Thanks Ed for your comments and corrections.Prelims
a) R34C8 = {69/78}
b) R67C2 = {39/48/57}, no 1,2,6
c) 24(3) cage at R1C1 = {789}
d) 9(3) cage at R6C8 = {126/135/234}, no 7,8,9
e) 21(3) cage at R7C3 = {489/579/678}, no 1,2,3
f) 8(3) cage at R8C7 = {125/134}
g) 19(3) cage at R8C9 = {289/379/469/478/568}, no 1
h) 30(4) cage at R8C5 = {6789}
1. 24(3) cage at R1C1 = {789}, locked for N1
1a. Max R12C3 = 11 -> min R1C4 = 2
2. 30(4) cage at R8C5 = {6789}, locked for N8
2a. 21(3) cage at R7C3 = {489/579} (cannot be {678} because R8C4 only contains 4,5), no 6, 9 locked for C3 and N7, clean-up: no 3 in R6C2
2b. R8C4 = {45} -> no 4,5 in R78C3
3. 19(5) cage at R3C1= {12349/12358/12367/12457/13456}, 1 locked for C1
4. 19(5) cage at R8C1 = {12358/12367/12457/13456}, 1 locked for N7
4a. Caged X-Wing for 1 in 19(5) cage and 8(3) cage at R8C7, no other 1 in R89
5. 45 rule on R12 1 outie R3C7 = 1 innie R2C2 + 2, no 1,2,9 in R3C7
6. 45 rule on R89 1 outie R7C3 = 1 innie R8C8 + 7 -> R7C3 = 9, R8C8 = 2, R67C8 = 7 = {16/34}, no 5
6a. Min R89C7 = 4 -> max R9C6 = 4
7. 9 in N9 only in 19(3) cage at R8C9 = {379/469}, no 5,8
7a. 8 in N9 only in R7C79, locked for R7, clean-up: no 4 in R6C2
8. 45 rule on C12 1 innie R5C2 = 1 outie R9C3 + 4, R5C2 = {56789}, R9C3 = {12345}
9. 45 rule on C89 1 outie R1C7 = 1 innie R5C8 + 3, R1C7 = {46789}, R5C8 = {13456}
10. 45 rule on R5 2 innies R5C19 = 10 = {19/28/37/46}, no 5
11. 45 rule on N7 3 remaining innies R7C12 + R8C3 = 17 = {278/368/458/467}
11a. 2,6 of {278/467} must be in R7C1 -> no 7 in R7C1
11b. 3 of {368} must be in R7C2 -> no 3 in R7C1
12. Hidden killer pair 6,7 in R7C789 and 19(3) cage at R8C9 for R7, 19(3) cage (step 7) contains one of 6,7 -> R7C789 must contain one of 6,7
12a. Hidden killer pair 6,7 in R7C123 and R7C789 for R7, R7C789 contains one of 6,7 -> R7C12 must contain one of 6,7
12b. R7C12 + R8C3 (step 11) = {278/368/467} (cannot be {458} which doesn’t contain 6 or 7), no 5, clean-up: no 7 in R6C2
12c. 2,6 only in R7C1 -> R7C1 = {26}
13. 45 rule on N7 2 remaining innies R7C12 = 1 outie R8C4 + 5
13a. R8C4 = {45} -> R7C12 = 9,10 -> R7C12 + R8C4 = [274/634/645], CPE no 4 in R7C456 + R8C12
13b. R7C456 = {123/125/135} (cannot be {235} which clashes with R7C12 + R8C4), 1 locked for R7, N8 and 29(6) cage at R6C3, no 1 in R6C34, clean-up: no 6 in R6C8 (step 6)
13c. 1 in N9 only in R89C7, locked for C7
13d. 4 in N8 only in R8C4 + R9C6, CPE no 4 in R8C7
13e. 5 in N8 only in R7C456 + R8C4, CPE no 5 in R6C4
14. R7C8 = {346}, 19(3) cage at R8C9 (step 7) = {379/469} -> combined cage R7C8 + 19(3) = 3{469}/{46}{379}, 3 locked for N9
15. 8(3) cage at R8C7 = {125/134}
15a. 2,3 only in R9C6 -> R9C6 = {23}
16. R8C4 = 4 (hidden single in N8), R8C3 = 8 (cage sum), R7C12 (step 13) = 9 = [27/63], no 4, clean-up: no 8 in R6C2
16a. 4 in N7 only in R9C123, locked for R9
17. Naked pair {15} in R89C7, locked for C7 and N9, R9C6 = 2 (cage sum)
17a. Naked triple {135} in R7C456, locked for R7 and 29(6) cage at R6C3, no 3,5 in R6C34 -> R7C2 = 7, R7C1 = 2 (hidden single in R7), R6C2 = 5, clean-up: no 6 in R5C2, no 1,3 in R9C3 (both step 8), no 8 in R5C9 (step 10), no 4 in R6C8 (step 6)
17b. Naked triple {468} in R7C789, locked for N9
17c. 7 in N1 only in R12C1, locked for C1, clean-up: no 3 in R5C9 (step 10)
18. 1 in N7 only in R89C2, locked for C2
18a. 12(3) cage at R2C2 = {246} (only remaining combination), locked for C2
18b. Naked pair {13} in R89C2, locked for N7
18c. 6 in N7 only in R89C1, locked for C1, clean-up: no 4 in R5C9 (step 10)
[I missed clean-up after steps 17 and 18a, using step 5, which would have eliminated 3 from R2C2 and 5 from R3C7 and later given naked triple {678} in R1C7 + R3C78, locked for N3, leading to a much simpler ending. OOPS! I should also have used step 5 clean-up after step 19 to eliminate 5 from R2C2.
This note has been expanded, after I worked through my steps again while looking at Ed's comments.]19. R7C456 = {135} (step 18a) -> 29(6) cage at R6C3 = {134579} (only remaining combination) -> R7C7 = 4, R6C34 = [79], R7C8 = 6, R6C8 = 1 (step 6), R7C9 = 8, clean-up: no 9 in R1C7 (step 9), no 9 in R34C8, no 9 in R5C1 (step 10)
19a. Naked pair {78} in R34C8, locked for C8
19b. 7 in N9 only in R89C9, locked for C9, clean-up: no 3 in R5C1 (step 10)
20. 16(3) cage at R6C5 = {268} (only remaining combination), locked for R6
21. 8 in C9 only in 25(5) cage at R3C9 = {24568} (only remaining combination, cannot be {12589} because R6C9 only contains 3,4, cannot be {13489} which clashes with 19(3) cage at R8C9) -> R6C9 = 4, R345C9 = {256}, locked for C9, R6C1 = 3, clean-up: no 7 in R1C7 (step 9), no 1 in R5C1 (step 10)
21a. 9 in N6 only in R45C7, locked for C7
22. 2,3 in C1 only in 19(5) cage at R3C1 = {12349/12358}
22a. R5C1 = {48} -> no 4,8 in R34C1
22b. 8 in N4 only in R5C12, locked for R5
23. 45 rule on N3 4 innies R2C7 + R3C789 = 20 = {2378/2567}
23a. 2 of {2378} must be in R3C9, 5 of {2567} must be in R3C9 -> R3C9 = {25}
23b. 2 of {2567} must be in R2C7 -> no 6 in R2C7
24. 6 in C9 only in R45C9, locked for N6
24a. 6 in R6 only in 16(3) cage at R6C5, locked for N5
25. 5 in C9 only in R34C9
25a. 45 rule on R1234 4(2+2) innies R34C19 = 17, R34C9 = {25/56} = 7,11 -> R34C1 = 6,10 = [51/19]
25b. Only permutations for R34C19 are [19]{25} (cannot be [51][25] which doesn’t total 17) -> R34C1 = [19], R5C12 = [48], R34C9 = {25}, R5C9 = 6, R1C2 = 9
25c. R9C3 = 4 (hidden single in N7)
25d. 9 in N3 only in R2C89, locked for R2
25e. R5C7 = 9 (hidden single in R5)
25f. 7 in N6 only in R4C78, locked for R4
26. 13(3) cage at R1C3 = {238/256}, no 7
26a. Hidden killer pair 3,5 in 13(3) cage and R3C3 for C3, 13(3) cage contains one of 3,5, which must be in C3 -> R3C3 = {35}, no 3,5 in R1C4
27. 13(3) cage at R4C3 = {148/238/346} (cannot be {256} which clashes with R4C2), no 5
27a. 1,2 of {148/238} must be in R4C3 -> no 1,2 in R4C45
27b. 4 in R4 only in R4C56, CPE no 4 in R3C5
[I think this CPE has been there for a while.]
28. 15(3) cage at R2C6 = {267/348/357} (cannot be {168} which clashes with R1C7, cannot be {258} which clashes with R6C7, cannot be {456} because 4,5 only in R2C6)
28a. 4,5 of {348/357} must be in R2C6, 7 of {267} must be in R2C6 (R23C7 cannot be {27} because R23C7 + R3C89 = {27}[85] totals more than 20) -> R2C6 = {457}
29. Consider combinations for 13(3) cage at R1C3 (step 27) = {238/256}
13(3) cage = {238} => R12C3 = {23}, locked for C3 => R5C3 = 1
or 13(3) cage = {256} => R3C3 = 3 => 3 in R4 only in 13(3) cage at R4C3 (step 27) = {238/346}, no 1
-> no 1 in R4C2
29a. Naked pair {26} in R4C23, locked for R4 and N4 -> R34C9 = [25], R5C8 = 3, R1C7 = 6 (step 9)
29b. Naked pair {78} in R4C78, locked for R4 and N6 -> R4C45 = [34], R4C3 = 6 (cage sum), R4C6 = 1
30. 13(3) cage at R1C3 (step 26) = {238} (only remaining combination) -> R1C4 = 8, R12C3 = {23}, locked for N1 -> R3C3 = 5, R12C1 = [78]
31. R3C3 = 5, R4C6 = 1, 9 in R3 only in R3C56 -> 31(6) cage at R3C3 = {135679} (only remaining combination) -> R4C7 = 7, R3C4 = 6, R3C56 = {39}, locked for R3 and N3, R23C7 = [38], R2C6 = 4 (cage sum)
and the rest is naked singles.