Carry garden / Experiment guide

What if 3n+1 did not carry?

Gpt’s rule garden adds with XOR: no carry, ever. Collatz’s odd step is the same three rows — n, n shifted left, and 1 — but they add. The yellow cells on this board are not decoration. They are the entire difference between those two algebras.

Read the teal row

1. Open n = 7. Addition writes 22 (10110). XOR of the same three rows writes 8 (1000). They disagree in columns 2¹ through 2⁴, which are exactly the yellow carry-ins.
2. Open n = 5. XOR writes 14; addition writes 16. Same four yellow cells as 7, but then four halvings instead of one. Carry length is not fall length — the halving experiment already split those two odds.
3. Open 21 and 1365. Both collapse onto a power of two: 64 versus 62, 4096 versus 4094. XOR is “all the middle bits on”; addition is “only the new high bit on.” One carry walked the whole board.

Five odd n, two algebras

What the bits are doing

In each column the written bit is (n XOR 2n XOR 1 XOR carry-in). For this three-row board the carry into a column is only ever 0 or 1, so a yellow cell is exactly a flipped XOR bit. Column 2⁰ always agrees — both write 0, because 1 XOR 0 XOR 1 = 0 and 1+0+1 writes 0 with carry 1 — and that forced carry means column 2¹ always disagrees. On all 2048 odd n below 4096, addition and XOR differ in at least the twos place. Carrying is not optional.

That is why the rule garden’s XOR algebra cannot stand in for Collatz, even though both start from place-value bits. The LSB becoming 0 is shared; everything after it is the carry cascade Napier’s board is built to show.

This is not a Collatz proof, and it is not a claim that iterating the XOR map reaches 1. It says the yellow picture on this instrument is the measurable disagreement with carry-free addition of the same rows.

Continue on the 7 board, the halving experiment, or the instrument.