Some of this ground has been covered before. I use google to search the forum.

site:yamahasynth.com montage polyphony

Returns:
https://www.yamahasynth.com/ask-a-question/awm2-polyphony-questions-and-observations
That's the first return - nothing official from Yamaha so maybe limited use.
https://www.yamahasynth.com/ask-a-question/about-polyphony
Second return - this one has information from Yamaha.
https://www.yamahasynth.com/ask-a-question/awm2-polyphony-test
Next return ... limited use but Yamaha does chime in
https://www.yamahasynth.com/ask-a-question/polyphony-sound-cut-out-and-release-time
This one describes polyphony in a way that more addresses some of your questions more directly.

... and so on. There's some research that can be done.

Somewhere scattered in the past messages are also discussions where PART number (position) of elements relates to priority if the engine needs to cut off an element from sounding in order to maintain polyphony maximums.

1. Do the number of elements affect polyphony? If those elements are spread out among multiple parts, would it change anything? The gist of the question is this: If I am going to create a complex sound, is there an advantage to try to stuff as much of that complexity with many elements into a single part vs fewer elements among multiple parts? If polyphony is limited by the number of parts vs the number of elements, it would seem like an inefficient “artificial” limit. You could conceivably play 8 times the musical wave forms using less parts. Is my thought process faulty?

Number of elements - if you just stop there do not affect polyphony. I know you may bop me on the side of the head and say I should assume an element means the element is in use - but it's important to realize that often elements are not in use even though they are in a PART. When constructing instruments from scratch - one goal is to conserve simultaneously activated elements in order to conserve polyphony. If one instrument (lets call it a one-PART instrument) uses 8 elements that all are active at the same time AND this instrument allows for chords (more than one note simultaneously played) - then each finger you add to your chord subtracts 8 units of polyphony. 8 fingers down is quite a big chunk of polyphony for only one PART. This isn't generally how Performances and PARTs are constructed. Lets take a piano (and make up one as a thought experiment). It can be one PART to make the description easy. It can use all 8 elements - but only ONE can sound at any given time (ignoring overlap from the release of the previous note). Element 1 can be the soft (ppp) sample and will only sound when you strike a key very softly. All the way through element 8 which is the very loudest sample (fff) and will only sound when you strike a key very hard. Then elements 2-7 are all levels between and have their own dedicated range of velocity and have samples that sound different for the different speeds you strike the keys. Even though there are 8 elements - only zero (not playing) or one element is sounding for each note struck. Chords are then 1:1 with the amount of fingers you're pressing at once since only one out of the 8 elements sounds at one time for each note.

There are also other ways to ensure elements do not sound simultaneously. You can use XA control which requires the assignable switches to match a certain pattern before elements will sound. Lets make up a goofy PART that uses all 8 elements - but the elements (all 8) will be OFF if the assignable switch 1 is ON. Now lets place this goofy PART into all slots 1-8 (PART1 through PART 8). So now we have the maximum number of elements assigned to all PARTs we can play with the piano keys at once. It's worst case. Turn ON Assignable Switch 1. This disables all elements from being activated. You can play as many keys as you want and will never run out of polyphony because even though you have assigned elements - a full stack - to all PARTs 1-8 - NONE of the elements are activated by policy. You can see that using switches is another way to manage if elements are using polyphony or not and why you can't just count the elements to determine polyphony. The same if you assigned the trigger velocity to be 127 for all elements. None would sound unless you started really mashing down on the keyboard. Or if you limited the elements to only sound in a certain range - they wouldn't use any polyphony if you were striking keys outside of the note limits.

HOW you shuffle around elements across PARTs is up to you - but there's other things that come into play when baking your own PART cakes from scratch. There's "only" 8 PART slots - so you may want to "play nice" and not consume too many PARTs. That is one consideration to limit the "width" of your creation. On the other end are nitty-gritty details on what's available to a single PART. Like perhaps how effects act on elements - and what your vision is of the effect sharing. Or how some parameters of a PART force peanut-butter spreading on ALL elements of a single PART - and you may not want to have those parameters the same for all elements (forcing you to either grow to another PART - or compromise). Or simply you run out of elements and just have to grow to another PART (or optimize your element usage and refactor the PART you're using). As long as when you move an element from one PART to another PART you don't change the triggering of the element or masking of the element (switches, etc) - then it doesn't matter which PART your element lives in for this one consideration. The only thing that will happen moving an element from one PART to another will be a change in its priority. But that's an exception case when things go "bad" and polyphony is exceeded. Ideally, you construct a PART or PARTs that "play nice" and don't cause polyphony issues so what the priority is doesn't matter. But you can plan ahead for polyphony being exceeded and maybe place "gimmick" elements in a lower priority position. Like extra stuff you can do without. Maybe like a thud of a piano key when you press it is expendable so you can place this in a lower priority PART (assuming you're using more than one PART for your creation). PART1 is highest priority and it goes down from there.

Yamaha made a comment on this here: https://www.yamahasynth.com/ask-a-question/montage-sound-clipping-issue
... found with the same kind of google search.

Polyphony isn't limited by the PARTs used. It's limited by active oscillators (elements). Say you have 7 PARTs that are all used only to control external gear (local keyboard turned off - or many other ways to stop these PARTs from activating elements on the local tone generator) - then have an 8th PART that's an organ PART that is polyphony hungry (organs - ones that simulate drawbars with elements - use up a lot of polyphony because these have multiple elements sounding at once to blend a mixed outcome). The 7 PARTs are not going to affect the organ PART at all. They don't consume any polyphony. They would if they had activated elements. It's about active elements. Not PARTs. You have to deconstruct a Performance (and PARTs within) to see how many active elements you can have at once.

2. This question is kind of related to #1 - but the description above outlines how pianos generally conserve elements so each note struck doesn't consume more than one element (for the most part). The 1-PART piano just has either less velocity levels or less note ranges or both. If you start picking apart the difference between the 1-PART pianos and multi-PART pianos, you'll see what's going on. A good breakdown (for MODX - couldn't find Montage but it's not important) of CFX Stage vs. CFX Concert is here:

https://www.yamahasynth.com/synths/performance-basics-i-mastering-modx

You see that CFX stage uses 5 velocity ranges for the low-to-mid range notes and then uses 2 velocity ranges for the upper range. That's 7 elements and the last (8th) element is a key-off noise. It's only used for the low-to-mid range notes (and one velocity level = any velocity). That's how this piano was constructed. Any note while you're holding down a key only uses one element at a time. When you lift up, a short percussive element fires off and quickly dies (keyoff triggered sound).

The CFX Concert (4 PARTs) is constructed with more detail. I'll just copy it verbatim from the article:

PART 1 only has data across the body of the instrument up to the note “G5” and is a 6-way velocity switch through to a maximum velocity of “92”. It cannot be used alone – it only makes sense used with these other components.
PART 2 only has data across the keys up to G5 and is a 3-way velocity switch starting at higher velocities “93” through “127”. It cannot be used alone – it only makes sense as part of these other components.
PART 3 only has data above G#5 and is an 8-way velocity switch in the undampered area of the piano.
PART 4 only has the Key Off Sound – the dampers and felts falling back in place on notes up to G5.

You can see that although you may be able to "pack" 2 velocity levels from PART to PART 1 - you'd still have one left over that couldn't fit. Maybe you could pack the key-off noise with the 3-way velocity switch of PART 2 - since there's room there to pack this Performance tighter into 3 PARTs. The benefit would not be polyphony (except priority) but would be conserving PARTs. I would probably just delete the key-off noise to conserve PARTs myself (if needed) rather than packing more tightly. The "nice" thing about having the key-off noise in a high PART is that it's the first to get cut off when polyphony is running out (in this group). There may be another reason you can dig out for having the PARTs segmented like this. Those other considerations do not relate to polyphony - so it's not worth digging here.

3. If two different parts use different ranges of the keyboard, do you get the same polyphony as if those parts were overlapped? Followup question: if those ranges were made at the element level vs the common level would it make a difference?

If elements are forced not to sound simultaneously due to key range (split) - then this would conserve polyphony vs. if you layered. Layering takes more polyphony if the elements at all layers are activated at the same time (and not otherwise limited by button masking or velocity range or ...). And this is another case where I have to be technical and say that just because notes overlap doesn't mean they sound. Overlapping is a left-and-right range on the keyboard. There's some range where it starts and some range it ends. Lets say you have a Montage 6 and you don't get access to all notes because the keyboard is smaller (less notes in the physical range). And lets say the overlap is entirely in a range below your lowest note. You'll never "hit" that range of overlap - so overlap unto itself doesn't eat up polyphony. You have to activate the elements. Here - you'd have to press the [OCTAVE - ] button a few times to get the piano keys to reach that overlap before you can activate those layered elements.

If an element is activated or not has rules. Masking (XA control), note limit/range, velocity limits, etc. If an element isn't active - it doesn't necessarily matter where you set your limits. But the different levels of limits allow for different granularity in different dimensions. Your note limit on the PART is a fairly coarse kind of range when element level is more surgical. It doesn't really apply so much to polyphony. The rules apply and ranges superimpose. So it's more about understanding what each range does to understand if a trigger source is going to fire off the element or not.

4. Do effects like delay impact polyphony?

Assumption is no.

5. Once the polyphony is maxed out is there a priority of parts within the performance that cut off before others?

Covered with link to BM before.

6. When transitioning from one performance to another, can the sustained overlap break the polyphony?

I know what you mean, but you don't really break polyphony. The polyphony limit is a feature like having a Mini Coupe means I can only carry so much gear. It's not broken if I can't fit a big band's worth of gear into my car. I just have too much of an expectation of what I can transport if that's the goal. But yes - if you're using SSS and elements are active then they subtract from polyphony. And if you add more active elements on top - then those active elements count too. If this exceeds polyphony or not depends on how hungry the held notes were and how many new elements are thrown on top when triggering new notes from the new Performance. There's nothing that says - not having defined the entire Performance(s) - polyphony can never be exceeded. It always can be exceeded if you have a goal to do that. And if you have a goal not to exceed polyphony - you can always manage that as well. But this also comes down to what you're choosing to play. Maybe a tradeoff. Maybe no need. Depends.

7. If one of the parts in a performance is muted, is polyphony increased? What if the muted part is actually a scene of a performance, is polyphony affected when that scene is selected?

Increased? If you hit the mute button I think this generally gives you polyphony back. When the mute button is active - you'll see when you mash down on keys that elements do not activate (turn green). If you have an organ (that sounds forever as long as a key is down) - press MUTE on that PART then mash down a key - then press MUTE (to unmute) then the organ won't start sounding. The element(s) are not activated when the MUTE is on. If you rolled the volume for the PART down to 0 - then this could "silence" the PART - but the elements would still be active. So these would consume polyphony even though you couldn't hear the elements.

Is the element on? Is the element's green circle lit? No - then no polyphony consumed. There's lots of ways to mask including mute.

8. In the worst case scenario if 8 parts are being used simultaneously, how low can the polyphony go?

The worst case is you have release=127 so the notes "never" fade out. In other words, the oscillators keep running even when you're playing the next note. Worst case is you have 8 of these elements activated in each PART and 8 PARTs = 64 elements activated at once (for every one piano key pressed). Because you've defined each of these elements to sound no matter which key is pressed and no matter which velocity and are not otherwise masked (mute, arpeggio, XA control, etc). Worst case is you have arms that can press down every key of an 88-note keyboard at once. It's not really worst case - you can use a DAW to get a "worser" case (and use all 16 PARTs). With less than all fingers in one hand you could've maxed out the 8-PARTs of polyphony easily. An arm or two would max it out worse. Worse in the sense that more notes would be affected negatively. Just because you can do something doesn't mean you should.

You're not going to find any factory presets that are this kind of "worst" case. But you have to analyze where you stand for any given recipe. Sounds are not designed to mix and max. They strove to mix and match. Organs with slider control and like Performances can get you in trouble fast.

Thank you, Jason!