Reports & output

A RenormalizedEquation renders itself in four formats, with three orthogonal views.

Quick access

eq = spde.renormalize()

print(eq.summary())                  # terse: one line per counterterm
print(eq.report())                   # full text report
print(eq.render("markdown"))         # "text" | "markdown" | "json" | "latex"
eq.save("kpz", outdir="output")      # writes kpz.txt, kpz.md, kpz.json, kpz.tex (+ .pdf if latexmk is installed)

Every format contains: the parsed equation, each divergent tree \(\tau\) drawn (in the paper’s convention — \(\circ\) noise, \(\bullet\) integration node, dotted edge = derivative kernel) with its homogeneity \(|\tau|\), symmetry factor \(S(\tau)\), free constant \(k_\tau\) and elementary differential \(F(\tau^*)\), and the assembled renormalized family.

Programmatic access, bypassing the renderer:

eq.counterterms          # flat list of Counterterm(tree, homogeneity, symmetry_factor, elem_diff, constant)
eq.per_component[0]      # the same, per equation of a system
eq.counterterm_rhs(0)    # the assembled Σ k_τ/S(τ)·F(τ*) as one SymPy expression
eq.all_trees             # the raw divergent-tree tuple, before Υ-zero trees are dropped

Canonical and reduced views

Three keyword flags on report / render / save / to_json control how much of the BPHZ theory is folded into the constants:

eq.report()                                  # free constants k_τ only (the default)
eq.report(canonical=True)                    # + the BPHZ section: k_τ = h(S'_- τ)
eq.report(reduced=True)                      # + all exact identities folded in
eq.report(reduced=True, symmetric=False)     # …for a noise NOT symmetric under x → −x

Default (free constants). The family with free \(k_\tau\) — correct for every renormalization character, no probabilistic input assumed. This is the theorem-level output.

canonical=True. Adds, per tree, the canonical (BPHZ) constant as an exact twisted-antipode combination \(k_\tau = h(S'_-\tau)\) of elementary-expectation symbols \(h(\sigma)\), each \(h(\sigma)\) spelled out as an \(\varepsilon\)-regularized Wick integral. Constants that provably vanish (Gaussian parity, root \(X^n\), pure-kernel total derivative) or duplicate another are marked but left in place — the display stays valid for any centered Gaussian noise. (This section is off by default because the twisted antipode grows quickly on deep trees — KPZ at \(\beta_0 = -\tfrac32\) produces \(\sim\)1400-term antipode forests.)

reduced=True (implies the canonical section). The same constants with the exact identities substituted: provable zeros dropped, duplicates merged — plus, when symmetric=True, the spatial-reflection identity (odd total spatial-derivative order on the kernels ⇒ \(h(\sigma) = 0\)). For KPZ this collapses the canonical family to Hairer’s single diverging constant: \(\partial_t u = \Delta u + (\partial_x u)^2 - C + \xi\).

symmetric is an assumption about your noise

The reflection identity is valid only for a noise whose law is invariant under \(x \to -x\) (white noise, symmetric mollifications). It defaults to True. If your noise is anisotropic, pass symmetric=False: the drift-type counterterms then genuinely survive, and the report says so explicitly (reduction_assumes_symmetric_noise: false in the JSON). The other reductions (parity, root \(X^n\), total derivative) are noise-independent and are applied in the reduced view regardless.

Side-by-side example (KPZ at \(\beta_0 = -\tfrac32 - \kappa\)): canonical view (PDF) — 8 constants, zeros marked; reduced view (PDF) — one constant, Hairer’s \(C_\varepsilon\).

The JSON format

eq.to_json(...) / eq.save(...) emit a machine-readable document: the parsed equation, per-component counterterms (tree as a nested dict, homogeneity as exact \((\text{std}, \text{kap})\) rationals, \(S\), \(F(\tau^*)\) as a SymPy string, constant name), and — with canonical/reduced — the character polynomials and the reduction flags. For the full algebraic structure (coproducts, antipode, group), see structure_json instead.

Drawing individual trees

from counterterms.render import shorthand, ascii_art, forest

shorthand(t, sig)    # one-line:  ●(I'[∘])²  style
ascii_art(t, sig)    # multi-line terminal drawing
forest(t, sig)       # LaTeX (forest package) code, as used in the PDF reports

Cosmetic nondeterminism

The RULE line ordering in text/LaTeX reports depends on Python’s hash seed; content is otherwise identical run to run. Diff JSON, not text, if you need stable comparisons.