Kill Switches

Three kill switches. All live.

You know where the joints are. The argument shows you how to break them.

KS-S.1 — ℤ₂ identification

Status: LIVE — HARD.

The identification of the algebraic ℤ₂ (from Axiom S) with π₁(SO(3)) (from the topology of the rotation group) is the load-bearing step.

It rests on the argument that under faithful embedding (EH, proven AP20), the algebraic ℤ₂ maps to the only intrinsic ℤ₂ in the rotation structure.

Test: Demonstrate a different embedding of the algebraic ℤ₂, equally consistent — one that does not correspond to π₁(SO(3)). If so, the derivation of the double cover from the axioms fails.

Current status: the embedding appears unique given that σ is order-reversing (→ time-reversal) and N = 3 (→ SO(3) has π₁ = ℤ₂). In N ≠ 3 dimensions, SO(N) has different fundamental groups, and the argument would change. But N = 3 is derived (AP10).

KS-S.2 — Minimum spin selection

Status: LIVE — HARD.

The assignment of the minimum half-integer spin (j = ½) to ε depends on the minimum principle in Axiom B.

Test: If "minimum viable splinter" does not uniquely select the smallest representation — if there is a structural reason for ε to carry j = 3/2 or higher — then spin ½ is not derived.

Current status: the minimum principle appears to select the minimum representation straightforwardly. The electron is observed to have spin ½.

KS-S.3 — Exchange–involution bridge

Status: LIVE — HARD.

The spin-statistics derivation requires that exchange symmetry of identical particles on the AP09 Hilbert space is correctly represented by the paired/unpaired σ-action inherited from the record algebra.

Test: Demonstrate a consistent AP09-compatible construction in which the paired/unpaired classification does not fix the exchange sign (or fixes it differently). If so, the spin-statistics mapping and Pauli derivation fail.

Current status: within the AP09 Hilbert-space framework, the exchange operator acts on the internal ℤ₂ degree of freedom, and the paired/unpaired distinction determines its eigenvalue (±1).