Foundation Relational Qm3 From Jcost Relational Qm3 Cert

The certificate's scope

RelationalQM3Cert is a formal object in the Recognition Science library, a machine-checked collection of theorems. The certificate bundles three proved statements about a function called domainCost, which measures the cost of recognition: the cost of comparing a quantity to itself is zero, the cost is never negative for positive inputs, and a certain threshold value derived from the golden ratio is positive. Each statement is a theorem, verified by the library's kernel, and together they form a small package of facts about how this cost function behaves.

The certificate's name suggests a connection to relational quantum mechanics, the view that quantum states depend on the observer. The library's own documentation records that this was the intended direction: each observer's recognition system would define its own cost frame, and different observers would agree on ratios but not absolute values. That research note is explicit that this is where the idea was meant to go, not a result. The actual formal content proves nothing specific to relational quantum mechanics, because the cost function is defined simply as J(m/e) without any reference to observers, measurement contexts, or quantum states.

What would make this a theorem about its subject is a definition of m and e in the subject's own terms, tying those symbols to actual physical quantities. Without that, the certificate is a generic statement about a mathematical function, one that the library's documentation says is shared verbatim across 2383 sibling modules. The three proved facts are real, but they are not yet about relational quantum mechanics; they are about a cost function that could be applied to any positive real numbers.

In plain language, the certificate establishes that a particular cost formula has three basic mathematical properties. It does not establish that relational quantum mechanics follows from Recognition Science, nor that observers' perspectives are captured by this cost function. The gap between the certificate's name and its content is not hidden; the library's own notes call it out plainly. A reader should take the three theorems as proved and the relational quantum mechanics connection as an open research direction, not a derived consequence.

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 relational quantum mechanics follows from Recognition Science. The certificate does not define observer-dependent states or measurement contexts. The certificate does not claim that the golden ratio threshold has physical meaning in quantum mechanics.

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/Foundation/RelationalQM3FromJCost.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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