coreOperatorPairBudget
coreOperatorPairBudget computes the aggregate pair budget for a discrete incompressible Navier-Stokes flow by summing pairEventBudget over the derived pair events extracted from the core velocity and topology data. Discrete fluid modelers and Recognition Science researchers deriving Navier-Stokes operators from lattice primitives cite it when constructing the full operator surface from physical data alone. The definition is a direct one-line composition of indexedPairBudget applied to corePairEvent.
claimLet $op$ be a CoreNSOperator on a lattice with $N$ sites. The core operator pair budget is $B = total_{i=0}^{N-1} pairEventBudget(corePairEvent(op,i))$, where corePairEvent derives amplitude and stretch factor at each site from the divergence-free velocity field and its discrete gradients.
background
The module builds a concrete discrete incompressible Navier-Stokes operator on a finite three-direction lattice. CoreNSOperator stores only the physical data: LatticeTopology, spacing $h>0$, viscosity $ν>0$, divergence-free velocity VectorField, and transport flux; pair-budget and absorption fields are absent and derived later. corePairEvent extracts the PairEvent (amplitude, stretchFactor) at each site from the velocity gradient and Laplacian. indexedPairBudget sums the individual pairEventBudget values over the finite lattice window.
proof idea
One-line wrapper that applies indexedPairBudget to corePairEvent op.
why it matters in Recognition Science
This supplies the total derived pair budget required by the downstream theorems coreOperatorPairBudget_nonneg, core_pair_budget_absorbed, and core_stretching_absorbed. It completes the DerivedEstimates layer that turns the bare CoreNSOperator into the full IncompressibleNSOperator whose pair-budget fields are constructed rather than assumed. In the Recognition framework it supports lattice derivations of incompressible flow consistent with the forcing chain and recognition composition law.
scope and limits
- Does not prove nonnegativity or absorption of the budget.
- Does not define the full IncompressibleNSOperator structure.
- Does not address continuous-space Navier-Stokes limits.
- Does not treat compressible or viscous-free flows.
Lean usage
theorem budget_nonneg {n : ℕ} (op : CoreNSOperator n) : 0 ≤ coreOperatorPairBudget op := coreOperatorPairBudget_nonneg opformal statement (Lean)
178def coreOperatorPairBudget {siteCount : ℕ} (op : CoreNSOperator siteCount) : ℝ :=
proof body
Definition body.
179 indexedPairBudget (corePairEvent op)
180