Information Algorithmic Prob3 From Jcost Algorithmic Prob3 Cert

The certificate's scope

Algorithmic probability asks how likely a string of data is as the output of a program. The classical answer, the Solomonoff prior, sums over every program that produces the string, weighting each by 2 raised to the negative of its length in bits. Longer programs contribute less, so simple explanations dominate. The Recognition Science declaration AlgorithmicProb3Cert is a machine-checked bundle of three statements about a cost function, not a derivation of that prior.

The cost function is defined as J applied to the ratio of two real numbers, m and e. The three proved facts are elementary. First, when m equals e and neither is zero, the cost is exactly zero. Second, when both m and e are positive, the cost is never negative. Third, the golden ratio minus 3/2 is a positive number. Each statement is proved in a machine-checked library of formal theorems, and the certificate simply packages them together.

The document's own status note is blunt: the proof establishes only these three general facts about the cost function. It proves nothing specific to algorithmic probability, because the cost function is defined without any reference to programs, strings, or computation. The paragraph about the Solomonoff prior is a research note recording where the idea was meant to go, not a result. What would make this certificate a theorem about its subject is a definition of m and e in that subject's own terms, and that definition is absent.

In Recognition Science, the framework models a discrete record of events called a ledger, and the cost of recognition is forced by the uniqueness theorem for J. But this certificate does not connect that machinery to algorithmic probability. It is a placeholder, a template shared verbatim with thousands of sibling certificates, each waiting for its own subject-specific definitions. The honest takeaway: the certificate proves three trivial facts about a ratio, and the interesting claim about compression rates remains a hypothesis, not a theorem.

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

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]

What this page does not claim

The certificate does not prove any connection between J-cost and the Solomonoff prior. No compression-rate claim about J(phi) times |x| is established by this certificate. The certificate does not define m or e in terms of programs, strings, or computation.

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/AlgorithmicProb3FromJCost.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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