PIN v3: A Shared Coordinate System for Model Families including Compression and Design by Specification

PIN describes a family of neural architectures in which a model's weights are tied across the orbits of a finite group G, so that the deployed model is a compact quotient B rather than the full parameter set A. Version 1.1 presented PIN primarily as a compression technique applied to existing models, supported by a geometric argument for constant-time inference. This revision reverses that emphasis. Empirical testing shows that retrofitting a fold onto a conventionally trained network is lossy and inferior, while co-designing the network around a chosen G, the alternative proposed in v1.1 section 7.2 as an aside, is both more accurate and cheaper to produce. The constant-depth inference claim is withdrawn in its original form: the geometric argument supporting it contains a codimension error, and every mechanism derived from it fails under direct measurement. A corrected form survives and is stronger. Constraining the activation space to k spectral components makes traversal depth at most k by construction, with no reliance on high-dimensional concentration. Version 3.0 extends this in three directions. Scaling: with a fold-aware initialisation, accuracy is set by the number of stored values and is flat in width across a sixteenfold range, so folding's cost stays constant while its compression grows with the layer; in a transformer a folded model is eighteen times smaller and six points better than the model it compresses. Families: the members of a task family can share one body entirely, at seventeen times less storage for a thousandth of a point, and an additive predictor emits unseen members from a task descriptor, needing one more example task than it has factors. Design: these results combine into a procedure. Calibrate a storage curve once and it predicts what will be built to within a few thousandths, carries its own confidence band derived from its local slope, transfers to families and architectures it never saw, and supports choosing a design on a small model before deploying it on a large one.