lambda_rec_pos
The lemma shows that the recognition length lambda_rec is strictly positive for any BridgeData satisfying the three minimal positivity conditions on c, hbar and G. Workers deriving constants or area eigenvalues from the bridge anchors cite it to license square-root and division steps. The proof is a short term-mode chain that unfolds the definition then applies the standard Real positivity rules for multiplication, division and square root.
claimLet $B$ be a BridgeData record and let $H$ witness that $B$ satisfies the Physical assumptions (i.e., $0 < B.c$, $0 < B.hbar$, $0 < B.G$). Then $0 < lambda_rec(B)$, where $lambda_rec(B) = sqrt( B.hbar * B.G / (pi * (B.c)^3) )$.
background
Bridge.DataCore supplies the minimal data structure BridgeData holding the external anchors G, hbar, c together with display parameters tau0 and ell0. The predicate Physical B packages the three positivity hypotheses on these anchors that are reused by every subsequent analytic lemma. The recognition length itself is defined by lambda_rec B := sqrt( hbar * G / (pi * c^3) ). The module deliberately keeps its import closure small so that only this core and the dimensionless identity remain inside the certified surface.
proof idea
The term proof first unfolds lambda_rec, reducing the goal to positivity of the radicand. It then applies Real.sqrt_pos.mpr, which converts the claim into a division-positivity goal. Two applications of mul_pos discharge the numerator and denominator using the three fields supplied by the Physical hypothesis.
why it matters in Recognition Science
The result is invoked directly by Constants.G_pos (which derives G from lambda_rec) and by the uniqueness theorem lambda_rec_is_forced. It also supplies the positivity needed for the minimal-area eigenvalue in Quantum.AreaQuantization. Within the Recognition framework it closes the positivity step that precedes the bridge to the RS-native constants c=1, hbar=phi^{-5}, G=phi^5/pi and to the eight-tick octave.
scope and limits
- Does not claim positivity of lambda_rec in the absence of the Physical hypothesis.
- Does not produce a numerical value or bound for lambda_rec.
- Does not connect lambda_rec to the phi-ladder or J-cost function.
- Does not prove the dimensionless identity (c^3 lambda_rec^2)/(hbar G) = 1/pi.
formal statement (Lean)
68lemma lambda_rec_pos (B : BridgeData) (H : Physical B) : 0 < lambda_rec B := by
proof body
Term-mode proof.
69 -- λ_rec = √(ħ G / (π c³)) > 0 since all components positive
70 unfold lambda_rec
71 apply Real.sqrt_pos.mpr
72 apply div_pos
73 · exact mul_pos H.hbar_pos H.G_pos
74 · exact mul_pos Real.pi_pos (pow_pos H.c_pos 3)
75
76end IndisputableMonolith.BridgeData