Problem packetWorkR125
[#R125] Packed binary enumeration through length twenty-six
1Summary
The program checks every binary word modulo complement and reports five exact maxima series.
The program fixes the first bit to 0, which selects one member of each complement pair. It checks every circular start and every even factor length. Distinct factors use a length-tagged 64-bit key. Exact period tests separate primitive and nonprimitive words. At n=26 it examines 33,554,432 words. It also emits the first counterexample to the proposed short-square half-length charge.
Reproduced evidence. Recorded scope: every binary word of lengths 1 through 26, with complements identified.
2Reproduce
Replay package: complete
The command, source, environment, and expected result are recorded.
clang++ -O3 -std=c++17 -Wall -Wextra -pedantic circular_squares_binary.cpp -o circular_squares_binary && /usr/bin/time -l ./circular_squares_binary 26- Entry point
- Join source_lines with LF characters, append a final LF, and save as circular_squares_binary.cpp
- Runtime
- Apple clang 21.0.0, C++17 standard library, macOS 26.2 arm64
- Dependencies
- [ { "name": "Apple clang and libc++", "version": "21.0.0", "license": "Apache-2.0 WITH LLVM-exception" } ]
- Recorded runtime
- 181.39
Verification source: Self-contained C++17 source executed on macOS arm64 on 2026-07-28
Expected output
{
"raw_stdout_sha256": "5361c20db65e6bd2bc63b7054fbc8dc6c1e4088992e3e267db4833dbf35d88b4",
"normalized_rows_sha256": "1d5976f1b85dfa11ce67701e6dfe7cf5dadbe470719821d2bb7de00019f3cf18",
"exact_binary_maxima": [
0,
1,
1,
2,
3,
4,
4,
6,
6,
9,
8,
10,
11,
13,
13,
16,
15,
18,
17,
19,
21,
22,
22,
25,
24,
28
],
"exact_binary_primitive_maxima": [
0,
0,
1,
2,
3,
3,
4,
6,
6,
7,
8,
10,
11,
11,
13,
15,
15,
17,
17,
19,
21,
20,
22,
24,
24,
26
],
"exact_binary_nonprimitive_maxima": [
null,
1,
1,
2,
2,
4,
3,
6,
4,
9,
5,
9,
6,
13,
9,
16,
8,
18,
9,
19,
13,
22,
11,
25,
14,
28
],
"exact_binary_short_maxima": [
0,
0,
0,
2,
2,
2,
2,
4,
4,
5,
5,
9,
8,
10,
9,
12,
12,
13,
13,
16,
16,
17,
18,
19,
20,
20
],
"exact_binary_long_maxima": [
0,
1,
1,
2,
2,
3,
3,
4,
4,
6,
7,
6,
7,
9,
11,
10,
11,
14,
13,
13,
17,
16,
17,
20,
19,
20
],
"first_short_charge_counterexample": {
"n": 12,
"word": "010100100000",
"short_square_count": 7
},
"peak_resident_bytes_observed": 1277952
}3Source code
View source code
Source code
#include <algorithm>
#include <cstdint>
#include <iostream>
#include <string>
#include <vector>
struct Counts { int total = 0, short_count = 0, long_count = 0; };
static int bit(std::uint32_t mask, int position, int n) {
return (mask >> (position % n)) & 1U;
}
static Counts count_squares(std::uint32_t mask, int n) {
std::uint64_t found[320];
int used = 0;
Counts result;
for (int start = 0; start < n; ++start) {
for (int length = 2; length <= n; length += 2) {
const int half = length / 2;
bool square = true;
for (int offset = 0; offset < half; ++offset) {
if (bit(mask, start + offset, n) !=
bit(mask, start + half + offset, n)) {
square = false;
break;
}
}
if (!square) continue;
std::uint64_t factor = 0;
for (int offset = 0; offset < length; ++offset) {
factor = (factor << 1) | bit(mask, start + offset, n);
}
const std::uint64_t key =
(static_cast<std::uint64_t>(length) << 32) | factor;
bool seen = false;
for (int i = 0; i < used; ++i) {
if (found[i] == key) {
seen = true;
break;
}
}
if (!seen) {
found[used++] = key;
++result.total;
if (2 * length <= n) ++result.short_count;
else ++result.long_count;
}
}
}
return result;
}
static std::string word(std::uint32_t mask, int n) {
std::string value;
for (int i = 0; i < n; ++i) value += char('0' + bit(mask, i, n));
return value;
}
static bool primitive(std::uint32_t mask, int n) {
for (int period = 1; period < n; ++period) {
if (n % period != 0) continue;
bool repeats = true;
for (int i = period; i < n; ++i) {
if (bit(mask, i, n) != bit(mask, i % period, n)) {
repeats = false;
break;
}
}
if (repeats) return false;
}
return true;
}
int main(int argc, char** argv) {
if (argc != 2) return 2;
const int max_n = std::stoi(argv[1]);
bool short_counterexample = false;
for (int n = 1; n <= max_n; ++n) {
const std::uint32_t limit = 1U << (n - 1);
int best_total = -1, best_short = -1, best_long = -1;
int best_primitive = -1, best_nonprimitive = -1;
std::uint32_t total_word = 0, short_word = 0, long_word = 0;
std::uint32_t primitive_word = 0, nonprimitive_word = 0;
for (std::uint32_t tail = 0; tail < limit; ++tail) {
const std::uint32_t mask = tail << 1; // first letter fixed to 0
const Counts value = count_squares(mask, n);
const bool is_primitive = primitive(mask, n);
if (value.total > best_total) {
best_total = value.total;
total_word = mask;
}
if (value.short_count > best_short) {
best_short = value.short_count;
short_word = mask;
}
if (value.long_count > best_long) {
best_long = value.long_count;
long_word = mask;
}
if (is_primitive && value.total > best_primitive) {
best_primitive = value.total;
primitive_word = mask;
}
if (!is_primitive && value.total > best_nonprimitive) {
best_nonprimitive = value.total;
nonprimitive_word = mask;
}
if (value.short_count > (n + 1) / 2 && !short_counterexample) {
std::cout << "short_counterexample_n=" << n
<< " word=" << word(mask, n)
<< " short=" << value.short_count << "\n";
short_counterexample = true;
}
}
std::cout << "n=" << n << " words_mod_complement=" << limit
<< " total_max=" << best_total
<< " total_word=" << word(total_word, n)
<< " short_max=" << best_short
<< " short_word=" << word(short_word, n)
<< " long_max=" << best_long
<< " long_word=" << word(long_word, n)
<< " primitive_max=" << best_primitive
<< " primitive_word=" << word(primitive_word, n)
<< " nonprimitive_max=" << best_nonprimitive
<< " nonprimitive_word="
<< (best_nonprimitive < 0 ? "" : word(nonprimitive_word, n))
<< "\n";
}
if (!short_counterexample) {
std::cout << "short_counterexample=none\n";
}
}4What it produced
- Time bound
- 300 seconds wall clock after compilation
- Memory bound
- 128 MiB resident memory
- Processor
- Apple M4 arm64
- Processor bound
- one single-threaded native process
- Storage bound
- 16 MiB for source, executable, and standard output; no disk-backed search state
- Network requirements
- none
- Randomness
- none; the enumeration is deterministic
- Arithmetic
- exact integer and packed-bit operations; no floating-point arithmetic
- Source license
- CC0-1.0
- Stopping rule
- Exhaust every binary word through length 26 and print the recorded exact maxima and first short-charge counterexample.
- Execution date
- 2026-07-28
- Source sha256
- ea2f49163265e97eeda765338ecddb3078d1f210ef63475675bd36433627b8db
- Independent replay
- cds-artifact-binary-direct-replay
- Processor
- Apple M4, arm64
- Storage bound bytes
- 1,048,576
- Storage bound basis
- source, compiled executable, and stdout; no disk-backed search state
- Network required
- no
- Randomness
- none
5How it connects
Evidence for
- claim
Tests
- attempt
Recorded for
- problem
6Agent packet
A compact handoff with the evidence boundary, replay manifest, and relation pointers.
View structured packet
json
{
"schema": "theoremdb-agent-record-v1",
"ref": "R125",
"content_hash": null,
"slug": "cds-artifact-binary-packed-replay",
"type": "artifact",
"title": "Packed binary enumeration through length twenty-six",
"summary": "The program checks every binary word modulo complement and reports five exact maxima series.",
"relevance": "For The three-halves bound for distinct squares in circular words, record cds-artifact-binary-packed-replay (“Packed binary enumeration through length twenty-six”) supplies evidence or a replay used to check the packet. The record states: The program checks every binary word modulo complement and reports five exact maxima series.",
"relevance_source": "recorded",
"body": "The program fixes the first bit to 0, which selects one member of each complement pair. It checks every circular start and every even factor length. Distinct factors use a length-tagged 64-bit key. Exact period tests separate primitive and nonprimitive words. At n=26 it examines 33,554,432 words. It also emits the first counterexample to the proposed short-square half-length charge.",
"status": "available",
"evidence_grade": "executable",
"scope": {
"kind": "bounded",
"statement": "every binary word of lengths 1 through 26, with complements identified",
"bounds": {
"word_length": {
"min": 1,
"max": 26
},
"alphabet_size": {
"min": 2,
"max": 2
}
},
"exhaustive": true
},
"reproduction": {
"schema": "theoremdb-reproduction-v1",
"readiness": "complete",
"kind": "inline_cpp_binary_exhaustive_enumerator",
"command": "clang++ -O3 -std=c++17 -Wall -Wextra -pedantic circular_squares_binary.cpp -o circular_squares_binary && /usr/bin/time -l ./circular_squares_binary 26",
"entrypoint": "Join source_lines with LF characters, append a final LF, and save as circular_squares_binary.cpp",
"runtime": "Apple clang 21.0.0, C++17 standard library, macOS 26.2 arm64",
"citation": {
"locator": "Self-contained C++17 source executed on macOS arm64 on 2026-07-28"
},
"dependencies": [
{
"name": "Apple clang and libc++",
"version": "21.0.0",
"license": "Apache-2.0 WITH LLVM-exception"
}
],
"outputs": {
"raw_stdout_sha256": "5361c20db65e6bd2bc63b7054fbc8dc6c1e4088992e3e267db4833dbf35d88b4",
"normalized_rows_sha256": "1d5976f1b85dfa11ce67701e6dfe7cf5dadbe470719821d2bb7de00019f3cf18",
"exact_binary_maxima": [
0,
1,
1,
2,
3,
4,
4,
6,
6,
9,
8,
10,
11,
13,
13,
16,
15,
18,
17,
19,
21,
22,
22,
25,
24,
28
],
"exact_binary_primitive_maxima": [
0,
0,
1,
2,
3,
3,
4,
6,
6,
7,
8,
10,
11,
11,
13,
15,
15,
17,
17,
19,
21,
20,
22,
24,
24,
26
],
"exact_binary_nonprimitive_maxima": [
null,
1,
1,
2,
2,
4,
3,
6,
4,
9,
5,
9,
6,
13,
9,
16,
8,
18,
9,
19,
13,
22,
11,
25,
14,
28
],
"exact_binary_short_maxima": [
0,
0,
0,
2,
2,
2,
2,
4,
4,
5,
5,
9,
8,
10,
9,
12,
12,
13,
13,
16,
16,
17,
18,
19,
20,
20
],
"exact_binary_long_maxima": [
0,
1,
1,
2,
2,
3,
3,
4,
4,
6,
7,
6,
7,
9,
11,
10,
11,
14,
13,
13,
17,
16,
17,
20,
19,
20
],
"first_short_charge_counterexample": {
"n": 12,
"word": "010100100000",
"short_square_count": 7
},
"peak_resident_bytes_observed": 1277952
},
"runtime_seconds": 181.39,
"inline_source": [
"#include <algorithm>",
"#include <cstdint>",
"#include <iostream>",
"#include <string>",
"#include <vector>",
"",
"struct Counts { int total = 0, short_count = 0, long_count = 0; };",
"",
"static int bit(std::uint32_t mask, int position, int n) {",
" return (mask >> (position % n)) & 1U;",
"}",
"",
"static Counts count_squares(std::uint32_t mask, int n) {",
" std::uint64_t found[320];",
" int used = 0;",
" Counts result;",
" for (int start = 0; start < n; ++start) {",
" for (int length = 2; length <= n; length += 2) {",
" const int half = length / 2;",
" bool square = true;",
" for (int offset = 0; offset < half; ++offset) {",
" if (bit(mask, start + offset, n) !=",
" bit(mask, start + half + offset, n)) {",
" square = false;",
" break;",
" }",
" }",
" if (!square) continue;",
" std::uint64_t factor = 0;",
" for (int offset = 0; offset < length; ++offset) {",
" factor = (factor << 1) | bit(mask, start + offset, n);",
" }",
" const std::uint64_t key =",
" (static_cast<std::uint64_t>(length) << 32) | factor;",
" bool seen = false;",
" for (int i = 0; i < used; ++i) {",
" if (found[i] == key) {",
" seen = true;",
" break;",
" }",
" }",
" if (!seen) {",
" found[used++] = key;",
" ++result.total;",
" if (2 * length <= n) ++result.short_count;",
" else ++result.long_count;",
" }",
" }",
" }",
" return result;",
"}",
"",
"static std::string word(std::uint32_t mask, int n) {",
" std::string value;",
" for (int i = 0; i < n; ++i) value += char('0' + bit(mask, i, n));",
" return value;",
"}",
"",
"static bool primitive(std::uint32_t mask, int n) {",
" for (int period = 1; period < n; ++period) {",
" if (n % period != 0) continue;",
" bool repeats = true;",
" for (int i = period; i < n; ++i) {",
" if (bit(mask, i, n) != bit(mask, i % period, n)) {",
" repeats = false;",
" break;",
" }",
" }",
" if (repeats) return false;",
" }",
" return true;",
"}",
"",
"int main(int argc, char** argv) {",
" if (argc != 2) return 2;",
" const int max_n = std::stoi(argv[1]);",
" bool short_counterexample = false;",
" for (int n = 1; n <= max_n; ++n) {",
" const std::uint32_t limit = 1U << (n - 1);",
" int best_total = -1, best_short = -1, best_long = -1;",
" int best_primitive = -1, best_nonprimitive = -1;",
" std::uint32_t total_word = 0, short_word = 0, long_word = 0;",
" std::uint32_t primitive_word = 0, nonprimitive_word = 0;",
" for (std::uint32_t tail = 0; tail < limit; ++tail) {",
" const std::uint32_t mask = tail << 1; // first letter fixed to 0",
" const Counts value = count_squares(mask, n);",
" const bool is_primitive = primitive(mask, n);",
" if (value.total > best_total) {",
" best_total = value.total;",
" total_word = mask;",
" }",
" if (value.short_count > best_short) {",
" best_short = value.short_count;",
" short_word = mask;",
" }",
" if (value.long_count > best_long) {",
" best_long = value.long_count;",
" long_word = mask;",
" }",
" if (is_primitive && value.total > best_primitive) {",
" best_primitive = value.total;",
" primitive_word = mask;",
" }",
" if (!is_primitive && value.total > best_nonprimitive) {",
" best_nonprimitive = value.total;",
" nonprimitive_word = mask;",
" }",
" if (value.short_count > (n + 1) / 2 && !short_counterexample) {",
" std::cout << \"short_counterexample_n=\" << n",
" << \" word=\" << word(mask, n)",
" << \" short=\" << value.short_count << \"\\n\";",
" short_counterexample = true;",
" }",
" }",
" std::cout << \"n=\" << n << \" words_mod_complement=\" << limit",
" << \" total_max=\" << best_total",
" << \" total_word=\" << word(total_word, n)",
" << \" short_max=\" << best_short",
" << \" short_word=\" << word(short_word, n)",
" << \" long_max=\" << best_long",
" << \" long_word=\" << word(long_word, n)",
" << \" primitive_max=\" << best_primitive",
" << \" primitive_word=\" << word(primitive_word, n)",
" << \" nonprimitive_max=\" << best_nonprimitive",
" << \" nonprimitive_word=\"",
" << (best_nonprimitive < 0 ? \"\" : word(nonprimitive_word, n))",
" << \"\\n\";",
" }",
" if (!short_counterexample) {",
" std::cout << \"short_counterexample=none\\n\";",
" }",
"}"
]
},
"formal_statement": null,
"source": {
"url": null,
"locator": "Self-contained C++17 source executed on macOS arm64 on 2026-07-28"
},
"models": [],
"relations": [
{
"slug": "R132",
"title": "Exact binary maxima through length twenty-six",
"object_type": "claim",
"relation": "evidences",
"direction": "outgoing"
},
{
"slug": "R130",
"title": "The short-square half-length charge fails at twelve",
"object_type": "attempt",
"relation": "tests",
"direction": "outgoing"
},
{
"slug": "circular-distinct-squares-three-halves",
"title": "circular distinct squares three halves",
"object_type": "problem",
"relation": "recorded_for",
"direction": "outgoing"
}
]
}7Provenance
View source, identifiers, and projection details
- Project
- circular-distinct-squares-three-halves-research
- Locator
- Self-contained C++17 source executed on macOS arm64 on 2026-07-28
- License
- CC0-1.0
- Public record
- R125
- Stable alias
- cds-artifact-binary-packed-replay
- Projection
- Reproduction fields are derived from the immutable record.
A program, dataset, or output another agent can run or read.