Edit: Important note: Here, \(\Omega\) is some unspecified sufficiently large cardinal, not \(\omega_1\). \(^+\) is cardinal successor.
1)
\(B_0(\alpha,\beta) = \beta \cup \{0,1,\omega,\Omega,\Omega^+\}\)
\(B_{n+1}(\alpha,\beta) = \{\gamma<\Omega^{++}:\text{cof}(\gamma) \in B_n(\alpha,\beta)\} \cup \{\Psi(\gamma):\gamma \in B_n(\alpha,\beta)\cap\alpha\}\)
\(B(\alpha,\beta) = \cup_{n<\omega}B_n(\alpha,\beta)\)
\(\Psi(\alpha) = \min\{\beta<\Omega:\beta\notin B(\alpha,\beta)\}\)
2)
Let \(V[G_{\kappa \leftarrow \lambda}]\) be the Levy collapse of \(\lambda\) onto \(\kappa\), for regular \(\kappa\), \(\lambda\). Let \(B <^* A\) denote that all limit points of \(B\) are in \(A\).
\(o(A) = \sup\{o(B)+1: B <^* A; B \subseteq \sup A\}\)
\(o(\alpha) = \sup\{o(A)+1 : A \subseteq \alpha \cap \text{Reg}; \sup A = \alpha\}\)
\(\Psi_\Omega(\alpha) = \min\{\beta<\Omega:(o(\beta))^{V[G_{\Omega \leftarrow |\alpha|^+}]} \geq \alpha\}\)
3)
This bit isn't a notation. If I'm correct, the two above notations (previously undiscovered) have the potential to greatly simplify ordinal analyses above those that typically collapse a weakly compact cardinal. A greatly Mahlo cardinal can be used in its place, and if I'm correct this clocks in at a mere \(\Psi((\Omega^+)^2)\) in both notations. (The two notations aren't identical, but I think they catch up at multiples of \(\Omega^\omega\), so few worries there.)
The key is that we can collapse all ordinals (not just the ones we can express in our notation) up to \(\Omega^+\) (which are various levels of inaccessibility), hence allowing \(\Omega^+\) to actually, successfully, collapse to the least weakly Mahlo cardinal, and successive additions of \(\Omega^+\) corresponding to higher tiers of Mahloness. This goes in a different, simpler, direction compared to most OCFs that pass this point, and I hope it is accessible to the community here.
I haven't included the functions to collapse below regular cardinals as they are effectively clutter to the purpose of this post; if someone wants to make a post for them, possibly explaining this stuff better, they are welcome.
(PS: It is theoretically possible that the types of large cardinals required are inconsistent, although this seems unlikely given likely similarities to the Mitchell order. And no, I haven't proved stuff; I probably would have pushed this out much sooner if I hadn't bothered trying.)
PPS: Hnng. Can't change the title (to 'A new sort of OCF', or anything else).