Pith Number

pith:JVSFIXUR

pith:2026:JVSFIXUREWDEACXTXCBLELNL4R

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Iterative maps emerging from cohomological structure of primes

Marzena Ciszak

Prime gaps follow a distance-dependent iterative map whose cohomological equation is solved by the logarithmic integral function.

arxiv:2605.17622 v1 · 2026-05-17 · cond-mat.stat-mech · math.NT

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Claims

C1strongest claim

the solution to this cohomological equation turns out to be the logarithmic integral function.

C2weakest assumption

The analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure, where the deterministic functional relation holds for primes up to small decaying fluctuations.

C3one line summary

Prime gaps are described by an iterative map, and fluctuations encode a cohomological structure whose solution is the logarithmic integral.

References

28 extracted · 28 resolved · 0 Pith anchors

[1] ln(p(n+τ)) = (1 + 1 24)p(n)− 1 24(p(n) + 1

[2] ln(p(n)) +τ[ln(p(n) + 2πτ)− 1 24(ln(p(n)))2] +ε(16) 9 It is clearly seen that the functional relation holds: f(p(n+τ)) =f(p(n)) +g(p(n);τ) +ε(17) wheref(x) = 1 24(25x−(x+ 1

[3] Note that the functiongis positive and monotonically decreasing

[4] Solving this equation forπ c gives the local number of primes between two numbersx1 andx 2 (see Fig

[5] Narkiewicz, The Development of Prime Number Theory 2000

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:49.209959Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4d64545e912586400af3b882b22dabe4678309310f3a89da0f9ec2a98be9acbd

Aliases

arxiv: 2605.17622 · arxiv_version: 2605.17622v1 · doi: 10.48550/arxiv.2605.17622 · pith_short_12: JVSFIXUREWDE · pith_short_16: JVSFIXUREWDEACXT · pith_short_8: JVSFIXUR

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JVSFIXUREWDEACXTXCBLELNL4R \
  | 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: 4d64545e912586400af3b882b22dabe4678309310f3a89da0f9ec2a98be9acbd

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "aa26a16193484365d0faa7d42f10da27a37973a92adc42d73d905cf83063a23f",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cond-mat.stat-mech",
    "submitted_at": "2026-05-17T19:39:50Z",
    "title_canon_sha256": "7ed34c51a0e73041cb900326e729d83dadbeebfc64d179402ef896bb54b277c5"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17622",
    "kind": "arxiv",
    "version": 1
  }
}