The floor under the plateau

The plateau theorem says discovery within a fixed model class dies. Its companion says something sharper and stranger: where it dies depends on your language, and the difference between two observers’ plateaus is measurable. This post proves the floor theorem and shows why it turns an epistemological puzzle into an experiment.

The statement

Two observers with different model classes, M1 and M2, study the same data stream D generated by a process outside both classes. Each plateaus. The floor theorem says: their terminal description lengths differ by a computable quantity, and that quantity is the information in D’s structure that M2 can express and M1 cannot. Call it the class gap. Three consequences follow immediately.

First, the gap is non-negative and one-directional: the more expressive class never floors higher. Enrichment can only lower the terminal cost. Second, the gap is the number the resource theory of Paper 1 calls discoverable structure remaining: it is exactly how much novelty the weaker observer has left to mine by upgrading their language. Third, and this is the theorem’s teeth, the gap is measurable without knowing the truth. Each observer’s residual cost at their floor is computable from their own data and their own class. The difference is a fact about the two classes and the data, no reference to the generating process required.

The proof sketch

Write the terminal cost of observer i as the data’s compressible content under M i plus the incompressible remainder under M i. The remainders are both anchored to the same true structure; the compressible parts differ by what each class can absorb. Subtract: the true structure cancels, and what survives the subtraction is precisely the cost of encoding, in M2’s language, the structure that M1 had to store as raw residual. That encoding cost is finite, computable, and equals the class gap. The proof is the subtraction; the subtlety is the technical condition that both observers use consistent, prefix-free encodings, which is the standard MDL apparatus and is stated in the paper.

Why this matters more than the plateau

The plateau theorem is pessimistic: discovery dies. The floor theorem converts the pessimism into a diagnostic. Any field whose progress has stalled is in one of two states, genuinely exhausted or floored, and the theorem says the states are distinguishable by a concrete measurement: compute the residual floor under the current class, then compute it under a modest enrichment. If the floor drops, the field was floored, and the enrichment was the discovery. If it does not drop, the field has genuinely mined its language, and the next discovery must come from new data with new physics.

That is a decision procedure for research strategy, and it is the framework’s most practical output. It is also the quiet refutation of the strongest form of the ending-of-science claim: the famous arguments that all the big discoveries are made rest on extrapolating one plateau, and the floor theorem says a plateau is a property of a language, not of the world. The world’s own floor is K of D, which no observer reaches, and the distance between our floors and K of D is unknown and, by the theorem’s own logic, cannot be estimated from inside any single language.

The two theorems together are the framework’s most durable claim: discovery has a budget in any given language, the budget is computable, and the budget is not the world’s. Everything else in the computational series is built on that foundation or measures its consequences.

The next post counts the exits from the plateau, and there are exactly two: two ways past a plateau.