Paries atrii iam non sordidus est, sed mundus.

Because atrii is the genitive singular of atrium, and it means of the atrium.

So:

• paries = wall
• atrii = of the atrium

Together, paries atrii means the wall of the atrium.

This is a very common Latin way to show possession or relationship:

• dominus villae = the master of the house
• ianua horti = the gate of the garden
• paries atrii = the wall of the atrium

Paries is the subject because it is in the nominative singular form and it is the thing being described.

The sentence says that the wall is not dirty anymore, but clean. So the wall is what is something.

Structure:

• paries atrii = the wall of the atrium
• est = is
• non sordidus ... sed mundus = not dirty, but clean

So paries is the noun that the rest of the sentence is talking about.

They are masculine singular because they describe paries, and adjectives in Latin must agree with the noun they describe in gender, number, and case.

• paries is masculine singular nominative
• so sordidus is masculine singular nominative
• and mundus is also masculine singular nominative

Even though in English we do not change adjectives this way, Latin does.

For example:

• paries sordidus = a dirty wall
• ianua sordida = a dirty door
  because ianua is feminine
• cubiculum sordidum = a dirty bedroom
  because cubiculum is neuter

Latin often uses one est with two contrasting predicate adjectives.

So:

• Paries atrii iam non sordidus est, sed mundus.

means:

• The wall of the atrium is no longer dirty, but clean.

The verb est naturally applies to both parts:

• not dirty
• but clean

Latin could repeat the verb, but it usually does not need to:

• non sordidus est, sed mundus est

That would still make sense, but the shorter version is more natural here.

Literally, iam means something like now / already, and non means not.

Together, in a sentence like this, iam non is naturally understood as no longer:

• iam non sordidus est = it is now not dirty
  which idiomatic English expresses as
  it is no longer dirty

This is a very common Latin combination.

Compare:

• iam adest = he is here now / he is already here
• iam non adest = he is no longer here

Because non negates the idea that follows. Here it is negating sordidus:

• non sordidus = not dirty

Latin often places non directly before the word or phrase being negated. In this sentence, the point is that the wall is not dirty, but rather clean.

So the contrast is:

• non sordidus
• sed mundus

That is, not dirty, but clean.

Sed means but. It introduces a contrast.

So:

• non sordidus est, sed mundus = is not dirty, but clean

It contrasts two descriptions:

• the negative one that is rejected: dirty
• the positive one that is true: clean

This is a very common pattern in Latin:

• non servus, sed dominus = not a slave, but a master
• non tristis, sed laetus = not sad, but happy

Because Latin has no definite article and no indefinite article.

So Latin does not have separate words for:

• the
• a
• an

That means paries can mean:

• wall
• a wall
• the wall

You understand which one is meant from the context. In this sentence, English naturally translates it as the wall.

Yes. Mundus can be either:

1. an adjective meaning clean, or
2. a noun meaning world

Here it is clearly an adjective, because it matches paries and contrasts with sordidus.

So here:

• sordidus = dirty
• mundus = clean

The meaning world would not fit the sentence at all.

Latin word order is more flexible than English because the endings show the grammatical relationships.

English relies heavily on word order:

• The wall of the atrium is no longer dirty, but clean.

Latin can move words around more freely:

• Paries atrii iam non sordidus est, sed mundus.

This order is perfectly natural. It begins with the thing being talked about, paries atrii, and then gives the description.

The exact order can also create emphasis, but here it is mainly just standard, readable Latin.

Yes. Even though est comes after sordidus, the meaning is still is dirty.

Latin often puts the predicate adjective before the form of esse:

• sordidus est = is dirty
• mundus est = is clean

So do not translate word by word in a rigid English order. Instead, read the structure:

• paries atrii = the wall of the atrium
• iam non sordidus est = is no longer dirty
• sed mundus = but clean

That gives the natural English meaning.