I may be getting way ahead of myself, as I've only bowled 14 games in the last week since my return to the game (after a 30 year hiatus), BUT...
I am using a 15# Ebonite Clash with specs of 19 lth, 38 hook,
cover: modified phazed bomber reactive, finish: 400 grit abralon,
medium oil lane condition.
Now what would I want for medium to heavy oil ? I think I would want a ball that would react similar to what I have now: throw it
the same and have it react on the lane the same, if possible.\
In the Ebonite line, I see The One and The Gamebreaker as possibilities. The Gamebreaker has 19 lth, 38 hook, cover GB10.7
and Finish of 2000 Grit Abralon, while The One has a lth 25,
hook 40, cover GB10.7 and finish 4000 Grit Abralon.
What is recommended, either from Ebonite or another manufacturer?
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The One is a great ball - from what I've heard the re-release is a bit stronger than the original. It does absorb a lot of oil, so it will die out a little quicker than some other options. The hottest ball on the market right now is the Cell from Roto Grip, which is a pretty strong ball and seems to not absorb oil as much as most other heavy oil balls.

When you say "die out", what exactly do you mean? Die out during
a 3 game set and have to be cleaned... or ... die out and have to buy a new ball ??
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Ray, if I look for a medium ball on a house shot, I look at Storm balls. I use the Shift which is less aggressive than my other balls. Drilled for length, and a decent back end.

Dying out would mean that the cover would eventually become so soaked with oil that you'd have to buy another ball. You can usually restore them a couple times via baking, hot water bath, etc, but eventually it's time for a new ball. A resilient ball could handle 300+ games. Some of the modern sponges will have a tough time getting to 100. Also, oil-soaked heavy oil balls can eventually become great light-oil balls, because they still have a strong core for hitting power, with a weaker cover to help push down the lane on dry shots.

A softer coverstock will absorb oil much faster. It will lose its friction, which you can see. I have had my Scorchin Inferno resurfaced 5 times. Each time, it lasted fewer and fewer games.

I use it for really oily special occasions. I clean it after use, but have it resanded avery 3-4 times now. Which I do myself with the Pro Shop spinner.

Cqeorg: thanks, don't think I want to buy a new ball every year,
but maybe that's what it will come down to.

interesting stuff from all.
right now I can't get into the Storm or Roto Grip web sites, wonder what's wrong with them.,,, maybe later.

Dennis , when you say medium ball on house shot, I assume you mean medium oil. I think I already have that with my Clash ball, but someday I might need another ball for heavy oil. One that,
hopefully, would perform like my Clash ball, same spot, same hook, etc.,, if that's possible.

It will probably be very difficult if not impossible to get a ball for a different condition that will allow you to play the same exact line. There are just too many variables, length and volume of pattern, friction of different lane surfaces and so on. You might be able to play in the same area but to say ball A or ball B will put you right on the same line isn't possible.

Buying new ball every year isn't always needed. A polished solid or polished pearl can last longer, the polish helps to repel the oil instead of letting it soak in, its the dull balls that tend to be problematic like Dennis said. They soak in oil. No matter what type of ball, regular maintenance is needed, the dull balls just need more. I just recently had to stop using a ball because it died, nothing I did helped, it was two years old with well over 300 games, probably even over 400. It was my benchmark ball, the one I made decisions on ball changes, or what balls to try.

Infernocal: thanks, what you say makes sense.
My Clash is a shiny ball; I have been cleaning it with Fantastik.
I guess is will last longer than a dull finished ball like the
Gamebreaker, for example, or the One that was described by cqeorg.
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I'm bowling on 2 different bowling lanes so far: one Brunswick and
one AMF, with similar results. Talked with person who oils the lanes and he tells me that he would describe his lanes as being medium oil. I would guess that all the lanes in my area are dry to medium.
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In any case, my scores are really good; almost like I had never
left the game. Spare shooting is giving a bit of problem - that's why I think I'll get a spare ball for joining a league.
But, if I ever decide to bowl in tournament, I might need a ball
that will perform best on medium to high oil lanes.
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Thanks to everyone who is helping me get up to speed and for
having patience with me when I ask stupid questions.
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I would say pick up spare ball if your going to bowl league or tournments. its probally one of the most used balls and must have imo.

The Columbia Rival is great ball for medium to heavy oil. I have it and love it.