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Uncertainty can be measured in bits, and there is a hard limit to how much a signal can be compressed before information is lost.","tier":"system","weight":1,"status":"active","source_ids":["S1"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null}]},{"id":"C2","text":"The Source Coding Theorem states that the minimum average number of bits needed to encode a message equals its Shannon entropy, and no code can beat this limit without losing information.","tier":"system","weight":1,"status":"active","source_ids":["S1"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null}]},{"id":"C3","text":"The Noisy Channel Coding Theorem states that information can be transmitted through a noisy channel with arbitrarily low error if and only if the transmission rate stays below the channel capacity C = max_{p(x)} I(X;Y).","tier":"system","weight":1,"status":"active","source_ids":["S1"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null}]},{"id":"C4","text":"The statistical-mechanical formula for entropy H = -Σ p log p is an identity that measures both disorder in a gas and uncertainty in a communication channel, not merely an analogy.","tier":"system","weight":0.95,"status":"active","source_ids":["S1","S4"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null},{"type":"supported_by","target":"S4","source_type":"adjacent","hash":null}]},{"id":"C5","text":"Shannon entropy is observer-relative in a way that Boltzmann's physical entropy is not: the same signal has different Shannon entropy under different codebooks.","tier":"system","weight":0.9,"status":"active","source_ids":["S1"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null}]},{"id":"C6","text":"Shannon's theory did not prove that information erasure costs energy; Landauer proved the physical cost of erasing one bit (at least kT ln 2 of heat) thirteen years later in 1961.","tier":"system","weight":0.95,"status":"active","source_ids":["S1","S2"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null},{"type":"supported_by","target":"S2","source_type":"adjacent","hash":null}]},{"id":"C7","text":"Jaynes and the objective Bayesian camp argue that entropy is a measure of our ignorance, not a property of the world, and that the thermodynamic convergence is formal analogy rather than physical identity.","tier":"speculative","weight":0.6,"status":"active","source_ids":["S3"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S3","source_type":"rival","hash":null}]},{"id":"C8","text":"Shannon entropy measures average compressibility across an ensemble, whereas Kolmogorov complexity measures compressibility of a single object; the gap between ensemble and individual compressibility matters for some incompressible strings drawn from compressible distributions.","tier":"system","weight":0.85,"status":"active","source_ids":["S1","S5"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"S1","source_type":"primary","hash":null},{"type":"supported_by","target":"S5","source_type":"adjacent","hash":null}]}],"sources":[{"id":"S1","type":"primary","url":"https://miscsubjects.com/a/shannon-1948","title":"Shannon, C.E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379-423; 27(4), 623-656.","quote":"The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem.","summary":"The foundational paper of information theory. Proves source coding theorem, noisy channel coding theorem, and defines channel capacity. Establishes that information is physical and measurable in bits.","claim_ids":["C1","C2","C3","C4","C5","C6","C8"]},{"id":"S2","type":"adjacent","url":"https://miscsubjects.com/a/landauer-1961","title":"Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. 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Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung.","quote":"","summary":"Originated the statistical-mechanical entropy formula S = k log W that Shannon adapted into H = -Σ p log p for information theory.","claim_ids":["C4"]},{"id":"S5","type":"adjacent","url":"https://miscsubjects.com/a/kolmogorov-1965","title":"Kolmogorov, A.N. (1965). Three Approaches to the Quantitative Definition of Information.","quote":"","summary":"Defines algorithmic information content as the length of the shortest program that generates a string. 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