{
 "experiment": "identity_search_secp256k1_constants",
 "timestamp_utc": "2026-11-01T05:45:18+00:00",
 "command": "experiments/automated_discovery/identity_search.py",
 "cwd": "/Users/simon-pierreboucher/Desktop/UQO/Recherche/Recherche-UQO/mathematiques/beyond-square-root-barrier",
 "seed": null,
 "host": {
  "hostname": "MacBookPro-de-simon-pierre-2.local",
  "os": "Darwin 27.0.0 (27.0)",
  "machine": "arm64",
  "python": "3.12.13",
  "cpu": "Apple M5 Max",
  "cores": 18,
  "ram_gb": 48.0
 },
 "software": {
  "numpy": "2.5.4",
  "scipy": "1.18.1",
  "sympy": "1.14.0",
  "gmpy2": "2.3.2",
  "z3": "5.3.3",
  "pysat": "1.9.dev17",
  "ecdsa": "0.19.2",
  "coincurve": "21.0.0"
 },
 "params": {
  "spaces": [
   "Q1",
   "Q2",
   "Q3",
   "Q4",
   "Q5"
  ],
  "coeff_bound": 6
 },
 "measurements": {
  "candidates_tested": 58064,
  "identities_found": 22,
  "found": [
   {
    "space": "Q1",
    "id": "Q1:t^2+0*t*f+3*f^2=4p",
    "relation": "t^2 +0\u00b7t\u00b7f +3\u00b7f^2 = 4p",
    "classification": "known_reproduced",
    "explanation": "CM theory: 4p = t\u00b2 + 3f\u00b2 for j = 0 (Frobenius \u03c0 = (t + f\u221a\u22123)/2 \u2208 Z[\u03c9])"
   },
   {
    "space": "Q1",
    "id": "Q1:t^2+0*t*b1+3*b1^2=4p",
    "relation": "t^2 +0\u00b7t\u00b7b1 +3\u00b7b1^2 = 4p",
    "classification": "known_reproduced",
    "explanation": "b1 = \u2212f: the GLV vector's second coordinate is the CM conductor (4p = t\u00b2 + 3f\u00b2)"
   },
   {
    "space": "Q1",
    "id": "Q1:f^2+1*f*a1+1*a1^2=n",
    "relation": "f^2 +1\u00b7f\u00b7a1 +1\u00b7a1^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:f^2-1*f*a2+1*a2^2=n",
    "relation": "f^2 -1\u00b7f\u00b7a2 +1\u00b7a2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:f^2+1*f*b2+1*b2^2=n",
    "relation": "f^2 +1\u00b7f\u00b7b2 +1\u00b7b2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a1^2+1*a1*f+1*f^2=n",
    "relation": "a1^2 +1\u00b7a1\u00b7f +1\u00b7f^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a1^2-1*a1*b1+1*b1^2=n",
    "relation": "a1^2 -1\u00b7a1\u00b7b1 +1\u00b7b1^2 = n",
    "classification": "known_reproduced",
    "explanation": "n = N(a1 + b1\u00b7\u03c9): the GLV basis vector has Eisenstein norm n (GLV 2001 / Z[\u03c9] structure)"
   },
   {
    "space": "Q1",
    "id": "Q1:a1^2-1*a1*a2+1*a2^2=n",
    "relation": "a1^2 -1\u00b7a1\u00b7a2 +1\u00b7a2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:b1^2-1*b1*a1+1*a1^2=n",
    "relation": "b1^2 -1\u00b7b1\u00b7a1 +1\u00b7a1^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:b1^2+1*b1*a2+1*a2^2=n",
    "relation": "b1^2 +1\u00b7b1\u00b7a2 +1\u00b7a2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:b1^2-1*b1*b2+1*b2^2=n",
    "relation": "b1^2 -1\u00b7b1\u00b7b2 +1\u00b7b2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a2^2-1*a2*f+1*f^2=n",
    "relation": "a2^2 -1\u00b7a2\u00b7f +1\u00b7f^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a2^2-1*a2*a1+1*a1^2=n",
    "relation": "a2^2 -1\u00b7a2\u00b7a1 +1\u00b7a1^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a2^2+1*a2*b1+1*b1^2=n",
    "relation": "a2^2 +1\u00b7a2\u00b7b1 +1\u00b7b1^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:a2^2-1*a2*b2+1*b2^2=n",
    "relation": "a2^2 -1\u00b7a2\u00b7b2 +1\u00b7b2^2 = n",
    "classification": "known_reproduced",
    "explanation": "same for the second basis vector"
   },
   {
    "space": "Q1",
    "id": "Q1:b2^2+1*b2*f+1*f^2=n",
    "relation": "b2^2 +1\u00b7b2\u00b7f +1\u00b7f^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:b2^2-1*b2*b1+1*b1^2=n",
    "relation": "b2^2 -1\u00b7b2\u00b7b1 +1\u00b7b1^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q1",
    "id": "Q1:b2^2-1*b2*a2+1*a2^2=n",
    "relation": "b2^2 -1\u00b7b2\u00b7a2 +1\u00b7a2^2 = n",
    "classification": "known_reproduced",
    "explanation": "quadratic form u\u00b2 \u00b1 uv + v\u00b2 = n among {a1, b1, a2, b2, f}: all are Eisenstein norms of unit multiples / conjugates of \u03c0\u0304 \u2212 1 (CM theory, GLV lattice = ideal (\u03c0\u0304 \u2212 1))"
   },
   {
    "space": "Q2",
    "id": "Q2:beta^2+1*beta+1=0 mod p",
    "relation": "beta^2 +1\u00b7beta +1 \u2261 0 (mod p)",
    "classification": "known_reproduced",
    "explanation": "\u03b2 is a primitive cube root of unity (\u03b2\u00b2 + \u03b2 + 1 = 0)"
   },
   {
    "space": "Q2",
    "id": "Q2:lambda^2+1*lambda+1=0 mod n",
    "relation": "lambda^2 +1\u00b7lambda +1 \u2261 0 (mod n)",
    "classification": "known_reproduced",
    "explanation": "\u03bb is a primitive cube root of unity in Z/nZ"
   },
   {
    "space": "Q3",
    "id": "Q3:a2=+1*a1-1*b1",
    "relation": "a2 = 1\u00b7a1 -1\u00b7b1",
    "classification": "known_reproduced",
    "explanation": "second GLV vector = \u03c9\u00b7(first) up to sign: (a1,b1)\u00b7\u03c9 rotates the lattice (Eisenstein units)"
   },
   {
    "space": "Q3",
    "id": "Q3:b2=+1*a1+0*b1",
    "relation": "b2 = 1\u00b7a1 +0\u00b7b1",
    "classification": "known_reproduced",
    "explanation": "same"
   }
  ],
  "closed_form_glv_basis": {
   "a1": "64502973549206556628585045361533709077",
   "b1": "-303414439467246543595250775667605759171",
   "a2": "367917413016453100223835821029139468248",
   "b2": "64502973549206556628585045361533709077"
  },
  "closed_form_matches_libsecp256k1": true,
  "closed_form_statement": "a1 = (t\u2212f)/2 \u2212 1, b1 = \u2212f, a2 = (t+f)/2 \u2212 1, b2 = a1 with 4p = t\u00b2 + 3f\u00b2; lattice = ideal (\u03c0\u0304\u22121)Z[\u03c9]",
  "negative_controls": {
   "Q4_cross_modulus_hits": 0,
   "Q5_small_linear_hits": 0,
   "expected_chance_hits": 5.0145049098300383e-73
  },
  "unexplained": []
 },
 "elapsed_s": 0.156
}