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  "exactFormalStatement": "theorem legendreSym.quadratic_reciprocity {p q : ℕ} [Fact (Nat.Prime p)] [Fact (Nat.Prime q)] (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) : legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2))",
  "family": "Quadratic residues",
  "id": "library.mathlib.quadratic-reciprocity.v001",
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    "completeFormalType": "legendreSym.quadratic_reciprocity {p q : ℕ} [Fact (Nat.Prime p)] [Fact (Nat.Prime q)] (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) : legendreSym q ↑p * legendreSym p ↑q = (-1) ^ (p / 2 * (q / 2))",
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      "quadratic residues",
      "Legendre symbols",
      "reciprocity"
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      "alt": "An ivory theorem poster links distinct odd-prime medallions p and q with reciprocal arcs, displays the exact Legendre-symbol product and natural-division exponent, and gives the parity-controlled sign rule.",
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      "caption": "The reciprocal Legendre-symbol product has one exact parity-controlled sign, with p divided by 2 and q divided by 2 interpreted by natural-number floor division.",
      "description": "Green and blue prime medallions face each other across reciprocal arcs. A central formula gives the Legendre-symbol product and its sign exponent. Below it, the poster explicitly says natural division rounds down and separates the both-3-mod-4 minus case from the otherwise-plus case.",
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      "title": "Quadratic Reciprocity at a glance",
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      "transcript": "NUMBER THEORY · QUADRATIC RESIDUES\nQUADRATIC RECIPROCITY\np, q DISTINCT ODD PRIMES\nlegendreSym q p · legendreSym p q\n= (−1)^(p/2 · q/2)\np / 2 AND q / 2 USE NATURAL DIVISION\nROUND DOWN\nSIGN RULE\np ≡ 3 mod 4 AND q ≡ 3 mod 4 · −1\nOTHERWISE · +1\nRECIPROCAL QUESTIONS · ONE EXACT SIGN\nEXACT SCOPE\nLegendre symbols only, with primality, oddness, and distinctness."
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      "caption": "Quadratic residue information in two odd distinct primes is linked in both directions, with a parity-controlled sign.",
      "description": "Two large symmetric residue circles face each other across reciprocal gold arrows. Smaller paired circles below show one aligned exchange and one restrained red reversal cue, suggesting the parity-dependent sign without rendering notation.",
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      "title": "Quadratic residue reciprocity schematic",
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      "introduction": "Mathlib's checked route prepares odd-prime field inputs, invokes its finite-field quadratic-character reciprocity identity, cancels the squared nonzero character factor, and evaluates the remaining parity signs.",
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          "title": "Prepare the odd-prime inputs",
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          "description": "The proof applies quadraticChar_odd_prime to ZMod p and the odd prime q, using p ≠ q to keep the transported element nonzero. After rewriting the cardinality of ZMod p, this becomes the relation that the remaining rewrites normalize.",
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          "title": "Invoke quadratic-character reciprocity",
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          "equation": "h := quadraticChar_odd_prime ...",
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          "alt": "Layered versions of the recurring field wheels expose character structure while a doubled gold loop on the blue wheel closes at a noncentral bead.",
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          "caption": "Unfolding the Legendre symbols exposes multiplicative quadratic characters, and the squared nonzero character factor collapses to one.",
          "description": "The final rewrite chain expands both Legendre symbols, aligns the natural and integer casts, substitutes the reciprocity relation, rotates the product, and uses quadraticChar_sq_one with prime_ne_zero. This algebraic cancellation leaves only parity-controlled sign factors.",
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          "title": "Normalize and cancel the square",
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          "equation": "quadraticChar (ZMod q) p ^ 2 = 1",
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          "alt": "The full green and blue wheels close a symmetric reciprocal circuit above a smaller parity echo with one restrained red half-turn on its gold rosette.",
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          "caption": "Oddness evaluates the two character signs and leaves exactly the parity-controlled sign in quadratic reciprocity.",
          "description": "The proof uses χ₄'s odd-input evaluation for p and q together with quadraticChar_neg_one in ZMod q. The exponent rewrites combine the two parity contributions and close the exact Legendre-symbol product stated by quadratic_reciprocity.",
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          "sha256": "sha256:2743cc0f7e5fdea3aca06aeaa2de4db7996f48626998a33f24d5fc8e9e941b04",
          "title": "Evaluate the reciprocity sign",
          "width": 1536,
          "equation": "legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2))",
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      "title": "How quadratic characters yield reciprocity"
    },
    "whyLandmark": "Quadratic reciprocity is a central symmetry law of elementary number theory. It turns one quadratic-residue question into another, with an exact sign correction."
  },
  "nonClaims": [
    "Proof Atlas did not originate quadratic reciprocity or Mathlib's declaration.",
    "The displayed sign uses Mathlib's exact p / 2 * (q / 2) exponent form.",
    "The page does not claim a new proof or a stronger reciprocity law."
  ],
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    "locallyReproduced": "Exact upstream declaration replayed",
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  "summary": "For distinct odd primes, the two Legendre symbols are related by the quadratic-reciprocity sign.",
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  "title": "Quadratic Reciprocity",
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