Constants Planck Scale Matching J Curv Eq Boundary Quadratic

The curvature cost identity

In the Recognition Science framework, the cost of a recognition event is not a free choice. A proved theorem forces the cost function to be J(x) = (x + 1/x)/2 - 1. From that single function, the framework derives a chain of consequences: the golden ratio as the self-similar scale, an eight-tick recognition cycle, and three spatial dimensions. The declaration J_curv_eq_boundary_quadratic is a small but load-bearing link in that chain.

The declaration establishes an algebraic identity. It proves that the curvature cost J_curv(λ) = 2λ², defined as a curvature packet (a fixed amount of curvature spread over the faces of a cube) distributed over the 8 faces of the Q₃ hypercube, is exactly equal to another expression called the boundary curvature quadratic cost. In plain terms, the theorem says that two different ways of writing down the cost of curvature in the framework are the same quadratic form. This is a definitional equality, checked by the machine, not a physical discovery.

The identity matters because it connects the abstract cost function to a concrete geometric picture. The framework models a recognition event as a discrete record, and the cost of that event must balance against the cost of curvature. The theorem J_curv_eq_boundary_quadratic ensures that the curvature cost used in the Planck-scale matching discussion is the same as the boundary form that appears in the cost functional. Without this identity, the later step where the bit cost equals the curvature cost at equilibrium would rest on a mismatch.

What the declaration does not claim is just as important. It does not derive the π that appears in the recognition-to-Planck ratio. That factor enters through the definition of the gravitational constant G and through the definition of lambda_rec_SI, not through this theorem. The declaration also does not prove that the curvature packet is the correct physical model; it only proves an algebraic equality between two defined quantities. The physical bridge from recognition events to actual spacetime curvature remains open.

/-- The Planck-scale matching curvature functional agrees with the boundary
angle-defect J-cost quadratic form derived in `CurvatureCostForm`. -/
theorem J_curv_eq_boundary_quadratic (lam : ℝ) :
    J_curv lam = CurvatureCostForm.boundaryCurvatureQuadraticCost lam := by
  unfold J_curv
  rw [CurvatureCostForm.boundaryCurvatureQuadraticCost_eq]

/-- The Planck-scale matching curvature functional agrees with the boundary
angle-defect J-cost quadratic form derived in `CurvatureCostForm`. -/
theorem J_curv_eq_boundary_quadratic (lam : ℝ) :
    J_curv lam = CurvatureCostForm.boundaryCurvatureQuadraticCost lam := by
  unfold J_curv
  rw [CurvatureCostForm.boundaryCurvatureQuadraticCost_eq]

/-- **THE 0.564 ALGEBRAIC RATIO**:

λ_rec/ℓ_P = 1/√π ≈ 0.564.

This follows from `lambda_rec_SI` and `ell_P`; it is not an independent
derivation of the `π` in `Constants.G`. -/
theorem lambda_rec_over_ell_P :
    lambda_rec_SI / ell_P = 1 / sqrt Real.pi := by
  unfold lambda_rec_SI ell_P
  have hpic3_pos : Real.pi * c^3 > 0 := mul_pos Real.pi_pos (pow_pos c_pos 3)
  have hc3_pos : c^3 > 0 := pow_pos c_pos 3
  have hhG_pos : hbar * G > 0 := mul_pos hbar_pos G_pos
  have hhG_nonneg : hbar * G ≥ 0 := le_of_lt hhG_pos
  have hpi_nonneg : (0 : ℝ) ≤ Real.pi := le_of_lt Real.pi_pos
  rw [sqrt_div hhG_nonneg, sqrt_div hhG_nonneg]
  have h_c3_eq : sqrt (Real.pi * c^3) = sqrt Real.pi * sqrt (c^3) :=
    sqrt_mul hpi_nonneg (c^3)
  rw [h_c3_eq]
  have h_sqrt_c3_ne : sqrt (c^3) ≠ 0 := (sqrt_pos.mpr hc3_pos).ne'
  have h_sqrt_pi_ne : sqrt Real.pi ≠ 0 := (sqrt_pos.mpr Real.pi_pos).ne'
  have h_sqrt_hG_ne : sqrt (hbar * G) ≠ 0 := (sqrt_pos.mpr hhG_pos).ne'
  field_simp [h_sqrt_c3_ne, h_sqrt_pi_ne, h_sqrt_hG_ne]

What this page does not claim

The declaration does not prove that the curvature packet model is physically correct. The declaration does not derive the value of π or the gravitational constant G from first principles. The declaration does not establish the physical bridge from recognition events to spacetime curvature.

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/Constants/PlanckScaleMatching.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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