Primes as a Field on a Discrete Torus: Primorial Field Dynamics, Gap Composition Profiles, and an Information-Theoretic Reformulation -- PrimSpace v3.0
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866902057713139712 |
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| author | Tatai, László |
| author_facet | Tatai, László |
| contents | <p>We introduce Primorial Field Dynamics (PFD), a geometric and information-theoretic framework for studying the local distribution of prime numbers. Primes are embedded into a hierarchy of discrete tori ᵏ = ∏ ℤ/pᵢℤ indexed by the k-th primorial Mₖ. The central object is the prime preference field ρ(s), a non-uniform measure on the allowed torus states encoding where primes cluster beyond the classical sieve prediction.</p> <p>The framework introduces four structural objects: Liquid Media (divisibility threads as pressure fields), the Swaddle Effect (every prime p>3 is surrounded by multiples of 2 and 3), the Gap Composition Profile (GCP, a binary matrix encoding divisibility structure inside each prime gap), and an information-theoretic reformulation of the master equation P(prime at s) = μ(s)·S(s)·ρ(s).</p> <p>Six theorems are proved, including: the Row Entropy theorem (primes are informationally silent — H(β_row(p))=0 for all p>p_k), a discrete entropy spectrum theorem, and a corrected W-convergence theorem. The information decomposition H(ℙ) = I(β_row; ℙ) + H(ℙ|β_row) shows the sieve accounts for 57.4% of primality entropy; ρ(s) carries the remaining 42.6%.</p> <p>Numerical test of Conjecture Q4 (GCP singular values ↔ Riemann zeta zeros): at N=10⁸, covering 5,761,453 prime gaps, the GCP Fourier spectrum shows statistically significant amplitude excess at Riemann zeta zero positions (mean Z=2.52, max Z=10.67, p<10⁻²⁹). The signal grows with N and is strongest for small zeros (γ<35). This is empirical evidence consistent with Q4; mathematical proof remains open.</p> <p>All claims are explicitly marked: Theorem (proved), Conjecture (empirically supported), Observation (weak signal), or Open question. One corrected conjecture (the original W-convergence target was wrong; the correct limit is proved) is documented as a methodological note. </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19794151 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Primes as a Field on a Discrete Torus: Primorial Field Dynamics, Gap Composition Profiles, and an Information-Theoretic Reformulation -- PrimSpace v3.0 Tatai, László prime numbers discrete torus Fourier analysis number theory Riemann zeta prime distribution primorial Gap Composition Profile information theory Shannon entropy singular value decomposition random matrix theory swaddle effect liquid medium prime gaps modular arithmetic GUE Montgomery pair correlation primorial field dynamics independent researcher <p>We introduce Primorial Field Dynamics (PFD), a geometric and information-theoretic framework for studying the local distribution of prime numbers. Primes are embedded into a hierarchy of discrete tori ᵏ = ∏ ℤ/pᵢℤ indexed by the k-th primorial Mₖ. The central object is the prime preference field ρ(s), a non-uniform measure on the allowed torus states encoding where primes cluster beyond the classical sieve prediction.</p> <p>The framework introduces four structural objects: Liquid Media (divisibility threads as pressure fields), the Swaddle Effect (every prime p>3 is surrounded by multiples of 2 and 3), the Gap Composition Profile (GCP, a binary matrix encoding divisibility structure inside each prime gap), and an information-theoretic reformulation of the master equation P(prime at s) = μ(s)·S(s)·ρ(s).</p> <p>Six theorems are proved, including: the Row Entropy theorem (primes are informationally silent — H(β_row(p))=0 for all p>p_k), a discrete entropy spectrum theorem, and a corrected W-convergence theorem. The information decomposition H(ℙ) = I(β_row; ℙ) + H(ℙ|β_row) shows the sieve accounts for 57.4% of primality entropy; ρ(s) carries the remaining 42.6%.</p> <p>Numerical test of Conjecture Q4 (GCP singular values ↔ Riemann zeta zeros): at N=10⁸, covering 5,761,453 prime gaps, the GCP Fourier spectrum shows statistically significant amplitude excess at Riemann zeta zero positions (mean Z=2.52, max Z=10.67, p<10⁻²⁹). The signal grows with N and is strongest for small zeros (γ<35). This is empirical evidence consistent with Q4; mathematical proof remains open.</p> <p>All claims are explicitly marked: Theorem (proved), Conjecture (empirically supported), Observation (weak signal), or Open question. One corrected conjecture (the original W-convergence target was wrong; the correct limit is proved) is documented as a methodological note. </p> |
| title | Primes as a Field on a Discrete Torus: Primorial Field Dynamics, Gap Composition Profiles, and an Information-Theoretic Reformulation -- PrimSpace v3.0 |
| topic | prime numbers discrete torus Fourier analysis number theory Riemann zeta prime distribution primorial Gap Composition Profile information theory Shannon entropy singular value decomposition random matrix theory swaddle effect liquid medium prime gaps modular arithmetic GUE Montgomery pair correlation primorial field dynamics independent researcher |
| url | https://doi.org/10.5281/zenodo.19794151 |