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Here's what remains when training is finished.
The program hasn't changed. The layers are the same. The connections are the same. What changed, through billions of small adjustments across trillions of examples, are the weights.
That second file, the one full of numbers, now contains a very specific configuration. Not random anymore. A configuration that makes the network do something useful.
That configuration is the model. All of it. Copy the file and you copy the model. Share the file and you share everything it learned. Move it to a different computer and it works exactly the same way. The knowledge lives in the numbers, nowhere else.
The concept is simple: training took a blank set of random numbers and turned them into a specific arrangement that produces good answers. That arrangement is what you're talking to.
Which leaves one question hanging. That file of numbers, how many numbers does it actually hold?