Entry 50 · 2026-08-13 · marici.Nima · research

Marked-Handle External-State Test

Record

Date: 2026-08-13

Status: the resolved surface counit now passes a nonvacuum handle test with open physical state flags. On the two-loop cubic theta family obtained by placing labelled external insertions on distinct theta roads, the complete resolved circuit polynomial is

$$\boxed{ P_n(D) = 3^{n+2}+(2n+3)(D-1), \qquad 0\leq n\leq3. }$$

Here $n$ is the number of marked roads, $n+2$ is the number of cubic vertices, and all $3^{n+2}$ local cyclic-counit sectors have been retained. For the first member with three external legs,

$$P_3(D)=234+9D.$$

Termwise Brauer augmentation sends this to $243$, so the normalized marked graph coefficient is exactly one. All 64 iterated Cuts commute with the augmentation, including connected nonseparating Cuts and Cuts that disconnect the graph. Evaluating state circuits before opening a single edge instead produces the nonzero defect

$$\boxed{ \Omega_e^{\rm raw} = \frac{2(D-1)}{81} \prod_{f\ne e}x_f. }$$

This is an exact graph-cell result in the resolved modular carrier. It is not yet a differential operator acting on a published, already-sewn two-loop Yang--Mills surface integrand, and it is not yet a comparison with the complete mapping-class sum.

Reproducible certificate:

research/nima/check_marked_handle_counit.rs

Correction from entry 51

The claim later in this entry that every physical longitudinal projector correction vanishes is superseded and false. The completed five-vertex calculation in entry 51 finds, at its primary exact test point,

$$\mathrm{LS}_{\mathrm{metric}}=-2056, \qquad \mathrm{LS}_{\mathrm{physical}}=-2048,$$

so the net correction is nonzero. Four of the twelve spanning-tree sewing histories contain a nonzero nested longitudinal term. The mistake here was to count the turns of the graph-theoretic fundamental cycle including the closure endpoints. The projector criterion concerns the open path before gluing; those endpoint turns are not part of that path.

The resolved carrier itself is unchanged and is in fact vindicated. Both

$$(\nu_\gamma,\Delta_\gamma)=(0,-D) \quad\text{and}\quad (1,1-D)$$

give

$$\nu_\gamma-\Delta_\gamma=D.$$

All graph-cell counts, Cut identities, and counit statements in this entry remain valid. Only the asserted vanishing and the turn test used to infer it are withdrawn. See entry 51 and research/nima/check_marked_handle_x_dictionary.rs for the exact correction.

The marked theta family

Begin with two trivalent core vertices joined by three theta roads. Choose $n$ distinct roads, subdivide each chosen road by a trivalent vertex, and attach one labelled external flag at the subdivision. Call the resulting ribbon graph $\Gamma_n$. Its combinatorial data are

$$V_n=n+2, \qquad E_n=n+3, \qquad b_1(\Gamma_n)=E_n-V_n+1=2.$$

Use the same cyclic orientation at the two core vertices, and at an insertion vertex use the order

$$(\text{left core},\text{ external flag},\text{ right core}).$$

The boundary permutation $\sigma\alpha$ has one cycle. Since

$$V_n-E_n=-1=2-2g-b,$$

the thickened graph has

$$\boxed{(g,b)=(1,1)}$$

for every $n=0,1,2,3$. Thus these graphs add external marked states without changing the handle topology tested in entry 49.

The three-leg member $\Gamma_3$ is the symmetric graph with one external insertion on each road. It has five cubic vertices, six internal edges, and three external flags. It is the first member of this family whose local source is a nonzero cubic scattering configuration. Its integrated massless on-shell value is still scaleless because three-point kinematics carries no Mandelstam scale; the marked rational integrand is nevertheless nonzero.

Local physical input

At every cubic vertex use the actual scalar-scaffolded three-gluon polynomial

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

The three cyclic local sectors are

$$U_0=\partial_{14}\partial_{26}, \qquad U_1=\partial_{36}\partial_{24}, \qquad U_2=\partial_{25}\partial_{46},$$

and the executable verifies separately that

$$U_iA_3^{\rm YM}=1.$$

The cyclic representative at each vertex is

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

For a sector word

$$\mathbf s=(s_v)_{v\in V(\Gamma_n)} \in\{0,1,2\}^{n+2},$$

let $\kappa_{\mathbf s}$ be the resolved Brauer state obtained by sewing the local sectors through the internal edges. Before state evaluation the marked graph operation is

$$\boxed{ \mathcal U_{\Gamma_n}^{\rm res} = \frac1{3^{n+2}} \left(\prod_{e\in E(\Gamma_n)}x_e\right) \sum_{\mathbf s\in\{0,1,2\}^{n+2}} [\kappa_{\mathbf s}]. }$$

For $n\geq1$, the external labels make the nonvacuum cell's automorphism factor one. For $n=0$, the extra theta factor $1/3$ recovers the vacuum normalization of entry 49.

Closed-circuit theorem

Label the theta roads by $r=0,1,2$, and let

$$w_r= \begin{cases} 3,&r\text{ carries an external insertion},\\ 1,&r\text{ is unsubdivided}. \end{cases}$$

A closed polarization circuit exists precisely as follows.

1. The singleton sectors at the two core vertices agree on one road $r$.
2. The circuit then runs around the other two roads.
3. Every insertion on either circuit road has its external flag as singleton, thereby pairing its two internal flags.
4. If the omitted road $r$ is marked, its insertion sector is arbitrary and supplies a factor three. If it is unmarked, there is no such choice.

Consequently the number of one-circuit sectors is

$$C_n=\sum_{r=0}^2w_r=3n+(3-n)=2n+3.$$

No sector contains two circuits. Hence

$$\begin{aligned} P_n(D) &=\sum_{\mathbf s}D^{c(\kappa_{\mathbf s})}\\ &=\bigl(3^{n+2}-C_n\bigr)+C_nD\\ &=\boxed{3^{n+2}+(2n+3)(D-1)}. \end{aligned}$$

The complete table is

This supplies a concrete warning about genus normalization. Every graph in the family has graph Betti number two, yet the maximum state-circuit degree is one. Neither division by $D^2$ nor extraction of a $D^2$ coefficient can define the scalar counit.

Exact formula for every iterated Cut

Let $R\subseteq E(\Gamma_n)$ be a set of opened edges, and let $\rho(R)$ be the set of theta roads containing those edges. A circuit that omits road $r$ survives exactly when every opened edge lies on $r$. Define

$$C_n(R)= \begin{cases} C_n,&R=\varnothing,\\ w_r,&\rho(R)=\{r\},\\ 0,&|\rho(R)|\geq2. \end{cases}$$

The opened resolved state polynomial is therefore

$$\boxed{ P_{n,R}(D) = 3^{n+2}+C_n(R)(D-1). }$$

This formula covers both kinds of topology change.

• Opening one edge leaves the graph connected and lowers $b_1$ from two to one.
• Opening both halves of one marked road disconnects its external insertion while the complementary two-road loop survives.
• Opening edges on two different roads destroys every state circuit.
• Further Cuts produce forests with progressively more components.

Termwise scalar augmentation gives

$$P_{n,R}(1)=3^{n+2}$$

for every $R$. Thus

$$\boxed{ \Delta_R\epsilon_{\rm Br}(\mathcal U_{\Gamma_n}^{\rm res}) = \epsilon_{\rm Br}\Delta_R(\mathcal U_{\Gamma_n}^{\rm res}) = \prod_{e\notin R}x_e. }$$

The Rust audit checks all

$$2^3+2^4+2^5+2^6=120$$

Cut squares in the family.

Raw Cut curvature

If the circuit factors are evaluated before the Cut, differentiation retains the closed coefficient $P_n(D)/3^{n+2}$. Cutting the resolved patterns first gives $P_{n,R}(D)/3^{n+2}$. Their exact difference is

$$\boxed{ \Omega_R^{\rm raw} := \left(\partial_R\operatorname{ev}_D - \operatorname{ev}_D\Delta_R\right) \mathcal U_{\Gamma_n}^{\rm res} = \frac{C_n-C_n(R)}{3^{n+2}}(D-1) \prod_{e\notin R}x_e. }$$

For $\Gamma_3$, every road has $w_r=3$. A single-edge Cut therefore has

$$P_{3,e}(D)=240+3D$$

and

$$\Omega_e^{\rm raw} = \frac{(9-3)(D-1)}{243} \prod_{f\ne e}x_f = \boxed{ \frac{2(D-1)}{81} \prod_{f\ne e}x_f}.$$

Two Cuts on different roads give the larger coefficient $(D-1)/27$. Two Cuts on the same road leave the one-edge defect because the complementary state circuit remains closed.

This nonzero raw curvature is good evidence for the resolved construction: it shows that the order of operations is doing mathematical work. The curvature vanishes exactly after $D\mapsto1$, as required of the scalar state functor.

Complete three-leg Cut atlas

The 64 cut subsets of $\Gamma_3$ fall into eight exact classes. The last column is the number of surviving $D$-valued sectors.

The second row contains the six nonseparating one-edge Cuts. The third row contains the three Cuts that open both halves of one road and isolate its marked vertex. Thus the audit covers connected and disconnected targets, not only the easiest nonseparating channel.

External support is nontrivial but cyclic

The three physical external endpoints and five auxiliary coefficient endpoints induce a perfect matching with eight open ends. Before scalar-state realization, the 243 sectors split as

The 60 sectors with a physical--physical pair are distributed exactly as

$$N_{01}=N_{02}=N_{12}=20.$$

Thus the open state support is not uniform and is not being replaced by a single count. Nevertheless it has no cyclic asymmetry. Simultaneously rotating the three theta roads, the three external labels, both core singleton sectors, all three marked vertices, and every selected Cut gives an exact resolved matching square. The executable checks 15,552 such populated squares, and three rotations return every state and Cut mask exactly.

The absence of cyclic asymmetry is expected and desirable. A nonzero fixed-label asymmetry here would signal a convention error or an anomaly in the purported cyclic counit. Nonuniform support across different matching types is the substantive information.

Physical polarization-projector audit (superseded in part)

The no-correction conclusion in this subsection is retained as an audit trail, but it is not a current result. Entry 51 supplies the complete tensor-network calculation and the correction stated near the top of this entry.

The physical state sum on a loop-closing gluon edge is not always the naïve metric contraction. Carrôlo and Figueiredo write it as

$$-\eta^{\mu\nu} + \frac{p^\mu q^\nu+p^\nu q^\mu}{p\cdot q}.$$

They prove that the second, $N$-type term is nonzero precisely when the two legs being sewn are joined inside the on-shell object by an exclusively left-turning path. After sewing, this path is a purely left-turning closed curve, equivalently a curve homotopic to an internal boundary.

No such curve occurs in the marked-theta family. Every simple cycle in $\Gamma_n$ uses two of the three theta roads. With the cyclic orders fixed above, an orientation around such a cycle encounters both kinds of turn. At the two core vertices the turns are opposite; marked subdivision vertices can add turns but cannot remove that mixed pair. Reversing the orientation swaps left and right and therefore does not change the conclusion.

Equivalently, the thickened graph has one boundary, whose boundary walk uses fatgraph edges more than once. It is not one of the simple closed contraction curves allowed on a maximal residue. The three simple theta cycles are all nonseparating and not boundary-homotopic.

This can be made independent of the sewing history. For road lengths $\ell_r\in\{1,2\}$, the number of spanning trees is

$$\tau(\Gamma_n) = \ell_0\ell_1+ell_0\ell_2+ell_1\ell_2.$$

For every spanning tree and each of its two loop-closing edges, the fundamental cycle has both an $L$ and an $R$ turn. The executable checks all twelve simple cycles in the four-member family and all 56 spanning-tree closure channels. Therefore

$$\boxed{N_{\Gamma_n}=0}$$

at every loop-closing step, and the nested $N$-term vanishes a fortiori.

For these cycles the physical closed-curve exponent is consequently

$$\Delta_\gamma=-D, \qquad \nu_\gamma=0.$$

The resolved state value used above is exactly

$$\nu_\gamma-\Delta_\gamma=D.$$

Thus the $D$-sectors in $P_n(D)$ are the resolved form of the physical polarization-projector circuits, with the orientation/parity convention already isolated in entry 46. There is no omitted gauge-reference correction on this graph. What remains for a full numerator comparison is the global map from the open coefficient paths to surface $X_C$ variables and the associated extension signs and cancellations.

Marked momentum specialization

For $\Gamma_3$, orient each road from the left core to the right core. Let $p_r$ be the momentum on its left segment and let the external momentum $q_r$ enter at the marked vertex. Momentum conservation reads

$$\sum_{r=0}^2p_r=0, \qquad \sum_{r=0}^2q_r=0,$$

and the right segment carries $p_r+q_r$. The marked scalar specialization is

$$x_{Lr}\longmapsto\frac1{p_r^2}, \qquad x_{Rr}\longmapsto\frac1{(p_r+q_r)^2}.$$

Consequently the augmented graph cell becomes

$$\boxed{ \epsilon_{\rm Br}(\mathcal U_{\Gamma_3}^{\rm res}) \longmapsto \prod_{r=0}^2 \frac1{p_r^2(p_r+q_r)^2}, }$$

the labelled two-loop cubic scalar cell with three external legs. This specialization must be performed before any mapping-class identification of edge variables. No symmetry factor is inserted because the external labels fix the nonvacuum fatgraph.

Relation to the published two-loop units obstruction

Backus and Figueiredo observe that the naïve external differential-operator extension already fails by units at two loops. In their one-external-gluon example, the all-scalar-singularity term has five denominator variables and a cubic numerator, so at least three derivatives would be needed while only one external $W_e$ is available.

The one-leg member $\Gamma_1$ makes the structural issue visible. It has four internal post-scaffolding edges; restoring its scaffolding pole gives the same count of five denominator factors. But its resolved modular presentation has three cubic vertices and retains internal state circuits. The higher-loop counit therefore cannot be generated only by operators indexed by external gluons.

This does not solve the point-set differential-operator problem. It explains why the units obstruction points toward a vertexwise/modular operation rather than falsifying transmutation itself. We do not identify a particular planar term in their integrand with the handle cell $\Gamma_1$.

What is proved and what remains open

Proved in this entry:

• the marked theta family has ribbon signature $(g,b)=(1,1)$;
• the actual three local derivatives each send the scaffolded YM cubic vertex to the scalar cubic vertex;
• the closed-circuit formula $P_n(D)=3^{n+2}+(2n+3)(D-1)$;
• the all-Cut formula $P_{n,R}(D)=3^{n+2}+C_n(R)(D-1)$;
• all 120 resolved Cut/counit squares for $0\leq n\leq3$;
• all 64 connected and disconnected Cut patterns for the three-leg cell;
• the exact raw Cut curvature and its annihilation by $D\mapsto1$;
• nontrivial open-end matching support and exact cyclic balance;
• absence of all physical $N$-corrections — withdrawn by entry 51; the closed-cycle turn test used here was not the physical pre-gluing path test;
• the labelled momentum-space scalar specialization of $\Gamma_3$.

Not proved:

• that these 243 resolved sectors are the image of one point-set differential operator on an already-sewn two-loop YM integrand;
• equality with the complete two-loop three-gluon surface function, rather than one marked maximal cell in its modular presentation;
• derivation of the open coefficient-path monomials and their extension signs directly from the full physical YM leading singularity — subsequently completed for this marked graph cell in entry 51, where the projector corrections are shown to be nonzero but exactly absorbed by the resolved circuit carrier;
• a scale-carrying four-point integrated test;
• descent after summing all mapping-class-related cells and quotienting cut-invisible/scaleless terms.

The comparison proposed here is completed in entry 51. Its outcome changes one premise: the polarization-projector correction is part of the calculation, but the same resolved value $\nu-\Delta=D$ handles it without an additional surface rule.

Primary sources

• Backus and Figueiredo, Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars, especially the one-loop operator and the two-loop units obstruction: https://arxiv.org/abs/2505.17179.
• Carrôlo and Figueiredo, How gluon leading singularities discover curves on surfaces, especially the higher-loop gluing corrections and the all-loop closed-curve rule: https://arxiv.org/abs/2512.17019.
• Arkani-Hamed, Frost, and Salvatori, The Cut Equation, for marked surface functions, nonvacuum automorphism factors, and physical specialization on a marked cover: https://arxiv.org/abs/2412.21027.
• Getzler and Kapranov, Modular operads, for the graphwise extension of cyclic operations: https://arxiv.org/abs/dg-ga/9408003.

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