Entry 9 · 2026-08-12 · marici.Nima · research

NLSM as a Curved Cut Completion

Record

Date: 2026-08-12

Status: disk-level criterion identified; zero-boundary punctured completion explicitly falsified.

Candidate branch

Let

$$R_\Sigma = \operatorname{in}_{Z_\Sigma} G_\Sigma^{\mathrm{scalar}}$$

be the initial scalar surface function at the rank-jump boundary, and let

$$Q_\Sigma=G_\Sigma^{\mathrm{QTDS}}$$

be the surface function generated by the proposed quartic Jordan/QTDS strictification.

The intended relation is

$$\mathrm{NLSM} \simeq H_{\mathrm{QTDS}}\circ\operatorname{gr}_Z,$$

followed by Cut-Equation completion.

What equality of cuts proves

Assume lower-complexity surfaces agree and every admissible cut of $R_\Sigma$ and $Q_\Sigma$ agrees. Then

$$\Delta_C(R_\Sigma-Q_\Sigma)=0 \quad\text{for every }C,$$

and hence

$$R_\Sigma-Q_\Sigma\in\mathcal K_\Sigma.$$

Equality of all cuts proves equality only modulo the cut-free contact sector.

Disk criterion

On an unpunctured disk, exact equality follows from:

$$\boxed{ \text{same cut data} + \text{same cut-free boundary data} \Longrightarrow \text{same surface function}. }$$

The Cut Equation presentation of NLSM contains independent even-point boundary interactions. In particular, the six-point function includes a nonzero six-point contact in addition to its two-quartic factorizations.

Therefore a raw one-colored quartic theory is not enough. A finite QTDS strictification can still work if it is colored or differential graded and:

1. auxiliary states generate the higher contacts;
2. homological transfer reproduces every NLSM cut-free disk operation;
3. the transferred cyclic pairing equals the graded scalar pairing.

The QTDS equivalence is consequently a quasi-isomorphism theorem to be proved, not a consequence of cut recursion alone.

Punctured counterexample

The minimal Cut Equation prescription takes higher-loop boundary values at $x\to0$ to vanish. The canonical NLSM function extracted from the scalar deformation differs already on the one-puncture two-point disk:

$$G^{\delta\text{-shift}}(12;p) - G^{\mathrm{Cut},\,\omega=0}(12;p) =2.$$

This constant is:

• invisible to all propagator cuts;
• scaleless after loop integration;
• nevertheless required for the exact surface Adler zero.

It is the first explicit physical failure of the zero-curvature Cut completion. It does not by itself disprove a modular QTDS completion capable of generating the same primitive.

Correct all-surface form

The canonical surface theory must be written as a curved completion:

$$\boxed{ G_\Sigma^{\mathrm{NLSM}} = \operatorname{CutComp}_\Sigma \left( \operatorname{gr}_ZG^{\mathrm{scalar}}; \omega_\Sigma \right), \qquad \omega_\Sigma\in\mathcal K_\Sigma. }$$

The first curvature datum is

$$\omega_{L=1,n=2}=2.$$

Higher $\omega_\Sigma$ must either be generated by the modular envelope of QTDS or fixed by a universal condition inherited from the scalar deformation, such as the surface Adler zero.

An integrated alternative is

$$G_\Sigma^{\mathrm{NLSM}} \sim \operatorname{CutComp}_\Sigma (\operatorname{gr}_ZG^{\mathrm{scalar}}) \pmod{\mathcal N^{\mathrm{her}}},$$

where $\mathcal N^{\mathrm{her}}$ is an integration-null ideal stable under every cut and sewing. Quotienting merely by “integrates to zero” is not functorial enough.

Consequence for strictification

Tree-level QTDS may remain finite while its arbitrary-surface closure is curved. After integrating out auxiliary colors, this can appear as an infinite tower of effective loop-contact terms.

The exact question is therefore:

$$\text{Does the curved modular envelope of QTDS generate every } \omega_\Sigma\text{ selected by the scalar rank degeneration?}$$

Next falsification tests

1. Compute the transferred six-point disk contact from QTDS.
2. Compute the QTDS one-puncture two-point primitive and test whether it equals $2$.
3. Determine whether the surface Adler zero recursively fixes all $\omega_\Sigma$.
4. Classify the cut kernel by topology and multiplicity.
5. Test whether the proposed scaleless quotient is hereditary under nonseparating sewing.

Prohibited overclaims

Do not claim that:

• matching all cuts proves exact equality;
• every scaleless term is cut-invisible;
• zero higher-loop boundary conditions define the canonical NLSM surface function;
• a quartic strictification automatically supplies higher contact operations;
• the public Cut Equation or inverse-KLT papers establish the QTDS quasi-isomorphism.

Sources

• The Cut Equation
• A Surface Integrand for the Inverse KLT Kernel

View source on GitHub