Pith Number

pith:O6OMS6LP

pith:2026:O6OMS6LPARNFEGV6ZVWQS26CPM

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Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

Vugar Ismailov

Deep KANs achieve universal approximation on compact sets precisely when they include one fixed non-affine continuous edge function.

arxiv:2604.23765 v2 · 2026-04-26 · cs.LG · cs.NE · math.FA

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\pithnumber{O6OMS6LPARNFEGV6ZVWQS26CPM}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We show that deep KANs in which all edge functions are either affine or equal to a fixed continuous function σ are dense in C(K) for every compact set K⊂R^n if and only if σ is non-affine.

C2weakest assumption

The edge functions are continuous real-valued functions and the KAN architecture follows the standard Kolmogorov-Arnold representation with summation at nodes; the proofs rely on this continuity and the specific layered structure without additional restrictions on width or activation placement.

C3one line summary

Deep KANs with edge functions restricted to affine maps plus one fixed non-affine continuous function σ are dense in C(K) for any compact K if and only if σ is non-affine.

Receipt and verification

First computed	2026-05-20T00:05:45.162277Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

779cc9796f045a521abecd6d096bc27b17dc796d1aa26f53f6291cc4917c7d55

Aliases

arxiv: 2604.23765 · arxiv_version: 2604.23765v2 · doi: 10.48550/arxiv.2604.23765 · pith_short_12: O6OMS6LPARNF · pith_short_16: O6OMS6LPARNFEGV6 · pith_short_8: O6OMS6LP

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/O6OMS6LPARNFEGV6ZVWQS26CPM \
  | 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: 779cc9796f045a521abecd6d096bc27b17dc796d1aa26f53f6291cc4917c7d55

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "fa1a088d861ff1b9fa1c6d1ab6bf1ecc6a20e2efff365766b79d2c1a00cb6956",
    "cross_cats_sorted": [
      "cs.NE",
      "math.FA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-04-26T15:31:51Z",
    "title_canon_sha256": "919220f8bd93ac8b977161f914e63724eb3d3f3ff8ab69b6ade6f3006efbe920"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.23765",
    "kind": "arxiv",
    "version": 2
  }
}