**Auteur** : Équipe 4NK

# Collatz — Modular register K_M and finite coverage audit (Option A)

## Objective

Provide a deterministic, versionable materialization of the modular coverage obligation (H2)/(I1):

- build a finite register \(\mathcal{K}_M\) containing only clauses decidable at palier \(2^M\) (i.e. \(m(c)\le M\));
- run a finite coverage audit on the instrumented domain \(L\) (the C3 witness domain), producing an equality-of-sets style artefact (covered vs uncovered on \(L\)).

This does not claim a global coverage of \(S_M\); it is an auditable step “indexed by artefacts”.

## Impacts

- Adds new versioned artefacts under `docs/artefacts/collatz/register_K/`.
- Does not modify any existing certificates, C3 artefacts, or universal clause extraction logic.

## Modifications

### Script

- `applications/collatz/collatz_k_scripts/collatz_build_register_K_modular.py`
  - reads `clauses_universelles.json` (Option A extracted clauses),
  - filters eligible clauses with `stable_modulus_power <= M` into \(\mathcal{K}_M\),
  - computes \(N^\*=\max_c N(c)\) on eligible clauses,
  - audits coverage on the finite C3 domain \(L\) (record `n` values), producing covered/uncovered sets and diagnostics.

### Artefacts

For each palier \(M\):

- `docs/artefacts/collatz/register_K/palier2pM/register_K_mod2pM.json`
- `docs/artefacts/collatz/register_K/palier2pM/register_K_mod2pM.md`
- `docs/artefacts/collatz/register_K/palier2pM/audit_register_K_mod2pM.json`
- `docs/artefacts/collatz/register_K/palier2pM/audit_register_K_mod2pM.md`
- `docs/artefacts/collatz/register_K/palier2pM/non_eligible_clauses_mod2pM.json`
- `docs/artefacts/collatz/register_K/palier2pM/non_eligible_clauses_mod2pM.md`
- `docs/artefacts/collatz/register_K/palier2pM/manifest_register_K_mod2pM.json`
- `docs/artefacts/collatz/register_K/palier2pM/manifest_register_K_mod2pM.md`

## Deployment

No deployment. Local scripts only; artefacts are intended to be committed.

## Analysis / Reproduction

Example for palier \(2^{15}\):

```bash
python3 applications/collatz/collatz_k_scripts/collatz_build_register_K_modular.py \
  --palier 15 \
  --horizon-k 5 \
  --repo-root /home/ncantu/code/algo
```