WEBVTT 00:00:00.250 --> 00:00:04.330 X means dot and Y means stripe on our original cards: A 00:00:04.352 --> 00:00:08.065 has only a dot, B both, C only a stripe, and D 00:00:08.185 --> 00:00:08.609 neither. 00:00:09.153 --> 00:00:13.320 A selected signature lists answers to the chosen formula slots. 00:00:14.000 --> 00:00:18.904 Keep its headers distinct from the fixed X,Y valuation annotations. 00:00:20.054 --> 00:00:24.372 Begin with the one-test menu X and all four valuations. 00:00:25.012 --> 00:00:31.585 Refining the observations means adding available tests, not declaring their outcomes true. 00:00:32.041 --> 00:00:37.109 The carrier remains fixed, and the old coordinate keeps its identity in 00:00:37.109 --> 00:00:39.242 the enlarged indexed list. 00:00:40.316 --> 00:00:43.333 Add Y as a second available test. 00:00:43.922 --> 00:00:48.763 The old three-open topology is contained in the new six-open 00:00:48.773 --> 00:00:49.558 topology. 00:00:50.110 --> 00:00:57.413 Every old certificate construction remains available, and new combinations may become expressible. 00:00:57.956 --> 00:01:02.184 Finer means more opens, not smaller state cards. 00:01:03.338 --> 00:01:08.037 The old generators are among the new generators, so any old union 00:01:08.091 --> 00:01:11.095 of finite intersections is still open. 00:01:11.545 --> 00:01:14.070 This proves topology refinement. 00:01:14.574 --> 00:01:20.240 An added formula need not create a strictly finer topology, because its 00:01:20.262 --> 00:01:22.808 region might already have been open. 00:01:23.662 --> 00:01:29.327 If two longer signatures agree, deleting the new coordinates leaves equal old 00:01:29.348 --> 00:01:30.163 signatures. 00:01:30.747 --> 00:01:35.276 Classes may split when tests are added, but cannot merge. 00:01:35.839 --> 00:01:42.665 Similarly a comparison preserving every new positive coordinate must preserve every old 00:01:42.686 --> 00:01:43.136 one. 00:01:44.047 --> 00:01:48.528 The support is the set of selected slots answered true. 00:01:48.868 --> 00:01:53.730 Our information arrow goes from a support to one containing all of 00:01:53.740 --> 00:01:53.991 it. 00:01:55.090 --> 00:02:00.398 The refined information preorder is a subrelation of the old preorder. 00:02:00.925 --> 00:02:04.061 Some one-way comparisons may disappear. 00:02:04.603 --> 00:02:09.149 The arrow direction is the same as before: from a smaller positive 00:02:09.149 --> 00:02:11.460 support to a support containing it. 00:02:12.504 --> 00:02:16.540 Symmetric distance adds weights at all differing slots. 00:02:17.190 --> 00:02:21.921 Directed loss adds them only for true-to-false changes from the 00:02:21.931 --> 00:02:22.642 source. 00:02:23.193 --> 00:02:27.468 The labelled coordinate weights are part of the chosen presentation. 00:02:28.598 --> 00:02:34.674 Distances can only increase under indexed extension when the old coordinates keep 00:02:34.674 --> 00:02:35.694 their old weights. 00:02:36.354 --> 00:02:39.706 New coordinates add nonnegative contributions. 00:02:40.320 --> 00:02:45.196 If we also lower old weights, that argument no longer proves a 00:02:45.249 --> 00:02:47.273 nondecreasing distance. 00:02:48.118 --> 00:02:53.044 These are weak inequalities: a particular pair may agree on every added 00:02:53.044 --> 00:02:55.683 coordinate and keep the same distance. 00:02:56.118 --> 00:03:00.892 A general refinement theorem does not promise that every pair becomes more 00:03:00.892 --> 00:03:04.192 distinguishable or numerically farther apart. 00:03:05.063 --> 00:03:08.796 For a new comparison, start from X, Y with unit weight on 00:03:08.796 --> 00:03:10.014 each coordinate. 00:03:10.285 --> 00:03:15.598 Add their conjunction as a third indexed observation, also with unit weight. 00:03:16.094 --> 00:03:19.611 Its truth region is B alone, already an open. 00:03:20.220 --> 00:03:24.811 The topology and its qualitative order therefore stay unchanged. 00:03:25.719 --> 00:03:30.708 Before the addition, B to A loses only Y, costing one. 00:03:31.233 --> 00:03:36.018 After the addition, B also loses the newly indexed conjunction. 00:03:36.623 --> 00:03:41.505 That is an additional coordinate contribution even though its region added no 00:03:41.538 --> 00:03:42.317 new open. 00:03:43.392 --> 00:03:48.084 With unit weight on the new coordinate, the directed loss becomes two. 00:03:48.612 --> 00:03:52.584 Topological redundancy is not numerical redundancy. 00:03:53.183 --> 00:03:58.293 The same distinction applies to literal repeated coordinates with their own weights. 00:03:59.427 --> 00:04:04.918 The topology remembers which events are open, but generally does not reconstruct 00:04:04.918 --> 00:04:07.067 the chosen list or its weights. 00:04:07.711 --> 00:04:12.062 Two presentations can have the same opens and different distances. 00:04:12.687 --> 00:04:16.854 This is a feature to label, not a contradiction to repair by 00:04:16.865 --> 00:04:18.563 changing the topology. 00:04:19.418 --> 00:04:24.659 Throughout these refinements, no state has been removed and no observed result 00:04:24.659 --> 00:04:25.303 asserted. 00:04:25.851 --> 00:04:31.219 Learning that Y is true would be a different operation, restricting possible 00:04:31.219 --> 00:04:32.282 valuations. 00:04:32.887 --> 00:04:37.451 We will reset the two operations to the same starting point next. 00:04:38.491 --> 00:04:42.424 Does adding the conjunction strictly refine the topology? 00:04:42.987 --> 00:04:45.949 Can it nevertheless change directed loss? 00:04:46.569 --> 00:04:51.560 Explain with B as source and A as destination under unit weights. 00:05:10.027 --> 00:05:15.018 The conjunction region was already open, so there is no strict topology 00:05:15.050 --> 00:05:15.890 refinement. 00:05:16.410 --> 00:05:21.557 Its indexed coordinate supplies an extra loss from B to A, increasing 00:05:21.557 --> 00:05:23.348 that distance from one to two. 00:05:24.012 --> 00:05:27.914 Qualitative sameness does not force numerical sameness.