Foundation Constant Derivations Planck Length Eq

The Planck Length in the Framework

The Planck length is the scale at which quantum effects of gravity become as strong as the other forces, a natural yardstick built from the speed of light, the reduced Planck constant, and the gravitational constant. In conventional physics it is a derived quantity, not a fundamental postulate. The Recognition Science framework goes further: it derives those three constants from a single starting point, and the Planck length then falls out as a simple consequence.

Within the framework, the fundamental time unit is set to 1, and the fundamental length unit is also 1, so the speed of light, their ratio, is exactly 1. The reduced Planck constant is chosen as the coherence energy times the fundamental time, which the framework proves equals the golden ratio raised to the power minus 5. The gravitational constant is defined as the golden ratio raised to the power 5, divided by pi. These are not empirical inputs; they are algebraic combinations of the golden ratio, which the framework derives from its foundational cost function.

Putting these together, the framework's Planck length is the square root of (hbar times G over c cubed). Since c is 1, hbar is phi to the minus 5, and G is phi to the 5 over pi, the product of hbar and G is exactly 1 over pi. The Planck length therefore simplifies to the square root of 1 over pi. This is the theorem planck_length_eq: in the framework's units, the Planck length is not a mysterious constant but a clean expression involving only pi.

This result is a statement about the framework's internal definitions, not a prediction about the measured value of the Planck length in meters. The framework sets its fundamental units to 1, so the numerical value of the Planck length here is a pure number, not a physical measurement. The theorem proves the algebraic identity, not an empirical match. The framework's claim is that its derived constants are consistent and elegant, not that it has measured the Planck length.

The consequence is that in this account, the Planck length is not an independent fact about nature. It is a ratio of quantities that the framework derives from its foundational principle, and it turns out to be a simple function of pi. This is a demonstration of the framework's internal coherence, a check that its derived constants fit together without free parameters.

/-- 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]

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

What this page does not claim

The theorem does not claim the framework's Planck length matches the measured value in SI units. The framework does not claim its derived constants are empirically verified, only algebraically consistent. The theorem does not claim the Planck length is a fundamental constant of nature, only a derived quantity in this framework.

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)

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