I don't really want to rain on your parade here, so by all means keep it up, this is a well-written TL, but keep these kinds of things in mind.
At the beginning, I have to thank you since I'm glad to see some (constructive) critizism at last, and be sure that I'm already adapting some imperfect sections of the scenario following already written elements.
I just want to make clear that that isn't how things have worked historically. Metallurgy and materials science in general is something that spent the long millennia moving at more or less a steady pace, with any noticable speed-up being due to the discovery of specific technologies. It only really took off to any great degree in the 17th century, on a base of materials development fifteen hundred years in advance of anything the Romans had available
They can't just magic up fifteen hundred years of advances in materials technology. There's no relationship here with theoretical physics -- physics had little to do with materials advancement until the late 19th century. Up until then, it was all slow, careful trial and error broken up by the occasional major invention (whether a tool or a technique).
You are somewhat right that the need for better materials can drive the development of better materials -- that is more or less how it happened IOTL. However, you cannot ignore more than a 1000 years of development in order to get from there to here.
I have to confess that I don't read enough metallurgy (law, physics and economy are simply to interesting

), but I was wery very intrigued by
ComradeHuyles's post - if steel is possible that early in Africa, why isn't it possible in Rome? It hasn't to be developed in 10 years - but I don't think hundred or two hundred years are that implausible.
Not to mention that one of the primary uses of coal prior to the widespread adoption of steam engines in the powering of industry (which happened after the First Industrial Revolution in Britain) was as coke in the making of iron. The Romans have no need whatsoever for the use of coke in this process: They have more wood with which to make charcoal than they could ever possibly use. No big need for coal -> no big need to pump water out of bell pits -> no real drive for a non-flowing water dependent pump.
And? Did I said that Rome will discover the steam engine before being able to make appropriate metals? Until now, they have only the Pneumatic Automaton, a precursor of the atmospheric engine, itself completly inefficient and wasteful - the Romans are not going to make the jump within 10 years, but I'm not thinking it will take 1000 years.
And that's just the downstream issues of metals and machine production. There are also issues of consumer demand and industrial scale financing to work out. It's telling that relatively advanced financial systems have existed since time immemorial (including in ancient Rome -- although the governing class never did like finance much themselves), but it wasn't until the late 17th century in an increasingly economically sophisticated Britain that we finally see the invention of the bank note and the final arrival of a permanent, circulating paper currency that can expand and contract to meet the demands of the economy.
I don't think that it is hard to teach the Romans the concept of paper currency - after all they knew promissory notes, and paper currency is only an advancment of these. What they need is a method duplicate paper currency (a printing press) - let's see how fast they can develop that.
But may I rise the question to whom the issue of paper money should be permitted? State would be most logical - but is this rational from a capitalist point of view?
If you re-oriented the entire timeline of Roman history around getting them to industrialize, then made them get just right coin flips coming up to keep them safe over the course of centuries, you could have them industrialized perhaps sometime in the late 1st millennium.
First millenium...

This isn't what I wanted.
But not prior, and not without completely changing what it meant to be Roman.
Being Roman meant much more than being a bigoted conservative - it also meant to adopt innovations if a Roman thought they were going to help the Empire.
A patent law is going to encourage quite a bit of innovation. I'm also interested to see how the concept of patronage(imperial or otherwise) impacts technological development. Will future emperors or senators, or even just wealthy individuals sponsor and build further centers of learning?
They have a center of learning now, they don't need a second one. I'm thinking of letting fashion promoting technological development (which senator has the best automaton in his mansion?).
Also, how has anything changed in terms of territorial expansion(I noticed a mention of Germania)?
Germania will be as hard as OTL, but since the Varian disaster was more or less a lucky coincidence for the Germanic tribes, and not a fundamental weakness of the Roman military, I don't think history will repeat itself in this point. Without Arminius having Varus falling, by luck, into his trap (and this trap requires a fair amount of luck too), ultimate Roman domination in Germania Magna is only a matter of time.
And what's the state of Succession after Agrippa?
That's up to Agrippa who will decide it only after Romulus' death. His sons (Gaius and Lucius as OTL), Tiberius (as a relative of Romulus) and Drusus are still possible.