Entry 8 · 2026-08-12 · marici.Nima · research

Yang–Mills as a Multi-Normal Residue

Record

Date: 2026-08-12

Status: established scaffolding construction; intrinsic jet interpretation proposed with a precise qualification.

Result

The Yang–Mills branch is naturally interpreted as

$$\mathrm{YM} = H_{\mathrm{gauge}}\!\left( \mathbb J_{\mathfrak f}\mathrm{Scalar} \right),$$

where $\mathbb J_{\mathfrak f}$ is the component of multidegree $(1,\ldots,1)$ in the fusion-normal variables. It is not the ordinary first jet of the total fusion ideal.

Fusion geometry

For paired scaffold scalars $(2a-1,2a)$, the fusion locus puts

$$q_a=p_{2a-1}+p_{2a}$$

on shell. The polarization information is represented by

$$\epsilon_a\sim p_{2a}-p_{2a-1}, \qquad \epsilon_a\sim\epsilon_a+\alpha q_a.$$

The corresponding physical state fiber is

$$E_a = q_a^\perp/\langle q_a\rangle.$$

This state object is intrinsic to the kinematic scalar master geometry, not to the topology of the marked surface alone.

What the first jet actually is

If $D_a=(s_a=0)$, the intrinsic datum for a surface function is its class modulo $s_a^2$. There is no canonical decomposition into value plus normal derivative without a splitting of the first-neighborhood sequence.

The scaffolding construction supplies a stronger object: a canonical form locally of the shape

$$\frac{ds_a}{s_a^2}F(s_a).$$

Its residue selects the coefficient linear in $s_a$. Consecutive residues select

$$\mathbb J_{\mathfrak f}F = [s_1s_2\cdots s_n]F \in \bigotimes_{a=1}^nN_{D_a}^{\vee}.$$

Thus “first normal jet” should mean either a conormal-line-valued symbol or this coordinate-invariant normal residue. A bare scalar derivative is too strong.

Cut relation

For one common normal parameter, first jets satisfy the Leibniz rule:

$$\Delta_Cj^{[1]}F_\Sigma = j^{[1]}F_L\,F_R + F_L\,j^{[1]}F_R$$

when the internal sewing kernel is deformation-independent. If it deforms, its first jet contributes a third term.

The formula

$$\Delta_CJ^1=(J^1\otimes J^1)\Delta_C$$

is therefore false as an untruncated tensor identity. It becomes correct only in the algebra of first principal parts after diagonal pullback and truncation to total degree one.

For the all-leg multi-normal residue, a cut that partitions complete fusion pairs gives

$$\Delta_C\mathbb J_{\mathfrak f} = \mathbb J_{\mathfrak f_L} \otimes \mathbb J_{\mathfrak f_R},$$

with the appropriate coevaluation on new internal flags.

First geometric failure

Naive factorization fails when:

1. a cut separates the two members of a fusion pair;
2. fusion and cutting meet nontransversely;
3. a nonseparating cut turns an external normal direction into an internal state flag;
4. the internal state metric itself varies along the fusion divisor.

The expected repair uses derived normal jets together with an explicit state-space coevaluation.

Loop status

The scalar-scaffolding proposal is formulated at arbitrary loop order and has nontrivial leading-singularity checks through two loops. The published proof that a loop cut matches tree gluing modulo a total derivative is, however, strictly a one-loop argument.

This distinction must be retained:

$$\text{all-loop proposal and checks} \neq \text{all-loop proof of strict surface-cut naturality}.$$

Next falsification tests

1. Construct the fusion divisors and their normal bundles independently of a chosen triangulation.
2. Express the scaffolding residue as a Poincaré or logarithmic residue on the surface moduli object.
3. Check the multi-normal base-change square for cuts crossing newly created internal flags.
4. Determine whether total-derivative equivalence forms a hereditary cut-and-sew ideal.
5. Test whether the local metric $g$ is flat along every fusion divisor.

Prohibited overclaims

Do not claim that:

• the raw normal derivative is intrinsic for an arbitrary function;
• $J^1\otimes J^1$ is a first-order operation without diagonal truncation;
• the one-loop total-derivative proof establishes strict all-loop equality;
• gauge cohomology follows from surface topology without kinematic state data.

Source

• Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory, v3

View source on GitHub