planck_relation_satisfied
Researchers verifying internal consistency of Recognition Science constant derivations cite this result to confirm the Planck relation holds for the derived ħ. It equates the expression built from the fundamental tick duration together with CODATA values of G and c to the CODATA ħ. The proof is a term-mode reduction that unfolds the derived definition, rewrites using the squared tick equality, and applies field simplification after establishing non-zero conditions.
claimThe derived reduced Planck constant satisfies $ħ_derived(τ_0, G_CODATA, c_CODATA) = ħ_CODATA$, where $τ_0$ is the fundamental tick duration and the CODATA values are the reference measurements for $G$ and $c$.
background
The module derives physical constants from Recognition Science primitives. It adopts CODATA reference values: $c = 299792458$ m/s, $ħ = 1.054571817 × 10^{-34}$ J s, and $G = 6.67430 × 10^{-11}$ m³ kg^{-1} s^{-2}. The fundamental time unit $τ_0$ is defined as the duration of one tick in RS-native units. Upstream lemmas establish that $τ_0$, $G_CODATA$, and $c_CODATA$ are positive and nonzero, enabling algebraic manipulations in the real numbers.
proof idea
The proof unfolds the definition of the derived ħ. It rewrites using the equality for the square of the tick duration. It then asserts that $G$, $c$, and $π$ are nonzero using the supplied lemmas and applies field simplification to verify the identity.
why it matters in Recognition Science
This theorem is recorded as proven in the derivation_status definition, which aggregates all consistency checks for the constants. It contributes to validating the RS-native expressions for ħ = φ^{-5} and G = φ^5 / π against empirical values. The result supports the forcing chain steps that fix the constants without additional hypotheses.
scope and limits
- Does not derive the numerical values of the constants from first principles.
- Does not account for experimental uncertainties in the CODATA inputs.
- Does not prove relations for other fundamental constants such as the fine structure constant.
- Does not address potential modifications from quantum gravity effects.
formal statement (Lean)
119theorem planck_relation_satisfied :
120 hbar_derived tau0 G_codata c_codata = hbar_codata := by
proof body
Term-mode proof.
121 unfold hbar_derived
122 rw [tau0_sq_eq]
123 have hG : G_codata ≠ 0 := G_codata_ne_zero
124 have hc : c_codata ≠ 0 := c_codata_ne_zero
125 have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
126 have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
127 field_simp
128