ell_P_pos
Positivity of the Planck length follows from the definitions of hbar, G, and c in Recognition Science units. A researcher verifying consistency of the lambda_rec derivation from the J-cost extremum would cite this to confirm the length scale is well-defined. The proof is a direct term-mode chain that unfolds the square-root expression and applies standard positivity rules for multiplication, powering, division, and square roots.
claimLet ell_P denote the Planck length defined by ell_P = sqrt( hbar G / c^3 ). Then ell_P > 0.
background
The PlanckScaleMatching module derives the recognition wavelength from equating bit cost J_bit = J(phi) to curvature cost over the eight faces of the Q3 hypercube, yielding lambda_rec = sqrt( hbar G / (pi c^3) ) = ell_P / sqrt(pi). The Planck length ell_P is the companion scale without the pi factor. Upstream results supply positivity of c (the voxel-per-tick speed), hbar = phi^{-5}, and G = lambda_rec^2 c^3 / (pi hbar), together with the energy-conservation certificate that licenses the underlying Hamiltonian invariance.
proof idea
The term proof unfolds the definition of ell_P, then applies the square-root positivity rule, followed by the division-positivity rule. It discharges the numerator via multiplication positivity of hbar and G, and the denominator via positivity of c raised to the third power.
why it matters in Recognition Science
This result anchors the constants in the Planck gate identity lambda_rec = ell_P / sqrt(pi) that realizes Conjecture C8 of the Discrete Informational Framework Paper. It supplies the required positivity after the phi-forcing chain (T5-T8) and the face-averaging step that introduces the factor 1/pi. The declaration sits in the constants domain and is a prerequisite for any downstream use of the recognition wavelength in mass or energy formulas.
scope and limits
- Does not compute a numerical value for ell_P.
- Does not derive the pi factor from face averaging.
- Does not address SI-unit conversion.
- Does not extend to quantum-gravity corrections.
formal statement (Lean)
195theorem ell_P_pos : ell_P > 0 := by
proof body
Term-mode proof.
196 unfold ell_P
197 apply sqrt_pos.mpr
198 apply div_pos
199 · exact mul_pos hbar_pos G_pos
200 · exact pow_pos c_pos 3
201
202/-- **THE PLANCK GATE IDENTITY**:
203
204λ_rec = √(ℏG/(πc³)) = ℓ_P / √π
205
206This follows from the face-averaging principle applied to the
207curvature extremum. -/