Geometric Learning Paths: Random-Envelope Diagnostics for Sequential Explanation Dynamics

We propose Geometric Learning Paths (GLP), an exploratory framework for comparingevidence streams by the geometry of the learning process they induce over a space of explanations.Given a stream X = x1x2 . . ., an explanation family E, a sequential scoring rule, and a softupdate rule, GLP maps each prefix X≤n to a belief state µn ∈ ∆(E). The resulting curveγX : n 7→ µn is studied through path functionals such as belief entropy, speed, acceleration, andturning angle. We introduce a random-envelope diagnostic: compare F(γX, n) against quantilebands generated by null streams, and define an escape score measuring the fraction of time thepath leaves the null envelope. Toy experiments with low-order Markov explanation families showthat the method separates simple periodic structure, morphic/deterministic structure, arithmeticsparsity, and streams that are null-like under the tested model class. Three-null controls - fairBernoulli, density-matched Bernoulli, and first-order Markov matched nulls - show that someapparent escapes disappear under stronger controls while others persist. The contribution isintended as a methodological note and reproducibility scaffold, not as a proof of randomness ornormality for any mathematical constant.