Certified Cross-Dictionary Feature Matching via Co-Activation Geometry

Feature matching across dictionaries is central to tracking concepts across layers, models, and fine-tunes, but common matchers return correspondences without a certificate. For non-negative codes evaluated on shared samples, the cross co-activation relation $K_{12}(a,b)={\max }_i\min (Z_1[i,a],Z_2[i,b])$ defines a filtered relation whose two feature-side Dowker complexes have identical persistent homology for every pair of codes, even when widths and ambient activation spaces differ. Restricting the induced maps to mutual matches yields a certified partial correspondence with zero round-trip error; its metric distortion ${\delta }_{\mathrm{mut}}$ is reported together with coverage. Across twelve synthetic width–sparsity configurations, every committed match is correct (precision $1.000$) and ${\delta }_{\mathrm{mut}}=0$; additional samples increase coverage with an approximately inverse-linear law in samples per feature. Released GPT-2 dictionaries occupy a different regime: max–min scores saturate with sample count, so coverage must be calibrated empirically rather than extrapolated from the synthetic law. At 8192 tokens, adjacent-layer coverage falls from $0.563$ at layers $0\to 1$ to a minimum of $0.134$ at $9\to 10$, and agreement with correlation and decoder-cosine matchers falls with it from $1\to 2$ onward; the embedding transition $0\to 1$ is the one place where the relational matcher commits most and the baselines agree with it least. The method converts feature matching from an unqualified assignment into an auditable correspondence whose precision, coverage, and geometric distortion are explicit.