{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-p-adic-valuation-integers",
  "semantic_family_id": "sf-p-adic-valuation-integers",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "整數p進賦值"
  },
  "summary": {
    "zh-Hant": "對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。"
  },
  "statement": {
    "zh-Hant": "prime p≥2；a≠0時以|a|=p^k*u且p不整除u的唯一k定義v_p(a)，負a合法並先取絕對值。固定可序列化sentinel `infinity` 表示v_p(0)。exact families核factor extraction、v_p(ab)=v_p(a)+v_p(b)與v_p(a+b)≥min；只有v_p(a)≠v_p(b)時保證sum equality。nonprime base不屬p-adic valuation，含零identity依infinity convention分開。"
  },
  "formula": {
    "tex": "V(p,a)=k",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>V</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "p",
            "glyph": "p"
          },
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "k",
        "glyph": "k"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "4ddf17806dcdf758d282883018fce78259183f0434fa91c2d7eefa9bd91ca1a3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "p是exact prime且p≥2。"
    },
    {
      "id": "a2",
      "zh-Hant": "非零輸入的factor extraction以absolute value進行。"
    },
    {
      "id": "a3",
      "zh-Hant": "零值唯一採字串sentinel `infinity`。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "integer_p_adic_valuation_operator",
      "role_zh": "整數p進賦值算子",
      "scope": "representative_formula"
    },
    {
      "token": "p",
      "role": "prime_base",
      "role_zh": "質數底",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "integer_input",
      "role_zh": "整數輸入",
      "scope": "representative_formula"
    },
    {
      "token": "k",
      "role": "valuation_or_infinity",
      "role_zh": "賦值或infinity",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-prime-number",
      "title_zh": "質數",
      "reason_zh": "prime base。"
    },
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除",
      "reason_zh": "最高整除次數。"
    },
    {
      "id": "mko-unique-prime-factorization",
      "title_zh": "整數唯一質因數分解",
      "reason_zh": "存在唯一性背景。"
    },
    {
      "id": "mko-integer",
      "title_zh": "整數",
      "reason_zh": "signed input與zero。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "negative normalization。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "prime base與zero convention先固定。",
      "product與sum laws在exact integer domain重算。",
      "unequal valuations才保證sum equality。"
    ],
    "common_misconception_zh": "把負整數視為無效、對composite base仍稱p-adic，或宣稱所有和都有等號。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "bounded integer extraction不取代一般valuation theory或p-adic completion。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入p-adic absolute value、completion與DVR。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-p-adic-valuation-integers.md 整理；2026-10-15 脈衝 R05（使用者手動接續 2026-11-18 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}