docs: unify algorithm explanation and update teaching video

show/video/check_lower35.py

@@ -75,8 +75,36 @@ def main():
     direct = {h for h in range(M)
               if all(first_output(lower+(h << 35), d) % 10 == 0 for d in terms)}
     assert analytic == direct and direct
-    assert 2 + (1 << 18)*((8190+4) % M) == 524290
-    assert 4965*4 % M == 3476
+    assert sorted(ts) == list(range(3383, 3440, 5))
+    assert 4965*3383 % M == 2995
+    assert lower == 218767 + (40302 << 18)
+    world = (lower + (2995 << 35) - coordinate_term(51069, 419124)) & MASK48
+    assert world == 15959335054446
+    # Use the film's X-first indexing and check all 25 accepted outputs.
+    terms = [(coordinate_term(51069+i%5, 419124+i//5)-coordinate_term(51069, 419124)) & MASK48
+             for i in range(25)]
+    expected = [1114693760,660285140,2105382940,47575410,1796905160,
+                2074383590,1410817530,1734889590,2009625200,440710690,
+                1664890290,537178720,1699969350,2115254790,688097620,
+                1842255790,1387847170,1302453480,1264531310,344660,
+                1757217670,204531470,2132545510,1606583470,569937090]
+    for i, term in enumerate(terms):
+        output = first_output(lower+(2995 << 35), term)
+        assert output == expected[i] and output % 10 == 0 and output < (1 << 31)-8
+    # Check every row of the selected candidate group and its two constant pieces.
+    root, base = 218767, 40206
+    masks = []
+    for i, term in enumerate(terms[1:], 1):
+        diffs = [(first_output(root+((base+16*j) << 18), term) -
+                  first_output(root+((base+16*j) << 18), 0)) % (1 << 31) for j in range(16)]
+        masks.append(sum(1 << j for j, value in enumerate(diffs) if value % 10 in (0, 8)))
+        if i == 1:
+            assert diffs == [1693075028]*8 + [411302484]*8
+    assert [masks[i-1] for i in (1,3,4,8,9)] == [0xff,0xfffc,0xfff8,0xfff0,0xffc0]
+    combined = 65535
+    for mask in masks:
+        combined &= mask
+    assert combined == 0xc0
     print(f'OK: {len(cases)*M} single-cell identities; '
           f'25-cell interval intersection equals all {M} high-bit trials ({len(direct)} candidates).')