Skip to content

The algebraic layer

Everything in this page is optional for the headline use case — the renormalized family needs no Hopf algebra. But the full BHZ structure is built, property-tested, and exposed, for when you want the objects themselves: coproducts, the twisted antipode, characters, the group.

The regularity structure \((T, T^+)\)

from counterterms import build_regularity_structure

rs = build_regularity_structure(spde, gamma=1)   # γ-truncated
  • rs.model_basis — the decorated trees spanning the model space \(T\) up to homogeneity \(\gamma\); rs.grades() groups them by homogeneity, rs.homogeneities() lists the grading.
  • rs.divergent — the negative subspace (the counterterm carriers).
  • rs.positive_basis() — the trees generating \(T^+\).
  • rs.recentering(t) — the coproduct \(\Delta : T \to T \otimes T^+\) (positive renormalization: Taylor recentring of the model). Returned as an exact tensor sum {(left, right): coefficient}.
  • rs.structure_coproduct(b) — \(\Delta^+ : T^+ \to T^+ \otimes T^+\), and rs.structure_antipode() — the antipode of the structure group’s Hopf algebra.

Note

The polynomial sector \(\{X^k\}\) is carried implicitly as node decorations, not as standalone basis vectors — consumers counting graded dimensions should account for this.

The renormalization structure and the BHZ character

from counterterms import build_renormalization

rn = build_renormalization(spde)
  • rn.coaction(t) — \(\delta : U \to U^- \otimes U\), the extraction–contraction coaction (cut out divergent subforests; the contracted node keeps an extended decoration; Taylor \(\mathfrak{e}'\) terms recentre the cut edges).
  • rn.coproduct(t) — \(\delta^- : U^- \to U^- \otimes U^-\).
  • rn.twisted_antipode(t) — the negative twisted antipode \(S'_-\), the recursive Bogoliubov-style subtraction (tex 5034). Note \(S'_-\) is not a Hopf antipode — it maps into the free algebra \(\mathbb{R}[U]\).
  • rn.h_symbol(σ) — the elementary-expectation symbol \(h(\sigma)\).
  • rn.bhz_character(t) — the exact polynomial \(k_\tau = h(S'_- t)\) in the \(h\)-symbols.
  • rn.canonical_character(t) — the same with the centered-Gaussian parity rule applied (odd-noise trees ⇒ 0). Symbols that survive here are not guaranteed nonzero — the further provable reductions live in the reduced report view.

The cointeraction between \(\delta\) and the recentering \(\Delta\) (tex 5717) — the compatibility that makes renormalization commute with recentring — is verified by the test suite over the whole equation corpus, including the singular \(\beta_0 = -\tfrac32\) case, where a naive reading of the paper’s recentring actually fails (see notes/cointeraction_singular_noise.md for that story).

The renormalization group \(G^-\)

from counterterms import build_renormalization_group

G = build_renormalization_group(spde)
chi = G.character({tree: value, ...})   # a character from tree values (multiplicative on forests)
G.convolve(chi1, chi2)                  # the group product (χ₁ ⊗ χ₂) ∘ δ⁻
G.inverse(chi)                          # χ ∘ S  (antipode inverse)
G.identity()                            # the counit

Group axioms (associativity, unit, inverse) are property-tested, including on multi-component forests. G.admissible() — the analytic subgroup \(G^-_{\mathrm{ad}}\) of characters compatible with the kernel’s vanishing moments — is deliberately unbuilt and raises NotImplementedError (it needs the \(\Pi\)-map, an analytic object).

Wick structure of the canonical constants

from counterterms.renorm.scheme import expectation, NoiseLaw, WHITE_NOISE

exp = expectation(tree, sig, law=WHITE_NOISE)
str(exp)     # the ε-regularized integral: heat kernels ∂^p K against covariances C(z_i − z_j)

expectation writes \(h(\sigma) = \mathbb{E}[\Pi^\zeta\sigma](0)\) as an explicit Isserlis / Wick-pairing sum: one \(\partial^p K\) per kernel edge, one covariance factor per matched noise pair, within-noise-type pairing only, odd counts ⇒ 0. The integrals are emitted, not evaluated — evaluation is the deliberately unbuilt analytic half (BPHZ.numeric_character raises until an IntegralEvaluator exists).

JSON export

from counterterms.render import structure_json

doc = structure_json(spde)   # one self-contained document

Exports the whole structure machine-readably: the basis trees (as nested dicts, exact homogeneities), the divergent set, the coproducts \(\Delta, \delta^-\) as tensor sums, antipode values, and the character polynomials — enough to reconstruct or cross-check the algebra in another system.