I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (propositions about subsets of $\mathbb{Q}[0,n]^k:=\{x\in\mathbb{Q}\;|\;0\leq x\leq n\}^k$ for $n,k\in\mathbb{N}$) and the large cardinals in the subtle/$k$-SRP range.

An example of the sort of statement he considers is the following (from this document):

Definition 1.$S\subseteq\mathbb{Q}[0,n]^k$ is anemulatorof $E\subseteq\mathbb{Q}[0,n]^k$ if each element of $S^2$ is order equivalent to some member of $E^2$ (considered as subsets of $\mathbb{Q}[0,n]^{2k}$), where $x,y\in\mathbb{Q}[0,n]^k$ are said to be order equivalent if for any $1\leq i,j\leq k$, $x_i<x_j\leftrightarrow y_i<y_j$.

Definition 2.$A\subseteq\mathbb{Q}[0,n]^k$ isstableif and only if for all $p<1$, $\langle p,1,\ldots,k-1\rangle\in A\leftrightarrow\langle p,2,\ldots,k\rangle\in A$.

Example statement (called "MES" by Friedman).For all $k$, and all finite $E\subseteq\mathbb{Q}[0,k]^k$, there is a stable, $\subseteq$-maximal emulator of $E$.

Apparently MES implies $\mathrm{Con}(\mathsf{SRP})$ (where $\mathsf{SRP}$ is $\mathsf{ZFC}$+"there is a $k$-SRP cardinal" as an axiom scheme in $k$), so somehow there is a lot of strength happening in stable maximal emulators. And the relation to $k$-SRP cardinals, specifically, makes me think that emulators must be hiding a Ramsey condition under the hood, but I haven't been able to spot it yet.

**How can I see what is happening with stable maximal emulators of finite subsets of $\mathbb{Q}[0,n]^k$ ($n,k>2$) that their guaranteed existence implies the consistency of the existence of large cardinals?** [Edited to reflect JDH's correction in the comments.]

(For a reference on $k$-SRP cardinals, see Friedman, Harvey. *Subtle cardinals and linear orderings*. (English summary)
Ann. Pure Appl. Logic 107 (2001), no. 1-3, 1–34.)

"...their guaranteed existence implies the existence of large cardinals"is not right, because no arithmetic statement can imply the existence of large cardinals. You mean to imply theconsistencyof the existence of the large cardinals. $\endgroup$formulatingstatements of this kind, less with k-SRP cardinals specifically. There's clearly something that leads Friedman to choose stable maximal emulators (or various other similar objects) as good objects for bumping up consistency strength, but I have no clue what the guiding idea is. $\endgroup$Finite Functions and the Necessary Use of Large Cardinals, which Friedman published around the same time. Broadly speaking the argument there involves using a strong finitary Ramsey's theorem and compactness to build a structure that has the ability to code some basic set theory and has a special cofinal indiscernible sequence of elements. This indiscernible sequence then gives you enough reflection to show that the interpreted model of basic set theory satisfies ZFC. I would guess that the story in this paper is broadly similar. $\endgroup$4more comments