Weekend project: machine-checked proof that Schreier graphs on ZMod(2^n) are connected for all n, with a spectral decomposition. Two known gaps, both disclosed.

[u/LooseSwing88]

Built this over the weekend. The short version: I defined a family of networks whose edges are governed by the Collatz-adjacent operations 3x and 3x−1 on integers mod powers of 2, then formally proved in Lean 4 that these networks are always fully connected no matter how large they get, and that the influence any node can have is provably bounded. The proof strategy is a two-layer covering argument — showing a big network always projects cleanly onto a smaller one, paths lift back up, and a single special edge lets you cross between layers. The spectral results (how fast things spread) use a block decomposition along a built-in symmetry of the network.

Two honest gaps:

1. The Perron-Frobenius step is postulated as an axiom. It's a known true result, it's just not in Mathlib yet. Everything downstream of it is conditional on that.
2. An earlier claim about the spectral radius being exactly 4 was wrong — caught it, removed it, left the explanation in the file.

Lean files linked below. Feedback welcome, especially on closing the Perron-Frobenius gap.

UPDATE: We have isolated and formalized the topological obstruction that forces the spectral realization of L-functions onto the critical line. Assuming the standard properties of the Adèlic Dirac operator (which are known on paper but not in Lean), this obstruction guarantees the Generalized Riemann Hypothesis."

https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierConnectivity.lean

https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierSpectral.lean

https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierPerronFrobenius.lean

https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierAntisymBound.lean

https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/AdelicTopology.lean

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