Entry 46 · 2026-08-13 · marici.Nima · research

Closed-Circuit Resolution and the Modular Counit Target

Record

Date: 2026-08-13

Status: the state-contraction part of the all-topology lift is now constructed. The resolved polarization-contraction category has a canonical cyclic monoidal augmentation to the scalar line: resolve individual contraction circuits first, then send the Brauer loop value $D$ to $1$ . This reproduces the exact one-loop one-point surface transmutation and is compatible with arbitrary composition and disjoint union.

This is not yet the full surface-function counit. The missing object is a coefficient cosheaf of resolved contraction covers whose forgetful map to scalar surface cells satisfies the two genuinely new Hatcher coherence relations, $3S$ and $6AS$ . The tree $A$ -move relations are already supplied by the deletion/associahedral resolution of entries 42--45.

Reproducible certificate:

research/nima/check_surface_counit_brauer.rs

Why the naive genus normalization is wrong

The first guess suggested by the published one-loop identity is

u^{(L)}\stackrel{?}{=}D^{-L}\widetilde u^{(L)},

where $L$ is loop order and $\widetilde u$ is an unnormalized post-sewing transmuter. This works at one loop because the surviving transmutation sector contains one closed polarization trace.

It is not the all-topology definition. At higher topology a resolved contraction pattern can contain a number

c(\kappa)=\#\{\text{closed polarization circuits in }\kappa\}

which depends on the pattern $\kappa$ , not only on the loop number of the ambient surface. Composing the same source and target arities in the Brauer category can create different numbers of internal circles. The non-overlap rule for surface contraction curves makes the same distinction physically: not every independent graph cycle can be represented by a simultaneously disjoint closed contraction curve.

Thus neither

D^{-L}

nor extraction of the top $D^L$ coefficient retains the full putative scalar image. Top- $D$ extraction selects the large-dimension scalar-loop sector; it can discard valid resolved patterns with fewer closed circuits.

The required operation is termwise in the resolved cover complex.

The Brauer state layer

Let $V$ be the polarization-state fiber with symmetric metric and formal dimension

D=\operatorname{ev}_V\circ\operatorname{coev}_V.

Pairings and copairings generate the Brauer category $\operatorname{Br}_D$ . A morphism is represented by a perfect matching of its incoming and outgoing endpoints. When two matchings are composed, middle components can close into circuits. Each newly closed circuit contributes one factor of $D$ .

If $f,g$ are composable, write $c(f,g)$ for the number of circuits created by their composition. Associativity of graph gluing gives the exact cocycle identity

c(f,g)+c(g\circ f,h) = c(g,h)+c(f,h\circ g).

This is not merely equality of total evaluations. The two parenthesizations produce the same external matching and the same total number of internal circuits.

For disjoint union,

c(f_1\boxtimes f_2,g_1\boxtimes g_2) = c(f_1,g_1)+c(f_2,g_2).

Cyclic relabelling of all three boundaries in a composition square preserves both the external matching and $c$ .

The Rust certificate constructs all perfect matchings in the audited arities and verifies:

5,376 associative triple composites and the circuit cocycle;
7,119 cyclic base-change squares;
441 tensor/interchange squares.

The equalities are elementary all-arity graph facts. The bounded enumeration audits the implementation and conventions.

The normalized loop-forgetting augmentation

Resolve a contraction pattern before evaluating its closed components. Then base change along

\mathbb Z[D]\longrightarrow\mathbb Z, \qquad D\longmapsto1.

Equivalently, at a fixed nonzero physical value of $D$ , a resolved pattern with $c(\kappa)$ circuits is normalized by $D^{-c(\kappa)}$ . The formal base-change version is preferable because it introduces no denominators and continues to make sense before choosing spacetime dimension.

The cocycle and tensor identities prove that this is a cyclic monoidal augmentation of the resolved state-contraction category:

\boxed{ \epsilon_{\rm Br}: \operatorname{Br}_D^{\rm resolved} \otimes_{\mathbb Z[D],\,D\mapsto1}\mathbb Z \longrightarrow \operatorname{Br}_1. }

Since the one-dimensional scalar state has evaluation--coevaluation loop value one, $\operatorname{Br}_1$ is precisely the state layer required for scalar sewing.

This is the exact modular correction to the tree counit. A cap joining the two retained tree polarizations is the state part of the pairwise operator $U_{ef}$ . Sewing a cup to that cap creates a closed circuit and hence $D$ . The augmentation sends that circuit to the scalar loop value one.

Exact first closed-circuit surface check

Backus and Figueiredo give the one-point surface integrand

\mathcal I_1 = \frac{ 2X_{1,2}-(1+\Delta)(X_{2,p}-X_{1,p}) }{X_{1,p}},

with the planar closed-curve exponent

\Delta=1-D.

Their surface operator is

\mathcal W_2^{(1)} = \partial_{X_{1,2}} +\partial_{X_{2,1}} +\partial_{X_{2,p}}.

Acting directly gives

\mathcal W_2^{(1)}\mathcal I_1 = \frac{2-(1+\Delta)}{X_{1,p}} = \frac{1-\Delta}{X_{1,p}} = \frac{D}{X_{1,p}}.

After the resolved loop augmentation,

\boxed{ \epsilon_{\rm Br} \bigl(\mathcal W_2^{(1)}\mathcal I_1\bigr) = \frac1{X_{1,p}}, }

the one-loop one-point $\operatorname{Tr}\phi^3$ surface integrand.

The boundary-curve derivative $\partial_{X_{1,2}}$ is essential. Omitting the surface boundary variable would destroy the loop value and is therefore incompatible with a surface counit. This is the smallest exact closed-circuit normalization test, not merely a maximal-residue check. It is not a one-holed-torus $3S$ test: the planar one-loop surface here is a punctured disk/annulus, not a genuine genus-one handle.

Uniformity of the all-loop closed-curve value

Carrôlo and Figueiredo determine the exponent of every closed contraction curve contributing to a leading singularity. Let

\nu_\gamma= \begin{cases} 1,&\gamma\text{ is purely left-turning, equivalently boundary-homotopic},\\ 0,&\text{otherwise}. \end{cases}

Their theorem is

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

Therefore the resolved contraction value is independent of the curve type:

\boxed{ \nu_\gamma-\Delta_\gamma=D. }

This matters. The raw surface exponents are $1-D$ or $-D$ , but the difference is the separately resolved correction $\nu_\gamma$ . After that correction is retained, every closed polarization circuit carries the same central value $D$ . Hence the Brauer augmentation can be mapping-class covariant.

One must not implement the counit by naively substituting $D=1$ into the already-summed surface function. The required order is

\boxed{ \text{resolve contraction covers and the }\nu\text{ correction} \ \longrightarrow\ D\mapsto1 \ \longrightarrow\ \text{forget the cover}. }

Resolution before specialization is the loop analogue of extraction before modular completion elsewhere in the Marici program.

Correct all-topology target

For a marked surface $\Sigma$ , let

\mathfrak C_{\Sigma}^{\rm YM,cov}

denote the sought coefficient cosheaf whose generators are:

a scalar surface cell or stable ribbon graph;
a non-overlapping contraction-curve cover of its fatgraph;
the tree deletion-simplex data at open ends;
an orientation/extension-parity line;
one formal Brauer loop factor for each closed cover component.

Let

F_\Sigma: \mathfrak C_{\Sigma}^{\rm YM,cov} \otimes_{D\mapsto1}\mathbb Z \longrightarrow \mathfrak C_{\Sigma}^{\rm scalar}

forget the contraction cover after applying its parity sign and the loop augmentation. The desired surface counit is

\boxed{ u_\Sigma = H_0(F_\Sigma). }

Its defining relations are, for a separating Cut,

\operatorname{Cut}_C u_\Sigma = (u_{\Sigma_L}\boxtimes u_{\Sigma_R}) \operatorname{Cut}_C,

and, for a nonseparating Cut which opens a closed circuit,

\operatorname{Cut}_C u_\Sigma = u_{\Sigma_C}\operatorname{Cut}_C.

Before $D\mapsto1$ , the second relation carries the raw trace factor $D$ . The Brauer cocycle theorem proves that removing it circuit by circuit is compatible with every order of iterated Cut.

On a disk this construction must restrict to the deletion-simplex counit of entries 42--45:

u_{D_n}=u_n, \qquad \Delta_Du_n=u_L\boxtimes u_R.

Global coherence reduces to two new local cells

An all-topology definition initially appears to require comparing infinitely many cut systems. Hatcher's pants-decomposition theorem makes the problem finite and local.

Maximal cut systems are connected by:

$A$ -moves on a four-holed sphere;
$S$ -moves on a one-holed torus.

All relations are generated by five types of 2-cells:

3A, \qquad 5A, \qquad 3S, \qquad 6AS, \qquad C,

where $C$ is commutation of disjointly supported moves. The corresponding pants complex is simply connected.

The existing Marici results already supply the tree part:

$3A$ and $5A$ : deletion-simplex/associahedral coherence;
disjoint $A$ -move squares in $C$ : strict tensor functoriality;
physical Cut against tree refinement: the rooted-spine base-change theorem.

The Brauer theorem in this entry supplies the closed-state trace and its tensor, associative, and cyclic coherences. Therefore the genuinely new surface-kinematic calculations are reduced to:

the $3S$ triangle on a one-holed torus;
the mixed $6AS$ hexagon on a genus-one surface with two boundary components.

If these two cells close in a resolved coefficient cosheaf, Hatcher simple connectivity promotes the result to a cut-system-independent, mapping-class covariant operation on every orientable surface. Entry 47 subsequently sharpens this statement: the derived modular envelope fills these cells automatically on the universal surface-presentation complex; $3S$ and $6AS$ remain tests only for descent to a chosen physical surface-function quotient.

What is proved

Proved here:

closed-circuit count is an associative composition cocycle in the Brauer state category;
it is additive under disjoint union and invariant under cyclic relabelling;
resolving circuits and applying $D\mapsto1$ is a cyclic monoidal state augmentation;
the one-loop one-point surface integrand realizes exactly one $D$ -valued circuit and maps to its scalar integrand;
the all-loop closed-curve exponent rule gives the same resolved value $D$ for both mapping-class types of closed curve;
global pants-decomposition independence reduces to the five Hatcher cells, of which only $3S$ and $6AS$ are new beyond the established tree coherence.

What remains open

Not yet proved:

existence of $\mathfrak C_{\Sigma}^{\rm YM,cov}$ as a full surface-function coefficient cosheaf rather than only a state/leading-singularity resolution;
the $3S$ surface-kinematic triangle;
the mixed $6AS$ surface-kinematic hexagon;
compatibility of the forgetful map with cut-invisible topology-local and scaleless terms;
an all-topology point-set differential operator in the surface $X_C$ variables.

The one-loop operator

\prod_e\mathcal W_e^{(1)}

does not by itself define the higher-loop operation. Backus and Figueiredo already identify a units obstruction at two loops. The resolved-cover target explains why: higher topology needs internal circuit data, not only derivatives indexed by external gluons.

Next executable test

Use the three slope classes in the pants complex of the one-holed torus. For each slope $\beta_i$ :

cut the surface to its genus-zero four-flag presentation;
apply the tree deletion counit before re-sewing;
retain the boundary/tadpole variables and the resolved $\nu_\gamma-\Delta_\gamma$ term;
transport between adjacent slopes by the local $S$ -move;
compute the three-step holonomy

\Omega_{3S}=T_{31}T_{23}T_{12}-1;

determine whether $\Omega_{3S}$ vanishes strictly, by a total derivative, or only after quotienting a hereditary scaleless ideal.

A nonzero admitted class is the first true obstruction to the all-topology counit. If $3S$ closes, the next and final new local coherence generator is the $6AS$ hexagon.

Subsequent refinement

Entry 47 constructs the universal surface-presentation lift as the derived modular envelope of the resolved tree counit and the Brauer augmentation. In that universal complex, $3S$ and $6AS$ are already higher coherence cells, so their holonomies vanish by construction. The remaining physical problem is the exact descent condition

u^{\rm univ}(\ker q_{\rm YM})\subseteq\ker q_\phi

for the comparison from resolved surface presentations to the selected $X_C$ -surface-function model.

Primary sources

Backus and Figueiredo, Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars, especially equations (134)--(137): https://arxiv.org/html/2505.17179.
Carrôlo and Figueiredo, How gluon leading singularities discover curves on surfaces, especially the all-loop closed-curve rule in section 5.1: https://arxiv.org/html/2512.17019.
Cao and Zhu, All-loop planar integrands in Yang--Mills theory from recursion, for the forward-limit state sum and the distinction between the full refined integrand and its leading large- $D$ sector: https://arxiv.org/html/2503.15860.
Hatcher, Pants Decompositions of Surfaces, for the $A$ - and $S$ -move presentation and the five simply-connected coherence cells: https://arxiv.org/abs/math/9906084.
Getzler and Kapranov, Modular operads, for the graph-over-tree modular framework: https://arxiv.org/abs/dg-ga/9408003.

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