The Sign-Inversion Layer: Transformation-Based Alignment as the Missing Multiplicative Operator

Contemporary alignment methods are subtractive: RLHF refusal, inference-time rejection (HCRS, WP-02), and weight-level abliteration (PT-CDS, WP-06) all remove or suppress an undesired direction. We close the gap identified in AAMT Foundations Paper IV by giving the missing Generative operator M a concrete, differentiable form: the Sign-Inversion Layer, which reflects a shadow (adversarial / low-coherence) direction's energy across the coherence surface onto its constructive counterpart via paired-negative multiplication (−a)(−b) = ab. We prove three theorems: (1) refusal/abliteration is the zero-gain special case (g≡0), so transformation-based safety strictly generalizes removal-based safety; (2) the layer is polarity-restoring; and (3) it is energy-conserving on the treated subspace, consistent with the TERA conservation law (WP-01). We instantiate M at three sites — a residual-stream layer, an inference-time transformation of HCRS (turning rejection into repair), and a denoising-repair step for diffusion language models — and map the classic failure modes (hallucination, adversarial vulnerability, mode collapse) onto the absence of M as predicted by Foundations Paper IV §6. A provisional patent is filed (Sign-Inversion Layer, USPTO).