Murgu Conjecture Vicious Redundancy (MCVR)

Historic Structural Record – Logical Image
Prepared as a neutral reading of the framework presented on h-intel.com

1. Origin

MCVR arises by examining the classical Collatz process on the negative integers and transferring the resulting dynamics back into the positive integers via the operator 3x − 1.

It is therefore the exact mirror image of the classical (3x + 1) map, expressed in the same formal language:

• G6 modular partition (LET1 / LDN / LET2)
• Murgu inverse generators
• Logical Dead Nodes as closures

2. Core Structural Change

The single sign change (+1 → −1) destroys the unicity property claimed for classical Collatz.

3. The Roots and Leaf Effects

The documents list ten primary roots, organised into two attracting cycles (leaf effects):

Short cycle
5 ↔ 7

Long cycle (closed form)
(91 → 17 → 25 → 37 → 55 → 41 → 61 → 91)

Once a trajectory enters either cycle it remains there forever. The sequence of values is finite, yet the iteration continues to infinity. These are the first fully explicit Logical Infinity Redundant Cycles exhibited inside a Collatz-style system that still uses the Murgu inverse-method and G6 coordinate language.

Infinity itself is not a root. The dynamics stay trapped in finite periodic orbits.

4. Role of the Formal Tools

• The inverse generators remain valid but reverse their operational roles relative to the classical case.
• LDNs continue to act as closures.
• The generators no longer converge to a single arrowhead; they feed the multiple roots and the two leaf-effect cycles.

5. Scientific Importance Claimed by the Framework

1. MCVR is elementary to define yet completely solved inside a known finite-root domain.
2. It supplies a transparent laboratory in which the structural ingredients of unicity can be switched off by a single sign change.
3. It isolates the phenomenon of Logical Infinity Redundant Cycles — finite sets that repeat forever under iteration.
4. Because such cycles can appear in other discrete dynamical systems, computational frameworks must introduce escape routines (cycle detection, iteration limits, structural recognition of known leaf effects).

6. Historic Status

As presented on h-intel.com (ISBN-978-1-63972-055-2), MCVR is offered as an independent conjecture and as a completion / contrast to the classical Collatz analysis carried out via Murgu Table2To3. It stands as a documented example of functional divergence with explicit infinite redundant cycling.