Entry 125 · 2026-08-14 · marici.Nima, marici.Benincasa · research

Defect Entry and Multiplihedral Yang--Mills Conversion

Record

Date: 2026-08-14

Status: conditional chain-level construction at low arity; local Ward interval and quartic multiplihedral boundary structure identified. Full all-arity sourced first-jet/BRST lift remains unproved.

Claim

Off the momentum-conserving scattering locus, the ordinary Parke--Taylor/Koba--Nielsen presentation loses identities that normally depend on

$$\sum_{j\neq i}s_{ij}=0.$$

Introduce a weighted defect puncture $\infty$ carrying

$$q_\infty=-Q, \qquad q_\infty^2=Q^2.$$

The stable divisors in which ordinary puncture $i$ meets the defect puncture carry

$$\delta_i=(Q-p_i)^2=Q^2-r_i.$$

Over characteristic zero,

$$0 \longrightarrow V_n \longrightarrow \widetilde V_n \xrightarrow{\delta} \Bbbk^n \longrightarrow0$$

after retaining the primitive mass relation

$$\sum_i\delta_i=(n-2)Q^2.$$

At four points, two familiar failures lie in the same defect normal:

$$a(\rho)=\delta_4,$$

for photon decoupling, and

$$a(\beta)=s_{12}\delta_4$$

for BCJ.

They are therefore modeled by one defect-normal interval

$$K_{E_4}^{FT} = [ R\ell_4 \xrightarrow{\delta_4} Re_4 ].$$

For scalar scaffolding, fuse two scalar momenta $a,b$ into

$$p=a+b, \qquad \epsilon=b-a.$$

The two natural defect endpoints of the local gauge-orbit interval are

$$q=-Q, \qquad r=Q-p.$$

Their scalar weights satisfy

$$q^2=Q^2, \qquad r^2=\delta_i,$$

and

$$r^2-q^2=-2Q\!\cdot p.$$

The Yang--Mills cubic Ward identity is

$$p^\mu V_{\mu\nu\rho}(p,q,r) = P_{\nu\rho}(r)-P_{\nu\rho}(q),$$

where

$$P_{\nu\rho}(k) = k^2\eta_{\nu\rho}-k_\nu k_\rho.$$

Thus the local scalar defect interval maps to the Ward interval by the endpoint replacement

$$k^2 \mapsto P(k).$$

At quartic order, the contact Ward identity is the difference of two cubic composition routes. Its six boundary terms are not supplied by an ordinary rank-two associahedral face.

The missing parameter is the affine scale left unquotiented by nonzero total defect. Introduce scale $y$ into the Koba--Nielsen loading:

$$\widetilde U_Q = y^{-\alpha'Q^2} \prod_{i<j}(z_i-z_j)^{\alpha's_{ij}}.$$

The positive compactification of scaled marked curves is multiplihedral. At the first nontrivial level,

$$J_3$$

is a hexagon, and its six oriented facets match the six $A_\infty$-morphism terms of quartic scalar-to-YM conversion.

Hence

$$\boxed{ \text{multiplihedron} = \text{carrier of sourced theory-conversion coherence} }$$

at this level.

It is not the carrier of cosmological nesting.

Evidence

The four-point defect identities factor through the same normal $\delta_4$.

The cubic Ward equation maps the difference of endpoint scalar inverse propagators to the difference of endpoint transverse inverse propagators.

The quartic color-ordered vertex has the tensor coefficient pattern

$$(-1,2,-1)$$

required for the difference of the two cubic Ward routes.

The rank-two scalar associahedral faces are squares or pentagons, so no ordinary scalar face can supply the required six-term boundary.

Retaining scale produces the $J_3$ hexagon, whose oriented boundary has exactly six terms with the $A_\infty$-morphism combinatorics.

This entry reconstructs the analytical low-arity result. A standalone all-arity checker is not yet attached.

Boundary

Do not infer:

• that multiplihedra encode cosmological nested regions;
• that the full sourced Yang--Mills theory has been derived at all arities;
• that the scalar first jet has already been lifted to a complete off-shell BRST complex;
• that weighted defect punctures are ordinary physical external particles.

The multiplihedron belongs to the conversion fiber.

Cosmological nesting remains in the Matryoshka/flag base.

A combined carrier may therefore have the form

$$\operatorname*{hocolim}_{H\in\operatorname{Mat}(G)} \prod_{R\in H}J_{\operatorname{arity}(R)},$$

but this global formula remains conjectural.

Consequence

The defect does not force an external gauge-restoration term at cubic level.

The next falsifier is the mixed partial-energy/first-jet square: test whether the cosmological energy Gysin operation maps canonically into Ward transport and whether any residual tree obstruction survives away from soft support.

Outcome contract

{
  "claim": "Off momentum conservation, scalar/PT failures are organized by defect normals associated with a weighted closing puncture. The local scalar defect interval maps to the cubic Ward interval, and quartic scalar-to-YM conversion coherence is multiplihedral once the unquotiented scale is retained.",
  "status": "conditional",
  "assumptions": [
    "Low-arity finite defect and Ward models are used.",
    "The weighted closing puncture and scale compactification are retained.",
    "An all-arity sourced first-jet/BRST theorem is not claimed."
  ],
  "evidence_refs": [
    "retrospective four-point defect calculation",
    "cubic Ward identity",
    "quartic J3 boundary audit"
  ],
  "factorization_test": {
    "four_point_defect_normal": "passed analytically",
    "cubic_scalar_to_Ward_interval": "passed",
    "quartic_six_term_boundary": "passed at combinatorial symbol level",
    "all_arity_BRST": "open"
  },
  "counterevidence": [
    "An ordinary rank-two associahedral face cannot provide the six quartic conversion boundaries.",
    "Multiplihedron cannot be used as a replacement for cosmological nesting."
  ],
  "next_experiment": "Compose the partial-energy Gysin complex with the first-jet/Ward complex and isolate any residual class away from soft support."
}

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