counterterms

A symbolic engine for the renormalization of singular SPDEs. You describe a subcritical singular stochastic PDE; it returns the family of renormalized equations — the original PDE plus the tree-indexed counterterms the theory prescribes — following Bailleul & Hoshino, "A tourist's guide to regularity structures and singular stochastic PDEs" (arXiv:2006.03524), which packages the BHZ/BCCH algebra.

⚠️ Personal research library — no guarantees. A one-person research project, not production software. It is validated against the paper where possible (see Validation), but it may be wrong, incomplete, or break on unfamiliar inputs. No warranty, no stability promises, no support. Check the output against the mathematics before trusting it.

It is symbolic

The engine computes what to renormalize and the structure of the renormalization — which counterterms appear, with which tree combinatorics, symmetry factors, and elementary differentials. It works in exact symbolic arithmetic (SymPy + an ordered ring $\mathbb{Q}\oplus\mathbb{Q}\kappa$ for homogeneities).

It does not compute the numeric values of the renormalization constants (those require Gaussian/Wick integrals, deferred), and it is not a solver: no model construction, no analytic estimates, no convergence, no time-stepping. The free constants are the complete symbolic answer — a solution of a singular SPDE is the family indexed by the renormalization group.

What it handles

Equations (subcritical, semilinear, parabolic):

• KPZ and generalized KPZ, $f(u)\zeta + g(u)(\partial_x u)^2$
• PAM and generalized PAM, $f(u)\zeta$
• coupled systems of equations, and several independent noises
• $\Phi^4_2$, $\Phi^4_3$ — supercritical, handled via the da Prato–Debussche lift

Generality: scalar or systems; one or many noises; 2nd-order parabolic operator (general operator order is allowed with a warning); the noise regularity $\beta_0$ is user-supplied and subcriticality ($\beta_0 > -\text{order}$) is checked, so the counterterm set is finite.

Assumptions (enforced — out-of-scope inputs are rejected with a clear error): the nonlinearity is affine in the noise; $g$ is at most quadratic in $\partial u$ (Assumption D2); derivative factors satisfy $|p|_\mathfrak{s} \le 1$.

What you get

• the renormalized family — the original PDE plus one counterterm $\frac{k_\tau}{S(\tau)}F(\tau^*)$ per negative-homogeneity tree, with free constants $k_\tau$;
• the canonical (BPHZ) character $k_\tau = h(S'_-\tau)$ — symbolic, as polynomials in the elementary expectations $h(\sigma)$, with the centered-Gaussian parity rule applied;
• a reduced view that folds in the exact identities (vanishing & duplicate constants, and — for a spatially-symmetric noise — the reflection identity);
• typeset reports in text / Markdown / JSON / LaTeX→PDF, with the trees drawn;
• the underlying algebra: the regularity structure $(T, T^+)$, the extraction/recentering coproducts, the twisted antipode, the BHZ character, the renormalization group $G^-$, and a JSON export of the whole structure.

Install

uv; SymPy is the only runtime dependency.

uv sync
uv run pytest          # 187 tests, ~10s

Quickstart

from sympy import Derivative, Function, Rational
from counterterms import SPDE, Noise, Parabolic, Unknown, kappa

u  = Unknown("u", dim=1)
xi = Noise("xi", regularity=Rational(-1) - kappa)              # noise regularity −1 − κ
f, g = Function("f"), Function("g")

# generalized KPZ:  (∂t − Δ + 1) u = f(u) ξ + g(u) (∂x u)²
spde = SPDE(operator=Parabolic(dim=1, mass=1), unknown=u, noises=[xi],
            rhs=f(u.field) * xi.symbol + g(u.field) * Derivative(u.field, u.x[0])**2)

print(spde.renormalize().summary())     # the five gKPZ counterterms, each with |τ|, S(τ), F(τ*)

The renormalized equation it produces is

$$(\partial_t - \Delta + 1) u = f(u)\zeta + g(u,\partial u) + \sum_{\tau \in \mathcal{B}, |\tau|<0} \frac{k_\tau}{S(\tau)} F(\tau^*)(u,\partial u).$$

The same spde also yields the canonical (BPHZ) character — the symbolic constant $k_\tau = h(S'_-\tau)$ the twisted antipode prescribes (odd-noise trees vanish by parity; the $h(\sigma)$ stay free symbols, since their Wick-integral values are out of scope):

from counterterms import build_renormalization

rs = build_renormalization(spde)
for t in rs.divergent:
    print(rs.canonical_character(t))    # exact antipode combination in the h-values

More runnable scripts in examples/ — start with uv run python -u examples/01_renormalized_equation.py.

Example output

eq.save() writes the report as text / Markdown / JSON and a typeset LaTeX → PDF. Both reports below are for KPZ, $(\partial_t - \Delta) u = (\partial_x u)^2 + \xi$ at $\beta_0 = -3/2 - \kappa$: the parsed equation, every divergent tree $\tau$ (drawn in the paper's convention — $\circ$ noise, $\bullet$ integration node, dotted = derivative kernel) with its homogeneity $|\tau|$, symmetry factor $S(\tau)$, free constant $k_\tau$ and elementary differential $F(\tau^*)$, the assembled renormalized family, and the canonical (BPHZ) constants $k_\tau = h(S'_-\tau)$.

• docs/kpz_canonical.pdf (canonical=True) — the BPHZ constants for a general centered-Gaussian noise. Each $k_\tau$ is a polynomial in the elementary expectations $h(\sigma)$; entries that provably vanish ($=0$) or duplicate another ($=h_j$) are marked but left in place. Correct for any noise (a non-symmetric mollifier genuinely keeps a $\partial_x u$ drift counterterm).
• docs/kpz_reduced.pdf (reduced=True) — the same constants fully reduced for a spatially-symmetric (e.g. white) noise: provable zeros and o-duplicates folded in, plus the spatial-reflection identity $x\to-x$, collapsing KPZ to a single diverging constant, $\partial_t u = \Delta u + (\partial_x u)^2 - C + \xi$ (Hairer's result). The reduction states its symmetric-noise assumption and is never claimed for an anisotropic noise.

---

Under the hood

Everything public is re-exported from the top-level counterterms package. The pipeline is SPDE (DSL) → structural rule → decorated trees with |τ|<0 → S(τ) + F(τ*) → assembled family, optionally composed with the renormalization coproducts and characters.

Input & the renormalized equation

You want…	Call	Module
Write the SPDE (DSL)	Unknown, Noise, Parabolic, SPDE	equation/dsl.py
Derive the renormalized family	SPDE(...).renormalize() → RenormalizedEquation	api.py, renorm/equation.py
Access each counterterm (tree, $\lvert\tau\rvert$, $S(\tau)$, $F(\tau^*)$, $k_\tau$)	eq.counterterms, eq.per_component	renorm/equation.py
Render a report (text / markdown / json / latex; canonical, reduced)	eq.summary(), eq.render(fmt, …), eq.save(…)	render/report.py, render/latex.py
$\Phi^4_2 / \Phi^4_3$ (supercritical, $\beta_0 \le -\text{order}$)	daprato_lift(spde).renormalize()	equation/daprato.py

The trees and the rule

You want…	Call	Module
The divergent trees $\mathcal{B}_{<0}$	generate_counterterms(sig)	equation/generate.py
The structural rule from the nonlinearity	build_context(spde) → (sig, base, unknowns)	equation/dsl.py
Subcriticality check ($\beta_0 > -\text{order}$)	check_subcritical(sig) (auto in build_context)	equation/rule.py
Decorated trees: canonical form, $S(\tau)$, homogeneity	DecoratedTree, tree, red_node	trees/tree.py
The elementary differential $F(\tau^*)$ ($\Upsilon$-map)	elem_diff(t, comp, base, sig)	renorm/nonlinearity.py
Draw a single tree (shorthand / ascii / forest)	shorthand, ascii_art, forest	render/tree.py

The algebraic structure (regularity & renormalization)

You want…	Call	Module
The regularity structure $(T, T^+)$, graded basis	build_regularity_structure(spde)	structures.py
Recentering $\Delta : T \to T \otimes T^+$, structure coproduct $\Delta^+$	delta_plus(t, sig[, project_left])	trees/coproducts.py
Extraction–contraction $\delta$ / $\delta^-$ (and the cointeraction)	delta_minus, delta_minus_group	trees/coproducts.py
Negative twisted antipode $S'_-$	twisted_antipode(t, sig)	trees/coproducts.py
Renormalization structure + symbolic BHZ character $k = h \circ S'_-$	build_renormalization(spde) → .bhz_character, .canonical_character	structures.py
The renormalization group $G^-$ (convolution, antipode inverse)	build_renormalization_group(spde)	structures.py
Generic, basis-agnostic Hopf ops (convolve / antipode / comodule)	convolve, antipode, comodule_action	core/hopf.py
Wick-pairing structure of the canonical constants (symbolic)	expectation, NoiseLaw, BPHZ, FreeConstants	renorm/scheme.py
JSON export of the whole structure	structure_json(spde), export_structure, tree_to_dict	render/export.py

Foundations

You want…	Call	Module
The ordered homogeneity ring $\mathbb{Q} \oplus \mathbb{Q}\cdot\kappa$	Homogeneity, kappa, Scaling	core/homogeneity.py
Jet variables $u^c_k$	jet, is_jet, jet_parts	core/jets.py
The Signature (the parametric vocabulary everything threads)	Signature	core/signature.py
The Symbol protocol (basis seam)	Symbol	core/symbol.py

Out of scope (rejected with clear errors)

Non-polynomial supercritical equations (e.g. sine-Gordon, which needs Wick exponentials); noise nonlinearities not affine in the noise; $g$ more than quadratic in $\partial u$; singular derivative factors $|p|_\mathfrak{s} &gt; 1$; quasilinear or non-parabolic operators. And — the analysis/probability wall — no numeric constant values (the Wick integrals are emitted symbolically by renorm/scheme.py but not evaluated), no model construction, estimates, convergence, or solving.

Status

Phases 1–3 complete and green; Phase 4 partially built.

• Phase 1–2 — the renormalized family from an SPDE, for scalar/systems, one/many noises, general operator order.
• Phase 3 — the coproducts (the cointeraction holds including $\beta_0 = -3/2$), RegularityStructure $(T, T^+)$, the generic core/hopf layer, subcriticality, the twisted antipode + BHZ character, the group $G^-$.
• Phase 4 (partial) — daprato_lift ($\Phi^4_{2/3}$), the canonical-character symbolic half + Wick parity, the reduced view, the JSON export, and the seams for the unbuilt analytic pieces (evaluating the divergent integrals).

See ROADMAP.md and CHANGELOG.md.

Validation

The paper is the only oracle (no reference implementation exists). The suite reproduces the gKPZ example's exact five counterterms ($\beta_0 = -1$) and the full strongly-conforming-tree table at $\beta_0 = -3/2$ (43 trees, tex 6024–6163) — matching the latter caught and fixed a real tree-generation bug. The renormalized KPZ / PAM / gPAM / $\Phi^4$ equations are cross-checked against the published results (Hairer; BCCH). Plus an independent symmetry-factor cross-check and benchmark counts. See notes/validation.md.

Documentation

• examples/ — runnable quickstart scripts (start here).
• ENTRYPOINTS.md — a guided reading order through the source.
• notes/use_cases.md — what it can and cannot solve, honestly.
• notes/initial_plan.md — the mathematics (pipeline, conventions, scope).
• notes/architecture.md — the module structure and design.
• notes/cointeraction_singular_noise.md, notes/phase4_plan.md, notes/validation.md — deep dives.
• references/ — pointer to the source paper (arXiv:2006.03524), cited by line number throughout.

License

MIT © 2026 Pavel Ievlev. The Bailleul–Hoshino paper is not included (it is the authors' copyright) — see references/ for the arXiv link.