Locally constrained immune-interaction fronts at tumor–immune interfaces are associated with immunotherapy response in hepatocellular carcinoma

Background. Spatial transcriptomics can reveal that the structural character of tumour–immune interactions—not merely cell-type abundance—may determine immunotherapy response. Existing spatial methods quantify co-localisation rather than the interaction field geometry, leaving locally constrained immune fronts analytically invisible. Methods. We developed an operator-regime framework that models tumour–immune interface programmes as antisymmetric edge fields on spatial kNN graphs and decomposes them via discrete Hodge operators into gradient and non-gradient (coexact) components. The framework further characterises interfaces via noncommutative commutator analysis, operator-regime state transitions, and local-versus-global spatial predictability. Results. In a matched pre/post immunotherapy HCC Visium cohort (11 patients, 22 sections), responder interfaces showed significantly stronger local spatial predictability (median local R2 0.352 vs. 0.110; p = 0.033) while resisting global spectral reconstruction— the hallmark of locally constrained interaction fronts. Immune–immune commutators were elevated in every responder above every non-responder (p < 0.012), independent of the universal tumour–myeloid backbone. Therapy drove all non-responders to the fragmented KTS S1 state versus organised states in responders (Fisher p = 0.015). An organised non-responder (HCC5NR) with supra-responder geometry but tumour–myeloid-dominated algebra confirmed that spatial organisation alone is insufficient: the immune-programme algebra layer was required for correct classification. Pan-cancer validation (26 sections, six cancer types) confirmed interface-localised coexact enrichment across tissue architectures (sign test p= 1.49 ×10−8). Cross-platform spatial transfer, tested independently of clinical-response claims, was supported in an independent cSCC CosMx single-cell cohort (333 FOVs; fixed-interface null collapse from 17.95 to ≈1.00), establishing that locally constrained operator fronts are not platform-specific. Conclusions. Immunotherapy response in HCC is associated with locally constrained operator-front geometry characterised by immune-programme algebraic organisation, not by immune abundance or global spatial structure alone. The framework provides a biologically grounded spatial analysis layer that proximity or co-localisation methods cannot replicate.