Constants Planck Scale Matching J Bit Eq Phi Minus

The bit cost identity

The golden ratio phi, the number roughly equal to 1.618, solves the equation phi squared equals phi plus one. A direct consequence is that its reciprocal equals phi minus one. The Recognition Science declaration J_bit_eq_phi_minus establishes that the framework's cost of a single recognition event, evaluated at this self-similar scale, takes the simple algebraic form phi minus three-halves.

In the framework, a recognition is a discrete event of matching an input to a stored pattern, and its cost is a measure of how expensive that match is. The framework's central cost function is J(x) = (x + 1/x)/2 - 1. The declaration J_bit_eq_phi_minus proves, as a formal theorem, that J(phi) = phi - 3/2. It derives this by substituting the reciprocal identity 1/phi = phi - 1 into the cost function and simplifying. This is a purely algebraic identity; it does not depend on any physical measurement.

The theorem also establishes a numerical bound: J(phi) lies strictly between 0.11 and 0.12. This value, approximately 0.118, is called the bit cost. The declaration is part of a larger chain in the framework's machine-checked library of formal theorems, where this bit cost is later balanced against a curvature cost to determine a recognition wavelength, and that wavelength is compared with the Planck length.

What the declaration does not claim is equally precise. It does not claim that the golden ratio itself is derived from the cost function; the golden ratio is taken as the self-similar scale. It does not claim that the bit cost has any physical meaning outside the framework's own definitions. It does not claim that the later comparison with the Planck length is a proof of a physical constant; that comparison is a separate algebraic identity, lambda_rec_SI / ell_P = 1 / sqrt(pi), and its physical interpretation remains a matter of framework modeling, not a theorem about nature.

/-- Using φ + 1/φ = φ + (φ - 1) = 2φ - 1 (from φ² = φ + 1 ⟹ 1/φ = φ - 1).
    Therefore J_bit = (2φ - 1)/2 - 1 = φ - 3/2.

    **Note**: This is exact. 1/φ = φ - 1 (from φ² = φ + 1).
    So φ + 1/φ = 2φ - 1.
    J_bit = (2φ - 1)/2 - 1 = φ - 3/2 ≈ 1.618 - 1.5 = 0.118. -/
theorem J_bit_eq_phi_minus : J_bit_val = phi - 3/2 := by
  unfold J_bit_val J
  -- Key identity: 1/φ = φ - 1 (from φ² = φ + 1)
  have h_inv : phi⁻¹ = phi - 1 := by
    have hphi_ne : phi ≠ 0 := phi_pos.ne'
    have hsq : phi^2 = phi + 1 := phi_sq_eq
    have : phi * phi = phi + 1 := by rw [← sq]; exact hsq
    field_simp at this ⊢
    linarith
  rw [h_inv]
  ring

/-- **Numerical Bound**: J_bit ≈ 0.118.
    Since 1.61 < φ < 1.62, we have 0.11 < J_bit < 0.12. -/
theorem J_bit_bounds : 0.11 < J_bit_val ∧ J_bit_val < 0.12 := by
  rw [J_bit_eq_phi_minus]
  constructor
  · have h := phi_gt_onePointSixOne
    linarith
  · have h := phi_lt_onePointSixTwo
    linarith

What this page does not claim

The golden ratio is not derived from the cost function; it is taken as the self-similar scale. The bit cost has no physical meaning outside the framework's own definitions. The comparison with the Planck length is not a proof of a physical constant; it is an algebraic identity within the framework's model.

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)

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