latency_per_pair_pos
The theorem establishes positivity of per-pair latency in Z-matched transceiver meshes. Engineers modeling RS_PAT_042 phantom-cavity networks cite it to confirm non-zero (Z, Θ)-channel delay. Proof is a one-line wrapper that unfolds the constant definition and applies norm_num.
claim$0 < 0.07$, where $0.07$ is the dimensionless coefficient for per-pair (Z, Θ)-channel latency given by $ℏ_C / (2 · ΔE)$ and independent of distance.
background
The Z-Matched Recognition-Transceiver Mesh module models a network of N Z-matched phantom-cavity transceivers whose aggregate throughput equals N times the single-node throughput T_node. Each (Z, Θ)-channel is independent and distance-decoupled, so pairwise latency remains constant per the upstream Foundation.ZThetaSpatialDecoupling result. The sibling definition records latency_per_pair : ℝ := 0.07 as the coefficient for ℏ_C / (2 · ΔE) ≈ 0.07 µs.
proof idea
The proof is a one-line wrapper. It unfolds the definition of latency_per_pair to expose the constant 0.07, then applies norm_num to discharge the strict inequality.
why it matters in Recognition Science
This theorem supplies the latency_pos field required by the master certificate zMatchedTransceiverMeshCert and is invoked directly by pairwiseLatency_pos. It completes the engineering derivation that aggregate throughput scales linearly with node count while per-pair latency stays constant, matching the module's stated goals of linear T(N) and distance-independent channels. No open scaffolding remains in the supplied material.
scope and limits
- Does not derive the numerical value 0.07 from first principles.
- Does not prove distance independence of latency.
- Does not address throughput scaling beyond the per-pair case.
- Does not supply a concrete falsifier for the mesh model.
Lean usage
theorem pairwiseLatency_pos (d : ℝ) : 0 < pairwiseLatency d := latency_per_pair_posformal statement (Lean)
47theorem latency_per_pair_pos : 0 < latency_per_pair := by
proof body
Term-mode proof.
48 unfold latency_per_pair; norm_num
49
50/-! ## §2. Aggregate throughput -/
51
52/-- Aggregate throughput at `N` nodes. -/