1 Introduction	3
1.1 Observable Physics of Information	3
1.2 Statistical Interpretation: Hierarchical Independences and Dependences Structures	3
1.3 Statistical Physics Interpretation: K-Body Interacting Systems	4
1.4 Machine Learning Interpretation: Topological Deep Learning	5
2 Information Cohomology	6
2.1 A Long March through Information Topology	6
2.2 Information Functions (Definitions)	7
2.3 Information Structures and Coboundaries	10
2.3.1 First Degree ($k=1$)	12
2.3.2 Second Degree ($k=2$)	12
2.3.3 Third Degree ($k=3$)	12
2.3.4 Higher Degrees	13
3 Simplicial Information Cohomology	13
3.1 Simplicial Substructures of Information	13
3.2 Topological Self and Free Energy of K-Body Interacting System-Poincaré-Shannon Machine	14
3.3 k-Entropy and k-Information Landscapes	18
3.4 Information Paths and Minimum Free Energy Complex	19
3.4.1 Information Paths (Definition)	19
3.4.2 Derivatives, Inequalities and Conditional Mutual-Information Negativity	20
3.4.3 Information Paths Are Random Processes: Topological Second Law of Thermodynamics and Entropy Rate	21
3.4.4 Local Minima and Critical Dimension	23
3.4.5 Sum over Paths and Mean Information Path	24
3.4.6 Minimum Free Energy Complex	25
4 Discussion	27
4.1 Statistical Physics	27
4.1.1 Statistical Physics without Statistical Limit? Complexity through Finite Dimensional Non-Extensivity	27
4.1.2 Naive Estimations Let the Data Speak	28
4.1.3 Discrete Informational Analog of Renormalization Methods: No Mean-Field Assumptions Let the Objects Differentiate	29
4.1.4 Combinatorial, Infinite, Continuous and Quantum Generalizations	29
4.2 Data Science	29
4.2.1 Topological Data Analysis	29
4.2.2 Unsupervised and Supervised Deep Homological Learning	30
4.2.3 Epigenetic Topological Learning—Biological Diversity	31
5 Conclusions	32
References	33