Papers · Preprint
Six Birds Verified XII: The Geometry of Access
How much an observer can read, whether its readings fit, and where they fail, all fixed by linear algebra.
In plain words
An observer reads parts of a system through linear probes and knows how the parts overlap. Three questions follow. How much can it read? Do its readings fit together? Where do they fail to? For finite dimensional rational state spaces, the declared access answers all three. What can be read at one place has a dimension, the dimension of the span of the probes, not their number. Ten copies of one probe read no more than one. A proposed global picture reproduces the compatible local readings exactly when it hides nothing and has the right dimension.
Some misfits are invisible pair by pair. Ask three readings to satisfy y minus x equals 0, z minus y equals 0 and z minus x equals 1. Any two can be met; all three cannot. Such a misfit is a class in a first cohomology group and shows up only around a loop. When parts are joined by transports, it is governed by holonomy: what a value turns into after a trip around the loop. A twisted loop trades global readings for solvable constraints, and the trade is exact. Readings can agree everywhere and still come from no single state, because of directions the probes cannot see.
On signed graphs that grow with a threshold, access rank equals points minus components exactly below a critical scale, the first one with an unbalanced loop, if there is one. Rank is not continuous, but two explicit gates bound it. Growth of access always stops, yet without a certificate a pause of any length can still end. All results are formalized in Lean 4.
What it shows
- Access rank is the dimension of the probe span, independent of coordinates and probe count.
- A candidate whole is exact when it is separated and of the right dimension; each local family has one of four statuses.
- Pairwise invisible misfits are first cohomology classes, carried on graphs by holonomy.
- Signed access rank matches connectivity exactly below a critical scale.
- Rank under perturbation is bounded by two gates; for Krylov rules the first plateau is final.
What it does not claim
It says nothing about spacetime, gravity or physical dimension. Only the rational case is formalized, rank is not claimed to be continuous, and the holonomy here is linear transport around graph loops only. The repair atlas covers declared failure types, not every failure.
Cite
Tsiokos, I. (2026). Six Birds Verified XII: The Geometry of Access. Zenodo. https://doi.org/10.5281/zenodo.23097938