# JOURNAL — phase `parse`

## Approach

Recursive-descent parser with standard precedence climbing:
- `expr` handles `+` and `-` (lowest precedence), left-associative via iteration
- `_term` handles `*` and `/` (higher precedence), left-associative via iteration
- `_unary` handles unary `-` recursively (right-recursive so `--5` works)
- `_primary` handles numbers and parenthesized subexpressions

Empty input detected early (before `_Parser` is constructed) by checking if the token list
is empty or contains only EOF. This avoids a confusing error path through `_primary`.

## Gate verification results (local)

- D1: `1+2*3` → `BinOp('+', Num(1), BinOp('*', Num(2), Num(3)))` ✓
- D2: `8-3-2` → `BinOp('-', BinOp('-', Num(8), Num(3)), Num(2))` ✓
- D2: `8/4/2` → `BinOp('/', BinOp('/', Num(8), Num(4)), Num(2))` ✓
- D3: `(1+2)*3` → `BinOp('*', BinOp('+', Num(1), Num(2)), Num(3))` ✓
- D4: `-5` → `Unary('-', Num(5))` ✓
- D4: `-(1+2)` → `Unary('-', BinOp('+', Num(1), Num(2)))` ✓
- D4: `3 * -2` → `BinOp('*', Num(3), Unary('-', Num(2)))` ✓
- D5: all 5 error cases raise `ParseError` ✓
- D6: 30 tests, 0 failures ✓