Continuous Endpoint-Compression Words From Discrete Freeness to Continuous Fibers

This paper extends the discrete endpoint-compression calculus of the Generative Mechanism Atlas to the continuous family Ma(q)=a/(1+q). The determinant obstruction that structured the discrete core disappears under continuation, while positivity persists across all words. The key structural change is the loss of freeness: at length four, an explicit positive one-parameter fiber appears, proving non-faithfulness of the continuous parametrization. The family also admits a boundary-value interpretation as a midpoint share observable of positivity-preserving second-order ODEs. The discrete pair {M1,M2} emerges as the arithmetic skeleton of a broader analytic family whose surviving invariant is positivity.