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DURAND, SERGE · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.5281/zenodo.17771075

1️⃣ Description – PDF File: The Dual Nature of Prime Gaps.pdf This preprint investigates the statistical structure of prime gaps through a dual lens: Random Matrix Theory on the one hand, and the Hardy–Littlewood singular series (Prime Cosmic Web Law, PCWL) on the other. On the “spectral” side, the primes are treated as energy levels and their nearest–neighbour spacings are analysed using modern tools from quantum chaos (level–spacing distribution and r–statistic). For primes up to 10610^6106, the normalized gaps yield an r–parameter of r=0.3857r = 0.3857r=0.3857, in perfect agreement (within 0.08%) with the Poisson prediction r≈0.386r \approx 0.386r≈0.386. This confirms that the primes, as a point process on the line, exhibit no level repulsion and behave locally like independent random points. On the “arithmetic” side, we study bounded prime constellations of the form (p,p+2k)(p, p+2k)(p,p+2k) (Polignac gaps) up to X=1010X = 10^{10}X=1010. The empirical counts for gaps 2,4,6,…,3002,4,6,\dots,3002,4,6,…,300 are compared to the Hardy–Littlewood prediction involving the twin prime constant C2C_2C2 and the singular series S(2k)\mathfrak{S}(2k)S(2k). The results show agreement at the level of 0.003%0.003\%0.003% across 50 distinct gaps, including the predicted factor 222 for multiples of 3, 4/34/34/3 for multiples of 5, and 2×4/32 \times 4/32×4/3 for mixed gaps such as 30. Together, these findings reveal a dual nature of prime gaps: spectrally, the primes are Poisson–random (no strong correlations between neighbouring gaps), while arithmetically, the distribution of specific gaps is governed with remarkable precision by the singular series. This reconciles the apparent paradox between randomness (Poisson statistics of gaps) and rigidity (Hardy–Littlewood/PCWL intensities), and provides one of the strongest numerical confirmations to date of the quantitative Polignac–Hardy–Littlewood framework. 2️⃣ Description – ZIP bundle (data + code + figures) File: polignac_duality_proof_bundle.zip This archive is the full companion data and code bundle for the preprint “The Dual Nature of Prime Gaps: Spectral Randomness and Arithmetic Rigidity”. It contains all numerical evidence, scripts, and figures required to reproduce the main results of the article. The bundle includes:– Prime tables up to 10610^6106 (primes_up_to_1e6.csv) and the corresponding consecutive gaps (prime_gaps_up_to_1e6.csv).– Summary statistics for the level–spacing analysis (prime_spacing_stats_1e6.csv), including the normalized spacings and the r–parameter used to verify Poisson behaviour.– Singular–series data for small gaps (singular_series_small_gaps.csv) and a table comparing Hardy–Littlewood predictions to actual counts of Polignac gaps up to 101010^{10}1010 (hl_vs_real_1e10_selected_gaps.csv).– Three high–resolution figures: the prime spacing distribution versus Poisson and GUE (fig1_prime_spacing_distribution_1e6.png), the singular–series intensities for small gaps (fig2_singular_series_small_gaps.png), and the theory–versus–data comparison for selected gaps at 101010^{10}1010 (fig3_hl_vs_real_selected_gaps_1e10.png).– The analysis scripts (run_real_probe.py, polignac_qutip_analyzer_v2.py) and auxiliary metadata files (bundle_metadata.json, README_polignac_duality_bundle.txt, checksums_sha256.txt) documenting the experimental setup and providing SHA–256 checksums for all assets. This package is intended to make the study fully transparent and reproducible. Researchers can rerun the probes, verify the Poisson nature of prime level spacings, and independently confirm the quantitative agreement between the Hardy–Littlewood / PCWL predictions and the observed distribution of Polignac gaps up to 101010^{10}1010.

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preprint

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https://doi.org/10.5281/zenodo.17771075

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https://openalex.org/W7108079933

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https://openalex.org/W7108079933

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https://doi.org/10.5281/zenodo.17771075

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The Dual Nature of Prime Gaps

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preprint

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2025

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2025-11-30

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DURAND, SERGE

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https://doi.org/10.5281/zenodo.17771075

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green

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https://doi.org/10.5281/zenodo.17771075

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Mathematics, Randomness, Prime (order theory), Poisson distribution, Twin prime, Bounded function, Series (stratigraphy), Dual (grammatical number), Random matrix, Point process, Discrete mathematics, Real line, Singular value, Prime number, Random variable, Statistical physics, Poisson point process, Statistical inference, Combinatorics, Sign (mathematics), Pure mathematics, Matrix (chemical analysis), Prime number theorem, Conjecture, Distribution (mathematics), Rigidity (electromagnetism), Independent and identically distributed random variables, Almost prime

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