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Scope & limitations

The project’s policy is explicit errors over silent wrong output, and honest documentation of what is unbuilt or imperfect. This page is the complete list, including known defects.

In scope

  • Scalar equations and coupled systems; one or several independent noises.
  • Second-order parabolic \(L\) (the proven case); other orders run with a warning (see below).
  • Subcritical noise, \(\beta_0 > -\text{order}\), checked rule-based on every input.
  • Nonlinearities affine in the noise; \(g\) at most quadratic in \(\partial u\) (Assumption D2); derivative factors with \(|p|_\mathfrak{s} \le 1\).
  • Supercritical polynomial additive-noise equations (\(\Phi^4_2\), \(\Phi^4_3\)) via the da Prato–Debussche lift.

Rejected, by design

  • Supercritical equations not liftable here (sine-Gordon — needs Wick exponentials).
  • Nonlinearities not affine in the noise (\(\xi^2\), \(f(\xi)\)).
  • \(g\) more than quadratic in \(\partial u\); factors \(\partial_t u\) or \(\partial^2_x u\) in the nonlinearity (\(|p|_\mathfrak{s} > 1\)).
  • Quasilinear or non-parabolic operators (not expressible in the DSL).

The analytic wall (deliberately unbuilt)

The engine is symbolic. It stops, on purpose, exactly where analysis/probability starts:

  • No numeric constant values. The canonical constants are exact polynomials in elementary-expectation symbols \(h(\sigma)\), each spelled out as an \(\varepsilon\)-regularized Wick integral — never evaluated. BPHZ.numeric_character raises until an IntegralEvaluator is plugged in.
  • No models, estimates, convergence, or solving.
  • \(G^-_{\mathrm{ad}}\) (the admissible subgroup — needs kernel moments and the \(\Pi\)-map) raises NotImplementedError.
  • \(\Phi^4\) canonical values are structure-only: the lift treats Wick powers as independent noises, so the free family is exact but canonical constants ignore the true correlations among \(:\!X^k\!:\).

Known sharp edges

None affects the golden-tested regime (order 2, the validated equations).

Operator order > 2

Schauder/admissibility is proven only for 2nd-order parabolic \(L\); the engine warns and computes the combinatorics anyway. The tree enumeration is complete for any order (the node-decoration cap scales with \(-\beta_0\), and the generator does not prune on intermediate homogeneity sums — both were audit findings, fixed with regression tests in tests/test_rule.py), but treat order ≠ 2 output as exploratory: the underlying analytic theory is unverified there.

Asymmetric systems and the raw tree basis

Gradient-degree budgets are shared across components (a max over equations), so a system in which only one equation has a gradient nonlinearity can over-generate trees in the raw basis (all_trees, and hence spurious always-zero entries in the canonical constant list). The renormalized equations themselves are unaffected — the spurious trees have \(F(\tau^*) = 0\) and are dropped.

Smaller notes

  • Multi-noise covariances share one display symbol: with several noises of different laws, all covariances render as the single symbol \(C\) in the Wick integrands. The pairing combinatorics (within-type only) are correct; only the label conflates. The API currently cannot express distinct laws anyway.
  • Subcriticality compares standard parts only: a hypothetical noise at \(\beta_0 = -\text{order} + \kappa\) (infinitesimally subcritical) is rejected although the ordered-ring comparison would accept it. One-sided — it never wrongly accepts; no physical noise sits there.
  • symmetric=True is the default on the reduced report view. It is an assumption about your noise (reflection invariance), stated in-band in every report; pass symmetric=False for anisotropic noise. In \(d \ge 2\) the reflection reduction uses total spatial parity only — conservative: it never wrongly zeroes, but misses per-axis reductions.
  • Report text ordering is hash-seed dependent (the RULE lines); JSON is the stable format.

Why trust any of it

Because the scope above is enforced by tests, the golden cases are re-derived from the paper rather than recorded from the engine, and the codebase has been audited — including adversarially, by independent re-derivation of the core formulas. See Validation.