> reality itself is stepped and in fact does instantaneously jump between flat levels

That's why I said "at least, at the resolutions we tend to sample at", and "ignoring microscopic effects like phonons"

> reality itself is stepped and in fact does instantaneously jump between flat levels. Its called wave function collapse.

That's pretty-much right, but that phenomenon is "quantisation". "Wave-function collapse" is a different (but related) thing.

Values like energy are "quantised", meaning they can only occur in certain amounts/steps (for energy these are called "energy levels"). A classic example is a "particle in a box" ( https://en.wikipedia.org/wiki/Particle_in_a_box ), which acts a bit like a guitar string: each "energy level" is like a resonant frequency of a string. These are discrete, since (a) frequency is related to wavelength, and (b) resonance only occurs when a half-integer multiple of the wavelength fits inside-the-box/along-the-string (i.e. half a wavelength; a full wavelength; one and a half wavelengths; etc.).

Guitar strings can also vibrate in more complicated ways; which can be described as adding together several of the resonant frequencies (possibly with different amplitudes). This "adding together" creates a "superposition", where the different waves can interfere, to produce the complicated vibrations we see/hear. The same is true for quantum systems: their behaviour can be a complicated adding-together and interference-between multiple energy levels (possibly with different amplitudes).

SPOILER ALERT: Describing a complicated wave by adding together a bunch of simple, discrete waves is called a Fourier series; and it's exactly what the parent was talking about for audio sampling!

Wave-function collapse is a separate thing: this adding-up and interference perfectly describes the behaviour of quantum systems; but we've never actually measured a system to be in such a mixed-state. Instead, every measurement shows a particular energy level (or, more generally, "eigenstate of the measurement operator"). We don't know why, but one explanation is that mixtures "collapse" when measured, with the probability of each outcome being the square of its amplitude. (Although that can't be the full picture, since it's observer-dependent; e.g. see https://en.wikipedia.org/wiki/Wigner%27s_friend )

> I think you're not familiar with quantum mechanics

I have a Masters degree in Physics ;)

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Regarding the issue of discretely sampling a sound: you still seem to be conflating "discrete" with "stepped". Let's stick with the "particle in a box" example above, which is actually a good model of sound waves in a solid (where the particles are called "phonons" https://en.wikipedia.org/wiki/Phonon ). This system has discrete energy levels; meaning that two neighbouring states have no states "in-between". Yet each of those energy levels is a smooth, continous function over space; in fact, they're perfect sine waves!

This is the heart of the confusion: if we sample a system at time/position 0 (let's call that sample S0), and we sample it at time/position 1 (to get S1), we can model that system using any function f we like, as long as f(0) = S0 and f(1) = S1.

(NOTICE: I'm only using two samples, not "sampling to infinite precision")

A "stepped" function (AKA piece-wise interpolation, AKA nearest neighbour interpolation, AKA Voronoi cells) would be something like:

f(t) = (t > 0.5)? S1 : S0

That fits the constraints, but it's a terrible model of a sound wave (and a terrible model of a quantum state), since it doesn't vary smoothly like the real thing. Its 2D equivalent, f(x, y) = ..., is the "little squares" model of pixels.

A better model would approximate the smoothly-varying shape of the sound wave (or quantum system). For example, we can eliminate the sudden jumps using linear interpolation:

f(t) = (S1-S0)*t + S0

Notice that both of these definitions only use two samples (S0 and S1). Yet we can sample these functions at any point t we like. This is how we can go from "input pixels" (point samples S0 and S1) to "output pixels". Say we want to drive a loudspeaker, whose position can be adjusted at evenly spaced times like -1.0, -0.9, -0.8, ..., 1.9, 2.0. We can choose the position of the loudspeaker at each time by using our function f, i.e. f(-1.0), f(-0.9), ..., f(1.9), f(2.0)

NOTE: The values of t will almost-always be translated/shifted by some amount. For example, if we record some audio then play it back, the time 't=0' of the recorded samples is different to that of the playback samples, since the playback occurs at a later time. Likewise, if we sample some light at position (x, y) on our camera sensor in Hawaii, those will be different to the (x, y) positions on our computer display in New York. We may also decide to stretch/compress the scale, although that's less common for audio!

If we use the "stepped" definition of f, our loudspeaker will be stuck at position S0 for a while, then quickly move to position S1, then stay stuck there for a while.

If we use the "linear" definition of f, our loudspeaker will start at position -S1 + 2S0, then gradually move to position 2S1 - S0 (passing through position S0 at time t=0, and passing through position S1 when time t=1).

The linear definition only takes two samples into account, so it doesn't work well if we have many samples; we hit a "sharp corner" as we pass through each sampled point. In the loudspeaker example, its speed will jump (AKA it has a discontinuous derivative). We can smooth-out such corners by using an equation involving a few more samples, e.g. higher-degree polynomials, or sine/cosine waves (AKA Fourier series, which the parent was alluding to), etc. For nice visuals see https://en.wikipedia.org/wiki/Interpolation#Example

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SIDE NOTE: In normal Quantum Mechanics, mixed states can have any Real numbers as their amplitudes. Hence we can transition from one energy state to another in a smooth, continuous way; e.g. starting at 100% A + 0% B, smoothly decreasing/increasing the amplitudes of A/B, and ending up at 0% A + 100% B. That's unrelated to this discussion though. Also it requires that no measurements occur during the process; e.g. see https://en.wikipedia.org/wiki/Quantum_Zeno_effect