Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius R→∞ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as N→∞ flattens the remaining curvature, yielding SMGR→∞IGN→∞CS.
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The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.