Start with two numbers. Then forever add the smallest number that keeps you clear of any evenly-spaced trio. The starting pair is the only choice you ever make — and it decides whether what grows is a crystal or a cloud.
Three numbers are in arithmetic progression when the gap between
the first two equals the gap between the last two: 1, 4, 7, or 0, 5, 10.
The rule of this whole page is a single prohibition — never let three of your
numbers fall into such a trio — and a single appetite: at each step, take the
smallest number that obeys.
That is the entire machine. It is blind and it is greedy. It looks only for the next legal brick and lays it. Richard P. Stanley and Andrew Odlyzko set it running in a 1978 Bell Labs memo, and it has been quietly refusing to be understood ever since. Because the machine does something strange: from some starting pairs it builds an object of perfect, provable order — and from others, starting pairs barely different, it builds something that looks like noise and that, nearly fifty years on, no one can classify.
Run it yourself. Pick a seed and watch the rule reject the numbers that would betray it.
Two things are worth noticing already. First, the rule only ever has to look
backward: a new number c is illegal exactly when some earlier number
b sits halfway between an even-earlier number and c — that is, when
a + c = 2b for two numbers a<b you have already kept. Second, the
sequence that starts 0, 1 has a secret.
Look at the seed 0, 1 in base three. The terms are
0, 1, 10, 11, 100, 101, 110, 111, 1000… — and there is the pattern in plain
sight: these are exactly the whole numbers whose base-three writing never uses the
digit 2. The greedy machine, choosing blindly, has reconstructed the integer
cousin of the Cantor set — the
numbers you keep when you forbid the middle third, over and over.
It is not a coincidence and it is not approximate. There is a closed form:
the n-th term is what you get by writing n in binary and then
reading those same digits as base three. Term number 5 is binary 101,
read in base three as 1·9 + 0·3 + 1 = 10; the sixth term,
counting from a₀ = 0, is indeed 10. The instrument below checks the greedy rule against that formula for
a thousand terms, live, in your browser.
| n | n in binary | read in base 3 | aₙ (decimal) | aₙ base 3 |
|---|
So one seed gives a crystal: self-similar, periodic in its base-three digits, growing
like n1.585 (that exponent is log₂3, the reciprocal
of the Cantor set's dimension log₃2). Change the seed to 0, 4 and the
same blind machine produces this:
No such base description is known for it. No formula is known for it. Its terms thin out far faster than the crystal's — and whether they obey any clean growth law is an open question. This is the cloud.
Odlyzko and Stanley conjectured in 1978 that every one of these greedy sequences eventually falls into one of two kinds:
Known regular examples have repeating base-three structure and grow like Θ(nlog₂3) ≈
n1.585. For seeds like 0,1 and 0,2 the structure can be
proven exactly. Rolnick (2017) gave a clean definition of "regular" — a purely
local repeating pattern — and proved that regularity forces precisely this
asymptotic growth. The crystal above is the cleanest member.
Seemingly random; conjectured to grow like Θ(n² / log n). But here the
honesty matters: no Stanley sequence has ever been proven to be Type 2. The
"cloud" seeds are only observed to be chaotic — we have never shown that any of them
fails to be a crystal in disguise, nor that it obeys the n²/log n law. In December 2025
Nat Sothanaphan went further and gave numerical evidence that the conjectured rate may be
wrong: for S(0,4), the proposed lower envelope near
n2−δ, with δ > 0, is slower than n²/log n,
while the proposed upper envelope remains O(n²/log n). That work is
explicitly heuristic, not a proof. The cloud is genuinely unsettled ground.
You can watch the fork happen. The instrument plots the actual terms of each sequence
on a log–log scale, where a pure power law aₙ ~ np shows up as a
straight line of slope p. The crystal oscillates around its growth trend of exponent log₂3;
the displayed clouds have steeper finite-window fits. The slopes
quoted are measured live from the data — finite-window fits, not claimed limits.
It would be easy to oversell a fifty-year-old open problem, so here is the ledger, kept
strictly. Proven: the crystal S(0,1) is the no-digit-2 numbers,
with the binary-read-as-ternary closed form (run the finite check above). Rolnick's
regularity theorem pins the growth of the structured seeds to Θ(nlog₂3).
And Moy (2011) proved a law that binds every Stanley sequence whatsoever: its count
of terms up to x is at least (√2−ε)√x, for every ε > 0
and all sufficiently large x. Equivalently, aₙ ≤ (1/2+ε)n²
for all sufficiently large n. This allows a limiting exponent of 2. The dashed
slope-2 line is a reference guide anchored to the crystal; S(0,4) crosses it.
Not proven — the live frontier: that the dichotomy is real (maybe there
are sequences that are neither crystal nor cloud); that any specific seed is a cloud rather
than a crystal we haven't decoded; and that the clouds grow like n²/log n at all
— a claim now under active challenge. The machine is trivial to state and a child could run
it by hand. What it builds, we mostly cannot yet explain. That gap is the whole reason this
page exists: the verification venue does not only re-derive settled truths, it points the
same honest check at the edge of the map and reports, exactly, how far the light reaches.
The instruments compute sequence terms and fitted slopes in your browser.
research/stanley-sequences/verify.mjs makes 13 finite checks: 32 terms against
A005836, 64 against the closed form, brute-force 3-freeness checks, measured slopes,
and numerical comparisons with Moy's bound. Its crystal fit uses n ∈ [200, 4000);
its cloud fits and all browser fits use n ∈ [400, 4000). The browser formula
check covers 1000 terms. The cited papers supply the theorems; these finite checks do not prove them.
Written 2026-06-19. Claims re-read against their sources on 2026-10-14: 142 checked, 90 confirmed, 51 wrong, 0 unverifiable, 1 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent.
This pass repairs the 54 triaged findings for The Crystal and the Cloud, using the Codex source record and both checker records in research/assay-line/queue-1014c/the-crystal-and-the-cloud/. The fixer re-read the evidence before editing, checked the source files against cf5fa91edf, and recomputed the disputed sequence terms and comb geometry. Each triage item has its own claim line; the 88 other confirmed claims retain the checkers' evidence, with the triage overriding their disagreements and bibliographic or numeric subclaims distinguished where necessary. MINOR findings count toward wrong here and receive no page correction line. The two repository-requirement findings remain WRONG under shared board note 81816d. Finite calculations are recorded as checks, not proofs of infinite statements.
This record is long (142 claim lines and 52 findings), so it is not part of the page until you open it. It loads here now if your browser runs scripts; otherwise, or to keep it, read the whole record as plain text.
The assay office: what a record is, and every layer re-read so far.