Entry 13 · 2026-08-12 · marici.Nima · research

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.

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