Twisted-Cohomology Reconstruction of J

Record

Date: 2026-08-12

Status: all-multiplicity reconstruction theorem at generic tree kinematics. Direct scalar-geometric descent through the Parke–Taylor quotient is proved in entry 14; resonant boundary naturality and the surface lift are separate remaining gaps.

Verdict

The inverse-pairing part of

[ \mathsf J_n

I_{{\rm scalar},n}^{-1} \operatorname{gr}R A{{\rm scalar},n} ]

is not an additional dynamical ansatz. On generic massless kinematics it is the musical inverse of a perfect twisted-intersection pairing and therefore reconstructs a unique cohomology class.

The argument separates into two statements:

  1. Reconstruction theorem. A coherent scalar grade covector uniquely determines a representation-independent half-object, and its periods identify it with ([({\rm Pf}’A_n)^2]).
  2. Scalar descent theorem. The order-by-order alternating scalar limits annihilate the KK/BCJ relations by a direct residue/contact induction followed by primary-relation closure.

Both statements are now established at generic genus-zero kinematics. Entry 14 supplies the second. Commutation with resonant boundary gluing is deliberately not folded into descent; its correct nearby-cycle formulation is recorded in entry 13.

Correctly typed spaces

Let (K_n) be the field of rational functions on generic massless (n)-particle kinematics and let

[ H_n^+

H^{n-3}(\mathcal M_{0,n},\nabla_{\omega_n}), \qquad H_n^-

H^{n-3}(\mathcal M_{0,n},\nabla_{-\omega_n}). ]

Both have dimension ((n-3)!). Their twisted-intersection pairing is perfect:

[ I_n:H_n^-\otimes H_n^+\longrightarrow K_n. ]

Thus the precise flat map is

[ I_n^\flat:H_n^+\xrightarrow{\sim}(H_n^-)^*. ]

The plus and minus spaces should not be silently identified. A choice of Parke–Taylor bases (B_-\subset H_n^-) and (B_+\subset H_n^+) gives the biadjoint scalar matrix

[ m_{\alpha\beta}

I_n\bigl({\rm PT}^-\alpha,{\rm PT}^+\beta\bigr). ]

For actual independent BCJ-sized bases this matrix is invertible at generic abstract kinematics. Merely selecting ((n-3)!) orderings does not guarantee a basis, and special Gram loci or physical poles can destroy invertibility. The full matrix indexed by all cyclic orderings is singular and must not be inverted or replaced by an undeclared pseudoinverse.

The scalar associated-grade family

For an even cyclic order

[ \alpha=(\alpha_1,\ldots,\alpha_{2r}), ]

the cyclic polygon has a canonical alternating two-colouring, unique up to exchanging the two colours. In the coordinates belonging to this order, shift planar diagonals by

[ \widetilde X_{e,e}=X_{e,e}+\delta, \qquad \widetilde X_{o,o}=X_{o,o}-\delta, \qquad \widetilde X_{o,e}=X_{o,e}. ]

Define

[ a_{R,2r}(\alpha)

\lim_{\delta\to\infty} \delta^{2r-2} A^{\delta}_{\operatorname{Tr}(\Phi^3),2r}(\alpha). ]

Exchanging the two colours sends (\delta\mapsto-\delta). Since (2r-2) is even, the selected grade is unchanged. A one-step cyclic rotation also exchanges the colours, so the construction is intrinsic to a bipartite even cyclic polygon rather than to a chosen label called “even.”

The source construction calls this a large-(\delta), low-energy kinematic shift. “Rank-jump associated grade” is Marici’s geometric interpretation and is not source terminology.

At odd multiplicity set (a_{R,2r+1}=0), as required by the NLSM sector and the reduced-Pfaffian representative.

The descent criterion

Let (\mathcal C_n) be the free vector space generated by cyclic orderings and let

[ q_n:\mathcal C_n\longrightarrow H_n^- ]

send an ordering to its Parke–Taylor class. The scalar values define a half-object covector if and only if

[ a_{R,n}(\ker q_n)=0. ]

Equivalently, the scalar grades must obey the cyclic, reflection, KK, and BCJ relations appropriate to the Parke–Taylor quotient. Four points cannot meaningfully test this because (\dim H_4^-=1). Six points, where (\dim H_6^-=6), is the first nontrivial descent test.

This formulation isolates the possible failure precisely: an order-dependent family of valid planar amplitudes need not define a covector on twisted cohomology.

Entry 14 proves that this particular scalar family passes the criterion at every multiplicity. Scalar factorization and Adler zeros imply photon decoupling and fundamental BCJ directly; cyclic symmetry plus the relabeled fundamental relation then generate the full KK/BCJ ideal.

Reconstruction theorem

Assume the descent criterion. For any pair of BCJ-sized Parke–Taylor bases define

[ \boxed{ \mathsf J_n

\sum_{\beta\in B_+} {\rm PT}^+\beta (m^{-1})^{\beta\alpha} a{R,n}(\alpha) } ]

with (\alpha\in B_-) summed.

This is the coordinate formula for

[ \mathsf J_n=(I_n^\flat)^{-1}a_{R,n}. ]

It is basis independent. Indeed, under changes of bases (P_-) and (P_+), the pairing matrix and period vector transform as

[ m\longmapsto P_-^{\mathsf T}mP_+, \qquad a_R\longmapsto P_-^{\mathsf T}a_R, ]

so the reconstructed coordinates transform by (P_+^{-1}), exactly as coordinates of the same element of (H_n^+).

This proof uses only perfect pairing and descent. No CHY representative is chosen.

Identification with the Pfaffian square

Two established all-multiplicity amplitude statements now meet:

  1. the alternating scalar grade gives the colour-ordered NLSM tree amplitude for every cyclic order;

  2. the CHY formula gives

    [ A_n^{\rm NLSM}(\alpha)

    I_n!\left( {\rm PT}^-_\alpha, [({\rm Pf}’A_n)^2] \right). ]

Therefore, for every (\alpha) in a basis of (H_n^-),

[ I_n({\rm PT}^-_\alpha,\mathsf J_n)

I_n!\left({\rm PT}^-_\alpha,[({\rm Pf}’A_n)^2]\right). ]

Perfectness of (I_n) implies

[ \boxed{ \mathsf J_n

[({\rm Pf}’A_n)^2] \in H_n^+. } ]

This is stronger than agreement after pairing with one Parke–Taylor factor. Equality of a full basis of periods separates cohomology classes.

The compact rational function ((\operatorname{Pf}’A)^2) need not itself be the preferred logarithmic representative on (\overline{\mathcal M}_{0,n}). Any massless twisted class has a logarithmic Parke–Taylor expansion. All-multiplicity polynomial DDM numerators for the NLSM give such expansions, although their individual numerator representatives retain generalized-gauge or reference-order freedom. That freedom changes the representative, not (\mathsf J_n)’s class.

Factorization before pairing

There are three distinct factorization claims.

Scalar covector

On an allowed pole (X_{o,e}=0), the shifted scalar amplitude already factorizes into two lower even-point shifted amplitudes. Taking the associated grade gives

[ \operatorname*{Res}{X_D=0}a{R,n}

a_{R,L},a_{R,R} ]

with the inherited normalization and ordering. Forbidden parity channels have no leading pole.

CHY half-object

For a worldsheet degeneration parameter (\tau), the reduced Pfaffian obeys in every allowed odd-particle channel

[ {\rm Pf}’A_n \longrightarrow \tau^{p/2} {\rm Pf}’A_{n_L+1} {\rm Pf}’A_{n_R+1}. ]

Consequently,

[ ({\rm Pf}’A_n)^2 \longrightarrow \tau^p ({\rm Pf}’A_{n_L+1})^2 ({\rm Pf}’A_{n_R+1})^2. ]

Its leading term vanishes in the forbidden channels. Hence the class identified above has a factorizing half-integrand representative before attachment of a Parke–Taylor factor.

Scalar-derived gluing square

The stronger Marici statement is that the scalar comparison map and index raising commute with a canonical boundary map. This statement lives in a Laurent or nearby-cycle associated grade, not in the naive specialization of the generic pairing at (s_D=0). At the physical pole the twist is resonant and the residue of the full (n)-point BAS matrix is rank deficient.

Let (\Delta_D^\pm) be channel residue maps including the plumbing normal/orientation line. The pairing compatibility to prove is

[ \operatorname*{Res}_{s_D=0} I_n(b,a)

\epsilon_D (I_L\otimes I_R) (\Delta_D^-b,\Delta_D^+a), ]

where (\epsilon_D) retains boundary-orientation and Koszul conventions. In a channel-adapted basis the compatible block behaves as

[ m_n \sim \frac{\epsilon_D}{s_D}(m_L\otimes m_R)+O(1), ]

so its inverse block behaves as

[ m_n^{-1} \sim s_D,\epsilon_D^{-1} (m_L^{-1}\otimes m_R^{-1})+O(s_D^2). ]

The factor of (s_D) cancels the pole of the amplitude covector. Only after passage to this induced channel quotient is the shorthand square meaningful:

[ \operatorname{Res}D (I_n^\flat)^{-1}a{R,n} \stackrel{?}{=} \bigl((I_L^\flat)^{-1}\otimes(I_R^\flat)^{-1}\bigr) \operatorname{Res}D^*a{R,n}. ]

In twisted cohomology it follows once the following comparison data are installed:

  1. a residue map (H_n^+\to H_L^+\otimes H_R^+) with fixed orientation and plumbing weight;
  2. factorization of compatible Parke–Taylor residues spanning (H_L^-\otimes H_R^-);
  3. the standard boundary-localization identity for the intersection pairing.

The proof is then separation by the perfect lower-point pairing: both sides pair equally with every product Parke–Taylor class. One must never replace this argument by inverting (\operatorname{Res}_D m_n) on the full (n)-point space. The CHY Pfaffian degeneration verifies the conclusion in the on-shell genus-zero category.

What remains unproved is that the scalar master or surface-function theory itself supplies this comparison map canonically. This is precisely the distinction between CHY factorization of the identified class and intrinsic scalar-boundary naturality.

Born–Infeld and special Galileon consequences

After the identification of (\mathsf J), write (\mathsf J^+\in H_n^+) and its twist-reversed partner (\mathsf J^-\in H_n^-). The standard CHY pairings give

[ I_n(\mathsf G^-,\mathsf J^+)

\mathrm{Born!-!Infeld}, ]

and

[ I_n(\mathsf J^-,\mathsf J^+)

\int d\mu_n,({\rm Pf}’A)^4

\mathrm{special\ Galileon}. ]

The word “symmetric” here means symmetry after exchanging the two twists, or after declaring a twist-reversal involution. The global BAS/KLT kernel contracts ordering indices. Internal physical state sewing remains data of each half: scalar coevaluation for (\mathsf J), transverse polarization coevaluation for (\mathsf G), and the appropriate Killing metrics for biadjoint colour. No fourth half-integrand is introduced. This closes the (\mathsf J) row at the level of genus-zero CHY amplitudes. The Born–Infeld entry still inherits the established gauge-half and pairing inputs; the (\mathsf J) input is no longer conditional on an unproved descent. This does not yet prove the surface/Cut-Equation lift or Jordan strictification at the half-object level.

Epistemic boundary

Established from primary constructions and linear algebra:

  • perfectness and dimension of the generic twisted-cohomology pairing;
  • inverse BAS/KLT reconstruction in a BCJ-sized basis;
  • direct all-order scalar descent through the Parke–Taylor KK/BCJ quotient, by entry 14;
  • all-order scalar-grade equality with colour-ordered NLSM trees;
  • the CHY representative ((\operatorname{Pf}’A)^2);
  • its allowed-channel building-block factorization;
  • Born–Infeld and special-Galileon pairings.

Not yet intrinsic to scalar boundary geometry:

  • a canonical scalar-to-CHY comparison functor intertwining all boundary maps;
  • a half-object interpretation of the Jordan/QTDS strictification;
  • the surface/Cut-Equation realization.

Sources checked

Decision

Promote the inverse-pairing, scalar-descent, and CHY-identification steps to a tree-level theorem. Do not spend the next research cycle re-deriving KLT inversion or ordering relations. Concentrate on the Jordan/QTDS action on the half-class, while YM and Frost test functorial compatibility with resonant boundary gluing and surface cuts.