The fourth world is broken up into a couple of sections; a southern section, a middle section, and an eastern section. You begin in the southern section, where you can collect a Speed Potion, and find the Konami Logo which summons Konamiman to heal any damage you have sustained.
In the middle section, more items are found along the top row of screens. You will have to be a little more diligent in order to find the various doors and items hidden on some of the screens, as they may be hidden below the red circles. You can typically access any of these circles that are positioned south of the traversable paths. While you're there, look for the Rock Case, a Hyper Bomb, and another Speed Potion.
Once you reach the eastern section, head south first in order to collect the Artificial Heart that is hidden in the lowest screen. Then return to the top in order to reach the hideout of this world's boss. Don't miss the invincibility granting Star Rock one screen earlier.
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Artficial Heart location
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Rock Case location
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Konami Logo location
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Warp Zone to World 6
Boss
The boss of World 4 is a gigantic Scallop. This Scallop moves back and forth across the top of the screen. When it opens its shell, it fires balls of energy in three directions. Once the balls fans out, they each break apart into three smaller balls of energy that proceed to bounce around the room for a period of time before disappearing. Your best bet is to occupy the upper portion of the room and toss rocks at the Scallop safely from a distance.
Once the boss is defeated, you'll want to return to the southern section of this world, and access the warp zone closest to the starting location of the world in order to advance to World 6.