To disambiguate terminology: By "dual space" I mean the standard meaning in QM; we go from a complex vector space V to a space $V^{\vee }$ of linear maps $V\to C$, with the bra/ket duality as a special case. Perhaps I should avoid using "dual" but this explains my earlier usage.

By "complex conjugate space" (sometimes abbreviated as "conjugate space") I mean specifically the $\overline{V}$ construction, a completely formal mathematical operation. (To avoid confusion I could say "complex conjugate space")

Since "complex conjugate space" is an entirely formal construction, it doesn't necessarily have a physical meaning. Or it could have multiple physical meanings as different isos with V.

OP doesn't try to go lower-level than Hilbert space; what follows is my attempt to engage with the level you're talking about:

A "duality" like thing is "polarity of representation of $U(1{)}_{\pm }$" as implied by the category $\left[U\right(1{)}_{\pm },Vec{t}_R]$. Where to simplify in OP, I'm always using the polarity between V and $\overline{V}$ in the representation, but this is not strictly necessary.

The correspondence with the SO(n) model might be: When you are representing an element of $\left[BSO\right(n),Vec{t}_R]$ as a complex vector space ($B$ means "delooping groupoid"), you are picking out a sub-group of SO(n) iso to $U\left(1\right)\cong SO\left(2\right)$. To get to a Hilbert space you clearly have to at least pick out a $U\left(1\right)$ subgroup.

Another idea is to find a functor $\left[U\right(1{)}_{\pm },BO\left(n\right)]$. This does not necessarily get you an O(2) subgroup, because J and $J^{-1}$ in $U(1{)}_{\pm }$ need not map to the same group element of O(n). (Hence spinors and so on)

If you have a representation $\left[BO\right(n),Vec{t}_R]$ and a groupoid functor $\left[U\right(1{)}_{\pm },BO\left(n\right)]$ you can trivially compose to get a polar representation $\left[U\right(1{)}_{\pm },Vec{t}_R]$. That gives you, as the positive component, something like a Hilbert space, and as the negative component, something formally iso to its complex conjugate space (though with a better physical interpretation).

At this point you can use the formal isos: $\overline{V}\cong V^{\vee }$ (Riesz representation), and the iso from $\overline{V}$ to the polar negative of your Hilbert-like space (where polar negation is from the $\left[U\right(1{)}_{\pm },Vec{t}_R]$ representation). This gives you nice physical interpretations of bra/kets, density matrices, observables, and so on, through formal isos.

(I am not sure how much I'm understanding or how much is connecting; feel free to ignore irrelevant detail)

In terms of goals of OP, I was taking standard QM notation at face value, and trying to peel back one layer of the onion: why antilinear-linear inner product, POVM observables as Hermitians implying quadratic forms, quadratic density matrices, etc? And I think real structure + conjugate spaces basically explains these dualities in the Hilbert space standard formulation, although that doesn't mean this is the deepest symmetry layer (it's not).

field strength tensor

This is not an area I understand well, but, either the field values are expressed as a real number or complex number. If real: OP framework doesn't apply, or applies at a lower level; perhaps applies indirectly through CPT. If complex: Then I'm guessing similar principles will apply, where for example you can "make a complex number real" as $\overline{\lambda }\lambda =|\lambda {|}^2$, yielding an inner product that looks like the Hilbert space inner product. The thing to look for would be, are complex conjugates showing up all over the place in the standard math? If so, it may perhaps be modeled with real structure & complex conjugate spaces.

we should not expect to be able to find dual spaces by looking at mirror images

For Hilbert space I'm applying Riesz representation for that; standard machinery for bra-ket. Not sure how far the principle generalizes. (Much of OP could work without Riesz representation, e.g. we can still define $A^{\otimes }(\overline{u}\otimes v)=⟨\overline{u}\otimes Av⟩$ just fine, it's just that $(-{)}^{\otimes }$ might not be an invertible map.)

We can identify $v_1\wedge v_2=v_1\otimes v_2-v_2\otimes v_1$

This is looking a bit like the 'swap' real structure on $\overline{V}\otimes V$. The main difference is that it's dealing with real rather than complex numbers. With the swap real structure, the 'imaginary component' would be written as $(\overline{u}\otimes v)\mapsto \frac{1}{2}(\overline{u-v}\otimes v-u)$. This of course looks similar to $u\wedge v$.

Here's one place where the analogy with symmetric group permutations breaks down. Say our Hilbert space is $H=H_1\otimes H_2$. Now our inner product (written to be bilinear) is $⟨\overline{a_1\otimes a_2},b_1\otimes b_2⟩=⟨\overline{a_1},b_1⟩⟨\overline{a_2},b_2⟩$. We can see this as quadri-linear in $\overline{a_1},\overline{a_2},b_1,b_2$. Notice with 'n-linearity' we always have n even, and the elements come in pairs. We pair $\overline{a_1}$ with $b_1$ and $\overline{a_2}$ with $b_2$.

What the standard Hilbert space framework is going to do is, by bundling $H_1\otimes H_2$, make the $H_1$ and $H_2$ bilinearity explicit, with the $H_1$ and $\overline{H_1}$ bilinearity implicit (through the inner product and bra/kets and so on). This is why we end up with quadratics all over the formalism: Born rule, inner product, density matrices, POVMs.

So for your analogy to hold, it's critical that $n$ be even. And what the 'swap' real structure on $\overline{H}\otimes H$ is going to do is swap elements with their conjugated elements ($\overline{H_1}$ with $H_1$, and $\overline{H_2}$ with $H_2$). This is going to be a specific involutive permutation.

So I guess if you wanted to get into deeper foundations, then a question to ask would be "why do the tensor products that POVM observables are linear in factor into components that come in pairs with their complex conjugate spaces (or something iso to their complex conjugate spaces)?". Which, again, OP isn't very much about.

The bilinear form is from the inner product. The inner product is generally defined as 'antilinear in first argument, linear in second'. If replacing the first argument with complex conjugate space, it is now 'linear in first argument, linear in second argument', i.e. bilinear. This is of course for a single Hilbert space (and its complex conjugate). When having multiple Hilbert spaces tensored together, the 'tensor product of Hilbert spaces' yields an inner product for that tensored space. That is implicitly multilinear, although decomposes as usual for tensor product, into multiple bilinearities.

1. Here's an evalution claiming it isn't even true at Anthropic in terms of lines of code (I haven't checked independently)

2. The kind of thing I don't want to count is that, for example, it would be easy to write a script that generates many many lines of code. It's just not interesting to count "how are most lines of code written" globally. "Code actually merged at particular big tech companies" is much closer and I expect that was less than 90 percent at the time.

Ah sorry. Edited.

1. With 10K lines it's not impressive to write 10K lines. I'm thinking of writing software that would be 10K lines if done by humans.

2. "bug-free" & "reliably" is a high bar!

Really appreciate the exchange, helped me clarify my thoughts.

As a brief note for potential future exploration (no need to reply if you don't want to):

A crux seems to be that with chroma world, I think experience is of difference between chroma, assuming chromatic realism (real quiddites). Whereas the Russelian monist wants the experience to be of identity of internal chroma. This is perhaps a disagreement on the phenomenology / epistemology relationship.

I think the analogy with Newtonian mechanics (if absolute positions are real, then experience is of relative positions, even if located at absolute positions) is pretty strong; chroma world "just" has different physical symmetries.

On my end I think I could continue this line of exploration by studying Deleuze (difference as more ontologically/metaphysically primary than identity), Dennett (more clarity on whether I agree/disagree with his basic picture), and Russell as a primary source.

Yeah I think it might be more productive, if I wanted to make progress on this, to look at the math rather than the metaphysical back-and-forth.

I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).

Nope, because they didn't write down the system. (This might be quibbling. I mean it doesn't write down that formal system anywhere.)

Let me move to a different example, why do you think our universe instantiates PA

Because mathematicians have written down the rules of PA. And they have programmed computers to use PA.

Do you think that our universe then either instantiates Con(ZFC) or doesn't?

Our universe instantiates formal system such as ZFC + Con(ZFC), because mathematicians can write them down. (It also instantiates inconsistent formal systems such as "ZFC + axiom of choice is false".)

Backtracking a bit. I reject axiom 6 (which is not to say I actively disbelieve it, but that I don't actively believe it) and I'm giving an idea of why. If you could logically prove axiom 6, you wouldn't need it to be an axiom. So you're going to have some trouble convincing me of it on logical grounds. Rather it's more metaphysical and so on.