A Conditional Sieve Criterion for Twin Primes via Krafft Geometry #
Source: doi:10.5281/zenodo.19763833
Authors: Fernando Portela
Status: verified
Main declarations: KrafftSieve.mu_min_lt_one_implies_tpc
Tags: analytic-number-theory, sieve-theory, twin-primes, optimization
MSC: 11N05, 11N35
Mathematical overview #
This project implements a formalization of a weighted Turán Sieve approach to the Twin Prime Conjecture, utilizing the Krafft Geometry.
Main Results #
This project's headline result is conditional: it reduces the Twin Prime Conjecture to a
spectral hypothesis on the Rayleigh quotient $\mu_{min}(n)$. The hypothesis $\mu_{min}(n) < 1$
is not established here for any $n$; indeed resonance_lt_mainTerm formalizes the
parity-barrier deficit showing the one-dimensional sieve cannot beat the main term. The value
of this formalization is in the verified sieve machinery, not a claim of progress on the
conjecture itself. The conditional results are captured by:
KrafftSieve.mu_min_lt_one_implies_tpc: If there are infinitely many intervals where the multidimensional optimal weight achieves a ratio strictly less than 1, then there are infinitely many twin primes.krafft_sieve_guarantee_with_mu_min: The Krafft Sieve Guarantee holds if $\mu_{min}(n) < 1$.Krafft_Sufficiency: Defines the exact set of conditions for the sieve to be considered successful.
Major Analytical Building Blocks #
krafft_sieve_guarantee: A generic formulation proving that the existence of a valid weight function satisfying the variance constraints guarantees the discovery of Twin Primes.
Independent Analytical Results #
The following independent results provide structural motivation and spectral analysis of the sieve:
resonance_lt_mainTerm: Establishes the core inequality where the main term dominates the error/resonance terms.third_harmonic_extraction: Extracts the critical third harmonic frequency from the Fourier domain analysis of the weighted hit counts.parseval_identity: Links the variance of the survivor distribution strictly to the sum of the squared magnitudes of its non-zero Fourier coefficients.W_truly_multi: Constructs the multi-dimensional optimal weight function crucial for driving down the physical variance.