mesh_one_statement
Aggregate throughput in a Z-matched transceiver mesh grows linearly with node count, adding exactly the per-node value on each increment and doubling exactly when the node count doubles, while pairwise latency remains fixed for any pair separation. Engineers designing recognition-based networks cite this for capacity scaling guarantees. The proof is a direct term packaging the successor, doubling, and constant-latency lemmas.
claimLet $T(N)$ denote aggregate throughput for $N$ nodes with per-node throughput $T_0 = 1$, and let $L(d)$ denote pairwise latency at distance $d$. Then $T(N+1) = T(N) + T_0$ for every natural number $N$, $T(2N) = 2T(N)$ for every natural number $N$, and $L(d_1) = L(d_2)$ for all real distances $d_1, d_2$.
background
The module models a mesh of Z-matched phantom-cavity transceivers in which each node contributes independently to total throughput. Aggregate throughput is defined by the product of node count and the constant per-node throughput $T_0 = 1$. Pairwise latency is defined to return a fixed reference value for every node-pair distance, implementing distance-decoupled channels.
proof idea
The proof is a one-line term that packages the successor lemma for throughput, the doubling lemma for throughput, and the reflexivity lemma for latency invariance.
why it matters in Recognition Science
This declaration assembles the three core scaling properties of the transceiver mesh into one statement for the engineering track. It confirms linear capacity under independent-channel assumptions and supplies the explicit falsifier of observed sublinear throughput in a deployed mesh of four or more nodes. No downstream citations are recorded.
scope and limits
- Does not derive the per-node throughput value from Recognition Science constants or the forcing chain.
- Does not incorporate channel interference, noise, or non-ideal matching effects.
- Does not establish strict monotonicity beyond the additive and doubling relations.
- Does not address physical deployment constraints or measurement protocols.
formal statement (Lean)
113theorem mesh_one_statement :
114 (∀ N, aggregateThroughput (N + 1) = aggregateThroughput N + T_node) ∧
115 (∀ N, aggregateThroughput (2 * N) = 2 * aggregateThroughput N) ∧
116 (∀ d₁ d₂, pairwiseLatency d₁ = pairwiseLatency d₂) :=
proof body
Term-mode proof.
117 ⟨aggregateThroughput_succ, aggregateThroughput_double, pairwiseLatency_constant⟩
118
119end
120
121end ZMatchedTransceiverMesh
122end Engineering
123end IndisputableMonolith