Physics Quark Confinement3 From Jcost

A cautionary module

Quark confinement is the observation that isolated quarks never appear in nature; they exist only bound inside protons, neutrons, and other hadrons. The energy needed to pull one free is often described by a string tension, a constant with units of energy per length, conventionally written σ and measured near 0.18 GeV². A module in the Recognition Science library, Quark_Confinement3_FromJCost, takes its name from an attempt to derive this number from the framework's cost function, but the file itself establishes only three general facts about a domain cost, none of them specific to quarks.

The module defines domainCost, a discrete record of the recognition cost of comparing two quantities m and e, as J(m/e), where J is the framework's unique cost function. It then proves three theorems: the cost is zero when m equals e, the cost is never negative for positive inputs, and the number φ − 3/2 is positive. These are true for any positive real inputs, so they apply equally to prices, lengths, or masses. The file's own docstring says plainly that the Lean code proves nothing specific to quark confinement, because the definition never connects m and e to quark physics.

The research note inside the file records the intended physical step: identify m and e with quark masses and a scale, then compute the string tension. The note's arithmetic shows the first guess, σ = J(φ) × (0.21 GeV)², lands at 0.0052 GeV², about 34 times too small. A revised guess, σ = φ² × (0.21 GeV)², gives 0.115 GeV², which is consistent with the measured 0.18 GeV² within a factor of about 1.5. That second line is a research note, not a theorem; it is a hypothesis with a named falsifier: any measurement of the string tension that disagrees with φ² Λ²_QCD would falsify it.

What the module does establish, in plain language, is a cautionary lesson about the framework's library: a machine-checked collection of formal theorems can carry a physics-flavored name while proving only generic mathematics. The three proved facts are real, but they would hold for any positive numbers, and the leap from the cost function to the quark sector remains entirely unformalized. A reader who wants the physics must look elsewhere; this file is a record of an intended derivation, not the derivation itself.

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]

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

What this page does not claim

The module does not derive the string tension from the cost function. The module does not prove that quarks are confined. The φ² Λ²_QCD formula is a research note, not a proved theorem.

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/Quark_Confinement3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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