Information Data Compression3 From Jcost Data Compr3 Cert

The certificate's scope

Data compression is the art of representing information in fewer bits. Claude Shannon's 1948 source coding theorem sets the limit: for a message source, the average code length can be brought arbitrarily close to the entropy H(X), the theoretical minimum. Practical schemes like Huffman coding land within one bit of that limit. The gap between what a code achieves and the entropy is the overhead.

In Recognition Science, a ledger (a discrete record of events) carries a forced cost function J(x) = (x + 1/x)/2 - 1. The framework's library, a machine-checked collection of formal theorems, proves this cost vanishes when the argument equals 1, is never negative on positive inputs, and has a specific positive threshold at phi - 3/2, where phi is the golden ratio. The declaration DataCompr3Cert packages these three facts into a single certificate object.

The certificate itself is deliberately modest. It proves that for a cost defined as J(m/e), the cost is zero when m equals e, nonnegative for positive m and e, and that the threshold phi - 3/2 is positive. These are general facts about the cost function, not results about compression. The documentation states this plainly: the proof establishes nothing specific to data compression, because the quantities m and e are not defined in compression terms. The research note records where the idea was meant to go, not a result that exists.

What would make this a theorem about compression is a definition of m and e in that subject's own terms, for example message length and encoded length. Without that bridge, the certificate is a reusable template, shared verbatim across 2383 sibling modules, awaiting a subject-specific interpretation. The framework's larger claim, that an optimal encoding uses J(phi) bits of overhead per symbol beyond Shannon entropy, remains a research direction, not a proved statement.

theorem cert_inhabited : Nonempty DataCompr3Cert := ⟨cert⟩

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)

What this page does not claim

The certificate does not prove that any encoding scheme achieves J(phi) overhead per symbol. The certificate does not define m and e in terms of message length and encoded length. The certificate does not establish a result about data compression, only about the cost function J.

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/DataCompression3FromJCost.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.

Derived articles

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