Directed distance: the cost of losing observations

Our cards use X for dot and Y for stripe: A has only a dot, B both, C only a stripe, and D neither. Their bits are ordered X,Y, with one for true and zero for false. These are complete semantic answers. Symmetric disagreement counts any changed selected bit; we will now pay attention to which endpoint starts with the true answer. A symmetric difference counts every disagreement. Directed loss asks a narrower question: which selected formulas are true at the source but false at the destination? Label source and destination explicitly. For our first comparison, give each coordinate weight one; we will change those weights afterward. From A to B, the true X coordinate remains true. Y becomes true, but gaining an observation is not a loss. The directed loss from A to B is therefore zero. Reverse the same endpoints. X remains true, but Y changes from true to false. Under unit weights that one lost coordinate contributes one. Zero in one direction and one in the reverse direction demonstrate asymmetry. Restore weights one for X and three for Y. Compare A with C. From A to C, X is lost and Y is gained. The lost X coordinate contributes one. The gained Y coordinate contributes zero. Thus the directed loss is one. From C to A, Y is lost and X is gained. Only the Y weight contributes, giving three. Both comparisons used the same carrier and the same weights; the change comes solely from reversing the source mask. The indicator is one exactly at a true-to-false change. The formula sums those weighted losses. It uses complete semantic profiles; an unfinished positive-certificate transcript does not supply every bit required for this calculation. Suppose a coordinate is true initially and false finally. At an intermediate point it is either false, so it was lost on the first leg, or true, so it is lost on the second. Every overall loss is counted on at least one leg. Multiply each coordinate inequality by its positive weight and add. This proves the directed triangle inequality. We have not assumed symmetry. A nonnegative function with zero self-distance and this triangle rule is called a quasi-pseudometric here. Our information order says that v precedes w when every selected positive true at v stays true at w. With positive weights, directed loss is zero exactly when no selected formula true at v becomes false at w. That is inclusion of the source support in the destination support, precisely the specialization direction fixed in Lecture 14. Two different valuations can still have zero loss in both directions when their signatures agree. A one-way zero does not force equal signatures: A to B already showed that distinction. The word pseudo allows this possible failure of point separation. At a disagreeing coordinate, one endpoint is true and the other false. The weight appears in exactly one of the two directed losses. At an agreeing coordinate it appears in neither. Adding the two directed losses therefore recovers weighted Hamming disagreement. For A and C our weighted losses were one and three, whose sum is four. Symmetrization recovers the number but discards which endpoint lost which coordinate. Compute directed loss from A to B and from B to A with weights one and three. Which direction has zero loss? Explain why a gained coordinate does not contribute. A loses nothing on the way to B. B loses the Y coordinate on the way to A, costing three. A gain is excluded by the true-at-source, false-at-destination condition. Their symmetric distance is the sum, three. Directed loss adds quantitative information to the support-inclusion order. The zero relation already recovers that order. Next we will use small forward balls to recover exactly the positive observation topology, with the same argument orientation.