Action Euler Jaction Quantum

The Euler-J action quantum

In classical physics, action is a quantity with units of energy times time, and its numerical value depends on the unit system you choose. In Recognition Science, the framework defines a discrete ledger, a record of recognition events, and assigns a cost to each event. The Euler-J action quantum is a specific product of two numbers that the framework's machine-checked library of formal theorems proves must equal exactly one.

The first factor is the Euler characteristic of a sphere, a topological invariant that equals 2 for any surface shaped like a sphere, regardless of size. The framework derives this value from a discrete version of the Gauss-Bonnet theorem applied to the elementary recognition cell. The second factor is the recognition cost of squaring the golden ratio φ, where φ ≈ 1.618. The framework's cost function J(x) = (x + 1/x)/2 - 1, forced by five plain conditions, gives J(φ²) = 1/2. Since φ² = φ + 1, the algebra is direct: φ² + 1/φ² = 3, so J(φ²) = 3/2 - 1 = 1/2.

Multiplying these two forced numbers, 2 times 1/2, gives exactly 1. This product, called the Euler-J action quantum, is a theorem in the framework's library, proved without free parameters or additional assumptions. The theorem establishes that the bare number 1, which an earlier postulate had merely asserted as the action of a closed recognition loop, is actually the product of two independently derived facts: one combinatorial (the Euler characteristic) and one cost-functional (the J-cost).

In plain language, the module shows that a particular normalization, one unit of action per sector, is not arbitrary but follows from deeper structure. The theorem does not claim that φ² is the only closed loop, nor that every loop carries this action. It is a clean numerical fact about two named quantities, kept on record because it provides a non-arbitrary basis for counting capacity quanta in the framework's holographic models.

/-- **The Euler-J action quantum.** `χ(S²) · J(φ²) = 1`: the Gauss-Bonnet Euler
characteristic of the recognition cell's bounding sphere, times the J-cost of squaring
the golden ratio, is exactly one. Both factors are independently forced by the kernel
for unrelated reasons (one combinatorial, one cost-functional); their product is `1`
with no free parameter. -/
theorem euler_J_action_quantum :
    (Constants.LambdaRecDerivation.euler_S2 : ℝ) * Cost.Jcost (Constants.phi ^ 2) = 1 := by
  rw [Jcost_phi_sq_eq_half]
  simp [Constants.LambdaRecDerivation.euler_S2]

/-- The recognition cost of `φ²` is exactly `1/2`.

Proof: `φ² = φ + 1` (`Constants.phi_sq_eq`), and `(φ+1)·(2-φ) = 1` follows from the same
identity by pure algebra, so `(φ²)⁻¹ = (φ+1)⁻¹ = 2 - φ`. Hence
`φ² + (φ²)⁻¹ = (φ+1) + (2-φ) = 3`, and `J(φ²) = 3/2 - 1 = 1/2`. -/
theorem Jcost_phi_sq_eq_half : Cost.Jcost (Constants.phi ^ 2) = 1 / 2 := by
  have hsq : Constants.phi ^ 2 = Constants.phi + 1 := Constants.phi_sq_eq
  have hpos : (0 : ℝ) < Constants.phi := Constants.phi_pos
  have hne : (Constants.phi ^ 2 : ℝ) ≠ 0 := by positivity
  have hsq4 : (Constants.phi ^ 2) ^ 2 = (Constants.phi + 1) ^ 2 := by rw [hsq]
  unfold Cost.Jcost
  field_simp
  nlinarith [hsq, hsq4]

/-- **The Euler-J action quantum.** `χ(S²) · J(φ²) = 1`: the Gauss-Bonnet Euler
characteristic of the recognition cell's bounding sphere, times the J-cost of squaring
the golden ratio, is exactly one. Both factors are independently forced by the kernel
for unrelated reasons (one combinatorial, one cost-functional); their product is `1`
with no free parameter. -/
theorem euler_J_action_quantum :
    (Constants.LambdaRecDerivation.euler_S2 : ℝ) * Cost.Jcost (Constants.phi ^ 2) = 1 := by
  rw [Jcost_phi_sq_eq_half]
  simp [Constants.LambdaRecDerivation.euler_S2]

What this page does not claim

The theorem does not claim that φ² is the unique closed recognition loop. The theorem does not claim every closed loop carries this exact action. The module does not derive the value of the fine-structure constant.

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/Action/EulerJActionQuantum.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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