Why quantum math is unclassical

For me, the important thing about quantum mechanics is the equations, the mathematics.  If you want to understand quantum mechanics, just do the math.  All the words that are spun around it don't mean very much.  It's like playing the violin.  If violinists were judged on how they spoke, it wouldn't make much sense.

— Freeman Dyson, in an interview with Onnesha Roychoudhuri, Salon, 2007.

Put aside all metaphysical questions about what sort of universe could be described by quantum mechanics.  Given that quantum mechanics is a recipe for making predictions about the physical world, and that those predictions are rather peculiar by classical standards, what is it about the recipe that causes these peculiarities?

In this post, I'm going to try to vastly simplify the recipe while still producing those peculiarities:  I'm going to build a toy cosmos, a really tiny system with really simple rules that, on their face, have almost none of the specific structure of quantum mechanics; yet, if it works out right, the system will still exhibit certain particular effects whose origins —whose mathematical origins— I want to understand better.  Here's my list of effects I want:

• Nondeterminism.
• Quantum interference.
• Disappearance of quantum interference under observation.
• Quantum entanglement.

I've tried this before, more than a decade ago, but my perspective has recently changed from my explorations of co-hygiene.  A little after the turn of the millennium I was studying a 1988 MIT AI Lab memo by Gary Drescher, "Demystifying Quantum Mechanics:  A Simple Universe with Quantum Uncertainty", and wanted to use a similar technique to explore some specific peculiarities of quantum math.  I used an even simpler toy cosmos than the 1988 memo had, which I could because my goals were narrower than Drescher's.  I eventually put my results up on the web through my WPI CS Department account (2006), though I didn't feel right at the time about making it a WPI CS Department tech report (a decision I eventually came to regret, after I'd got my doctoral hood and left, and it was too late).  But, nifty though the 2006 paper was in some ways, I now feel it didn't go far enough in simplifying the simple universe.  At the time I wanted to keep the "quantum" math similar enough to actual quantum mechanics to retain its look-and-feel, so that the reader would still think, yes, that is like quantum mechanics.  Now, though, I really want to strip away almost all the structure of quantum mechanics; because I'm now very interested to know which consequences of quantum mechanics are caused by which parts of the mathematical model from which they flow.

The result, with most of the instrument missing, won't be recital-quality violin; not even musical, really.  But I hope to learn from it a bit of how the instrument works.

Contents
Classical toy cosmos
Quantum toy cosmos
Nondeterminism
Interference
Observation
Entanglement
Why

Classical toy cosmos

A quantum view of a cosmos can only be constructed relative to a classical view.  So we have to start with a classical toy cosmos.

The instantaneous state of this cosmos consists of just two boolean —true/false— variables, a and b; so there are only four possible states for the cosmos to be in, which we call TT, TF, FT, FF (listing a then b).  Time advances discretely from one moment  t  to the next  t+1, and we're allowed to apply some experimental apparatus across that interval that determines how the state at  t+1  depends on the state at  t.  There are just three kinds of experimental apparatus, each of which has two variants depending on whether it's focused on  a  or  b:

• set v: causes the variable to be true in the next state.
• clear v: causes the variable to be false in the next state.
• copy v: causes the value of the variable in the old state to become the value of both variables in the next state.

Nothing changes unless explicitly changed by the apparatus.

For example, from state TF, here are the states produced by the six possible experiments:

TF		→		set a	→		TF
TF		→		set b	→		TT
TF		→		clear a	→		FF
TF		→		clear b	→		TF
TF		→		copy a	→		TT
TF		→		copy b	→		FF

Quantum toy cosmos

A quantum state of the cosmos consists of a vector indexed by classical states; that is,  q = ⟨w_s⟩  where s varies over the four classical states of the cosmos (in order TT, TF, FT, FF).

We understand a quantum state to determine a probability distribution of classical states of the toy cosmos; for quantum state  q, we denote the probability of classical state  s  by  p_s(q).

As always when reasoning about quantum mechanics — but this bears repeating, to keep the concepts straight — we, as physicists studying the mathematics of the situation, are not observers in the technical sense of quantum theory.  That is, we are not part of the toy cosmos at all.  We can reason about the evolution of the quantum state of the toy cosmos; how an experiment changes the probabilities from time  t  to time  t+1, from  p_s(q_t)  to  p_s(q_t+1); and our reasoning does not alter the system.  Observation is one of the possible processes within the toy cosmos, which we will eventually get around to reasoning about, below.

What sorts of values, though, are the weights  w_s  within the quantum state?

In current mathematical physics, one would expect these weights to be what's called a gauge field — one of those terms that doesn't mean much to outsiders but, to those in the know, carries along a great deal of extra baggage.  We don't want that baggage here; and it's worth a moment just to consider why we don't want it.

In classical Lagrangian mechanics, one considers the evolution of a system as a path through the system's classical state-space (where points in the space are classical states of the system).  A function called a Lagrangian maps points in the state-space to energies.  The action of the system is the line integral along this path.  The principle of least action says that from a given state, the system will follow the path that minimizes the action.  One solves for this minimal path using a mathematical technology called the calculus of variations.  And Noether's theorem (yeah, yeah, Noether's first theorem) says that each differentiable invariant of the action — each symmetry of the action — gives rise to a conservation law.

In recent quantum physics, the system state — the range of points in the state-space — consists of a classical state together with what I've called here a "weight"; that's the wavy part of the wave function.  While part of that weight can be perceived more-or-less directly as probability (traditionally, probability proportional to the square of the amplitude of a complex number), the rest of it can't be perceived; but its symmetries give rise to conservation laws which in turn come out as classes of particles.  Photons, gluons, and whatnot.  The weights form a gauge field, the invariances that give rise to the conservation laws are gauge symmetries, etc.

Physicists tend to ground their thinking in an imagined "real world"; a century or so of quantum mechanics hasn't really dimmed this attitude, even if the "real world" now imagined is Platonic such as a gauge field.  The attitude has considerable merit imho (leading, e.g., to the profound change I've noted in my view of λ-calculus, which was after all originally an exercise in formalist meta-mathematics, essentially a manipulation of syntax deliberately disregarding any possible referent); but the attitude does seem to make physicists especially vulnerable to mistaking the map for the territory.  That is, in treating the gauge field as if it were "really there", the physicist may forget to distinguish between a mathematical theory that successfully describes observable features of reality, and mathematics that is "known" to underlie reality.  The Lagrangian (as I pointed out in an earlier post) isn't some magic deeper level of reality, it's just whatever works to cause the principle of least action to give the right answer; and Noether's theorem, profound as it is, points out the physical consequences of a mathematical structure that was devised in the first place from the physical world, with the mathematical structure thus serving as essentially a catalyst to reasoning.  Physicists, lacking a traditional classical-style model of reality, observe (say) a force and construct a gauge theory for it which they then think of as a theorized "real thing" (not necessary a bad attitude), reason through Noether's theorem to a class of particles, look for them using massive devices such as the Large Hadron Collider, and when they observe the phenomenon they predicted, then treat the particle as "known" and even take some properties of the gauge field as "known".  The chain of reasoning is so long that even the question of whether the observed particle "exists" is somewhat open to interpretation; and the gauge field is even more problematic.

More to the immediate point, the purpose of this post calls for avoiding the entire baggage train attached to the term "gauge", in pursuit of a minimal mathematical structure giving rise to the specifically named peculiar behaviors of quantum mechanics.

Taking a semi-educated stab at minimality, let's have just three possible weights:  a neutral weight, and two polar opposites.  Call the neutral weight 0 (zero).  One might call the other two 1 and −1, but really the orientation of those has to do with multiplication, and we're not going to have any sort of multiplication of weights by each other, so to avoid implying any particular orientation, let's unimaginatively call them left and right.  Two operations are provided on weights.  Unary negation, −w, transforms left to right, transforms right to left, and leaves 0 unchanged.

In the classical toy cosmos, each experiment determined, given the classical state  s  at time  t, the resulting classical state  s'  at time  t+1.  In the quantum version, each experiment determines, for each possible classical state  s  at time  t, and each possible classical state  s'  at time  t+1, what contribution does weight  w_t,s  make to weight  w_t+1,s'.  Each weight at time  t+1  is simply the sum of the contributions to that weight from each of the weights at time  t.  This requires, of course, that we sum a set of weights; let the sum of a set of weights be whichever of left or right there are more of amongst the arguments, or zero if there are the same number of left and right arguments.  This summation operation —for which we'll freely use the usual additive notation— is, btw, not at all mathematically well-behaved; commutative, but not associative since, for example,

left + left + (right + right)  =  left
left + (left + right) + right  =  0
(left + left) + right + right  =  right.

The ill-behavedness however is a bit moot, because in the six possible experiments of our toy cosmos, no sum will ever have more than two non-zero addends, and non-associativity only happens when there are at least three non-zero addends.

We understand a zero weight to mean that classical state is not possible at that time; and assign equal probabilities to all non-zero-weighted classical states in the quantum state.  Presumably, for all possible experiments, a zero weight at time  t  contributes zero to each weight at time  t+1. 

It remains to define, for each experiment, the contribution of each weight before the experiment to each weight after the experiment.  We'll write  s  for a classical state before,  s'  after; before weight  w_s, after weight  w'_s', and contribution of the former to the latter  w_s→s'.  We have  w'_s' = Σ_s w_s→s'  (that is, each after-weight is the sum of the contributions to it from each of the before-weights).  We'll mainly represent these transformations by tables, rather that depending on all this elaborate notation.

Consider any  set/clear v  experiment.  Before-state s contributes nothing to any after-state that changes the non-v variable. If s already has v with the value called for, only the contribution to s'=s can be non-zero, w'_s→s = w_s.  If s doesn't have the value of v called for, it contributes its weight to the state with v changed, and also contributes the negation of its weight to the unchanged state.  In all,

			set a →
TT		wTT		wTT + wFT
TF		wTF		wTF + wFF
FT		wFT		−wFT
FF		wFF		−wFF

			set b →
TT		wTT		wTT + wTF
TF		wTF		−wTF
FT		wFT		wFT + wFF
FF		wFF		−wFF

			clear a →
TT		wTT		−wTT
TF		wTF		−wTF
FT		wFT		wTT + wFT
FF		wFF		wTF + wFF

			clear b →
TT		wTT		−wTT
TF		wTF		wTT + wTF
FT		wFT		−wFT
FF		wFF		wFT + wFF

Follow the same pattern for a  copy v  experiment, adjusting which values are changed.

			copy a →
TT		wTT		wTT + wTF
TF		wTF		−wTF
FT		wFT		−wFT
FF		wFF		wFT + wFF

			copy b →
TT		wTT		wTT + wFT
TF		wTF		−wTF
FT		wFT		−wFT
FF		wFF		wTF + wFF

This has, btw, all been constructed to avoid awkward questions when interpreting quantum states probabilistically by guaranteeing that each experiment, operating on a predecessor quantum state with at least one non-zero weight, will always produce a successor quantum state with at least one non-zero weight.

Demonstrating the intended quantum effects is —if it can be done— then just a matter of assembling suitable compositions of experiments.

Nondeterminism

The fundamental difference between quantum state and classical state is, always, that any observed state of reality is classical.  Quantum state evolves deterministically — we've just specified precisely how it evolves through each experiment — and our difficulty is that we see no way to interpret the probability distributions of quantum mechanics as deterministic evolution of classical states.

Interference

The effect to be demonstrated is that a sequence of two experiments produces a probability distribution that doesn't compose the probability distributions of the two individual experiments.

Suppose we  set a  and then  clear a.  To be clear on what's going on, we start from a pure state, that is, a quantum state in which only one classical state is possible.  If that pure state has a=true, the quantum state after  set a  would be unchanged, so the final probability distribution would be just that of the second experiment,  clear a.  So choose instead a pure starting state with a=false.

			set a →		clear a →
TT				wFT		−wFT
TF
FT		wFT		−wFT
FF

Here, the second experiment produces a quantum state at time  t+2  where the weight on classical state FT is the sum of the weights on states TT and FT at time  t+1; and since the first experiment has left those two as polar opposites, they cancel,  w_FT − w_FT = 0, so the outcome of the sequence of two experiments is pure state TT.  Even though each of the experiments individually, when applied to a pure state where the value isn't what the experiment seeks to make it, would produce a probability distribution between two possible classical result states.

Observation

In the standard two-slit experiment, electron wave interference disappears when we observe which slit the electron goes through.  So, to disrupt the interference effect we've just demonstrated, put a  copy a  in between the other two operations, to observe, within the toy cosmos, the intermediate classical state of the system.

			set a →		copy a →		clear a →
TT				wFT		wFT		−wFT
TF
FT		wFT		−wFT		wFT		wFT
FF						−wFT		−wFT

Here, the final experiment gives a time  t+3  weight for FT that is the sum of the time  t+2 weights for TT and FT, but now they have the same sign so they don't cancel.

Interestingly, although this does spoil the interference pattern from the previous demonstration, it doesn't produce the crisp "classical" probability distribution that we expect observation to exhibit in a similar scenario in real-world quantum mechanics.  In my 2006 paper, I did get a crisply classical distribution; but there, the transformation of weights by the  copy v  operation was itself deterministic, assigning zero weight to those classical outcomes in which the value was not copied.  I defined the copy transformation differently this time because it had always bothered me that the 2006 paper did not guarantee that an experiment could not result in an all-zero quantum state.  My best guess, atm, as to why this zero-outcome problem doesn't ordinarily arise in full-blown quantum mechanics is that it has to do with the overall coherence provided by the wave equation, a structural component of quantum mechanics entirely omitted here.  At least, I've never heard of this particular anomaly arising in full-blown quantum mechanics; though full-blown quantum mechanics does have anomalies of its own that seem no less alarming if perhaps more sophisticated, such as infinities that may crop up causing renormalization problems in quantum gravity.

Conceivably, this may be a clue that the presence of a wave equation is profoundly fundamental to the overall structure of quantum mechanics.  Identifying the deep structural role of a wave equation, independent of the details of any particular wave equation, would seem to be another exercise for another day — though possibly not all that distant a day, given the sorts of questions I've been asking regarding co-hygiene.

At any rate, the intervening  copy a  experiment does alter the probability distribution of values of a despite the fact that the classical effect of the experiment on a pure classical state never alters the value of a.

Entanglement

The idea of entanglement, in its strongest sense, is that things done to one variable affect the other variable.  Loosely, we want to perform experiments on one variable that don't touch the other variable, yet alter the probability distribution of the other variable.  There is so little mathematical structure left in our toy cosmos, that there aren't a lot of options to consider for demonstrating this effect.  The only operations that don't touch one variable are set/clear of the other variable.  Asymmetric handling of states can be derived from the fact that the set-clear sequence we used to demonstrate interference only causes interference on a pure start state if a=false.  So, suppose we run our  set-clear  on an initial quantum state with a correlation between a and b.

			set a →		clear a →
TT				wFT		−wFT
TF		wTF		wTF		−wTF
FT		wFT		−wFT
FF						wTF

The two starting weights never get added to each other, so it doesn't matter for this sequence whether they have the same polarity, as long as they're both non-zero.  In the start state, the probability of  b=true  is 1/2, as is the probability of  a=true; in the final state, the probability of  b=true  is 1/3, while the probability of  a=true  is 2/3.

Why

Our toy cosmos deliberately leaves out most complications of quantum mechanics.  We do require, in order that the theory be at all quantum-y, to be able to understand the mathematical model as describing a probability distribution of possible perceived classical states; to understand the quantum state as being partitioned into elements associated with particular classical states; and to understand each of these elements as contributing to various elements of the successor quantum state.  That leaves the question of what sort of information a quantum state associates with each classical state; that is, what is the range over which each weight varies; and then, of course, what are the rules by which a given experiment transforms predecessor quantum state to successor quantum state.  In order to exhibit interference, it seems there must be a way for weights to cancel each other out during the summation process, and in this post I've deliberately taken the simplest sort of weight I could imagine that would allow canceling.

The resulting toy cosmos does exhibit the quantum interference effect, and clearly the demonstration of this effect does rely on weights canceling during summation.

Nondeterminism —relative, that is, to classical states— arises, potentially, when a single predecessor classical-state contributes non-zero weight to more than one successor classical-state.  Interference arises (given the cancellation provided for), again potentially, when a single successor classical-state receives non-zero contributions from more than one predecessor classical-state.

The quantum interference effect depends crucially on the fact that weights are holistic.  That is, a weight is assigned to a classical state of the entire cosmos; it isn't a characteristic of any particular feature within the classical state of the cosmos.  This is why observation within the toy cosmos disrupts interference:  once the particular part of the cosmos we're manipulating (variable a in our demonstration) is "observed" by another part of the cosmos (variable b in our demonstration), the classical state of the cosmos as a whole may differ because of what the observer saw, so that interference does not occur.  (Tbh, this point was more clearly exhibited in the 2006 paper, where observation was absolute — as it is in the full-blown quantum mechanics of our physical world; but it is still there to be found in the toy cosmos of this blog post.)

Entanglement was something I really wanted to understand in 2006; curiously, in 2018 I'm finding it less interesting than observation.  An experiment can cause interference amongst the successors of one classical-state and not amongst the successors of another classical state, so that, in the quantum successor-state, successors of one classical-state are collectively more probable than successors of another classical-state.  If the experiment only manipulates one variable (a) without affecting the other (b), this difference in probabilities of successor states can mean a difference in probabilities of values of the unmanipulated variable (b).

These latter two points are somewhat murkier from the above demonstrations than they were from the 2006 paper; the murkiness is apparently due to my decision in this blog post to define the  copy v  operation as something that might or might not change the state, rather than something that always changes the state in the 2006 paper; and that decision was made here due to considerations of avoiding possible quantum zero-states.  As noted earlier, this seems to be something to do with the absence, from this immensely simplified mathematical structure, of a wave equation that would ward off such anomalies.

It seems, then, that I went into this blog post seeking to clarify minimal structure needed to produce certain quantum effects; and confirmed that those effects could still be produced by the chosen reduced structure; but the structure became so reduced that the demonstrations were less clear than in the 2006 paper, and questions arose about what other primal characteristics of quantum mechanics may have already been lost due to evisceration of internal structure of the transformation of quantum state, i.e., the "wave equation" which has been replaced above by ad hoc tables specifying the successor weights for each experiment.

Sapience and the limits of formal reasoning

Anakin:      Is it possible to learn this power?
Palpatine:  Not from a Jedi.

— Star Wars: Episode III – Revenge of the Sith, George Lucas, 2005.

In this post I mean to tie together several puzzles I've struggled with, on this blog and elsewhere, for years; especially, on one hand, the philosophical implications of Gödel's results on the limitations of formal reasoning (post), and on the other hand, the implications of evidence that sapient minds are doing something our technological artifacts do not (post).

From time to time, amongst my exploratory/speculative posts here, I do come to some relatively firm conclusion; so, in this post, with the philosophical implications of Gödel.  A central notion here will be that formal systems manipulate information from below, while sapiences manipulate it from above.

As a bonus I'll also consider how these ideas on formal logic might apply to my investigations on basic physics (post); though, that will be more in the exploratory/speculative vein.

As this post is mostly tying together ideas I've developed in earlier posts, it won't be nearly as long as the earlier posts that developed them.  Though I continue to document the paths my thoughts follow on the way to any conclusions, those paths won't be long enough to meander too very much this time; for proper meandering, see the earlier posts.

Contents
Truth
Physics

Truth

Through roughly the second half of the nineteenth century, mathematicians aggressively extended the range of formal reasoning, ultimately reaching for a single set of axioms that would found all of logic and mathematics.  That last goal was decisively nixed by Gödel's Theorem(s) in 1931.  Gödel proved, in essence, that any sufficiently nontrivial formal axiomatic system, if it doesn't prove anything false, cannot prove itself to be self-consistent.  It's still possible to construct a more powerful axiomatic system that can prove the first one self-consistent, but that more powerful system then cannot prove itself self-consistent.  In fact, you can construct an infinite series of not-wrong axiomatic systems, each of which can prove all of its predecessors self-consistent, but each system cannot prove its own self-consistency.

In other words, there is no well-defined maximum of truth obtainable by axiomatic means.  By those means, you can go too far (allowing proofs of some things that aren't so), or you can stop short (failing to prove some things that are so), but you can't hit the target.

For those of us who work with formal reasoning a lot, this is a perplexing result.  What should one make of it?  Is there some notion of truth that is beyond the power of all these formal systems?  And what would that even mean?

For the question of whether there is a notion of objective mathematical truth beyond the power of all these formal systems, the evident answer is, not formally.  There's more to that than just the trivial observation that something more powerful than any axiomatic system cannot itself be an axiomatic system; we can also reasonably expect that whatever it is, we likely won't be able to prove its power is greater axiomatically.

I don't buy into the notion that the human mind mystically transcends the physical; an open mind I have, but I'm a reductionist at heart.  Here, though, we have an out.  In acknowledging that a hypothetical more-powerful something might not be formally provable more powerful, we open the door to candidates that we can't formally justify.  Such as, a sapient mind that emerges by some combination of its constituent parts and so seemingly ought to be no more powerful than those parts, but... is.  In practice.  (There's a quip floating around, that "In theory, there is no difference between theory and practice. But, in practice, there is.")

A related issue here is the Curry-Howard correspondence, much touted in some circles as a fundamental connection between computation and logic.  Except, I submit it can't be as fundamental as all that.  Why?  Because of the Church-Turing thesis.  Which says, in essence, that there is a robust most-powerful sort of computation.  In keeping with our expectation of an informal cap on formal power, the Church-Turing thesis in this general sense is inherently unprovable; however, specific parts of it are formally provable, formal equivalence between particular formal models of computation.  The major proofs in that vein, establishing the credibility of the general principle, were done within the next several years after Gödel's Theorems proved that there isn't a most-powerful sort of formal logic.  Long story short:  most-powerful sort of computation, yes; most-powerful sort of formal logic, no; therefore, computation and formal logic are not the same thing.

Through my recent post exploring the difference between sapient minds and all our technological artifacts, I concluded, amongst other things, that  (1) sapience cannot be measured by any standardized test, because for any standardized test one can always construct a technological artifact that will outperform sapient minds; and  (2) sapient minds are capable of grasping the "big picture" within which all technology behaves, including what the purpose of a set of formal rules is, whether the purpose is achieved, when to step outside the rules, and how to improvise behavior once outside.

A complementary observation about formal systems is that each individual action taken —each axiomatic application— is driven by the elementary details of the system state.  That is, the individual steps of the formal system are selected on a view looking up from the bottom of the information structure, whereas sapience looks downward from somewhere higher in the information structure.  This can only be a qualitative description of the difference between the sapient and formal approaches, for the simple reason that we do not, in fact, know how to do sapience.  As discussed in the earlier post, our technology does not even attempt to achieve actual sapience because we don't know, from a technical perspective, what we would be trying to achieve — since we can't even measure it, though we have various informal ways to observe its presence.

Keep in mind that this quality of sapience is not uniform.  Though some cases are straightforward, in general clambering up into the higher levels of structure, from which to take a wide-angle view, may be extremely difficult even with sapience, and some people are better at it than others, apparently for reasons of both nature, nurture, and circumstance.  Indeed, the mix of reasons that lead a Newton or an Einstein to climb particularly high in the structure are just the sort of thing I'd expect to be quite beyond the practical grasp of formal analysis.

What we see in Gödel's results is, then, that even when we accept a reductionist premise that the whole structure is built up by axioms from an elementary foundation, for a sufficiently powerful system there are fundamental limits to the sorts of high-level insights that can be assembled by building strictly upward from the bottom of the structure.

Is that a big insight?  Formally it says nothing at all.  But I can honestly say that, having reached it, for the first time in <mumble-mumble> decades of contemplation I see Gödel's results as evidence of something that makes sense to me rather than evidence that something is failing to make sense to me.

Physics

In modern physics, too, we have a large-scale phenomenon (classical reality) that evidently cannot be straightforwardly built up by simple accretion of low-level elements of the system (quanta).  Is it possible to understand this as another instance of the same broad phenomenon as the failure, per Gödel, to build a robust notion of truth from elementary axioms?

Probably not, as I'll elaborate below.  However, in the process I'll turn up some ideas that may yet lead somewhere, though quite where remains to be seen; so, a bit of meandering after all.

Gödel's axiomatic scenario has two qualitative features not immediately apparent for modern physics:

• Axiomatic truth appears to be part of, and therefore to evolve toward, absolute truth; the gap between the two appears to be a quantitative thing that shrinks as one continues to derive results axiomatically, even though it's unclear whether it shrinks toward zero, or toward some other-sized gap.  Whereas, the gap between quantum state and classical state is clearly qualitative and does not really diminish under any circumstances.
• The axiomatic shortfall only kicks in for sufficiently powerful systems.  It's not immediately clear what property in physics would correspond to axiomatic power of this sort.

The sapience/formalism dichotomy doesn't manifest the same way for different sorts of structure; witness the aforementioned difference between computational power and axiomatic power, where apparently one has a robust maximum while the other does not.  There is no obvious precedent to expect the dichotomy to generate a Gödel-style scale-gap in arbitrary settings.  Nonetheless; might there still be a physics analog to these features of axiomatic systems?

Quantum state-evolution does not smooth out toward classical state-evolution at scale; this is the point of the Schrödinger's-cat thought experiment.  A Gödel-style effect in physics would seem to require some sort of shading from quantum state-evolution toward classical state-evolution.  I don't see what shading of that sort would mean.

There is another possibility, here:  turn the classical/quantum relationship on its head.  Could classical state-evolution shade toward quantum state-evolution?  Apparently, yes; I've already described a way for this to happen, when in my first post on co-hygiene I suggested that the network topology of spacetime, acting at a cosmological scale, could create a seeming of nondeterminism at comparatively small scales.  Interestingly, this would also be a reversal in scale, with the effect flowing from cosmological scale to small scale.  However, the very fact that this appears to flow from large to small does not fit the expected pattern of the Gödel analogy, which plays on the contrast between bottom-up formalism and top-down sapience.

On the other front, what of the sufficient-power threshold, clearly featured on the logic side of the analogy?  If the quantum/classical dichotomy is an instance of the same effect, it would seem there must be something in physics corresponding to this power threshold.  Physics considered in the abstract as a description of physical reality has no obvious place for power in a logical or computational sense.  Interestingly, however, the particular alternative vein of speculation I've been exploring here lately (co-hygiene and quantum gravity) recommends modeling physical reality as a discrete structure that evolves through a dimension orthogonal to spacetime, progressively toward a stable state approximating the probabilistic predictions of quantum mechanics — and it is reasonable to ask how much computational power the primitive operations of this orthogonal evolution of spacetime ought to have.

In such a scenario, the computational power is applied to state-evolution from some initial state of spacetime to a stable outcome, for some sense of stable to be determined.  As a practical matter, this amounts to a transformation from some probability distribution of initial states of spacetime, to a probability distribution of stable states of spacetime that presumably resembles the probability distributions predicted by quantum mechanics.  As it is unclear how one chooses the initial probability distribution, I've toyed with the idea that a quantum mechanics-like distribution might be some sort of fixpoint under this transformation, so that spacetime would tend to come out resembling quantum mechanics more-or-less-regardless of the initial distribution.

The spacetime-rewriting relation would also be the medium through which cosmological-scale determinism would induce small-scale apparent nondeterminism.

Between inducing nondeterminism and transforming probability distributions, there would seem to be, potentially, great scope for dependence on the relative computational power of the rewriting relation.  With such a complex interplay of factors at stake, it seems likely that even if there were a Gödel-like power threshold lurking, it would have to be deduced from a much better understanding of the rewriting relation, rather than contributing to a basic understanding of the rewriting relation.  Nevertheless, I'm inclined to keep a weather eye out for any such power threshold as I move forward.