Foundation Constant Derivations

How constants arise

In physics, the speed of light, Planck's constant, and the gravitational constant are usually measured inputs: the theory takes their values from experiment and builds on them. Recognition Science (RS) makes a different choice. It starts from a single forced cost function, J(x) = (x + 1/x)/2 - 1, proved unique from five plain conditions, and then derives the constants as ratios of quantities that come out of that foundation. The module ConstantDerivations in the framework's machine-checked library of formal theorems spells out that derivation chain in Lean 4, with no free parameters and no RS-specific axioms.

The chain runs level by level. The unique cost function forces the golden ratio φ = (1 + √5)/2 as the self-similar fixed point, and that ratio in turn forces three spatial dimensions and an eight-tick recognition cycle. From those, the module defines a fundamental time τ₀ = 1 and a fundamental length ℓ₀ = 1, and then the constants fall out as simple ratios. The speed of light is c = ℓ₀/τ₀ = 1, Planck's constant is ℏ = φ⁻⁵, and the gravitational constant is G = φ⁵/π. The module proves these identities directly: c_rs_eq_one, ℏ_rs_eq, and G_rs_eq are all theorem declarations in the library, each checked by the kernel. The Planck length, built from these, becomes √(1/π).

The key theorem, all_constants_from_phi, bundles the results: c = 1, ℏ is a φ-power, G·π = φ⁵, G·ℏ = 1/π, and the Planck length is √(1/π). Every one of these is a proved identity, not a fitted value. The module also defines E_coh = φ⁻⁵ as the coherence energy, and J_bit = log φ as the cost of a single recognition bit. The fine-structure constant α appears in the narrative as approximately 1/137, but that is a geometric seed with corrections, not a proved identity; the module does not derive α exactly.

What this establishes in plain language is that the constants are not arbitrary. Within the framework, they are ratios of framework-native quantities, all algebraic in φ. The module does not claim these match measured values; it claims they are forced by the foundation. The comparison to experiment, where it exists, is a separate empirical check. The derivation is a theorem; the match to the real world is a question the framework leaves open.

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

/-- G = φ⁵/π in RS-native units. -/
theorem G_rs_eq : G_rs = φ_val ^ 5 / Real.pi := rfl

/-- **ALL CONSTANTS FROM φ** (Family-A canonical values)

    In RS-native units:
    - c = 1 (definition of causal coherence)
    - ℏ = φ⁻⁵ (IR gate scale)
    - G = φ⁵/π (curvature extremum; the π is the holographic closure factor)
    - α ≈ 1/137 × correction (geometric seed)

    ℏ is a pure φ-power; G carries the physical π, so the honest algebraic
    statement is that `G·π` is the φ-power `φ⁵`. Consequences: `G·ℏ = 1/π`,
    `ℓ_P = √(1/π)`. φ is forced by the self-similarity equation from the
    unique cost J. -/
theorem all_constants_from_phi :
    -- c = 1
    c_rs = 1 ∧
    -- ℏ = φ⁻⁵
    (∃ n : ℤ, ℏ_rs = φ_val^n) ∧
    -- G·π = φ⁵ (G = φ⁵/π)
    (∃ n : ℤ, G_rs * Real.pi = φ_val^n) ∧
    -- G · ℏ = 1/π
    G_rs * ℏ_rs = 1 / Real.pi ∧
    -- Planck length = √(1/π)
    planck_length_rs = Real.sqrt (1 / Real.pi) :=
  ⟨c_rs_eq_one, ℏ_algebraic_in_φ, G_pi_algebraic_in_φ, G_ℏ_product, planck_length_eq⟩

What this page does not claim

The module does not derive the fine-structure constant α exactly; it only appears in a narrative as approximately 1/137. The derived constants are not claimed to match measured values; the comparison to experiment is outside the module's theorems. The module does not prove that the constants are physically realized; it proves they are forced by the framework's foundation.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: