I tiakina i:
| Kaituhi matua: | |
|---|---|
| Hōputu: | Recurso digital |
| Reo: | Ingarihi |
| I whakaputaina: |
Zenodo
2026
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| Ngā marau: | |
| Urunga tuihono: | https://doi.org/10.5281/zenodo.19794151 |
| Ngā Tūtohu: |
Tāpirihia he Tūtohu
Kāore He Tūtohu, Me noho koe te mea tuatahi ki te tūtohu i tēnei pūkete!
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Rārangi ihirangi:
- <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>