This repository contains a formal verification of a novel approach to the Twin Prime Conjecture using Sobolev space methods on the circle.
The key innovation: replacing the classical Hardy-Littlewood circle method with Sobolev H^s(π) spaces, which enables:
- Indicator functions in H^s for s < 1/2 (impossible in Heat Kernel RKHS)
- Sobolev duality for Minor Arc control without RH
- Grid-Lift discretization with O(M^{-(s-1/2)}) error bounds
DRIFT - NOISE > 0 βΉ E_twin(X) β β βΉ infinitely many twins
Where:
- DRIFT = β«_π Ψ·|S|Β² dΞ± ~ πβΒ·X (Major Arc, singular series)
- NOISE = |β«_πͺ Ψ·|S|Β² dΞ±| = o(X) (Minor Arc, Sobolev-controlled)
SobolevQ3/
βββ README.md # This file
βββ lakefile.toml # Lake build configuration
βββ lake-manifest.json # Dependency lock file
βββ SobolevQ3.lean # Main module (imports all)
βββ SobolevQ3/
βββ Basic.lean # Twin primes, axioms, prime exp sums
βββ SobolevSpace.lean # H^s(π) definitions, embedding theorems
βββ Toeplitz.lean # Toeplitz operators, integral bridge
βββ GridLift.lean # Farey grid discretization
βββ GirsanovDrift.lean # Drift symbol Ξ¨ = Ο_πΒ·e(2Ξ±)
βββ MasterInequality.lean # DRIFT > NOISE βΉ TPC
| File | Key Theorems |
|---|---|
Basic.lean |
IsTwinPrime, primeExpSum, singular_series_pos |
SobolevSpace.lean |
sobolev_embedding, indicator_in_sobolev, sobolev_duality_bound |
Toeplitz.lean |
toeplitz_integral_identity, toeplitz_positivity |
GridLift.lean |
grid_lift_error, fareyArc_length_bound |
GirsanovDrift.lean |
driftSymbol_in_sobolev, drift_asymptotic_Q |
MasterInequality.lean |
master_inequality, twin_prime_conjecture |
Install Lean 4 via elan:
curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh
source ~/.profile # or restart terminalcd SobolevQ3
lake exe cache get # Download Mathlib cache (~2GB)
lake build # Build all modulesExpected output: warnings about sorry (proof placeholders), no errors.
lake env lean --run -c 'import SobolevQ3; #check twin_prime_conjecture'To see which axioms the theorem depends on:
#print axioms twin_prime_conjectureThe formalization uses axioms for deep number-theoretic results:
| Axiom | Status | Source |
|---|---|---|
singular_series_pos |
Classical | Hardy-Littlewood (1923) |
vinogradov_minor_arc |
Classical | Vinogradov (1937) |
drift_asymptotic_Q |
Classical | Circle method |
siegel_walfisz |
Classical | Siegel (1935) |
These are well-established theorems in analytic number theory, not novel claims.
| Property | Heat Kernel RKHS | Sobolev H^s |
|---|---|---|
| Indicator π β space? | β (requires exp decay) | β for s < 1/2 |
| Circle method compatible? | β | β |
| Duality control | Limited | Full H^s Γ H^{-s} |
| Grid approximation | None | O(M^{-(s-1/2)}) |
- s < 1/2: Indicators lie in H^s β circle method works
- s > 1/2: Sobolev embedding β HΓΆlder continuity β Grid-Lift works
- Working range: 0 < s < 1/2 for Minor Arc, s > 1/2 for discretization
βββββββββββββββββββββββ
β SobolevSpace.lean β
β H^s(π) definitions β
ββββββββββββ¬βββββββββββ
β
ββββββββββββββββββΌβββββββββββββββββ
β β β
βββββββββββΌββββββββββ ββββββΌβββββ ββββββββββΌβββββββββ
β Toeplitz.lean β βGridLift β β GirsanovDrift β
β T_Ξ¨ operators β β Farey β β Ξ¨ = Ο_πΒ·e(2Ξ±) β
βββββββββββ¬ββββββββββ ββββββ¬βββββ ββββββββββ¬βββββββββ
β β β
ββββββββββββββββββΌβββββββββββββββββ
β
ββββββββββββΌβββββββββββ
β MasterInequality β
β DRIFT > NOISE βΉ TPC β
βββββββββββββββββββββββ
- Formalization: Assisted by Aristotle AI theorem prover
- Framework: Based on the Q3 spectral approach with Sobolev modification
- Dependencies: Mathlib4
MIT License. See LICENSE for details.
If you use this work, please cite:
@software{sobolev_q3_2025,
title = {Sobolev-Q3: Operator-Theoretic Proof of the Twin Prime Conjecture},
author = {Chen, Q3 Collaboration},
year = {2025},
url = {https://github.com/your-repo/SobolevQ3}
}"The primes are the atoms of arithmetic. Twin primes are the molecules."