all_constants_from_phi
Recognition Science fixes the speed of light, reduced Planck constant, gravitational constant and Planck length as algebraic expressions in the golden ratio phi within its native units. A physicist deriving the constant sector from the composition law would cite this result to eliminate free parameters. The proof is a one-line constructor that assembles five prior lemmas on each component.
claimIn RS-native units, $c = 1$, there exists an integer $n$ such that $h = phi^n$, there exists an integer $m$ such that $G = phi^m$, $G h = 1$, and the Planck length equals 1.
background
The module derives physical constants from the Recognition Science foundation. The composition law forces the unique cost J(x) = (x + x^{-1})/2 - 1. Self-similarity fixes phi as the golden ratio. The eight-tick octave then requires three spatial dimensions, yielding fundamental scales tau_0 = 1 tick and ell_0 = 1 with coherence energy phi^{-5}. Constants appear as ratios of these scales. Upstream results include the definition of G_rs as phi^5 and the tick as the unit time quantum.
proof idea
The proof is a one-line wrapper that applies the conjunction of c_rs_eq_one, ℏ_algebraic_in_φ, G_algebraic_in_φ, G_ℏ_product and planck_length_eq_one. These lemmas are established earlier in the module from the phi-ladder and the forcing chain.
why it matters in Recognition Science
This result closes the constant sector by showing every fundamental constant is algebraic in phi, which is forced by J-uniqueness and the self-similar fixed point. It supports the claim that the entire physics follows from the composition law with no free parameters. The module narrative links it to the eight-tick cycle forcing D = 3 and the mass formula on the phi-ladder.
scope and limits
- Does not compute numerical approximations for alpha or other constants.
- Does not address quantum field theory or particle masses beyond the constant sector.
- Does not prove uniqueness of the RS-native units choice.
- Does not derive the fine-structure constant explicitly.
formal statement (Lean)
270theorem all_constants_from_phi :
271 -- c = 1
272 c_rs = 1 ∧
273 -- ℏ = φ^(-5)
274 (∃ n : ℤ, ℏ_rs = φ_val^n) ∧
275 -- G = φ^5
276 (∃ n : ℤ, G_rs = φ_val^n) ∧
277 -- G × ℏ = 1
278 G_rs * ℏ_rs = 1 ∧
279 -- Planck length = 1
280 planck_length_rs = 1 :=
proof body
Tactic-mode proof.
281 ⟨c_rs_eq_one, ℏ_algebraic_in_φ, G_algebraic_in_φ, G_ℏ_product, planck_length_eq_one⟩
282
283/-! ## The Derivation Narrative -/
284
285/-- **THE CONSTANT DERIVATION NARRATIVE**
286
287 1. The composition law (d'Alembert) is the foundation.
288 2. It uniquely determines J(x) = ½(x + 1/x) - 1.
289 3. Self-similarity under J forces φ = (1+√5)/2.
290 4. The eight-tick cycle (2^D = 8) forces D = 3.
291 5. These determine the fundamental scales:
292 - τ₀ = 1 (fundamental tick)
293 - ℓ₀ = 1 (fundamental length)
294 - E_coh = φ^(-5) (coherence quantum)
295 6. The constants follow:
296 - c = ℓ₀/τ₀ = 1
297 - ℏ = E_coh · τ₀ = φ^(-5)
298 - G = φ^5 (curvature extremum)
299 - α ≈ 1/137 (geometric + gap-45)
300
301 **No free parameters.** The entire constant sector is determined
302 by the composition law. -/