The mathematics¶

This page explains, from scratch, what the engine computes and why. It follows Bailleul & Hoshino’s tourist guide (arXiv:2006.03524); parenthetical references like (tex 4337) are line numbers in the paper’s LaTeX source, which is how the codebase cites its oracle throughout.

1. Why singular SPDEs need renormalization¶

Consider a semilinear parabolic equation driven by a very rough noise \(\zeta\):

\[(\partial_t - \Delta + 1)\, u = f(u)\,\zeta + g(u, \partial u).\]

If \(\zeta\) has Hölder regularity \(\beta_0 < 0\) (e.g. spacetime white noise in one space dimension has \(\beta_0 = -\tfrac32 - \kappa\)), the solution \(u\) is only as regular as the equation allows, and products like \(f(u)\zeta\) or \((\partial_x u)^2\) are ill-defined: the sum of the regularities of the factors is negative, so classical (Young / paraproduct) multiplication fails.

The renormalization strategy: mollify the noise (\(\zeta_\varepsilon = \zeta * \rho_\varepsilon\)), solve the now-classical equation, and try to take \(\varepsilon \to 0\). The limit does not exist — but it does exist if you first subtract diverging counterterms from the equation. The theory of regularity structures (Hairer; Bruned–Hairer–Zambotti; Bruned–Chandra–Chevyrev–Hairer) identifies exactly which counterterms, and this engine computes them symbolically.

The answer has a remarkable shape: one counterterm per decorated tree of negative homogeneity,

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

where the constants \(k(\tau)\) are free — each choice of them is one member of the family of renormalized equations (the paper’s Theorem “ThmRenormPDEs”, the BCCH formula). This display is the engine’s output.

2. Subcriticality¶

The whole machine only works when the noise is not too rough relative to the smoothing of the operator: subcriticality requires \(\beta_0 > -\text{order}\) (for the heat operator, \(\beta_0 > -2\)) (tex 5485). Subcriticality guarantees that iterating the equation gains regularity, so only finitely many trees have negative homogeneity and the counterterm sum is finite. The engine checks this rule-based condition on every input and rejects supercritical equations (for polynomial additive-noise cases like \(\Phi^4_3\), it offers the da Prato–Debussche lift instead).

3. Decorated trees: the vocabulary of the expansion¶

Iterating the mollified equation (Picard iteration) produces a sum of explicit stochastic integrals; each integral is encoded by a decorated tree (tex 3963–3970):

\(\circ\) (noise node) — an occurrence of the noise \(\zeta\). Multiple independent noises get distinct node types \(\circ_j\).
\(\bullet\) (integration node) — a point where the nonlinearity \(g\) is evaluated.
Edges \(\mathcal{I}_p\) — a convolution with the kernel \(\partial^p K\) of \(L^{-1}\); the multi-index \(p\) records derivatives hitting the kernel. For systems, the edge also carries which equation’s kernel it is — component identity lives on the edge type, not the node.
Node decorations \(X^n\) — polynomial factors \(x^n\) from Taylor expansion (recentring).

For example, in KPZ the tree \(\bullet\!\!\Rightarrow\!\!\circ\,\circ\) (an integration node with two derivative-kernel edges to noises) encodes \(\big(\partial_x K * \zeta\big)^2\) — the famous ill-defined square.

Two trees are the same iff they are isomorphic as decorated combinatorial trees; the engine uses a canonical form for equality, dictionary keys, and symmetry counting.

4. Homogeneity: which trees diverge¶

Each tree gets a homogeneity \(|\tau|\) — the parabolic Hölder regularity of the stochastic object it encodes — computed by three rules:

\[|\Xi| = \beta_0, \qquad |\mathcal{I}_p \tau| = |\tau| + \text{order} - |p|_\mathfrak{s}, \qquad |X^n| = |n|_\mathfrak{s},\]

multiplicativity over subtrees, with the parabolic scaling \(\mathfrak{s} = (2, 1, \dots, 1)\) (time counts double) (tex 5276–5283, 2213, 1153). Kernel edges gain the Schauder order (2 for the heat operator) and lose \(|p|_\mathfrak{s}\) per derivative.

Homogeneities are exact elements of the ordered ring \(\mathbb{Q} \oplus \mathbb{Q}\kappa\), where \(\kappa > 0\) is an infinitesimal: white noise sits just below the rational threshold, and critical trees sit at homogeneity \(-k\kappa\) — negative, but only by an infinitesimal. Using floats here would silently misclassify them; the engine never does.

A tree needs a counterterm exactly when \(|\tau| < 0\) — then the associated stochastic object diverges as \(\varepsilon \to 0\) and must be recentred by a constant. The set \(\mathcal{B}_{<0}\) of such trees is what the engine enumerates. It includes the bare noises \(\circ^n\) themselves (in KPZ these produce the \(k(\circ)\) and \(k(\circ_1)\) terms); the bare \(\bullet\) has homogeneity \(0\) and is excluded.

5. The rule: which trees can occur at all¶

Not every decorated tree appears in the expansion of a given equation — the nonlinearity dictates the possible branching. From the right-hand side the engine derives a rule (tex 5306–5340): e.g. for gKPZ, an integration node may carry at most one noise edge (the nonlinearity is affine in \(\zeta\)) and at most two derivative-kernel edges (\(g\) is quadratic in \(\partial u\) — Assumption D2, with the bound on the total gradient degree per node). Trees conforming to the rule (“strongly conforming”, tex 4555) with negative homogeneity are exactly the counterterm carriers.

Validation anchor: for gKPZ at \(\beta_0 = -\tfrac32 - \kappa\) the paper tabulates all 43 strongly-conforming trees in six homogeneity rows (tex 6028–6163); the engine reproduces the table row for row.

6. The coefficient: \(S(\tau)\) and the elementary differential \(F(\tau^*)\)¶

Each divergent tree contributes the counterterm \(\dfrac{k(\tau)}{S(\tau)} F(\tau^*)(u, \partial u)\):

The symmetry factor \(S(\tau)\) counts the tree’s internal symmetries (tex 3982):

\[S(\tau) = n! \; \prod_j S(\sigma_j)^{m_j}\, m_j!\]

over the distinct child branches \(\sigma_j\) with multiplicities \(m_j\) (and the root decoration’s multi-index factorial \(n!\)). Dividing by \(S(\tau)\), never by anything else, is a load-bearing convention (tex 4915).

The elementary differential \(F(\tau^*)\) — the “Υ-map” — turns the tree back into a function of the solution (tex 4337):

\[F(\tau^*) = \Big(\prod_i F(\tau_i^*)\Big)\cdot \Big(D^n \prod_i \partial_{p_i}\Big) F(b^*),\]

recursively over the branches: each child branch contributes its own factor, and the base symbol of the node (\(F(\circ_j^*) = f_j\), \(F(\bullet^*) = g\)) is differentiated once in the argument slot matching each child’s edge decoration \(p_i\), then \(n\) more times by the total derivative \(D_i = \sum_k u_{k+e_i} \partial_{u_k}\) (tex 4316). Two subtleties the engine gets right and tests:

the slot derivatives \(\partial_{p_i}\) are applied before \(D^n\) — they do not commute;
\(D^n\) is a genuine iterated total derivative (Faà di Bruno applies).

A worked example of why both \(S\) and \(\Upsilon\) must be individually right: in gKPZ, the tree with one derivative edge to a noise gives \(\Upsilon = \partial_{u_1}[g\,u_1^2]\cdot f = 2fg\,\partial_x u\) with \(S = 1\) — the paper’s explicit \(2k(\tau_3)\); the tree with two identical derivative edges gives \(\Upsilon = \partial^2_{u_1}[g\,u_1^2]\cdot f^2 = 2f^2 g\) with \(S = 2\), so the factor 2 cancels — the paper’s plain \(k(\tau_5)\) (tex 6004–6011). The engine reproduces both.

7. Free constants, the renormalization group, and BPHZ¶

The family with free \(k(\tau)\) is the complete structural answer: the set of admissible renormalizations forms a group \(G^-\) (characters on the negative trees under a convolution product), and each group element gives one renormalized equation. The engine builds \(G^-\) explicitly — see the algebraic layer.

Among all choices, the BPHZ (canonical) character is the distinguished one: it recentres each divergent object by its expectation at the origin. Algebraically (tex 5034):

\[k^\zeta = h^\zeta \circ S'_-,\]

where \(h^\zeta(\sigma) = \mathbb{E}\big[\Pi^\zeta \sigma\big](0)\) are the elementary expectations (divergent Gaussian integrals — heat kernels against noise covariances) and \(S'_-\) is the negative twisted antipode, a recursive Bogoliubov-style subtraction operator on the tree Hopf algebra. The engine computes \(S'_-\) exactly and emits each canonical constant as an exact polynomial in the symbols \(h(\sigma)\), with the integrals written out in \(\varepsilon\)-regularized form — it does not evaluate them (that is the analytic wall; see Scope).

Several \(h(\sigma)\) vanish or coincide for provable, purely structural reasons, and the engine knows these identities:

Gaussian parity: an odd number of noise vertices ⇒ \(h = 0\) (Isserlis / Wick).
Root polynomial decoration: \(h(X^n \tau) = 0\) for \(n \neq 0\) (tex 5083).
Pure-kernel total derivative: a noiseless tree with a derivative kernel leaf integrates to 0.
Spatial reflection (only for a spatially symmetric noise): odd total spatial derivative order on the kernels ⇒ \(h = 0\). This is the identity that collapses KPZ’s canonical family to Hairer’s single diverging constant.

The first three are noise-independent and always safe; the reflection identity is gated behind an explicit symmetric flag — see Reports & output.

8. The full algebraic picture (Phase 3)¶

Behind the pipeline sits the two-Hopf-algebra structure of BHZ, all of which the engine builds and property-tests over its equation corpus:

the recentering coproduct \(\Delta : T \to T \otimes T^+\) (positive renormalization — Taylor recentring of the model), and the structure group;
the extraction–contraction coproduct \(\delta^- \) (negative renormalization — cutting out divergent subforests, with the contracted node inheriting an extended decoration and the Taylor \(\mathfrak{e}'\) recentring terms);
their cointeraction compatibility (tex 5717) — verified over the whole corpus including the singular \(\beta_0 = -\tfrac32\) case;
the twisted antipode \(S'_-\) and the renormalization group \(G^-\).

None of this is needed to state the renormalized family (the free constants are), but it is what makes the canonical character computable and the theory self-consistent, and it is exposed programmatically — see the algebraic layer.