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I used quite a lot of short forcing chains in my solving path; I expect there are equivalent "fish" for most of them. In some cases I kept the forcing chains fairly short, for clarity, and then started them again to make further eliminations.

Cages cannot contain consecutive numbers, even if they aren’t adjacent. There is no other condition about non-consecutive numbers.

Prelims

a) R4C56 = {15/24}
b) R67C8 = {18/27/36}, no 4,5,9
c) R67C9 = {29/38/47}, no 1,5,6
d) 11(3) cage at R1C1 = {137/146}, no 2,5,8,9
e) 12(3) cage at R1C8 = {138/147/246}, no 5,9
f) 12(3) cage at R2C1 = {138/147/246}, no 5,9
g) 11(3) cage at R5C1 = {137/146}, no 2,5,8,9
h) 13(3) cage at R6C1 = {139/148/157/247}, no 6
i) 14(3) cage at R8C4 = {149/158/248/257}, no 3,6
j) 18(3) cage at R8C7 = {279/369/468}, no 1,5
k) 13(3) cage at R8C9 = {139/148/157/247}, no 6
l) 13(3) cage at R9C4 = {139/148/157/247}, no 6
m) 23(4) cage at R1C4 = {2579}
n) 17(4) cage at R1C5 = {1358}
o) 25(5) cage at R6C6 = {13579}

1. 11(3) cage at R1C1 = {137/146}, 1 locked for R1 and N1
1a. 12(3) cage at R2C1 = {246} (only remaining combination), locked for N1
1b. Naked triple {137} in 11(3) cage, locked for R1 and N1

2. Naked pair {59}, in R23C3, locked for C3, N1 and 23(4) cage at R1C4 -> R12C4 = [27], R3C2 = 8

3. Naked pair {58} in R1C56, locked for R1 and N1
3a. Naked pair {13} in R2C56, locked for R2 and N2

4. Naked pair {46} in R1C89, locked for N3 -> R1C7 = 9
4a. R1C89 = {46} = 10 -> R2C9 = 2, clean-up: no 9 in R67C9
4b. Naked pair {58} in R2C78, locked for R2 and N3 -> R23C3 = [95]
4c. R3C1 = 2 (hidden single in N1)

5. 11(3) cage at R5C1 = {137/146}, 1 locked for R5 and N4

6. 13(3) cage at R6C1 = {139/148/157}, 1 locked for C1 and N7

7. 45 rule on N9 3 innies R7C789 = 14 = {158/257} (cannot be {248} because no even numbers in R7C7) -> R7C7 = 5, R2C78 = [85], clean-up: no 7 in R6C9
7a. R7C7 = 5 -> R7C89 = [18/27], R6C8 = {78}, R6C9 = {34}
7b. Killer pair 1,2 in R7C8 and 13(3) cage at R8C9, locked for N9, clean-up: no 7 in 18(3) cage at R8C7
7c. 2 in N9 only in R79C8, locked for C8

8. 18(3) cage at R8C7 = {369/468}
8a. 8,9 only in R8C8 -> R8C8 = {89}
8b. 18(3) cage at R8C7 = {369/468}, 6 locked for C7

9. 13(3) cage at R8C9 = {139/148/247}
9a. 2 of {247} must be in R9C8 -> no 7 in R9C8
9b. 7 in N9 only in R789C9, locked for C9
9c. 9 of {139} must be in R89C9 (R89C9 cannot be {13} which clashes with R3C9), no 9 in R9C8

10. Naked quad {1379} in R2678C6, locked for C6, clean-up: no 5 in R4C5
10a. Naked quad {1379} in R2678C6, 7 locked for 25(5) cage at R6C6, no 7 in R6C7

11. 14(3) cage at R8C4 = {158/248/257} (cannot be {149} which clashes with 13(3) cage at R9C4), no 9

12. 13(3) cage at R9C4 = {148/157/247} (cannot be {139} which clashes with 25(5) cage at R6C6, ALS block), no 3,9

13. 6 in N8 only in R7C45, locked for R7

14. 14(3) cage at R8C4 (step 11) = {158/248/257}, 13(3) cage at R9C4 (step 12) = {148/157/247}
14a. All cells of each of these cages “see” each other -> combined cage 14(3) + 13(3) = {124578}, 1 locked for N8
14b. 25(5) cage at R6C6 = {13579}, 1 locked for R6

15. Double hidden killer pair 1,2 in combined cage 14(3) at R8C4 + 13(3) cage at R9C4 (step 14a) = {124578}, 13(3) cage at R8C9 (step 9) = {139/148/247} and R8C123 + R9C2 for R89, combined cage contains both of 1,2, 13(3) cage contains one of 1,2 -> R8C123 + R9C2 must contain one of 1,2
15a. Hidden killer pair 1,2 in R7C123 and R8C123 + R9C2 for N7, R8C123 + R9C2 contains one of 1,2 -> R7C123 must contain one of 1,2
15b. Killer pair 1,2 in R7C123 and R7C8, locked for R7
15c. 2 in N8 only in R8C5 + R9C56, CPE no 2 in R9C3

16. 14(3) cage at R8C4 (step 11) = {158/248/257}
16a. 1 or 5 of {158} must be in R8C5, 2 of {248/257} must be in R8C5 -> R8C5 = {125}

17. 13(3) cage at R9C4 = {148/157/247}
17a. 7 of {157/247} must be in R9C5 -> no 2,5 in R9C5
17b. 5 of {157} must be in R9C6 -> no 5 in R9C4

18. Combined cage 14(3) at R8C4 + 13(3) cage at R9C4 (step 14a) = {124578}, 7 locked for R9

19. 5 in C9 only in R45C9
19a. Hidden killer triple 7,8,9 in R45C9, R7C9 and 13(3) cage at R8C9 for C9, R45C9 cannot contain more than one of 8,9, R7C9 = {78}, 13(3) cage cannot contain more than one of 7,8,9 -> R45C9 = {58/59}, 13(3) cage must contain one of 7,8,9 in C9, no 8 in R9C8

20. R1C9 = 6 (hidden single in C9), R1C8 = 4

21. 13(3) cage at R8C9 (step 9) = {139/247} (cannot be {148} which clashes with R67C9), no 8
21a. 7 of {247} must be in R8C9 -> no 4 in R8C9

22. 45 rule on N9 2 outies R6C89 = 11 = [74/83], R67C9 = [38/47] -> R6C8 = R7C9
22a. 8 in N9 only in R7C9 + R8C8 -> 8 in R68C8, locked for C8

23. Consider combinations for 13(3) cage at R8C9 (step 21) = {139/247}
13(3) cage = {139} => R7C8 = 2, R7C7 = 7, naked pair {13} in R39C8, locked for C8
or 13(3) cage = {247} => R7C8 = 1
-> no 1 in R4C8
23a. 1 in N6 only in R46C7, locked for C7

[Looking at the same combinations a different way]
24. Consider combinations for 13(3) cage at R8C9 (step 21) = {139/247}
13(3) cage = {139} => R7C8 = 2, R7C7 = 7 => R3C7 = 7 (hidden single in R3)
or 13(3) cage = {247} => R89C7 = {36}, locked for C7 => R3C7 = 7
-> R3C7 = 7
24a. Naked quad {1234} in R456C7 + R6C9, locked for N6

25. 14(3) cage at R8C4 (step 11) = {158/248/257}, 13(3) cage at R8C9 (step 21) = {139/247}
25a. Consider combinations for 13(3) cage at R9C4 (step 17) = {148/157/247}
13(3) cage = {148/247}, 4 locked for R9 and N8
or 13(3) cage = {157} => R78C6 = {39}, R6C67 = [71], R6C8 = 8, R7C8 = 1, 13(3) cage at R8C9 = {247} => R9C9 = 4
-> no 4 in R9C1237
25b. 14(3) cage at R8C4 (step 11) = {158/248/257}
25c. R9C3 = {78}, no 8 in R8C4

26. Combined cage 14(3) at R8C4 + 13(3) cage at R9C4 (step 14a) = {124578}, 4 locked for N8, 8 locked for R9
26a. 4 in R7 only in R7C123, locked for N7

27. Consider combinations for 13(3) cage at R6C1 (step 6) = {139/148/157}
27a. 13(3) cage = {139/157}, killer pair 3,7 in R1C1 + 13(3) cage, locked for C1 => naked pair {46} in R25C1, locked for C1
or 13(3) cage = {148}, locked for C1 => R2C1 = 6, naked pair {37} in R15C1, locked for C1
-> no 3,4,6,7 in R49C1

28. Consider placements for R9C3
28a. R9C3 = 7 => R8C45 = 7 = [52]
or R9C4 = 8 => R9C5 = 7 (hidden single in R9), R78C6 = {39}, R6C67 = [71], R6C8 = 8, R7C8 = 1, R9C8 = 2 (hidden single in N9) => R8C5 = 2 (hidden single in N8)
-> R8C5 = 2, clean-up: no 4 in R4C6
28b. R8C5 = 2 -> R8C4 + R9C3 = 12 = [48/57]
28c. 1 in N8 only in 13(3) cage in R9C4, locked for R9
28d. 2 in R6 only in R6C23, locked for N4

29. 13(3) cage at R8C9 (step 21) = {139/247}
29a. R9C8 = {23} -> no 3 in R89C9
29b. 1 of {139} must be in R8C9 -> no 9 in R8C9

30. 13(3) cage at R9C4 (step 17) = {148/157}
30a. Killer triple 4,5,9 in R9C1, 13(3) cage and R9C9, locked for C9
[Alternatively R9C278 = {236} (hidden triple in R9).]

31. Consider combinations for 13(3) cage at R9C4 (step 30) = {148/157}
13(3) cage = {148} => R9C9 = 9, R8C8 = 8, R7C9 = 7
or 13(3) cage = {157}
-> no 7 in R7C56

[Looking at the same combinations a different way]
32. Consider combinations for 13(3) cage at R9C4 (step 30) = {148/157}
13(3) cage = {148} => R8C6 = 7 (hidden single in N8)
or 13(3) cage = {157} => R6C6 = 7 (hidden single in C6), R6C8 = 8, R7C8 = 1, R8C9 = 7
-> 7 in R8C69, locked for R8
32a. 13(3) cage = {148} => R8C6 = 7 (hidden single in N8), R8C9 = 1, R7C8 = 2, R6C8 = 7
or 13(3) cage = {157} => R6C6 = 7 (hidden single in C6)
-> 7 in R6C68, locked for R6

33. 13(3) cage at R6C1 (step 6) = {139/148/157}
33a. Consider combinations for 13(3) cage at R9C4 (step 30) = {148/157}
13(3) cage = {148} => R9C3 = 7, R5C5 = 7 (hidden single in C5) => 11(3) cage at R5C1 = {146} => naked pair {46} in R25C1, locked for C1 => 13(3) cage at R6C1 = {139}
or 13(3) cage = {157} => R9C1 = 9, R9C3 = 8, R9C9 = 4, R6C9 = 3, R6C8 = 8 (step 22) => 13(3) cage at R6C1 = {157}
-> 13(3) cage at R6C1 = {139/157}, no 4,8
[Cracked, the rest is fairly straightforward.]

34. R25C1 = {46} (hidden pair in C1)
34a. 11(3) cage at R5C1 = {146} (only remaining combination, cannot be {137} because R5C1 only contains 4,6), locked for R5 and N4
34b. R4C8 = 6 (hidden single in N6)
34c. R3C6 = 6 (hidden single in C6)

35. R9C6 = 4 (hidden single in C6), R8C4 = 5, R9C3 = 7 (cage sum), R9C9 = 9, R8C8 = 8, R78C9 = [71]
35a. Naked triple {369} in R8C123, locked for R8 and N7 -> R9C2 = 2

and the rest is naked singles.

Maybe I ought to go a bit higher because of the number of short forcing chains I used, but I'll stick to a rating of Hard 1.5 for my walkthrough.

a contraction move is significantly quicker than using a forcing chain. That would require the elimination of 8 from R9C8 first, and possibly other eliminations.