# Exact reference API

Install the included `src/noulith/` package with `python -m pip install -e .`.
The functions below are existing upstream entry points.

| Function in `noulith.third` | Meaning / boundary |
|---|---|
| `all_components(m)` | Star, triangle, and twin descriptors; theorem domain m>=6 |
| `component_equations(m, component)` | Exact integer matrix in ascending logarithmic edge order |
| `intersection_dimension(m, components)` | Dimension computed from stacked exact equations |
| `component_membership(m, q)` | Components containing a coefficient point under the implemented relations |
| `certify_scalar(m, q)` | Exact rational certificate, m>=6 and abs(q+1)<1/20 |
| `predicted_rank(m, q)` | Third-deficit rank from classification conditions |
| `direct_block_rank(m, q, r)` | Independent all-subset calculation; exponential in m |
| `optimal_calibration_subfamily(m)` | Attaining selection of twin outlier pairs, m>=6 |
| `calibration_classes(m, selected_twins)` | Equivalence classes of edge scales constrained by the selected twin subspaces |
| `example_point(m, kind, permutation=None)` | Nearby rational example point |
| `q_from_exponents(exponents, base=Rational(1001,1000))` | Rational coefficients from directed antisymmetric integer exponents |

Named examples: `origin`, `star`, `triangle`, `twin`, `uniform_star`, `star_twin`,
`edge`, `double_star`, `generic`.

## Verify a raw witness

```python
from noulith.algebra import TwistedAlgebra
from noulith.third import example_point, certify_scalar

m = 6
q = example_point(m, "generic")
certificate = certify_scalar(m, q)
assert certificate.visible
blocks = certificate.full_partition(m)
algebra = TwistedAlgebra(m, q)
raw = algebra.isolated_response(blocks)

assert len(blocks) == m - 3
assert raw == algebra.isolated_unit(blocks) * certificate.normalized_response
assert raw != 0
```

This checks the actual sparse algebra, including its crossing factor. It should not be
replaced by a rounded near-zero test.

## Scope

The local analytic theorem and strict real-rational implementation neighborhood are
distinct. Passing an arbitrary dictionary to a low-level routine does not extend a theorem.
A selected calibration pair represents an entire twin-subspace constraint, not a detector
sample. Do not apply the third-deficit theorem at m=5 or the generic core formula at r=0.