Pairwise Trace Sectors and the Cyclic Transmutation Counit

Record

Date: 2026-08-13

Status: all-arity tree-level field-theory theorem, conditional only on two published scalar-scaffold inputs: the Backus–Figueiredo pair-transmutation theorem and even-label multilinearity of the canonical scaffolded Yang–Mills amplitude. The support identities were independently enumerated through seven points. The relation to the complete Dong–Su–Yang graph coframe is verified at four and five points but remains conditional at general multiplicity.

Reproducible certificate:

research/nima/check_transmutation_counit_all_arity.rs

Result

The amplitude-level lowering operation has a reference-free all-arity form. It is not necessary to choose a cubic graph, a Parke–Taylor basis, or a metric adjoint.

Let

[ E_n={2,4,\ldots,2n}, \qquad O_n={1,3,\ldots,2n-1}, ]

and write

[ \mathcal W_e

\sum_{j\notin{e,e\pm1}} \partial_{X_{e,j}}. ]

For distinct (e,f\in E_n), define

[ P_{ef}

\prod_{g\in E_n\setminus{e,f}}\mathcal W_g, \qquad V_{ef}

\partial_{X_{e,f}}P_{ef}. ]

Backus and Figueiredo prove at tree level and at leading low energy that

[ P_{ef}A_n^{\rm YM}

X_{e,f}A_n^{\operatorname{Tr}\phi^3}. ]

Consequently,

[ V_{ef}A_n^{\rm YM}

A_n^{\operatorname{Tr}\phi^3}. ]

The new point is that (V_{ef}) has a universal, much smaller representative. Set

[ B_g

\sum_{\substack{o\in O_n\o\notin{g-1,g+1}}} \partial_{X_{g,o}}, ]

and

[ U_{ef}

\partial_{X_{e,f}} \prod_{g\in E_n\setminus{e,f}}B_g. ]

Then

[ \boxed{ U_{ef}A_n^{\rm YM}

A_n^{\operatorname{Tr}\phi^3} } ]

for every unordered pair ({e,f}\subset E_n).

Thus every edge of the complete graph on the even scaffold labels carries the same scalar transmutation counit.

Exact proof of the odd-target reduction

The canonical scaffold amplitude is multilinear in every gluon polarization. In scalar variables this implies the stronger support statement used explicitly by Dong–Su–Yang: in each term an even label occurs in at most one (X)-coordinate. Equivalently, any differential monomial containing two derivatives whose coordinates share the same even label annihilates the amplitude.

Expand (V_{ef}). Suppose the (\mathcal W_g) factor chooses an even target (h).

• If (h=e) or (h=f), that label already occurs in (\partial_{X_{e,f}}).
• If (h\notin{e,f}), then (h) is itself one of the sources of another (\mathcal W_h) factor.

In either case the differential monomial contains two coordinates sharing even label (h), so it vanishes on (A_n^{\rm YM}). Every surviving choice therefore sends every remaining even source to an allowed odd target. Those choices are exactly the monomials in (U_{ef}). Hence

[ V_{ef}-U_{ef} \in \operatorname{Ann}(A_n^{\rm YM}), ]

which proves the boxed identity.

This is an amplitude-quotient statement. It does not assert equality of the two differential operators on arbitrary functions.

Support counts

Each (\mathcal W_g) has (2n-3) possible targets. Each restricted (B_g) has (n-2) possible odd targets. Since (V_{ef}) contains (n-2) (\mathcal W)-factors,

[ #\text{ raw choices in }V_{ef} =(2n-3)^{n-2}, ]

while

[ \boxed{ |\operatorname{supp}U_{ef}| =(n-2)^{n-2}. } ]

The monomials are distinct: an even–odd coordinate remembers its unique even source. Supports belonging to distinct pairs ({e,f}) are disjoint because each (U_{ef}) monomial contains exactly one even–even coordinate, namely (X_{e,f}).

The certificate enumerates every raw choice for every pair through (n=7), checks the closed counts with overflow-safe arithmetic through (n=25), and verifies exact covariance under rotation by two scalar labels.

A second theorem: vertex plus incident edges

The pair sectors expose a stronger decomposition of the full Backus–Figueiredo transmuter. For an omitted even label (e), define

[ T_e

\prod_{g\in E_n\setminus{e}}\mathcal W_g, \qquad R_e

\prod_{g\in E_n\setminus{e}}B_g. ]

Any multi-affinity-surviving term in (T_e) is of precisely one of two kinds.

1. Every source chooses an odd target. These terms comprise (R_e).
2. Exactly one source (f) chooses the only even label not already used as a source, namely (e). All other sources choose odd targets. These terms comprise (U_{ef}).

No other even target can occur, and two sources cannot both target (e). Therefore

[ \boxed{ T_e \equiv R_e+\sum_{f\in E_n\setminus{e}}U_{ef} \pmod{\operatorname{Ann}(A_n^{\rm YM})}. } ]

Backus–Figueiredo transmutation gives

[ T_eA_n^{\rm YM}=A_n^{\operatorname{Tr}\phi^3}, ]

whereas every one of the (n-1) incident edge operators gives the same scalar amplitude. It follows immediately that the previously unidentified all-odd vertex sector obeys

[ \boxed{ R_eA_n^{\rm YM}

-(n-2)A_n^{\operatorname{Tr}\phi^3}. } ]

Its support has size

[ |\operatorname{supp}R_e|=(n-2)^{n-1}. ]

The complete surviving support of (T_e) therefore has size

[ (n-2)^{n-1}+(n-1)(n-2)^{n-2} =(2n-3)(n-2)^{n-2}. ]

The Rust certificate verifies the disjoint support decomposition for every omitted label through seven points.

Complete-graph incidence form

Let (Q) be the signless vertex–edge incidence matrix of (K_n):

[ Q_{e,{a,b}}

\begin{cases} 1,&e\in{a,b},\ 0,&e\notin{a,b}. \end{cases} ]

Collecting the operators into vectors gives

[ T=R+QU \qquad \text{in } \operatorname{Diff}/\operatorname{Ann}(A_n^{\rm YM}). ]

The scalar augmentation has weights

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

Since each row of (Q) contains (n-1) ones, the incidence equation is numerically exact:

[ -(n-2)+(n-1)=1. ]

This is the first nontrivial finite algebra seen directly among the lowering operators. The reference labels do not disappear by an unexplained cancellation; they organize into the vertices and edges of (K_n), while the canonical amplitude quotient collapses every normalized representative to one counit class.

Manifestly cyclic representatives

Let (e_i=2i), with (i) cyclic modulo (n). Several reference-free operators now follow without further amplitude input:

[ \epsilon_n^{\rm cycle}

\frac1n\sum_{i=1}^{n}U_{e_i,e_{i+1}}, ]

[ \epsilon_n^{\rm edge}

\frac{2}{n(n-1)} \sum_{1\leq i<j\leq n}U_{e_i,e_j}, ]

[ \epsilon_n^{\rm vertex}

-\frac{1}{n(n-2)} \sum_{i=1}^{n}R_{e_i}, ]

and

[ \epsilon_n^{W}

\frac1n\sum_{i=1}^{n}T_{e_i}. ]

For (n\geq3), all four send (A_n^{\rm YM}) to (A_n^{\operatorname{Tr}\phi^3}), are invariant under cyclic rotation, and represent the same class in (\operatorname{Diff}/\operatorname{Ann}(A_n^{\rm YM})).

The cycle representative is particularly economical: it uses the Hamiltonian cycle selected by the planar order rather than averaging over all edges of (K_n).

Location of the Dong–Su–Yang coframe

Dong–Su–Yang conventionally fix

[ \partial_{X_{2,2n}} \qquad\text{and}\qquad \partial_{X_{1,4}}. ]

The first derivative selects the edge sector (U_{2,2n}). The second fixes one odd target in that sector. The resulting fixed slice contains

[ (n-2)^{n-3} ]

monomials. Their graph rules select a much sparser Catalan family of size

[ \mathcal C_{n-2} ]

inside that slice.

The hierarchy is therefore

[ (2n-3)^{n-2} \longrightarrow (n-2)^{n-2} \longrightarrow (n-2)^{n-3} \longrightarrow \mathcal C_{n-2}, ]

corresponding respectively to:

1. the raw pair transmuter;
2. its universal odd-target representative;
3. a fixed trace/reference slice;
4. the planar cubic-diagram coframe.

At four points the sizes are

[ 25\longrightarrow4\longrightarrow2\longrightarrow2, ]

and at five points they are

[ 343\longrightarrow27\longrightarrow9\longrightarrow5. ]

The complete published four- and five-point DSY coframes were checked monomial by monomial to lie in the stated slice. The paper proves its general graph rules only for specified families, so this ledger does not promote the all-graph Catalan selection to a theorem.

Interpretation

This changes the operator-algebra picture in three ways.

First, lowering is an augmented trace operation, not presently a metric adjoint of the scalar-to-YM jet. Its universal primitive is the pairwise trace sector (U_{ef}).

Second, cyclic-reference independence is now an all-arity theorem at amplitude level. It does not rely on the bounded four- and five-point cancellations of entry 41.

Third, the planar DSY graph operators are best viewed as a sparse normal form for one edge of a larger, fully symmetric incidence object. The complete graph is supplied by the choice of the two polarizations left for the final trace; the planar ordering then selects a Hamiltonian cycle and a Catalan chart.

This is precisely the sort of small native grammar sought by Marici:

[ \text{gauge first-jet object} \xrightarrow{\text{pairwise trace counit}} \text{scalar object}, ]

with the apparently large graph family resolving one morphism rather than defining many unrelated operations.

What is proved and what is not

Proved here, from the cited published inputs:

• (U_{ef}A_n^{\rm YM}=A_n^{\operatorname{Tr}\phi^3}) for every pair and all tree multiplicities;
• the exact support counts and pair-sector disjointness;
• the vertex–edge incidence decomposition of (T_e);
• (R_eA_n^{\rm YM}=-(n-2)A_n^{\operatorname{Tr}\phi^3});
• manifestly cyclic averaged representatives and their equality in the canonical amplitude quotient.

Not proved:

• that every DSY cubic graph has the proposed derivative at arbitrary multiplicity;
• that the Catalan graph family is a canonical basis before quotienting by the Yang–Mills annihilator;
• that this counit is adjoint to the scalar fusion jet under an intrinsic gauge-sector metric;
• that (U_{ef}) has a canonical chain-level lift;
• that the lower-point counits obtained after a physical Cut obey the required tensor/coaction law.

Constant-coefficient differentiation in even-containing coordinates commutes formally with extraction of a residue in an independent odd–odd physical channel. This is weaker than factorization naturality: after the cut one must still identify how the retained trace pair and all remaining even labels distribute across the two factors.

Next falsification target

The next sharp question is the cut coaction of the edge class.

For a physical odd–odd channel (D), determine whether there is a canonical formula of the form

[ \operatorname{Cut}D,U{ef}^{(n)}

\sum_{\sigma} U_{e_Lf_L}^{(L,\sigma)} \otimes U_{e_Rf_R}^{(R,\sigma)}, ]

where (\sigma) accounts for the internal polarization/trace allocation and where the formula is compatible with the complete-graph incidence relation.

A positive result would promote the amplitude counit to a factorization counit. A failure would locate precisely where the lowering operation requires extra chain or state-sum data.

Primary sources

• Backus and Figueiredo, Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars, arXiv:2505.17179, especially equations (68), (108)–(109), and the all-arity low-energy proof in section 7.3: https://arxiv.org/abs/2505.17179.
• Dong, Su, and Yang, On differential operators for scalar-scaffolded gluons, arXiv:2512.15882v2, especially equations (2.16)–(2.18), section 3, and the stated scope of the factorization proofs: https://arxiv.org/abs/2512.15882.