README.en.md

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Collatz Reverse Tree Interactive Visualizer

An interactive browser visualizer for the cycle-pruned reverse Collatz tree rooted at 1. It supports adjustable depth, raw-number and (m,k) labels, forward-trajectory inspection, and PNG export of the fitted viewport.

Correctness note: An earlier version of this README incorrectly claimed that the reverse subtrees rooted at 5 and 32 were isomorphic, described the recurrence below as having a 21-step cycle, and gave an incorrect (m,k) formula for an odd predecessor. Those claims are corrected here and covered by automated tests.

Table of Contents

• What the project does
  • Features
• Mathematical facts used by the visualizer
  • Odd-predecessor and forking condition
  • Multiples of 3
  • Correct (m,k) reverse rules
• Correction: the subtrees rooted at 5 and 32 are not isomorphic
• Correction: largest-prime-factor recurrence and the 233 cycle
• Usage
• Automated verification
• Repository structure
• Limitations
• References
• Research-integrity note
• Related reading

What the project does

Collatz reverse tree.html builds a breadth-first reverse tree from 1 using the ordinary, unaccelerated Collatz map

[ C(x)= \begin{cases} 3x+1, & x\text{ odd},\ x/2, & x\text{ even}. \end{cases} ]

For a current value n, the possible reverse predecessors are:

1. 2n, which is always valid;
2. (n-1)/3, when it is a positive odd integer.

The visualizer deliberately omits the reverse edge from 4 to 1. Without this exception, the reverse structure contains the familiar cycle 1 → 4 → 2 → 1 and is not a rooted tree. The page therefore displays a cycle-pruned rooted tree, not the complete reverse directed graph.

Features

• Generate the cycle-pruned reverse tree to a selected depth.
• Switch labels between n and (m,k), where [ m=\lfloor n/6\rfloor,\qquad k=n\bmod 6. ]
• Click a node to display its actual forward Collatz trajectory back to 1.
• Highlight the nodes and edges in that trajectory.
• Export the graph after fitting it into the current canvas viewport.

Mathematical facts used by the visualizer

Odd-predecessor and forking condition

An odd reverse predecessor exists exactly when

[ \frac{n-1}{3}=2q+1 ]

for some non-negative integer q. Rearranging gives

[ n=6q+4, ]

so, in the complete reverse graph,

[ n\text{ has an odd predecessor}\iff n\equiv4\pmod 6. ]

Every positive integer also has the even predecessor 2n. Therefore a node has two reverse predecessors exactly when n ≡ 4 (mod 6).

There is one display-specific exception: at n=4, the odd predecessor is 1, and that edge is omitted to break the root cycle. Thus node 4 has only one displayed child in this cycle-pruned tree.

Multiples of 3

If n=3q, then

[ \frac{n-1}{3}=q-\frac13 ]

is not an integer. A multiple of 3 therefore has no odd reverse predecessor.

Correct (m,k) reverse rules

Write

[ n=6m+k,\qquad 0\le k<6. ]

For the always-valid even predecessor,

[ 2n=12m+2k, ]

so its coordinates are

[ \left(2m+\left\lfloor\frac{k}{3}\right\rfloor,;2k\bmod6\right). ]

An odd predecessor exists only for k=4. In that case,

[ \frac{n-1}{3}=2m+1. ]

Its (m,k) coordinates must be computed from the resulting integer:

[ \left( \left\lfloor\frac{2m+1}{6}\right\rfloor, (2m+1)\bmod6 \right). ]

Equivalently, if m=3q+r with r∈{0,1,2}, the odd predecessor has coordinates

[ (q,2r+1). ]

For example, 16=6·2+4 has odd predecessor 5, whose coordinates are (0,5).

Correction: the subtrees rooted at 5 and 32 are not isomorphic

The earlier README inferred infinite structural isomorphism from a finite-depth visual similarity. That inference was false.

Apply the same reverse-branch word EEEOEEOEO, where E(x)=2x and O(x)=(x-1)/3 when valid:

5  → 10 → 20 → 40 → 13 → 26 → 52  → 17  → 34  → 11
32 → 64 → 128 → 256 → 85 → 170 → 340 → 113 → 226 → 75

The corresponding values 11 and 75 are not congruent modulo 6:

11 ≡ 5 (mod 6)
75 ≡ 3 (mod 6)

After one more even reverse step, they become 22 and 150:

22  ≡ 4 (mod 6)  → has an odd predecessor and forks
150 ≡ 0 (mod 6)  → has no odd predecessor

Accordingly, the level counts of the two rooted reverse trees first differ at depth 11:

Depth from subtree root	T5 nodes	T32 nodes
9	9	9
10	12	12
11	15	14
12	19	17

This counterexample disproves the former isomorphism claim. It also illustrates why finite visual agreement and modulo-6 observations are insufficient to establish an infinite tree isomorphism.

Correction: largest-prime-factor recurrence and the 233 cycle

The associated investigation considered the recurrence

[ F(n)=10\operatorname{LPF}(n)+(\operatorname{LPF}(n)\bmod10), ]

where LPF(n) is the largest prime factor of n.

Under this exact definition, the orbit starting at 233 is a 26-state cycle, not a 21-step cycle:

233 → 2333 → 23333 → 233333 → 6611 → 6011 → 60111
→ 66799 → 9977 → 9077 → 3133 → 2411 → 24111 → 477
→ 533 → 411 → 1377 → 177 → 599 → 5999 → 8577 → 9533
→ 95333 → 136199 → 194577 → 8211 → 233

In particular, 9533 is prime, so

[ F(9533)=10·9533+3=95333, ]

not 233.

This repository verifies the cycle above, but it does not claim or prove that every starting integer enters this cycle. Earlier unsupported statements about experiments up to 10^9 and pathway probabilities have been removed because the corresponding data and reproducible analysis were not present in the repository.

Usage

1. Download or clone the repository.
2. Open Collatz reverse tree.html in a modern browser with internet access. The page currently loads vis-network from a version-pinned CDN.
3. Enter a depth and click Generate / Refresh Tree.
4. Toggle (m,k) labels if desired.
5. Click a node to inspect and highlight its forward trajectory to 1.
6. Click Export PNG (Fitted View) to save the graph as currently fitted into the canvas.

The depth input is limited to 30. With the current cycle-pruned generator, depth 8 has 17 nodes, depth 12 has 47 nodes, and depth 27 has 1,748 nodes.

Automated verification

The mathematical core is in collatz-math.js, which is shared by the browser page and the Node.js tests.

Run:

node --test tests/collatz-math.test.js

The tests check:

• every generated reverse edge maps back under the forward Collatz rule;
• the special root-cycle pruning convention;
• (m,k) round trips and the corrected odd-predecessor example;
• known node counts at selected depths;
• the explicit counterexample to T5 ≅ T32;
• the complete 26-state cycle under the stated largest-prime-factor recurrence.

Repository structure

├── Collatz reverse tree.html   # Interactive visualizer
├── collatz-math.js             # Shared mathematical core
├── tests/
│   └── collatz-math.test.js    # Automated mathematical checks
└── README.md

Limitations

• The Collatz conjecture remains unproved; this visualizer is not a proof.
• The visualizer shows a finite, cycle-pruned tree and cannot justify claims about infinite subtree equivalence from appearance alone.
• JavaScript Number arithmetic is exact only for integers up to 2^53-1; the UI depth cap keeps the current root-1 tree within that range, but any future extension should preserve explicit safe-integer checks.
• PNG export captures the fitted canvas viewport; it is not an arbitrarily scalable full-graph renderer.
• vis-network is loaded from the internet, so the page is not yet fully offline.

References

• Lagarias, J. C. (1985). The 3x + 1 problem and its generalizations. American Mathematical Monthly, 92(1), 3–23.
• Andrei, S., & Kudlek, M. Some results on the Collatz problem.

Research-integrity note

The purpose of these corrections is to distinguish proven facts, computational observations, disproved conjectures, and open questions. Finding a counterexample to an earlier conjecture is part of the mathematical process and is more informative than retaining a claim contradicted by computation.

Related reading

More projects, write-ups, and experiments live on my personal site: keng0nion.github.io.