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In the book "Advanced race guide", the racial trait "Slow Speed" says:

If the race is Medium, its members’ speed is never modified by armor or encumbrance.

Does that negate heavy armors' penalty to x3 (instead of x4) the normal speed when running?

Or is it just a way of saying that the normal speed would remain 20 feet with or without armor?

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The Slow Speed trait does not negate the penalty to run distance for heavy armor.

From the PFSRD on movement (emphasis mine):

Run (×3)
Moving three times speed is a running pace for a character in heavy armor (about 7 miles per hour for a human in full plate).

Run (×4)
Moving four times speed is a running pace for a character in light, medium, or no armor (about 12 miles per hour for an unencumbered human, or 9 miles per hour for a human in chainmail.)

The distance you move when you run is based on your speed, but is not itself a speed.

From the PFSRD on encumbrance (emphasis mine):

Encumbrance by Weight: If you want to determine whether your character’s gear is heavy enough to slow him down more than his armor already does, total the weight of all the character’s items, including armor, weapons, and gear. Compare this total to the character’s Strength on Table: Carrying Capacity. Depending on the character’s carrying capacity, he or she may be carrying a light, medium, or heavy load. Like armor, a character’s load affects his maximum Dexterity bonus to AC, carries a check penalty (which works like an armor check penalty), reduces the character’s speed, and affects how fast the character can run, as shown on Table: Encumbrance Effects. A medium or heavy load counts as medium or heavy armor for the purpose of abilities or skills that are restricted by armor. Carrying a light load does not encumber a character.

Notice that reducing the character's speed and reducing their run distance are listed separately.

From the PFSRD on armor (emphasis mine):

Medium or heavy armor slows the wearer down. The number on Table: Armor and Shields is the character’s speed while wearing the armor. Humans, elves, half-elves, and half-orcs have an unencumbered speed of 30 feet. They use the first column. Dwarves, gnomes, and halflings have an unencumbered speed of 20 feet. They use the second column. Remember, however, that a dwarf’s land speed remains 20 feet even in medium or heavy armor or when carrying a medium or heavy load.

Notice that it says nothing about dwarves getting their speed times four when running.

TLDR: "Speed" and "distance moved when taking the Run action" are different game mechanics. The Slow Speed trait only negates penalties to the former, and not the latter.

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