Holonomy with Memory in Six Birds Theory: Predictive Quotients, Witnesses, and Fixed-Support Route Transport

In plain words

Imagine two paths that arrive at the same place. From where you stand, they are the same. But one carries a memory the other does not, and that memory changes what happens next. The paper builds a careful frame for this. At every interface it defines two ways to group histories. One by what is seen now. One by what they predict later.

The prediction grouping is always at least as fine as the present grouping. When the two agree, the interface is flat and the past does not matter. When they differ, there is a witness: a pair of histories that match now and split later. Moving forward along the allowed steps is always well defined on the prediction grouping, and only sometimes on the present one.

None of this is left as words. A finite memory wheel is checked in Lean 4. Exact finite models, run with rational numbers, show the strict gap and the loop asymmetry. A benchmark sorts small systems into regimes, from flat to coherent, and a small search finds new coherent cases while a control search finds none.

What it shows

• Future equivalence refines present equivalence, and the gap has a witness.
• Transport is generic exactly at flat interfaces.
• Lean proofs and exact finite benchmarks show the frame is not empty.

What it does not claim

It does not derive quantum amplitudes, interference or the Born rule. It works at the level of groupings and finite models.

Cite

Tsiokos, I. (2026). Holonomy with Memory in Six Birds Theory: Predictive Quotients, Witnesses, and Fixed-Support Route Transport. Zenodo. https://doi.org/10.5281/zenodo.22335923