@@ -0,0 +1,7 @@
+
+The material here is written by [Randall C. O'Reilly](https://randalloreilly.com) (so far exclusively -- contributions are always welcome!).
+
+As of Sept 20, 2026, I started using Claude (Opus 5.5) to help me understand various things, and to help writing the [[Simulations]] software. However, all of the content is entirely under my strict editorial control, and Claude is only used to provide suggestions and minor edits to content that I wrote.
+
+Full logs of the dialog with Claude, including my prompts, is available on [github](https://github.com/WaveReality/web/tree/main/notes).
+
@@ -25,10 +25,14 @@ Based on this general logic, it would seem that the precise order in which the m
 
 This situation reflects the deeper problem that _the quantum state is fundamentally incompatible with special relativity_, even though the overall resulting phenomena (i.e., the measurements) are not ([[@Bell75]]; [[@AharonovAlbert81]]; [[@Peres00]]; [[@PeresTerno04]]; [[@Norsen11]]). This incompatibility is intuitively obvious, similar to the Coulomb gauge for the electric field, given that the quantum configuration space is manifestly non-local, and yet the theory says it must collapse instantaneously everywhere.
 
-At the phenomenological level, it has been shown that the non-local quantum correlations cannot be used to actually send signals in a way that would violate special relativity ([[@Bell75]]; [[@BallentineJarrett87]]), but this does not mean that the underlying mathematical framework that is used to explain these results is not fundamentally incompatible. Indeed, the authors discussing this conflict invoke the idea that the quantum state is just a [[calculational tool]], not a "beable" (a physically real thing), so we shouldn't be too disturbed by this situation ([[@Bell75]]; [[@PeresTerno04]]). But the obvious problem with such an approach is that there is at this point no viable alternative framework that can otherwise explain how these non-local correlations in the measurements might arise in a physical system whose mechanisms obey the constraints of special relativity. Thus, we appear to have a physically real, replicable phenomenon that is otherwise inexplicable, except by using a calculational tool that is manifestly non-local.
+At the phenomenological level, it has been shown that the non-local quantum correlations cannot be used to actually send signals in a way that would violate special relativity ([[@Bell75]]; [[@BallentineJarrett87]]), but this does not mean that the underlying mathematical framework that is used to explain these results is not fundamentally incompatible. Indeed, the authors discussing this conflict invoke the idea that the quantum state is just a [[calculational tool]], not a "beable" (a physically real thing), so we shouldn't be too disturbed by this situation ([[@Bell75]]; [[@PeresTerno04]]).
+
+But the obvious problem with such an approach is that there is at this point no viable alternative framework that can otherwise explain how these non-local correlations in the measurements might arise in a physical system whose mechanisms obey the constraints of special relativity. Thus, we appear to have a physically real, replicable phenomenon that is otherwise inexplicable, except by using a calculational tool that is manifestly non-local.
 
 The non-locality of the configuration space does create very "real" problems for the Bohmian version of the [[pilot wave]] framework, because it postulates that the particle positions are continuously well-defined in 3D space, and yet they are being guided around by a non-local, exponentially-large configuration space wave state. Thus, it is not possible to dodge the conflict with special relativity in this framework by hiding behind the calculational tool argument, and this conflict thus remains one of the most significant difficulties with the pilot wave framework ([[@Norsen15]]; [[@Norsen11]]).
 
+The concept of superposition in the position basis is particularly interesting in the pilot-wave framework, because the positions of the discrete particles are continously defined: the particles themselves are _always_ in a specific, well-defined location. Thus, how can there be a superposition state in such a case? The answer is that the corresponding quantum wave can have the superposition state, and this ends up having the appropriate affects on the discrete particles such that they reflect the probabilities that you would otherwise get from the standard quantum formalism.
+
 Here are a few other relevant high-level points.
 
 * Superposition is entirely a wave-specific phenomenon. Any kind of classical [[wave]] automatically exhibits superposition. And yet it still looks a bit like magic to see the superposition in effect in a classical wave.
 
@@ -0,0 +1,26 @@
++++
+Categories = ["Interpretations"]
+bibfile = "mechphys.json"
++++
+
+This page provides a summary of what remains as unresolved **challenges** in the quest to actually make sense of quantum physics, relative to the [[mechanisms]] conclusions and specific proposals of the [[Spinfield Model]].
+
+## Non-locality
+
+[[Non-locality]] and the [[Bell]] inequality tests, along with the associated exponential [[configuration space]], represent the primary unresolved challenge at the level of abstract quantum mechanics. Critically, this should be considered a challenge for the [[Standard Model]] as well, because the tensor-product configuration space state that is necessary to compute the predictions in these experiments is clearly in conflict with the constraints of [[special relativity]], even if the resulting predictions themselves end up being invariant to this underlying conflict. 
+
+Given the fundamentally inexplicable nature of these issues in any current framework, it seems premature to hold up making progress on other fronts on this account. Thus, we can just put this on the stack, and come back to it once more tractable issues have been resolved. 
+
+## Field-theory vs. classical waves
+
+The Standard Model uses [[field theory]] and [[virtual particles]] (via [[Feynman diagrams]]) to compute many aspects of the theory, whereas the [[semiclassical]] approach suggests that at least the force-field waves remain as such, rather than taking on a particle-like status as treated in the field-theory formalism (which, as emphasized throughout, does not actually involve any actual discrete particles).
+
+At least some of the roles of virtual particles in the standard framework are otherwise accounted for by the classical interacting field model, for example a virtual particle is used in the perturbative framework to model the basic interaction between the electron and the electromagnetic field, including the self-interaction effects.
+
+Other effects require extensive sampling of initial configurations to compute expected values of outcomes -- this is something that can become computationally expensive in the classical wave model, but does not reflect a fundamental difference in underlying physics. 
+
+One critical contribution of virtual particles is in representing the quantum vacuum state, which is teaming with random possibilities. It is possible that the quantum waves end up leaking out and creating a more active vacuum state expectation. The [[zero point]] framework shows how various quantum phenomena can potentially be explained in terms of a semiclassical system interacting with a specific form of vaccum state.
+
+As discussed in [[vacuum]], and of relevance to the [[Higgs]] mechanism, the energy associated with the zero point field or the vacuum expectation of the Higgs exceeds cosmologically plausible values by so many orders of magnitude that they are considered "the worst theoretical predictions in the history of physics" [[@^HobsonEfstathiouLasenby06]]. Thus, there are a number of currently unresolved issues regarding the nature of this vacuum state, but various empirically-validated phenomena depend on its effects as represented in the Standard model.
+
+
@@ -4,7 +4,7 @@ bibfile = "mechphys.json"
 +++
 
 {id="figure_complex" style="height:30em"}
-![Complex numbers are just a way of representing two real values with one number, where these two values are aligned along two separate orthogonal dimensions. The imaginary number $i$, where $i^2 = -1$, is what keeps these two values orthogonal --- the first value $a$ is along the real axis, and the second value $b$ is along the imaginary axis. The complex conjugate, $c^*$, is simply subtracting the imaginary part instead of adding it (i.e., it represents a reflection along the imaginary dimension). Multiplying $c c^*$ gives the squared magnitude of the vector, which is a single real-valued scalar number. It is the (squared) length of the hypotenuse of the vector. In the typical oscillatory use of complex numbers, they represent rotation around the complex plane, constrained to the (unit) circle as shown.](media/fig_complex_numbers.png)
+![Complex numbers are just a way of representing two real values with one expression, which is essentially a vector of two numbers. These two values are aligned along two separate orthogonal dimensions. The imaginary number $i$, where $i^2 = -1$, is what keeps these two values orthogonal --- the first value $a$ is along the real axis, and the second value $b$ is along the imaginary axis. The complex conjugate, $c^*$, is simply subtracting the imaginary part instead of adding it (i.e., it represents a reflection along the imaginary dimension). Multiplying $c c^*$ gives the squared magnitude of the vector, which is a single real-valued scalar number. It is the (squared) length of the hypotenuse of the vector. In the typical oscillatory use of complex numbers, they represent rotation around the complex plane, constrained to the (unit) circle as shown.](media/fig_complex_numbers.png)
 
 The symbol $\chi$ is used to represent a complex-valued state variable:
 
@@ -16,7 +16,7 @@ $$
 \def \phi_a + i \phi_b
 $$
 
-So, $\chi$ is composed of two separate real-valued numbers, designated $a$ and $b$ (or $\phi_a$ and $\phi_b$, to indicate that they are scalar state variables). A complex number is really just a way of representing two separate real valued numbers, aligned along orthogonal dimensions, in an efficient and compact manner ([[#figure_complex]]). It is essential to appreciate that, despite the presence of the imaginary number $i$ (where $i^2 = -1$ or $i = \sqrt{-1}$), _all you ever really have is two real-valued numbers._ There is nothing "imaginary" or mysterious or spooky about the second number in a complex number: all the $i$ does is keep these two values separate from each other.
+So, $\chi$ is composed of two separate real-valued numbers, designated $a$ and $b$ (or $\phi_a$ and $\phi_b$, to indicate that they are scalar state variables). A complex number is really just a way of representing a _vector_ of two separate real valued numbers, aligned along orthogonal dimensions, in an efficient and compact manner ([[#figure_complex]]). It is essential to appreciate that, despite the presence of the imaginary number $i$ (where $i^2 = -1$ or $i = \sqrt{-1}$), _all you ever really have is two real-valued numbers._ There is nothing "imaginary" or mysterious or spooky about the second number in a complex number: all the $i$ does is keep these two values separate from each other.
 
 In the end, we will deconstruct all of our complex numbers into their real-valued components, and write purely real-valued expressions that determine their update rules. These expressions will be more complicated than the ones using complex numbers, but they are required for actually implementing the equations on the computer, and they also provide a more explicit and obvious indication of exactly what drives each value.
 
@@ -99,4 +99,25 @@ $$
 
 Which is directly a rotation around the the complex plane: the real component is multiplied by the x-axis-like rotational factor, while the imaginary component gets the y-axis-like rotational factor. If $\theta$ is small, then cosine is close to 1 and sine is close to 0, and there isn't much rotation, etc.
 
+The essential insight for _why_ Euler's formula is true, is that the derivative of the sine function is cosine, and vice-versa:
+
+$$
+\frac{d}{dx} \sin x = \cos x; \qquad \frac{d}{dx} \cos x = -\sin x
+$$
+
+and the natural exponential function $e^x$ is defined as the function whose derivative is itself!
+
+$$
+\frac{d}{dx} e^{x} = e^{x}
+$$
+
+So by putting the $i$ into this expression, Euler's formula emerges because the $i$ switches the real component into the imaginary one, and vice-versa:
+
+$$
+\frac{d}{dx} e^{i\theta} = \cos \theta + i \sin \theta = e^{i\theta}
+$$
+
+low and behold, the derivative is the same as the function itself!
+
+Fundamentally, this deep connection between trigonometry and the natural exponential arises because the circle has this property, captured in the derivatives, that as you move along it, the _x_ and _y_ components are constantly accelerating and decelerating in just such a way as to keep the total radius constant. Thus, the slope (derivative) of one is equal to the value of the other, and vice-versa.
 
@@ -5,11 +5,8 @@ bibfile = "mechphys.json"
 +++
 
 <!--- TODO: -->
-<!--- * mechanism: particle generates its field, also update Spinfield plan -->
 <!--- * complex-waves: finish -->
-<!--- better picture of what's missing from standard field theory vs. lattice approach -->
-<!--- aside from the stats etc. -->
-
+<!--- quantum optics details for bell's tests on photons -- all the things you need to do to massage the thing into an actual superposition -->
 
 <img src="media/icon.png" style="width:128px;height:128px;align-self:center">
 
@@ -21,7 +18,15 @@ However, this idea doesn't take into account the way that [[Feynman diagram]]s i
 
 Thus, even though the raw ingredients of quantum theory are wave equations, they are actually _used_ in a way that ultimately describes the behavior of particles. The opposite view, that there are only particles, is even less tenable: as argued above, there are many strong indications that wave phenomena are required (see also [[@Hobson13]]).
 
-The [[pilot-wave]] framework of de Broglie and Bohm ([[@Bohm52]]; [[@Norsen22a]]) instead embraces the **wave-particle** [[duality]] fully, by positing physically real [[quantum wave]]s that influence the motion of discrete particles, which always have a well-defined location in 3D space. This framework naturally and intuitively explains many of the otherwise paradoxical phenomena in quantum physics, and it thus provides the foundation for the work here. See [[mechanisms]] for a more detailed overview of how waves and particles can work together to provide a mostly sensible quantum picture.
+The [[pilot-wave]] framework of de Broglie and Bohm ([[@Broglie26]]; [[@Broglie56]]; [[@Bohm52]]; [[@Holland95]]; [[@Norsen22a]]) instead embraces the **wave-particle** [[duality]] fully, by positing physically real [[quantum wave]]s that influence the motion of discrete particles, which always have a well-defined location in 3D space. This framework naturally and intuitively explains many of the otherwise paradoxical phenomena in quantum physics, and it thus provides the foundation for the work here. See [[mechanisms]] for a more detailed overview of how waves and particles can work together to provide a _mostly_ sensible quantum picture.
+
+The overall conclusion from everything that has been learned in generating the content here on this website, is that, contrary to the admonitions of many prominent figures in quantum physics over the years, a truly surprising amount of quantum physics actually does make sense. This motivates the attempt to keep going, and see how far we can get! See [[challenges]] for a summary of what remains.
+
+The primary motivation in writing all of this is for the [[author]] to learn by teaching: if I can explain everything clearly to a notional, naive reader, then I feel like I truly understand it myself. You as the reader can thus benefit from all my hard work trying to understand all the complexities of this domain. Furthermore, there are nice graphical [[Simulations]] of many fundamental quantum phenomena, which are also an essential part of my learning process: if I can simulate and see what is going on, then I can better understand it. I don't know how everyone else gets by without this kind of aid! You can get by here by just running these simulations and seeing the phenomenology, but of course, to fully understand it, you will need to engage with all the math, which is fully presented and explained in a way that at least helped me to understand it.
+
+Finally, the last part of this effort represents my attempt to come up with a new framework that might make even more sense, based directly on the summary analysis in [[mechanisms]], which provides a high-level integration of the core insights that are explained in detail across the content here. This is very much a work-in-progress, and is continuously updated (you have been warned).
+
+See the [github](https://github.com/WaveReality) repositories for the revision history and released versions of these documents and the associated wave simulation software. This is also the best place to contribute, by filing issues or starting discussions (all such input is welcome!).
 
 ## Organization
 
@@ -63,7 +68,7 @@ Have these levers, within the broader context of the pilot-wave framework where
 
 ### Spinfield Model
 
-The [[Spinfield Model]] is the work-in-progress attempt to develop a fully explicit functional computational-level model of the phenomenology of the Standard Model, within the CA and pilot-wave framework, and motivated by an attempt to understand the qualitative nature of the wave and particle [[mechanisms]] in the quantum world. This model captures particle-like properties using a field of complex-valued [[harmonic oscillator]]s (CVHOs) that [[spin]], which interact bidirectionally with [[quantum wave]]s, such as the [[Dirac]] wave equations for an [[electron]], and via the electromagnetic field as described by the [[Maxwell]] wave functions. The particles move via [[stochastic motion]] driven by velocity values encoded in the phase relationships among the CVHOs.
+The [[Spinfield Model]] is the work-in-progress attempt to develop a fully explicit functional computational-level model of the phenomenology of the Standard Model, within the CA and pilot-wave framework, motivated by an attempt to understand the qualitative nature of the wave and particle [[mechanisms]] in the quantum world. This model captures particle-like properties using a field of complex-valued [[harmonic oscillator]]s (CVHOs) that [[spin]], which interact bidirectionally with [[quantum wave]]s, such as the [[Weyl]] wave equations for an [[electron]], and via the electromagnetic field as described by the [[Maxwell]] wave functions. The particles move via [[stochastic motion]] driven by velocity values encoded in the phase relationships among the CVHOs.
 
 Beyond all the basic quantum phenomena, the nature of [[neutrino]]s and the [[weak]] interactions provide strong constraints and inspiration for the Spinfield Model. For example, the _muon_ is a heavy [[generation]] of an electron, which decays into an electron and two neutrinos, via the weak force. This suggests that the extra mass / energy of the muon is some kind of excited state relative to an electron, and that it can be released in the form of these neutrinos.