L-EFM: A Complete Spectral Suite for Prime-Based Theorems and the Riemann Hypothesis
Description
Executive Summary: L-EFM Spectral Suite for Prime-Based Theorems
The provided research presents a comprehensive, executable spectral framework designed to quantify major prime-number theorems and provide a deterministic proof of the Riemann Hypothesis (RH). Central to this work is the Laplace-extended Euler-Fourier-Mellin (L-EFM) operator, which acts as a unified mathematical lens for analyzing prime structures through the principles of Arithmetic Spectral Theory (AST).
The L-EFM Spectral Framework
The L-EFM operator is defined on a scale-invariant state space and utilizes an infinite product over all primes to construct its operator symbol. This framework transitions prime number theory from qualitative or asymptotic descriptions to quantitative, numerical analysis. A critical discovery within this framework is the Universal Spectral Coherence Constant of 0.5, which emerges as a fundamental invariant across all prime subsets and structures when measured at the critical line.
Proof of the Riemann Hypothesis
The paper claims a formal proof of the Riemann Hypothesis through a mechanism called the Spectral Trap. The core logic is as follows:
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Admissibility: The L-EFM operator is shown to be admissible in the Gelfand-Shilov dual space only when its growth factor is zero.
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The Trap: Numerical testing across the critical strip (from 0.1 to 0.9) demonstrates that the operator exhibits catastrophic "spectral escape" at any point other than the critical line ($\sigma = 0.5$).
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Conclusion: Because admissibility is forced at $\sigma = 0.5$, the non-trivial zeros of the Riemann zeta function must satisfy $Re(s) = 1/2$.
Quantification of Prime-Based Theorems
The suite provides exact numerical coherence values for six landmark mathematical theorems and conjectures, all evaluated at the critical line with a deterministic seed:
| Theorem/Conjecture | Spectral Result |
| Dirichlet’s Theorem | Identical coherence (0.5) for all residue classes; the difference between classes is zero. |
| Prime Number Theorem | Constant coherence (0.5) with spectral corrections that align with the asymptotic convergence of $\pi(x)$ to $li(x)$. |
| Chebyshev’s Bias | The bias vanishes in the spectral domain (0.00 magnitude), suggesting it is a numerical rather than a spectral phenomenon. |
| Hardy-Littlewood | All prime pairs and k-tuples exhibit an identical spectral coherence of 0.5. |
| Polignac’s Conjecture | Even gap sizes produce a constant coherence of 0.5 with no observed decay pattern. |
| Cramér’s Conjecture | Results show a Cramér ratio of 0.468712, consistent with the conjecture’s predicted growth form. |
Reproducibility and Auditability
A cornerstone of this research is the philosophy that "the code is the proof." To ensure scientific rigor, all results are:
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Deterministic: Hardcoded with seed 123 for exact repetition.
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Cryptographically Auditable: The entire suite of results is verified by a SHA-256 hash, ensuring that any user running the provided GitHub code will generate the exact same data and tables.
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Open Source: The complete implementation is available as a Jupyter Notebook for public verification and extension to other zeta functions.
Files
LEFM_fixed.pdf
Files
(247.8 kB)
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