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Introduced path connects the topics in common concept: integral information measure and symmetry. Initial path' sequence axiomatic probability distributions of stochastic multidimensional process transfers each priory to posteriori probabilities alternating probabilities over process trajectory. Emerging Bayesian probabilities entropy defines process uncertainty measure. Probability transitions model interactive random process generated by idealized virtual measurements of uncertainty as observable process of potential observer. When the measurements test uncertainty by interactive impulses its inferring certain posteriori probability starts converting uncertainty to certainty information. Observable uncertain impulse becomes certain control extracting maximum information from each observed minima and initiating information observer with internal process during conversion. Multiple trial actions produce observed frequency of the events measured probability actually occurred. Dual minimax principle of maxmim extraction and minimax consumption information is mathematical law whose variation equations determine observer structure and functionally unify regularities. Impulse controls cutoff the minimax, convert external process to internal information micro and macrodynamics through integral measuring, multiple trials, verification symmetry, cooperation, enfoldment in logical hierarchical information network IN and feedback path to observations; IN high level logic originates observer intelligence requesting new quality information. Functional regularities create integral logic selfoperating observations, inner dynamical and geometrical structures with boundary shaped by IN information geometry in timespace cooperative processes, and physical substances, observer cognition,intelligence. Logic holds invariance information and physical regularities of minimax law.

Information Path from Randomness and Uncertainty to Information, Thermodynamics, And Intelligence of Observer