So the three who are traveling head toward the pillars of light, but as they leave the room...

The internal storyline progression tracking variable is now 0x4A.

They go back to find out what he wants.

Old man: No need to rush, is there?
Try dropping in on the other side
of the door up there, too.

OLD MAN: Don't be in such a rush!
Before you go, take a peek inside the
room behind me!

The internal storyline progression tracking variable is now 0x4B.

So they go through the door, which wouldn't open before. There's another room, with a strange creature in the middle. "Merry Specchio" starts upon talking to it.

: Whatcha want?
Me? I'm Specchio.
God of war!
I watch wars in various eras from here.
You, how do I look?

kahran042 has brought to my attention that TVTropes notes a possible connection with the Italian word specchio, meaning "mirror". The pronunciation checks out, and the meaning fits, so I think it's likely intentional, and have switched to this spelling.

The player gets a choice of 強そう (looks strong) and 弱そう (looks weak). The choice doesn't actually affect anything, even the dialogue, but it's there anyway.

Specchio: Is that so.
My appearance is your strength.
I look strong if you're strong,
and I look weak if you're weak.

SPEKKIO: I see.
Let's put it this way.
If you're strong, I look strong.
I'f you're weak, I look weak.

Specchio: Hm?
You guys have it.
Power of the heart...
I get it, that's why the old man
outside let you in here.
Long before you guys were born...
a kingdom prospered through magic.
Everyone in that world used magic.
But that kingdom grew addicted to
magical power and perished...
After that, no one could use magic any more.
Except for the Demons, anyway.
But you guys've got it. Strength of the heart.
Magic's the heart's strength, its power.
Magic's made up of the four powers of
Aether, Dark, Fire, and Water.

I probably don't need to point out that the history lesson got mangled in the NA version.

As mentioned earlier, 魔族 doesn't seem to have the same connotations as "demon", but I can't think of anything in English that's any better. "Fae" maybe?

SPEKKIO: You, with the punk hairdo!
You're "Lightning."

SPEKKIO: The one with the ponytail is
"Water."

SPEKKIO: The one with goofy glasses is
"Fire."

Specchio: This hugeass doll...
You, you're not a living thing.
You've got a strong heart too, but magic's
a no-go since you don't carry the blood
of the ancient magic peoples.
But that "laser" weapon of yours has
got some serious destructive ability.
It resembles Dark power.

Robo can't use magic because he's not descended from the ancient magic-users, not because Specchio can't get a feel for him.

Specchio: Like so, it's not just magic,
but all things that are made up of
the balance of these four.
Silently pray, "I wanna use magic,"
while you circle clockwise along
the edge of this room three times,
starting from where the door is!
Careful not to turn to butter.

SPEKKIO: Not just magic, but
EVERYTHING is based on the balance
of these 4 powers.
Think "MAGIC," and, starting
from the door, walk clockwise along
the walls of my room 3 times.
Don't lose track, now.

I've finally found the story behind バターにならない (not become butter). It comes from a children's book called The Story of Little Black Sambo, which includes a part where tigers chase each other around and around a tree until they melt into butter. Apparently the notion of turning to butter if you run around in circles too much has caught in to some extent in Japan.

The internal storyline progression tracking variable is now 0x4C.

It seems odd, but... whatever. They do as instructed, then report back to Specchio.

"Fanfare 2" plays.

SPEKKIO: Ipso facto, meeny moe...
MAGICO!

Special thanks to lebenspieler for introducing me to a Buddhist sutra, known in English as the Heart Sutra, that appears to be the source of the apparent gibberish. The full title in normally translated as "The Heart of the Perfection of Wisdom", and part of the first line in Japanese reads 般若波羅蜜多時 (hannya haramitta ji). It's most commonly read aloud during funerals, I'm told, making it potentially more rote than not. Further complicating matters is that it's basically classical Chinese as interpreted through Japanese phonetics. As lebenspieler suggests, that makes it something similar to the ceremonial use of Latin in the Catholic Church. It's therefore not especially unexpected that Specchio would mangle it, especially given that this is one of the sillier parts of the game. In particular, Specchio's ハラヘッタミタ～イ seems to mean something more like "looks like I'm hungrrrrry" than anything about seeking enlightenment.

Crono raises his arms as his clothes and hair ripple as though in a strong updraft. What looks like a small cyclone circles him, with accompanying wind noises. He pumps his fist in his victory pose, 2-15 ファンファーレ３ (Fanfare 3) plays, and his first spell is added to his skill list.

If present, Marle closes her eyes and clasps her hands in front of her, as her ponytail flies straight up and her clothes ripple in the same imaginary wind. Water splashes around her. She leaps into her victory pose, "Fanfare 3" plays again, and her first spell is added to her skill list.

If present, Lucca stands on her tiptoes, hands clasped in front of her, and chants. Flame appears around her. She stands with her gaze and one arm towards the sky, "Fanfare 3" plays again, and her first spell is added to her skill list.

If present, Robo starts to celebrate, but is interrupted by a "fail" noise.

All characters who just learned magic resume learning additional skills normally.

The internal storyline progression tracking variable is now 0x4D.

Specchio: So, I bet you're gushing with
new power. Wanna try it out?

If they agree, it's time to fight Specchio... "Merry Specchio" plays.

Specchio is a tough opponent. He uses powerful magic, and can't be hit by physical attacks. But now, the party has magic of their own. Crono can call down thunder from the heavens, Marle can freeze things, and Lucca can toss fireballs around. Even so, don't be surprised by a loss when first trying his strength. If the party wins, he gives them a prize. If they lose, it doesn't matter. Specchio's form changes periodically as the game progresses depending on the party leader's level. Each form is harder than the last, but gives a better prize for winning (limit one prize per form per playthrough).

Specchio's weaker forms (under level 40) also have very high physical defense, limiting the usefulness of physical/magical combos like Blaze Wheel. The magical part will work as usual, but the physical portion will contribute essentually nothing. However, his two strongest forms, including the level 40 to 98 form you're likely to be the most familiar with if you've cleared the game, not only have normal physical defense but also have elevated magic defense. This allows the physical portions of any mixed combos to inflict full damage, while hampering the effect of purely magical attacks.

Anyway, here are the prizes. Each prize may be won at most once per playthrough.
Levels  1 to  9: one Magic Capsule
Levels 10 to 19: one Magic Capsule and five Ethers
Levels 20 to 29: one Magic Capsule and five Middle Ethers
Levels 30 to 39: one Magic Capsule and five High Ethers
Levels 40 to 98: one Magic Capsule, one Speed Capsule, one Power Capsule, and ten Elixirs
Maximum level: ten Magic Capsules, ten Speed Capsules, ten Power Capsules, and ten Last Elixirs

Specchio: If you make new teammates,
you're bringing them here.
You guys are VERY interesting.

SPEKKIO: You guys are too much!
Bring newcomers to meet me!

They return to the main room. "The Farthest Reaches of Time" plays again.

Old Man: Oh? Just as I thought.
All the people of the distant past had
such powers.
Ah yes, the distant past...
It looks as though you guys are about to
go and perpetrate something unreasonable,
but perhaps you should first try returning
to your own era...?
Less haste, more speed...
Being here makes you quite familiar with
the meaning of such words as those...
If anything happens, come any time.
I shall be a guide to time for you.

OLD MAN: Well, well!
People who lived long ago ALL
enjoyed such powers...
Now, I know you are itching to go
ripping back and forth through time,
but first you need to return to your
era.
And you must hurry.
The longer you remain here, the
harder it will be to change that
which must be changed...
Stop by whenever you're in the
area!

More mix-ups in the NA version. The old man isn't saying to hurry. In fact, he's saying almost the opposite. Translated from the J-J dictionary on Goo, 急がば回れ means "When in a hurry, rather than a dangerous shortcut, taking the long but safe main road is faster in the end. An admonition to take the safe and steady way." In other words, his point appears to be that, instead of trying something brave but foolish right away, they should take time to investigate and prepare. "Less haste, more speed" is a comparable English proverb that says you'll reach your objectives more quickly if you don't rush them.

And so they head for the room where they arrived. The three pillars of light are now active and display information on where they go.

But first, each character not in the current party has special dialog if spoken to here:

She also pouts.

She leaps with excitement, and "Crono and Marle ~Distant Promise~" starts playing as the party select screen appears.

Lucca: Don't you need my brain
power?

She also turns away.

Lucca: See!
You DO need me!

She takes hold of her glasses, and "Fanfare 1" starts playing as the party select screen appears.

Robo: PLEASE do not HESITATE to call
when you have need of me.

He also slumps over.

He pumps his arms, and "Robo's Theme" starts playing as the party select screen appears.

If the player is curious enough to take a closer look at the bucket anyway, the man explains why to avoid it.

Old Man: That is connected to A.D. 1999...
To the time called The Day of Lavos...
You could try going if you would like
to watch the destruction of the world...
but you might be destroyed yourselves.

OLD MAN: That Gate leads to 1999 A.D...
It leads to the "Day of Lavos"...
Go there only if you're looking to
achieve a shorter life span...
Lavos will help you leave this mortal
coil.

Clearing the game at this point leads to the alternate ending "Good Night".

The positioning of the pillars, and the pairs that appear later, suggest that the same Gate links Mystery Mountain and Medina Village, but the one from Promethe Dome doesn't have another end anywhere in normal time. An alternate explanation is that a single Gate somehow connects all three times and places.

Medina Village is an unfamiliar name, but at least the era is correct. The party enters that Gate...

There's one more thing you can do first, and it's extremely easy to miss. Return to Promethe Dome and run back and forth in front of the Gate a few times. For no discernable reason, you'll hear a save point noise, and a Power Capsule will appear there.

Scene: A normal house. Two of the goblin-like creatures that the party knows as the cannon fodder of the Magus's army sit at a table. With a series of weird noises, a bureau opens, revealing a Gate, from which Crono and the others emerge. The goblins are understandably surprised. "Silent Light" plays.

The internal storyline progression tracking variable is now 0x4E, starting the next chapter.

Access to magic and a temporal hub will help the party make progress, as will the information available here, but while all of that will prove exceedingly useful, it only affects them, and has no historical impact on its own.