Information Shannon Entropy Max Rs

Maximum entropy

Shannon entropy, introduced by Claude Shannon in 1948, quantifies the average amount of information produced by a source. For a set of n symbols, each equally likely, the entropy reaches its maximum value of H_max = log2(n) bits. This means a system that must distinguish among n possibilities needs at least log2(n) binary choices, or bits, to do so. The formula is the foundation of data compression: no lossless scheme can represent n equally likely symbols using fewer than log2(n) bits on average.

The maximum grows slowly with n. Doubling the number of symbols adds exactly one bit of maximum entropy. This logarithmic scaling is why information can be stored and transmitted efficiently even as message spaces grow large. The famous special case n = 2 gives H_max = 1 bit, the basic unit of digital information, while n = 8 gives 3 bits and n = 256 gives 8 bits, one byte.

In Recognition Science, the framework models recognition as a discrete record of events, a ledger, and derives a forced cost function J(x) = (x + 1/x)/2 - 1 for any recognition event. The framework's machine-checked library of formal theorems proves three general facts about this cost when it is applied to a ratio m/e: the cost vanishes when m equals e, it is nonnegative for positive inputs, and the quantity phi - 3/2 is positive, where phi is the golden ratio. The library proves nothing specific to Shannon entropy, because the module defines its cost as J(m/e) without connecting m and e to information-theoretic quantities.

The research note attached to the module records an intended direction: at an alphabet size of n = phi^k, the maximum entropy becomes k times log2(phi), roughly k times 0.694 bits. This would mean recognition systems with alphabet sizes following the golden-ratio power ladder maximize information efficiency. That statement is a research note, not a proved result, and it remains open until the module defines m and e in terms of actual information-theoretic quantities.

What the module does establish is a reusable certificate: a packaged collection of the three proved facts about the cost function, available for any future theory that supplies a meaningful definition of m and e. The certificate is inhabited, meaning the facts are consistent and the structure exists. This is a small but honest step: it prepares the mathematical ground without pretending the information-theoretic connection is already made.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem cert_inhabited : Nonempty ShannonEntropyMaxCert := ⟨cert⟩

What this page does not claim

The golden-ratio maximum entropy formula is a proved result. The module establishes any connection between the cost function and Shannon entropy. Recognition Science derives the value of Shannon entropy from first principles.

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/Information/Shannon_Entropy_Max_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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