Cold — the series
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Below zero
Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.
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How hot a day gets
A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.
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How hot a real position is
Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.
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What a game actually produces
Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.
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One fight makes a board a fight
The rung below found 16 per cent of the components a played game produces to be hot, against 53 per cent of the catalogue they are drawn from, and predicted that the share of hot boards would be much larger. Taking the same play-outs and tallying at the board gives 32 per cent — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster on the same boards.
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The obstacle was the catalogue
The rung below could not measure the early game because its regions were too large for the catalogue, and asked for a bracket rather than a value. No bracket is needed: a twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was cataloguing every shape rather than sweeping the positions a board actually reaches — and the sweep says a board is hot four times in five three moves in, and cools when it breaks up.
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Eight squares, and no hotter
The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.
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The ceiling was a plateau
Three halves of a move looked like a ceiling on a Domineering region's temperature: it held at eight squares, at nine and at ten, and the rise that had been a quarter every two sizes stopped. At eleven squares four regions reach seven quarters — and they contain the hottest eight-square shapes and are hotter than them, so the extra material is not cold.
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A description, and not a detector
The rung below noticed that the count of shapes attaining the hottest temperature grew across a plateau and collapsed at the step, and proposed it as a way to read a plateau off a single size. The growth is exact — five plateaus, no exception — and the rule is impossible: five orbits precede a rise at seven squares and no rise at eight.