How Much Investiture is Lost When Compressing Feruchemical Attributes?

[Trusk'our]

For example, if I were to have access to Feruchemical chromium and I say “well, considering that Fortune is a normal attribute that people have that is just so weak it isn't noticeable, I probably need to tap it at like 50 times the normal amount”, how much time would I need to spend storing at 50% my normal capacity in order to gain one second of time tapping at that speed? I know that there likely isn’t a straight answer that can be given right now, but I’m the kind of guy who is super frustrated by not knowing the numerical value of what my magic can do, or at the very least get a general idea of what needs to happen to make what I want to do make work.

[Karger]

Well sadly the answer will very between Feruchemists based on experience, aptitude and situation.  We don't actually know much about Fortune itself.  I have a theory about it here

but that is just some guesswork I came up with based on some vague hints.  Brandon has been coy about canonizing anything.  I do believe that coppermind storage can be used at a high degree of efficiency(something like 90% although I can't find the WoB atm).  So in order to gain 50 times the normal amount of fortune for one second you could probably accomplish that in a few minutes.  Unfortunately the effects would probably not show themselves in any significant way although you would catch the attention of every shard in the system and maybe of the spren or gods if you were on Roshar and Nalthis respectively.

[CryoZenith]

It's probably likely that because of the nature of how they work, copperminds are much more efficient and less lossy than other types of metalminds, and aren't a good barometer for the rest of them.

[therunner]

We have a general answer here (but quite an old one, however I assume the basic principle 'more compression -> greater loss' still holds)

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Sporkify

This is more towards the whole physics stuff, but is Feruchemy really balanced? If it gives diminishing returns, wouldn't this end up as a net loss of power?

Brandon Sanderson

It doesn't diminish. Or, well, it does—but only if you compound it. You get 1 for 1 back, but compounding the power requires an expenditure of the power itself. For instance, if you are weak for one hour, you can gain the lost strength for one hour. But that's not really that much strength. After all, you probably weren't as weak as zero people during that time. So if you want to be as strong as two men, you couldn't do it for a full hour. You'd have to spend some energy to compound, then spend the compounded energy itself.

In more mathematical terms, let's say you spend one hour at 50% strength. You could then spend one hour at 150% strength, or perhaps 25 min at 200% strength, or maybe 10min at 250% strength. Each increment is harder, and therefore 'strains' you more and burns your energy more quickly. And since most Feruchemists don't store at 50% strength, but instead at something like 80% strength (it feels like much more when they do it, but you can't really push the body to that much forced weakness without risking death) you can burn through a few day's strength in a very short time if you aren't careful.

Footnote: This question was asked when fueling Feruchemy with Allomancy had only been seen in Rashek. As such, the term compounding is used purely to reference tapping at a higher rate than can be stored.
Hero of Ages Q&A - Time Waster's Guide (Oct. 15, 2008)

Couple years later he answered a question based on this, so we could hope it still holds.

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Lance Alvein (paraphrased)

Does the loss during the withdrawal of large amounts of attribute depend on the rate of original storage?

Brandon Sanderson (paraphrased)

No.

Hal-Con 2012 (Oct. 30, 2012)

So based on the WoB it is not additive, i.e. divide the time by 50, but multiplicative which will be bad.

If you store x seconds at 50% capacity, then you could have x seconds at 150%, but only x/2-something at 200% and x/4-something at 250%. So ignoring the loss for the moment, it seems that tapping and compressing the attribute is exponentially inefficient going 1/(2^n) with n being the difference in percentage divided by storing factor (e.g. (250-150)/50) .
if you would want to get to 50x the normal amount of attribute, you need to tap for 5000%, so the n here would be n = (5000-150)/50 = 97. So to get 1 second at 50 times the normal level, you would need 2^97 seconds of storing at 50%. That is roughly 31*10^21 years. Sadly that is a bit too long for any practical purposes, and we have still ignored the inefficiency factor.

Of course it is possible that the math is not as bad, and as you go up it stops being so punishing. However as far as I know, this is the only numbers we have. And generally it seems that feats like tapping at such high compressions factors are accessible only to compounders, who would not be limited by time (up to a point of course).

EDIT: Heh, realized that I have actually not answered the question exactly, so this is only covers how the attribute*time compressions seems to work (or have worked). The loss itself (the something factor) is not something we have raw numbers for, however based on his comments, it is something that also grows with more compression, so it will be increasingly relevant at higher compression factors.