Foundation Constant Derivations All Constants From Phi

The constant chain

The golden ratio, usually written φ, is the number that solves r² = r + 1, about 1.618. It appears in pentagons, in the Fibonacci sequence, and in a machine-checked library of formal theorems called Recognition Science. That library contains a theorem named all_constants_from_phi, and the theorem is a chain of exact identities, not an approximation. It says the framework's own versions of three physical constants are all algebraic in φ.

The first identity is the simplest: the framework's speed of light c equals 1. This is not a claim about meters and seconds. It is a definitional choice within the framework, where the unit of length and the unit of time are both set to 1, so their ratio is 1. The second identity is that Planck's constant ℏ equals φ⁻⁵, roughly 0.09017. The theorem states this as an exact power: there exists an integer n such that ℏ equals φ to the n, and the proof shows n is −5. The third identity is that the gravitational constant G times π equals φ⁵, so G is φ⁵/π. A fourth identity follows from the third: G times ℏ equals 1/π. The theorem also derives a Planck length of √(1/π).

These are not empirical measurements. They are consequences of the framework's starting definitions, checked by a computer. The definitions themselves, such as setting the fundamental time τ₀ to 1 and the coherence energy E_coh to φ⁻⁵, are choices, not derivations. What the theorem proves is that given those choices, the constants take the stated algebraic forms. The proof is a series of algebraic rewrites, each one verified by the machine, so the chain from definitions to identities is airtight within the framework.

In Recognition Science, the point is that these constants are not free parameters. They are ratios of framework-native quantities, and those quantities are all powers of φ. The library's narrative describes a chain: a composition law forces a unique cost function, which forces φ as a self-similar fixed point, which leads to the constants. The theorem all_constants_from_phi is the formal end of that chain for c, ℏ, and G.

The theorem does not include the fine-structure constant α. The library's own docstring lists α as approximately 1/137, from a geometric seed plus corrections, but that is not part of all_constants_from_phi. The exact value of α remains an open problem in the framework. The theorem also does not claim that the framework's c, ℏ, and G match the measured SI values; they are framework-internal definitions, and the identities are about those definitions, not about laboratory results.

/-- c = 1 in RS-native units. -/
theorem c_rs_eq_one : c_rs = 1 := by
  unfold c_rs ℓ₀ τ₀
  norm_num

/-- G · π = φ⁵. The bare `G` is not a pure φ-power (the physical `π` is present);
    the honest algebraic statement is that `G·π` is the φ-power `φ⁵`. -/
theorem G_pi_eq_phi5 : G_rs * Real.pi = φ_val ^ (5 : ℤ) := by
  unfold G_rs
  exact div_mul_cancel₀ _ Real.pi_ne_zero

/-- Planck length = √(1/π) in RS-native units (Family A, `ℓ_P² = 1/π`). -/
theorem planck_length_eq : planck_length_rs = Real.sqrt (1 / Real.pi) := by
  unfold planck_length_rs
  rw [c_rs_eq_one]
  simp only [one_pow, div_one]
  rw [mul_comm, G_ℏ_product]

What this page does not claim

The theorem does not derive the fine-structure constant α; its exact value remains open. The identities do not claim that the framework's c, ℏ, and G match the measured SI values in kilograms, meters, and seconds. The theorem does not prove that the definitions of τ₀, ℓ₀, and E_coh are forced by the framework; they are choices.

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/Foundation/ConstantDerivations.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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