21. Nonlinear packet truncation with an explicit remainder
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Polynomial interactions were expanded exactly in Proposition 5. A computational model may retain only some ordered interaction tuples. It then needs a bound on what was omitted, rather than an assumption that packetwise evaluation is exact.
Let \(P_i\) be an orthogonal resolution of a finite-dimensional real Hilbert space. Suppose
with bounded multilinear maps. For each ordered tuple \(\boldsymbol i=(i_1,\ldots,i_m)\), let \(\beta_{m,\boldsymbol i}\) bound the norm of \(B_m\) restricted to \(\operatorname{im}P_{i_1}\times\cdots\times\operatorname{im}P_{i_m}\). Let \(\mathcal I_m\) be the retained tuple set, and let \(F_{\mathcal I}\) retain the constant term and exactly those interactions.
Proposition 20: Packet-interaction truncation bound
Put \(a_i=\|P_ix\|\). Then
On the ball \(\|x\|\leq R\), a valid Lipschitz bound for \(F_{\mathcal I}\) is
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Proof
Subtract the retained terms from the exact multilinear expansion. Bound each omitted term by its restricted multilinear norm and sum. For the Lipschitz estimate, telescope the difference between two \(m\)-fold evaluations by replacing one argument at a time. Each changed argument has norm at most \(\|x-y\|\) after projection, and each unchanged argument has norm at most \(R\). There are \(m\) terms for each retained tuple. \(\square\)
If the declared polynomial itself approximates a nonpolynomial map with remainder at most \(\varepsilon_R\) on the ball, add \(\varepsilon_R\) to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.
For a retained state \(z=Px\) and omitted state \(q=(I-P)x\), dropping every term that uses \(q\) gives the coarser majorant
The tuple-level result is finer because it can distinguish a harmless omitted packet from a strongly coupled one. Ordered tuples must be counted with their multiplicities unless symmetry has explicitly combined them.
A prediction guarantee follows by inserting these defect and Lipschitz bounds into Proposition 18, provided the true and predicted trajectories remain in the declared domain. That domain condition cannot be inferred from a small initial error. One possible construction is to assume that the true map preserves a closed convex set \(K\) and to project every approximate output back onto \(K\). Euclidean projection is nonexpansive and fixes the true output, so it does not increase the one-step approximation bound. This is a declared modification of the predictor, not an identity for an unmodified recurrence.
For the scalar quadratic \((u+v)^2\), retaining only \(u^2+v^2\) omits \(2uv\). The packet bound is \(2|u||v|\) and is exact in magnitude. Small energy in one packet matters in proportion to both its amplitude and the cross-interaction coefficient.