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Turns out it'd be a dissapointingly small sphere:
US annual defence budget = $800 billion
2% = $16 billion
Obsidian cost per kg = $5
Total kg in budget = 3.2 billion kg
Density of obsidian = 2.6 g/cm3 = 2600 kg/m3
Total volume of sphere = 3.2b/2600 = 1230769 m3
Volume of sphere = 4/3 π r^3
Radius = (3V/4π)^(1/3) = 66.48 m
The sphere would only stand at 133m tall, I propose we instead utilise the entire defence budget for a much more skyscraper like 490m tall orb
Nobody said it had to be a solid sphere, how else would you get it to emit the ominous hum without attuning its natural frequency by carefully designing the thickness of the obsidian layer?
ignores engineering and construction cost. but we can assume that all people involved would work for free, because its a massive honour
Obsidian Orb is really Roko's Basilisk. Allowing you to live is your payment for building it.
Well that's assuming it's completely solid and not hollow. Hollow would probably be pretty huge, although the structural rigidity might not be great. Maybe we make a giant obsidian 3D printer and print it at like 10-15% infill.
An obsidian 3d print is less crazy than you might think. It's essentially rapidly cooled lava.
Need something to hold the lava, then pressurize it to squeeze it through a nozzle that that has attached cooling units.
I think you just described a volcano.
We're in luck! They already sell that nozzle!
Simple: bucket of lava, bucket of water, repeat.
Grabs a bucket. Quick, to the lava depository!
less crazy than you might think
Using a planet for a 3d printer's nozzle still sounds pretty crazy.
You're assuming one year of budget. I take it as 2% per year and something of that size would likely take 10+ years to build out.