Foundation Wave Particle Duality3 From Jcost Wpduality3 Cert

The certification structure

In physics, wave-particle duality is the observation that light and matter sometimes behave as waves and sometimes as particles. A familiar picture: an electron passes through two slits and lands in an interference pattern, a wave effect, yet each detector click arrives at a single point, a particle effect. Recognition Science (RS) is a framework that starts from a different object, a ledger, a discrete record of recognition events, and derives physical structure from the forced cost of maintaining that record. The declaration WPDuality3Cert belongs to a family of modules that try to connect that ledger to duality.

The classical mathematics here is small. The framework defines a cost function J(x) = (x + 1/x)/2 - 1, which is zero when x equals 1 and positive otherwise. The module defines a domain cost as J(m/e), the cost of a ratio between two quantities m and e. Its three theorems are: this cost is zero when m equals e, it is nonnegative for positive m and e, and the number phi - 3/2 is positive, where phi is the golden ratio, about 1.618. The certificate WPDuality3Cert is a structure that packages these three facts, and the module proves the certificate exists.

The framework's own documentation is explicit about the limit. The module proves three general facts about J(m/e), and it proves nothing specific to wave-particle duality, because domainCost is defined as J(m/e) without any reference to the subject. The research note records where the idea was meant to go, not a result. The prose suggests a mapping: J(path_interference/path_detection) equals 0 at pure interference and 1 at pure detection, making complementarity a continuous transition in J-cost between wave and particle limits. That mapping is an intention, not a proved theorem.

In Recognition Science, then, the certificate establishes only the three algebraic facts, and the duality interpretation remains a research note. The machine-checked library of formal theorems, a collection of kernel-verified statements, contains the certificate as a valid object, but validity here means the three facts hold, not that the facts describe an electron. A reader who wants the classical content gets the cost function's basic behavior; a reader who wants duality gets a labeled hypothesis about how the framework might model it.

The consequence is a clean separation: the mathematics is checked, and the physics is not. What the certificate changes is the epistemic status of the three facts, they are proved, while the duality claim stays a research note. That distinction is the whole point of the declaration.

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]

theorem cert_inhabited : Nonempty WPDuality3Cert := ⟨cert⟩

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

What this page does not claim

Wave-particle duality is a proved consequence of the cost function. The certificate identifies m and e with any physical quantities. The golden ratio threshold has any observed physical meaning.

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/WaveParticleDuality3FromJCost.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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