Here's the best I can do. These can be bought at the Guild Shop for Grade. The price varies, but most of them are at least 8,000 Grade. I left the Japanese in there for the name so you can find them.
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Level Jewel
Each time you replay, your levels and stats will be carried over.
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Arte Jewel
Each time you replay, your artes and mastery will be carried over.
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Style Jewel
Each time you replay, your styles and abilities will be carried over.
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Gald Jewel
Each time you replay, all Gald in your possession will be carried over.
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Grade Jewel
Each time you replay, all Grade in your possession will be carried over.
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Item Jewel
Each time you replay, items (excluding cooked food) will be carried over.
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Friendship Jewel
Each time you replay, your friendship with your allies will be carried over.
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Awakening Jewel
Each time you replay, your Awakening percentage will be carried over.
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Strategy Jewel
Each time you replay, your strategies and settings will be carried over.
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Library Jewel
Each time you replay, your Library data will be carried over.
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Guild Jewel
Each time you replay, your Guild Rank will be carried over.
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Map Jewel
Each time you replay, your map exploration data will be carried over.
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Capacity Jewel 1
During a replay, you can carry up to 30 of each item.
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Capacity Jewel 2
During a replay, you can carry up to 60 of each item.
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Capacity Jewel 3
During a replay, you can carry up to 99 of each item.