lambda_rec_from_Jbit
The definition sets the recognition wavelength λ_rec to the positive square root of half the bit cost J_bit_val. Researchers deriving Planck-scale matching from the ledger curvature extremum in Recognition Science would cite this step. It follows by direct algebraic rearrangement of the quadratic relation J_curv(λ) = 2λ² set equal to J_bit_val.
claimThe recognition wavelength is defined by $λ_{rec} := √(J_{bit}/2)$, where $J_{bit} := J(φ)$ is the bit cost at the golden ratio and the curvature cost satisfies $J_{curv}(λ) = 2λ²$.
background
In the PlanckScaleMatching module the bit cost J_bit_val is introduced as J(φ) with φ the golden ratio, equivalently cosh(ln φ) - 1, representing the fundamental cost of a single ledger bit transition. The curvature cost J_curv(λ) is defined as 2λ² by distributing a ±4 curvature packet over the eight faces of the Q₃ hypercube. The module derives λ_rec from equating these two costs at equilibrium as the third step in the chain leading to Conjecture C8.
proof idea
This is a direct definition that applies the square-root operation to J_bit_val / 2. It depends only on the upstream definition of J_bit_val and the standard square-root function from Mathlib.
why it matters in Recognition Science
The definition supplies the candidate λ_rec verified by the downstream theorem extremum_condition that J_curv(λ_rec) equals J_bit_val and by extremum_unique that the solution is unique for positive λ. It completes the third link in the PlanckScaleMatchingCert structure and connects the unique cost functional J to the Planck length via the subsequent face-averaging step that introduces the factor 1/π. It touches the open question of restoring SI dimensions from RS-native units.
scope and limits
- Does not establish the physical validity of the curvature packet axiom.
- Does not include the face-averaging factor of 1/π.
- Does not compute the numerical ratio to the Planck length ℓ_P.
- Does not address higher-order corrections beyond the leading extremum.
formal statement (Lean)
148noncomputable def lambda_rec_from_Jbit : ℝ := sqrt (J_bit_val / 2)
proof body
Definition body.
149
150/-- λ_rec_from_Jbit > 0 since J_bit > 0. -/