N × M belt balancers

Three blueprint books (blueprint book strings, Factorio 2.0.77, base game without Space Age), one per belt tier:

Each file is the current version; earlier versions stay in the git history. The sections below describe the blue book; the differences in the red and yellow books are in Red and yellow books.

• Root book Blue Belt balancer with 24 sub-books (1 … 24), one per number of input belts.
• Each sub-book holds N to 1 … N to 24 (sub-book 2 has 25 items because it keeps both variants, 2 to 3 (Long) and 2 to 3 (Wide)).
• Total: 577 blueprints, covering the 576 N × M pairs.

Defective originals (replaced)

These 16 blueprints from the original book are not correct balancers (e.g., 4 to 5 merges two loopbacks onto a single belt and delivers 1.0 / 0.75 / 0.75 / 0.75 / 0.75 with all 4 inputs full):

2 to 5, 3 to 5, 3 to 7, 4 to 5, 5 to 2, 5 to 3, 5 to 4, 5 to 6, 5 to 7, 6 to 5, 6 to 7, 7 to 3, 7 to 5, 7 to 6, 7 to 8, 8 to 7

Library blocks (the approved originals + Raynquist's book, https://github.com/raynquist/balancer) are physically composed, always flowing north, with contiguous inputs/outputs:

• LOOP: an (n+L → m+L) block with L outputs fed back into the inputs along the sides (e.g., 24→24 = 32-32 core + 8 loops).
• STACK: n→k at the bottom and g copies of (k/g → m/g) on top (e.g., 12→24 = 12-12 + 12×(1→2)).
• SQ + K1: an n × n square balancer followed by a square core with loops and pruned inputs (when there is no common divisor).
• N > M: an N>M library block, DSTACK (the dual of STACK), or an N≤M design with the flow reversed — accepted only when the reversal creates no sideload.
• Long generated straight runs use express underground belts to reduce the entity count.

1. Flow simulation with backpressure (scripts/deep_verify.py): 6.8 million input/output patterns in total; 577/577 passed (output balanced under any input, input balanced under any output, total throughput = min(N,M) belts, contiguous ports, 0 sideload or unpaired underground belt warnings).
2. Automated in-game test (headless Factorio 2.0.77, isolated data in scripts/ingame/): each blueprint is imported, built, fed by loaders and measured in 9 phases (everything on; half/one third/random/a single input; half/one third/random/a single output). Result: 577/577 built with no collision and no unpaired underground belt; worst difference between outputs (or between inputs) = 4 items in 2,700; maximum throughput shortfall 0.15%.
3. Third-party checker (tzwaan/factorio_balancers, in scripts/xcheck/xcheck.py): 576 PASS, 0 FAIL (1 to 1 has no splitter and the tool does not analyze it). The same tool confirms the 16 defective originals and passes 143/143 of Raynquist's.
4. The full book string was imported into the game: 24 sub-books, 577 blueprints with entities.

• median 453 entities, mean 690, maximum 2,248 (17 to 18, 48 × 84 tiles).
• Pairs with N and M between 17 and 24 and no common divisor are the largest (two stacked 32-32 cores). They are correct but large; they are not SAT-optimized layouts like Raynquist's.

cd scripts
export FBTIER=blue        # blue | red | yellow
python3 gen.py            # generates every pair
python3 build_book.py     # assembles the book and overwrites blueprints/belt-balancers/$FBTIER-belt.txt
python3 deep_verify.py ../blueprints/belt-balancers/$FBTIER-belt.txt
python3 export_tests.py ../blueprints/belt-balancers/$FBTIER-belt.txt && ./run_ingame.sh 3400 && python3 analyze_ingame.py

The generator reads its inputs from scripts/sources/ (not tracked by git; see scripts/README.md).

Same structure (24 sub-books × N to 1..24), generated by the same pipeline with FBTIER=red / FBTIER=yellow:

• red-belt.txt — red belt (fast), underground belts up to 7 tiles.
• yellow-belt.txt — yellow belt, underground belts up to 5 tiles.

Differences from the blue book:

• The library blocks are converted to the belt tier and are used only if every underground belt stays paired (Raynquist's 16-16 and 32-32 use 9-tile tunnels and cannot be used).
• The red 16 × 16 core comes from Raynquist's "16-16 red"; the yellow 16 × 16 core and both 32 × 32 cores are built by doubling (two smaller cores + N splitters + a de-interleaving router packed in bands, scripts/weave.py), all verified.
• That is why the large blueprints (N, M ≥ 17) are bigger than the blue ones: red up to 3,605 entities, yellow up to 4,876.

Red

• origin: {'original': 49, 'raynquist': 18, 'generated': 494, 'raynquist-fix': 16}
• generated: median 529 entities, maximum 3,605 (18 to 19, 56 × 175 tiles)
• verification: simulation 577/577; in-game test 577/577 in 9 phases (maximum difference 4 items per 60 s window); third-party checker 576 PASS / 0 FAIL.

Yellow

• origin: {'original': 49, 'raynquist': 11, 'generated': 501, 'raynquist-fix': 16}
• generated: median 1,036 entities, maximum 4,876 (17 to 18, 56 × 206 tiles)
• verification: simulation 577/577; in-game test 577/577 in 9 phases (maximum difference 4 items per 60 s window); third-party checker 576 PASS / 0 FAIL.
• note: on the yellow belt, the 18 largest loops (17..23 to 17..24) take more than 45,000 ticks to fill; with a 180,000-tick warm-up the measured throughput reaches 99.97%.