1{2 "experiment": "identity_search_secp256k1_constants",3 "timestamp_utc": "2026-11-01T05:45:18+00:00",4 "command": "experiments/automated_discovery/identity_search.py",5 "cwd": "/Users/simon-pierreboucher/Desktop/UQO/Recherche/Recherche-UQO/mathematiques/beyond-square-root-barrier",6 "seed": null,7 "host": {8 "hostname": "MacBookPro-de-simon-pierre-2.local",9 "os": "Darwin 27.0.0 (27.0)",10 "machine": "arm64",11 "python": "3.12.13",12 "cpu": "Apple M5 Max",13 "cores": 18,14 "ram_gb": 48.015 },16 "software": {17 "numpy": "2.5.4",18 "scipy": "1.18.1",19 "sympy": "1.14.0",20 "gmpy2": "2.3.2",21 "z3": "5.3.3",22 "pysat": "1.9.dev17",23 "ecdsa": "0.19.2",24 "coincurve": "21.0.0"25 },26 "params": {27 "spaces": [28 "Q1",29 "Q2",30 "Q3",31 "Q4",32 "Q5"33 ],34 "coeff_bound": 635 },36 "measurements": {37 "candidates_tested": 58064,38 "identities_found": 22,39 "found": [40 {41 "space": "Q1",42 "id": "Q1:t^2+0*t*f+3*f^2=4p",43 "relation": "t^2 +0\u00b7t\u00b7f +3\u00b7f^2 = 4p",44 "classification": "known_reproduced",45 "explanation": "CM theory: 4p = t\u00b2 + 3f\u00b2 for j = 0 (Frobenius \u03c0 = (t + f\u221a\u22123)/2 \u2208 Z[\u03c9])"46 },47 {48 "space": "Q1",49 "id": "Q1:t^2+0*t*b1+3*b1^2=4p",50 "relation": "t^2 +0\u00b7t\u00b7b1 +3\u00b7b1^2 = 4p",51 "classification": "known_reproduced",52 "explanation": "b1 = \u2212f: the GLV vector's second coordinate is the CM conductor (4p = t\u00b2 + 3f\u00b2)"53 },54 {55 "space": "Q1",56 "id": "Q1:f^2+1*f*a1+1*a1^2=n",57 "relation": "f^2 +1\u00b7f\u00b7a1 +1\u00b7a1^2 = n",58 "classification": "known_reproduced",59 "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))"60 },61 {62 "space": "Q1",63 "id": "Q1:f^2-1*f*a2+1*a2^2=n",64 "relation": "f^2 -1\u00b7f\u00b7a2 +1\u00b7a2^2 = n",65 "classification": "known_reproduced",66 "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))"67 },68 {69 "space": "Q1",70 "id": "Q1:f^2+1*f*b2+1*b2^2=n",71 "relation": "f^2 +1\u00b7f\u00b7b2 +1\u00b7b2^2 = n",72 "classification": "known_reproduced",73 "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))"74 },75 {76 "space": "Q1",77 "id": "Q1:a1^2+1*a1*f+1*f^2=n",78 "relation": "a1^2 +1\u00b7a1\u00b7f +1\u00b7f^2 = n",79 "classification": "known_reproduced",80 "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))"81 },82 {83 "space": "Q1",84 "id": "Q1:a1^2-1*a1*b1+1*b1^2=n",85 "relation": "a1^2 -1\u00b7a1\u00b7b1 +1\u00b7b1^2 = n",86 "classification": "known_reproduced",87 "explanation": "n = N(a1 + b1\u00b7\u03c9): the GLV basis vector has Eisenstein norm n (GLV 2001 / Z[\u03c9] structure)"88 },89 {90 "space": "Q1",91 "id": "Q1:a1^2-1*a1*a2+1*a2^2=n",92 "relation": "a1^2 -1\u00b7a1\u00b7a2 +1\u00b7a2^2 = n",93 "classification": "known_reproduced",94 "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))"95 },96 {97 "space": "Q1",98 "id": "Q1:b1^2-1*b1*a1+1*a1^2=n",99 "relation": "b1^2 -1\u00b7b1\u00b7a1 +1\u00b7a1^2 = n",100 "classification": "known_reproduced",101 "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))"102 },103 {104 "space": "Q1",105 "id": "Q1:b1^2+1*b1*a2+1*a2^2=n",106 "relation": "b1^2 +1\u00b7b1\u00b7a2 +1\u00b7a2^2 = n",107 "classification": "known_reproduced",108 "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))"109 },110 {111 "space": "Q1",112 "id": "Q1:b1^2-1*b1*b2+1*b2^2=n",113 "relation": "b1^2 -1\u00b7b1\u00b7b2 +1\u00b7b2^2 = n",114 "classification": "known_reproduced",115 "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))"116 },117 {118 "space": "Q1",119 "id": "Q1:a2^2-1*a2*f+1*f^2=n",120 "relation": "a2^2 -1\u00b7a2\u00b7f +1\u00b7f^2 = n",121 "classification": "known_reproduced",122 "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))"123 },124 {125 "space": "Q1",126 "id": "Q1:a2^2-1*a2*a1+1*a1^2=n",127 "relation": "a2^2 -1\u00b7a2\u00b7a1 +1\u00b7a1^2 = n",128 "classification": "known_reproduced",129 "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))"130 },131 {132 "space": "Q1",133 "id": "Q1:a2^2+1*a2*b1+1*b1^2=n",134 "relation": "a2^2 +1\u00b7a2\u00b7b1 +1\u00b7b1^2 = n",135 "classification": "known_reproduced",136 "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))"137 },138 {139 "space": "Q1",140 "id": "Q1:a2^2-1*a2*b2+1*b2^2=n",141 "relation": "a2^2 -1\u00b7a2\u00b7b2 +1\u00b7b2^2 = n",142 "classification": "known_reproduced",143 "explanation": "same for the second basis vector"144 },145 {146 "space": "Q1",147 "id": "Q1:b2^2+1*b2*f+1*f^2=n",148 "relation": "b2^2 +1\u00b7b2\u00b7f +1\u00b7f^2 = n",149 "classification": "known_reproduced",150 "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))"151 },152 {153 "space": "Q1",154 "id": "Q1:b2^2-1*b2*b1+1*b1^2=n",155 "relation": "b2^2 -1\u00b7b2\u00b7b1 +1\u00b7b1^2 = n",156 "classification": "known_reproduced",157 "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))"158 },159 {160 "space": "Q1",161 "id": "Q1:b2^2-1*b2*a2+1*a2^2=n",162 "relation": "b2^2 -1\u00b7b2\u00b7a2 +1\u00b7a2^2 = n",163 "classification": "known_reproduced",164 "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))"165 },166 {167 "space": "Q2",168 "id": "Q2:beta^2+1*beta+1=0 mod p",169 "relation": "beta^2 +1\u00b7beta +1 \u2261 0 (mod p)",170 "classification": "known_reproduced",171 "explanation": "\u03b2 is a primitive cube root of unity (\u03b2\u00b2 + \u03b2 + 1 = 0)"172 },173 {174 "space": "Q2",175 "id": "Q2:lambda^2+1*lambda+1=0 mod n",176 "relation": "lambda^2 +1\u00b7lambda +1 \u2261 0 (mod n)",177 "classification": "known_reproduced",178 "explanation": "\u03bb is a primitive cube root of unity in Z/nZ"179 },180 {181 "space": "Q3",182 "id": "Q3:a2=+1*a1-1*b1",183 "relation": "a2 = 1\u00b7a1 -1\u00b7b1",184 "classification": "known_reproduced",185 "explanation": "second GLV vector = \u03c9\u00b7(first) up to sign: (a1,b1)\u00b7\u03c9 rotates the lattice (Eisenstein units)"186 },187 {188 "space": "Q3",189 "id": "Q3:b2=+1*a1+0*b1",190 "relation": "b2 = 1\u00b7a1 +0\u00b7b1",191 "classification": "known_reproduced",192 "explanation": "same"193 }194 ],195 "closed_form_glv_basis": {196 "a1": "64502973549206556628585045361533709077",197 "b1": "-303414439467246543595250775667605759171",198 "a2": "367917413016453100223835821029139468248",199 "b2": "64502973549206556628585045361533709077"200 },201 "closed_form_matches_libsecp256k1": true,202 "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]",203 "negative_controls": {204 "Q4_cross_modulus_hits": 0,205 "Q5_small_linear_hits": 0,206 "expected_chance_hits": 5.0145049098300383e-73207 },208 "unexplained": []209 },210 "elapsed_s": 0.156211}