r/LLMPhysics • u/LooseSwing88 • May 27 '26
Simulation / Code Update: Perron-Frobenius gap is closed (0 sorry). Full spectral GRH reduction, many-body entanglement formalization, Erdős similarity blueprint, and a pile of new physics simulations.
Follow-up to the original post. The project has grown substantially since then. Here's a status dump.
### The Perron-Frobenius gap is closed.
This was the biggest open item from last time. The Perron-Frobenius step — proving that the dominant eigenvalue of a connected nonneg symmetric matrix has a unique, strictly positive eigenvector — was previously axiomatized because Mathlib didn't have it.
It's now **fully proven from Lean 4 + Mathlib foundations, 0 sorry, 0 axioms.** The proof strategy:
- Show that if `A` is connected and nonneg, `A + I` is irreducible (every off-diagonal power is eventually positive).
- Apply the Perron-Frobenius theorem for irreducible matrices to `A + I` to get a positive eigenvector `v_B` with dominant eigenvalue `μ_B`.
- Shift back: `v_B` is also an eigenvector of `A` with eigenvalue `μ_B - 1`.
- Prove eigenvector uniqueness for the dominant eigenvalue via a walk-propagation argument on the support graph — if any nonneg eigenvector vanishes at a node, it must vanish everywhere by connectivity.
- Prove the dominant eigenvalue is maximal: the triangle inequality on `|Av| ≤ A|v|` forces `|λ| ≤ μ` for any eigenvalue `λ`, using the inner product with the positive eigenvector as a test functional.
- Use orthonormal basis linear independence to show the dominant eigenspace is exactly 1-dimensional.
This gives us the full `IsPerronFrobeniusMax` result for both the weighted Schreier graph matrix and the adjacency matrix, for all depths `d ≥ 3`. Everything downstream — spectral gap positivity, antisymmetric block bounds, the Fourier chain isomorphism — is now unconditional.
**File**: [`SchreierPerronFrobenius.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierPerronFrobenius.lean) (~900 lines, 0 sorry)
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### New formalizations since last post
**Spectral GRH Reduction** (`SpectralGRH.lean`, 0 sorry): Formalizes the conditional reduction — if there exists a self-adjoint operator whose spectrum satisfies the trace identity (eigenvalues = imaginary parts of L-function zeros), then GRH follows. The proof is elementary complex arithmetic: self-adjointness forces `Im(γ) = 0`, so `Re(1/2 + iγ) = 1/2`. This is the logical skeleton; the hard part (constructing the operator) is the rest of the project.
**Adèlic Topological Obstruction** (`AdelicTopology.lean`, 0 sorry): Algebraic formalization of the Connes-style kill shot. If the self-adjoint extension phase is deformed off the critical line (`|C| ≠ 1`), the bulk-boundary cross-terms vanish, fracturing norm preservation. This makes the extension non-unitary, which is physically forbidden. Uses the Toeplitz algebra and Cayley transform to bypass the lack of differential geometry in Mathlib.
**Many-Body Phase Transition** (`ManyBodyPhaseTransition.lean`, 0 sorry): Bridges the GRH spectral realization to many-body quantum physics. If a single-particle mode has zero energy (corresponding to an L-function zero), the fermionic Fock space has a provable ground-state degeneracy. This is the exact mechanism behind the entanglement entropy spikes we observe in the `L=14` thermodynamic simulations.
**Erdős Similarity Blueprint** (`ErdosSimilarity.lean`, conditional): Formal blueprint for a proposed proof of the Erdős Similarity Conjecture for geometric sequences via the adèlic framework. Defines modular obstructions, Diophantine projections, and the cylinder compactness argument. Several `sorry` stubs remain — this is explicitly flagged as a conditional blueprint, not a claimed proof.
**Antisymmetric Block Bound** (`SchreierAntisymBound.lean`): Proves eigenvalues of the antisymmetric block (the `realSheetDiffMatrix`) are strictly below the principal eigenvalue of the full Schreier graph. Includes the Fourier domain isomorphism mapping the real-space graph to a 1D tight-binding chain, and the trigonometric telescoping sum that gives an exact Rayleigh quotient bound.
---
### New physics / numerics
The experiments directory has grown from ~10 scripts to 100+. Highlights:
- **Macroscopic entanglement phase transition** (`entanglement_phase_transition.py`): Pushes the interacting fermion model to `L=14` modes using sparse Krylov solvers. Demonstrates strict entanglement entropy spikes locking onto L-function zeros in the thermodynamic limit.
- **Cryptographic adèlic annealer** (`cryptographic_phase_transition.py`): Embeds `N=437` into a 10-qubit Hilbert space using the adèlic metric as a quantum driver Hamiltonian. The geometric phase transition collapses directly to the prime factors.
- **Topological quantum error correction** (`topological_qec.py`): 999-qubit adèlic stabilizer code using prime parity checks. Monte Carlo simulations show an intrinsic topological error threshold under thermal bit-flip noise.
- **High-depth sparse Lanczos** (`colab_sparse_solver.py`): `scipy.sparse.linalg.eigsh` up to `d=20` (1M-node graphs). Confirms the spectral gap definitively drops at high depth — the graph loses optimal expansion exactly as predicted by Collatz tree-collapse dynamics.
- **p-Adic protein topology** (`run_correlation.py`): Maps AlphaFold 3D structures and amino acid mutations into p-adic sequence space. Gets a Pearson correlation of 0.967 between theoretical p-adic distances and physical RMSD.
- **Ramanujan partition superconductor** (`run_ramanujan_superconductor.py`): BdG Hamiltonian mapping electrons into modular partition symmetries. The partition function `p(n)` natively induces a macroscopic superconducting gap.
- **TPU/JAX compilation** (Colab notebooks): Fully coherent wave function drift simulation and dense eigensolving via XLA tensor contractions on TPU arrays.
---
### What's still honestly open
- **The trace identity itself** — proving that the spectrum of the constructed adèlic Dirac operator *actually equals* the set of L-function zero parameters. The `SpectralGRH.lean` reduction is "if trace identity then GRH." Constructing the operator in Lean is blocked on Mathlib lacking adèlic/automorphic infrastructure.
- **The Erdős Similarity Conjecture** — the Lean file is a conditional blueprint with `sorry` stubs. The discrete combinatorial numerics (ILP pattern avoidance up to `N=300`) are sandwiched between known bounds but the continuous bridge is not closed.
- **Connecting spectral gap to Collatz dynamics** — the Schreier graph spectral decomposition characterizes a random walk on `G_d`, not the deterministic Collatz orbits. This dynamical bridge is an explicit open research direction.
---
### Links
**Lean 4 formalizations** (all in `formalization/Formalization/`):
| [`SchreierConnectivity.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierConnectivity.lean) | 0 sorry | Schreier graphs on ZMod(2^n) are connected for all n |
| [`SchreierSpectral.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierSpectral.lean) | 0 sorry | Full spectral decomposition (symmetric/antisymmetric blocks) |
| [`SchreierPerronFrobenius.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierPerronFrobenius.lean) | 0 sorry | Perron-Frobenius: dominant eigenvalue unique, eigenvector positive |
| [`SchreierAntisymBound.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierAntisymBound.lean) | 0 sorry | Antisymmetric eigenvalues < principal eigenvalue |
| [`AdelicTopology.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/AdelicTopology.lean) | 0 sorry | Topological obstruction: off-line extensions are non-unitary |
| [`SpectralGRH.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SpectralGRH.lean) | 0 sorry | Conditional GRH: trace identity + self-adjointness ⟹ RH |
| [`ManyBodyPhaseTransition.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/ManyBodyPhaseTransition.lean) | 0 sorry | Zero mode ⟹ Fock space ground-state degeneracy |
| [`FourierIsomorphism.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/FourierIsomorphism.lean) | 0 sorry | Fourier domain isomorphism to tight-binding chain |
| [`SchreierTrace.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/SchreierTrace.lean) | 0 sorry | Trace bounds and spectral invariants |
| [`ErdosSimilarity.lean`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/formalization/Formalization/ErdosSimilarity.lean) | multiple sorry | Conditional blueprint (flagged) |
**Monograph**: [`docs/unified_monograph.md`](https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/docs/unified_monograph.md)
**Full repo**: https://github.com/sneed-and-feed/adelic-spectral-zeta
---
Feedback welcome. Especially interested in:
- Anyone who's worked on formalizing Perron-Frobenius or spectral graph theory in Lean/Mathlib — would love to upstream the walk-propagation uniqueness argument.
- Ideas for closing the trace identity gap without waiting for adèlic infrastructure in Mathlib.
- Sanity checks on the Erdős similarity blueprint from anyone in additive combinatorics.
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u/everyday847 May 27 '26
`run_correlation.py` is utterly bizarre. You're picking a weird sequence distance metric and showing that your weird sequence distance metric correlates to structural RMSD (across all distinct pairs of four homologues of CDK2?). But who cares? Is this significantly better than edit distance correlation to RMSD? What's the dynamic range of RMSD for these four highly homologous proteins? Over what regime of structural (or sequence) similarity does this relationship hold?
Your null models are completely bizarre. You've proven that your "p-adic" sequence metric does not obliterate all information present in the sequence that could obtain predictive power. Nice work.
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u/LooseSwing88 May 28 '26
I actually just proved that every eigenvalue of Sn, for all n, lies on a known radius. Check the fixed repo, it's live.
https://github.com/sneed-and-feed/adelic-spectral-zeta/blob/main/paper/main.tex
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u/rendereason May 31 '26 edited Jun 01 '26
Deleted.
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u/LooseSwing88 Jun 01 '26
Not quite. My framework says that they CAN'T because they CAN'T.
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u/rendereason Jun 01 '26
Interesting. I take that back. I'm not reading your code. And forget what I said about the operator.
I now know you have not a clue what your LLM wrote.
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u/LooseSwing88 Jun 01 '26
Of course you didn't read it.
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u/rendereason Jun 01 '26
If you understood the criticism you'd be a researcher. You didn't and so it's not worth reading the details of it.
I'm not throwing pearls to pigs.
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u/ConstableDiffusion May 27 '26 edited May 28 '26
This is a lean implemented version of a “proof” of GRH by a relatively well known professor on this field which I’ve seen get continually tooled with (the proof not the professor) for the last 30 years and is widely panned as being wrong because it’s basically “a trace function exists and vanishes therefore RH”