Mathematics Information Theory From Rs Pos Entropy

Positive entropy

In information theory, Shannon entropy measures the uncertainty of a probability distribution. A coin that always lands heads has zero entropy; a fair coin has one bit. The Recognition Science framework re-expresses this quantity through its cost function J, which assigns a nonnegative number to every positive real r. The framework's machine-checked library of formal theorems proves two boundary facts about this cost: the cost of certainty is zero, and the cost of any uncertain outcome is strictly positive.

The second fact is the declaration pos_entropy. It states that for any positive real number r that is not equal to 1, the cost J(r) is greater than zero. In plain language: if an outcome is not certain, it carries positive cost. The proof is immediate from the framework's earlier result Jcost_pos_of_ne_one, which itself follows from the defining properties of J. The declaration also appears inside a certificate structure that bundles the five Shannon axioms, the minimum-entropy fact, and the positive-entropy fact into one machine-checked package.

The classical context gives this its meaning. Shannon's 1948 axioms for entropy are continuity, maximality, additivity, symmetry, and subadditivity; the framework's library counts exactly five such axioms and identifies the number five with a structural dimension. The minimum-entropy theorem Jcost 1 = 0 says a certain outcome costs nothing. The positive-entropy theorem extends this: every non-certain outcome costs something. Together they give the cost function the same shape as entropy: zero at certainty, positive elsewhere.

What pos_entropy does not claim is narrower than it sounds. It does not assert that all uncertain outcomes cost the same amount, nor does it give the numerical value of the cost for any particular r. It only establishes the strict inequality 0 < J(r). The declaration also does not prove the Shannon capacity theorem or the additivity of entropy; those remain separate results in the library. The theorem is a boundary condition, not a full theory of information.

/-- 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

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

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

What this page does not claim

The declaration does not give the numerical value of J(r) for any r other than 1. The declaration does not prove additivity or the Shannon capacity theorem. The declaration does not assert that all uncertain outcomes carry equal cost.

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)

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