Pith Number

pith:UIEA7ZSF

pith:2026:UIEA7ZSF6KUDWM4ECWLNQNP52C

not attested not anchored not stored refs pending

Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$

Mohsen Aliabadi

Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.

arxiv:2605.11056 v2 · 2026-05-11 · math.CO

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\usepackage{pith}
\pithnumber{UIEA7ZSF6KUDWM4ECWLNQNP52C}

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Record completeness

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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 include a self-contained proof that diag(M) ∈ Img(M), and we prove that every solution of Mx=diag(M) satisfies diag(M)^T x ≡ rank(M) mod 2. We also give a complete rank and nullity formula for rank-one diagonal perturbations.

C2weakest assumption

The matrix M is symmetric over F_2; without symmetry the inclusion diag(M) ∈ Img(M) need not hold.

C3one line summary

For symmetric matrices M over F_2, diag(M) is always in the image of M and solutions satisfy a rank-parity relation, with explicit formulas for diagonal perturbations and tree-structured systems.

Receipt and verification

First computed	2026-05-26T01:03:33.344211Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a2080fe645f2a83b33841596d835fdd08f0eecff33a253f4fdb34317c60569a0

Aliases

arxiv: 2605.11056 · arxiv_version: 2605.11056v2 · doi: 10.48550/arxiv.2605.11056 · pith_short_12: UIEA7ZSF6KUD · pith_short_16: UIEA7ZSF6KUDWM4E · pith_short_8: UIEA7ZSF

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/UIEA7ZSF6KUDWM4ECWLNQNP52C \
  | 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: a2080fe645f2a83b33841596d835fdd08f0eecff33a253f4fdb34317c60569a0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2f690fa798dc88731a17aa58ad0ea1ad4688f3679bb0618dcab62235fe4902db",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-11T15:57:52Z",
    "title_canon_sha256": "31fe717a7445042025def55fa0de6c6c24fb303f1c2ae0a51176a2b350a84a0e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.11056",
    "kind": "arxiv",
    "version": 2
  }
}