aggregateThroughput_double
Aggregate throughput of a Z-matched transceiver mesh doubles exactly when node count doubles. RS engineers modeling scalable networks cite this to confirm additive scaling. The term-mode proof unfolds the definition as N times T_node, casts to reals, and reduces via ring normalization.
claimFor every natural number $N$, $T(2N) = 2 T(N)$, where $T(N) := N · T_{node}$ denotes aggregate throughput of the $N$-node mesh.
background
The Z-Matched Recognition-Transceiver Mesh module treats a network of $N$ Z-matched phantom-cavity transceivers whose $(Z, Θ)$-channels are independent. Aggregate throughput is defined as the product of node count and per-node throughput $T_{node}$. Per-pair latency equals the constant 0.07, independent of separation. Upstream results include the J-cost structure from PhiForcingDerived, the ledger factorization from DAlembert.LedgerFactorization, and the gauge content from SpectralEmergence.
proof idea
Unfold aggregateThroughput. Push the natural-number cast to reals. Apply the ring tactic to obtain the identity.
why it matters in Recognition Science
The result is invoked inside mesh_one_statement to bundle doubling with successor additivity and constant latency, and inside zMatchedTransceiverMeshCert to certify the mesh. It supplies the doubling clause of the module's linearity claim for track J9, consistent with the self-similar fixed point phi.
scope and limits
- Does not model channel interference or crosstalk.
- Does not derive the numerical value of $T_{node}$ from forcing axioms.
- Does not extend to non-integer or defective node counts.
- Does not incorporate bandwidth saturation or recognition defects.
Lean usage
theorem mesh_example (N : ℕ) : aggregateThroughput (2 * N) = 2 * aggregateThroughput N := aggregateThroughput_double Nformal statement (Lean)
74theorem aggregateThroughput_double (N : ℕ) :
75 aggregateThroughput (2 * N) = 2 * aggregateThroughput N := by
proof body
Term-mode proof.
76 unfold aggregateThroughput; push_cast; ring
77
78/-! ## §3. Distance-decoupled latency -/
79
80/-- Distance-decoupled latency: returns `latency_per_pair` regardless
81of node-pair separation `d`. -/