lambda_rec_dimensionless_id_physical
The lemma packages the dimensionless identity for the recognition length under the minimal Physical positivity assumptions on bridge data. Analysts bridging Recognition Science constants to certified calculations would cite it to avoid repeating positivity checks. The proof is a one-line term wrapper that extracts the three positivity fields from the Physical hypothesis and feeds them to the core dimensionless identity lemma.
claimLet $B$ be BridgeData and assume the Physical predicate on $B$, i.e., $c>0$, $ħ>0$, $G>0$. Then $(c^3 λ_{rec}^2)/(ħ G)=1/π$, where $λ_{rec}=√(ħ G/(π c^3))$.
background
The Bridge.DataCore module supplies the minimal certified data structure for external constants and the associated recognition length. BridgeData is a record containing real numbers G, ħ, c, τ₀, ℓ₀ with no axioms attached. The Physical structure asserts the three positivity conditions c>0, ħ>0, G>0 that are reused across analytical lemmas. The recognition length is defined by lambda_rec B := √(ħ G/(π c³)). The core lemma lambda_rec_dimensionless_id proves the identity under explicit positivity hypotheses; the present lemma simply packages that result behind the Physical interface.
proof idea
The proof is a term-mode one-liner that applies lambda_rec_dimensionless_id to the bridge data B together with the three positivity fields extracted from the Physical hypothesis H.
why it matters in Recognition Science
This declaration closes the packaging of the dimensionless identity for λ_rec inside the certified bridge layer. It supplies the physical-assumption wrapper that downstream bridge and astrophysics lemmas can import without importing the full Core subsystem. The identity itself is the direct algebraic consequence of the definition of λ_rec and is a prerequisite for any Recognition Science calculation that normalizes lengths to the Planck scale.
scope and limits
- Does not derive numerical values of G, ħ or c from Recognition Science axioms.
- Does not extend to negative or zero values of the constants.
- Does not connect λ_rec to the phi-ladder or J-cost function.
- Does not address measurement protocols or extended data helpers.
formal statement (Lean)
63lemma lambda_rec_dimensionless_id_physical (B : BridgeData) (H : Physical B) :
64 (B.c ^ 3) * (lambda_rec B) ^ 2 / (B.hbar * B.G) = 1 / Real.pi :=
proof body
Term-mode proof.
65 lambda_rec_dimensionless_id B H.c_pos H.hbar_pos H.G_pos
66
67/-- Positivity of λ_rec under physical assumptions. -/