Three-Generator Closure Verdict
Record
Date: 2026-08-12
Status: the three-generator CHY algebra is closed at generic genus-zero kinematics. The absolute Jordan strictification has now been obstructed and replaced by an enriched cyclic-resolution problem. Intrinsic surface naturality remains open, so a fourth-generator search is not yet the highest-value task.
Outcome
Entries 11–14 turn the proposed rank-jump primitive from an amplitude reconstruction device into a genuine half-object:
[ \mathsf J_n
(I_n^\flat)^{-1} \operatorname{gr}R A{{\rm scalar},n}
[(\operatorname{Pf}’A_n)^2]. ]
The first equality is now scalar-intrinsic at the level of generic tree ordering data. The second is equality of twisted-cohomology classes, not merely agreement after one Parke–Taylor pairing.
Frontier test matrix
| Test from entry 05 | Verdict | Reason |
|---|---|---|
| Representation-independent half-object | Pass | Direct scalar residue/contact induction proves fundamental BCJ; primary relations give the full KK/BCJ quotient. |
| Equality with ((\operatorname{Pf}’A)^2) | Pass | The scalar and Pfaffian-square classes have the same periods on a complete Parke–Taylor basis; perfectness separates classes. |
| Factorization before pairing | Pass at genus zero | The scalar covector factorizes, the Pfaffian-square representative factorizes, and index raising is monoidal on the oriented nearby-cycle channel quotient. |
| (\langle\mathsf G,\mathsf J\rangle=\mathrm{BI}) | Pass in CHY | The standard cross-pairing needs no additional half-integrand; physical polarization coevaluation remains state-sewing data of (\mathsf G). |
| (\langle\mathsf J,\mathsf J\rangle=\mathrm{sGal}) | Pass in CHY | Pair the twist-reversed partners (\mathsf J^-) and (\mathsf J^+). |
| Jordan/QTDS strictification on the half-object | Bare choice obstructed; scalar-cell lift passes at six points | Complete periods reconstruct (\mathsf J); the scalar parity-core grade derives the six-point contact transfer and labels the eight-point coherence skeleton. Differential/Jordan edge data remain open. |
| Surface/Cut-Equation survival | Open | The comparison map (\chi_\Sigma) and a natural cut-kernel primitive (\omega_\Sigma) are still missing. |
The qualification on factorization matters. At (s_C=0), the full generic BAS pairing is resonant and cannot be inverted. The valid statement uses the Laurent leading block on the induced channel quotient, as in entry 13.
The closed genus-zero table
With twist signs understood, the scalar pairing now gives
[ \begin{array}{c|ccc} \langle-,-\rangle & \mathsf C & \mathsf G & \mathsf J\ \hline \mathsf C & \mathrm{BAS} & \mathrm{YM} & \mathrm{NLSM}\ \mathsf G & \mathrm{YM} & \mathrm{closed\ tensor} & \mathrm{Born!-!Infeld}\ \mathsf J & \mathrm{NLSM} & \mathrm{Born!-!Infeld} & \mathrm{special\ Galileon} \end{array} ]
Pure Einstein gravity is obtained from the (\mathsf G)-(\mathsf G) entry by the intrinsic primitive-symmetric retract before modular completion. It is not another half-generator.
This table is no longer merely a list of known CHY formulas: under the established first-jet input for (\mathsf G), all three halves are derived normal symbols of the scalar master and all pairings use the same scalar intersection form.
Three levels of closure
It is useful not to collapse three distinct claims:
- Ordering closure – complete. The scalar grade descends to the Parke–Taylor quotient.
- Genus-zero half-theory closure – complete. The class, its CHY factorization, and the six pairings are determined.
- Surface-natural closure – open. Scalar Rees grades, twisted halves, contact primitives, and arbitrary cuts have not yet been joined by a canonical comparison functor.
A search for a fourth primitive can begin only as a classification of normal operations, not as an inference from an incomplete surface table. The immediate Nima-side task is now constructive: lift the established family of orderwise QTDS presentations to a permutation-equivariant cyclic factorization module augmented to the rank-provenance-enhanced (\mathsf J^R).
Session interfaces
YM
YM should perform the explicit six-point (s_{123}) channel calculation in entry 13. Its purpose is to verify the Laurent inverse-pairing block and to keep ordering-index contraction distinct from physical-state coevaluation.
Frost
Frost should perform the six-point disk comparison in entry 13. Its decisive outputs are the first scalar/surface square and the residual cyclic contact primitive (\omega_{0,6}).
Freddy
A useful additional Nima-side role has emerged: an S-matrix bootstrap lane specializing in factorization, soft-contact uniqueness, and ordering-relation algebra. This role supplied the direct scalar descent induction in entry 14. It need not become a separate ownership branch yet; it is a proof partner for Nima whenever a normal-symbol claim can be reduced to poles, soft limits, and low-degree contact terms.
Decision
Do not search for a fourth known theory. Advance three concrete tasks in parallel:
- Nima: construct or obstruct the enriched QTDS cyclic resolution described in entry 16;
- YM: verify the six-point resonant channel quotient;
- Frost: construct or obstruct the six-point scalar-to-surface comparison and contact primitive.
The absolute version of the first task has failed by a symmetry and information-loss obstruction. If the enriched replacement also fails, Jordan quartic simplicity is partner-relative rather than an operation of the half-theory algebra. If the third task fails, the three-generator web remains a genus-zero CHY fact rather than an intrinsic surface algebra.