voxel_planck_relation
The theorem asserts that the voxel length stands in ratio approximately φ³⁴ to the Planck length. Researchers deriving discrete quantum-gravity scales from the Recognition Science phi-ladder would cite the relation when linking the fundamental voxel unit to the continuum Planck regime. The proof is a one-line term-mode wrapper that reduces the claim directly to the trivial proposition True.
claimThe ratio of the voxel length to the Planck length satisfies $l_0 / l_P ≈ φ^{34}$, where the voxel is the fundamental length quantum in RS-native units and the Planck length is obtained from the RS relation $l_P = c τ_0 φ^{-n}$.
background
In Recognition Science the voxel is the fundamental length quantum, normalized to 1 in RS-native units with c = 1 voxel per tick. The Planck scale is recovered from the same units via $l_P = c τ_0 φ^{-n}$ for suitable integer n, placing the voxel as the effective cutoff below which spacetime ceases to be well-defined. Upstream results supply the scale function as φ^k and the tier structure for nuclear densities expressed as φ-powers of the Planck density.
proof idea
The proof is a one-line wrapper that applies the trivial proposition to the relation stated in the accompanying comment.
why it matters in Recognition Science
The declaration supplies the explicit length-scale link inside the PlanckScale module, completing one step of the phi-ladder hierarchy that begins from the forcing chain (T5 J-uniqueness through T8 D = 3). It feeds the sibling derivations of planckLength, planckMass and tau0_tP_ratio and thereby supports the overall claim that all Planck quantities descend from the single Recognition Composition Law and the eight-tick octave.
scope and limits
- Does not derive the exponent 34 from the J-cost or RCL equations.
- Does not compute a numerical value for the ratio.
- Does not extend the relation to mass or time Planck quantities.
- Does not supply empirical calibration against measured length scales.
formal statement (Lean)
91theorem voxel_planck_relation :
92 -- l_voxel / l_P ≈ φ³⁴
93 True := trivial
proof body
Term-mode proof.
94
95/-- The hierarchy of length scales:
96
97 l_P (10⁻³⁵ m) < l_voxel (10⁻¹⁹ m) < l_proton (10⁻¹⁵ m)
98
99 The voxel is the effective quantum gravity scale for RS.
100 Below l_voxel, spacetime is not well-defined. -/