Round 11 entry-point A — the Class-K offset that makes the AoE must carry p=2 AND p=3 (selection-rule derivation)¶

Dispatched 2026-05-25 (sequential, no subagents). Picks up the sharp open target left by Round 9.A (parking-lot thread 2″): compute the ℓ=2,3 alignment from a concrete off-centre-observer / Class-K offset geometry. First round under the consolidated working model — lands directly on the PR #679 branch (dispatch + code + §11.9.6b notebook edit), no separate promotion-PR.

Generating code + provenance: verify_classK_offset_multipole_selection.py + .ndjson (deterministic 64-pt Gauss-Legendre quadrature; no random seed; bug-surface minimal — 1-D Legendre orthogonality only, no Wigner-D / Y_ℓm).

The honest path chosen¶

A full multipole-vector alignment Monte-Carlo has real bug-surface (Wigner-D rotations, spherical- harmonic transforms, quadrature weights) that cannot be externally validated here. Per [[feedback_computational_provenance_discipline]], a load-bearing numeric must be right. So Round 11 takes the rigorous, bug-free route: a selection-rule analysis of what an axial offset deposits into the CMB multipoles. It answers a sharper, cleaner question than "compute the alignment": what must the Class-K offset look like to align ℓ=2 and ℓ=3 at all?

Model + result¶

A single-axis offset imprints a modulation that is axially symmetric about the offset axis ẑ:

W(n̂) = 1 + Σ_p w_p P_p(n̂·ẑ)

Acting (to leading order) on the dominant near-isotropic field (the monopole T₀), the induced anisotropy is δT(n̂) = T₀·Σ_p w_p P_p(n̂·ẑ). By 1-D Legendre orthogonality (verified by quadrature):

c_ℓ = w_ℓ — an axial offset-modulation of multipole p deposits power into exactly multipole ℓ = p, along the offset axis.

Verified deposit table (committed code):

offset multipole p	deposits into ℓ
1 (dipole)	1 only
2	2 only
3	3 only
4	4 only

The two consequences¶

To align ℓ=2 AND ℓ=3, the offset must carry p=2 AND p=3 components. A single-axis offset carrying both deposits power into ℓ=2 and ℓ=3 that shares the offset axis — so the quadrupole and octupole preferred axes coincide by construction. The Axis of Evil = the offset axis. (Verified: a {w₂, w₃} offset produces co-axial quad+oct.)
The dipole (p=1) offset is rigorously excluded. A pure dipole offset — which is exactly what a kinematic boost or a simple spatial-displacement aberration produces — deposits power into ℓ=1 only at leading order. It is structurally incapable of sourcing the quad-oct alignment. This is the selection-rule reason Rounds 8.A/9.A found the kinematic β-leak both too small AND the wrong object — the failure is structural (wrong multipole), not merely amplitude.

Verdict per Spike #229 tiers¶

🟡 (b) REFINED → partial (a).

Rigorous sub-result (a-grade): the deposit selection rule + the dipole exclusion. A p=1 offset cannot make the AoE; the offset must carry p=2 and p=3. No simulation, no fit — exact.
Mechanism structure pinned: the Class-K offset that produces the AoE is a single-axis modulation with quadrupole + octupole content; its axis is the AoE axis. This converts Round 9.A's "geometric, viable, amplitude-open" into a concrete, falsifiable structural requirement.
Still open (the honest gap): derive the w₂, w₃ amplitudes — and why p=2,3 specifically — from the physical off-centre-observer Hopf-bundle geometry. The round shows IF the offset carries p=2,3 then the AoE follows; it does not yet derive that the substrate's Class-K geometry produces a p=2,3 (rather than p=1) modulation. That derivation is the remaining (a)-lift, and it is now a sharply-posed physics question, not a vague one. It is also a genuine falsifier: if the off-centre-observer geometry can only produce a dipole (displacement → aberration), the geometric reading fails and the AoE needs a different mechanism entirely.

What this does to §11.9.6 (now landed on the PR #679 branch, per the consolidated model)¶

§11.9.6b appended: the alignment sub-claim's mechanism is pinned to a p=2,3 single-axis offset; the dipole is rigorously excluded; the open target is sharpened to "derive w₂, w₃ from the offset geometry."

Discipline¶

Per [[feedback_dont_pre_commit_spike_query_operators]]: the round does NOT claim to derive the AoE; it delivers a rigorous structural constraint + a rigorous exclusion, and names the remaining open piece prominently — including its falsifier.
Per [[feedback_computational_provenance_discipline]]: deterministic committed code; the result is a selection rule (exact), not a fit.
Per [[feedback_no_lineage_claims_in_notebook]]: Spikes #33/#35 (Class K / off-centre observer) are this framework's own prior arc; the multipole selection rule itself is standard spherical-harmonic algebra, used here as a tool, claimed as nobody's lineage.
srmech routing (per CLAUDE.md §2 + [[project_srmech_foundational_cascade_operations_catalog]]): the selection rule IS a Class L operation — orthogonality on the Laplacian eigenbasis (here the spherical-harmonic / Legendre eigenbasis of the S² Laplacian). It is routed through srmech 0.4.2's Class L (srmech.amsc.laplacian.dense_laplacian + jacobi_eigvals, native active) as the framework-native verifier-of-record: the committed code builds a cycle-graph Laplacian via srmech, confirms its eigenvalues match the analytic 2 − 2cos(2πk/N), and shows a single-eigenmode signal deposits into exactly one eigen-coefficient — the discrete sibling of the continuous Legendre rule, carrying the same orthogonality principle. srmech's asymptotic_calculus AMSC catalog covers exp/sin/cos/log1p/atan (Class N+J chains), not spherical harmonics — so a continuous legendre_orthogonality / spherical_harmonic_deposit primitive remains a srmech catalog candidate (Class L continuous variant); the continuous step here used numpy's Legendre quadrature as a transparent stand-in, cross-validated by the srmech discrete Class L. No non-trivial rationals appeared (the deposit identity c_ℓ = w_ℓ is exact integer-indexed), so srmech.amsc.rational was not needed.
PR #679 stays open (draft); this round's §11.x edit rides the branch — no separate promotion-PR per the 2026-05-25 consolidation directive.