The Big Picture

This chapter puts the tour's pieces side by side: how they fit together, how solid each one is, and what is still open. It is not a proof. The Collatz conjecture is open.

A map of the ideas

Click any node to see what it says and how solid it is:

Known mathematics Proved here (elementary) Verified by computation Heuristic Open

Click any node to see what it says and how solid it is.

The two questions

The conjecture can fail in two ways: an orbit could loop, or it could escape to infinity. Neither has been ruled out.

Loops

Piece	Status
Negative gap ($3^S > 2^E$) gives no positive cycle	Elementary, known
No cycle with $S=5$, $E=8$	Enumerated: none of the 35 parity words that start at an odd value (91 counting every starting point)
No cycle with $S=41$, $E=65$	My own meet-in-the-middle search; already implied by the literature
No nontrivial $m$-cycles for $m \leq 91$	Known: Simons and de Weger (2005), Hercher (2023)
Any nontrivial cycle has $E/S$ extremely close to $\log_2 3$ and is enormous	Known: Eliahou (1993)
No nontrivial cycles at all	Open

See No Loops? for the details.

Escape

Piece	Status
The multiplier $3^s/2^{k-s}$ of every drop is at most $2^{-\beta(s)}$, with $\beta(s) = 1 - \{s \log_2 3\}$	Elementary: the multiplier of a drop is below 1. Exactly $2^{-\beta(s)}$ in every case checked ($n \leq 10^7$); that it always is would follow from Terras's coefficient stopping time conjecture, which is open. The number itself shrinks by a little less, because of the $+1$s: $5 \to 4$ loses 0.32 bits, and $\beta(1) = 0.415$
$v_2(m+1)$ countdown: every climb ends at a member of Set₃	Elementary, proved
$v_2(m-1)$ countdown ends every run of weak drops: in a deep drop if $v_2(m-1)$ is even, in a new climb if it is odd	Elementary
Bounces before the first deep drop $\leq (B+3)/4$	Computed for every odd $m \leq 5 \times 10^6$; over a whole orbit it fails ($m = 2919$)
Bits read faster than generated	The Finite Fuel bookkeeping; not measured, not proved
Every orbit eventually falls below its start	Open

How the pieces fit together

Every Collatz drop is a place where the halvings outpace the triplings. The carry propagation of $+1$ creates deterministic countdowns: every climb ends in a step down, and every run of weak drops ends. These are facts about the current value, not about the starting value. Natural numbers have finite binary expansions, and the Finite Fuel picture guesses that this is what stops the bounces; the only evidence offered is a count of bounces before the first deep drop, for odd starting values up to $5 \times 10^6$. Averaged over residue classes, an odd step together with the halvings that follow it multiplies a number by $3/4$ (geometric mean).

The gap between this picture and a proof is the gap between on average and for every orbit. Averages over residue classes hold just as well for the negative integers, where the same rule has cycles at $-1$, $-5$ and $-17$. Anything that closes the gap has to use the fact that a positive integer's bits run out — and that is the part nobody knows how to do.

A physics analogy

Click any row to expand the analogy. It is a mnemonic for the Finite Fuel picture, not a mechanism and not evidence; the two rates in it are that picture's bookkeeping constants, not measurements.

Speed of light c

↔

Carry propagation: 1.92 bits/bounce

+

Particle velocity v < c

↔

Orbit growth: 0.51 bits/bounce

+

Finite energy E = mc²

↔

Finite binary expansion: B bits

+

Event horizon

↔

Position B: all zeros beyond

+

Hawking radiation

↔

~0.51 new bits per bounce from growth

+

Heat death of universe

↔

Bit budget exhausted → deep drop

+

Trivial zeros of ζ(s)

↔

Cycles at negative integers

+

Summary table:

Physics	Collatz
Speed of light	Carry propagation: ~1.92 bits/bounce
Particle velocity	Orbit growth: ~0.51 bits/bounce
Finite energy ($E = mc^2$)	Finite binary expansion ($B$ bits)
Event horizon	Position $B$: all zeros beyond
Heat death	Bit budget exhausted → deep drop
Hawking radiation	The ~0.51 bits of growth per bounce
Trivial zeros of $\zeta$	Cycles at negative integers

The role of each ingredient

• $\log_2 3$ irrational → halvings and triplings never cancel exactly → the gap $2^E - 3^S$ is never zero
• Base-6 rotation → a picture of orbits as a slightly wobbly irrational rotation (Shakibaei Asli) → a step counter, which says nothing about size
• $+1$ carry propagation → deterministic countdowns → every climb ends in a forced step down (from the current value, not below the start)
• Finite binary expansion → a finite bit budget → the intuition for why positive integers might differ from $-1$

Explore further

• Affine Orbit Structure — the piecewise-linear structure of orbits (Terras, Everett)
• Bit Destruction — the $\beta(s)$ picture of drops
• Mixing Modulo Powers of Two — how destinations spread over residue classes
• The Cycle Equation: Small Cases — the first few cases worked through by hand

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