diff --git a/MyIA.AI.Notebooks/SymbolicAI/Lean/ANALYSE/ANALYSE-04-PFR-Primitives-Python.ipynb b/MyIA.AI.Notebooks/SymbolicAI/Lean/ANALYSE/ANALYSE-04-PFR-Primitives-Python.ipynb index c6ffbaf47ac..fa02f7d49c0 100644 --- a/MyIA.AI.Notebooks/SymbolicAI/Lean/ANALYSE/ANALYSE-04-PFR-Primitives-Python.ipynb +++ b/MyIA.AI.Notebooks/SymbolicAI/Lean/ANALYSE/ANALYSE-04-PFR-Primitives-Python.ipynb @@ -5,10 +5,10 @@ "id": "3fd55e7c", "metadata": { "papermill": { - "duration": 0.004399, - "end_time": "2026-10-16T23:17:52.000689+00:00", + "duration": 0.006553, + "end_time": "2026-10-23T22:00:24.235820+00:00", "exception": false, - "start_time": "2026-10-16T23:17:51.996290+00:00", + "start_time": "2026-10-23T22:00:24.229267+00:00", "status": "completed" }, "tags": [] @@ -69,10 +69,10 @@ "id": "e44d9187", "metadata": { "papermill": { - "duration": 0.003269, - "end_time": "2026-10-16T23:17:52.007718+00:00", + "duration": 0.006131, + "end_time": "2026-10-23T22:00:24.248209+00:00", "exception": false, - "start_time": "2026-10-16T23:17:52.004449+00:00", + "start_time": "2026-10-23T22:00:24.242078+00:00", "status": "completed" }, "tags": [] @@ -110,16 +110,16 @@ "id": "8b72f4f5", "metadata": { "execution": { - "iopub.execute_input": "2026-10-16T23:17:52.016169Z", - "iopub.status.busy": "2026-10-16T23:17:52.015903Z", - "iopub.status.idle": "2026-10-16T23:17:52.028306Z", - "shell.execute_reply": "2026-10-16T23:17:52.027866Z" + "iopub.execute_input": "2026-10-23T22:00:24.263621Z", + "iopub.status.busy": "2026-10-23T22:00:24.263129Z", + "iopub.status.idle": "2026-10-23T22:00:24.282661Z", + "shell.execute_reply": "2026-10-23T22:00:24.281194Z" }, "papermill": { - "duration": 0.018415, - "end_time": "2026-10-16T23:17:52.029325+00:00", + "duration": 0.028896, + "end_time": "2026-10-23T22:00:24.283908+00:00", "exception": false, - "start_time": "2026-10-16T23:17:52.010910+00:00", + "start_time": "2026-10-23T22:00:24.255012+00:00", "status": "completed" }, "tags": [] @@ -211,10 +211,10 @@ "id": "c69d894c", "metadata": { "papermill": { - "duration": 0.005437, - "end_time": "2026-10-16T23:17:52.041830+00:00", + "duration": 0.005455, + "end_time": "2026-10-23T22:00:24.295702+00:00", "exception": false, - "start_time": "2026-10-16T23:17:52.036393+00:00", + "start_time": "2026-10-23T22:00:24.290247+00:00", "status": "completed" }, "tags": [] @@ -250,16 +250,16 @@ "id": "feb6d5f4", "metadata": { "execution": { - "iopub.execute_input": "2026-10-16T23:17:52.051781Z", - "iopub.status.busy": "2026-10-16T23:17:52.051334Z", - "iopub.status.idle": "2026-10-16T23:17:54.129206Z", - "shell.execute_reply": "2026-10-16T23:17:54.128137Z" + "iopub.execute_input": "2026-10-23T22:00:24.308291Z", + "iopub.status.busy": "2026-10-23T22:00:24.307809Z", + "iopub.status.idle": "2026-10-23T22:00:27.235958Z", + "shell.execute_reply": "2026-10-23T22:00:27.235071Z" }, "papermill": { - "duration": 2.084772, - "end_time": "2026-10-16T23:17:54.130090+00:00", + "duration": 2.936577, + "end_time": "2026-10-23T22:00:27.236884+00:00", "exception": false, - "start_time": "2026-10-16T23:17:52.045318+00:00", + "start_time": "2026-10-23T22:00:24.300307+00:00", "status": "completed" }, "tags": [] @@ -365,10 +365,10 @@ "id": "0fc19933", "metadata": { "papermill": { - "duration": 0.003959, - "end_time": "2026-10-16T23:17:54.137756+00:00", + "duration": 0.004034, + "end_time": "2026-10-23T22:00:27.245240+00:00", "exception": false, - "start_time": "2026-10-16T23:17:54.133797+00:00", + "start_time": "2026-10-23T22:00:27.241206+00:00", "status": "completed" }, "tags": [] @@ -381,15 +381,125 @@ "**Leçon** : la primitive a une **puissance incomparable** sur son terrain (groupes abéliens, `F₂ⁿ`, entropie appliquée à la combinatoire additive), mais l'**utiliser hors de ce terrain** serait du « mapping affine » — c'est-à-dire plaquer une structure sans la mériter. Le carnet ANALYSE-03 lui-même s'en garde : la preuve PFR ne s'applique qu'à `F₂ⁿ`, pas à un objet métrique général.\n" ] }, + { + "cell_type": "markdown", + "id": "b8737d0d", + "metadata": { + "papermill": { + "duration": 0.003889, + "end_time": "2026-10-23T22:00:27.253062+00:00", + "exception": false, + "start_time": "2026-10-23T22:00:27.249173+00:00", + "status": "completed" + }, + "tags": [] + }, + "source": [ + "### 1.4 Test 6 — l'inégalité triangulaire, mesurée\n", + "\n", + "L'énoncé de 1.1 liste le **triangle** `d[X;Z] ≤ d[X;Y] + d[Y;Z]` parmi les invariants — c'est le seul de la liste qui n'était pas encore mesuré, alors qu'il n'est pas une évidence de la formule : l'inégalité triangulaire entropique de Ruzsa est un théorème du terrain, démontré par Tao, *Sumset and inverse sumset theory for Shannon entropy* (arXiv:0906.4387), théorème 1.10, équation (16), sous la forme `distR(X, Z) ≤ distR(X, Y) + distR(Y, Z)`. Le test suivant la mesure sur trois triplets de `ℤ/8ℤ` choisis pour discriminer — égalité exacte, saturation, triplet tendu — en réutilisant `d_ruzsa` telle quelle, sans réimplémentation." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "21e9830c", + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-23T22:00:27.264551Z", + "iopub.status.busy": "2026-10-23T22:00:27.264186Z", + "iopub.status.idle": "2026-10-23T22:00:27.272394Z", + "shell.execute_reply": "2026-10-23T22:00:27.271389Z" + }, + "papermill": { + "duration": 0.015728, + "end_time": "2026-10-23T22:00:27.273164+00:00", + "exception": false, + "start_time": "2026-10-23T22:00:27.257436+00:00", + "status": "completed" + }, + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Test 6 : l'inégalité triangulaire d[X;Z] <= d[X;Y] + d[Y;Z], sur Z/8Z\n", + "\n", + "Triplet 1 -- égalité exacte : X = Y = Z uniformes sur A = {0,2,4,6}\n", + " d[X;Y] = 0.0000 d[Y;Z] = 0.0000 d[X;Z] = 0.0000\n", + " inégalité : 0.0000 <= 0.0000 marge = +0.0000 bit OK\n", + "\n", + "Triplet 2 -- saturation sans colinéarité : X, Y sur A ; Z sur A + 1\n", + " d[X;Y] = 0.0000 d[Y;Z] = 0.0000 d[X;Z] = 0.0000\n", + " inégalité : 0.0000 <= 0.0000 marge = +0.0000 bit OK\n", + "\n", + "Triplet 3 -- tendu : Y dégénéré en 0, X uniforme sur Z/8Z, Z sur A\n", + " d[X;Y] = 1.5000 d[Y;Z] = 1.0000 d[X;Z] = 0.5000\n", + " inégalité : 0.5000 <= 2.5000 marge = +2.0000 bit OK\n" + ] + } + ], + "source": [ + "# 1.4 — Test 6 : l'inégalité triangulaire d[X;Z] <= d[X;Y] + d[Y;Z], mesurée.\n", + "# d_ruzsa (cellule 1.2) est réutilisée telle quelle : trois distances par\n", + "# triplet, l'inégalité lue sur trois témoins qui discriminent.\n", + "n8 = 8\n", + "A = {0: 0.25, 2: 0.25, 4: 0.25, 6: 0.25} # uniforme sur le sous-groupe A = {0,2,4,6}\n", + "A_plus = {1: 0.25, 3: 0.25, 5: 0.25, 7: 0.25} # uniforme sur le translaté A + 1\n", + "u8 = {k: 1 / n8 for k in range(n8)} # uniforme sur Z/8Z tout entier\n", + "degen0 = {0: 1.0, **{k: 0.0 for k in range(1, n8)}} # dégénéré en 0\n", + "\n", + "def lit_triplet(nom, X, Y, Z, n):\n", + " dxy, dyz, dxz = d_ruzsa(X, Y, n), d_ruzsa(Y, Z, n), d_ruzsa(X, Z, n)\n", + " borne, marge = dxy + dyz, dxy + dyz - dxz\n", + " print(nom)\n", + " print(f\" d[X;Y] = {dxy:.4f} d[Y;Z] = {dyz:.4f} d[X;Z] = {dxz:.4f}\")\n", + " print(f\" inégalité : {dxz:.4f} <= {borne:.4f} marge = {marge:+.4f} bit\"\n", + " f\" {'OK' if dxz <= borne + 1e-9 else 'VIOLEE'}\")\n", + "\n", + "print(\"Test 6 : l'inégalité triangulaire d[X;Z] <= d[X;Y] + d[Y;Z], sur Z/8Z\")\n", + "print()\n", + "lit_triplet(\"Triplet 1 -- égalité exacte : X = Y = Z uniformes sur A = {0,2,4,6}\", A, A, A, n8)\n", + "print()\n", + "lit_triplet(\"Triplet 2 -- saturation sans colinéarité : X, Y sur A ; Z sur A + 1\", A, A, A_plus, n8)\n", + "print()\n", + "lit_triplet(\"Triplet 3 -- tendu : Y dégénéré en 0, X uniforme sur Z/8Z, Z sur A\", u8, degen0, A, n8)" + ] + }, + { + "cell_type": "markdown", + "id": "8eedd2e8", + "metadata": { + "papermill": { + "duration": 0.003774, + "end_time": "2026-10-23T22:00:27.283259+00:00", + "exception": false, + "start_time": "2026-10-23T22:00:27.279485+00:00", + "status": "completed" + }, + "tags": [] + }, + "source": [ + "### Lecture du résultat — Test 6\n", + "\n", + "- **Triplet 1** (`X = Y = Z` uniformes sur le sous-groupe `A = {0,2,4,6}`) : les trois distances valent `0` — l'inégalité est une **égalité exacte** (`0 ≤ 0 + 0`).\n", + "- **Triplet 2** (`X`, `Y` sur `A` ; `Z` sur le translaté `A + 1`) : les trois distances valent encore `0` alors que `X` et `Z` n'ont **pas la même loi** — l'inégalité est **saturée sans colinéarité** des lois. C'est le phénomène du test 2 : deux uniformes sur des translatés d'un même sous-groupe sont à distance nulle, et la borne ne peut pas descendre plus bas.\n", + "- **Triplet 3** (`Y` dégénéré en `0` entre `X` uniforme sur `ℤ/8ℤ` et `Z` uniforme sur `A`) : `d[X;Y] = ½H(X) = 1.5` bit et `d[Y;Z] = ½H(Z) = 1.0` bit — la borne vaut `2.5` bits, et la mesure rend `d[X;Z] = 0.5` bit : l'inégalité tient avec une **marge de 2.0 bits**. Un test qui ne rendrait que `0.5 ≤ 2.5` n'apprendrait rien ; les trois témoins ensemble disent où l'inégalité colle et où elle respire.\n", + "\n", + "Le contrôle négatif est déjà dans le carnet : le test de limite (b) de 1.3 montre que sans inversibilité de `x ↦ x − y`, `d` devient **négatif** — la triangularité meurt avec la structure de groupe, comme les autres invariants. La ligne d'énoncé du triangle en 1.1 est désormais adossée à sa mesure." + ] + }, { "cell_type": "markdown", "id": "ebc41c85", "metadata": { "papermill": { - "duration": 0.003629, - "end_time": "2026-10-16T23:17:54.145458+00:00", + "duration": 0.003737, + "end_time": "2026-10-23T22:00:27.290946+00:00", "exception": false, - "start_time": "2026-10-16T23:17:54.141829+00:00", + "start_time": "2026-10-23T22:00:27.287209+00:00", "status": "completed" }, "tags": [] @@ -412,20 +522,20 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "id": "a31f41c6", "metadata": { "execution": { - "iopub.execute_input": "2026-10-16T23:17:54.157105Z", - "iopub.status.busy": "2026-10-16T23:17:54.156731Z", - "iopub.status.idle": "2026-10-16T23:17:54.161709Z", - "shell.execute_reply": "2026-10-16T23:17:54.160635Z" + "iopub.execute_input": "2026-10-23T22:00:27.300017Z", + "iopub.status.busy": "2026-10-23T22:00:27.299654Z", + "iopub.status.idle": "2026-10-23T22:00:27.304756Z", + "shell.execute_reply": "2026-10-23T22:00:27.303937Z" }, "papermill": { - 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