Primitive Half-Object Pairing Web
Problem
The familiar CHY exceptional-theory table can be written as pairings of a small number of half-integrands. Marici’s stronger conjecture is that the relevant half-objects are not merely a convenient CHY list: each is produced intrinsically by a normal operation on one scalar master geometry, and all are paired by the same scalar-derived kernel.
The CHY table itself is not the new claim. The new claim is common scalar provenance, representation independence, and factorization naturality of each half before pairing.
Candidate generators
The inherited candidates are
[ \mathsf C=\operatorname{PT}, ]
the ordering or color primitive;
[ \mathsf G
H_{\rm gauge}!\left(J_F^1\mathrm{Scalar}\right) \simeq \operatorname{Pf}’\Psi, ]
the gauge first-jet primitive; and
[ \mathsf J
I_{\rm scalar}^{-1} \operatorname{gr}R A{\rm scalar}, ]
the proposed rank-jump or Jordan primitive.
The target CHY representative for the third generator is
[ [\mathsf J]
\left[(\operatorname{Pf}’A)^2\right] ]
in the appropriate scattering-equation or twisted-cohomology quotient. Equality as raw rational functions is neither required nor generally the correct invariant statement.
Type requirement for raising the scalar index
The notation (I_{\rm scalar}^{-1}\operatorname{gr}R A{\rm scalar}) is meaningful only after its spaces are fixed. At multiplicity (n), let (\mathcal H_n) denote the proposed scalar-derived half-object space and let
[ I_n^\flat:\mathcal H_n\longrightarrow\mathcal H_n^* ]
be the map induced by the scalar pairing. If
[ a_{R,n}=\operatorname{gr}R A{{\rm scalar},n}\in\mathcal H_n^*, ]
then the intrinsic candidate is
[ \mathsf J_n=(I_n^\flat)^{-1}(a_{R,n}). ]
This construction is basis independent only if (I_n^\flat) is nondegenerate on the declared quotient and (a_{R,n}) descends to its dual. If the pairing is inverted before quotienting null directions, (\mathsf J_n) can depend on a generalized inverse or an ordering basis and is not yet an intrinsic object.
Predicted symmetric pairing table
With (\langle-,-\rangle_I) denoting the common scalar-derived pairing, the proposed table is:
| First half | Second half | Target theory |
|---|---|---|
| (\mathsf C) | (\mathsf C) | biadjoint scalar theory |
| (\mathsf C) | (\mathsf G) | Yang–Mills |
| (\mathsf C) | (\mathsf J) | NLSM |
| (\mathsf G) | (\mathsf G) | closed tensor or NS–NS sector |
| (\mathsf G) | (\mathsf J) | Born–Infeld |
| (\mathsf J) | (\mathsf J) | special Galileon |
Pure Einstein gravity is not the unqualified (\mathsf G)-self-pairing. It is selected from the doubled gauge sector by the intrinsic primitive-symmetric idempotent (\operatorname{PrimSym}_g^2).
What factorization naturality must mean
It is insufficient that
[ \langle\mathsf C,\mathsf J\rangle_I ]
reconstruct the NLSM amplitude. A half-object can differ by a null or pairing-invisible term and still give the same result against one chosen partner.
For every physical boundary divisor (D), the desired object needs a canonical gluing law of the form
[ \operatorname{gl}D(\mathsf J_n) \simeq \sum{\text{internal data}} \mathsf J_L\otimes\mathsf J_R, ]
with the correct boundary weight, sign, normalization, and internal pairing. This equation belongs in the relevant line bundle, local system, or cohomology group; naive multiplication of two raw CHY functions may have the wrong worldsheet weight. The exact gluing target is therefore part of the construction, not a cosmetic detail.
Equivalently, raising an index with (I^{-1}) must commute with scalar boundary gluing. Full amplitude factorization after pairing does not prove this stronger square commutes.
Consequences if the third generator exists
If (\mathsf J) is intrinsically defined and its CHY class is ([(\operatorname{Pf}’A)^2]), the standard CHY representatives make the Born–Infeld and special Galileon entries plausible immediately. Marici still owes a monoidality proof: the cross- and self-pairings must use no extra normal data and must inherit the same boundary gluing.
The Jordan/QTDS presentation supplies an additional compatibility test. Entry 16 resolves the bare-class version negatively: the universal class does not contain enough asymmetric or target data to select an order, polarity, or Jordan pair. QTDS is established as a presentation of each ordered period. A stronger half-object statement now means constructing a permutation-equivariant Jordan-colored resolution over all cyclic-order fibers, with an augmentation to (\mathsf J) before any partner is attached.
Failure modes
The three-generator claim fails in its strong form if any of the following occurs:
- (\mathsf J) depends on an ordering basis, generalized inverse, gauge choice, or CHY representative with no canonical equivalence class;
- the scalar associated grade determines only the paired NLSM amplitude and not a vector in a half-object space;
- the proposed class agrees with ((\operatorname{Pf}’A)^2) only after contraction with (\operatorname{PT});
- its boundary residues cannot be expressed through lower-point (\mathsf J) objects with the scalar gluing map;
- Born–Infeld or special Galileon requires additional structure not already present in (\mathsf G), (\mathsf J), and (I_{\rm scalar}^{-1}).
In those cases,
[ I_{\rm scalar}^{-1}\operatorname{gr}R A{\rm scalar} ]
may remain a useful amplitude-reconstruction device without being a primitive normal symbol.
Fourth-generator gate
Do not search for a fourth primitive by guessing another known theory. A fourth generator becomes a legitimate question only after (\mathsf C), (\mathsf G), and (\mathsf J) are intrinsic and closed under the common pairing. It should then be sought by classifying primitive normal operations that produce new factorization-compatible half-objects.
Decision
Adopt the three-generator table as a sharply falsifiable conjecture. Treat the existence and factorization of (\mathsf J), rather than reproduction of the NLSM amplitude, as the closing condition for the third row and column.