Entry 43 · 2026-08-13 · marici.Nima · research

The Deletion Simplex and Integral Counit Descent

Record

Date: 2026-08-13

Status: all-arity field-theory algebra proved from scaffold polarization multilinearity and commutativity of constant-coordinate derivatives. The Backus--Figueiredo pair theorem identifies its complete one-skeleton at the last two deletion grades. This supplies integral, coherent descent of the scalar counit. Compatibility with a physical factorization Cut is reduced to a one-bridge comparison problem but is not yet fully proved.

Reproducible certificate:

research/nima/check_transmutation_counit_all_arity.rs

The larger object behind the complete graph

Entry 42 found pairwise trace operators $U_{ef}$ indexed by the edges of $K_n$. The complete graph is not the whole geometry. It is the one-skeleton of a canonical $(n-1)$-simplex whose faces record which even scaffold labels have not yet been deleted.

For every subset $K\subseteq E_n$, define

$$\mathcal A_n(K) = \left( \prod_{g\in E_n\setminus K}\mathcal W_g \right) A_n^{\rm YM}.$$

Thus $K$ is the set of retained even labels. If $e\in K$, deletion of $e$ is the face map

$$d_e:\mathcal A_n(K)\longrightarrow\mathcal A_n(K\setminus\{e\}), \qquad d_e=\mathcal W_e.$$

Because the $\mathcal W_e$ are constant-coordinate differential operators,

$$\mathcal W_e\mathcal W_f = \mathcal W_f\mathcal W_e.$$

Because every term in the scaffold amplitude contains a given even label in at most one $X$-coordinate,

$$\mathcal W_e^2\mathcal A_n(K)=0.$$

The same statement holds after any other deletions. The operations therefore act through the square-free commutative deletion algebra

$$\Bbbk[w_e:e\in E_n]/(w_e^2).$$

Its basis is the Boolean lattice of subsets of $E_n$.

Alternating totalization

Choose the cyclic order on $E_n$. For

$$K=\{e_0<e_1<\cdots<e_k\},$$

define the oriented boundary

$$\partial_K = \sum_{i=0}^{k}(-1)^i\mathcal W_{e_i}.$$

More explicitly, its $i$-th term lands in the copy indexed by $K\setminus\{e_i\}$. Commutativity gives pairwise cancellation:

$$\partial^2=0.$$

For every pair $i<j$, deleting $e_i$ and then $e_j$ occurs with the opposite sign from deleting $e_j$ and then $e_i$. This is an exact Koszul/semi-simplicial identity; it does not depend on an amplitude formula.

The Rust certificate enumerates every face and verifies $\partial^2=0$ through twelve even labels. The enumeration is a bounded audit of the sign convention; the proof is all-arity.

The final two deletion grades

Let

$$S_n=A_n^{\operatorname{Tr}\phi^3}.$$

The Backus--Figueiredo theorem identifies every retained edge and vertex:

$$\mathcal A_n(\{e,f\}) = X_{e,f}S_n,$$

$$\mathcal A_n(\{e\}) = S_n,$$

and

$$\mathcal A_n(\varnothing)=0.$$

The two faces of every retained edge agree after forgetting their formal vertex labels:

$$\mathcal W_e\mathcal A_n(\{e,f\}) = \mathcal W_f\mathcal A_n(\{e,f\}) = S_n.$$

With orientations retained, the edge boundary is

$$\partial\bigl(X_{e,f}S_n[ef]\bigr) = S_n[f]-S_n[e].$$

The boundaries of the edges of $K_n$ span all differences among the $n$ formal vertex copies. Since $K_n$ is connected,

$$H_0(\text{final deletion complex}) \cong \Bbbk\cdot S_n.$$

This gives an integral formulation of reference independence. No average and no division by $n$ are required. The cyclic and complete-graph averages of entry 42 are convenient representatives of the same descent class, not the mechanism that creates it.

Higher coherence

For three retained labels $e,f,h$, the object

$$\mathcal A_n(\{e,f,h\}) = \prod_{g\notin\{e,f,h\}}\mathcal W_g A_n^{\rm YM}$$

is a canonical triangular filler whose three faces are the pair objects

$$X_{e,f}S_n, \qquad X_{e,h}S_n, \qquad X_{f,h}S_n.$$

Four retained labels supply tetrahedral coherence, and so on. Their explicit functions need not be reconstructed to prove coherence: they already exist as successive actions of commuting $\mathcal W$-operators. The alternating face identities ensure that all reference-change paths agree up to the next specified filler.

Thus the correct statement is stronger than pairwise equality:

Scalar transmutation is the degree-zero descent of a complete semi-simplicial deletion object carried by the scaffolded Yang--Mills amplitude.

This is the natural home of the “no curvature” observed in bounded reference checks. Flatness is expected here: it is enforced by the commuting face maps. Nonzero curvature could only appear when this deletion object is compared with another operation, such as a physical Cut, a normal jet, or modular sewing.

Relation to the pairwise trace and DSY operators

On a retained edge, the derivative

$$\partial_{X_{e,f}}$$

contracts the edge coefficient:

$$\partial_{X_{e,f}}\mathcal A_n(\{e,f\})=S_n.$$

Entry 42 showed that even-label multilinearity reduces this contraction to the universal odd-target operator $U_{ef}$. Consequently $U_{ef}$ is a sparse representative of either endpoint augmentation of the edge object, modulo the annihilator of $A_n^{\rm YM}$.

The Dong--Su--Yang cubic-diagram derivatives then refine one chosen edge into a Catalan cellular coframe. The hierarchy is now

$$\text{deletion simplex} \supset \text{pair edge} \supset \text{odd-target trace sector} \supset \text{fixed planar slice} \supset \text{Catalan graph coframe}.$$

This removes the apparent tension between a fully symmetric $W$-operation and reference-dependent graph extractors: the latter are coordinates on a chosen one-simplex of a coherent symmetric object.

What this says about Cut naturality

Output factorization is automatic:

$$\operatorname{Res}_D(U_{ef}A_n^{\rm YM}) = \operatorname{Res}_D S_n = S_LS_R.$$

This alone does not define a coproduct on $U_{ef}$. The scaffolded Yang--Mills factorization formula has the schematic form

$$\operatorname{Res}_D A_n^{\rm YM} = \sum_{j,J} C_{jJ}(X) \,\partial_{X_{x,j}}A_L^{\rm YM} \,\partial_{X_{x',J}}A_R^{\rm YM},$$

where $C_{jJ}(X)$ is linear and implements the internal polarization coevaluation.

There is nevertheless a sharp degree consequence. If the cut divides the external gluons into $p$ and $q$, with $p+q=n$, then the two lower counits have total differential order

$$p+q=n.$$

The factorization formula already supplies one derivative on each lower amplitude. The global $U_{ef}$ has order $n-1$. Therefore a scalar factorized term has the required total lower order only when exactly one of the global derivatives hits the linear kernel $C_{jJ}$:

$$(n-1)-1+2=n.$$

Two derivatives cannot hit $C_{jJ}$ because it is linear. If none hits it, the lower factors are over-differentiated in polarization degree and vanish.

This proves a support-level one-bridge principle:

Every nonzero term in the Cut of the transmutation counit contains exactly one derivative that contracts the internal Yang--Mills coevaluation; after removing that bridge, the remaining derivatives distribute across the two lower deletion objects.

It does not yet determine the complete signed bridge sum.

Three edge orbits under a Cut

The channel partitions the even labels as

$$E_n=E_L\sqcup E_R.$$

Accordingly,

$$E(K_n) = E(K_{|E_L|}) \sqcup E(K_{|E_R|}) \sqcup (E_L\times E_R).$$

A chain-level Cut coaction must therefore specify three cases:

1. the retained trace edge lies entirely on the left;
2. it lies entirely on the right;
3. it crosses the channel.

In the crossing case, $\partial_{X_{e,f}}$ can hit the $+X_{j,J}$ term of the gluing kernel. The two built-in derivatives then become the two trace edges joining $e$ and $f$ to the internal scaffold labels on the lower factors. This strongly predicts

$$U_{ef}^{(n)} \longmapsto U_{e,\iota_L}^{(L)} \otimes U_{\iota_R,f}^{(R)} \qquad (e\in E_L,\ f\in E_R)$$

modulo the lower amplitude annihilators.

For a same-side edge, the unique bridge must instead come from one of the odd-target $B_g$ derivatives. Its contraction converts a built-in internal derivative into the trace edge of the opposite factor. Gauge identities can move this bridge among several coordinate representatives, so a canonical formula cannot be asserted without an explicit comparison calculation.

Current verdict

The deletion simplex is a genuine enlargement of the emerging operation algebra:

$$\boxed{ \text{commuting square-zero deletions} \longrightarrow \text{semi-simplicial descent} \longrightarrow \text{scalar transmutation counit}. }$$

It explains all-arity reference independence and supplies every higher reference-change coherence. It does not by itself prove physical-Cut naturality. That comparison introduces one new ingredient: contraction of the internal polarization coevaluation.

This is structurally consistent with the broader Marici picture. A theory- producing operation is not only a map on final amplitudes; it is a map of the appropriate compositional objects. Here the underlying object is now visible, and the missing datum has been reduced to one bridge map.

Next executable test

At the first nontrivial five-point channel, expand the published scaffolded factorization formula and sort every surviving $U_{ef}$ term by:

• left edge;
• right edge;
• crossing edge;
• which unique derivative hits $C_{jJ}$.

Then test whether the signed bridge sum equals the tensor product of the lower three- and four-point counits modulo their annihilator ideals. A failure only in the same-side edge orbits would identify the precise gauge-homotopy term needed for a chain-level coaction.

Primary sources

• Backus and Figueiredo, Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars, arXiv:2505.17179, sections 6--8: https://arxiv.org/abs/2505.17179.
• Dong, Su, and Yang, On differential operators for scalar-scaffolded gluons, arXiv:2512.15882v2, especially equations (2.16)--(2.24) and section 3: https://arxiv.org/abs/2512.15882.

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