Boundary Pairing and Surface Lift Handoff

Record

Date: 2026-08-12

Status: corrected genus-zero boundary square; surface comparison and contact primitive remain open.

Division of labor

Entry 11 identifies the generic twisted-cohomology class (\mathsf J). Boundary naturality is a separate problem with two interfaces:

  • YM owns the Laurent/nearby-cycle behavior of the scalar/BAS pairing and physical internal state coevaluation.
  • Frost owns the lift from scalar Rees grades to surface functions, including the cut kernel and contact primitives.

Neither interface should be hidden inside the notation (I^{-1}).

Exact genus-zero square

For a separating divisor or cut (C), use dual half-spaces and retain the plumbing normal line. The intended square is

[ \begin{array}{ccc} H_\Sigma^+ & \xrightarrow{\ I_\Sigma^\flat\ } & (H_\Sigma^-)^\vee\ \downarrow\rho_C^+ && \downarrow B_C\ H_L^+\otimes H_R^+ & \xrightarrow{\ I_L^\flat\otimes I_R^\flat\ } & (H_L^-\otimes H_R^-)^\vee, \end{array} ]

where the transpose boundary map is defined by

[ (B_Ca)(u_L\otimes u_R)

\operatorname*{Res}_{C} a!\left(\gamma_C^-(u_L\otimes u_R)\right). ]

Here (\gamma_C^-) glues test half-objects and (\rho_C^+) takes the residue of the raised half-object. Pairing compatibility is

[ B_CI_\Sigma^\flat

(I_L^\flat\otimes I_R^\flat)\rho_C^+. ]

Whenever the relevant maps are invertible on the induced channel quotient, this becomes

[ \rho_C^+(I_\Sigma^\flat)^{-1}

\bigl((I_L^\flat)^{-1}\otimes(I_R^\flat)^{-1}\bigr)B_C. ]

Orientations, Koszul signs, the plumbing weight, and the normal line remain explicit. A scalar formula appears only after trivializing those lines.

Resonance qualification

The generic ((n-3)!)-dimensional pairing cannot simply be specialized and inverted at (s_C=0). The twist is resonant there and the residue matrix has lower rank in the full (n)-point space.

In a channel-adapted basis,

[ m_n \sim \begin{pmatrix} s_C^{-1}\epsilon_C(m_L\otimes m_R)+O(1)& *\ & \end{pmatrix}, ]

and hence

[ m_n^{-1} \sim \begin{pmatrix} s_C\epsilon_C^{-1}(m_L^{-1}\otimes m_R^{-1})+O(s_C^2)&O(s_C)\ O(s_C)&* \end{pmatrix}. ]

The (s_C) factor cancels the amplitude covector’s pole. Index raising is monoidal on this associated channel block, not by inversion of (\operatorname{Res}_C m_n).

Three-column scalar-to-surface target

The full target is

[ \operatorname{gr}{Z}\mathcal A\Sigma^{\rm scalar} \xrightarrow{\ \chi_\Sigma\ } (H_\Sigma^-)^\vee \xrightarrow{\ (I_\Sigma^\flat)^{-1}\ } H_\Sigma^+, ]

with vertical maps

[ \Delta_C^{\rm Rees}, \qquad B_C, \qquad \rho_C^+. ]

The second square is cyclicity or boundary compatibility of the perfect pairing. The first square is the missing scalar comparison theorem:

[ B_C\chi_\Sigma \stackrel{?}{=} (\chi_L\otimes\chi_R)\Delta_C^{\rm Rees}. ]

For a normal grade (r), the Rees cut uses graded convolution rather than a same-grade tensor product:

[ \Delta_C\operatorname{gr}^{r}

\sum_{a+b+c=r-w_C} \operatorname{gr}^{a}\otimes \operatorname{gr}^{c}\eta_C\otimes \operatorname{gr}^{b}. ]

Derived normal data may be required when scalar degenerations are not transverse.

Cut-kernel obstruction

Even if every cut in the three-column diagram commutes, cuts do not determine a unique surface function. The solution set

[ \left{ \widehat{\mathsf J}\Sigma: \mathbf\Delta\Sigma\widehat{\mathsf J}_\Sigma

\text{prescribed cuts} \right} ]

is a torsor for

[ \mathcal K_\Sigma

\bigcap_C\ker\Delta_C. ]

Thus a mapping-class- and sewing-compatible primitive

[ \omega_\Sigma\in\mathcal K_\Sigma ]

or a natural splitting of the total-cut sequence is additional required data. The known punctured-disk constant term already rejects the zero-primitive choice.

This is not a defect of the on-shell class ([({\rm Pf}’A)^2]). It is the expected loss of local contact information under cuts.

Internal state sewing

The BAS/KLT kernel contracts ordering indices. Channel state coevaluation is separate data of the half-object:

  • (\mathsf J): scalar coevaluation;
  • (\mathsf G): physical coevaluation on (q_I^\perp/\langle q_I\rangle);
  • biadjoint colour: internal Killing metrics.

These are not new primitive half-integrands, but omitting them makes the gluing formula ill typed.

Concrete ownership tests

YM: six-point (s_{123}) channel

  1. Form the (6\times6) BAS matrix in complementary KLT bases.

  2. Construct channel residue maps (R_{123}^\pm).

  3. Verify

    [ \operatorname*{Res}{s{123}=0}m_6

    (R_{123}^-)^{\mathsf T} (m_4\otimes m_4) R_{123}^+ ]

    with the selected orientation convention.

  4. Verify the leading (s_{123}(m_4^{-1}\otimes m_4^{-1})) inverse block.

  5. Apply it to the NLSM grade covector and recover (\mathsf J_4\otimes\mathsf J_4).

  6. In the (\mathsf G)-(\mathsf J) pairing, show that the only additional contraction is the physical transverse-polarization coevaluation.

Frost: six-point disk comparison

  1. Compute (R_6=\operatorname{in}Z G^{\rm scalar}{D_6}).
  2. Verify the first comparison square on each allowed (3|3) channel and zero residue on forbidden channels.
  3. Raise the compatible channel covector and check (\rho_C\mathsf J_6=\mathsf J_4\otimes\mathsf J_4).
  4. Subtract the Cut-Equation completion from (R_6); the remainder must define a cyclic, ordering-compatible primitive (\omega_{0,6}).

Plumbing dependence, failure of iterated-cut coassociativity, a singular induced channel pairing, or a non-covariant primitive falsifies intrinsic surface naturality.

Decision

Treat genus-zero CHY factorization of ((\operatorname{Pf}’A)^2) as established. Treat pairing monoidality as established only on the correctly oriented nearby-cycle channel quotient. Keep the scalar comparison (\chi_\Sigma) and the cut-kernel primitive (\omega_\Sigma) as separate open Frost problems.