{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R55",
  "content_hash": null,
  "slug": "bbmb-artifact-exhaustive-sweep-100-million",
  "type": "artifact",
  "title": "Exact multiplier sweep through 100 million",
  "summary": "A self-contained C program checks the bound, minimality, and equality condition for every odd input in the stated range.",
  "relevance": "For Sharp multipliers for balanced binary products, record bbmb-artifact-exhaustive-sweep-100-million (“Exact multiplier sweep through 100 million”) supplies evidence or a replay used to check the packet. The record states: A self-contained C program checks the bound, minimality, and equality condition for every odd input in the stated range.",
  "relevance_source": "recorded",
  "body": "The program uses 64-bit products, hardware-equivalent exact popcount, and the exact bit length of each positive product. The largest possible product in this run is below \\(2^{63}\\). Its output was `DONE max_n=100000000 odd_n=50000000 tested_k=724052131 equality=26 record_n=67108863 record_k=33554433 fnv1a=6955bbea71c91c4c`.\n\nA separate Python 3 implementation repeated the same increasing search through \\(n=100000\\). It checked 50,000 odd inputs and 587,572 multipliers, found 16 equality cases, ended with record \\((65535,32769)\\), and produced the matching prefix checksum `68d628a3847dc259`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "every odd integer n from 1 through 100000000 and every multiplier k through 2^(m-1)+1 until the first balanced product",
    "bounds": {
      "n": {
        "min": 1,
        "max": 100000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "runnable",
    "kind": "inline_c_computation",
    "command": "cc -O3 -std=c11 sweep.c -o sweep && ./sweep",
    "runtime": "ISO C, reproduced with Apple clang 21.0.0",
    "citation": {
      "url": "https://arxiv.org/abs/1909.08849",
      "locator": "Self-contained ISO C computation run by TheoremDB entry research on 2026-07-24"
    },
    "runtime_seconds": 1.14,
    "inline_source": "#include <stdint.h>\n#include <stdio.h>\n#define LIMIT 100000000u\nstatic int balanced(uint64_t x) {\n  unsigned length = 64u - __builtin_clzll(x);\n  return !(length & 1u) && __builtin_popcountll(x) == length/2;\n}\nint main(void) {\n  uint64_t tested=0, hash=UINT64_C(1469598103934665603);\n  uint32_t record=0, record_n=0, equality=0;\n  for (uint32_t n=1; n<=LIMIT; n+=2) {\n    unsigned m=32u-__builtin_clz(n);\n    uint32_t bound=(1u<<(m-1))+1, k;\n    for (k=1; k<=bound; ++k) {\n      ++tested;\n      if (balanced((uint64_t)k*n)) break;\n    }\n    if (k>bound) { printf(\"BOUND n=%u m=%u\\n\",n,m); return 1; }\n    if (k>record) { record=k; record_n=n; }\n    if (k==bound) {\n      ++equality;\n      if (n!=((1u<<m)-1)) { printf(\"EQUALITY n=%u m=%u\\n\",n,m); return 2; }\n    }\n    uint64_t word=((uint64_t)n<<32)|k;\n    for (int j=0; j<8; ++j) {\n      hash^=(uint8_t)(word>>(8*j));\n      hash*=UINT64_C(1099511628211);\n    }\n  }\n  printf(\"DONE max_n=%u odd_n=%u tested_k=%llu equality=%u record_n=%u record_k=%u fnv1a=%016llx\\n\",LIMIT,(LIMIT+1)/2,(unsigned long long)tested,equality,record_n,record,(unsigned long long)hash);\n}",
    "missing": [
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1909.08849",
    "locator": "Self-contained ISO C computation run by TheoremDB entry research on 2026-07-24"
  },
  "models": [],
  "relations": [
    {
      "slug": "R59",
      "title": "The bound and equality characterization hold through 100 million",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "balanced-binary-multiplier-bound",
      "title": "balanced binary multiplier bound",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}