lambda_rec_unique_root
The theorem shows that the curvature functional K vanishes exactly at the recognition length lambda_rec for any positive real lambda. Researchers deriving constants non-circularly in Recognition Science would cite this uniqueness to close the balance condition before defining G. The proof unfolds K and lambda_rec then solves the resulting quadratic equation via arithmetic tactics and factorization.
claimFor any positive real number $lambda$, the curvature functional $K(lambda) := lambda^2 / lambda_rec^2 - 1$ satisfies $K(lambda) = 0$ if and only if $lambda = lambda_rec$.
background
The LambdaRecDerivation module formalizes a non-circular derivation of the recognition length lambda_rec using only normalized bit cost (=1) and curvature cost (=2 lambda^2) from the Q3 Gauss-Bonnet normalization. Here lambda_rec is identified with ell0, the fundamental length unit set to 1 in RS-native units. The curvature functional K is the algebraic restatement K(lambda) = lambda^2 / lambda_rec^2 - 1, retained for verification compatibility; G is then defined as a consequence via G := pi lambda_rec^2 c^3 / hbar.
proof idea
The tactic proof unfolds K, lambda_rec and ell0, then applies simp to reduce to lambda^2 = 1. It splits the biconditional with constructor. The forward direction derives (lambda-1)(lambda+1)=0, applies mul_eq_zero, and concludes lambda=1 via linarith in both cases. The reverse direction substitutes the hypothesis and closes with ring.
why it matters in Recognition Science
This uniqueness result feeds the parent theorem lambda_rec_is_forced, which asserts existence and uniqueness of a positive root for K(lambda)=0. It completes the non-circular step in the Constants module, where lambda_rec equals ell0 in the eight-tick cycle and G follows directly. The result aligns with the recognition length being the voxel scale before further constants are derived.
scope and limits
- Does not establish any numerical value for lambda_rec beyond its equality to ell0.
- Does not apply when lambda is non-positive.
- Does not derive the curvature cost functional from the J-cost or Recognition Composition Law.
- Does not address the phi-ladder or mass formula.
Lean usage
rw [(lambda_rec_unique_root lambda hlambda).mp hK]formal statement (Lean)
100theorem lambda_rec_unique_root (lambda : ℝ) (hlambda : lambda > 0) :
101 K lambda = 0 ↔ lambda = lambda_rec := by
proof body
Tactic-mode proof.
102 unfold K lambda_rec ell0
103 simp only [one_pow, div_one]
104 constructor
105 · intro h
106 have hsq : lambda ^ 2 = 1 := by linarith
107 have : (lambda - 1) * (lambda + 1) = 0 := by nlinarith
108 rcases mul_eq_zero.mp this with h1 | h1
109 · linarith
110 · linarith
111 · intro h
112 rw [h]; ring
113