papalex

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Last Notes

2026-05-09 22:58:09 GMT

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thanks for the correction! I made an embarrassing confusion (with a good resolve):
1. PA is not ℵ₀-categorical: there are countable models that are not isomorphic.
2. PA admits an initial model in ZFC: you are right, and I was wrong. I confused the two points.
This also means that my “interpretation 2” is at best morally fine. There is a canonical model and that is what it is. However, deconfusing all this led me to understand the following, which is anyways a way better formulation of the two interpretations fused to one, and I hope this time is (closer to) correct:
Not all minimal inductive sets in all models are equal: which is the crucial point for the discussion. Given some model of ZFC M and create a new model M’ by extension by some new element c. Add axioms c ∈ ω and c ≠ 0, c ≠ S(0), c ≠ S(S(0)),… (the latter one supposedly by completeness to deal with the infinitely many) So we bluntly force a new element into ω. Then, by definition every inductive set contains c for else c \not\in ω violating M’. Thus, by definition we have a model in which the standard PA via minimal inductive set is not the standard ω_0 (though this difference is only seen externally by observing that ω is not isomorphic to ω_0).
This is a countable model which is not isomorphic but, and that was my earlier confusion, the standard naturals do uniquely map into this set. So there is an initial model but the standard PA model in such nonstandard ZFC models can be a nonstandard PA model as seen from the outside.

2026-05-08 23:19:20 GMT

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after all, I have the impression there are two possible meanings of whether “a model of PA in a nonstandard model of ZFC is standard or not” (likely there is only one by some definition, if anyone knows that: please drop a reference 🙏). However, in either the solution to the problem becomes quite tautological.
First, the notion of standard model in a theory is anchored by the theory. In ZFC for example the axiom of infinity is that anchor, but there is no initial model of PA in ZFC. Standard is more like the informal „common“ rather than the formal „canonical“, in that it’s us identifying this axiom as the anchor, but there is not a canonical rule it seems to identify the anchor in an arbitrary theory.
Now, if we change from „model of PA in ZFC“ to „model of PA in __model of__ ZFC“ I wonder what’s the appropriate adaptation of the notion of standard. Here are the two options I see:
1/n

2026-05-08 06:51:38 GMT

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You are right and this shows where the IST approach actually fails and why it’s not a nonstandard model in the model theoretic sense (it’s just not the standard theory). Being nonstandard in IST does not mean to be nonstandard in the model theoretic sense. It is just an additional predicate we can choose to add or not to add to ZFC. But crucially, being an unbounded integer in IST means being unbaunded against “standard” integers. But is does not mean unbounded in the conventional sense. So we cannot compare these things which, as you write, collapses the argument. That’s right.
So the question: can any of the argument be recycled for an actual nonstandard model or was @npub1nf4…nqe4 right all along tha the quote must be wrong?

2026-05-07 17:27:29 GMT

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Indeed, here are several levels at play causing confusion. I'll have a try to deconfuse:
1. In any model of ZFC we have a standard model of PA defined via the axiom of infinity. Call this inductively defined set ℕ.
2. Any nonstandard model of ZFC has a meta theory strictly richer than ZFC. This meta theory is the axiomatic (or theoretical) shadow of the nonstandard structure of this model. Reasoning in this meta theory is in a sense reasoning about the actual structure of the model.
3. In the standard model of ZFC ℕ only contains finite sets (or bounded numbers).
4. IST is conservative over ZFC, thus every model of IST is by reduct also a model of ZFC. Reasoning in the IST metatheory is just the god's eye view onto the models structure as visible from outside ZFC.
5. In IST by idealization every infinite set contains non-standard elements. Thus, in IST ℕ contains infinite sets (or unbounded numbers) and conversely this means that ℕˢᵗ is finite and thus not the same as ℕ in any meaningful way (neither in ZFC nor in IST).
6. Thus, the standard model of PA in a non-standard model of ZFC (via IST) is not isomorphic to the standard model of PA in the canonical ZFC model—because idealization forces ℕ itself to contain unbounded naturals.

2026-05-07 13:31:14 GMT

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This I understand somewhat as the statement that "one cannot construct the standard (i.e. intended model of) N in *V but any model of PA in *V will have nonstandard elements."
Which to me would still be in line with the original quote. That is, to get the standard N one would have to adopt some meta theory (if that is even allowed?) but the standard (i.e. intended) N is definitely not a standard model of PA in *V in any meaningful way.
What am I missing?

2026-05-07 09:33:17 GMT

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I think I am a bit confused and likely just miss a key definition somewhere, so I'll just ask a naive question:
From N in V you have constructed 'N in *V. You argue that 'N contains N (in a specific sense) but also non-standard numbers (represented by non-constant sequences such as [(0,1,2,3,...)]). Then you observe that 'N is the initial model of PA in *V from which you conclude that ('N,*V) is not the example we want.
But as I understand it, you just showed that 'N is richer than N, in particular it contains non-standard elements. And to reduce to only elements of the type [(a,a,a,...)] which correspond to N wihtin 'N would correspond to a non-initial model, right?
So why isn't this the example you want?

2026-05-06 12:26:23 GMT

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I must admit that I am not an expert and might have misunderstood the assumptions of the quote. In particular, I read the ZFC part in the quote more as an example rather than an assumption. Though I also don't know prior to googling whether "non-standard model of set theory" implicitly excludes IST?
Either way, I would enjoy to learn more about any of these things and am happy to accept if my answer was missing the point, if that is the case?
Ignoring any nitty gritty interpretations of the original quote. You wrote "in a non-standard model of ZFA". What actually does this include? I would assume anything with ZFA axioms plus whatever other non-standard axioms goes, which I think IST would fulfill, no?

2026-05-06 11:30:46 GMT

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Indeed, with a linguistic glitch: In IST, the standard natural numbers are standard in the sense of the standard predicate. However, the standard predicate is not standard, so the standard natural numbers in IST are non standard in the standard sense 🤯

2026-05-06 11:08:09 GMT

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In internal set theory (IST) the idealization principle (I) implements the intuition that we can only test a finite number of objects at a time. A consequence of (I) is that any infinite set must contain a non-standard element and in particular there exists a non-standard number in ℕ (see [1] end of pg. 5). So to get rid of the infinite numbers in ℕ we would have to add further aximos (invoking the external standard predicate), making the standard model of PA ℕˢᵗ a non-standard model of PA within IST.
I think the intuition is somewhat natural at first and then makes an interesting twist: We do not tell maths what it may do, but only make axioms about what we can check, this gives us the idealization principle: For any property we can always only check the property on finite elements, which is sort of a pragmatic filter of what we mean by that property. However, the real work of (I) is then that it asserts: If (see LHS of (I)) for any finite standard set 𝑥' there exists a 𝑦(𝑥') such that the property holds (made the 𝑥' dependence explicit), then (I) asserts that there also exists a fixed 𝑦 such that the property holds wrt all standard 𝑥.
So the real work of this axiom is in making 𝑦 independent of 𝑥'. I think this is a really nice way of making ∞ and related infinitesimals operational.
So the short form intuition could be:
IST enforces that the sequence 𝑆ₙ={0,1,...,n} with 𝑆ₙ<𝑆ₙ₊₁ must have a supremum with 𝑆ₙ<S_ω for all 𝑛∈ℕˢᵗ and the identification 𝑛≃𝑆ₙ survives this limit. the full set ℕ is the "completion" that the sequence necessitates, because no standard bound suffices.
[1]: https://web.math.princeton.edu/~nelson/books/1.pdf

2026-05-04 21:38:28 GMT

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love it, in particular the constant contradictions. First he defines the measure of consciousness via “if, after lengthy conversation, you cannot tell whether it is a human”, and then he ignores that there surely is no single human who can not only write one but like five or so different sonatas in a few minutes? This destroys the whole operational meaning of the test, if you suddenly say “I know it’s not human but this is super human”. No! You (that is he) just proved that Claude strictly speaking fails the Turing test and any “but it’s even better!” just confirms that the Turing test is completely besides the point for actually discussing consciousness, which means his entire argument collapses in the viscous circle of confirming his own beliefs by — tada: his own beliefs.

2026-03-26 23:00:48 GMT

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To add some details on the “complicated” side of things:
In standard QM, we work with a complex Hilbert space where every vector has a clear physical meaning. Moving to QFT, we shift to Fock space, which aggregates these states. We replace individual particle states with field operators (ladder operators) that create the vacuum excitations. But we still work with a conventional complex Hilbert space.
However, there is no single "correct" Fock space here. Thanks to Haag’s Theorem, there is an infinite choice of unitarily inequivalent representations, meaning the way we define our particles is fundamentally linked to the state of the vacuum.
Then, things get messy when we move to gauge theory (which apparently we need to describe nature). We start by quantizing the fields, but this introduces massive redundancy because we quantized redundant degrees of freedom. The raw Fock space gets flooded with "unphysical" states that don't satisfy gauge symmetry. In many cases, these raw spaces contain negative-norm states, making a direct probabilistic interpretation impossible. To recover a physical Hilbert space, we have to "mod out" the gauge orbits and ghosts (aka undo the mistake of quantizing too much).
Finally, what I did not yet mention: transitioning to relativistic QM (aka Dirac equation) forces us to switch from complex scalar fields to slices of spinor bundles, which accounts to space time geometry.

2026-03-26 21:55:43 GMT

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ah, sorry. Of course the soup was of a different taste. That was unintended sloppyness on my side.

2026-03-26 07:54:22 GMT

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And while John is of course right about his diagnosis about the SM, I think in the particular case of this post one could make a mathsy phrasing of what the "soup of quarks" means. Let me have a try at it.
Original: "protons and neutrons act like bags full of a soup [...] In the quark condensate, the clockwise spinning virtual quarks are entangled with counterclockwise spinning virtual antiquarks."
Mathsify: "SM Quarks are a local excitation in a 𝔰𝔲(3)⊕𝔰𝔲(2)⊕𝔲(1)-gauge 𝔰𝔬(3,1)-spinor bundle with respect to the fundamental representation. But they do not appear as isolated excitation because the Hamiltonian / Lagrangian (aka dynamics) prohibits this (strong coupling). Protons and Neutrons are a fairly stable form of a collective excitation. One can expand it in terms of representations using three fundamental representations (three quarks) but the dynamics makes things complicated, due to frustration of constraints (quark confinement versus charge repulsion etc). This has the effect of adding a whole bunch of virtual representations to the mix (i.e. the decomposition into three fundamental representations is not that clean but really there is a bit more than just that, and this "bit more" is not a clean representation but a superposition of things => the condensate). This has the effect of both, breaking the symmetry between the different Weyl sectors of the spinor bundle and of increasing the mass from the Higgs mass of the three quarks."
This is still only somewhat mathsy. I would be curious how to make this even sharper @npub1nf4…nqe4?

2026-03-26 07:01:43 GMT

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Let me add another metaphor which is somewhat the same as @npub1p89…vv2h 's just in different clothing: Often the physics thinking works on the very basic level while the maths thinking is on a more complex level, which comes with heavier machinery than necessary. But the heavy machinery has its value in something else than bare necessity: It is the precision tool to assure the statements are understood exactly right (=non-ambiguous) in a most economic (=few words) way.
I want to make a maths analogy which I hope does fit (and I am aware that some here are lightyears more experienced than me in this topic, please be gentle):
one can do a surprising lot of things in maths without the heavy machinery of category theory but using fairly simple stuff, just constructively. For example, to define real numbers via Dedekind cuts one does not need the category theory notion of limits and to extend functions from finite stages to the full reals one does not need Kan extensions, comma categories etc to formulate and proof any of these things (of course one implicitly uses them, but "trivially" so).
But: the category theory language can be very useful for precision in the written text (i.e. if its not written as Agda program), and it is a very economic way of transporting the information content human to human (conditioned on the CT understanding). But this is at a price which may be precisely the price of (non_ambiguity)+(economy).

2026-01-15 08:25:47 GMT

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I hope not to spam this thread, though felt this quote from [Adlam] (page 27) regarding the Poincare-Einstein debate is too insightfull not to share:
*"Poincare often refers to the network of conventions used in science as a framework; so for instance he claims that "space is another framework which we impose on the world." The fact that all his scientific work presupposes the existence and reality of space therefore reflects a deliberate choice to work within the traditional framework of science. Indeed, his commitment to retaining conventions whenever possible makes it clear that he does not see it as the place of the scientist to question this framework; as Stein observes, "the basic mathematical presuppositions of physics were seen by Poincare as defining a framework within which it is the task of the theoretical physicist to fit all phenomena." (unpublished). [...] He recognised that this explanatory framework is a freely chosen convention, but he believed that the role of science is to construct theories within this framework, not to investigate the nature of the framework itself."*
==> What this highlights:
Einstein did not merely introduce special relativity. He introduced a whole mindset for physics. The totality of Einsteins 1905 papers best reflect this revolutionary momentum in full. We can and should acknowledge Poincare (and Lorentz) for deriving the math of SR. But when we acknowledge Einsteins derivation of SR, I claim, in fact we talk about more than just the math -- we talk about the paradigm shift that Poincare rejected.

2026-01-15 00:45:20 GMT

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this is an even clearer way for putting it, thanks for that. I must admit I forgot most of those details. Clifford’s program feels to me a mix of fascinating ahead of his time ideas, together with other just not yet ripe ideas. Fans can pick one side, skeptics the other, almost a win win 😅 But that reminds me at a todo from long ago for future me: digging deeper into the Clifford program to understand his reasoning and differentiate it from the confusion stemming from the time (and lack of internet) of his days. Superficially his ideas are fascinating. Though I did not find the time to deep dive.
@npub14a5…9fse

2026-01-15 00:25:46 GMT

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Along similar lines, I think Clifford and Sylvester were great thinkers of their time. The core difference however to Einsteins curvature is that they were thinking about curved space, not curved space-time. Which might very well be the reason why Cliffords program did not succeed. From the distance this might seem like a minor difference, and reading Cliffords letter is fascinating and displays his genius (essentially predicting relativity and quantum). Yet, they had missed one key point: relativity (which might be partially in Cl's thinking) and the constancy of c.

2026-01-15 00:14:21 GMT

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This is a fascinating issue. And I don't think the "one idea by many minds" captures the core. On the one side, yes, there are ideas floating around, and most ideas will be in multiple minds. So there must be some truth to it. But: My experience tells me that it is often the details that matter. To the point: as @npub1nf4…nqe4 writes, Poincare had the math, but he was trapped in his metaphysics. And this is the key to the question. Einstein took relativity as an empirical operational fact and from this derived SR. Poincare, on the other hand, wanted to explain relativity from sth more fundamental. So he was rejecting that Einsteins axiomatic reasoning explains anything and he rather focused on the contracting electron as explanation for relativity. Here I would argue two points:
1. Poincare fooled himself, because he had to believe the contracting electron story in order to explain relativity thus assuming the premise (since the contracting electron story only was selected BECAUSE it had the right predictions). In a sense, he derived the maths, but being caught in the microscopic picture he was not able to distill the structural essence of relativity. Or with [Adlam] "it seems inaccurate to say that he had any significant intimations of special relativity before Einstein’s 1905 papers".
2. Einstein in contrast was transparent about the assumptions and in fact introduced a new way for physics: Operational axioms + the maths ==> A principle that later lead Heisenberg to Matrix mechanics and where Einstein took what was Poincare's seat.
[Adlam] https://arxiv.org/pdf/1112.3175

2025-12-23 22:28:39 GMT

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@npub1nf4…nqe4 thanks for the TLDR of their mapping, now I don't need to filter the paper for that info.
@npub12cu…d38p The fun thing is that in this system one can calculate the system at arbitrary future times (you're also right that there are no shortcuts, but that is inevitably encoded in the word "computation"). However, this goes besides the point of undecidability, and this is i.m.o. the crux about all the misunderstandings. It is not "undecidable" to calculate the trajectory (and its less clear what that even is supposed to mean). Its the decission problem, whether the trajectory has or doesn't have a certain property, which is undecidable.
And in this phrasing the undecidability is suddenly not-surprising-at-all even to the lay person: If the property can take arbitrarily long to realize (like an arbitrarily long periodic path), how should we ever decide that we have waited sufficiently arbitrarily long to tell...
That said, I find that sentence about "undecidable trajectories" in the papers abstract borderline misleading (and yet I totally understand that its sexy to write). And I find the debate about undecidability in general often covered in a bit too much of miracle cloths.

2025-10-15 10:35:23 GMT

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Ah, of course you’re right! Unital composition algebra implies division algebra only over the reals. So that was too impulsive 😅
And this makes more clear why this is an interesting question. Namely, what’s the effect of the idempotent structure that emerges when going from octonions to bioctonions on the Moufang identities?

2025-10-15 09:51:30 GMT

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As for the Moufang identities. Since
> * The invertible elements in any alternative ring or alternative algebra form a Moufang loop.*
The answer should be yes, bioctonions (which are just octonions and hence a unital composition algebra over \(\mathbb{C}\) and thus invertible except for 0) form a Moufang loop. So the identities hold.
See second example in: https://ncatlab.org/nlab/show/Moufang+loop

2025-10-14 19:43:30 GMT

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The equation holds and is in fact super straight forward (one might say trivial) to prove from alternative algebra.
To avoid confusion, the amendment equation reads:
(𝑥𝑥*)𝑦−𝑥(𝑥*𝑦)=(𝑦𝑥)𝑥*−𝑦(𝑥𝑥*)
However, this follows immediately from the fact that in any alternative algebra 𝔸 and 𝑎,𝑏,𝑐∈𝔸 it holds:
(𝑎𝑏)𝑐−𝑎(𝑏𝑐)=(𝑐𝑎)𝑏−𝑐(𝑎𝑏)
or using the associator:
[𝑎,𝑏,𝑐]=[𝑐,𝑎,𝑏]
since
[𝑎,𝑏,𝑐]=−[𝑎,𝑐,𝑏]=[𝑐,𝑎,𝑏]
See for example https://en.wikipedia.org/wiki/Alternative_algebra#The_associator
Thus, since the octonions and the bioctonions are alternative, your amendment is correct by setting 𝑎=𝑥,𝑏=𝑥*,𝑐=𝑦.
PS: @npub1nf4…nqe4 either you edited your post, or more likely my phone did not render the maths correctly. Now on my computer it look the same 👍

2025-10-14 17:59:22 GMT

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This is interesting. It makes me curious. My intuition tells me that this could work. Will have a go in a bit.
Btw, @npub1nf4…nqe4 your equations seems to miss a conjugation on the second x on the RHS compared to @npub1689…nwtu ’s version (or it’s a displaying issue).

2025-10-14 08:38:08 GMT

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I did not check the very long outputs in detail (i.e. the 𝑥𝑥* and the (𝑥𝑥*)𝑦−𝑥(𝑥*𝑦) outputs), but it looks good to me! So indeed, alternating but not "conjugate-alternating" or whatever the appropriate name for the last expression would be.

2025-10-14 07:33:19 GMT

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I‘m not sure about your conjugation (ie your second output). Using the conjugation by @npub1nf4…nqe4 you should have all 𝑋ᵢ↦−𝑋ᵢ* and only 𝑋₀↦𝑋₀* . In your code, however, 𝑋₃,𝑋₅ and 𝑋₆ behave akin to 𝑋₀.
What is your convention though, do you have 𝑒₁𝑒₂=𝑒₄ or 𝑒₃ ? In the latter case it seems that your code does not take into account that (𝑎𝑏)*=𝑏*𝑎* .

2025-10-13 23:08:43 GMT

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Ah, I just found your discussion with @npub1nf4…nqe4 , now I understand your questions 😅
I will check whether these guys are alternative tomorrow, here its way too late already. But reading your discussion it makes sense to me now that the alternative law seems to hold: If you don't do conjugations then the associator term seems to vanish (for the same reasons as in the real case). Which is not surprising as we only work over a different field now. But when we also introduce the conjugation, then we changed more than just the field! Because working over ℂ instead of ℝ would mean that we only conjugate the octonion part, but not the complex part. Because the complex conjugation is an operation on a real algebra (because it defines the real axis). Thus, terms involving the conjugation may differ from standard octonion reasoning while terms without conjugation (like the alternative laws) should carry over from octonions.
I hope for now this helps for some intuition

2025-10-13 22:38:10 GMT

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1) ((1+𝑖)𝑒₁)*=(−1+𝑖)𝑒₁ , in particular, there are hyperbolic elements, like (𝑖𝑒₁)*=𝑖𝑒₁. This follows immediately from the definition of the conjugation in @npub1nf4…nqe4 's original post.
2) I haven't checked that. I would guess that not, but my longer post above seems to indicate that it might be (at least right now I don't see how to break the identity 𝑎²𝑏=𝑎(𝑎𝑏)). If you tensor the quaternions to it it certainly ceases to be alternative, but that doesn't help here.
3) Nope, by 1) you can easily construct counter examples. In fact, 𝑥𝑥* (as defined in the long post above) is not real but 2+2𝑖𝑒₄. I think though that 𝑥𝑥* always gives you a non-imaginary number, i.e. a sum of only real and hyperbolic numbers.

2025-10-13 22:06:43 GMT

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Happy to help! I was also just writing down the TLDR version for the intuition but didn't have the place left in the post, so here we go:
Two things are crucial:
1) First \(y\) lies outside the quaternionic subalgebra spanned by \(x\) to obtain a non-trivial associator,
2) and second, the complex factor to each term in \(x\) counters -- due to complex conjugation -- the anti-symmetry of the associator.

2025-10-13 21:58:00 GMT

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Finally I had some time to pin things down. Turns out, writing things out explicitly yields intuition and an even simpler counter example.
Lets define
\[
\begin{split}
x = \sum_i x_{oi}\otimes x_{ci}\\
y = \sum_i y_{oi}\otimes y_{ci}
\end{split}
\]
with \(x_{oi}\in\mathbb{O}\) and likewise for \(y_{oi}\) and \(x_{ci}\in\mathbb{C}\) and likewise for \(y_{ci}\). Then, writing out \((xx^*)y\) becomes
\[
\begin{split}
(xx^*)y=\sum_{ijk}((x_{oi}x_{oj}^*)y_{ok})\otimes x_{ci}x_{cj}^*y_{ck}\\
= \sum_{ijk}[x_{oi},x_{oj}^*,y_{ok}]\otimes x_{ci}x_{cj}^*y_{ck} + \\
+ \sum_{ijk}(x_{oi}(x_{oj}^*y_{ok}))\otimes x_{ci}x_{cj}^*y_{ck}
\end{split}
\]
where we used the associator \([a,b,c]=(ab)c-a(bc)\). Thus, when the term with the associator can be made not to vanish we've found a counter example.
Lets use
\[
\begin{split}
[e_1,e_2,e_3]=-2e_6\\
[e_2,e_1,e_3]=2e_6
\end{split}
\]
together with the fact that \([x_{oi},x_{oj}^*,y_{ok}]\) is zero on the \(i=j\) diagonal.
Note, that in the pure octonionic case (i.e. all the \(\mathbb{C}\) variables are real) the term with the associator vanishes, although the associator is non-zero on the \(i\neq j\) off-diagonal. Thus, the reality of the complex coefficients is crucial for the vanishing of that term and we found the last hint how to break it using \(\mathbb{C}\). Thus, we can guess the counter example
\[
\begin{split}
x=e_1\otimes 1 + e_2\otimes i \\
y=e_3
\end{split}
\]
Checking
\[
\begin{split}
(xx^*)y=2e_3\otimes 1+2e_6\otimes i\\
\neq x(x^*y)=2e_3\otimes1 - 2 e_6\otimes i
\end{split}
\]
(Based on the convention https://math.ucr.edu/home/baez/octonions/conway_smith/fano.gif)

2025-10-13 18:34:27 GMT

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I had a few minutes. If there are no sign mistakes on my side (so don't trust me here) then this is a much simpler counter example:
\[x=e_1\otimes1 + e_2\otimes i\]
\[y=e_1\otimes1 + e_3\otimes i\]

2025-10-13 18:08:35 GMT

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I ask myself the same question. Though did not yet have the time to think about it. My guess for a simpler counter example would be a combination of x and y consisting of just two octonions each which form different quaternionic subalgebras within the octonions (usually all "exceptional" examples are of this type I feel). I started with three octonions mainly because I followed my gut feeling rather than systematic search and because I wasn't sure whether 1 and i as complex coefficients would be sufficient. But it is certainly simpler to check such an example, its just 12 terms compared to the 27 terms when multiplying out the three three-fold combinations :D

2025-10-13 16:36:44 GMT

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I fear that won’t work. If my calculation calculation is correct (which it may not be given my sloppiness), then it tells me that you get a counterexample with for example:
\(x=e_1\otimes 1 + e_2\otimes e^{i2\pi/3} + e_3\otimes e^{i4\pi/3}\) and \(y=e_4\otimes1+e_5\otimes e^{i2\pi/3}+ e_6\otimes e^{i4\pi/3}\).
I’m happy later to verify my calculation and update accordingly!
PS: the \(e_i\) are the octonionic basis vectors and I used the convention 1x2–>4.