Intertwiners

This is a serious and coherent project, and I want to engage it on its own terms rather than redirect it toward physics. What you're describing is a grammar of homology — a vocabulary whose forms are stable enough to survive the crossing from formal structure into everyday speech, and whose uses in the vernacular remain structurally faithful to their origin.

Let me try to name what you're doing, then offer some candidate machinery.

1. What kind of language you're actually after

You're not after analogy in the loose sense ("the atom is like a solar system"). You're after homology: the same abstract form appearing in different substrates, so that a word borrowed from one domain carries its structure intact into another. The word should not merely evoke the formal structure; it should preserve it under translation.

That is exactly what intertwiners do. An intertwiner is a map $\varphi :V\to W$ such that

$$\varphi ({\rho }_V(g)v)={\rho }_W(g)\varphi (v)\text{for all }g,v\mathrm{.}$$

It says: you can move between two representations, and the group action is preserved. That is the abstract form of faithful translation. A metaphoric bridge that is an intertwiner is one where the structure survives the crossing.

So your instinct is right: you are looking for intertwiners, but in the register of language rather than linear algebra.

2. The problem with vernacular metaphor

Ordinary metaphor is usually a homomorphism of connotation, not of structure. "Argument is war" carries affective and pragmatic structure, but it does not preserve the actual dynamics of argument under transformation. It is a useful lens, not a faithful map.

What you want is a vocabulary where the metaphor is structure-preserving. The vernacular word should behave under composition, negation, nesting, and inversion the way the formal concept does. That is a much stronger requirement, and it is why most vernacular metaphor fails as a bridge to higher-order phenomena.

3. Candidate formal structures worth carrying into the vernacular

Here are some whose forms are robust enough to survive the crossing:

Invariance. Something that does not change under a transformation. Vernacular use: "what stays the same when you look at it from a different angle." Already partly present in ordinary speech, but often confused with "unchanging." The formal version is sharper: invariance is always relative to a group of transformations.

Representation. A concrete realization of an abstract structure. Vernacular use: "the way a thing shows up in a particular medium." Useful for talking about how the same situation appears differently to different participants without collapsing into relativism.

Irreducibility. Cannot be decomposed further. Vernacular use: "you can't break this into independent parts without losing what it is." Useful for talking about situations where analysis destroys the phenomenon.

Tensor product. The space of joint states, larger than either alone. Vernacular use: "the space of what two things can be together, which is not just the sum of what each can be alone." This is one of the most useful and least vernacularized concepts.

Intertwiner. A structure-preserving map between two representations. Vernacular use: "a translation that keeps the shape." This is the concept you are reaching for.

Commutator. The failure of two operations to commute. Vernacular use: "the residue left when the order of doing things matters." This is extraordinarily useful in everyday life — most practical conflicts are commutator effects.

Clebsch–Gordan decomposition. How joint representations break into independent parts. Vernacular use: "the ways two things can combine, and what the combinations are." Useful for talking about relationships, teams, and mixtures.

Coherent state. A state that behaves as classically as possible while remaining quantum. Vernacular use: "the most thing-like version of something that isn't really a thing." Useful for talking about habits, roles, and stable patterns.

Symmetry breaking. A symmetric situation that settles into a less symmetric one. Vernacular use: "when a situation that could have gone many ways settles into one." Useful for talking about decisions, institutions, and history.

4. The homology criterion

For a vernacular word to be a faithful bridge, it should satisfy something like:

• Composition: if $A$ composes with $B$ formally, the vernacular terms should compose in the same way.

• Identity: there should be a vernacular term for the identity map (the "do nothing" translation).

• Inverse where applicable: if the formal map is invertible, the vernacular term should carry that.

• Functoriality: translations should compose, not merely accumulate.

• Naturality: the translation should not depend on arbitrary choices.

If a vernacular metaphor satisfies these, it is an intertwiner. If it fails them, it is a decoration.

5. Everyday domains where these forms already live

The reason this project is viable is that these structures already occur in everyday life; they are just not named with their formal forms:

• Promises are intertwiners between intentions and expectations.

• Rituals are coherent states of social systems.

• Contracts are tensor products of obligations.

• Conversations have Clebsch–Gordan structure: two people's frames combine into a joint frame that decomposes into shared and private parts.

• Habits are symmetry-broken states of behavioral space.

• Reputation is a representation: the same person, realized differently in different social media.

• Trust is an invariant under many transformations of circumstance.

• Misunderstanding is a failure of intertwining: the map does not preserve structure.

• Apology is an attempt to restore an intertwiner that was broken.

Each of these could be described with a vocabulary that carries the formal structure intact.

6. A suggested discipline

If you want this to be more than a poetic exercise, I'd suggest a discipline:

1. Pick a formal structure with a clean abstract definition.

2. Find its minimal vernacular form — the shortest everyday phrase that preserves the structure.

3. Test it against composition, identity, inversion, and naturality in everyday cases.

4. Discard it if it fails; the failure tells you what the vernacular cannot yet carry.

5. Keep a lexicon of terms that pass the test, with their formal counterparts.

The lexicon is the artifact. It is a dictionary of intertwiners between formal structure and vernacular speech.

7. Why this is worth doing

The vernacular is where most human reasoning actually happens. If the vernacular lacks words for invariance, irreducibility, tensor products, and commutators, then most human reasoning will lack those concepts. The formal vocabulary exists, but it does not cross into everyday speech without loss.

Your project is to build the crossing. Not to popularize physics, but to import its structural forms into the vernacular as reusable linguistic machinery. That is a real contribution, and it is the kind of thing that has happened before — "energy," "field," "feedback," "network," "information," and "system" all made this crossing, and each one changed how people think.

The ones you are reaching for — invariance, intertwining, tensor product, commutator, symmetry breaking — are the next candidates. They are already structurally mature. They are waiting for their vernacular forms.

If you want, I can help you build the lexicon: take one formal structure at a time, propose a vernacular term, and test it against everyday cases for homology.