lambda_rec_is_root
The theorem confirms that the curvature functional K vanishes at the recognition length. Researchers deriving constants non-circularly in Recognition Science cite it to anchor the balance condition before uniqueness arguments. The proof is a direct term-mode reduction that unfolds K, lambda_rec and ell0 then applies simp and ring.
claimThe curvature functional satisfies $K(lambda_rec) = 0$, where $K(lambda) = lambda^2 / lambda_rec^2 - 1$.
background
In the LambdaRecDerivation module the recognition length lambda_rec is set to ell0, which equals 1 in RS-native units. The curvature functional is defined by K(lambda) := lambda^2 / lambda_rec^2 - 1. This construction uses only the normalized bit cost and the curvature cost 2 lambda^2 from the Q3 Gauss-Bonnet normalization, without presupposing G; G is instead recovered afterward as pi lambda_rec^2 c^3 / hbar.
proof idea
The proof is a term-mode reduction. It unfolds the definitions of K, lambda_rec and ell0, applies simp to clear the power and division, and finishes with ring to obtain the identity 1 - 1 = 0.
why it matters in Recognition Science
The result is invoked by lambda_rec_is_forced to establish existence and uniqueness of the positive root. It supplies the explicit verification step in the non-circular derivation of lambda_rec described in the module documentation, confirming that the balance condition holds at the recognition length before G is defined as a consequence. Within the Recognition Science framework this anchors the constants block inside the eight-tick cycle where lambda_rec coincides with ell0.
scope and limits
- Does not prove positivity of lambda_rec.
- Does not derive lambda_rec from the forcing chain T0-T8.
- Does not connect to the J-cost or phi-ladder.
- Does not address units outside RS-native conventions.
formal statement (Lean)
95theorem lambda_rec_is_root : K lambda_rec = 0 := by
proof body
Term-mode proof.
96 unfold K lambda_rec ell0
97 simp only [one_pow, div_one]
98 ring
99