aggregateThroughput
Aggregate throughput for N nodes equals N times the per-node throughput T_node in a Z-matched transceiver mesh. Network engineers cite it when predicting linear capacity scaling for phantom-cavity arrays. The definition is a direct cast-and-multiply that enables immediate algebraic reductions in additivity and doubling lemmas.
claimThe aggregate throughput $T(N)$ of a mesh with $N$ nodes is $T(N) = N · T_0$, where $T_0$ is the dimensionless per-node throughput reference.
background
The Z-Matched Recognition-Transceiver Mesh module models networks of phantom-cavity transceivers whose (Z, Θ)-channels are independent and distance-decoupled. T_node is the constant 1 serving as the per-node throughput reference in dimensionless units. This setting rests on Foundation.ZThetaSpatialDecoupling, which supplies the constant pairwise latency used in the module's one-statement summary.
proof idea
The definition is a direct multiplication of the cast natural number N by the constant T_node. No lemmas are invoked; the body is an abbrev that unfolds immediately for ring and cast tactics in downstream proofs.
why it matters in Recognition Science
This definition supplies the base expression for mesh_one_statement, which packages linearity, doubling at 2N, and constant latency into a single engineering claim. It completes the J9 track derivation that aggregate throughput grows linearly with node count, matching the Recognition framework's prediction of distance-decoupled channels. No scaffolding remains.
scope and limits
- Does not model hardware implementation details or fabrication tolerances.
- Does not incorporate channel noise, interference, or finite bandwidth.
- Does not address non-ideal Z-matching or cavity losses.
- Does not extend to dynamic node addition or routing overhead.
Lean usage
theorem usage (N : ℕ) : aggregateThroughput (N + 1) = aggregateThroughput N + T_node := by unfold aggregateThroughput; ringformal statement (Lean)
53def aggregateThroughput (N : ℕ) : ℝ := (N : ℝ) * T_node
proof body
Definition body.
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