Is "accidental symmetries are the wildly ramified ones" more than a reading — does ramification theory predict an integrality relation between the charge an operator violates and its dimension?

Why this exists. DICTIONARY-AUDIT.md ranks §13.4 second among the dictionary's pre-loaded conclusions: the one entry the closure rung was credited with producing, labelled (D) "new dictionary entry", with a note written from it, and never tested. This probe asks what it would take for the identification to be more than a table of correspondences.

The identification as filed. A symmetry's ramification depth = the operator dimension at which it is first broken; the EFT filtration of theory space (operators of dimension ≤ d) = the ramification filtration of the outer action; C and CP are tame (broken at dimension 4), lepton number is wild at depth 5 (Weinberg operator), baryon number at depth 6.

What a filtration of this kind would have to carry. Written as a descending filtration of the accidental symmetry group — the subgroup of U(1)_B × U(1)_L left exact by all operators of dimension ≤ d — lepton number goes U(1) ⊃ ℤ₂ ⊃ 1 at d = 5, 6. Ramification filtrations are not arbitrary descending chains: their jumps obey integrality theorems (Hasse–Arf: the upper-numbering jumps of an abelian extension are integers), their quotients have fixed shapes (G₀/G₁ cyclic of order prime to p, Gᵢ/Gᵢ₊₁ elementary abelian p-groups), and the two numberings are related by the Herbrand function. If the identification is structural, some such constraint must have an EFT image that is true and not otherwise explained.

The candidate. In the Standard Model EFT there is an integrality relation of exactly this shape between the charge an operator carries and its dimension: (ΔB − ΔL)/2 ≡ d (mod 2) for every operator (Kobach, Phys. Lett. B 758 (2016) 455, arXiv:1604.05726 — recalled at sealing, verified 2026-09-28 before scoring; see the resolution). It links the amount of violation to the depth at which it appears, as Hasse–Arf links jumps to integers.

Sealed prediction (committed 2026-09-28 before reading the source or attempting the map). The identification does not derive the relation: the SMEFT parity comes from Lorentz invariance and hypercharge bookkeeping (fermion number against spinor-index count), and no ramification-theoretic integrality maps onto it with the same modulus. §13.4 is then a reading — (D) → (B) — and the note stands as a vocabulary, not a result.

Falsifier. A map, written out, under which the SMEFT relation (in its verified form) is the image of a ramification-theoretic constraint — Hasse–Arf integrality, the shape of the graded quotients, or a Herbrand-function identity — with the modulus 2 and the dimension d both accounted for, and not merely "both are integrality statements". If found, §13.4 earns its (D) and the note becomes a probe for the Herbrand ↔ UV/IR matching that §13 left open.

Cost. Reading and derivation, no compute: the SMEFT source, Serre's Local Fields ch. IV–V for the constraints, one page of mapping. This thread's; the owner scores.

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Source verified. Kobach, Baryon number, lepton number, and operator dimension in the Standard Model, Phys. Lett. B 758 (2016) 455, arXiv:1604.05726: an operator's dimension d has the parity of ΔB−ΔL over 2, and it survives any number of right-handed neutrinos. The recalled form was right.

Where it comes from (computed, scripts/p046_kobach_center.py). Every multiset of SM fields, Higgs and derivatives up to dimension 9 is kept if it passes a set of necessary conditions for being an invariant Lorentz scalar, and the relation is checked on all of them:

So the relation needs Lorentz spinor parity and both centre conditions — SU(3) triality and SU(2) doublet parity — and nothing else. Hypercharge can stand in for either centre condition, because Y = 0 forces both: the ℤ₆ that ties the centres to hypercharge (§40, "closure decides isogeny"). The relation is therefore a statement about the global form — the centre ℤ₃ × ℤ₂ of the gauge group and the spinor parity of the Lorentz group, i.e. the Γ ⊂ ℤ₆ × ℤ₂^F structure the dictionary filed at §13.3 and §40 — and not about a filtration by depth. The scan is finite (dimension ≤ 9) and uses necessary conditions on field content, so it is evidence for the mechanism, not a proof; a proof is a mod-2 linear identity among the per-field charges, which the table says exists.

Scored against the seal. Prediction: the identification does not derive the relation — held. The sealed reason ("Lorentz invariance and hypercharge bookkeeping") was half right: Lorentz yes; hypercharge only as a proxy for the centres. No ramification-theoretic constraint (Hasse–Arf integrality of jumps, the shape of graded quotients, Herbrand) maps onto a single ℤ₂ grading fixed by a centre element: the SMEFT relation has no jumps and no filtration in it. §13.4 → (B), and the note note-accidental-symmetries-as-ramification.md stands as vocabulary. The one thing the probe adds to the dictionary is the relocation: charge–dimension integrality in the SMEFT is a global-form (isogeny-rung) fact, a new instance for §40.