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:
- a cut separates the two members of a fusion pair;
- fusion and cutting meet nontransversely;
- a nonseparating cut turns an external normal direction into an internal state flag;
- 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
- Construct the fusion divisors and their normal bundles independently of a chosen triangulation.
- Express the scaffolding residue as a Poincaré or logarithmic residue on the surface moduli object.
- Check the multi-normal base-change square for cuts crossing newly created internal flags.
- Determine whether total-derivative equivalence forms a hereditary cut-and-sew ideal.
- 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.