total
plain-language theorem explainer
The total definition aggregates a scalar field by summing its values across all sites in a finite lattice window. Researchers formalizing discrete fluid dynamics or J-cost monotonicity within Recognition Science cite it when tracking integrated amplitudes before RMS or contribution splits. The implementation is a direct summation over the Fin-indexed function, which serves as the base aggregation step in the bookkeeping layer.
Claim. Let $f$ be a real-valued scalar field on a finite set of $N$ sites. Then the total is given by $N^{-1} $ no, wait: the sum $sum_i f(i)$.
background
The Discrete Vorticity module packages exact bookkeeping primitives for finite vorticity fields on a lattice window as part of the J-cost monotonicity program. It defines totals, RMS values, normalized amplitudes, and the three contribution fields (transport, viscous, stretching) so that the J-cost derivative can be decomposed exactly. The module documentation states that hard PDE inequalities are separated from this surface to allow incremental addition of lemmas. Upstream structures supply the J-cost from PhiForcingDerived.of and the ledger factorization from DAlembert.LedgerFactorization.of, which calibrate the convex function J(x) = (x + x^{-1})/2 - 1.
proof idea
This is a one-line definition that unfolds directly to the summation operator over the finite index set Fin siteCount.
why it matters
This aggregation primitive is used in downstream results such as energyConservationCert, hamilton_equations_from_EL, and space_translation_invariance_implies_momentum_conservation to establish conservation laws. It supports the exact decomposition of the J-cost derivative into transport, viscous, and stretching pieces required by the module. In the Recognition framework it underpins discrete approximations tied to the eight-tick octave and phi-ladder structures while feeding the Action and Hamiltonian formalisms.
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