Murgu Collatz Unicity

Infinite Upward • Unique Downward

1. Upward vs Downward – Simple Explanation

Upward = Preimages (many numbers can flow into a target)

Downward = Normal Collatz steps → always unique path

2. Example: LET2 = 5

A. Upward Connections

Exponent	Multiplier	Preimage	Type
1	2	3	LDN
3	8	13	LET1
5	32	53	LET2
7	128	213	LDN
9	512	853	LET1

B. How they reach 5 (Downward paths)

Preimage	Path to 5
3	3 → 10 → 5
13	13 → 40 → 20 → 10 → 5
53	53 → 160 → 80 → 40 → 20 → 10 → 5
213	213 → 640 → 320 → 160 → 80 → 40 → 20 → 10 → 5
853	853 → ... → 5

Unique continuation from 5: 5 → 16 → 8 → 4 → 2 → 1

3. Example: LET1 = 7

A. Upward Connections

k	Multiplier	Preimage	Type
0	4	9	LDN
3	16	37	LET1
6	28	65	LET2
9	40	93	LDN

B. How they reach 7 (Downward paths)

Preimage	Path to 7
9	9 → 28 → 14 → 7
37	37 → 112 → 56 → 28 → 14 → 7
65	65 → 196 → 98 → 49 → 148 → 74 → 37 → ... → 7
93	93 → 280 → 140 → 70 → 35 → ... → 7

Unique continuation from 7: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

4. Conclusion

Multiple numbers (whether LDN, LET1, or LET2) can reach the same target (5 or 7),
but each does so through its own unique path.

Once they arrive, the road ahead becomes strictly unique.

This is the essence of Murgu Collatz Unicity.

Based on Ion Murgu’s Table2To3 Framework