Phase 11 · nullspace audits, soft modes, protection gate

Identity Protection & Nullspace Audit

AT-0087 → AT-0094. A hidden free direction can make a shortcut look profound. This lab measures the rank and singular spectrum of every toy coupling, removes the exact null directions, and asks whether identity protection survives without them.

SeedRelational dimension d_R24Control identity dimension d_I12Damage tolerance (RMS)1.0e-2Benchmark target projection0.80Hand-set soft σ2.0e-2

Set the control identity dimension below the relational dimension and the control gains an exact null space of width d_R − d_I. Its identity-neutral relocation is then a fact about matrix shape, not about the model.

ΔI = K · ΔR, K = U Σ Vᵀ, damage = ‖K ΔR‖ / √d_I

Linearised identity response. Any right-singular direction with σ ≈ 0 is free: relocation along it costs no identity damage by construction.

Running the seeded protection sweep…