2026-10-20 WR

Rearrange the digits in ⟨125034⟩ to meet the rules below.

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

Jump(0,4) = 2

⟨⋯ Perm(2,3) ⋯⟩

⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 420

2nd → a, 0th → b, a+b=7

⛔Avoid

⟨⋯ ᵃb ⋯⟩, ab=6+7n

#125034_v2.15

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │ 4 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 4 │   │   │ 0 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │   │ 4 │   │ 2 │ 0 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │   │ 4 │   │ 2 │ 0 │ 5 │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │   │ 4 │ 3 │ 2 │ 0 │ 5 │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 4 │ 3 │ 2 │ 0 │ 5 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2026-10-20 WR
══════════════════════

Notation: if nth -> a, then we write [nth] = a.

By ✅「⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 420」, we have

(1) [4th] = 4 or 5.

If it is 5, then we would match ⛔「⟨⋯ ᵃb ⋯⟩, ab=6+7n」, which is not allowed. Therefore, [4th] = 4:

       ┌───┬───┬───┬───┬───┬───┐
       │5th│ 4■│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │ 4 │   │   │   │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ 0 │ 3 │   │
└───┴───┴───┴───┴───┴───┘

Then, in view of ✅「Jump(0,4) = 2」, plainly we have [1st] = 0:

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 4 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 4 │   │   │ 0 │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │   │ 3 │   │
└───┴───┴───┴───┴───┴───┘

Now, to match ✅「2nd → a, 0th → b, a+b=7」, we need

(2) {[2nd], [0th]} = {2,5}.

Combining this with ✅「⟨⋯ Perm(2,3) ⋯⟩」, we finish by

       ┌───┬───┬───┬───┬───┬───┐
       │ 5■│4th│ 3■│2nd│ 1■│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 4 │   │   │ 0 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │   │ 4 │   │ 2 │ 0 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │   │ 4 │   │ 2 │ 0 │ 5 │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │   │ 4 │ 3 │ 2 │ 0 │ 5 │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 4 │ 3 │ 2 │ 0 │ 5 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.15