Slavic
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Slavic is a universal constructor-based quadratic growth pattern constructed in December of 2019 by Pavel Grankovskiy. At the time it was the smallest such pattern.
Method
Its initial configuration is five copies of a single-channel recipe pointed at a block. Each recipe uses 0-degree slow salvos to build a syringe-based signal splitter plus some guard eaters, like so:
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Then it removes the zero-degree elbow and pushes its two new construction arm elbows a little farther away from the splitter, and builds and launches 2-engine Corderships along each branch, then shoots them down after a decent interval, to move the elbows a good distance away. Then the cycle repeats.
The result is that after four cycles of the recipe being used up, four splitters have been built in a diamond-shaped loop. The fifth copy of the recipe enters the loop and cycles around it indefinitely, sending new copies out through the increasing network of splitters to build more and more splitters.
Inevitably some of the descendants of the original four splitters will try to build new splitters on top of the existing ones. Whenever that happens, the guard blocks come into play, absorbing a newly created elbow block before it can start the construction process, and allowing the incoming recipe to be absorbed by the nearby guard eater.
The result is a pattern with quadratically-growing population that is nonetheless not quite a quadratic-growth replicator, because there is not a single initial unit that makes exact copies of itself. Instead, five copies of the single-channel recipe are needed to get started, and they need to be aimed at an initial elbow block.