The four paragraphs prove something else

Which top is the top found the last premise the crossover law needed. The walls of a thermograph above the crossover are held up by the options with the two largest mean values — exact on all 23,586 heights, where a ranking by temperature is wrong on a fifth of them — and it closed by saying the proof was now four paragraphs away:

What is left is to write the geometry: two walls, each the envelope of a couple of means-less-tax lines, meeting where the law says they meet. It is four paragraphs and this anchor has spent three rungs earning the right to attempt them.

The four paragraphs are writable. They are correct. And the theorem they prove is not the crossover law.

The slip is one sentence wide. The wall above the crossover is held by a top-two option by mean is a fact about which option attains the envelope. That option’s wall is its mean less the tax is a fact about what shape that option’s wall has, and it is true only above the option’s own temperature — below that the option is still a live fight and its wall still bends.

The wall that is straight, and why it does not count #

Half the proposed construction is free, and looking at why is the fastest way to see what the other half is up against.

The positions this ladder works on are of the form $\{n \mid A\}$ — a number for Left, an answer for Right — because that is the shape a sente position has. So the left wall is the number’s own wall shifted by the tax, which is a straight line from the bottom of the diagram to the top, and no argument is needed.

That the two walls are asymmetric is not an accident of the pool; it is what makes the pool a pool of sente positions. The whole difficulty is on one side, and a construction that treats the two walls alike has not noticed.

Why an option’s wall bends at all is the fact the whole page turns on, and it takes one paragraph. A thermograph is built by taxing: at ambient temperature $t$ every move costs $t$, so an option contributes its own wall shifted down by $t$. While $t$ is below the option’s own temperature, the option is still a position somebody would fight over, and its wall is still tracking the outcome of that fight — it bends. Once $t$ passes the option’s temperature the fight has been taxed out of existence and nothing is left but the long-run value, so the wall settles onto the straight line through its mean. A wall is straight exactly where its option has stopped being interesting.

That is why mean was the right ranking and temperature was not, which is the rung below’s finding read forward. High on a diagram every option contributes its mean less the tax, so the largest mean wins outright and the ranking by mean is exact. It is exact for the same reason the wall is straight there, and the region in which both hold is the region above every option’s temperature — not above the crossover.

The wall that straightens, and where #

The right wall becomes straight exactly at the answer’s temperature, on nineteen of the twenty-three positions, to the eighth. That is not a coincidence and it is the mechanism: the right wall is the answer’s left wall shifted up by the tax, so it is straight precisely when the answer’s is, and the answer’s is straight precisely above the answer’s own temperature.

So the construction’s second claim has a region, and the region is above the answer’s temperature — a quantity belonging to the answer.

The crossover is a quantity belonging to the position.

The band the law lives in #

Those two heights are not the same, and the gap between them is exactly where the crossover law has its content.

On eleven of the nineteen positions whose right wall straightens at all, the crossover sits strictly below the height at which it does — by as much as two and a quarter. So at the crossover, and for some way above it, the right wall is not a straight mean-less-tax line and the construction does not describe it.

That is the whole objection, and it is not a technicality about small positions. The crossover law says where a player stops answering a fight, and the answer to why there has to be an argument about the diagram at that height. At that height the diagram is bent.

On four of the twenty-three the right wall never straightens anywhere inside the position’s own thermograph. Those are exactly the positions whose answer is hotter than they are — $\{8 \mid \{6 \mid 0\}\}$ is worth a temperature of 2 and its answer is worth 3 — so the diagram closes below the height where the answer’s own fight would have been taxed away.

A construction with no region to be true in is not a construction that needs care at its edges.

Those four are also the sharpest reading of what the proposal got wrong, because they are not marginal cases. An answer hotter than the position it answers is the ordinary situation in a sente position that is worth calling sente at all: the point of answering is that the fight one is answering is bigger than the one being left. Sente is a fact about the rest of the board is where that is established, and it means the four exceptions are not an awkward corner of the pool — they are the shape the anchor exists to describe, and they are exactly where the proposed geometry has nothing to say.

What the lines do prove #

The four paragraphs are still worth writing, because they prove something, and what they prove is worth knowing.

Where the two straight mean-less-tax lines cross is $(M_L - m_R)/2$ — half the difference between the largest Left mean and the smallest Right mean — and on the nineteen positions where both walls do straighten, that is exactly the temperature.

It is never the crossover — not on one position of the twenty-three, and not nearly. On $\{20 \mid \{14 \mid \{8 \mid 0\}\}\}$ the lines meet at $11/2$ and the crossover is 3; on $\{24 \mid \{18 \mid \{12 \mid \{6 \mid 0\}\}\}\}$ they meet at $45/8$ and the crossover is again 3. The two quantities are not close and were never going to be, because one is where the diagram closes and the other is where a player changes their mind, and those are different events at different heights.

So the geometry is a proof of the mean–temperature identity: the temperature of a position is half the gap between the extreme option means, once every option’s own fight has been taxed out. That is a real theorem, it is the reason what is at stake can define a temperature as the height where two walls meet, and this anchor already had it.

And it fails on exactly the four positions whose walls never straighten, which is the same failure in a different currency: with the answer hotter than the position, the extreme means are not the quantities the diagram closes at.

What the proof would actually need #

The negative here is precise enough to say what the missing argument is about, which is the point of writing the geometry out.

An argument for the crossover law has to work at a height where the right wall is bent, and a bent wall is a wall attained by an option that is still fighting. So the object it needs is not a mean-less-tax line but the answer’s own thermograph, read at a height inside the answer’s temperature — and the law’s own statement, $t - \min(t_{\text{answer}}, t)/2$, is a statement about exactly those two temperatures.

Read that way the law is well posed and the proposed proof was pointing away from it. The $\min$ in the law is the tell: it distinguishes the case where the answer is colder than the position from the case where it is hotter, and the second of those is precisely the case where the right wall never straightens. The law already knows about the four exceptions; the geometry does not.

That reframes what the anchor’s three previous rungs bought. The premises an induction would need checked that the law survives translation, heating and cooling and holds five levels deep; the two numbers at the top established that the crossover depends on two temperatures and nothing below them; and the rung below found which options hold the walls. All three are about the region above the crossover, and every one of them is true. The law is about what happens at the crossover, and the ladder has been accumulating facts about the wrong side of it.

Why this was not visible three rungs ago #

A ladder that has spent three rungs assembling premises and finds them pointing at the wrong region is worth asking about, and the answer is a specific one rather than carelessness.

Every measurement this anchor has made was made above the crossover, because that is where the law’s premise was stated and because it is where the quantities are well behaved. The premises an induction would need checked translation, heating and cooling — all of which are operations that move a whole diagram, so a premise holding above the crossover before holds above it after. The two numbers at the top grouped positions by their two temperatures and found the crossover a function of them, which is a statement about the crossover’s value and not about the diagram near it. And the rung below measured which option holds each wall, at 23,586 heights, every one of them above the crossover.

So the region below the crossover has never been measured on this anchor at all, and the region between the crossover and the answer’s temperature — the band this page is about — has been measured only in the sense that the wall-holder census included it and reported which option was winning there. That census is still correct. What it does not say, and was never asked, is what shape the winning option’s wall had.

The proposal inherited the region from the measurements rather than from the argument. Above the crossover was the phrase every previous rung ended with, so the geometry was written for that region, and the geometry’s own requirement — above every option’s temperature — is a different and higher place. Nothing checked that the two coincided because nothing had ever needed both at once.

What the solver computed, and how #

Twenty-three positions of the form $\{n \mid A\}$ with $A$ a fight — the eleven the anchor’s own pool holds, and twelve more of the same shape, added so that the counts are not eleven.

For each, the thermograph is computed to a tax of forty by the ordinary recursion, and its two walls are sampled every eighth of a move. The crossover is found as it is throughout this anchor: the position is played against a coupon stack at every ambient temperature on a grid, and the crossover is the highest ambient at which the player still answers the fight rather than taking a coupon. The answer’s temperature is the largest temperature among the Right options, and the extreme means are the largest Left-option mean and the smallest Right-option mean, each read off that option’s own thermograph.

The straightening height of a wall is found by scanning upward: the lowest height from which the wall agrees with its mean-less-tax line, to the last eighth, all the way to the temperature. A wall that never agrees is recorded as never straightening rather than as agreeing somewhere high up — which is the distinction the four exceptions turn on, and a scan that stopped at the first agreement would have missed them.

Three things are asserted rather than reported. Every left wall must be straight throughout, since the asymmetry between the two walls is the finding rather than an observation. Every right wall that straightens must do so at its answer’s temperature, since that is the mechanism the page offers. And the walls that never straighten must belong exactly to the positions whose answer is hotter than they are.

Where the model stops #

Twenty-three positions of one shape. Every one is $\{n \mid A\}$ with a number on the left, which is what makes the left wall free and the asymmetry total. A position with a genuine fight on both sides would have two bent walls and the construction would fail on both — worse for the proposal, not better — but it has not been measured, and the anchor’s law is stated for this shape.

A grid of eighths. The straightening height is read off a sampled wall, so a wall that departs from its line by less than the sampling can see between grid points would be recorded as straight. The deviations found are large — up to two and an eighth — so nothing here is near the resolution, but the height at which straightening begins is only ever accurate to an eighth.

And the crossover itself is a measured quantity, found by playing against a coupon stack at quarter-move steps rather than derived. Every rung of this anchor rests on that, and it means the crossover sits strictly below the straightening height is a statement to a quarter of a move on one side and an eighth on the other.

Normal play throughout, and a thermograph is a normal-play object: the tax that builds it is cooling, and cooling is defined by the recursion this convention makes well-founded.

And the figures cannot show the one thing this page is about, which is a bent wall. Six tables of heights describe a diagram, and the object — a right wall curving away from its own asymptote below the answer’s temperature, with the crossover marked underneath the curve — is a picture. Reading a thermograph draws one and is the page a reader should have open beside this one; what neither of them draws is the bend and the crossover in the same frame, which is the figure this anchor still owes.

Where the ladder goes next #

The sente anchor has eleven rungs: what sente is, that double sente is a band, what the reverse costs, the two cases, the wall behind them, half of the smaller temperature, what the halving is a function of, the two numbers at the top, what an induction would need, which options hold the walls, and now what the geometry those buy actually proves.

The rung above is the bent wall itself. Everything this ladder has established is about heights where the answer has frozen, and the law is about a height where it has not — so the missing object is the right wall’s shape inside the answer’s temperature, which is the answer’s own thermograph read below its own top. That is a smaller and better-posed question than the one this page failed to answer: for a single position, compute the right wall at every height between the crossover and the answer’s temperature, and ask what function of the two temperatures describes it. The law says the crossover is $t - \min(t_{\text{answer}}, t)/2$, so a wall description on that band would give the law by setting the two walls equal, and it would give it on the four exceptions too, which the mean-difference construction cannot.

Two neighbours are worth the trip. Half of the smaller temperature is where the law’s halving was found and where the $\min$ first appeared, and it is worth reading beside a page whose finding is that the $\min$ was carrying the exceptions all along. And temperatures do not add is the standing warning about arithmetic on temperatures, which is what the mean-difference identity looks like until it is noticed that it holds only where every fight has been taxed away.