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Oh wow. That's a dang lot you've made me think about. I commend you for your eye for details, good sir.

That bedrock is shown to be far more significant, as you have presented in the instances above. I've never thought of that link between the ichor-made constitution and the bedrock. Your first and second quote I did know about, but I dismissed the bedrock to be nothing more than a metaphor. The third quote, in comparison, is more incriminating. I've finished the ruthless route, but definitely missed the detail of the artifact coming from the bedrock itself. Whatever that artifact was, it is likely related to ichor (also resembling the bedrock's obsidian) and had the power to produce an effect that violates the labyrinth's constitution. 

With this, let's make a theory and their implications: 

Hardened ichor is the format of the Labyrinth's constitution, and is found in the bedrock
By the gods' combined ichor the Labyrinth is created, a prison for those deemed eternally unworthy of respite. Through their ichor each god is allowed to leave a living legacy in the realm.

You can interpret this as their hardened ichor forming the contract/s that consequently made the labyrinth into existence. If Asterion's blood can harden as such, perhaps he can fashion something to exert his will upon the labyrinth. Though only being part divine, the volume of blood he'd have spilled over the years would suffice, no? He's been tortured over the years, wine and natural regeneration here and there.

Asterion of Crete, adopted son to King Minos, and every drop of his blasphemous blood is hereby sentenced to the Labyrinth.

If the constitution is indeed found at the bedrock or at the least can be interacted, accessed or affected within the labyrinth, perhaps Asterion can append/overwrite it with something of his own design.

If this is true, why hasn't Asterion tried to do this before? Perhaps the years of torture and servitude have made it so that the thought hasn't crossed his mind once to even start thinking it. He confessed himself that he was deserving and a 'wretch' as he describes it. Additionally, what if he can't because he hasn't truly contributed to the realm like the other gods?

In accordance with the hybrid's trial the Olympians staked each a drop of their ichor. One by one each deposited their power on the threadcutter's rhyton. From this shared bounty the realm is created to imprison the damned hybrid and all the Olympians deem guilty of the most reprehensible crimes against divine order.

Perhaps he needs to make the proper offering on the 'threadcutter's rhyton' to make his additions to the realm. I do not conclusively know what this thing is, but I have a morbid idea. It may be Asterion's severed head (idk if he needs to be beheaded to qualify), or something fashioned like it. I interpreted the threadcutter to be Asterion himself, and that's where it lead. Try and look up 'rhyton' and 'bull rhyton' and you'll see what I mean. 

Also, a very dark theory: In the ruthless ending where Dominikos takes Asterion's place as the Prisoner, what if it was actually Asterion exerting his power over the realm? Asterion is fixated on the repeating pattern he has seen in his life: 

In freeing a prisoner, the redeemer takes on his shackles.

What if Asterion (sub)consciously willed the realm to make Dominikos the new prisoner? After all, this on the route where Asterion did not express pity (he was consumed in outrage) on the Argos's act.

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It's definitely a team effort here, I wouldn't've started looking down this path without what you've already posited! 

Interesting idea about having Asterion be able to add his own addition to the constitution, I think a fully divine Asterion would be able to do that, yes. If we follow the theory that Asterion's birthmark coming back is a sign of his rekindling divinity because he believes in himself, I think he can build up to have this effect, and it explains why even if he understands the obsidian and bedrock, he has never done it before because it wouldn't work for his 'blasphemous blood'.
Speaking of said blasphemous blood, there may be a loophole here in this 2nd quote if he fully ascends to divinity somehow: pure divine ichor clearly can't be blasphemous, right? I'm not sure how we break where it calls out his name, but at least here's an idea that can handle the latter half of that sentence that would prevent him from leaving. 

Mmm, good point on how it specifically says rhyton. With what we know right now, the threadcutter is either Labrys, which we have the axe head of in storage (Where's the rest? What's it shaped like?), or Asterion himself, yes... maybe in a less morbid way it's the rest of Labrys or just one of his horns snapped off and since regrown? googles 'bull rhyton' Yikes, or maybe not.
Still, whatever the rhyton is, I'd agree that if Asterion can muster up more ichor than human blood and add it to the rhyton, he should be able to affect the realm in some way, and possibly amend the constitution at minimum. 

EDIT: Oh forgot the point you had on that one ruthless ending. I think it's just further reinforcing the theme of recursion, and it's also highlighting that the pelt still counts as Asterion in some ways for the realm's constitution. That's also the only ending where there's someone around for the pelt to graft on to with no Asterion around, so it might not be tied to Asterion's subconscioud but instead the laws of the realm just in general... or like, whatever the eff is going on with pelts in this game, lol, the Tapir God's nearly did the same to P.