What happens when an unnoticed creature rolls Stealth for initiative and fails the check?

Question

Suppose the adventurer Alice is moving stealthily and using the Avoid Notice exploration activity, and then tries to sneak up on an enemy Bob. Alice rolls a Stealth check for initiative versus Bob's Perception check for initiative, but she gets a lower result.

If you’re Avoiding Notice at the start of an encounter, you usually roll a Stealth check instead of a Perception check both to determine your initiative and to see if the enemies notice you (based on their Perception DCs, as normal for Sneak, regardless of their initiative check results).

Assume that Alice had been unnoticed by Bob, prior to the initiative roll.

At the start of Bob's first turn, does he consider Alice to be undetected, hidden, or observed? Does the outcome depend on circumstances, such as whether Alice's Stealth check was a critical failure or normal failure?

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For quick reference:

• Undetected = Bob knows Alice is nearby, but he doesn't know where she is. Bob is flat-footed to Alice. He can attempt Seek actions to make her hidden instead.
• Hidden = Bob knows Alice's location, but he does not perceive her directly. Bob is flat-footed to Alice. Depending on Bob's senses, he can attempt Seek actions to make her observed instead.
• Observed = Bob knows Alice's position and he can perceive her clearly.

Answer 1

As Sneak vs Perception DC

The player Avoiding Notice counts as Sneaking, with their initiative check as their Stealth roll. If their Stealth roll beats the enemy's Perception DC, the player remains undetected. If they fail, then they are hidden, and on a critical failure they are observed. This all depends on a few other things (e.g. having cover or concealment) - see the text of Sneak.

(Here I've trimmed up the text of Sneak to the parts that should be relevant to this situation)

Sneak:

If you’re undetected by a creature and it’s impossible for that creature to observe you ... for any critical failure you roll on a check to Sneak, you get a failure instead.

... You don’t get to roll against a creature if, at the end of your movement, you neither are concealed from it nor have cover or greater cover against it. You automatically become observed by such a creature.

Success You’re undetected by the creature during your movement and remain undetected by the creature at the end of it.

Failure A telltale sound or other sign gives your position away, though you still remain unseen. You’re hidden from the creature throughout your movement and remain so.

Critical Failure You’re spotted! You’re observed by the creature throughout your movement and remain so. If you’re invisible and were hidden from the creature, instead of being observed you’re hidden throughout your movement and remain so.