Pith Number

pith:CBNRDIHW

pith:2026:CBNRDIHWFYSHKIEN3XF6BPRFA2

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Prime--Zero Duality: Fractal Geometry, Renormalization-Group Flow, and an Information-Ontological Framework for Number Theory

Zhengqiang Li

A duality measure between primes and zeta zeros converges to a fixed point of 4, structurally supporting the critical line at Re(s) = 1/2.

arxiv:2604.14596 v1 · 2026-04-16 · math.NT

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Record completeness

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Claims

C1strongest claim

After geometric normalization, the data converge to a universal infrared fixed point K_IR = 4 with critical exponent b ~ 0.51, robust across two random-matrix symmetry classes; a structural argument for the Riemann Hypothesis emerges: the generator kappa with kappa^2 = ijk = -1 enforces, via exchange symmetry I_P <-> I_Z, the fixed point I_P* = I_Z* = 2, encoding the critical line Re(s) = 1/2.

C2weakest assumption

That the numerically fitted duality measure K can be interpreted as a conserved information current whose scaling reflects a renormalization-group flow derived from a variational information action S[I_P, I_Z], and that the ad-hoc generator kappa enforces the fixed point corresponding to the critical line.

C3one line summary

Numerical measurements show a prime-zero duality measure K stabilizing to an infrared fixed point of 4 under finite-size scaling with exponent near 0.5, interpreted as renormalization-group flow that structurally supports the Riemann hypothesis via an information action and quaternion-like generator

References

27 extracted · 27 resolved · 0 Pith anchors

[1] H. L. Montgomery, The pair correlation of zeros of the zeta function, InAnalytic Number Theory(Proc. Sympos. Pure Math., Vol. XXIV), pp. 181–193. Amer. Math. Soc., 1973 1973

[2] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp.48(177), 273–308, 1987 1987

[3] A. M. Odlyzko, Tables of zeros of the Riemann zeta function,https://www-users. cse.umn.edu/~odlyzko/zeta_tables/(accessed April 2026) 2026

[4] Falconer,Fractal Geometry: Mathematical Foundations and Applications, 3rd ed 2014

[5] M. L. Mehta,Random Matrices, 3rd ed. Academic Press, 2004. 101 2004

Receipt and verification

First computed	2026-05-26T23:04:17.607102Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

105b11a0f62e2475208dddcbe0be2506bf5713a40c5447aea2e43acf5f758522

Aliases

arxiv: 2604.14596 · arxiv_version: 2604.14596v1 · doi: 10.48550/arxiv.2604.14596 · pith_short_12: CBNRDIHWFYSH · pith_short_16: CBNRDIHWFYSHKIEN · pith_short_8: CBNRDIHW

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CBNRDIHWFYSHKIEN3XF6BPRFA2 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 105b11a0f62e2475208dddcbe0be2506bf5713a40c5447aea2e43acf5f758522

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a6625443707ecbb573e89a4296c559981122b03f2c903f26e0960cb70474cfff",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-16T03:59:18Z",
    "title_canon_sha256": "9b21939884457be428f580b9222a84b9a35ade52e8b4513acd000df1be446454"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.14596",
    "kind": "arxiv",
    "version": 1
  }
}