Physics Rs Physics Module 003

A template, not a result

Physics RS Module 003 appears, from its header note, to be about the neutron lifetime: it records that phi^17 times 0.246 seconds equals 878.5 seconds, close to the measured 878.4 seconds. That comparison is a research note, not a proved result. The machine-checked content of the module proves three general facts about a single function, and none of them mention neutrons.

The function is the cost, a measure of how expensive a recognition event is, defined as J(x) = (x + 1/x)/2 - 1. The module defines domainCost(m, e) = J(m/e), where m and e are real numbers. The three proved facts are: domainCost(r, r) = 0 for any nonzero r, meaning matching inputs cost nothing; domainCost(m, e) is nonnegative when both m and e are positive; and the number phi - 3/2 is positive, where phi is the golden ratio. These are simple consequences of the cost function's definition, verified in the machine-checked library of formal theorems.

What the module does not do is connect these facts to any specific physics. The definition of domainCost uses m and e as abstract real numbers, not as a mass and a charge or any other named physical quantity. The header note says the neutron lifetime was where the idea was meant to go, but the formal content stops at the general properties. The module is one of 2384 sibling modules that share this exact body; the shared part is stated once in a template called DomainCostTemplate.

In Recognition Science, this template structure is deliberate. The framework's library proves general facts about the cost function once, and individual modules instantiate those facts for particular subjects only when they define m and e in that subject's own terms. Module 003 has not done that for the neutron. Its certificate, RSPhysics003Cert, packages the three general facts into a single object, and the certificate is inhabited, meaning the facts hold. But the certificate says nothing about neutrons, and the neutron lifetime comparison in the header remains an unproved research note.

The lesson is precise: a module can carry a physics label and a numerical agreement without establishing anything about that physics. What Module 003 establishes is about the cost function itself: it vanishes at equality, it is nonnegative on positive inputs, and the golden ratio sits above 1.5. Those are real, machine-checked facts. The neutron lifetime is a separate claim that would need its own definition of m and e to become 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]

theorem cert_inhabited : Nonempty RSPhysics003Cert := ⟨cert⟩

What this page does not claim

The module does not prove the neutron lifetime value of 878.5 seconds. The module does not establish that m and e refer to any physical quantities. The module does not derive the golden ratio from neutron physics.

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/Physics/RS_Physics_Module_003.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: