Physics Dirac Equation From Jcost

The Dirac equation in recognition space

The Dirac equation is the relativistic wave equation for spin-1/2 particles like electrons. It was introduced by Paul Dirac in 1928 to reconcile quantum mechanics with special relativity. Its most striking feature is that it predicts the existence of antimatter: for every particle, the equation has a solution with the same mass but opposite charge. In its standard form, the wavefunction has four components, which encode the two spin states of the particle and the two spin states of its antiparticle.

The equation is usually written as (iγ^μ∂_μ - m)ψ = 0, where ψ is the four-component spinor, m is the particle mass, and γ^μ are 4×4 matrices. The four components are not arbitrary; they are forced by the structure of spacetime and the requirement of Lorentz invariance. In three spatial dimensions plus time, a spin-1/2 field needs exactly four components to represent both particle and antiparticle with their two spin orientations each.

In Recognition Science, the framework models physical structure from a single starting point: a discrete record of events called a ledger. The framework's central result proves that any cost function satisfying five plain conditions must take the form J(x) = (x + 1/x)/2 - 1. From this cost function, the framework derives a chain of consequences including the golden ratio as a self-similar scaling and the number of spatial dimensions.

Within this framework, the Dirac equation's four spinor components appear as a counting consequence. The framework models a two-dimensional recognition space, and the number of components is 2^D+1 = 2³ = 4 for D=2. This matches the standard four-component Dirac spinor. However, the machine-checked library of formal theorems that accompanies the framework, called DiracEquationFrom_JCost, does not prove this connection. It proves three general facts about the cost function: it vanishes when the two inputs are equal, it is nonnegative for positive inputs, and a certain threshold involving the golden ratio is positive.

The cost function is defined as J(m/e), the cost of recognition when mass m is compared to charge e. It is proved that this cost is zero when m equals e, and nonnegative when both are positive. A canonical threshold phi - 3/2 is also defined and proved positive. These are general facts about the cost function, not facts specific to the Dirac equation. The accompanying documentation states this plainly: it proves nothing specific to this subject, because the definition of the cost function does not reference the Dirac equation's own terms.

What the library does establish is a template. It shows how a cost function can be defined on a physical ratio, and that this cost has the basic properties one would want: zero at equality, nonnegativity, and a positive threshold. The step from these general properties to a theorem about the Dirac equation would require a definition of mass and charge in the framework's own terms. That definition is not present in this library. The four-component spinor remains a framework-level observation, not a proved theorem from the cost function.

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

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

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]

What this page does not claim

The library does not prove that the Dirac equation follows from the cost function. The four-component spinor is an observation, not a derived theorem from the cost function in this library. The framework does not derive the fine-structure constant from the cost function.

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

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