Mathematics Information Theory From Rs Min Entropy

Minimum entropy

In classical information theory, entropy measures uncertainty. The Shannon entropy H = -∑ pᵢ log pᵢ assigns a number to a probability distribution; it is zero exactly when one outcome has probability 1 and all others have probability 0. That is the minimum entropy state: no surprise, no information gain, nothing left to learn. The same zero point appears in Recognition Science, where the framework models a ledger, a discrete record of recognition events, and assigns each event a cost, a forced price of recognition. The framework's cost function J(x) = (x + 1/x)/2 - 1 is proved to be the unique function satisfying five plain conditions, and it vanishes exactly at x = 1.

What the framework's declaration min_entropy establishes is a small, precise fact: Jcost 1 = 0. In plain language, the cost of recognizing a certain outcome is zero. The declaration is a theorem in the machine-checked library of formal theorems, proved without axioms beyond the standard logical ones. It also proves the companion fact that for any probability r greater than 0 and not equal to 1, the cost J(r) is strictly positive. Together these two statements say that uncertainty carries positive cost and certainty carries none. The framework's library bundles these into a certificate that also counts the five Shannon axioms (continuity, maximality, additivity, symmetry, subadditivity) and confirms that number is 5.

What min_entropy does not claim is broader. It does not claim that the framework derives Shannon entropy from scratch; the docstring identifies Shannon's H with the average J-cost, but that identification is a definitional choice, not a proved theorem. It does not claim that the five axioms force the exact form of Shannon entropy; the count of five is a theorem, but the equivalence of the axiom list to the entropy formula is not in this declaration. It does not claim anything about the Shannon capacity theorem beyond the algebraic rearrangement C/B = log₂(1 + S/N) with S/N = J⁻¹; that is a model statement, not a proof of the classical capacity result. The declaration is a narrow, verified anchor: the zero of the cost function, nothing more.

/-- Minimum entropy: J = 0 (certain outcome). -/
theorem min_entropy : Jcost 1 = 0 := Jcost_unit0

/-- Positive entropy: J > 0 (uncertain outcome). -/
theorem pos_entropy {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem shannonAxiomCount : Fintype.card ShannonAxiom = 5 := by decide

What this page does not claim

The identification of Shannon entropy with average J-cost is a definitional choice, not a proved theorem. The five axioms are counted, but their equivalence to the entropy formula is not established by this declaration. The capacity formula C/B = log₂(1 + S/N) is a model statement, not a proof of the classical theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/InformationTheoryFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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