Process journal of learning by Attila Vajda

csillam-ének szövés

🌲🌊 Egy álom-toposz erdőtengerben
a Kis Fényfonó minden patakba
ugyanazt a csillam-éneket szőtte.

A Bölcs Bagoly suttogta:
„Ha a holdritmus és az ösvények jól illenek egymáshoz,
minden levélnek pontosan egy otthona lesz.”

Így senki sem tévedt el;
minden út egyetlen tisztásra vezetett,
és az erdő rendje láthatatlan dallamként őrizte a világot. 🦉✨

:
:
:
:

α↦β? εδώ α→α μόνο. 🧩

Lean4:

🐚:
∅ → seen ⊆ {0,…,n−1}
0 ↦ p[0] ↦ p²(0) ↦ ⋯
Accept iff |seen|=n ∧ cycle closes at 0. 🔄

∘ ⊢ ⇒ ⇔ ∀ ∃ λ ↦ ⟨⟩ ∩ ∪ ⊆ ∅ ✓

α β γ, ∘, ιδ
`写像・
全単射・
単射・
全射・
巡回・
証明・湖
・環
・道
・木・
海
🌲
🌊

Yes
— it maps onto the project's heart.
Your fable ("every leaf has exactly one home,
every path leads to one clearing")
is exactly bijectivity:

a permutation where nothing is lost and nothing collides.

That's precisely what the library proves.
The core result is the Müller–Cohen–Matthews
permutation theorem on F_{2^n}:

the map

  x ↦ L_m(x)·x^e   

      (with L_m the linearized trace   

          `∑_{i<m} x^{2^i})`

is a bijection when m is odd,

1<m<n, gcd(m,n)=1 —    
and this drives the proof   
 that Kasami power functions x^d are APN.`    


    Injective via h₁ : g∘f = id
    Surjective via h₂ : f∘g = id
    Bijective = Injective ∧ Surjective ✓

🐚 cycle-walk
(0 ↦ p[0] ↦ p²(0) ↦ …,
accept iff |seen|=n and it closes at 0)

is the finite/computational cousin:
on a finite set,

"visits everything once and returns home" ⟺ permutation.

In the field setting we instead use
the id-composition route above rather
than cycle enumeration.

same "aha"
(the moonlit order = a bijection),
two idioms — algebraic
(compose-to-identity) vs combinatorial (single full cycle).