Trivial answer: no, because you cannot center antimagic field on a pebble

Antimagic field can only be emanated from the caster him- or herself. You cannot cast it on other creatures or objects, the way you can with, e.g., light. Thus, the sequence you describe cannot take place.

Arcane archers¹ and master abjurers² can stick antimagic field onto other creatures, but not onto objects.

Trivial answer: no, the prismatic field destroys anything you try to bring in

Still, if you were an arcane archer or master abjurer, you could presumably tag some tiny-and-harmless critter with antimagic field (the master abjurer is much nicer about this process) and carry that instead of your pebble.

The violet veil is still going to just destroy it. It won’t make it to you, and the antimagic field will be annihilated along with the kitten (and you become a terrible person, if you were a master abjurer and not already).

Trivial answer: no, the antimagic field is in the prismatic sphere with you

OK, fine: assume an indestructable kitten, or at least one immune to the various effects of prismatic sphere.³

So you have your antimagic field-emanating kitten, you step out of the field (or never entered it, if you were a heartless bast— I mean, arcane archer), and you cast prismatic sphere. You now step towards the kitten, and the antimagic field is held at bay by your prismatic sphere. You step close enough to pick up the kitten,⁴ at which point your prismatic sphere is wholly enclosed inside an antimagic field. What happens?

The kitten is now inside your prismatic sphere. The prismatic sphere’s edge is a barrier through which the antimagic field cannot penetrate, but our amazing kitten can. Since the source of the antimagic field is now inside the prismatic sphere, it affects you, suppressing your magic, including the prismatic sphere, and freeing the antimagic field to extend to its full radius (rather than being limited by the inside of the prismatic sphere). Your prismatic sphere will return if and only if you step away from the kitten, allowing it to stay outside your prismatic sphere.

Real answer: antimagic field does not block magic, only suppresses it

Ultimately, the thing you were trying to do here is possible. The archmage’s Mastery of Shaping, for instance, could leave a 5-foot hole in the middle of the emanation that is centered on you, leaving you magical, with antimagic field around you. Unfortunately, that’s largely worthless, since magic can pass through an antimagic field. It doesn’t stop magic, just suppresses it while it’s in the antimagic field.

In game terms, antimagic field does not block the line of effect needed to target spells. So you are just as vulnerable to spells cast at you as you were without the antimagic field. Someone standing outside your shell can cast fireball at you, and the little red bead will disappear when it hits the shell – and reappear on your side of the shell, before hitting you in the face.⁵ Then it blows up, filling your hollow with fire, leaving your antimagic field-filled shell free of any flames, and yet more flames outside the shell.

All it really does is suppress the magic of those standing very close to you – which isn’t awful by any means, but it is a far cry from the invulnerability to spells you might imagine it is.

Also, casting prismatic sphere at this point does an interesting – but not particularly useful – thing. It limits the size of your antimagic field, basically defeating the Widen Spell metamagic you attempted to put on it. Instead of a 20-ft.-radius antimagic field with a 5-ft.-square hole in the middle, you end up with a 10-ft.-radius antimagic field with a 5-ft.-square hole in the middle, and a prismatic sphere around that. Even if you shrunk the prismatic sphere somehow, so it was inside the 5-ft.-square hole, your antimagic field won’t happen outside it.

If you do get a prismatic sphere with an antimagic field outside it, which can be done in more convoluted ways than I feel like getting into, spells can still go right through the antimagic field to attempt to cancel your prismatic sphere. But the prismatic sphere would not “wink out” because of the antimagic field around it, because antimagic field does not block magic, just suppresses it in the specific area covered by the field.⁶

1. ^{Arcane archers can shoot an arrow that sticks a spell that ordinarily can only emanate from you, onto a target creature, who emanates it instead of you. The class feature was almost certainly written specifically for antimagic field; there are very few other spells where this would be desirable, but for antimagic field it is very desirable.}

2. ^{A “master abjurer” here is an abjurer who takes the master specialist prestige class in Complete Mage – their major school esoterica class feature allows them to cast abjurations that are normally centered on you (e.g. antimagic field) as a touch-attack spell that emanates from the touched creature. Again, antimagic field was almost-certainly the spell they had in mind when they wrote this.}

3. ^{Arcane archer, that doesn’t make what you did any kind of okay.}

4. ^{For the purposes of this answer, I am ignoring the fact that antimagic field and prismatic sphere have exactly the same size, and assuming the antimagic field is larger; assume Widen Spell was applied to antimagic field if it makes you feel better.}

5. ^{And if you are an arcane archer, you deserve it.}

6. ^{Even if antimagic field did block magic, it’s far from clear to me that the ambient magical energy necessary for spells wouldn’t still exist in the hollow – after all, I can have air inside a container whose walls block air. This is a setting detail that is not addressed by the general rules, and you would have to ask your DM.}

Yes, but maybe it's an error.

The description of the spell as written does indeed say that if the indigo layer starts to, but does not succeed in, petrifying a creature, then prismatic wall ends.

But given that the other language in the spell description strongly implies that that the only way to end the spell is by successively bringing down each of the seven layers, that seems really incongruous.

So, here's a theory. Suppose that in an earlier draft of the spell, the indigo layer's effects read something like this:

6. Indigo. The creature is subjected to the effects of the flesh to stone spell.

At some point, someone realizes that flesh to stone requires a Constitution save, which is inconsistent with the other Dexterity saves that the wall requires. So the indigo layer's effects are spelled out by copy-pasting in text from flesh to stone:

A creature restrained by this spell must make another Constitution saving throw at the end of each of its turns. If it successfully saves against this spell three times, the spell ends. If it fails its saves three times, it is turned to stone and subjected to the petrified condition for the duration. The successes and failures don’t need to be consecutive; keep track of both until the target collects three of a kind.

This is nearly identical to the printed effects of the indigo layer of prismatic wall, in particular the contentious the spell ends wording.

I suspect that instead of the the spell ends, what is meant here is the creature is no longer restrained. Because there actually isn't any phrase the description of prismatic wall that specifies how long the affected creature is restrained for. Compare to other spells which impose the restrained condition:

Imprisionment:

The target is restrained until the spell ends,

Web

On a failed save, the creature is restrained as long as it remains in the webs or until it breaks free.

And, indeed, prismatic wall already imposes another condition, and it specifies a duration for that:

If another creature that can see the wall moves to within 20 feet of it or starts its turn there, the creature must succeed on a Constitution saving throw or become blinded for 1 minute.

But the most compelling reason to think that the spell ends is a copy/paste error is the text for the related spell prismatic spray. The effects for the indigo ray of that spell use the same language:

6. Indigo. On a failed save, the target is restrained. It must then make a Constitution saving throw at the end of each of its turns. If it successfully saves three times, the spell ends.

But prismatic spray has a duration of Instantaneous; it's already ended by the time the target is making a third saving throw. The phrase the spell ends has no real meaning in the context of an instantaneous spell.

So, my contention is that the intent of the effects for prismatic wall is that saving three times against the effects of the indigo layer means that the restrained condition on the creature is ended, but the prismatic wall is not dispelled.