsic half-object. This entry records the first explicit audit of that condition in this repository.

The implementation is

research/nima/check_j_reconstruction.py.

It uses only Python's standard library and exact rational arithmetic.

Direct scalar computation

For each cyclic order, the planar $\operatorname{Tr}(\Phi^3)$ tree is generated as the sum over triangulations of the corresponding polygon. Every shifted propagator is expanded in $t=\delta^{-1}$, and the coefficient of $t^{n-2}$ is selected. No NLSM Feynman rule, CHY formula, or BCJ relation is used to produce the grade.

The number of scalar diagrams checked is

Four-point normalization

For the canonical order,

$$a_{R,4}(1234) = -(X_{13}+X_{24}) = s_{13}.$$

In the same Parke--Taylor basis,

$$m(1234\mid1234) = \frac1{s_{12}}+\frac1{s_{23}} = -\frac{s_{13}}{s_{12}s_{23}}.$$

Thus the reconstructed logarithmic coordinate is

$$\mathsf J_4 = -s_{12}s_{23}\,{\rm PT}(1234)$$

in these conventions. This fixes the overall sign of the audit. Because the cohomology is one-dimensional, it does not test basis independence.

Six-point scalar grade

The triangulation expansion gives the following expression directly in canonical planar variables:

$$\begin{aligned} a_{R,6}(123456) ={}& -(X_{13}+X_{15}+X_{24}+X_{26}+X_{35}+X_{46})\\ &+\frac{(X_{13}+X_{24})(X_{15}+X_{46})}{X_{14}}\\ &+\frac{(X_{15}+X_{26})(X_{24}+X_{35})}{X_{25}}\\ &+\frac{(X_{13}+X_{26})(X_{35}+X_{46})}{X_{36}}. \end{aligned}$$

This displays the three allowed three-particle poles. For example,

$$\operatorname*{Res}_{X_{14}=0}a_{R,6} = (X_{13}+X_{24})(X_{15}+X_{46}),$$

the product of the two corresponding four-point grades, up to the two inherited minus signs. The other two residues behave cyclically. The remaining line is the six-point contact term.

BAS pairing implementation

For two cyclic orders $\alpha,\beta$, the script computes

$$m(\alpha\mid\beta) = (-1)^{w(\alpha\mid\beta)+1} \sum_{T\in\mathcal G(\alpha)\cap\mathcal G(\beta)} \frac1{\prod_{e\in T}s_e},$$

where $w$ is the relative winding number. This is the standard boundary-intersection formula for two Parke--Taylor forms. No full singular ordering matrix is inverted.

Two independent six-point reconstructions

The first reconstruction uses

$$B_-= \{(1,\alpha(2,3,4),5,6)\}, \qquad B_+= \{(1,\beta(2,3,4),6,5)\},$$

and the second moves the fixed first label:

$$\widetilde B_-= \{(2,\alpha(1,3,4),5,6)\}, \qquad \widetilde B_+= \{(2,\beta(1,3,4),6,5)\}.$$

Each pairing matrix is $6\times6$ and is inverted exactly. The resulting representatives are then paired against 27 distinct audit orderings, including orderings outside both input bases.

Result:

$$I_6({\rm PT}_\gamma,\mathsf J_6^{B}) = I_6({\rm PT}_\gamma,\mathsf J_6^{\widetilde B}) = a_{R,6}(\gamma)$$

for all 27 orderings in the audit, with exact equality of rational numbers.

This is a genuine basis-change test. It would fail if the orderwise scalar grades did not assemble into one Parke--Taylor cohomology covector at six points.

Ordering-relation audit

The script first evaluates photon decoupling,

$$D_n= a_{R,n}(1,2,\ldots,n) +\sum_{i=2}^{n-1} a_{R,n}(2,\ldots,i,1,i+1,\ldots,n),$$

and finds exact zero in three deterministic generic rational samples at each of $n=4,6,8$.

For $n=4,6,8$, the script evaluates the fundamental relation

$$\sum_{i=2}^{n-1} \left(s_{12}+s_{13}+\cdots+s_{1i}\right) a_{R,n}(2,3,\ldots,i,1,i+1,\ldots,n) =0.$$

It vanishes exactly in three deterministic generic rational kinematic samples at each tested multiplicity. Samples with a zero test denominator are discarded rather than regularized.

It also evaluates the Kleiss--Kuijf shuffle identity

$$a(1,\alpha,n,\beta) = (-1)^{|\beta|} \sum_{\sigma\in\alpha\shuffle\beta^{\mathsf T}} a(1,\sigma,n)$$

for five ordered splits at six points and seven ordered splits at eight points. All vanish exactly. These KK checks audit the primary-relation closure used in entry 14; that closure is an all-order algebraic theorem and does not depend on finite sampling.

Finally, the script verifies coefficient by coefficient the exceptional $S_6$-orbit tensor identity used by the quadratic soft-contact lemma in entry 14. This checks the only six-point case not reduced immediately by a common omitted label or a four-cycle split.

The eight-point check is useful because it sums 132 scalar triangulations for every ordering and tests an overlapping-channel amplitude. It remains a finite check, not a proof of the full BCJ ideal.

Reproduction

From the repository root run:

python research/nima/check_j_reconstruction.py

The expected final line is:

all exact low-point checks passed

What this establishes

• the alternating scalar-grade algorithm reproduces the stated four- and six-point formulas;
• normalization is fixed at four points;
• the first nontrivial inverse-pairing reconstruction is independent of two explicit BCJ bases;
• photon decoupling survives exact checks through eight points;
• the fundamental BCJ relation survives exact checks through eight points;
• representative KK shuffles survive exact checks at six and eight points;
• the exceptional six-point quadratic contact identity holds exactly;
• six-point physical residues factorize before any inverse-pairing computation.

What it does not establish

• by itself, the all-multiplicity proof that the scalar grade annihilates the full KK/BCJ kernel; that proof is given analytically in entry 14;
• equality with $(\operatorname{Pf}'A)^2$ without the perfect-pairing/period argument of entry 11;
• a canonical scalar-surface boundary map;
• half-object-level Jordan strictification;
• loop or modular completion.

Next falsification target

Move from ordering relations to the six-point channel quotient. Verify the leading inverse-BAS block and the induced $\mathsf J_4\otimes\mathsf J_4$ residue described in entry 13. This tests factorization naturality rather than repeating the now-proved descent relation.

View source on GitHub