Binary Order Thinning

The mathematical question

Begin with a finite partially ordered set. A Boolean label retains some comparisons and removes others, producing a thinner order on the same states. The paper asks when the original and thinned order complexes remain closely related, and exactly where a simple answer stops working.

What the current manuscript establishes

For height-at-most-two orders and a single Boolean label, the relative boundary has a graph-incidence form. This yields an exact rooted-forest criterion for vanishing relative homology and simplicial collapse. The manuscript also gives examples showing that two simultaneous Boolean coordinates or greater height can defeat that simple criterion.

Why it matters to The MIND

This gives a carefully bounded answer to one version of “what changes when an observer gains a distinction?” It is a foundation for observation refinement and a concrete result within Logical Homology.

Current boundary

The theorem concerns finite posets under specified operations. It does not establish that a biological or artificial mind has this structure, and its height and one-bit limits should remain visible when the result is described.