Action Euler Jaction Quantum Jcost Phi Sq Eq Half

The Euler-J action quantum

The golden ratio, φ, is the number that solves φ² = φ + 1, approximately 1.618. Squaring it gives about 2.618. The Recognition Science declaration Jcost_phi_sq_eq_half proves that a particular function, the recognition cost J, assigns to this squared value the exact number 1/2. This is a formal theorem in the framework's machine-checked library, meaning the proof is verified by a computer and relies on no unproven assumptions beyond the standard axioms of logic.

The theorem is not an isolated curiosity. It combines with another independently forced result: the Euler characteristic of a sphere, which is 2. Multiplying these two numbers, 2 times 1/2, gives exactly 1. This product, called the Euler-J action quantum, is the subject of the theorem euler_J_action_quantum. The significance is that the number 1, which a previous unproven postulate had simply asserted as the action of a closed recognition loop, turns out to be the forced product of two quantities the framework's kernel already derives for unrelated reasons. One factor is combinatorial, from discrete Gauss-Bonnet; the other is cost-functional, from the J-cost equation. No free parameter or calibration is involved.

What the declaration does not claim is equally precise. It does not claim that φ² is the unique closed recognition loop, nor that every closed loop carries this exact action. It is a clean numerical fact about two named quantities. The downstream use in the framework's holographic mode capacity cites it as the forced normalization that makes one capacity quantum per sector non-arbitrary, not as a general statement about all recognition loops.

/-- 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 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]

/-- **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 that every closed recognition loop carries this exact action. The theorem does not claim the number 1 is a physically measurable action in conventional units.

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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