Physics Tunneling3 From Jcost

The module's actual scope

In the Recognition Science framework, a ledger is a discrete record of events, and the cost of recognizing an event is a forced number, not a choice. The module physics-tunneling3-from-jcost defines a domain cost as Jcost (m / e), where Jcost is the framework's proved cost function and m and e are real numbers. Its name suggests a connection to quantum tunneling, but the Lean code proves nothing specific to that subject.

What the machine-checked library of formal theorems establishes are three general facts. First, the cost is zero when the two inputs are equal: domainCost r r = 0 for any nonzero r. Second, the cost is never negative for positive inputs. Third, the constant phi - 3/2, where phi is the golden ratio, is positive. These are properties of the cost function itself, not of any physical system.

The module also defines a structure called MuonCatFusion3Cert that packages these three facts, and shows it is inhabited. The docstring records a research note: one muon might catalyze about 150 fusions before decaying, and 150 is approximately phi^11, or about 199. That is an idea for where the mathematics could go, not a proved result.

In plain language, the module proves that its cost function behaves sensibly, and it documents a hope about muon-catalyzed fusion. The gap between the two is the missing definition of m and e in terms of muons and fusions. Without that, the module is a template, not a physics theorem.

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

This module does not prove that muon-catalyzed fusion works or that its rate is phi^11. The research note about 150 fusions is a hypothesis, not a measured or derived result.

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/Tunneling3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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