Residues of group actions — resolved by performing them, not by computing

The principle. When a quantity is determined only up to the action of a group G, no calculation fixes it. There are exactly two routes: physically implement a G-element and observe, or find an independent structure that breaks G. Computation is not one of them — the ambiguity is in the definition, not the arithmetic.

The worked case — the sign of J. Spectrally undetermined: minor spectra fix every modulus, hence |J|, but the residue is a ℤ/2 quotient by conjugation ([[N-011]], [[CASE-24]]). It is measurable only because nature lets you build both a neutrino and an antineutrino beam — the antineutrino beam is the group element U → U*. If you could only make neutrinos, δ_CP would be as inaccessible experimentally as it is spectrally: a residue with no way to perform its group element is simply not knowable.

The experimental-design consequence, from the ambiguity's structure. The Earth's matter also breaks CP, faking an asymmetry that flips sign with the mass ordering — so there are two unknown signs (δ_CP and the ordering) against one observable. That degeneracy is why two baselines are forced, not chosen: Hyper-K (295 km, matter effect small, δ clean) and DUNE (1300 km, matter effect large, ordering) break it only together. Computed: the vacuum asymmetry is ∓0.257 (DUNE) / ∓0.243 (Hyper-K) at δ = ±90°; the matter fake at δ = 0 is +0.226 (DUNE), flipping to −0.229 under inverted ordering. The complementarity of the two experiments is dictated by the shape of the ambiguity, not by engineering.

Three prior instances of the same shape — a quantity fixed only up to an involution, resolved by performing the involution: - the Hall-effect carrier sign (resolved by reversing the field / the current — performing the ℤ/2); - the matter–antimatter asymmetry (resolved by building the antiparticle — the case above); - chiral orientation (resolved by a parity-reversed apparatus). Three is a candidate recurring structure, not yet a filed pattern.

Connection to the certificate-direction axis (docs/rung-vocabulary-v4-part1.md §8.2). A residue of a group action is a ∀→∃ conversion that has no certificate: you cannot exhibit the witness, you can only apply the group element and look. This is the part that survives at all n. The bit count of the residue is a three-generation accident (ℤ/2 at n = 3, ≥ 2 bits at n = 4 — [[N-011]]); the group-action shape — no certificate, resolve by performing — does not depend on n.

The residue you cannot perform. Both routes assume the group element is performable. Almost always it is, and the reason is one physical fact: the weak interaction breaks the very symmetries whose residues matter, so nature hands us the G-element as a process. Parity — Wu's cobalt-60 (1957). Matter vs antimatter — the K_L semileptonic charge asymmetry, δ_L = 3.32×10⁻³, which is the Ozma answer: it tells a civilization we have never met which of their two charges is our electron, with no shared sample. Antimatter gravity — ALPHA-g (2023) watched antihydrogen fall. So CP violation is what makes "we are made of matter" a measurement, not a definition — the pivot of the baryon-asymmetry thread (Sakharov's three conditions on three rungs, docs/rung-vocabulary-v3.54.md).

The one genuine exception — cosmic topology beyond the horizon. Write spacetime as M̃/Γ, the universal cover modulo a deck group. The residue is Γ; performing an element means traversing an identification — leaving in one direction and returning from another. If Γ's scale exceeds the particle horizon, causality forbids the traversal outright: the group element is unperformable in principle, not merely in practice — the first residue here for which neither route is open. Planck's matched-circle test found no matched pairs, bounding the scale above ~0.985 of the 27,740 Mpc observable diameter; past 1.0 the identified circles never meet the last-scattering sphere and the test is blind by construction. So every measurement is a lower bound with no matching upper bound — the witness that would close it sits outside the light cone.

Edge to [[P-010]] (determined-but-uncomputable) — a distinction, not an identity. Run the mechanism test rather than assert the link. "Topology nontrivial" is Σ₁: exhibit one matched circle pair. "Topology trivial" is Π₁: no finite witness — you must rule out every pair. That is BB(745)'s exact shape: a determined quantity, one-ended, certifiable in one direction only. Shared theorem — the Σ₁/Π₁ certificate-direction axis (§8.2); both quantities are one-sided along it. Different mechanism — BB's obstruction is proof-theoretic (the halting problem, ZFC's strength); cosmic topology's is causal (the light cone). Same axis, different boundary, so an edge with a named distinction, not the identity it first looks like. Both leave a quantity that has a value nobody can have.