Three groups, and three yields

The mirror was the floor found Toppling Dominoes’ yield — the share of rows producing a value nothing smaller had produced — falling toward a half rather than toward nought, and identified the reason: a row and its mirror image are the same game, so half the rows can never contribute anything new. It closed with a conjecture and with the thing that would beat it:

The conjecture the sweep suggests is that each ruleset’s yield tends to the reciprocal of its symmetry group’s order … A ruleset whose yield settles below its predicted floor would be the first genuine value collision on this ladder, and would be worth more than the conjecture.

Three rulesets fall below. And the refutation does not need any of them individually.

Hackenbush strings, Shove and Push all have no value-preserving symmetry at all — a Hackenbush string read backwards is a different game, and so is a reversed Shove or Push strip. The conjecture therefore predicts a yield of one for all three. They give 1.000, 0.531 and 0.204.

A spread of eight tenths on an identical prediction is not a conjecture that needs its constant adjusted. It is a conjecture whose independent variable is not the one that governs, and no repair of the form the group is bigger than it looks is available: Hackenbush’s yield of exactly one means it has no identifying map whatever, so its group is trivial for certain, and any map found for Shove or Push would only widen the gap by lowering their predicted floors below the one Hackenbush is pinned at.

The conjecture, scored #

The symmetries are measured rather than eyeballed. Each candidate map — the mirror, and for Clobber the colour swap — is applied to every row of the sweep and the two values compared, so none means checked and absent rather than not noticed. The colour swap comes back as a negating map on Clobber, which is worth recording: it sends a value to its negative, so it relates positions without identifying values, and it does not belong in the group the conjecture counts.

A negating map is not nothing, and it is worth saying what it does buy. If the colour swap sends every Clobber row to a row worth the negative, then the ruleset’s value set is closed under negation — which is a real structural fact and is what the values nobody’s game produces uses when it compares rulesets by which values they reach. What it does not do is reduce the count of distinct values, because a value and its negative are two values. So it belongs in an account of what a ruleset produces and not in the group the conjecture divides by, and the sweep separates the two rather than lumping them.

Push and Shove have neither map. A Push strip reversed is a genuinely different game — the coins are pushed one way and the rule is not symmetric in the direction of travel — and the sweep confirms it rather than taking it on trust.

Where the mechanism is exact #

The conjecture is not a bad guess, and the reason is that it is exactly right about the ruleset it was found on.

Every one of Toppling Dominoes’ twenty-eight colliding values is a row and its reflection. Not one is a coincidence. So the rung below’s account of its ruleset is exact, and what failed is the generalisation rather than the observation.

That distinction is worth insisting on because the two are easy to conflate when a conjecture falls. Nothing measured on the rung below has been overturned: its sweep to twelve dominoes stands, its yield still tends to a half, and the mirror is still why. What has been overturned is a sentence that never appeared in its measurements — an extrapolation from one ruleset to five, offered as a conjecture and tested here as one.

That is worth separating carefully, because the two are easy to run together. This ruleset’s floor is its reversal symmetry is true and is now confirmed at the level of individual collisions. Every ruleset’s floor is its symmetry group is false.

Where the collisions are genuine #

In Push, 55 values are carried by more than one row and all 55 have a pair no mirror relates. In Shove, 120 of 120. Clobber has 30 colliding values of which 13 are genuine.

So there are 188 colliding values across the three failing rulesets and 175 of them are unexplained by any symmetry — which is what the rung below asked for, in larger numbers than it expected.

Clobber is the mixed case and the most informative of the three. Thirteen of its thirty collisions are genuine and seventeen are mirror pairs, so its symmetry is doing real work and is not doing all of it — which is what a ruleset looks like when the conjecture is partly right. Its yield of 0.041 against a predicted half is nonetheless the furthest below of the three, so the part the symmetry explains is a small part of a large fall.

Push’s 486 strips give 99 values and Shove’s give 258, on the same strips under a different rule — so the two rulesets differ by a factor of two and a half in what they preserve, with identical positions and identical symmetry groups. .....L and .L...L are both worth 5. The second strip has an extra coin near the left end, and it changes nothing about the value. There is no map of Push sending one to the other; they simply coincide.

The value being an integer is the tell. Both strips are worth 5 — a plain number, cold, with nothing at stake — and a number is exactly what a value looks like when the position has stopped having any structure worth recording. Nothing worth fighting over is where Shove’s values were found to be cold everywhere and decided by one coin, and a ruleset whose values are numbers is a ruleset with a great deal of room for two positions to arrive at the same one. So the collisions are not scattered arbitrarily; they are concentrated where the arithmetic is coarsest.

That is a hypothesis this page does not test and the rung above should. If the genuine collisions are mostly on cold positions, the low yields of Push and Shove have a shape, and it is the same shape their whole value theory has.

What the yield is actually about #

The most useful part of the refutation is the case the conjecture gets right for the wrong reason.

Hackenbush strings have a yield of exactly one, at every size measured, and the conjecture says this is because they have no symmetry. That is not the reason. Hackenbush is a numeral is the reason: a Hackenbush string is a binary numeral, read from the ground up, and two different strings are two different numerals and therefore two different numbers. The yield is one because the map from strings to values is injective by construction, and no amount of symmetry or its absence enters.

Shove has no symmetry either and its yield is 0.531. The difference between the two is not a difference of groups; it is that a Hackenbush string’s value is a transcription of the string and a Shove strip’s value is the outcome of a computation over it — and a computation is free to send different inputs to the same output.

So the yield measures how much of a position the value throws away, which is a fact about the ruleset’s arithmetic. Symmetry is one way for information to be discarded and it is not the only way, and on three of these five rulesets it is not the main one.

What this does to the anchor #

The anchor’s business is which values a ruleset can produce and how cheaply, and it now has a cleaner statement of what a yield is.

A yield near one means the ruleset is a notation. Hackenbush strings are the case: the position and the value carry the same information, and the ruleset is a way of writing numbers down. The values nobody’s game produces is where the census across rulesets was built, and it is a census over exactly this kind of variation.

A yield falling to a symmetry’s reciprocal means the ruleset is a notation up to that symmetry. Toppling Dominoes is the case, confirmed here at the level of collisions.

And a yield falling below means the ruleset genuinely loses information. Push loses four-fifths of it by six squares: 486 strips give 99 values. That is the interesting regime and it is where three of these five rulesets live.

Those three descriptions are a scale rather than three boxes, and reading a ruleset’s place on it is a useful thing to be able to do. A ruleset near the top is one where a reader can recover the position from the value, so the value theory is a re-encoding and the interesting questions are about the encoding. A ruleset near the bottom is one where the value is a genuine summary, most of the position has been discarded, and the interesting question is what was discarded — which is how long a row a value needs’s question asked backwards.

The anchor has been measuring yields for three rungs without a reading of what a yield is. It now has one, and it is not the reading the conjecture offered.

Why the conjecture was worth making #

A refuted conjecture is worth more when it was a good one, and this one was, for a reason worth naming.

It was a mechanism rather than a curve fit. The rung below did not observe a yield near a half and reach for a half; it identified why half the rows can never contribute — each is another’s mirror — and then asked what that reasoning gives elsewhere. That is the right way to generalise a measurement, and it is what makes the refutation informative: the mechanism is exact where it was found, so what has been learned is the mechanism’s scope rather than that a number was wrong.

And it was falsifiable in the sharpest way available. The rung below said what would beat it — a ruleset below its predicted floor — and named the consequence, that such a ruleset would exhibit the first genuine value collision on the ladder. Both halves happened, and the second is the part worth keeping.

What the refutation costs the anchor is nothing it had. No page on this ladder had been quoting the conjecture; it was one rung old and stated as a conjecture. What it gains is a sharper reading of the yield, a confirmed mechanism in one ruleset, and 175 collisions nobody had counted.

That is a good trade, and it is the ordinary outcome when a mechanism is generalised honestly: the generalisation fails, the mechanism survives inside its own ruleset, and the failure points at something the original measurement could not see.

What the solver computed, and how #

Five row-shaped rulesets — Hackenbush strings, Toppling Dominoes, Clobber rows, Shove and Push — each enumerated at every length up to six, which is 486 strips for Shove and Push, 4,096 rows for Hackenbush and comparable counts elsewhere.

Each candidate symmetry is tested rather than assumed: the map is applied to every row of a probe size, both values are computed and reduced to canonical form, and the map counts as value-preserving only if they agree on every row. The same test is run for negation, which is how Clobber’s colour swap is identified as a map that relates positions without identifying values.

The yield at each size is the number of distinct values divided by the number of rows. Collisions are counted at the top size: a value carried by more than one row is a collision, and it counts as genuine when two of its rows are not reflections of each other.

Three things are asserted rather than reported. Some ruleset must fall below its predicted floor, or the conjecture stands and this page has nothing to report. At least two rulesets must have a trivial group and their yields must differ by more than a fifth, since that comparison is the refutation. And Toppling Dominoes must have no genuine collision, since the page’s contrast depends on the mechanism being exact where it was found.

Where the model stops #

Six squares, and five rulesets. The yields are still falling at the top size on four of the five, so tends to is doing work no finite sweep can discharge — what is measured is where each yield has got to, not its limit. That is enough for the refutation, which only needs two predicted-equal yields to be unequal, and it is not enough to say where any of them settles.

Only mirror and colour swap are tried. A ruleset could have a value-preserving map nobody has thought of, and finding one would raise its group’s order and lower its predicted floor. That would rescue the conjecture for that ruleset and it cannot rescue the refutation: Hackenbush and Shove would both need new symmetries, and Hackenbush’s yield of exactly one rules out any map that identifies two of its strings.

And genuine means unexplained by the mirror, which is the only map these rulesets have. A collision explained by some map not tested here would still be a collision no symmetry in the conjecture’s sense accounts for, since the conjecture is stated over the group the sweep measures.

Normal play throughout, and a value here is a canonical form under the disjunctive-sum theory, which is what makes two positions with equal values genuinely interchangeable.

And the figures cannot show a collision. Six tables of counts describe two strips being worth the same thing, and the object — .....L and .L...L drawn as strips with their coins, side by side, both labelled 5 — is a picture this ruleset’s own generator draws elsewhere. The cliff a cut invents draws Push strips; what no page draws is two of them that coincide.

Where the ladder goes next #

The realisability anchor has eight rungs: the cheapest way to show a value, the birthday as a floor, which values no game produces, whether width accounts for the excess, the entry fee that was the cap, what a rate measures, what a yield’s fall ends at, and now what a yield is a fact about.

The rung above is the 175. Every colliding value in Push and Shove has two rows nothing relates, and not one of those pairs has been looked at — the sweep counts them and stops. Asking what the pairs have in common is the natural next question and it is well posed: for each, record what the two strips differ in, and whether the difference is a coin the value never consults. The Push witness suggests one answer immediately — a coin far from the action, in a place the rule cannot reach — and if that is the general shape then Push’s low yield has a mechanism as concrete as Toppling Dominoes’ mirror, and a better one, because it would be about the rule rather than about the drawing.

Two neighbours are worth the trip. Hackenbush is a numeral is why one of these five rulesets has a yield of exactly one, and it is the page that shows the reason is arithmetic rather than symmetry. And the entry fee was the cap is where a quantity on this ladder turned out to be an artefact of the sweep rather than a property of a ruleset, and it is the standing warning this page had to clear before its own numbers could be believed.