# nb-15 solutions — Quantities and dimensions

*Checkable against `law test packs/examples/language-demo/math02`
(42 checked, 42 passed), `law test
packs/examples/language-demo/boundaries` (21 checked, 21 passed),
`law engine check` on the math02 package and on the frozen refusal
snippets. Identifiers and code as written.*

## 1. `load_q(5 km)` versus `load_q(22 kW)` for `heavy_load()`

**Answer: 5 km gives TRUE_ONLY (melded scalar `5000/1`); 22 kW
gives NEITHER — no scalar is melded at all.**

`LoadScalar` divides each reading's magnitude by `magnitude(1 m)`
and applies `scalar_of`: 5 km becomes the Rational `5000/1`, which
clears the `(44/1)` bar in `HeavyLoad` — the rule fires. For
22 kW the suite shows `heavy_load()` NEITHER but pins no
`load_scalar` value itself; a scratch probe outside the suite
settles it — `truth(load_scalar(22/1))` given `load_q(22 kW)`
returns NEITHER while `load_q(22 m)` returns TRUE_ONLY — so the
metre-divisor meld accepted nothing (the silent `UNIT_UNKNOWN`
mode from the article's Limits). No rule derives `heavy_load()`
and the status is NEITHER, not false. The suite's test names say
exactly this: `scalar of five kilometres in metres` versus `load
scalar below the threshold is not derived`.

## 2. `load_q(5 km)` versus `load_q(0.5 km)` for `long_trip()`

**Answer: 5 km clears it (5000 m ≥ 1000); 0.5 km does not
(500 m < 1000). The explicit step is `unit_convert(magnitude(q),
"m")` in `TripLength`.**

The target unit `"m"` is written in the call — kilometres never
meet the metre threshold until converted. 5 km converts to 5000 m
(`kilometres converted to metres`, TRUE_ONLY); 0.5 km converts to
500 m (`short trip fails the threshold`, NEITHER). The judging rule
`LongTrip` is identical in both cases; only the converted scalar
differs.

## 3. The `convert` equality and its explicit factor

**Answer: it establishes `conv_ok()` (TRUE_ONLY) — 1 km equals
1000 m — and the factor 1000/1 must appear because the engine
performs no silent coercion.**

`convert(with_unit(1, "km"), 1000, 1, "m")` re-expresses the value
with a stated numerator and denominator; the equality against
`with_unit(1000, "m")` then checks two explicitly constructed
quantities. A bare `1 == 1000` comparison would be meaningless (and
dimensionless); the factor in the call is what makes the claim
checkable rather than assumed. Suite witness: the `unit conversion
by value` ok line.

## 4. What the `tag resolves` tests add

**Answer: they prove tag resolution works for locally declared
units, not just the SI registry.**

`declared unit tag resolves` (`with_unit(5, "permit")`) and
`derived tag resolves` (`with_unit(2, "permit_rate")`, where
`permit_rate = permit / permit`) run over `unit permit = 1`
declared in the boundaries package itself — no SI import involved.
The SI-side tests prove registry conversion; these two prove the
construct is general: any package can coin countable tags for
non-physical things (permits, rates) and compare them the same way.

## 5. The `temp02` refusal and the 0.2 alternative

**Answer: refused with `LDC-E2105` (undeclared) plus `LDC-E2403`
(computability not proved); a 0.2 modeller uses `unit_convert`,
`convert`, `quantity_of` and `with_unit`.**

`temp_convert` belongs to the 0.7 temperature profile, which this
implementation (`law 0.1.0`, `law.core/0.2`) does not execute — so
the checker reports the name unknown and the purity unprovable.
`angle_convert` fails identically. The refusal is a fact about this
profile, not about the language: stay inside the executable subset
(the `ConvOk`/`QBackOk`/`ExpoOk` rules are the positive pattern)
and leave newer-profile converters to nb-17.

## How to verify

```sh
law test packs/examples/language-demo/math02
```

Expected: `мир demo.northbridge.math02, units.si`, ok lines for
`scalar of five kilometres in metres`, `load scalar below the
threshold is not derived`, `kilometres converted to metres`,
`short trip fails the threshold`, `unit conversion by value`;
total `42 проверено, 42 прошли, 0 не прошли`.

```sh
law test packs/examples/language-demo/boundaries
```

Expected: ok lines `declared unit tag resolves` and `derived tag
resolves`; total `21 проверено, 21 прошли, 0 не прошли`.

```sh
law engine check packs/examples/language-demo/math02/package.law
```

Expected: `check OK`.

```sh
law engine check packs/examples/language-demo/boundaries/evidence/snippets/temp02.law.txt
```

Expected: `LDC-E2105` and `LDC-E2403` (likewise for
`angle02.law.txt` with `angle_convert`).