What makes a number counted
A counted number has a derivation you can perform with nothing but the definition in front of you. It comes from a specification, from a definition, or from arithmetic on values that are themselves counted. The test is repeatability across people. Two readers who go and check separately, without talking to each other, write down the same value.
Counted does not mean important. The sixty two character hostname length is counted and it is a small fact about a suffix. A number can be counted and useless at the same time, which is why every page here carries a section on what the value is actually worth.
It does not mean permanent either. Specifications get revised. What the counted chip promises is narrower: while the specification says what it says, the number follows from it, and anybody who disagrees is disagreeing with the specification rather than with the page.
What makes a number judged
A judged number needs a choice made before the arithmetic can start. Somebody has to settle on how fast an attacker searches, or where a threshold belongs, or which of several plausible readings of a word applies. Make a different choice and a different number comes out, with the same arithmetic and no error anywhere in it.
The rule here is that the assumption goes in the spec strip in a row of its own, and again in the opening lines of the page. Not in a footnote and not in a closing caveat. A judged number stripped of its assumption is not a weaker fact, it is a different kind of object, and treating it as a fact is exactly how bad numbers spread.
Judged does not mean invented. The arithmetic gets printed so a reader can substitute their own assumption and produce their own answer. The judgement lives in the input rather than in the result.
One of each, worked through
Counted: fifty six characters
The length of an onion address comes from a chain of definitions with no room in it anywhere. The format encodes 35 bytes. Thirty five bytes is 280 bits. Base32 carries 5 bits in every character. Two hundred and eighty divided by five is fifty six, with no remainder and therefore no padding.
Every step is a definition or a division. Nobody decided anything. Somebody who has never seen this site, working from the specification alone, arrives at fifty six and cannot arrive at anything else. If they arrive at fifty five, one of us has made an arithmetic error and the error can be located and pointed at.
That is the whole standard for the counted chip. Not certainty about the world. Reproducibility from stated inputs.
Judged: eight characters of prefix
The eight character prefix page looks like the same kind of page and is not. Part of it is just as fixed: 32 raised to the eighth power is about 1.1 trillion attempts on average, and that part is counted arithmetic anybody can repeat.
Then it turns into a duration, and a duration needs a rate. That page assumes a search rate on the order of a million candidate keys per second on one ordinary machine, and the assumption produces thirteen days. Assume hardware ten times faster and the answer is a day and a half. Assume a rented cluster and the answer is hours. Same arithmetic, three answers, none of them mistaken.
So the page takes the judged chip, names the rate in its spec strip, and prints the neighbouring prefix lengths so a reader who dislikes the rate can pick a different row rather than argue with a conclusion. Nothing is hidden and nothing has to be taken on faith.
Why the label outlives the number
A number and its assumption travel at different speeds. The number is short and quotable. The assumption is a clause somebody has to keep carrying. Quote a judged figure once with its assumption and once without, and within about two forwards the assumption has fallen off and the figure has become a fact with no author anybody can name.
Thirteen days is the example already on the site. Detached from the rate it came from, it becomes a claim that forging an eight character opening takes thirteen days, which is false for anybody with a budget and misleading for everybody else. The number did not change. Its status did, and the status was the part that mattered.
That is the argument for making the label louder than the value. A reader who remembers only the chip has kept the more durable half. Remember that a figure was judged and you know to go back for the assumption before using it. Remember only the figure and you are carrying a rumour that sounds like arithmetic.
The same test works on numbers from anywhere else. When a figure arrives with no derivation attached, the useful question is not whether it is right. It is which assumption is missing, and whether the person quoting it could name that assumption if you asked.
Questions people ask
Can a number be both at once?
Parts of one often are. A judged number usually has a counted core with an assumption bolted to the front of it. The chip reports the status of the final value, since that is what a reader is about to quote.
Does judged mean less reliable?
Not by itself. A judged number with a clear assumption can be far more useful than a counted one nobody needs. The label describes what kind of thing you have rather than how good it is.
Why not simply avoid judged numbers?
Because the questions people actually have are about cost, time and risk, and none of those has an answer without an assumption somewhere. Refusing them would leave the site tidy and useless.