| # Difficulty definition and quantitative audit
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| This document defines the difficulty fields used by Sudoku DLM Reasoning release
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| `1.1.0`. Difficulty is an operational, solver-relative measurement;
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| it is not a human Sudoku grade and is not defined by the number of clues.
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| ## 1. Provenance of the rating
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| For the Sudoku Extreme rows, the upstream `rating` column is copied into
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| `official_rating` and labeled `rating_type = tdoku_backtracks`. The converter
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| does not recompute the rating:
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| - Sudoku Extreme card: https://huggingface.co/datasets/sapientinc/sudoku-extreme
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| - Conversion code: https://github.com/hengyuf/diffusion-vs-ar/blob/ba0445f4d0808113bf08a9a9d8c7041086bfdf10/tools/prepare_sudoku_datasets.py
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| The upstream card describes the value as the number of backtracks required by
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| Tdoku. At public Tdoku commit `af426180dc53aef89b82868e7b3fdfcf42165654`, the counter is named
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| `num_guesses_` and is incremented whenever the DPLL solver expands a binary
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| configuration decision:
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| - Branch implementation: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L539-L559
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| - Branch-variable selection: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L473-L536
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| Operationally, `official_rating` is best interpreted as the number of search
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| decision nodes visited before the first solution, rather than literal failed
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| undo operations.
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| ## 2. Constraint propagation and the DFS root
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| Tdoku does not begin search from an empty board or from the first empty cell.
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| For a standard 81-character puzzle it:
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| 1. loads every given digit;
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| 2. eliminates incompatible row, column, box, band, and stack configurations;
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| 3. propagates those eliminations until the initialized state is consistent; and
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| 4. calls depth-first search on that propagated partial state.
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| The initialization and call path are visible in that pinned public source:
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| - Initialization: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L612-L634
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| - Solve entry point: https://github.com/t-dillon/tdoku/blob/af426180dc53aef89b82868e7b3fdfcf42165654/src/solver_dpll_triad_simd.cc#L680-L693
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| Propagation applies only logical consequences of the current givens or branch
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| assumption. It makes no new hypothesis and is not counted by the rating. It can
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| end in a complete solution, a contradiction, or a fixed point that still has
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| multiple configurations.
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| This matters because two puzzles can both have rating zero while requiring very
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| different amounts of propagation.
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| ## 3. What one branch means
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| For one digit in a three-row band (or a three-column stack), there are six
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| possible global placements across the three boxes. Tdoku represents these as
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| band configurations. When propagation leaves multiple configurations, the
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| branching heuristic selects:
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| 1. the unresolved band or stack with the fewest configurations; then
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| 2. the digit in that band or stack with the fewest configurations.
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| It then creates a binary decision:
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| - left branch: force the first remaining configuration;
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| - right branch: exclude that configuration and retain the others.
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| The counter increases once for this binary split. If the left branch fails and
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| the right branch immediately solves the puzzle, the count for that decision is
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| still one. A later split of the remaining configurations increases it again.
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| Therefore the rating is not the recursion depth, the number of filled cells, the
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| number of contradictions, or a human solving-step count.
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| ## 4. Bucket definitions
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| The source converter applies these thresholds:
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| | release label | operational rule | interpretation |
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| |---|---:|---|
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| | `original` | no Tdoku rating | separate easy collection; not numerically calibrated to Extreme |
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| | `r0` | rating = 0 | propagation resolves the puzzle without a search decision |
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| | `r1_4` | 1 <= rating < 5 | 1--4 visited decision nodes |
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| | `r5_19` | 5 <= rating < 20 | 5--19 visited decision nodes |
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| The upstream converter also defines `r20_49`, `r50_99`, and `r100_plus`, but
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| those buckets are outside this v1 candidate pool. Thus `r5_19` is an adjacent
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| held-out difficulty band, not the hardest part of the upstream corpus.
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| ## 5. Quantitative audit of the raw candidate pool
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| The following statistics were computed over all 1,415,420 rows included under
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| `raw/`. Rating and clue statistics use the full pool.
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| | bucket | rows | rating mean / median | clue mean / median | clue range |
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| |---|---:|---:|---:|---:|
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| | `original` | 101,000 | not rated | 33.813 / 34 | 29--37 |
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| | `r0` | 614,136 | 0 / 0 | 26.282 / 26 | 17--37 |
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| | `r1_4` | 588,453 | 1.976 / 2 | 25.604 / 26 | 17--31 |
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| | `r5_19` | 111,831 | 11.776 / 12 | 24.932 / 25 | 17--30 |
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| Clue count is not a substitute for this rating. Within `r1_4`, the Pearson
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| correlation between rating and clues is -0.035; within `r5_19` it is 0.002.
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| As an independent diagnostic, standard row/column/box naked-single propagation
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| was run over all 101,000 original rows and deterministic 20,000-row reservoir
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| samples of each Extreme bucket. The reservoir seeds were `20260816` for `r0`,
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| `20260817` for `r1_4`, and `20260818` for `r5_19`:
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| | bucket | rows checked | solved by naked singles |
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| |---|---:|---:|
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| | `original` | 101,000 | 100,959 (99.959%) |
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| | `r0` | 20,000 | 3,691 (18.455%) |
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| | `r1_4` | 20,000 | 0 |
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| | `r5_19` | 20,000 | 0 |
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| The 41 original rows that stalled under naked singles were all solved after
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| adding hidden-single propagation. This independent audit describes basic Sudoku
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| logic, not Tdoku's own internal propagation, which is configuration-based and
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| stronger.
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| ## 6. Released train and evaluation distributions
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| The three training buckets are equal in size, but exact ratings inside `r1_4`
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| retain the upstream proportions:
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| | exact rating | train rows |
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| |---:|---:|
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| | 1 | 29,584 |
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| | 2 | 16,809 |
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| | 3 | 10,244 |
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| | 4 | 8,899 |
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| The main `r1_4` test allocation instead uses 250 rows at each exact rating 1,
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| 2, 3, and 4. The `r5_19` test allocation uses 250 rows from each of `[5,8)`,
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| `[8,12)`, `[12,16)`, and `[16,20)`; `test_ood_confirm` uses 1,000 per range.
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| This supports per-rating or per-range reporting rather than only a bucket-wide
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| average.
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| ## 7. Source confounding
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| Difficulty and source collection are not independent in the raw pool:
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| - `r0`: 77.40% `puzzles1_unbiased`, 16.28% `puzzles0_kaggle`, and 6.30%
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| `puzzles2_17_clue`;
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| - `r1_4`: 85.02% `puzzles1_unbiased`, 9.70%
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| `puzzles4_forum_hardest_1905`, and 3.40% `01_file1`;
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| - `r5_19`: 68.19% `puzzles4_forum_hardest_1905`, 28.15% `01_file1`, and only
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| 2.18% `puzzles1_unbiased`.
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| Consequently, an `r5_19` result combines a Tdoku-rating shift, a held-out
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| training-exposure shift, and a source-distribution shift. It must not be
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| interpreted as a pure causal effect of search count.
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| Recommended reporting includes exact rating or subrange, source collection, and
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| clue count. A source- and clue-matched diagnostic set is appropriate when the
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| goal is to isolate the rating axis.
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| ## 8. Reproducibility limitation
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| The public Sudoku Extreme card does not pin the Tdoku commit, build flags,
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| command, permutation policy, or seed used to create its rating column. This
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| release therefore preserves `official_rating` as an upstream measurement rather
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| than claiming to reproduce it.
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| For a fully controlled future rating, pin a Tdoku commit and configuration,
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| record the branch count for the exact stored puzzle, and optionally measure its
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| mean and variance over Sudoku-preserving isomorphic transformations.
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