Murgu Table2To3 Framework

Structured Modular Analysis of the Collatz Conjecture

Positive Integers Domain Only

Abstract

The Murgu Table2To3 Framework uses modulo-6 partitioning and double-linear inverse functions to organize the Collatz dynamics. It highlights clear structure through LET1, LET2, LDN classes and Marker USA Murgu Arrows.

1. Domain of the Collatz Conjecture

The classic Collatz Conjecture is formally defined only for Positive Integers (ℕ⁺).

Negative integers produce multiple independent cycles, so the standard conjecture does not apply there.

2. Mod-6 Partitioning of Odd Numbers

ClassFormmod 6Role
LET11 + 6i1Active upward nodes
LDN3 + 6k3Logical Dead Nodes / Closures
LET25 + 6j5Active upward nodes

3. Collatz Functional Divergence Functions

C.E.-1: \((2(2k+2)(1+6i) - 1) = 3Q_i\)

C.E.-2: \((2(2l+1)(5+6j) - 1) = 3Q_j\)

These generate the Marker USA Murgu Arrows in the Table2To3 coordinate system.

4. LDN Rules

  1. LDNs (≡ 3 mod 6) have no upward inverse connections.
  2. Trajectories from LDNs reach 1 without hitting another LDN.
  3. Trajectories from LET nodes do not hit LDNs before 1.

5. Murgu Collatz Unicity

The inverse functions preserve unicity within LET1 and LET2 classes. Combined with LDN closures and the Infinity_Per_6 1D array, this supports structured convergence to the 4→2→1 cycle.

6. MCVR – Independent Conjecture

Extension to negative numbers or other domains creates a separate problem (MCVR) with different behavior.