Pith Number

pith:MQZS6MMC

pith:2026:MQZS6MMCUARYTWTMRQROPDN43Y

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ARE Method: Orbital Decompositions and Dihedral Cancellations for Determinants

Ramon Moya

Cyclic group actions on the symmetric group reorganize the full Leibniz expansion of the determinant into (n-1)! orbits of size n that preserve every term and expose explicit sign laws and geometric patterns.

arxiv:2605.13615 v1 · 2026-05-13 · math.RA · math.CO · math.GR

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\pithnumber{MQZS6MMCUARYTWTMRQROPDN43Y}

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Claims

C1strongest claim

The framework yields an exact reorganization of the Leibniz expansion preserving all n! terms while exposing hidden geometric and combinatorial structure. We further prove an impossibility theorem showing that no fixed-width direct extension of the classical Sarrus rule can capture all determinant terms for n >= 4.

C2weakest assumption

That the chosen right action by the cyclic group C_n on S_n produces orbits whose sign behavior and term coverage exactly match the Leibniz formula without omissions or overcounting once the rectification permutation is applied.

C3one line summary

A cyclic-orbit decomposition of the Leibniz formula yields sign laws, a rectification theorem, and a proof that no fixed-width Sarrus-style rule exists for n greater than or equal to 4.

References

21 extracted · 21 resolved · 1 Pith anchors

[1] Verallgemeinerte Sarrussche Regel [Generalized Sarrus rule] 1935

[2] Artin, Michael. Algebra. Englewood Cliffs, NJ: Prentice Hall, 1991 1991

[3] Combinatorics of Permutations (2nd ed.) 2012

[4] Eigenvalues and eigenvectors of symmetric centrosymmetric matrices 1976

[5] Abstract Algebra (3rd ed.) 2004

Receipt and verification

First computed	2026-05-18T02:44:17.983234Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

64332f3182a02389da6c8c22e78dbcde05f579f4a2bce6623567057413c7bc0d

Aliases

arxiv: 2605.13615 · arxiv_version: 2605.13615v1 · doi: 10.48550/arxiv.2605.13615 · pith_short_12: MQZS6MMCUARY · pith_short_16: MQZS6MMCUARYTWTM · pith_short_8: MQZS6MMC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MQZS6MMCUARYTWTMRQROPDN43Y \
  | 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: 64332f3182a02389da6c8c22e78dbcde05f579f4a2bce6623567057413c7bc0d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b3f000f50b5e502a47a7e3d8fde866aaf6daf063e479ce832e8c1855efa58ad2",
    "cross_cats_sorted": [
      "math.CO",
      "math.GR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.RA",
    "submitted_at": "2026-05-13T14:45:21Z",
    "title_canon_sha256": "7fe15dabb9efdf52d6a201cc500c4470e73674109cafc0c092a45826c0c967dc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13615",
    "kind": "arxiv",
    "version": 1
  }
}