WEBVTT

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Our cards use X for dot and Y for stripe: A has

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only a dot, B both, C only a stripe, and D neither.

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Their bits are ordered X,Y, with one for true and zero

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for false.

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These are complete semantic answers.

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Symmetric disagreement counts any changed selected bit; we will now pay attention

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to which endpoint starts with the true answer.

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A symmetric difference counts every disagreement.

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Directed loss asks a narrower question: which selected formulas are true at

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the source but false at the destination?

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Label source and destination explicitly.

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For our first comparison, give each coordinate weight one; we will change

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those weights afterward.

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From A to B, the true X coordinate remains true.

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Y becomes true, but gaining an observation is not a loss.

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The directed loss from A to B is therefore zero.

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Reverse the same endpoints.

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X remains true, but Y changes from true to false.

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Under unit weights that one lost coordinate contributes one.

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Zero in one direction and one in the reverse direction demonstrate asymmetry.

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Restore weights one for X and three for Y.

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Compare A with C.

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From A to C, X is lost and Y is gained.

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The lost X coordinate contributes one.

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The gained Y coordinate contributes zero.

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Thus the directed loss is one.

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From C to A, Y is lost and X is gained.

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Only the Y weight contributes, giving three.

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Both comparisons used the same carrier and the same weights; the change

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comes solely from reversing the source mask.

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The indicator is one exactly at a true-to-false change.

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The formula sums those weighted losses.

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It uses complete semantic profiles; an unfinished positive-certificate transcript does not

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supply every bit required for this calculation.

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Suppose a coordinate is true initially and false finally.

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At an intermediate point it is either false, so it was lost

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on the first leg, or true, so it is lost on the

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second.

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Every overall loss is counted on at least one leg.

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Multiply each coordinate inequality by its positive weight and add.

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This proves the directed triangle inequality.

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We have not assumed symmetry.

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A nonnegative function with zero self-distance and this triangle rule is

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called a quasi-pseudometric here.

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Our information order says that v precedes w when every selected positive

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true at v stays true at w.

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With positive weights, directed loss is zero exactly when no selected formula

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true at v becomes false at w.

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That is inclusion of the source support in the destination support, precisely

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the specialization direction fixed in Lecture 14.

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Two different valuations can still have zero loss in both directions when

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their signatures agree.

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A one-way zero does not force equal signatures: A to B

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already showed that distinction.

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The word pseudo allows this possible failure of point separation.

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At a disagreeing coordinate, one endpoint is true and the other false.

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The weight appears in exactly one of the two directed losses.

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At an agreeing coordinate it appears in neither.

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Adding the two directed losses therefore recovers weighted Hamming disagreement.

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For A and C our weighted losses were one and three, whose

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sum is four.

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Symmetrization recovers the number but discards which endpoint lost which coordinate.

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Compute directed loss from A to B and from B to A

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with weights one and three.

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Which direction has zero loss?

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Explain why a gained coordinate does not contribute.

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A loses nothing on the way to B.

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B loses the Y coordinate on the way to A, costing three.

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A gain is excluded by the true-at-source, false-at-destination

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condition.

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Their symmetric distance is the sum, three.

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Directed loss adds quantitative information to the support-inclusion order.

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The zero relation already recovers that order.

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Next we will use small forward balls to recover exactly the positive

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observation topology, with the same argument orientation.