Balance Constants, Majority Cycles, and the Gold Partition Conjecture through Fourteen Elements

Manuscript: arXiv:2607.23926 · Immutable Release 2.1 archive · Immutable Release 2 archive

Two exhaustive computations over all 1,338,193,159,771 unlabeled posets on fourteen elements. The first is a census of exact balance constants and linear-extension-majority cycles, extending by one order the complete census of De Loof, De Baets, and De Meyer. The second verifies Peczarski's Gold Partition Conjecture, extending his order-11 frontier from 2006.

All arithmetic is exact: no floating-point value is computed, compared, or stored in any decision path. Every extremal witness is recomputed from the witness string alone by a program that shares no algorithm with the census.

The repository contains:

• the manuscript;
• the exact GPC source and the production-equivalent released census source;
• portable build, sharding, and aggregation tools;
• independent small-order verifiers and a certificate classifier;
• compact result and build records.

Results

Balance constants at order 14:

quantity	value
least balance constant above 1/3	37/106
least over posets that are not ordinal sums	254/725, the ladder L₁₄,₁,₉
classes with balance exactly 1/3	128
classes with balance below 1/3	0
distinct values in (1/3, 9/25]	42, over 469 classes
values inside Peczarski's conjectured gap	0

Majority cycles, with orders 9–13 reproducing De Loof et al.:

order	any cycle	3	4	5	6	7	8
13	9,348,400	9,318,881	102,127	471	363	1	0
14	478,632,938	477,954,774	5,419,981	33,473	10,423	61	30

No order-14 class carries a cycle of length 9 or more.

Gold Partition certificates:

order	unlabeled posets	pair certificate	triple certificate	open
12	1,104,891,746	1,066,006,204	38,885,541	0
13	33,823,827,452	32,418,324,910	1,405,502,541	0
14	1,338,193,159,771	1,272,077,147,789	66,116,011,981	0

At each order, exactly one class is a chain. “Pair certificate” is the invariant union of the low-slave and half-balanced searches; their internal split depends on search order.

Build

The classifiers use the plugin interface of Brinkmann and McKay's genposetg, distributed with nauty and Traces. The dependency is not vendored here. The calculation used genposetg 1.1 from nauty 2.9.1. The release archive and its SHA-256 digest are:

https://pallini.di.uniroma1.it/nauty2_9_1.tar.gz
488fa906d10a372c72d2364c5dee48e0f7307004fbe52c2bce50c52de8cd873e

The program build requires Bash, GNU Make, GNU coreutils, GNU time, Python 3.10 or later, and GCC 13 or later. The default make target also builds the paper and therefore requires pdfLaTeX with TikZ:

curl -LO https://pallini.di.uniroma1.it/nauty2_9_1.tar.gz
printf '%s  %s\n' \
  488fa906d10a372c72d2364c5dee48e0f7307004fbe52c2bce50c52de8cd873e \
  nauty2_9_1.tar.gz | sha256sum -c -
tar xf nauty2_9_1.tar.gz
cd nauty2_9_1
./configure --enable-generic --disable-popcnt
make nautyS1.a
cd ..
export GENPOSETG_C="$PWD/nauty2_9_1/genposetg.c"
export NAUTY_INCLUDE="$PWD/nauty2_9_1"
export NAUTY_LIB="$PWD/nauty2_9_1/nautyS1.a"
make programs

The recipe above also keeps nauty generic. For a machine-specific nauty build, run its plain ./configure instead. This repository's default build is portable; set NATIVE=1 to enable -march=native -funroll-loops; the order-14 binary was built that way on AMD EPYC 7B13 processors with GCC 13.3.0. The recorded production binary SHA-256 applies to that native environment. Portable builds reproduce the results, but are not expected to reproduce the production binary byte for byte.

Verification

Run the complete release checks with:

# Uses the three nauty variables exported by the build block above.
make check

This command:

• generates all unlabeled posets through order 7 independently of genposetg, enumerates their linear extensions, and checks Peczarski's certificate conditions directly;
• compares the C classifier and a separate exact implementation on every poset of orders 8 and 9, using non-topological labels;
• runs the compiled GPC classifier through order 9;
• tests the full-pair classifier at order 8;
• compares the balance census against a brute-force reference that enumerates linear extensions directly, on every class at orders 5 through 7;
• checks the certificate classifier's pair/triple boundary against that reference on every poset through order 8, and against Peczarski's published Table I;
• tests both aggregators on malformed and inconsistent inputs;
• runs an end-to-end, multi-residue calculation.

The slower census differential extends the direct comparison through orders 8 and 9:

bash tests/check_census.sh --slow

The production source hashes are recorded in data/gpc-n14.txt. The controlled timing ablation of the bilateral screen is recorded in data/gpc-ablation.txt. The published majority-cycle cross-check is recorded in data/majority-regression.txt. The complete order-14 census aggregate, including every extremal witness, the full low-balance tail, and the equality representatives, is recorded in data/census-n14.txt.

Research atlas

Release 2.1 adds a complete, locally reproducible atlas of every unlabeled poset on seven or fewer elements: 2,450 classes in under 4 MB, one file per order. Its purpose is to let researchers test structural questions and data workflows before larger extracts are considered. Every row contains a stable label-independent identifier, exact pair probabilities, balance data, majority edges and cycle spectra, structural invariants, ordinal summands, a dual join, and the Gold Partition certificate class.

• Atlas documentation
• JSON Schema
• JSON Lines data and checksum-bearing manifests, by order: 1, 2, 3, 4, 5, 6, 7

Row counts are A000112(1) through A000112(7), and identifiers are unique across the whole range, so the files can be concatenated and used as one table. Ordinal summand identifiers point into the smaller orders: all 3,784 summand references resolve, 2,207 of them across files.

The prototype is a complete small-order universe, not a sample of order-14 classes. The order-14 shards retain aggregates and selected witnesses rather than one row per enumerated poset, so no full order-14 atlas is claimed.

First validate the complete shard inventory, completion metadata, payload seals, aggregate counters, and every retained witness:

python3 scripts/aggregate_census.py 14 16384 \
  /path/to/census-n14-shards --verify all > census-n14-check.txt

Then recompute the complete retained-witness statistics in Sections 4 and 6 of the paper:

python3 scripts/analyze_witness_archive.py \
  /path/to/census-n14-shards --jobs 8 --check-paper

This checks all 44,013 distinct cycle witnesses, the 180 retained low-tail witnesses, and the 68 equality representatives. It reports the exact retention rules separately; none of its witness proportions is a population estimate over poset classes.

Regenerate the three committed figure sources from their witnesses and the aggregate tail table with:

python3 scripts/make_census_figures.py paper/figures \
  --tail data/census-n14.txt --tail-order 14

Sharded calculations

A shard is one residue of a deterministic modulus partition:

scripts/run_shard.sh gpc 11 0 16 results/gpc-n11

Run residues 0,...,15, then aggregate:

scripts/aggregate.py gpc 11 16 results/gpc-n11

The runner records the exact binary and shard coordinates, synchronizes the output and metadata, and publishes the completion marker last. The aggregator requires every residue, rejects mixed binaries and incomplete runs, checks the known number of unlabeled posets and the certificate partition, and requires open=0.

src/balance_census.c is the census program. It computes, for every class, the exact balance constant, the low-tail membership, and the majority-cycle spectrum in both the full and the incomparable-pairs-only relation:

scripts/run_census_shard.sh 12 0 64 results/census-n12
scripts/aggregate_census.py 12 64 results/census-n12

src/majority_census.c is an earlier full-pair implementation retained for cross-checks.

Extending to order 15

The method extends to order 15 without modification. Scaling the measured order-14 production run by the ratio of unlabeled poset counts gives a projection of roughly 86,000 core-hours and, at the Spot prices recorded for the order-14 campaign, on the order of $1,000. An order-15 aggregate would be required to reproduce the known total 68,275,077,901,156 unlabeled posets with a single chain, a complete certificate partition, and no open classes.

These figures are a projection from a completed run, not the result of an order-15 pilot, and no order-15 computation has been performed. Nothing is claimed or predicted about what such a run would find.

Citation and license

The source and data archive is available from Zenodo. That is the concept DOI and always resolves to the latest version. The immutable archive for this release, Release 2.1, is 10.5281/zenodo.21865430, and the immutable archive for Release 2 and its order-14 census artifacts is 10.5281/zenodo.21696940; each computational data file records the immutable DOI of the release it describes. Citation metadata are provided in CITATION.cff. The local source is released under the MIT License; nauty and genposetg remain under their upstream terms.