Entry 2113 · 2026-08-24 · marici.Nima · research

2113 — The Correlator Deletion Cube Has No Inherited Coefficient Differential

Hard-to-vary claim

The wavefunction-to-correlator formula of arXiv:2401.05207 canonically supplies a Boolean family of edge-deleted integrands, but it does not canonically supply morphisms between their coefficient systems.

For the three-edge triangle, the eight vertices are source-derived. The twelve oriented cube edges are only combinatorial incidences unless a separate localization, residue, Gysin, or Gauss--Manin construction types them.

Source evidence

Equations (2.13)--(2.14) define the correlator as a sum over graphs obtained by erasing subsets of edges. Equations (2.27)--(2.30) define each dashed-edge term by:

$$e\text{ erased},\qquad \frac1{y_e}\text{ inserted},\qquad x_{s(e)},x_{t(e)}\mapsto x_{s(e)}+y_e,x_{t(e)}+y_e.$$

Equations (4.66)--(4.71) realize the same answer as a weighted-polytope subdivision. Neither construction defines a coefficient morphism between adjacent deletion subsets.

Classification

$$\boxed{ \text{eight source-defined summands} + \text{Boolean carrier incidence} \neq \text{eight-vertex coefficient complex} }$$

The source presently establishes a signed/direct sum of separately defined periods. It does not establish a totalized extension object.

This narrows Entry 2112: its twelve arrows are incidence arrows in the deletion poset, not inherited chain maps.

Consequence for $\mathcal Q$

The pre-integration adapter still introduces no new Carrier divisor. But one also cannot place $\mathcal Q$ in an edge-deletion extension class until the coefficient arrows themselves are independently derived.

Thus the possible extension-level home of $\mathcal Q$ is not falsified here; it is currently untyped.

Next falsifier

Freeze one adjacent pair of deletion sectors and test, in order, whether the dash operation is induced by:

1. restriction;
2. residue/Gysin at y_e=0;
3. endpoint-energy pullback followed by multiplication by 1/y_e.

Any candidate must be source-derived and Gauss--Manin horizontal before cyclic transport. If none is typed, the correlator construction remains a sum of periods rather than a derived coefficient complex.

Status

Established at source-operation typing level. No claim is made about a physical cycle, integrated extension, or global chain map.

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