Formalize C2 and add deterministic projective verification artefacts

**Motivations:** - Make Lemma C2 assertions citeable without terminal transcripts - Provide a deterministic check for “both-children residual” and projective projection to B12 **Root causes:** - C2 relied on narrative links between completion docs and B12 invariance without a machine-checkable artefact **Correctifs:** - Add a deterministic verifier that checks set equalities and projections from the existing completion MD sources **Evolutions:** - Version C2 verification artefacts in docs/artefacts - Add a run report profile (sha256 + metrics) for C2 projective verification - Update the proof plan with explicit C2 statements and references **Pages affectées:** - applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py - docs/artefacts/collatz/c2_projective/verification_c2_projective.json - docs/artefacts/collatz/c2_projective/verification_c2_projective.md - applications/collatz/collatz_k_scripts/collatz_generate_run_report.py - docs/collatz_run_report_2026-03-09_c2_projective.md - docs/collatz_run_report_format.md - docs/features/collatz_run_report_generator.md - docs/features/collatz_c2_projective_verification.md - applications/collatz/collatz_k_scripts/plan_lemmes_manquants_et_programme_de_preuve.md
2026-03-09 01:25:16 +01:00 · 2026-03-09 01:25:16 +01:00 · e3e68fe39c
commit e3e68fe39c
parent e12c544e7b
9 changed files with 658 additions and 4 deletions
--- a/applications/collatz/collatz_k_scripts/collatz_generate_run_report.py
+++ b/applications/collatz/collatz_k_scripts/collatz_generate_run_report.py
@ -119,6 +119,111 @@ class LocalH6Metrics:
    certificate_paliers: list[tuple[str, int]]


+@dataclass(frozen=True)
+class C2ProjectiveMetrics:
+    b12_size: int
+    both14_size: int
+    children15_size: int
+    residual15_size: int
+    children15_equals_residual15: bool
+    proj_both14_equals_b12: bool
+    both15_size: int
+    children16_size: int
+    proj_both15_equals_b12: bool
+
+
+def parse_c2_projective_metrics(verification_json: str) -> C2ProjectiveMetrics:
+    obj = read_json(Path(verification_json))
+    if not isinstance(obj, dict):
+        raise ValueError("Invalid C2 verification JSON: expected object")
+
+    b12 = obj.get("B12")
+    checks = obj.get("checks")
+    if not isinstance(b12, dict) or not isinstance(checks, dict):
+        raise ValueError("Invalid C2 verification JSON: missing 'B12' or 'checks'")
+
+    c1415 = checks.get("m14_to_m15")
+    c1516 = checks.get("m15_to_m16")
+    if not isinstance(c1415, dict) or not isinstance(c1516, dict):
+        raise ValueError("Invalid C2 verification JSON: missing check blocks")
+
+    def req_int(d: dict, k: str) -> int:
+        v = d.get(k)
+        if not isinstance(v, int):
+            raise ValueError(f"Invalid C2 verification JSON: {k} must be int")
+        return v
+
+    def req_bool(d: dict, k: str) -> bool:
+        v = d.get(k)
+        if not isinstance(v, bool):
+            raise ValueError(f"Invalid C2 verification JSON: {k} must be bool")
+        return v
+
+    return C2ProjectiveMetrics(
+        b12_size=req_int(b12, "count"),
+        both14_size=req_int(c1415, "both_parent_count"),
+        children15_size=req_int(c1415, "computed_children_count"),
+        residual15_size=req_int(c1415, "listed_residual_count"),
+        children15_equals_residual15=req_bool(c1415, "children_equals_listed_residual"),
+        proj_both14_equals_b12=req_bool(c1415, "projection_mod_4096_equals_b12"),
+        both15_size=req_int(c1516, "both_parent_count"),
+        children16_size=req_int(c1516, "computed_children_count"),
+        proj_both15_equals_b12=req_bool(c1516, "projection_mod_4096_equals_b12"),
+    )
+
+
+def write_c2_projective_run_report(
+    *,
+    output_path: Path,
+    report_title: str,
+    command: str,
+    git_commit: str,
+    sha_entries: list[Sha256Entry],
+    metrics: C2ProjectiveMetrics,
+    artefacts_dir: Path,
+) -> None:
+    lines: list[str] = []
+    lines.append("**Auteur** : Équipe 4NK")
+    lines.append("")
+    lines.append(f"# {report_title}")
+    lines.append("")
+    lines.append("## Contexte")
+    lines.append("")
+    lines.append("- **But du run** : vérifier C2 (complétion par frères ⇒ both ⇒ projection B12) sur des artefacts déterministes.")
+    lines.append("- **Assertion vérifiée** : égalités d’ensembles et égalité de projections modulo 4096 (voir métriques).")
+    lines.append("")
+    lines.append("## Code et reproductibilité")
+    lines.append("")
+    if git_commit:
+        lines.append(f"- **Commit Git** : `{git_commit}`")
+    if command:
+        lines.append("- **Commande** :")
+        lines.append("")
+        lines.append("```bash")
+        lines.append(command)
+        lines.append("```")
+        lines.append("")
+    lines.append("## Empreintes sha256 (scripts, artefacts)")
+    lines.append("")
+    lines.extend(format_sha256_list(sha_entries))
+    lines.append("")
+    lines.append("## Compteurs et métriques")
+    lines.append("")
+    lines.append(f"- |B12| = {metrics.b12_size}")
+    lines.append(f"- m=14: |both|={metrics.both14_size}, |children(m=15)|={metrics.children15_size}, |listed residual(m=15)|={metrics.residual15_size}")
+    lines.append(f"- m=14: children(m=15) == residual(m=15) : {metrics.children15_equals_residual15}")
+    lines.append(f"- m=14: both mod 4096 == B12 : {metrics.proj_both14_equals_b12}")
+    lines.append(f"- m=15: |both|={metrics.both15_size}, |children(m=16)|={metrics.children16_size}")
+    lines.append(f"- m=15: both mod 4096 == B12 : {metrics.proj_both15_equals_b12}")
+    lines.append("")
+    lines.append("## Chemins d’artefacts (versionnés)")
+    lines.append("")
+    lines.append(f"- ARTEFACTS : `{artefacts_dir}`")
+    lines.append("")
+    output_path.parent.mkdir(parents=True, exist_ok=True)
+    output_path.write_text("\n".join(lines) + "\n", encoding="utf-8")
+
+
 def _discover_base_noyau_path(artefacts_dir: Path) -> Path:
    candidates = sorted((artefacts_dir / "noyaux").glob("noyau_*_B12.json"))
    if len(candidates) != 1:
@ -692,7 +797,15 @@ def main() -> None:
    ap.add_argument(
        "--profile",
        default="extend_finale",
-        choices=["extend_finale", "validation_section7", "pipeline_d16_d17", "fusion_palier2p25", "local_H6", "local_H6_E1"],
+        choices=[
+            "extend_finale",
+            "validation_section7",
+            "pipeline_d16_d17",
+            "fusion_palier2p25",
+            "local_H6",
+            "local_H6_E1",
+            "c2_projective",
+        ],
        help="Report profile",
    )
    ap.add_argument("--pipeline-extend-log", default=None, help="Path to pipeline_extend.log (defaults to OUT/pipeline_extend.log)")
@ -708,6 +821,11 @@ def main() -> None:
        default="",
        help="For profile local_H6: path to the versioned artefacts directory (e.g. docs/artefacts/collatz/local_E1_palier2p13)",
    )
+    ap.add_argument(
+        "--c2-artefacts-dir",
+        default="",
+        help="For profile c2_projective: path to deterministic artefacts directory (e.g. docs/artefacts/collatz/c2_projective)",
+    )
    args = ap.parse_args()

    repo_root = Path(args.repo_root).resolve() if args.repo_root else Path.cwd().resolve()
@ -922,6 +1040,44 @@ def main() -> None:
        print(f"Wrote: {output_path}")
        return

+    if args.profile == "c2_projective":
+        command = args.command.strip()
+        if not command:
+            command = (
+                "python3 applications/collatz/collatz_k_scripts/collatz_generate_run_report.py "
+                "--profile c2_projective --scope c2_projective "
+                "--out-dir applications/collatz/out --docs-dir docs "
+                "--c2-artefacts-dir docs/artefacts/collatz/c2_projective"
+            )
+
+        artefacts_dir = Path(args.c2_artefacts_dir).resolve() if args.c2_artefacts_dir.strip() else (docs_dir / "artefacts" / "collatz" / "c2_projective")
+        verification_json = artefacts_dir / "verification_c2_projective.json"
+        verification_md = artefacts_dir / "verification_c2_projective.md"
+        metrics = parse_c2_projective_metrics(str(verification_json))
+
+        date_str = pick_report_date_from_mtime([verification_json, verification_md])
+        sha_paths: list[Path] = [
+            repo_root / "applications" / "collatz" / "collatz_k_scripts" / "collatz_verify_c2_projective.py",
+            repo_root / "applications" / "collatz" / "collatz_k_scripts" / "noyau_both_base_4096.md",
+            repo_root / "applications" / "collatz" / "collatz_k_scripts" / "complétion_minorée_m14_vers_m15.md",
+            repo_root / "applications" / "collatz" / "collatz_k_scripts" / "complétion_minorée_m15_vers_m16.md",
+            verification_json,
+            verification_md,
+        ]
+        sha_entries = compute_sha256_entries(sha_paths)
+        output_path = docs_dir / f"collatz_run_report_{date_str}_{args.scope}.md"
+        write_c2_projective_run_report(
+            output_path=output_path,
+            report_title=f"Rapport d’exécution — {args.scope}",
+            command=command,
+            git_commit=commit_hash,
+            sha_entries=sha_entries,
+            metrics=metrics,
+            artefacts_dir=artefacts_dir,
+        )
+        print(f"Wrote: {output_path}")
+        return
+
    raise ValueError(f"Unknown profile: {args.profile}")


--- a/applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py
+++ b/applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py
@ -0,0 +1,274 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+collatz_verify_c2_projective.py
+
+Deterministic verification artefact for Lemma C2 (projective reduction):
+
+From completion-by-brothers audit documents, verify that:
+- residual after completion equals the set of children of "both" parents
+- projections modulo 2^12 of B_m (both parents) match the fixed base B12
+
+Inputs (Markdown):
+- noyau_both_base_4096.md (contains B12 list)
+- complétion_minorée_m14_vers_m15.md (contains both-parent list at 2^14 + residual list at 2^15)
+- complétion_minorée_m15_vers_m16.md (contains both-parent list at 2^15)
+
+Outputs:
+- a short JSON summary (machine-checkable)
+- a short Markdown report (human-checkable)
+
+No terminal transcripts. Standard library only.
+"""
+
+from __future__ import annotations
+
+import argparse
+import json
+import re
+from dataclasses import dataclass
+from pathlib import Path
+
+
+def _read_text(path: Path) -> str:
+    return path.read_text(encoding="utf-8", errors="strict")
+
+
+def _extract_ints(text: str) -> list[int]:
+    return [int(x) for x in re.findall(r"\b\d+\b", text)]
+
+
+def _section_after_heading(md: str, heading_regex: str) -> str:
+    """
+    Return the Markdown slice starting at a heading matched by heading_regex (regex applied line-wise).
+    The slice ends at the next heading of same or higher level, or end of document.
+    """
+    lines = md.splitlines()
+    start = None
+    level = None
+    hr = re.compile(heading_regex)
+    for i, line in enumerate(lines):
+        if hr.search(line):
+            start = i + 1
+            m = re.match(r"^(#+)\s", line)
+            level = len(m.group(1)) if m else 1
+            break
+    if start is None or level is None:
+        raise ValueError(f"Cannot find heading matching regex: {heading_regex}")
+
+    end = len(lines)
+    for j in range(start, len(lines)):
+        m2 = re.match(r"^(#+)\s", lines[j])
+        if not m2:
+            continue
+        if len(m2.group(1)) <= level:
+            end = j
+            break
+    return "\n".join(lines[start:end]).strip()
+
+
+def _parse_list_under_heading(md: str, heading_regex: str) -> list[int]:
+    section = _section_after_heading(md, heading_regex)
+    return sorted(set(_extract_ints(section)))
+
+
+def _assert_all_odd(values: list[int], label: str) -> None:
+    bad = [x for x in values if x % 2 == 0]
+    if bad:
+        raise ValueError(f"{label}: expected odd residues only, found {len(bad)} even values (e.g. {bad[:10]})")
+
+
+@dataclass(frozen=True)
+class C2CheckResult:
+    palier_parent: int
+    palier_child: int
+    both_parent_count: int
+    computed_children_count: int
+    listed_residual_count: int | None
+    children_equals_listed_residual: bool | None
+    projection_mod_4096_equals_b12: bool
+
+
+def _lift_children(both_parents: list[int], parent_palier: int) -> list[int]:
+    shift = 1 << parent_palier
+    out = set()
+    for p in both_parents:
+        out.add(p)
+        out.add(p + shift)
+    return sorted(out)
+
+
+def _project_mod_4096(values: list[int]) -> set[int]:
+    return {int(x % 4096) for x in values}
+
+
+def _display_path(path: Path, repo_root: Path) -> str:
+    try:
+        rel = path.resolve().relative_to(repo_root.resolve())
+        return str(rel)
+    except ValueError:
+        return str(path)
+
+
+def verify_c2(
+    *,
+    noyau_both_base_4096_md: Path,
+    completion_m14_to_m15_md: Path,
+    completion_m15_to_m16_md: Path,
+    repo_root: Path,
+    output_json: Path,
+    output_md: Path,
+) -> None:
+    b12_md = _read_text(noyau_both_base_4096_md)
+    c1415_md = _read_text(completion_m14_to_m15_md)
+    c1516_md = _read_text(completion_m15_to_m16_md)
+
+    b12 = _parse_list_under_heading(b12_md, r"^##\s+Base projective B_12\b")
+    _assert_all_odd(b12, "B12")
+    if len(b12) != 192:
+        raise ValueError(f"B12: expected 192 residues, got {len(b12)}")
+
+    # m=14 -> m=15
+    both14 = _parse_list_under_heading(c1415_md, r"^###\s+Parents\s+«\s+both\s+»\s+au\s+palier\s+2\^14\b")
+    _assert_all_odd(both14, "Parents both (m=14)")
+    children15 = _lift_children(both14, 14)
+    _assert_all_odd(children15, "Computed children at m=15")
+
+    residual15 = _parse_list_under_heading(
+        c1415_md,
+        r"^###\s+Résidu\s+restant\s+au\s+palier\s+2\^15\s+après\s+complétion\s+«\s+one\s+»",
+    )
+    _assert_all_odd(residual15, "Listed residual at m=15")
+
+    c1415 = C2CheckResult(
+        palier_parent=14,
+        palier_child=15,
+        both_parent_count=len(both14),
+        computed_children_count=len(children15),
+        listed_residual_count=len(residual15),
+        children_equals_listed_residual=(set(children15) == set(residual15)),
+        projection_mod_4096_equals_b12=(_project_mod_4096(both14) == set(b12)),
+    )
+
+    # m=15 -> m=16
+    both15 = _parse_list_under_heading(c1516_md, r"^###\s+Parents\s+«\s+both\s+»\s+au\s+palier\s+2\^15\b")
+    _assert_all_odd(both15, "Parents both (m=15)")
+    children16 = _lift_children(both15, 15)
+    _assert_all_odd(children16, "Computed children at m=16")
+
+    # This completion file does not necessarily list the full residual after completion, so we only check the count.
+    c1516 = C2CheckResult(
+        palier_parent=15,
+        palier_child=16,
+        both_parent_count=len(both15),
+        computed_children_count=len(children16),
+        listed_residual_count=None,
+        children_equals_listed_residual=None,
+        projection_mod_4096_equals_b12=(_project_mod_4096(both15) == set(b12)),
+    )
+
+    summary = {
+        "inputs": {
+            "noyau_both_base_4096_md": _display_path(noyau_both_base_4096_md, repo_root),
+            "completion_m14_to_m15_md": _display_path(completion_m14_to_m15_md, repo_root),
+            "completion_m15_to_m16_md": _display_path(completion_m15_to_m16_md, repo_root),
+        },
+        "B12": {"count": len(b12), "min": min(b12), "max": max(b12)},
+        "checks": {
+            "m14_to_m15": c1415.__dict__,
+            "m15_to_m16": c1516.__dict__,
+        },
+        "ok": {
+            "m14_to_m15_children_equals_listed_residual": bool(c1415.children_equals_listed_residual),
+            "m14_projection_equals_b12": bool(c1415.projection_mod_4096_equals_b12),
+            "m15_projection_equals_b12": bool(c1516.projection_mod_4096_equals_b12),
+        },
+    }
+
+    output_json.parent.mkdir(parents=True, exist_ok=True)
+    output_json.write_text(json.dumps(summary, indent=2, ensure_ascii=False) + "\n", encoding="utf-8")
+
+    lines: list[str] = []
+    lines.append("**Auteur** : Équipe 4NK")
+    lines.append("")
+    lines.append("# Vérification déterministe — Lemma C2 (réduction projective)")
+    lines.append("")
+    lines.append("## Entrées")
+    lines.append("")
+    lines.append(f"- `noyau_both_base_4096.md` : `{_display_path(noyau_both_base_4096_md, repo_root)}`")
+    lines.append(f"- `complétion_minorée_m14_vers_m15.md` : `{_display_path(completion_m14_to_m15_md, repo_root)}`")
+    lines.append(f"- `complétion_minorée_m15_vers_m16.md` : `{_display_path(completion_m15_to_m16_md, repo_root)}`")
+    lines.append("")
+    lines.append("## Base projective B12")
+    lines.append("")
+    lines.append(f"- |B12| = {len(b12)} (attendu 192)")
+    lines.append(f"- min(B12) = {min(b12)}, max(B12) = {max(b12)}")
+    lines.append("")
+    lines.append("## Transition m=14 → m=15 (complétion par frères ⇒ résidu = enfants des both)")
+    lines.append("")
+    lines.append(f"- |both parents (m=14)| = {c1415.both_parent_count}")
+    lines.append(f"- |computed children (m=15)| = {c1415.computed_children_count} (attendu 2*|both|)")
+    lines.append(f"- |listed residual (m=15)| = {c1415.listed_residual_count}")
+    lines.append(f"- computed children == listed residual : {c1415.children_equals_listed_residual}")
+    lines.append(f"- projection both(m=14) mod 4096 == B12 : {c1415.projection_mod_4096_equals_b12}")
+    lines.append("")
+    lines.append("## Transition m=15 → m=16 (projection des both)")
+    lines.append("")
+    lines.append(f"- |both parents (m=15)| = {c1516.both_parent_count}")
+    lines.append(f"- |computed children (m=16)| = {c1516.computed_children_count} (attendu 2*|both|)")
+    lines.append(f"- projection both(m=15) mod 4096 == B12 : {c1516.projection_mod_4096_equals_b12}")
+    lines.append("")
+    lines.append("## Sorties")
+    lines.append("")
+    lines.append(f"- JSON : `{output_json}`")
+    lines.append(f"- Markdown : `{output_md}`")
+    lines.append("")
+
+    output_md.parent.mkdir(parents=True, exist_ok=True)
+    output_md.write_text("\n".join(lines) + "\n", encoding="utf-8")
+
+
+def main() -> None:
+    ap = argparse.ArgumentParser(description="Deterministic C2 verification (projective reduction)")
+    ap.add_argument(
+        "--repo-root",
+        default="",
+        help="Repository root used to display relative paths in outputs (defaults to current working directory)",
+    )
+    ap.add_argument(
+        "--noyau-both-base-4096-md",
+        default=str(Path(__file__).parent / "noyau_both_base_4096.md"),
+        help="Path to noyau_both_base_4096.md",
+    )
+    ap.add_argument(
+        "--completion-m14-to-m15-md",
+        default=str(Path(__file__).parent / "complétion_minorée_m14_vers_m15.md"),
+        help="Path to complétion_minorée_m14_vers_m15.md",
+    )
+    ap.add_argument(
+        "--completion-m15-to-m16-md",
+        default=str(Path(__file__).parent / "complétion_minorée_m15_vers_m16.md"),
+        help="Path to complétion_minorée_m15_vers_m16.md",
+    )
+    ap.add_argument(
+        "--output-dir",
+        default="docs/artefacts/collatz/c2_projective",
+        help="Output directory for deterministic artefacts (JSON + MD)",
+    )
+    args = ap.parse_args()
+
+    repo_root = Path(args.repo_root).resolve() if args.repo_root.strip() else Path.cwd().resolve()
+    out_dir = Path(args.output_dir)
+    verify_c2(
+        noyau_both_base_4096_md=Path(args.noyau_both_base_4096_md),
+        completion_m14_to_m15_md=Path(args.completion_m14_to_m15_md),
+        completion_m15_to_m16_md=Path(args.completion_m15_to_m16_md),
+        repo_root=repo_root,
+        output_json=out_dir / "verification_c2_projective.json",
+        output_md=out_dir / "verification_c2_projective.md",
+    )
+
+
+if __name__ == "__main__":
+    main()
+
--- a/applications/collatz/collatz_k_scripts/plan_lemmes_manquants_et_programme_de_preuve.md
+++ b/applications/collatz/collatz_k_scripts/plan_lemmes_manquants_et_programme_de_preuve.md
@ -215,8 +215,47 @@ Les artefacts versionnés montrent, pour les 60 états \(E\) de \(B_{12}\) :
 Ces constats sont des assertions d’audit (égalité d’ensembles sur artefacts) et ne remplacent pas la preuve des hypothèses (H4)–(H5).

 Lemme C2 (réduction de \(R_m\) vers un noyau projectif).
-Il existe un mécanisme (à préciser : scission par frères + règles de fermeture) montrant que tout élément de \(R_m\) possède, après un nombre borné d’étapes de relèvement et de normalisation, un représentant dans un noyau projectif de type \(B_{12}\) (ou dans une famille finie de noyaux projectifs).
-Objectif : réduire la preuve globale à une élimination de noyaux projectifs finis.
+On fixe un schéma de complétion par frères à chaque transition \(m\to m+1\) :
+
+- Pour un parent \(r\in R_m\), ses deux enfants au palier suivant sont \(r\) et \(r+2^m\).
+- On distingue :
+  - **cas one** : un seul enfant est survivant à la transition ; l’autre est fermé (couvert) ;
+  - **cas both** : les deux enfants sont survivants.
+
+On note \(B_m\subset R_m\) l’ensemble des parents “both”.
+
+### Énoncés cibles (formes à rendre exactes)
+
+- (C2.1) **Complétion par frères ⇒ résidu = enfants des both**.  
+  Après fermeture (au palier \(m+1\)) des enfants “one” par clauses de descente minorées (lemme de frère), le résidu restant au palier \(m+1\) est exactement :
+  \[
+  R_{m+1}^{\mathrm{comp}}=\{\,r,\ r+2^m\;:\; r\in B_m\,\}.
+  \]
+
+- (C2.2) **Projection projective**.  
+  Pour certains paliers (en pratique : dès \(m\ge 12\) dans les artefacts actuels), il existe une base projective fixe
+  \[
+  B_{12}\subset(\mathbb{Z}/2^{12}\mathbb{Z})^\times,\quad |B_{12}|=192,
+  \]
+  telle que
+  \[
+  B_m \bmod 2^{12} = B_{12}.
+  \]
+
+### Artefact déterministe (vérification citable)
+
+Une vérification déterministe (script + sortie courte) est versionnée dans `docs/artefacts/` :
+
+- script : `applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py`
+- sorties : `docs/artefacts/collatz/c2_projective/verification_c2_projective.{json,md}`
+- rapport d’exécution (sha256 + métriques) : `docs/collatz_run_report_2026-03-09_c2_projective.md`
+
+Ces artefacts valident (au niveau “audit”) les instances suivantes :
+
+- transition \(m=14\to 15\) : \(R_{15}^{\mathrm{comp}}=\{r,r+2^{14}:r\in B_{14}\}\) et \(B_{14}\bmod 4096 = B_{12}\),
+- projection \(m=15\) : \(B_{15}\bmod 4096 = B_{12}\).
+
+Objectif de C2 : utiliser (C2.1)–(C2.2) pour réduire l’analyse des survivants à une élimination par états sur \(B_{12}\) (C1), puis à un verrou final (C3).

 Lemme C3 (impossibilité d’orbite infinie sous contraintes).
 Sous un paquet explicite de contraintes arithmétiques \(P\) (congruences, bornes sur sommes de valuations sur des horizons, contraintes modulo \(3^k\), etc.) extraites comme invariants des survivants, aucune orbite de la dynamique accélérée \(U\) ne peut être infinie.
--- a/docs/artefacts/collatz/c2_projective/verification_c2_projective.json
+++ b/docs/artefacts/collatz/c2_projective/verification_c2_projective.json
@ -0,0 +1,37 @@
+{
+  "inputs": {
+    "noyau_both_base_4096_md": "applications/collatz/collatz_k_scripts/noyau_both_base_4096.md",
+    "completion_m14_to_m15_md": "applications/collatz/collatz_k_scripts/complétion_minorée_m14_vers_m15.md",
+    "completion_m15_to_m16_md": "applications/collatz/collatz_k_scripts/complétion_minorée_m15_vers_m16.md"
+  },
+  "B12": {
+    "count": 192,
+    "min": 27,
+    "max": 4095
+  },
+  "checks": {
+    "m14_to_m15": {
+      "palier_parent": 14,
+      "palier_child": 15,
+      "both_parent_count": 593,
+      "computed_children_count": 1186,
+      "listed_residual_count": 1186,
+      "children_equals_listed_residual": true,
+      "projection_mod_4096_equals_b12": true
+    },
+    "m15_to_m16": {
+      "palier_parent": 15,
+      "palier_child": 16,
+      "both_parent_count": 1101,
+      "computed_children_count": 2202,
+      "listed_residual_count": null,
+      "children_equals_listed_residual": null,
+      "projection_mod_4096_equals_b12": true
+    }
+  },
+  "ok": {
+    "m14_to_m15_children_equals_listed_residual": true,
+    "m14_projection_equals_b12": true,
+    "m15_projection_equals_b12": true
+  }
+}
--- a/docs/artefacts/collatz/c2_projective/verification_c2_projective.md
+++ b/docs/artefacts/collatz/c2_projective/verification_c2_projective.md
@ -0,0 +1,34 @@
+**Auteur** : Équipe 4NK
+
+# Vérification déterministe — Lemma C2 (réduction projective)
+
+## Entrées
+
+- `noyau_both_base_4096.md` : `applications/collatz/collatz_k_scripts/noyau_both_base_4096.md`
+- `complétion_minorée_m14_vers_m15.md` : `applications/collatz/collatz_k_scripts/complétion_minorée_m14_vers_m15.md`
+- `complétion_minorée_m15_vers_m16.md` : `applications/collatz/collatz_k_scripts/complétion_minorée_m15_vers_m16.md`
+
+## Base projective B12
+
+- |B12| = 192 (attendu 192)
+- min(B12) = 27, max(B12) = 4095
+
+## Transition m=14 → m=15 (complétion par frères ⇒ résidu = enfants des both)
+
+- |both parents (m=14)| = 593
+- |computed children (m=15)| = 1186 (attendu 2*|both|)
+- |listed residual (m=15)| = 1186
+- computed children == listed residual : True
+- projection both(m=14) mod 4096 == B12 : True
+
+## Transition m=15 → m=16 (projection des both)
+
+- |both parents (m=15)| = 1101
+- |computed children (m=16)| = 2202 (attendu 2*|both|)
+- projection both(m=15) mod 4096 == B12 : True
+
+## Sorties
+
+- JSON : `docs/artefacts/collatz/c2_projective/verification_c2_projective.json`
+- Markdown : `docs/artefacts/collatz/c2_projective/verification_c2_projective.md`
+
--- a/docs/collatz_run_report_2026-03-09_c2_projective.md
+++ b/docs/collatz_run_report_2026-03-09_c2_projective.md
@ -0,0 +1,46 @@
+**Auteur** : Équipe 4NK
+
+# Rapport d’exécution — c2_projective
+
+## Contexte
+
+- **But du run** : vérifier C2 (complétion par frères ⇒ both ⇒ projection B12) sur des artefacts déterministes.
+- **Assertion vérifiée** : égalités d’ensembles et égalité de projections modulo 4096 (voir métriques).
+
+## Code et reproductibilité
+
+- **Commit Git** : `e12c544e7bf8d7a6f9518cdd4959d0da07bccc07`
+- **Commande** :
+
+```bash
+python3 applications/collatz/collatz_k_scripts/collatz_generate_run_report.py --profile c2_projective --scope c2_projective --out-dir applications/collatz/out --docs-dir docs --c2-artefacts-dir docs/artefacts/collatz/c2_projective
+```
+
+## Empreintes sha256 (scripts, artefacts)
+
+- `/home/ncantu/code/algo/applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py`
+  - sha256: `4f515d649c44c4ffe55349b07c72e9e766f591f7596e1bcdf0f9447e6b94950d`
+- `/home/ncantu/code/algo/applications/collatz/collatz_k_scripts/noyau_both_base_4096.md`
+  - sha256: `d0abffeca9ce2356d13b31ee4325b68fd85cf061670a8524b8e60fd15aba881b`
+- `/home/ncantu/code/algo/applications/collatz/collatz_k_scripts/complétion_minorée_m14_vers_m15.md`
+  - sha256: `acc8667be8e93a33fadf45bacd5cd788ec05089d844616cf1762330ac0d41d67`
+- `/home/ncantu/code/algo/applications/collatz/collatz_k_scripts/complétion_minorée_m15_vers_m16.md`
+  - sha256: `956f8fd19570b7614705384995b52b6bf81399ca8dc36c3ea33bcbad3e4aff50`
+- `/home/ncantu/code/algo/docs/artefacts/collatz/c2_projective/verification_c2_projective.json`
+  - sha256: `b1b9ab9d5d4f9d4ee60e7efd43b853ac29d3d51a19c2d71bdac45c4340d01bf0`
+- `/home/ncantu/code/algo/docs/artefacts/collatz/c2_projective/verification_c2_projective.md`
+  - sha256: `8bd84cb2149d297487e0e0ff03ebe96ff27b07a5ab677407cb4b9a17a339c2ba`
+
+## Compteurs et métriques
+
+- |B12| = 192
+- m=14: |both|=593, |children(m=15)|=1186, |listed residual(m=15)|=1186
+- m=14: children(m=15) == residual(m=15) : True
+- m=14: both mod 4096 == B12 : True
+- m=15: |both|=1101, |children(m=16)|=2202
+- m=15: both mod 4096 == B12 : True
+
+## Chemins d’artefacts (versionnés)
+
+- ARTEFACTS : `/home/ncantu/code/algo/docs/artefacts/collatz/c2_projective`
+
--- a/docs/collatz_run_report_format.md
+++ b/docs/collatz_run_report_format.md
@ -86,6 +86,21 @@ Lors de la génération d’un rapport `local_H6`/`local_H6_E1`, un audit automa

 En cas de désaccord, la génération échoue (afin d’éviter une citation ambiguë d’artefacts).

+Pour la vérification C2 (complétion par frères ⇒ both ⇒ projection \(B_{12}\), artefacts déterministes versionnés) :
+
+```bash
+python3 applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py \
+  --repo-root /home/ncantu/code/algo \
+  --output-dir docs/artefacts/collatz/c2_projective
+
+python3 applications/collatz/collatz_k_scripts/collatz_generate_run_report.py \
+  --profile c2_projective \
+  --scope c2_projective \
+  --c2-artefacts-dir docs/artefacts/collatz/c2_projective \
+  --out-dir applications/collatz/out \
+  --docs-dir docs
+```
+
 ### Contexte

 - **But du run** : (énoncé court)
--- a/docs/features/collatz_c2_projective_verification.md
+++ b/docs/features/collatz_c2_projective_verification.md
@ -0,0 +1,52 @@
+**Auteur** : Équipe 4NK
+
+# Collatz — Vérification déterministe de C2 (réduction projective)
+
+## Objectif
+
+Rendre citable (sans transcript terminal) une vérification déterministe des assertions utilisées dans le lemme C2 :
+
+- **complétion par frères** : après fermeture des cas “one”, le résidu au palier suivant est exactement l’ensemble des **enfants** des parents “both” ;
+- **projection projective** : l’ensemble des parents “both” à certains paliers se projette modulo \(2^{12}\) sur une base fixe \(B_{12}\subset(\mathbb{Z}/2^{12}\mathbb{Z})^\times\).
+
+## Impacts
+
+- Ajout d’un artefact versionné dans `docs/artefacts/` et d’un rapport d’exécution dans `docs/`, alignés sur le format standard des rapports.
+
+## Modifications
+
+- Script déterministe :
+  - `applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py`
+- Artefacts versionnés (sortie du script) :
+  - `docs/artefacts/collatz/c2_projective/verification_c2_projective.json`
+  - `docs/artefacts/collatz/c2_projective/verification_c2_projective.md`
+- Rapport d’exécution (sha256 + métriques) :
+  - `docs/collatz_run_report_2026-03-09_c2_projective.md`
+- Extension du générateur de rapports :
+  - `applications/collatz/collatz_k_scripts/collatz_generate_run_report.py` (profil `c2_projective`)
+
+## Modalités d’analyse
+
+- Générer/mettre à jour les artefacts déterministes :
+
+```bash
+python3 applications/collatz/collatz_k_scripts/collatz_verify_c2_projective.py \
+  --repo-root /home/ncantu/code/algo \
+  --output-dir docs/artefacts/collatz/c2_projective
+```
+
+- Produire le rapport d’exécution (empreintes sha256 + métriques) :
+
+```bash
+python3 applications/collatz/collatz_k_scripts/collatz_generate_run_report.py \
+  --profile c2_projective \
+  --scope c2_projective \
+  --c2-artefacts-dir docs/artefacts/collatz/c2_projective \
+  --out-dir applications/collatz/out \
+  --docs-dir docs
+```
+
+## Modalités de déploiement
+
+Aucun déploiement.
+
--- a/docs/features/collatz_run_report_generator.md
+++ b/docs/features/collatz_run_report_generator.md
@ -23,8 +23,9 @@ afin d’éviter toute insertion de transcript terminal dans les documents math
 - Ajout d’un format standard : `docs/collatz_run_report_format.md`.
 - Ajout d’un générateur : `applications/collatz/collatz_k_scripts/collatz_generate_run_report.py`.
 - Ajout d’un exemple réel de rapport : `docs/collatz_run_report_2026-03-04_extend_D18_D21_resume_from_D20.md`.
- - Profils supportés : `extend_finale`, `validation_section7`, `pipeline_d16_d17`, `fusion_palier2p25`, `local_H6_E1`, `local_H6`.
+ - Profils supportés : `extend_finale`, `validation_section7`, `pipeline_d16_d17`, `fusion_palier2p25`, `local_H6_E1`, `local_H6`, `c2_projective`.
 - Pour `local_H6`/`local_H6_E1`, un audit automatique compare le **palier attendu** (artefacts) au **palier certifié** (certificats) et échoue en cas de désaccord.
+ - Pour `c2_projective`, le rapport cite les artefacts déterministes `docs/artefacts/collatz/c2_projective/verification_c2_projective.{json,md}`.

 ## Modalités d’analyse