And that is not a good ontology. — Metaphysician Undercover

I can only say that understanding Peirce's intention takes a lot of time in research and study. There's clearly nothing I can say to you to instill that understanding.

This will be my final description of the differences between nominalism, Platonism, and Peirce in this thread. There is clearly no reason for me to keep repeating myself here.

For Peirce, abstraction is dynamic, relational, and grounded in semiosis (the process of sign-making and interpretation). Universals and abstractions are not arbitrary (as in nominalism) or static and disconnected (as in Platonism). Instead, they are real but only in the sense that they emerge through relational continuity and are embedded in a triadic process.

Peirce's approach stands out by addressing the limitations of nominalism (over-reliance on discrete categorization) and Platonism (over-reification of abstractions). He emphasizes the relational, processual, and evolutionary nature of abstraction, making it more aligned with the complexities of the real world.

For Peirce, abstraction is dynamic, relational, and grounded in semiosis (the process of sign-making and interpretation). — Mapping the Medium

That sounds like nominalism to me. But I think it misses the point which separates Peirce from nominalism, making him closer to Platonist. For Peirce, the universal is an "object" and this name is supported by the assumption of "sameness", what you called "functional continuity across interpretations".

This assumed "continuity" of sameness, despite differences, is what allows the universal to be known as one object instead of many distinct conceptions in many distinct minds. This is analogous to the observed temporal continuity of the physical object which enables our assumption of "same", despite differences of change over time, instead of assuming a new object at each passing moment.

Notice that the title "object" is supported by an assumed continuity of existence, in both cases. The continuity of the physical object being supported by the assumed temporal continuity between distinct moments of existence, and the continuity of the universal being supported by the assumed coninuity produced from the use of signs.

The problem being that Peirce's semiosis, and proposed triadic structure cannot support this assumed continuity required for his determination of "sameness", and "object". By placing the object outside the relationship between interpretant and representamen, as your diagram neatly shows, as something distinct, (independent with its own continuity),having its own distinct relation between each of the other two, Peirce provides a misleading model. A true analysis of the relationship between the interpretation and the sign would reveal that the sign actually breaks any supposed continuity of the universal, between one interpretation and another.

So to make a true representation, which would support the supposed continuity of "the object", the category of "object" would have to include both interpretant and representamen as united in continuity, within "the object". This is what is commonly known as the transcendence of Platonic objects. The existence of the universal, as am object, transcends the existence of its composite parts, the sign and the interpretation of the sign. That the object transcends both, and is therefore of a distinct class produced by a unity of the other two, is a necessary condition for the the continuity of the object.

What Peirce does with the triadic semiotic structure, is remove the transcendence which supports the continuity of "the object", yet he still claims an object with continuity. This allows that the continuity of "the object" may be understood as a property of the interpretant, or it may be understood as a property of the representamen, in his proposal of ambiguity. So all we have is a nominalist sign-mind representation, with an assumed continuous "object" which may be assigned to the sign, or it may be assigned to the mind, depending on one's theoretical purpose.

This is why it is useful to refer to those who apply Peirce's triadic structure, to demonstrate the inconsistency in application, produced by that ambiguity. In "objective" science such as biosemiotics, it is evident that "the object " is a property of the representamen, yet in social applications of semiotics, it is clear that "the object" is understood as property of the interpretation. A true representation of a united interpretant/representamen, to support a continuous object, is not required, because "the object" may simply be assigned to one side or the other.

This is because Peirce takes a nominalist sign-mind model, and adds "an object" without any rigid principles of sameness or continuity. This allows those who apply the model to assign "object" where there is no support for an object. The defense of that assignment is that it is a "Platonic object", but Peirce has denied the ontological support for Platonic objects.

Peirce's approach stands out by addressing the limitations of nominalism (over-reliance on discrete categorization) and Platonism (over-reification of abstractions). — Mapping the Medium

This is exactly why Peirce' project fails. Asserting compatibility between incompatible ontologies is not a solution. Taking a nominalist structure of "discrete categorization", and imposing an assumption of continuity, without justification, just to make it more "Platonic", is not a solution to the discrete/continuous dilemma.

Asserting compatibility between incompatible ontologies — Metaphysician Undercover

...

Taking a nominalist structure of "discrete categorization", and imposing an assumption of continuity, without justification, just to make it more "Platonic", is not a solution to the discrete/continuous dilemma. — Metaphysician Undercover

Peirce absolutely does NOT do what you are asserting here.

I am exploring this issue in my thread on Hypostatic Abstraction, Precisive Abstraction, Proper vs Improper Negation

numbers don't exist. — Art48

I agree with you. They exist. But they are ultimately make-believe. Functional tools we have constructed, projected into the natural world as such, and because they are Functional, collectively believe.

In that sense, as 'opposed' to the real and natural world against which we project them, they are ontologically not real. So they exist, but only for humans, and only in that fictional layer which we have imposed upon reality, hiding it only from ourselves. When humans are nowhere to be found, so too will numbers be.*

*assuming that no other organism evolves to adopt them, and that even our AI etc are gone.