An Executable Proof of the Riemann Hypothesis via the L-EFM Spectral Trap

This paper details an executable, computational proof of the Riemann Hypothesis (RH) using the Laplace-extended Euler-Fourier-Mellin (L-EFM) operator within the framework of Arithmetic Spectral Theory (AST). Rather than a traditional symbolic proof, the work provides a deterministic implementation where the code itself serves as the proof, ensuring results are reproducible through a cryptographic SHA-256 hash and a fixed seed.

The L-EFM Spectral Trap Framework

The proof centers on the "spectral trap" mechanism, which dictates that the L-EFM operator is admissible in the Gelfand-Shilov dual space $S'$ if and only if the real part of its argument is $\sigma = 0.5$. The framework is built upon three pillars:

• Deterministic Prime Shift Operators: Primes are generated using a deterministic Sieve of Eratosthenes to ensure consistency.

• L-EFM Operator Symbol: This is constructed as an infinite product over primes. On the critical line ($\sigma = 0.5$), the operator is normalized to a magnitude of 1.

• Growth Lemma: The proof hinges on the requirement that for a distribution to belong to the $S'$ space, its growth exponent $\alpha$ must be zero. For the L-EFM operator, $\alpha = |\sigma - 0.5|$, meaning admissibility only occurs at the critical line.

Computational Validation

The executable proof utilizes three primary tests to validate the spectral trap:

1. Spectral Trap Test: Evaluates the operator across the critical strip (0.1 to 0.9). Numerical results show that $\alpha = 0$ and a normalized magnitude of 1.0 occur exclusively at $\sigma = 0.5$. All other values result in "spectral escape," where the operator fails admissibility.

2. Green-Tao Test: Using the Green-Tao theorem regarding prime arithmetic progressions, the implementation shows that these progressions exhibit high spectral coherence only on the critical line.

3. Boundary Spectral Escape Test: Tests at the extreme boundaries ($\sigma = 0.01$ and $\sigma = 0.99$) demonstrate catastrophic divergence, with magnitudes reaching up to $10^{244}$, further confirming that admissibility is restricted to $\sigma = 0.5$.

Formal Proof and Conclusion

The formal proof concludes that every non-trivial zero $\rho = \sigma + i\gamma$ of the Riemann zeta function must satisfy $\sigma = 1/2$. This is based on the requirement that any non-trivial zero must correspond to an admissible state in the $S'$ space. Because the computational and theoretical evidence shows that admissibility occurs only at $\sigma = 1/2$, all non-trivial zeros must lie on the critical line.

This approach represents a paradigm shift toward "executable science," where the documentation (the paper) and the proof (the code) are inextricably linked, allowing for independent, tamper-proof verification by any researcher.