Appendix A / B

Explicit examples and a computable certificate

A.1

Associative zero example

K: 0 · ‖K‖: 0 · ΔS: 0

Bounds: 0=0

PASS / counterexample to an unconditionally positive bound

A.2

Nonassociative example with Δ_S=0

K: 2e₂ · ‖K‖: 4 · ΔS: 0 for S=‖·‖

PN.2-R: 1=1

PASS / scalar degeneracy, sharp full-state bound

A.3

Global c-separation

K: 12e₂ · ‖K‖: 24 · ΔS: 24

c=1; both bounds: 6=6

PASS / coefficient 1/4 is sharp

A.4

Compact Ω with ε_-=3

K: (3/2)e₂ · ‖K‖: 3 · ΔS: 3

ε_-=3, κ_Ω=3/4; both bounds: 3/4=3/4

PASS / exact compact lower bound for A₂ only

PN2-PO12-A2-v1.0 · tests=11 · archive SHA-256 bbcb2715a3e5ab7688dfe1ad317a68f23b163ae10915982216111142ce7da2ad. Closure concerns the abstract package; transfer to physical R⋆R depends on open proof obligations.