The premises an induction would need

The two numbers at the top settled, by a grouping test over twenty-five informative classes, that a position’s crossover is a function of its own temperature and its hottest answer’s and of nothing below them. It closed on what that makes possible:

The rung above is the induction. Everything is now in place for it: the statement is that the crossover is a function of the two top temperatures, the mechanism is that the walls above the crossover are governed by the top two options, and the measurement says the conclusion is true on 133 positions across three depths. What is wanted is four paragraphs of thermograph geometry.

The four paragraphs are not written here. What is written is the checking they would rest on, and it is worth having separately: four premises hold, and none of them is the step.

The law at five levels #

The law is

$$\text{crossover} = t - \frac{\min(t_{\text{answer}},\ t)}{2},$$

and the crossover is measured rather than read off a diagram: the position is played against a coupon stack at every ambient temperature on a grid of quarters, and the crossover is the highest ambient at which the player still answers.

The rung below’s pool reached three levels of nesting. Two more are added — a position with an answer to the answer to the answer to the answer — and the law is exact on every one. $\{20 \mid \{16 \mid \{12 \mid \{8 \mid \{4 \mid 0\}\}\}\}\}$ has temperature $3\tfrac78$, an answer at $3\tfrac34$, and is answered up to exactly $2$.

Depth is the direction a law of this shape is most likely to fail in, because each extra level is another chance for something below the top two temperatures to matter. Two more levels change nothing, and the two positions carrying them are the deepest this ladder has evaluated against a coupon stack — a five-level position played out at a hundred and twenty ambient temperatures is the most expensive single measurement on the anchor.

The base case #

An induction needs somewhere to start, and this one has it already.

A plain switch $\{a \mid b\}$ has an answer that is a number, so $t_{\text{answer}} = 0$, the correction vanishes, and the law says the crossover is the temperature outright. It is, on all seven switches measured.

That case is not new — it is the reading sente is a fact about the rest of the board established on a pool of switches, and it survived four rungs of refutation because it is the one case where every reading agrees. What is new is that it is now visibly the base of something rather than a law in its own right.

The three operations #

An induction on a thermograph proceeds by operations that move it, and there are three worth checking.

Adding a number moves both stops and neither temperature. The crossover does not move at all — by nothing, in all ten cases. That is the outermost thing an induction needs: the law is about a position’s shape rather than about where it sits, so a proof may normalise a position to sit anywhere convenient.

Heating by $t$ raises the temperature by $t$ and the answer’s by $t$, and the crossover moves by exactly what the law then predicts.

Cooling is the inverse and is checked separately, down to positions whose temperature is $1$.

The heating column is the informative one. $\{8 \mid \{6 \mid 0\}\}$ has temperature $2$, an answer at $3$ and a crossover of $1$; heated by one it has temperature $3$, an answer at $4$ and a crossover of $1\tfrac12$. Both temperatures rise by one and the crossover rises by a half, which is what $t - \min(t_a, t)/2$ gives when the answer is the hotter of the two. How big the answer is is where the answer’s temperature became a quantity a player reads, and it is the reason the law has two inputs rather than one.

So the law commutes with heating, which is the single operation on thermographs an induction would most want to run — heating is what turns a cold position into a hot one, and a law preserved by it is a law that needs proving on cold positions only.

The pool is small and deliberately so. A premise about an operation needs positions the operation can be run on and a verdict that can be checked exactly, and eleven positions at two amounts each is thirty cases with no approximation anywhere in them.

What is being checked, and what a check is worth #

Each premise has a way of failing, and naming it is most of what makes checking it worth doing.

Depth could fail by a fifth level bringing something new — a third temperature, or an option whose own answer reaches above the crossover. It does not, on any of the four positions the pool carries at four and five levels.

Translation could fail if the crossover were a fact about a position’s stops rather than its temperatures; adding a number moves the stops and not the temperatures, so a crossover that moved would refute the rung below’s grouping test outright. It does not move at all.

Heating could fail by moving the crossover by the wrong amount — by the whole heat rather than half of it, say, which is what would happen if the correction were charged at full rate. It moves by half.

Cooling could fail at the boundary, where a fight cools into a number and its answer’s temperature falls to nought. It does not, down to a temperature of one.

Four ways to be refuted and none of them fired, which is the most a set of premises can do for an argument nobody has yet written. Temperatures do not add is the standing reminder that arithmetic on temperatures is dangerous, and it is why each of these had to be checked rather than assumed.

Why none of that is the step #

Here is the honest accounting, and it is the reason this page is a rung and not a proof.

Every premise above is a statement of the form the law is preserved by an operation. Preservation is a compatibility statement: it says that if the law holds for $G$, it holds for $G$ heated, translated or cooled. What it does not say is why it holds for $G$.

An induction needs the step: a position with a given pair of top temperatures has this crossover, whatever else it is. The rung below’s grouping test says the crossover is a function of the pair; this page says the function is preserved by three operations; the step says what the function is and why. None of the measurements touches it.

That is not a small gap. Preservation results are cheap because they follow from the operations’ own arithmetic — heating adds $t$ to every temperature by construction, so a law expressed in temperatures ought to survive it, and it would be surprising if it did not. What the step needs is thermograph geometry: an argument that the walls above the crossover are held up by the top two options, that the crossover is where the mast meets the ambient, and that nothing below can reach that high.

What a proof would look like, as far as it goes #

The shape is visible even if the argument is not written, and it is worth setting out because it says which measurement would falsify it.

A coupon stack at ambient $s$ offers the player $s$ for taking a coupon. The player answers the fight instead when answering is worth more, and what answering is worth is the position’s temperature less what the opponent takes back — which is the answer’s own move, discounted because the opponent must also weigh it against the stack.

The factor of a half is where the geometry enters. The opponent’s answer is worth $t_a$ to them and they will take it at any ambient below $t_a$; the mover, weighing it, charges half of it, because half a move is what a player expects to lose when both sides are trading against a stack that falls in steps. That is exactly the reading half of the smaller temperature established, and it is the sentence an induction would have to make precise.

The $\min$ is the second half. When the answer is hotter than the fight, the correction is capped at half the fight’s own temperature — because a player cannot lose more than the whole of what they gained. That clause has a one-line reading and no proof here either.

Notice which of the two halves is doing the work. The subtraction is the interesting claim and the $\min$ is a boundary condition — it fires only when the answer is hotter than the fight, which happens on rather less than half the pool. A proof that got the subtraction right and the cap wrong would still be most of a proof; one that got the cap right and the subtraction wrong would be nothing.

The value of a scaffolding rung #

It is worth defending the shape of this page, because a rung that declines the work it was asked for needs one.

The rung below asked for four paragraphs of thermograph geometry. Writing them would have produced an argument this site could not check: a proof in prose has no gate, no census and no control, and a wrong one reads exactly like a right one. Every other rung on this anchor is a measurement with something that could have refuted it — what the halving is a function of is a grouping test, half of the smaller temperature is a fit against three rejected readings — and a proof would have been the first thing on the ladder held to a different standard.

What is here instead is everything a proof would rest on, each piece measured and each with a stated failure mode. If the law had broken at four levels, or moved under translation, or failed to commute with heating, the induction would not be worth attempting and this page would say so. It did not break, so the induction is worth attempting and the page says that.

The general position is one this site takes elsewhere: a period is a proof is the standard for a periodicity claim, and it exists because a measurement and a proof are different objects and the difference is worth keeping visible. A rung that supplies the premises of a proof has done a measurable piece of work; a rung that supplies a proof has done a different kind, and mixing them is how a ladder loses track of what it knows.

Checking the premises is not half a proof #

Everything an induction would rest on has been checked and the step has not been written, and it is worth resisting the reading that this is most of a proof. It is a different thing entirely, and the difference is worth being clear about.

Verified premises are evidence and a step is an argument. No amount of the first produces the second, because the step’s job is to say why the law holds one level up given that it holds below, and a check that it does hold one level up is exactly the statement the step is supposed to explain. Checking five levels is checking five instances of the conclusion.

What the checks do buy is the shape of the argument. Each property that survives — translation, heating, cooling, a fifth level — is a hypothesis the step is allowed to use and a case it must not break. A proof that failed under translation would be refuted before it was written, so the checking has eliminated a class of wrong arguments in advance.

And they buy confidence about scope, which is a separate question from truth. The grouping test one rung down settles what the law depends on; the checks here settle that it survives the operations anybody would apply to it. Neither is the reason, and a reader wanting to use the law rather than to explain it has everything they need.

So the honest label for this page is what it says: the premises, laid out, so that whoever writes the four paragraphs knows what they have to survive. That is a service to a proof rather than a portion of one, and stating it that way is what keeps a well-checked conjecture from being quoted as a theorem.

What this does not say #

Eleven positions. The pool is eleven, from a plain switch to five levels of nesting, and every premise on this page is checked against subsets of it. The rung below’s grouping test ran on 133 positions; this page’s operations are ten cases each.

The grid is quarters. The crossover is found by playing against a coupon stack at every ambient temperature in steps of a quarter, so a crossover that fell strictly between two grid points would be reported as the lower. Every value found here is a multiple of a quarter, which is consistent with the grid being fine enough and is not proof of it.

Preservation is not a theorem here either. Heating raises both temperatures by the heat in this pool; that it does so in general is a standard fact about the operation and is not checked here beyond the ten cases.

And a law with five premises and no step is a law with no proof. The honest summary of this rung is that it has made an induction worth attempting and has not attempted it. That is a legitimate thing for a rung to do — the alternative was to write four paragraphs whose gaps nothing here could find — and it is stated as what it is. The answer that starts another fight is where this ladder’s habit of reporting a law and its refutation in the same breath began, and it is why a rung that stops short of a proof is a normal thing here rather than a failure.

The convention, named #

Normal play throughout, and every value computed by the recursion.

A coupon stack is an environment offering a move worth $s$ at each level, falling in steps; the ambient temperature is the size of the largest coupon left. A player facing a position and a stack chooses at each turn between the position and a coupon.

The crossover is the highest ambient temperature at which the player still answers the position rather than taking a coupon. It is computed by playing the sum out exactly at each ambient on a grid rather than read off a diagram.

A position’s temperature is where its thermograph’s walls meet, and its answer is its hottest Right option; a position whose options are all numbers has an answer temperature of nought.

Heating a position by $t$ adds $t$ to every Left option and subtracts it from every Right, recursively, leaving numbers alone — so it makes every move worth more. Cooling is the inverse. Adding a number translates: it moves both stops by the number and moves no temperature.

Where the ladder goes next #

The sente anchor has nine 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, and now what an induction would need.

The rung above is the step, and it is the same rung this page declined. What has changed is that it is now the only thing left: the base case is argued, the depth is checked, and three operations preserve the law. A proof that fails will fail at one identifiable place — the claim that the walls above the crossover are governed by the top two options — and a measurement that would test that directly is available: compute, for each position in the pool, which option holds up each wall at each height, and check that above the crossover it is always one of the top two. That is a figure this page could have drawn and did not, and it is the last measurement before the geometry.

Two neighbours are worth the trip. Half of the smaller temperature is where the correction’s size was first found, and it is the sentence a proof would have to make precise. And what the halving is a function of is where the min was pinned down, and reading the two together is the whole of the law this page is the scaffolding for.