Murgu Collatz Unicity

Infinite Upward Connections • Unique Downward Path to Unity

After 89 years — A new mathematical lens.

Simple Explanation: Up vs Down

Upward = Finding preimages (many numbers can flow into a target) — this is where infinity appears.

Downward = Normal Collatz procedure (3n+1 if odd, divide by 2 if even) — this path is always unique.

Example 1: LET2 Number = 5

A. Upward Connections (Preimages)

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 (Unique Downward Paths)

Preimage	Path until reaching 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

Example 2: LET1 Number = 7

A. Upward Connections (Preimages)

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 (Unique Downward Paths)

Preimage	Path until reaching 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

Conclusion – The Beauty of Murgu Unicity

Multiple different numbers (LDNs, LET1, LET2) can reach the same target (like 5 or 7),
each through its own unique history.

But from that target onward, there is only one single deterministic path to Unity.

This is the elegant truth revealed by Murgu Table2To3.

Based on Ion Murgu’s Table2To3 Framework and Collatz Unicity
Corrected & Verified • June 2026