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Decided by exhaustion — page 6

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.
Five certificates against any two of three decide. A table of every minimal balanced family on three players, what each demands of the game, and whether the grand coalition's value covers it — the complete test for whether a stable split exists. Applied

Five weighings and the question is closed

Searching the triangle of splits can only ever fail to find a stable one, which is not the same as there being none. Weighing five families of coalitions against the whole settles the question outright — and the family that fails is the proof that nothing survives.

 The splits of any two of three decide that nobody can out-argue. The triangle of all splits of a joint gain, with the splits marked at which every player's loudest complaint against every other is matched by an equally loud complaint back. Applied

An objection one player makes to another

The core lets a coalition object to everybody at once. Narrow it to one player objecting to one other, require every such objection to be met by an equally loud one coming back, and exactly one split survives — with no dictionary order anywhere in the argument.

 The most mass 2 deviations out, with only a mean and a variance. The distribution putting as much probability as possible outside a window 2 standard deviations wide, found by searching every triple of support points, with the quadratic certificate that bounds it drawn over. Probability

The bound is the answer to a search

Chebyshev's inequality is not a clever estimate that happens to be sharp. It is the exact answer to a maximisation over all distributions with a stated mean and variance, and the polynomial that proves nothing beats it is the certificate a search of that kind always produces.

 What one input can do, over 1024 cases. A table of functions of several inputs with the largest effect any single input has on each, the bound that effect implies, and the true tail probability — computed by enumerating every input. Probability

No single input can move it far

Independence was never the hypothesis doing the work. A quantity built from many separately drawn inputs concentrates whenever changing one of them moves it only a little — and that covers quantities which are not sums of anything and have no formula at all.

 Reducing abcabc to a standard form. The gluing word abcabc rewritten step by step into one of the classification's standard forms, with the move used and the two invariants recomputed at each step. Topology

Every word driven to a normal form

The classification is usually met as a statement: two numbers name the surface. The proof is a procedure — a short list of cut-and-reglue moves that drive any gluing word to one of the standard forms, with a measure that never rises to say why the procedure stops.

 4 surfaces with an edge, and the three numbers they need. Polygons whose gluing words leave some edges unpaired, each with its Euler characteristic, its sidedness and the number of boundary circles the unpaired edges form. Topology

The third number a surface needs

Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.

 The classical centres as three weights each. A table of triangle centres with the weights on the three corners that produce each, and the determinants that decide which triples of them are collinear. Geometry

A centre is three weights

Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.

 The smallest algebra that refutes each formula. A table of formulas against the smallest finite Heyting algebra refuting each, found by searching every order on a few points, with the formulas no such algebra refutes marked. Logic

Refutable in something small

A formula that is not a theorem of the constructive system fails in some finite algebra, and the algebra can be found by search. That single property is what makes the propositional logic decidable — and the predicate version loses the property and the decidability with it.

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