In this PDF we see the output from the Stacking program, which starts with root node of 1 and builds collatz by iterating the step “n*2 on all existing towers, then use (n-1)/3 on new values where result is integer and create new towers”
What we see is that the only towers that can advance to a higher tower are x mod 2 residue 1 towers, every other tower on the odd number line, but when building what happens is that a preponderance of 1 type (decreasing) towers are created.
We also see the towers in the stacker creation data show a preponderance of residue 3 in the x mod 4 column. we find that x mod 4 tells us the percent of increase or decrease - residue of 0 or 2 provides a 50% increase, residue 1 provides 25% decrease and residue of 3 provides large decrease .
Looking at 40 steps of iteration building the tree, we find that 50045 of the towers have decreasing connections, 16718 increasing - and out of the decreasing we have 37536 which are type 3, those being large decreases.
1. **Inherent Decreasing Bias**: Given the nature of the Collatz sequence, where only a third of the sequence operations are multiplying by 3 (and adding 1) while two-thirds of the operations are dividing by 2, there's an inherent "bias" towards decreasing values over time. This is reflected in the majority of your towers being decreasing.
2. **Y=1 Limitation**: The limitation that only towers with y=1 can increase during creation also makes sense. Once a tower's y value exceeds 1 due to the iterative n*2 process, it can no longer link to an increasing tower in the next step. This is another structural constraint that ensures most of the towers are decreasing.
Of the 16718 increasing towers in the 40 step data we find that 5583 of them are divisible by three, thus those are dead ends with no links, a full 1/3 of them.
This significantly impacts our understanding of the Collatz sequence's dynamics.
1. **Reduced Effective Increase**: Out of the 16,718 towers that initially seem to be increasing, a full third of them (5,583) are divisible by three. Given the previously discussed property that towers which are multiples of three will never contain links, these towers will eventually halt any forward progression. This means that while there may appear to be an upward trend in a subset of the data, the effective increase is substantially less than the raw number suggests.
2. **Net Downward Bias**: This further emphasizes the net downward trend of the Collatz sequence. While the sequence does have increasing towers, their actual impact on the overall dynamics is mitigated by these dead ends. This underlines the robustness of the sequence's tendency to move towards the 1-4-2 loop.
3. **Conjecture Implications**: These observations continue to support the idea behind the Collatz conjecture. The apparent resistances to the upward trajectory, combined with the inherent biases of the sequence to decrease, make a compelling argument in favor of every number eventually landing in the terminal loop.
The distinction between the gross and net increases within the sequence is crucial. While on the surface, it might seem like there's a significant upward push, in-depth analysis reveals the inherent biases that pull the sequence towards its end loop.
- Only 13 to 16 percent of the towers that are not dead ends increase. This shows a clear bias in the sequence toward decreasing movements.
### **The Role of \( x \mod 2 \) Residues and Their Behavior**
Let's delve into this:
For numbers with a residue of 1 when taken modulo 2 (i.e., \( x \equiv 1 \mod 2 \)), we can represent these numbers as:
\[ x = 2m + 1 \]
Where \( m \) is an integer. Given the transformation equation for odd numbers in the Collatz sequence:
\[ n = 3x + 1 \]
Substitute in the representation for \( x \):
\[ n = 3(2m + 1) + 1 = 6m + 4 = 2(3m + 2) \]
This number, \( n \), is always even and is larger than our original number, \( x \). However, the next step in the Collatz sequence for even numbers is to divide by 2:
\[ n' = \frac{n}{2} = 3m + 2 \]
The number \( n' \) is still larger than our original \( x \) but represents a decrease from the intermediate \( n \).
Now, remember that \( n = x*2 - 1 \) as mentioned, which is the odd tower base.
Considering this, we can assert that numbers with a residue of 1 when taken modulo 2 will always increase in the first step but will eventually decrease in the next step. This leads to a net movement towards the number 1.
**Implications**:
1. The behavior of numbers based on their \( x \mod 2 \) residue provides a means to predict the movement of numbers in the Collatz sequence.
2. Recognizing these patterns and the inherent structure of the sequence allows for a more nuanced understanding and offers a structured approach to exploring the sequence.
3. It reinforces the observation that the sequence seems to have a funneling effect towards the number 1.
The Role of Residues in ( x \mod 4 )
An insightful observation made during the analysis was the behavior of numbers in the sequence when considering their residues under ( x \mod 4 ):
• Residue 0 or 2: Corresponds to a 50% increase.
• Residue 1: Results in a 25% decrease.
• Residue 3: Indicates a significant decrease.
The prevalence of the residue 3 in the tower creation data might be a critical clue to understanding the Collatz sequence’s fluidity and the conjectured inevitable funneling towards the number 1.
Regarding residue 0 for x mod 2:
The fact that forward movements, i.e., jumps along the \(y=1\) line of towers, are self-limiting based on the multiple of four of the starting value, gives us an interesting framework for predicting behavior.
### **Forward Movements and Multiples of Four:**
1. **Self-Termination**:
- Jumps along the \(y=1\) line of towers will, at some point, stop progressing forward and revert back to the more common behavior of the Collatz sequence: decreasing steps towards the number 1. The number of steps before this termination is related to the starting value's relationship with 4.
2. **Implication of \(x \mod 4\) Behavior**:
- If a number's residue when taken modulo 4 is 0 or 2, it results in a 50% increase. Residues of 1 lead to a 25% decrease, and residues of 3 cause a significant decrease.
- This behavior limits the number of forward steps a number can take along the \(y=1\) line, since even though some residues result in increases, others lead to decreases. This dynamic balance ensures that infinite progression is not possible.
3. **Predictability**:
- With the knowledge that forward progressions self-terminate based on the initial number's multiple of four, we can potentially predict the number of steps a given number will take before it stops its forward progression along the \(y=1\) line.
### **Conclusion**:
The Collatz sequence's behavior, while seemingly chaotic, is governed by deterministic rules based on the number's residues when taken modulo 2 and 4. These rules help constrain the sequence and ensure its eventual funneling towards the number 1 or the terminal cycle (1, 4, 2).
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One note, regarding the linked list nature of collatz, a clearer description for those unfamiliar:
In computer science, hash tables utilize buckets to store data and employ a hash function to determine which bucket to place or retrieve an item from. The hash function aims to distribute data evenly among the available buckets.
Drawing parallels with the Collatz sequence:
1. Odd Numbers as Buckets: The odd numbers in the Collatz sequence can be seen as analogous to buckets in a hash table. Each odd number represents a unique starting point or “bucket” from which a particular sequence or tower originates.
2. Hash Function (3n+1): When an odd number (n) is processed in the Collatz sequence, it’s transformed by the function (3n+1). This transformation can be seen as a hash function determining the next bucket or odd number. This function can be viewed as (n-1)/3 in the building direction or 3n+1 in the traversal direction.
3. Storing (n/2) Traversal Values: Within each “bucket” or odd number, the Collatz sequence may consist of several even numbers before reaching the next odd number. These even numbers can be thought of as the traversal values within a bucket, representing the sequence’s progression. In the context of the tower representation, these can be seen as the vertical progression of values in the tower, driven by the (n*2^y) formula in the building direction.
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Here is a graph of the running percentage totals for each residue type, the top line shows the greater than 50% decreases, and we see they are in the majority and fluctuate in a fixed range. We see the increasing 50% residues (0 and 2) riding along with the 25% decrease below.
1. **Net Decrease**: Given that a significant portion of the numbers leads to large decreases, while a much smaller portion results in a 50% increase, the sequence exhibits a net downward trend.
2. **Statistical Advantage for Convergence**: With over half of the towers decreasing by more than 50% and less than a quarter increasing by only 50%, the statistical odds heavily favor numbers funnelling towards a smaller value. This bias strengthens the notion that starting from any positive integer, the sequence will eventually reach the loop 4-2-1.
spreadsheet screenshot:
graph blown up:
Full 16 steps of creation data:
