Pith Number

pith:M6RWMEXS

pith:2025:M6RWMEXSHB2ACVMUHQRAWUZQOM

not attested not anchored not stored refs pending

Spectral results for free random variables

Brian C. Hall, Ching-Wei Ho

If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.

arxiv:2510.03382 v5 · 2025-10-03 · math.OA · math-ph · math.MP · math.PR

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\pithnumber{M6RWMEXSHB2ACVMUHQRAWUZQOM}

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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

✓ Portable graph bundle live · download bundle · merged state

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

Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.

C2weakest assumption

The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph).

C3one line summary

A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:25.268741Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395

Aliases

arxiv: 2510.03382 · arxiv_version: 2510.03382v5 · doi: 10.48550/arxiv.2510.03382 · pith_short_12: M6RWMEXSHB2A · pith_short_16: M6RWMEXSHB2ACVMU · pith_short_8: M6RWMEXS

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/M6RWMEXSHB2ACVMUHQRAWUZQOM \
  | 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: 67a36612f238740155943c220b5330731c9ea3eb2fefd43a5e4aaf315acac395

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1e62d94852568a00a048d935dffdba6c7b8cd2b06d4714bdc073b1090eec1647",
    "cross_cats_sorted": [
      "math-ph",
      "math.MP",
      "math.PR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OA",
    "submitted_at": "2025-10-03T15:52:14Z",
    "title_canon_sha256": "655d18f8204514639ec085112d547fe2440c89f4b1ee7eeff67f42aae9a2fb17"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2510.03382",
    "kind": "arxiv",
    "version": 5
  }
}