A latent interaction landscape inferred from Micro-C encodes chromatin regulatory identity and conformational diversity beyond contact frequency

Chromatin contact maps report population-averaged frequencies that obscure the physical principles of genome folding. Here we show that a sparse, signed interaction landscape - a matrix of Lagrange multipliers,{lambda} , inferred by maximum entropy from Micro-C contact probabilities at nucleosome resolution, encodes regulatory and structural information beyond contact statistics. Sign decomposition reveals context-dependent associations with chromatin state. Attractive interactions ({lambda}-) are enriched at active regulatory elements, whereas repulsive interactions ({lambda}+) identify genomic positions whose contacts are depleted relative to the polymer reference, including Polycomb-associated domains. Top-ranked{lambda} bins show substantially greater enrichment for regulatory chromatin marks than O/E-normalised bins across active, architectural, and repressive classes, demonstrating that{lambda} preferentially identifies regulatory regions over contact-based rankings. Structural information is hierarchically encoded: the strongest 2% of interactions recover domain boundaries, the top 10% reproduces average contact statistics, yet reconstructing the conformational free-energy landscape, including dominant structural sub-populations and occupancies requires the full interaction spectrum. Nucleosome perturbation experiments show that{lambda} detects positional information contact maps largely miss, and targeted removal of CTCF/RAD21-associated{lambda} interactions selectively disrupts loop architecture without altering global polymer statistics. Together, these results establish{lambda} as a physically interpretable intermediate layer linking chromatin contact maps to three-dimensional structural ensembles.