Punctured-Torus Hostile Test of the Resolved Surface Counit

Record

Date: 2026-08-13

Status: the smallest genuine handle test closes coefficient by coefficient, without reconstructing the answer from Cut-equation uniqueness. On the once-punctured torus, two cyclic three-point counits sew through the theta fatgraph in nine resolved sectors. Three sectors contain one closed polarization circuit and six contain none. Termwise Brauer augmentation sends all nine sectors to the scalar state and gives

[ G_{T_{1,1}}^\phi=\frac13x_{11}^3. ]

The nonseparating orbit Cut gives the annulus function (x_{11}^2) directly. By contrast, evaluating the (D)-circuits before cutting fails to commute with the Cut by

[ \frac{2(D-1)}9x_{11}^2. ]

Thus this example does not merely agree with the general theorem. It exposes the obstruction that the resolved order of operations removes.

Reproducible certificate:

research/nima/check_punctured_torus_counit.rs

Why this is the first hostile topology

The planar one-loop one-point test of entry 46 contains a closed polarization circuit, but its surface is a punctured disk or annulus. It has no handle and does not support the one-holed-torus (3S) relation.

The first scalar surface in the published Cut-equation examples that does both jobs is the torus with one puncture. It has a single theta fatgraph. The three edges of a marked theta presentation lie in one mapping-class orbit, and the ribbon automorphism group has order three. Consequently its scalar surface function is

[ \boxed{ G_{T_{1,1}}^\phi

\frac1{3}x_{11}^{3}. } ]

Cutting one representative of the orbit opens the handle to the annulus. The published Cut equation reads

[ \partial_{x_{11}}G_{T_{1,1}}^\phi

G_{\rm annulus}^\phi

x_{11}^{2}. ]

We use these formulas only as target data. The calculation below derives both coefficients from the resolved tree sectors and their sewings; it does not use the uniqueness argument of entry 48.

The local three-point input

In scalar-scaffold variables, the three-gluon amplitude is

[ \begin{aligned} A_3^{\rm YM} ={}&X_{14}X_{26}+X_{36}X_{24}+X_{25}X_{46}\ &-X_{25}X_{36}-X_{14}X_{36}-X_{14}X_{25}. \end{aligned} ]

Its three cyclic pairwise-counit sectors are

[ U_0=\partial_{X_{14}}\partial_{X_{26}}, \qquad U_1=\partial_{X_{36}}\partial_{X_{24}}, \qquad U_2=\partial_{X_{25}}\partial_{X_{46}}. ]

They obey, separately,

[ \boxed{ U_iA_3^{\rm YM}=1, \qquad i=0,1,2. } ]

Rotation by two scaffold labels cyclically permutes both the amplitude and the three sectors. Over the rational coefficient ring appropriate to a surface function, the manifestly cyclic local representative is therefore

[ u_3^{\rm cyc}=\frac13(U_0+U_1+U_2). ]

The integral deletion resolution of entry 43 is still the chain-level source. The average is used here because the target theta graph already carries the rational symmetry factor (1/3).

The resolved theta sewing table

The state part of (U_i) pairs two of the three polarization flags and sends the remaining flag into the coefficient strand. Call that remaining flag the singleton of sector (i). Sew two three-point vertices along the three theta edges, and denote the resolved result for local sectors ((i,j)) by (\kappa_{ij}).

There are two cases.

  1. If (i=j), the common singleton makes a through coefficient strand, while the other two theta edges close into one polarization circuit.
  2. If (i\ne j), all three theta edges belong to the through strand and no polarization circuit closes.

Thus the circuit number is exactly

[ \boxed{ c(\kappa_{ij})=\delta_{ij}. } ]

The raw Brauer evaluation table is

[ \left( \begin{array}{ccc} D&1&1\ 1&D&1\ 1&1&D \end{array} \right). ]

This is already a useful falsifier. The number of state circuits is not the graph Betti number and is not constant across resolved sectors.

Let (a,b,c) denote the three marked theta edges. Including the two local cyclic averages and the theta automorphism factor gives the resolved surface coefficient

[ \boxed{ \mathcal U_{\Theta}^{\rm res}

\frac1{27}x_ax_bx_c \sum_{i,j=0}^{2}[\kappa_{ij}]. } ]

If the state patterns are evaluated prematurely, this becomes

[ \operatorname{ev}D(\mathcal U{\Theta}^{\rm res})

\frac{3D+6}{27}x_ax_bx_c

\frac{D+2}{9}x_ax_bx_c. ]

The resolved augmentation instead acts on each pattern before forgetting it:

[ \begin{aligned} \epsilon_{\rm Br}(\mathcal U_{\Theta}^{\rm res}) &= \frac1{27}x_ax_bx_c\sum_{i,j}1\ &= \boxed{\frac13x_ax_bx_c}. \end{aligned} ]

After mapping-class quotient,

[ x_a=x_b=x_c=x_{11}, ]

this is the published scalar surface function (x_{11}^3/3). The equality has been obtained by explicit local differentiation and state sewing, not by integrating its Cut.

The nonseparating Cut, before and after resolution

Open theta edge (k). A closed polarization circuit survives precisely when both local singleton sectors equal the opened edge:

[ \boxed{ c_k(\kappa_{ij})

\delta_{ik}\delta_{jk}. } ]

For each marked edge, the raw opened table therefore contains one (D)-valued sector and eight circuit-free sectors. The orbit Cut sums the three edge representatives. After termwise augmentation its coefficient is

[ \frac1{27} \sum_{k=0}^{2}\sum_{i,j=0}^{2}1

\frac{27}{27} =1. ]

The two uncut theta edges give (x_{11}^2), so directly

[ \boxed{ \Delta_{x_{11}} \epsilon_{\rm Br}(\mathcal U_{\Theta}^{\rm res})

x_{11}^2

G_{\rm annulus}^\phi. } ]

The factor three in the orbit Cut cancels the theta automorphism factor. This is the coefficient-level orbit–stabilizer mechanism behind (\partial_x(x^3/3)=x^2).

The obstruction seen by premature state evaluation

The same calculation before (D\mapsto1) does not form a Cut square.

Evaluate first and then differentiate the orbit variable:

[ \partial_x \left( \frac{3D+6}{27}x^3 \right)

\frac{D+2}{3}x^2. ]

Cut the resolved patterns first and then evaluate their remaining circuits:

[ \frac{3D+24}{27}x^2

\frac{D+8}{9}x^2. ]

Their difference is

[ \boxed{ \left( \partial_x\operatorname{ev}_D

\operatorname{ev}D\Delta_x \right) \mathcal U{\Theta}^{\rm res}

\frac{2(D-1)}9x^2. } ]

This is nonzero for the physical polarization dimension. It vanishes exactly after the scalar state augmentation (D\mapsto1). Therefore:

The resolved state cover is not dispensable bookkeeping. Cutting can open a circuit, so a circuit evaluated before the Cut cannot be recovered from the already-summed coefficient.

This gives an explicit reason that a global genus normalization or a naïve substitution on an already-sewn function cannot define the surface counit.

Coefficient-level (3S) covariance

Take primitive slopes (a,b\in\mathbb Z^2) with (\det(a,b)=1), and set

[ c=a+b. ]

The three slopes form a Farey triangle. Transport around its three (S)-moves simultaneously permutes:

  • the three marked theta edges;
  • the singleton sector at the left vertex;
  • the singleton sector at the right vertex;
  • the edge selected by a Cut.

Both tables are invariant under every simultaneous permutation:

[ \delta_{ij}

\delta_{\rho(i)\rho(j)}, ]

[ \delta_{ik}\delta_{jk}

\delta_{\rho(i)\rho(k)}, \delta_{\rho(j)\rho(k)}. ]

The second displayed product is ordinary multiplication; it records the same single surviving circuit after relabelling. Hence the complete coefficient carrier, not merely its augmented scalar value, is transported consistently.

For the oriented chart frames

[ F_a=(a,b), \qquad F_b=(b,-a), \qquad F_c=(c,-a), ]

the transition matrices still obey

[ F_a^{-1}F_c,F_c^{-1}F_b,F_b^{-1}F_a=1. ]

Thus there is no (3S) holonomy in the explicitly populated coefficient table. This is stronger than the earlier empty chart check: the state and Cut coefficients have now been carried around the triangle.

Marked physical specialization

The mapping-class-quotiented surface function has only one variable (x_{11}). As emphasized in the Cut-equation paper, a general nonplanar surface function is not by itself a conventional loop integrand: distinct propagators in one mapping-class orbit need distinct momentum assignments.

The physical specialization must therefore be made on the marked cover before the orbit variables are identified. With theta-edge momenta (P_a,P_b,P_c),

[ x_a\mapsto\frac1{P_a^2}, \qquad x_b\mapsto\frac1{P_b^2}, \qquad x_c\mapsto\frac1{P_c^2}, ]

the augmented coefficient becomes

[ \boxed{ \frac1{3P_a^2P_b^2P_c^2}. } ]

For a vacuum routing one may take (P_c=-(P_a+P_b)). This is scaleless after massless loop integration, but the marked rational integrand identity is exact. No additional contact monomial is generated because the counit acts on the resolved coefficient carrier and leaves the three propagator variables untouched.

This also identifies a necessary refinement of the slogan “physical specialization”: beyond the planar limit it is a map from the marked MCG cover, not from the one-variable surface-function quotient.

What this proves and what it does not

Proved directly in this entry:

  • all three local three-point counit sectors give the scalar cubic vertex;
  • the complete nine-sector theta sewing table;
  • the pattern-dependent circuit count (c(\kappa_{ij})=\delta_{ij});
  • the scalar coefficient (1/3) without Cut-equation reconstruction;
  • the nonseparating torus-to-annulus Cut coefficient (1);
  • the raw pre-augmentation Cut defect (2(D-1)/9);
  • covariance of the populated coefficient and Cut tables under (3S);
  • exact marked momentum specialization of the scalar image.

Not proved:

  • existence of one differential operator acting on an already-sewn two-loop Yang–Mills vacuum integrand and producing the scalar integrand;
  • derivation of the resolved nine-sector carrier from such a post-sewing operator rather than from the established local tree counits;
  • a nonvacuum genus-one example with external physical gluon states;
  • survival of a strict point-set representative after forgetting the marked cover and imposing momentum homology.

The first two omissions are the strictification problem already isolated in entry 48. The third is now the next genuinely stronger physical test. The vacuum theta graph tests the modular state algebra, the nonseparating Cut, the automorphism factor, and (3S), but not an external S-matrix element.

Executable evidence

The Rust certificate verifies:

  • exact action of the three second-order transmutation sectors on (A_3^{\rm YM});
  • cyclic invariance of the six-term three-point polynomial;
  • all nine closed theta sewings;
  • all twenty-seven opened-edge sewings;
  • 162 simultaneous slope/state/Cut covariance squares;
  • 308 oriented Farey (3S) chart triangles.

The finite enumeration audits signs, normalization, circuit opening, and orbit factors. The Kronecker-delta formulas above are the all-sector proof.

Primary sources

  • Arkani-Hamed, Frost, and Salvatori, The Cut Equation, especially the annulus and punctured-torus surface functions, the symmetry-factor discussion, and the warning that nonplanar surface functions are not already conventional integrands: https://arxiv.org/abs/2412.21027.
  • Carrôlo and Figueiredo, How gluon leading singularities discover curves on surfaces, for the resolved contraction-curve interpretation of gluon state sums and the all-loop closed-curve rule: https://arxiv.org/abs/2512.17019.
  • Backus and Figueiredo, Scaffolding Residues and Transmutations, for the tree and one-loop transmutation operators used in entries 42 and 46: https://arxiv.org/abs/2505.17179.
  • Hatcher, Pants Decompositions of Surfaces, for the one-holed-torus (3S) relation: https://arxiv.org/abs/math/9906084.