vmal/3-digit-arithmetic-scratchpad-traces

Contents

Split	Rows
train	100,000
validation	4,000
test	4,000
total	108,000

Splits are prompt-disjoint — no expression appears in more than one split, and commutative swaps and trace keys are de-duplicated across splits to prevent split leakage.

Operation	Rows
×	32,000
÷	32,000
+	22,000
−	22,000

Operands lie in [−999, 999]. Division answers use a fixed DDD.ddd form (round-half-up to three decimals); division by zero is an atomic <nan>.

Schema

One JSON object per line. Key fields:

Field	Description
prompt	the question, e.g. 842/37=
text	the full serialized string: prompt + scratchpad + answer
operation	+, -, *, /
effective_operation	canonical magnitude op after sign planning (add/sub/mul/div)
result_sign	positive / negative
state	the normalized operands the scratchpad works over
steps	list of structured step records (each with a serialized string)
internal_answer	canonical internal answer (e.g. 022.757, or <nan>)
natural_answer	human-readable answer (e.g. 22.757)
scenario	difficulty/skill bucket (e.g. rounded_ge_one_up, high_column_sum)
split	train / val / test
encoded_tokens	actual row length including <bos> and <eos> (max 162)

The scratchpad grammar

Every row begins with the original expression and then a normalized arithmetic plan:

<prompt><effective-op><result-sign><state><six-digit-state>...<ans><answer>

<effective-op> is one of <add>, <sub>, <mul>, or <div>, and <result-sign> is <pos> or <neg>. For signed + and - expressions, the effective magnitude operation can differ from the prompt operator. For example, adding operands with opposite signs uses the subtraction grammar after their magnitudes are ordered.

Addition (<add>)

The six-digit state contains two zero-padded, three-digit magnitudes in reversed (units-first) order:

<state><left-reversed:3><right-reversed:3>

Exactly three <step> records follow, one each for the units, tens, and hundreds columns. Every record has five digits:

<step><left-digit><right-digit><carry-in><output-digit><carry-out>

The answer is a four-digit, units-first magnitude; the fourth position holds the final carry:

457+386=<add><pos><state>754683<step>76031<step>58141<step>43180<ans>3480

Here, <ans>3480 decodes to 843.

Subtraction (<sub>)

The state contains the larger magnitude followed by the smaller magnitude, both zero-padded to three digits and reversed:

<state><larger-reversed:3><smaller-reversed:3>

Exactly three five-digit <step> records describe the units, tens, and hundreds columns:

<step><minuend-digit><subtrahend-digit><borrow-in><output-digit><borrow-out>

The answer is a four-digit, units-first magnitude. Its sign comes from <result-sign>:

312-748=<sub><neg><state>847213<step>82060<step>41030<step>73040<ans>6340

Here, <neg> and <ans>6340 decode to -436.

Multiplication (<mul>)

The state contains the two zero-padded magnitudes in reversed, units-first order. Five output columns are processed from least significant to most significant. Across those columns the grammar contains nine term records, distributed 1, 2, 3, 2, 1, and five column-finalization records:

<step><left-digit><right-digit><product:2><running-total:3>
<col><output-digit><carry-out:2>

Each <step> therefore has seven digits and each <col> has three. The answer is a six-digit, units-first product:

213*145=<mul><pos><state>312541<step>3515015<col>501 … <col>300<ans>588030

Here, <ans>588030 decodes to 30885.

Division (<div>)

Division retains ordinary most-significant-digit-first order. The state is the zero-padded numerator followed by the zero-padded divisor:

<state><numerator:3><divisor:3>

For a nonzero divisor, long division has seven positions: three integer positions, three retained decimal positions, and one guard position used for round-half-up. Each position is decomposed into three records:

<step><remainder-in:3><incoming-digit><partial-dividend:4>
<qmul><quotient-digit><quotient-product:4>
<rem><remainder-out:3>

The fixed answer format is DDD.ddd:

842/37=<div><pos><state>842037<step>00080008<qmul>00000<rem>008<step>00840084<qmul>20074<rem>010 … <ans>022.757

Division by zero contains no arithmetic records and ends with the atomic <nan> answer:

12/0=<div><pos><state>012000<ans><nan>

How it was generated (and verified)

The complete dataset was generated deterministically with seed 20260722. Operands and sign patterns were sampled according to per-operation scenario quotas so that each split contains its requested mix of identity cases, carry/borrow patterns, multiplication column-sum bands, and exact or rounded quotients.

Every accepted row passed the same structured validation pipeline:

1. Parse and validate both signed operands and the requested operation.
2. Construct the canonical magnitude operation, result sign, and normalized six-digit state.
3. Generate every structured arithmetic record and the serialized scratchpad.
4. Reconstruct the internal answer from the step records and compare it with the exact arithmetic result.
5. Verify the serialized form, fixed field widths, token count, scenario label, and row metadata.

The test split was generated first, followed by validation and training. Prompts, canonical trace keys, sign-equivalent traces, and commutative swaps were reserved before sampling later splits. The final integrity audit confirmed:

• 108,000 rows and 108,000 unique prompts;
• zero prompt overlap across train, validation, and test;
• zero canonical-trace overlap across splits;
• zero swapped-commutative leakage into held-out splits;
• every row successfully verified and within the declared token limit.

License

MIT.