diff --git a/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.md b/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.md
deleted file mode 100644
index 0eb1a1ac..00000000
--- a/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.md
+++ /dev/null
@@ -1,179 +0,0 @@
----
-
-excalidraw-plugin: parsed
-tags: [excalidraw]
-
----
-==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
-
-
-# Excalidraw Data
-
-## Text Elements
-## Embedded Files
-0c2ad4dae5ed514b2fd2099d68d0760c69fbce4d: $$[1^{1}]$$
-
-f9c478059e1cce4e96a58e13baac6bfbb4f93eff: $$[1^{1}X^{1}$$
-
-214cc7890168fcc3649c7581bd637e267a6de7bf: $$[1^{1}2^{2}$$
-
-84fa363149df51fc341b5998c6b4e67ed3dbdb8e: $$[1^{3}$$
-
-f49bf0b1178f855291cee484fe1921917621c6c0: $$[1^{1}3^{2}$$
-
-c6ae105adef2251de3956c588d95fb1ed29d572b: $$[1^{2}2^{2}$$
-
-f7ab87077d1c10f7309f5bbce5c714d5f25bef0f: $$[1^{2}2^{3}$$
-
-539b82e2e9bc43b1e2665050071a453e72116434: $$[2^{1}X^{2}$$
-
-082b26651855e6e47f735b3481c3b45c1678ff69: $$2^{1}X^{\neq 2}$$
-
-76483ecab4b1eb14f16fcd978d387e8ac0f6344a: $$[2^{3}$$
-
-32a8965b6d15d90d3898edfdb7969dbd9a92f3c8: $$[3^{1}X^{3}$$
-
-cc5e37cfba585550da4471dd85852d6a38354787: $$[3^{1}X^{\leq 2}$$
-
-b752624b6df8d46597c3242c793dafa003ae9422: $$[3^{2}X^{3}$$
-
-848fe42b60268d03c05e522d05a9dd988119d57b: $$[3^{2}X^{\leq 2}$$
-
-98ffd4aa77ffc7c0ea17203a781a2cd621186d8e: $$[n^{1}$$
-
-8aef2acc0c00b47668a81c160807e1ab6d4b03ce: $$[1^{2}X^{1}$$
-
-%%
-## Drawing
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-
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-
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-
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-```
-%%
\ No newline at end of file
diff --git a/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.svg b/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.svg
deleted file mode 100644
index ddfefd1f..00000000
--- a/Excalidraw/Drawing 2026-03-30 01.05.52.excalidraw.svg
+++ /dev/null
@@ -1,2 +0,0 @@
-
\ No newline at end of file
diff --git a/Excalidraw/demo_théorème_du_début_2.excalidraw.md b/Excalidraw/demo_théorème_du_début_2.excalidraw.md
new file mode 100644
index 00000000..fd26bb98
--- /dev/null
+++ b/Excalidraw/demo_théorème_du_début_2.excalidraw.md
@@ -0,0 +1,183 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+0c2ad4dae5ed514b2fd2099d68d0760c69fbce4d: $$[1^{1}]$$
+
+f9c478059e1cce4e96a58e13baac6bfbb4f93eff: $$[1^{1}X^{1}$$
+
+214cc7890168fcc3649c7581bd637e267a6de7bf: $$[1^{1}2^{2}$$
+
+84fa363149df51fc341b5998c6b4e67ed3dbdb8e: $$[1^{3}$$
+
+f49bf0b1178f855291cee484fe1921917621c6c0: $$[1^{1}3^{2}$$
+
+539b82e2e9bc43b1e2665050071a453e72116434: $$[2^{1}X^{2}$$
+
+082b26651855e6e47f735b3481c3b45c1678ff69: $$2^{1}X^{\neq 2}$$
+
+76483ecab4b1eb14f16fcd978d387e8ac0f6344a: $$[2^{3}$$
+
+32a8965b6d15d90d3898edfdb7969dbd9a92f3c8: $$[3^{1}X^{3}$$
+
+cc5e37cfba585550da4471dd85852d6a38354787: $$[3^{1}X^{\leq 2}$$
+
+b752624b6df8d46597c3242c793dafa003ae9422: $$[3^{2}X^{3}$$
+
+848fe42b60268d03c05e522d05a9dd988119d57b: $$[3^{2}X^{\leq 2}$$
+
+98ffd4aa77ffc7c0ea17203a781a2cd621186d8e: $$[n^{1}$$
+
+8aef2acc0c00b47668a81c160807e1ab6d4b03ce: $$[1^{2}X^{1}$$
+
+f297bb8c586c081e6495ec2b90fc2134c5756fa0: $$[1^{2}X^{\neq 1}$$
+
+%%
+## Drawing
+```compressed-json
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+
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+
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+
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+
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+
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+
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+
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+
+AB8gBQ0xQQBwAH2Hk1wgJSD/AfwIAA==
+```
+%%
\ No newline at end of file
diff --git a/Excalidraw/demo_théorème_du_début_2.excalidraw.svg b/Excalidraw/demo_théorème_du_début_2.excalidraw.svg
new file mode 100644
index 00000000..ed7b428f
--- /dev/null
+++ b/Excalidraw/demo_théorème_du_début_2.excalidraw.svg
@@ -0,0 +1,2 @@
+
\ No newline at end of file
diff --git a/Mathématiques pour non spécialistes – suite look-and-say.md b/Mathématiques pour non spécialistes – suite look-and-say.md
index afe7581c..94182391 100644
--- a/Mathématiques pour non spécialistes – suite look-and-say.md
+++ b/Mathématiques pour non spécialistes – suite look-and-say.md
@@ -114,5 +114,28 @@ Le début arrivera toujours sur l'un de ces cycles :
### Théorème du début — Démonstration
3) Simplifier les cas (en assimiler certains)
+![[attachments/Pasted image 20260330021624.png]]
+La version de Conway :
-![[attachments/Capture d’écran 2026-03-29 à 22.50.04.png]]
\ No newline at end of file
+![[attachments/Capture d’écran 2026-03-29 à 22.50.04.png]]
+
+
+---
+## Théorème du découpage
+
+> [!definition] Découpage
+> Quand les descendants de $L$ et $R$ n'interfèrent jamais dans $LR$, c'est-à-dire :
+> $\forall n,\quad (LR)_{n} = L_{n}R_{n}$
+>
+> On dit que $LR$ se **découpe** en $L \cdot R$
+
+
+--
+## Théorème du découpage
+
+
+> [!definition] Découpage trivial
+> $[\;\;]\cdot R$ ou $L \cdot [\;\;]$
+
+> [!definition] Atome
+> - **atome** : morceau sans découpage non trivial
diff --git a/attachments/Pasted image 20260330021624.png b/attachments/Pasted image 20260330021624.png
new file mode 100644
index 00000000..a32471d7
Binary files /dev/null and b/attachments/Pasted image 20260330021624.png differ
diff --git a/désintégration audioactive.md b/désintégration audioactive.md
index a6c7f938..e912cb98 100644
--- a/désintégration audioactive.md
+++ b/désintégration audioactive.md
@@ -60,10 +60,12 @@ header-auto-numbering:
> - i Il est évident que cela arrive lorsque le dernier chiffre de $L_{n}$ est toujours différent du premier chiffre de $R_{n}$ (ou bien quand l'une des deux est vide)
> ---
> - def On appelle **trivial** un découpage du type $[\;]\cdot L$ ou $L\cdot [\;]$
+^def-decoupage
> [!definition] Atome
> Les **atomes** (ou *éléments*) sont les chaînes qui ne possèdent pas de découpage non trivial.
> - source:: [[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=181&selection=221,11,241,7&color=note|(John Horton Conway, 1987)]]
+^def-atome
- i toute chaîne est **composée** d'un certain nombre d'éléments. On dit que cette chaîne **comprends** lesdits éléments.
- source:: [[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=181&selection=243,5,260,9&color=note|(John Horton Conway, 1987)]]
@@ -211,7 +213,7 @@ header-auto-numbering:
> > - en assimilant $[3^{1}X^{1}$ et $[3^{1}(\leq 2)^{2}$ aux deux cas $[3^{1}X^{3}$, $[3^{1}X^{\leq 2}$
> > - en assimilant $[3^{2}(\leq 2)^{\leq 3}$ aux cas $[3^{2}X^{3}$, $[3^{2}X^{\leq 2}$
> > - en assimilant $[1^{3}2^{2}$ et $[1^{3}X^{1}$ au seul cas $[1^{3}$
-> > - en séparant $[1^{2}2^{\leq 3}$ en $[1^{2}X^{1}$ (qui est déjà listé), $[1^{2}2^{2}$, $[1^{2}2^{3}$
+> > - en assimilant $[1^{2}2^{\leq 3}$ à $[1^{2}X^{1}$ (qui est déjà listé) et $[1^{2}X^{\neq 1}$
> > - en séparant $[2^{1} X^{\leq 2}$ en $[2^{1}X^{2}$ et $[2^{1}X^{\neq 2}$
> > - en ajoutant $[1^{1}3^{2}$
> > Cela nous donne la liste suivante :
@@ -220,8 +222,7 @@ header-auto-numbering:
> > - $[1^{2}X^{1}$
> > - $[1^{1}2^{2}$
> > - $[1^{1}3^{2}$
-> > - $[1^{2}2^{2}$
-> > - $[1^{2}2^{3}$
+> > - $[1^{2}X^{\neq 1}$
> > - $[1^{3}$
> > - $[2^{1}X^{2}$
> > - $[2^{1}X^{\neq 2}$
@@ -234,7 +235,21 @@ header-auto-numbering:
> >
> > Cela nous permet d'atteindre le schéma original de Conway :
> > ![[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=186&rect=12,345,377,408|schéma original de Conway p.186]]
+^theoreme-debut
+> [!proposition]+ théorème du découpage
+> Un chaîne de $\geq 2$ jour $LR$ se découpe en $L \cdot R$ seulement dans ces cas :
+>
+> | L | R |
+> | --------- | --------------------------------------------------------------------------------------------- |
+> | $n]$ | $[m$ |
+> | $2]$ | $[1^1X^1$ ou $[1^{3}$ ou $[3^{1}X^{\neq 3}$ ou $[n^{1}$ |
+> | $\neq 2]$ | $[2^{2} 1^{1}X^{1}$ ou $[2^{2}1^{3}$ ou $[2^{2}3^{1}X\neq 3$ ou $[2^{2}n^{(0 \text{ ou } 1)}$ |
+> avec $n \geq 4$ et $m \leq 3$
+> ou bien quand l'un des deux est vide ($L = [\;\;]$ ou $R = [\;\;]$)
+> > [!démonstration]- Démonstration
+> > Cela suit directement du [[désintégration audioactive#^theoreme-debut|téorème du début]] appliqué à $R$, et du fait que le dernier chiffre de $L$ est constant
+ ^theoreme-decoupage
## Tableau des éléments