Validation

There is no reference implementation of this pipeline anywhere — the source paper is the only oracle. The validation strategy is therefore: pin the paper’s worked examples as golden tests, property-test the algebraic laws, and cross-check the classical equations against the published literature. uv run pytest runs all of it (187 tests, ~10 s).

Tier 1 — the paper’s own worked examples (exact)

The gKPZ five counterterms (tex 6004–6012). For \((\partial_t - \Delta + 1)u = f(u)\zeta + g(u)(\partial_x u)^2\) at \(\beta_0 = -1-\kappa\), the paper displays the renormalized equation explicitly. The test asserts exact multiset equality of \((|\tau|, S(\tau), F(\tau^*))\) triples. This is a stringent check of the symmetry factor and Υ-map simultaneously, because the factor 2 behaves differently in two terms: it survives in \(2k\,f g\,\partial_x u\) (one derivative edge, \(S=1\)) and is cancelled in \(k\,f^2 g\) (two identical edges, \(S=2\)) — both must be individually right for the display to reproduce, and it does, term for term.

The 43-tree table (tex 6028–6163). At \(\beta_0 = -\tfrac32 - \kappa\) the paper tabulates every strongly-conforming gKPZ tree in six homogeneity rows, with \((1, 2, 6, 2, 23, 9)\) trees per row. The engine reproduces the table row for row — and building this test caught (and fixed) a genuine tree-generation undercount, which is what golden tests are for.

The \(\Phi^4_3\) lift (tex 2026–2034): the remainder equation \(-v^3 - 3v^2X - 3vX^2 - X^3\) with \(X^k \in \mathcal{C}^{(-k/2)^-}\), line for line.

Coproduct examples (tex 6168–6205): the paper’s worked \(\delta^-\) expansions, including the term with combinatorial coefficient 2, reproduce exactly.

These goldens were hand-derived from the paper, not recorded from engine output — the distinction between a validation and a tautology. (An independent audit re-derived them from the tex and confirmed.)

Tier 2 — the classical renormalized equations (literature, structural)

Running the engine on the standard models and applying the appropriate exact reductions reproduces the published results:

Model	Published result	Engine after reduction
KPZ	one diverging constant \(-C_\varepsilon\) (Hairer)	8 counterterms → parity + reflection → one constant
PAM (d=2)	\(u(\xi - C_\varepsilon)\)	4 → root-\(X^n\) + parity → \(u \cdot \text{const}\)
gPAM (d=2)	\(-C_\varepsilon f(u)f'(u)\) (BCCH)	same, exactly \(f f'\)
\(\Phi^4_2\)	mass renormalization \(+3C\varphi\)	mass term \(\propto v\) survives
\(\Phi^4_3\)	two diverging mass constants	two mass constants \(\propto v\); gradient terms vanish

Property tests — the algebraic laws

The Phase-3 layer is tested by verifying the laws, parametrized over a corpus of six equations including the singular \(\beta_0 = -\tfrac32\):

• counits, coassociativity of \(\Delta^+\) and \(\delta^-\), the comodule conditions;
• the cointeraction between extraction and recentering (tex 5717) — uncapped, over all divergent KPZ trees;
• the twisted antipode’s characterizing (Dyson–Salam) relation (tex 5034), checked numerically in exact rationals;
• the group axioms of \(G^-\) (associativity, unit, inverse), including on multi-component forests;
• homogeneity stability of every coproduct.

Independent cross-checks

• \(S(\tau)\) is verified against a brute-force automorphism count (permutations, not the product formula) — so the golden \(S\) values don’t test the formula against itself.
• The Υ-map’s non-commuting order of operations (\(\partial_p\) before \(D^n\)) is tested on a tree where the wrong order gives a different answer.
• The reflection identity in the reduced view emerges, independently, with the paper’s own admissibility statement \(h(X^n\tau) = 0\) (tex 5083) as a by-product — an unplanned consistency success.
• The repository has been through code-quality and mathematical audits, the latter with independent adversarial re-derivation of every load-bearing formula (see notes/math_audit.md and notes/validation.md in the repository).

What validation does not cover

Anything behind the analytic wall (numeric constants, models, convergence) — there is nothing to validate because nothing is computed. The remaining sharp edges listed in Scope & limitations sit outside the test corpus's reach; the generation defects an audit found there (order > 2 enumeration, partial-sum pruning) have been fixed and are now pinned by regression tests in tests/test_rule.py.