Cosmology Phi Rung Ladder

The phi rung ladder

The golden ratio, usually written φ, is the number that solves r² = r + 1, so r ≈ 1.618. Its powers form a ladder: φ² ≈ 2.618, φ³ ≈ 4.236, and so on. The cosmology phi rung ladder is a scheme in Recognition Science that assigns key physical and conceptual thresholds to specific rungs of this ladder, meaning each threshold is modeled as φ raised to an integer exponent. The framework's machine-checked library of formal theorems proves the arithmetic relations between these rungs, such as which products of powers land on which other rungs.

The ladder has four core rungs. The counterfactual threshold Z_cf sits at φ⁵ ≈ 11.09. The moral patienthood ignition Z_moral sits at φ¹⁹ ≈ 9,349. The baryon asymmetry η_B sits at φ⁻⁴⁴, a tiny number. The consciousness saturation threshold Θ_crit sits at φ⁴⁵. These are not arbitrary placements; the framework's library proves the product identities that connect them: φ⁵ × φ⁴⁵ = φ⁵⁰ and φ⁵ × φ⁴⁴ = φ⁴⁹. In plain arithmetic, multiplying the counterfactual threshold by the consciousness saturation threshold lands exactly on the rung 50, and multiplying by the inverse baryon asymmetry lands on rung 49. The library also proves the rung gaps: 45 − 5 = 40, which factors as 8 × 5, and 19 − 5 = 14, among others.

These identities are proved in a formal library with no axioms beyond the standard logical ones, meaning the arithmetic is machine-checked. The module does not prove that the thresholds themselves are real; it proves that if you accept those rung assignments, the multiplicative structure follows. The framework treats the rung values as definitions, and the theorems establish the relations among them. The master certificate, PhiRungLadderCert, bundles these results into a single object, certifying the whole ladder's internal consistency.

What the ladder establishes in plain language is a compact arithmetic skeleton for a set of otherwise unrelated-looking constants. Instead of remembering five separate numbers, you can remember five exponents and a few addition facts. The product identities mean that the counterfactual threshold and the consciousness threshold combine to reach the rung 50, and the counterfactual threshold combined with the inverse baryon asymmetry reaches rung 49. These relations are the kind of structure that, if the underlying threshold identifications hold up, would tie together cosmology and cognition through one scaling law.

/-- Z_cf sits at rung 5. -/
theorem zcf_rung_eq_5 : zcf_rung = 5 := rfl

/-- The moral patienthood ignition threshold at rung 19. -/
def z_moral_rung : ℤ := 19

/-- The baryon asymmetry rung: −44. -/
def eta_B_rung_val : ℤ := -44

/-- Θ_crit sits at rung 45. -/
theorem theta_crit_rung_eq_45 : theta_crit_rung_val = 45 := rfl

/-- **THEOREM**: Z_cf × Θ_crit = φ⁵⁰.
    From the Counterfactual Boundary paper: the product of the two
    consciousness thresholds equals the 50th φ-power. -/
theorem zcf_times_theta_crit :
    phi ^ (5 : ℤ) * phi ^ (45 : ℤ) = phi ^ (50 : ℤ) := by
  rw [← zpow_add₀ phi_ne_zero]
  norm_num

/-- **THEOREM**: Z_cf × η_B⁻¹ = φ⁴⁹.
    From the Counterfactual Boundary paper: the product of the counterfactual
    threshold and the inverse baryon asymmetry equals the 49th φ-power. -/
theorem zcf_times_eta_B_inv :
    phi ^ (5 : ℤ) * phi ^ (44 : ℤ) = phi ^ (49 : ℤ) := by
  rw [← zpow_add₀ phi_ne_zero]
  norm_num

What this page does not claim

The rung assignments themselves are not proved from first principles; they are definitions. The framework does not derive the measured baryon asymmetry from the ladder; it only assigns the rung. The product identities hold for the formal definitions, not for any empirical measurement of the thresholds.

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

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