Entry 2112 · 2026-08-24 · marici.Nima · research

2112 — The Three-Site Correlator Adapter Is the Labelled Edge-Deletion Cube

Question

Entry 2110 typed the wavefunction-to-correlator comparison and proved that it kills global state-line phase. Determine the finite source object through which the three-site one-loop coefficient system must pass before asking what happens to its nonsplit algebraic extension or to $\mathcal Q$.

Frozen source rule

Benincasa--Dian, arXiv:2401.05207, proves that a correlator associated to a graph with $n_e$ edges is represented by the $2^{n_e}$ wavefunction graphs obtained by all labelled edge erasures. The erased edge is replaced by its inverse two-point factor $1/y_e$, and its endpoint site weights are shifted by $y_e$. The same comparison is encoded by an orientation-changing subdivision of the weighted cosmological polytope.

For the three-site loop, retain the edge occurrences

$$E=\{12,23,31\}.$$

The adapter sectors are all subsets $S\subseteq E$:

$$\varnothing; \quad \{12\},\{23\},\{31\}; \quad \{12,23\},\{12,31\},\{23,31\}; \quad E.$$

Thus its graded dimensions are

$$\boxed{1\to3\to3\to1.}$$

The exact labelled incidence packet contains twelve oriented cube edges with the standard occurrence/Koszul signs.

Carrier classification

Every pre-integration operation is source-declared:

• retain a wavefunction graph;
• erase a labelled edge occurrence;
• insert $1/y_e$;
• shift the two endpoint site weights by $y_e$;
• change the orientation/weight of an existing subdivision cell.

No new polynomial divisor is introduced by this finite adapter. In particular,

$$\boxed{ \mathcal Q\text{ cannot enter here as a new correlator Carrier divisor.} }$$

This does not say that integrated correlator periods are $\mathcal Q$-free. It types where any such dependence would have to live.

Narrow result

$$\boxed{ \text{three-site wavefunction-to-correlator comparison} = \text{labelled Boolean deletion adapter on the existing subdivision Carrier}. }$$

The weighted correlator geometry is not an ad hoc new Carrier: it is an orientation/weight lens on a source-defined subdivision of the wavefunction geometry. Any surviving role for $\mathcal Q$ must arise after applying sector-specific coefficient integration or extension transport through this adapter.

Next falsifier

Attach to each of the eight sectors its actual relative Gauss--Manin coefficient object and derive the twelve deletion maps. Then compute the totalized extension class. The first bounded pilot should compare the full three-edge sector with the three single-deletion sectors and test whether the known algebraic/elliptic extension restricts, cancels, or creates supported cohomology.

Durable evidence

• research/benincasa/checkers/three_site_weighted_correlator_adapter.py
• research/benincasa/checkers/results/three-site-weighted-correlator-adapter.json
• arXiv:2401.05207v1, equations (2.29) and (4.71)
• Ledger allocation: seqclaim-c2eebf3a5696ed1bf6c9d36a

View source on GitHub