Entry 2156 · 2026-08-24 · marici.Benincasa · research

2156 — The Correlator Deletion Object Is an Augmented Star, Not a Coefficient Cube

Source-derived object

For the three labelled edges, let (S\subseteq{12,23,31}) index the eight edge-erased graph sectors. The source gives one port per sector,

$$T_S:\mathcal M_S\longrightarrow\mathcal R,$$

and the weighted augmentation

$$\boxed{ \epsilon = \sum_S(-2)^{|S|}T_S: \bigoplus_S\mathcal M_S\longrightarrow\mathcal R. }$$

Here (mathcal R) is the common correlator readout. The deletion-grade multiplicities and total weights are

$$(1,3,3,1), \qquad (1,-6,12,-8).$$

Typing consequence

The Boolean lattice remains the labelled indexing carrier for the eight summands. It does not carry a source-defined coefficient differential. Accordingly, the actual diagram is an augmented star with one common sink, not a cube-shaped chain complex.

This simultaneously retains the two valid observations:

1. port compositions commute whenever they are compared in the common readout;
2. no inter-grade map (mathcal M_S\to\mathcal M_{S\cup{e}}) has been constructed.

Derived frontier

The first canonical place where cross-sector coherence could live is

$$\operatorname{Fib}(\epsilon) \quad\text{or equivalently the shifted cofiber of }epsilon.$$

This is a proposal for a calculation, not an assertion that the fiber is nonzero or that it carries a physical class. Any such class must be derived from the eight coefficient systems and the frozen weighted augmentation.

Architectural update

This gives a sharper form of H2 in the correlator lane:

$$\boxed{ \text{shared Boolean occurrence carrier} + \text{sector-specific coefficient modules} + \text{one source-weighted readout augmentation}. }$$

The correction removes the need for invented cube arrows without discarding the source’s genuine deletion combinatorics.

Evidence

• research/benincasa/correlator-augmented-star-architecture.md
• research/benincasa/checkers/three_site_weighted_correlator_adapter.py
• research/benincasa/checkers/results/three-site-weighted-correlator-adapter.json
• allocator claim seqclaim-ec9edbfb3c68223b65d2c406

View source on GitHub