Entry 44 · 2026-08-13 · marici.Nima · research

Cut-Saturated Counit and Equivariant Torsion

Record

Date: 2026-08-13

Status: two all-arity tree-level field-theory results. First, the scaffold transmutation counit is group-like under every physical Cut after passage to the Cut-evaluation quotient. This is the correctly typed weakest factorization statement and identifies the remaining problem as comparison of two raw resolutions; it is not yet strong tensor-level naturality. Second, a unit counit has no integral cyclically invariant representative in the lattice generated by the pair, vertex, and full-deletion operators. The obstruction has order $n$ for odd $n$ and $n/2$ for even $n$.

Reproducible finite certificate for the orbit arithmetic:

research/nima/check_transmutation_counit_all_arity.rs

Why the ordinary annihilator quotient is too small a statement

Entries 41--43 produced the all-arity scalar counit class

$$u_n=[U_{ef}] \in Q_n, \qquad Q_n= \operatorname{Diff}^{\rm pol}_n/ \operatorname{Ann}(A_n^{\rm YM}),$$

where

$$U_{ef}A_n^{\rm YM}=A_n^{\operatorname{Tr}\phi^3}$$

Here $\operatorname{Diff}^{\rm pol}_n$ is generated by derivatives in the even-containing, polarization-type scaffold coordinates. These vector fields are tangent to every odd--odd physical pole. For every even-scaffold pair $\{e,f\}$, equality in $Q_n$ controls the final amplitude. It does not, by itself, say how a representative distributes over the two factors of a physical Cut.

The reason is elementary. On a channel $D=(X_{i,j}=0)$, scalar-scaffolded Yang--Mills factorization is

$$\operatorname{Res}_D A_n^{\rm YM} = \sum_{k\in L,\,m\in R} C_{km}(X) \frac{\partial A_L^{\rm YM}}{\partial X_{k,x_L}} \frac{\partial A_R^{\rm YM}}{\partial X_{m,x_R}},$$

with

$$C_{km}(X)=X_{k,m}-X_{k,j}-X_{m,i}$$

or its gauge-equivalent second form. Applying such a channel-tangent differential operator uses Leibniz on the coevaluation kernel as well as on the two lower amplitudes. Consequently a null operator at $n$ points can become a nonzero-looking sum of lower operator tensors whose total evaluation cancels.

Thus

$$D\equiv D'\pmod{\operatorname{Ann}(A_n^{\rm YM})}$$

does not imply equality term by term in a chosen free tensor product of lower differential-operator algebras.

The Cut-evaluation quotient

Apply a raw operator $D$ to the displayed factorization formula and collect all Leibniz terms. Denote the resulting lower-factor expression by

$$\Delta_D^{\rm raw}(D).$$

This expression includes the internal polarization coevaluation and the derivatives that hit its linear kernel. Let

$$\operatorname{CutEv}_D$$

be evaluation of such expressions on $A_L^{\rm YM}\boxtimes A_R^{\rm YM}$, followed by the internal state contraction, and define

$$K_D^{\rm cut}=\ker(\operatorname{CutEv}_D).$$

If $N A_n^{\rm YM}=0$, then

$$0=\operatorname{Res}_D(NA_n^{\rm YM}) = \operatorname{CutEv}_D \bigl(\Delta_D^{\rm raw}N\bigr).$$

Therefore

$$\Delta_D^{\rm raw} \operatorname{Ann}(A_n^{\rm YM}) \subseteq K_D^{\rm cut}.$$

This proves that the Cut descends canonically as

$$\boxed{ \Delta_D^{\rm ev}: Q_n \longrightarrow \mathcal B_D/K_D^{\rm cut}, }$$

where $\mathcal B_D$ is the lower-factor bridge-expression space. The class is independent of which gauge-equivalent representation of the gluing kernel is used, because those representations have the same Cut evaluation.

This is the minimal quotient in which a statement about the Cut of an amplitude-quotient operator is actually typed.

A factorization-compatible lift already exists

Cheung--Shen--Wen define the ordered transmuter

$$\mathcal T[\alpha] = \mathcal T_{\alpha_1\alpha_n} \prod_{r=2}^{n-1} \mathcal T_{\alpha_{r-1}\alpha_r\alpha_n},$$

with a trace seed and successive insertion operators. Their tree induction tracks the completeness relation across a factorization channel. For every planar channel compatible with $\alpha$, it gives, in the physical factorization quotient,

$$\boxed{ \Delta_D^{\rm CSW}\mathcal T[\alpha] = \mathcal T[\alpha_L,I_L] \boxtimes \mathcal T[I_R,\alpha_R] }$$

after the internal trace state is created; incompatible channels vanish. Equations (72)--(78) in their factorization proof are the local insertion identities that generate this formula.

After expressing polarization contractions in a fixed scaffold gauge chart while holding momentum invariants fixed, let $\widetilde{\mathcal T}[\alpha]$ denote this transmuter in the same coordinate differential algebra as $U_{ef}$. Backus--Figueiredo explicitly show that repeated $\mathcal W$-operations are not literally the same raw operator as $\mathcal T[\alpha]$, already at four points. Nevertheless both tree field-theory operators produce the same normalized planar scalar amplitude. Hence

$$U_{ef}-\widetilde{\mathcal T}[\alpha] \in \operatorname{Ann}(A_n^{\rm YM}).$$

The same equality holds for the lower-point counit classes. Applying the descended Cut now gives

$$\begin{aligned} \Delta_D^{\rm ev}[U_{ef}] &= \Delta_D^{\rm ev} [\widetilde{\mathcal T}[\alpha]]\\ &= [\widetilde{\mathcal T}[\alpha_L,I_L] \boxtimes \widetilde{\mathcal T}[I_R,\alpha_R]]\\ &= [u_L\boxtimes u_R]. \end{aligned}$$

Therefore

$$\boxed{ \Delta_D^{\rm ev}u_n=u_L\boxtimes u_R. }$$

This is the all-arity weak factorization theorem for the counit class. Passage to $K_D^{\rm cut}$ is deliberately coarse: equality there follows whenever the complete Cut evaluations agree. Its role is to type exactly what follows from amplitude equality and to prevent it from being mistaken for a raw coproduct.

There is nevertheless additional content in the comparison. The standard trace/insertion transmuter supplies a known decomposable, factorization- compatible representative in polarization variables, whereas the scaffold deletion operators supply a different amplitude-quotient representative with an intrinsic simplex of reference changes. The unresolved object is now the explicit Cut-kernel homotopy comparing those two resolutions.

Weak versus strong Cut naturality

The proved target is the Cut-evaluation quotient. A stronger statement would identify it with the plain tensor product of the two ordinary amplitude quotients:

$$\mathcal B_D/K_D^{\rm cut} \stackrel{?}{\simeq} Q_L\boxtimes Q_R.$$

Equivalently, after isolating the unique coevaluation bridge one would need to prove that every null lower-factor combination is generated separately by the left and right annihilators:

$$K_D^{\rm cut} \stackrel{?}{=} \operatorname{Ann}(A_L)\boxtimes\operatorname{Diff}^{\rm pol}_R + \operatorname{Diff}^{\rm pol}_L\boxtimes\operatorname{Ann}(A_R),$$

with the shared channel relations and gauge-state coend treated correctly. This equality is not established. Entangled gauge or shared-kinematic null relations may enlarge the left-hand side.

Accordingly:

• derived/evaluation factorization is proved;
• a strict local bridge representative remains open.

The proposed five-point expansion is still useful, but its role has changed. It no longer decides whether the counit class factorizes. It determines whether the scaffold deletion representative itself admits a canonical, local, separately quotiented coproduct and identifies the explicit homotopy relating it to the CSW lift.

The one-bridge theorem revisited

Entry 43 proved by degree and multilinearity that every nonzero term in $\Delta_D^{\rm raw}U_{ef}$ has exactly one global derivative hitting the linear kernel $C_{km}$. The present result fixes the total class of the remaining signed bridge sum:

$$\left[ \sum_{\text{one-bridge terms}} \right] = [u_L\boxtimes u_R] \quad\text{in}\quad \mathcal B_D/K_D^{\rm cut}.$$

Thus the three edge orbits of entry 43 cannot produce a genuine amplitude- level anomaly. Any discrepancy among the left, right, and crossing formulas is necessarily a representative-level Cut-kernel element. This locates the remaining datum as a gauge homotopy, not a failure of scalar transmutation.

Integral cyclic obstruction

There is a separate and exact symmetry issue. Let

$$M_n= \mathbb Z\langle U_{ef},T_e,R_e\rangle_{\rm free}$$

be the free reference-generator lattice of pair and vertex representatives, before imposing amplitude-annihilator relations, with augmentation

$$\epsilon(U_{ef})=1, \qquad \epsilon(T_e)=1, \qquad \epsilon(R_e)=-(n-2).$$

Let $C_n$ rotate the $n$ even scaffold labels. Unordered edges of cyclic distance $d$ form one orbit. Its size is

$$|\mathcal O_d| = \begin{cases} n,&d<n/2,\\ n/2,&n\text{ even and }d=n/2. \end{cases}$$

The vertex orbit always has size $n$. Hence the augmentation of every cyclically invariant integral combination lies in

$$\boxed{ \epsilon(M_n^{C_n})=g_n\mathbb Z, \qquad g_n= \begin{cases} n,&n\text{ odd},\\ n/2,&n\text{ even}. \end{cases} }$$

For $n\geq3$, $g_n>1$. Therefore no integral, unit-normalized, cyclically invariant counit representative exists in this lattice.

This can be stated cohomologically. For

$$0\longrightarrow\ker\epsilon \longrightarrow M_n \xrightarrow{\epsilon}\mathbb Z \longrightarrow0,$$

the connecting class

$$\delta(1)\in H^1(C_n,\ker\epsilon)$$

has order $g_n$. It is the obstruction to an invariant integral section of the augmentation. The conclusion is specifically about this natural presentation lattice. It does not forbid a new invariant primitive outside the span of the published reference-labelled generators.

The Rust certificate enumerates the edge orbits through $n=25$, verifies their partition, and checks the displayed augmentation ideal exactly.

Optimal cyclic representatives over characteristic zero

After adjoining $1/g_n$, the obstruction disappears. For odd $n$, every fixed cyclic distance gives

$$\epsilon_n^{(d)} = \frac1n \sum_{i\in\mathbb Z/n} U_{i,i+d}.$$

For even $n$, the antipodal orbit is smaller and gives the most economical dihedrally invariant edge representative:

$$\boxed{ \epsilon_n^{\rm antipodal} = \frac{2}{n} \sum_{i=0}^{n/2-1} U_{i,i+n/2}. }$$

It uses $n/2$ pair sectors rather than $n$ adjacent sectors or $\binom n2$ complete-graph sectors.

Alternatively one may adjoin a new fixed generator $b_n$ with

$$\epsilon(b_n)=1.$$

Geometrically this is a barycentric strictification: the rational average is the barycenter in the old coordinates, while $b_n$ treats that barycenter as a primitive fixed point. Producing $b_n$ as an intrinsic local scalar- surface operation, rather than merely adjoining it formally, is a meaningful construction problem.

Interpretation

The absence of asymmetric support or nonzero curvature found in the bounded audits was expected and useful. The deletion faces commute, so the ordinary descent object is flat. The first nontrivial obstruction lives one categorical level higher:

$$\text{flat deletion resolution} \quad+\quad \text{cyclic equivariance} \quad\Longrightarrow\quad \text{integral torsion }\delta(1).$$

Thus symmetry is strict on the counit class but only homotopy-coherent on its reference-labelled integral representatives. This is not a defect in the physical scalar amplitude. It says that the native operation is a resolved counit rather than one distinguished coordinate differential operator.

The same lesson applies to the larger Marici operation algebra. One should distinguish:

1. a physical class;
2. its factorization class in the Cut-saturated quotient;
3. a local chain representative;
4. an equivariant strict representative.

Equal amplitudes settle only the first. Entries 42--44 now settle the second for scalar transmutation and exhibit the complete deletion resolution; the third and fourth contain the remaining gauge-homotopy and torsion data.

Scope

The results here concern generic tree-level field theory.

• At finite $\alpha'$, repeated scaffold $\mathcal W$-operations produce shifted string integrals and are explicitly not the same construction as the standard CSW transmuter. No string-level group-like deletion counit is asserted.
• A loop/surface extension requires a well-defined surface Cut coend and treatment of closed-curve state sums. The Backus--Figueiredo loop relation is not promoted here beyond its published scope.
• The all-arity result is a class statement in the Cut-evaluation quotient, not a preferred off-shell or point-set chain map.

Next falsification target

At the first nontrivial five-point channel, compute

$$\Omega_D = \Delta_D^{\rm raw}U_{ef} - U_L\boxtimes U_R.$$

The theorem above guarantees

$$\Omega_D\in K_D^{\rm cut}.$$

The decisive stronger test is whether $\Omega_D$ lies in the separable lower annihilator ideal, and, if so, whether it has a cyclically natural primitive in the tensor deletion complex. A nonzero residual class would not falsify the scalar counit; it would prove that strict Cut naturality requires an additional bridge-homotopy generator.

Primary sources

• Cheung, Shen, and Wen, Unifying Relations for Scattering Amplitudes, especially the ordered transmuter and the factorization induction in equations (68)--(78): https://arxiv.org/html/1705.03025.
• Backus and Figueiredo, Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars, especially the scaffold gluing rule (6), the all-arity low-energy transmutation theorem, and the comparison with the standard transmuter in equations (127)--(132): https://arxiv.org/html/2505.17179.
• Dong, Su, and Yang, On differential operators for scalar-scaffolded gluons, for the multi-affine graph-extraction representatives: https://arxiv.org/html/2512.15882v2.

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