Theorem(non-normal) — the spectral-dichotomy extension

0. The idea (why the symmetric split was the wrong tool)

TheoremV = ½‖P_M x‖²splits state space by the symmetric part A_s = ½(A+Aᵀ) (orthogonal eigenbasis) and runs a Lyapunov energy V = ½‖P_M x‖² on the active modes. For non-normal A the cross-term P_M A P_N ≠ 0 (the §1.1 obstruction) breaks V̇ ≤ 2αV, and worse — §2 already notes the A_s-null manifold isn't even invariant under the real flow. So "collapse onto the A_s-null manifold" is genuinely false for non-normal A, not merely unproven. The A_s split is an artifact that is exact only when A is normal (A = A_s).

The fix: split by A's own spectrum (the oblique Riesz projector), not A_s's.

1. Theorem (non-normal contraction dichotomy)

Let ẋ = Ax, A ∈ ℝⁿˣⁿ (possibly non-normal). Fix δ > 0 with no eigenvalue on the line Re λ = −δ (a spectral gap). Let Π_M be the Riesz spectral projector onto M = ⊕{generalized eigenspaces with Re λ < −δ}, N the complementary invariant subspace (Re λ ≥ −δ), Π_N = I − Π_M.

(a) Invariance — the cross-term vanishes. AM ⊆ M, AN ⊆ N, so Π_M A = A Π_M and hence Π_M A Π_N = 0. The obstruction that defeats the §1 energy proof is identically zero in A's own spectral basis. (Verified: ‖Π_M A − A Π_M‖ ≤ 3e-11 overrandom non-normal A.)

(b) Active decay with bounded transient. A_M := A|_M (the active block in an orthonormal basis of M, A_M = BᵀAB) has spectral abscissa < −δ — Hurwitz. The Lyapunov equation A_Mᵀ P + P A_M = −I has a unique P ≻ 0; with V(ξ)=ξᵀPξ, V̇ = −‖ξ‖² ≤ −V/λ_max(P), so

‖Π_M x(t)‖ ≤ √(cond(P)) · e^{−t / (2 λ_max(P))} · ‖Π_M x(0)‖ → 0.

The prefactor √cond(P) is the transient overshoot: for non-normal A_M it can be ≫ 1 but is always finite, and is equivalently bounded by the Kreiss constant K(A_M) ≤ sup_t‖e^{tA_M}‖ ≤ e·n·K(A_M). The active modes always die; non-normality shows up only as a bounded overshoot, never as a failure to contract.

(c) Dichotomy — fate decided purely by the slow block. Let β = max{Re λ : λ ∈ spec(A_N)} (the slow-block spectral abscissa).

• β > 0 ⟹ a mode in the open RHP ⟹ ‖x(t)‖ → ∞ (generic x0): divergence.
• β ≤ 0 and A_N semisimple on the imaginary axis ⟹ sup_t‖e^{tA_N}‖ < ∞ ⟹ ‖Π_N x(t)‖

bounded; with (b), x(t) → N: collapse onto the center/slow manifold (the generalized 42-state).

By (b) the active part always decays, so only sign(β) decides the fate — no third option: no trajectory both keeps the active part alive and avoids the collapse/diverge dichotomy.

Caveat (the honest edge). A defective (Jordan) eigenvalue exactly on Re λ = 0 gives polynomial growth — a measure-zero, non-generic boundary; conservatively classify it as non-collapse. And Π_M is ill-conditioned when an eigenvalue sits within ~GAP of the split line −δ; choose δ in a spectral gap.

Relation to Theorem 1. For normal A, A_s = A, the Riesz projector is the orthogonal A_s-projector, and this reduces to TheoremA_s = Aexactly. So T1-NN is a strict generalization, and TheoremA_s = Ais its normal-A special case.

2. Evidence (machine-checked, 2026-06-26)

python experiments/prove_t1_nonnormal_dichotomy.py → VERIFIED (600 random non-normal A, d∈{4,5,6,8}). pytest tests/test_cio_sde.py → 42 passed (thesrc/sigma0/loop_lm.py::_stability_gatesT1 tests included). The shipped certificate is src/cio_sde/collapse.py::dichotomy_certificate; it is surfaced at decode time via src/sigma0/loop_lm.py::_stability_gates (the "dichotomy" field), reported alongside the #768 stability gates.

3. What was done (handoff completed 2026-06-26)

3.1 Sweep rewritten with the correct numerics (experiments/prove_t1_nonnormal_dichotomy.py). The earlier check evolved the FULL propagator e^{tA}·Π x0 (which amplifies projector roundoff in the unstable direction, ratios ~1e38) and used −max Re λ as the envelope rate. Both fixed: each block is evolved by its REDUCED dynamics in an orthonormal basis (c(t)=e^{tAᴹ}c₀, d(t)=e^{tAᴺ}d₀, no leakage), the active envelope uses the valid Lyapunov bound √cond(P)·e^{−t/(2λ_max(P))}, and the fate is validated by the SIGN of the slow block's empirical late-time growth rate (robust to β's magnitude) rather than fixed ratio thresholds.

3.2 dichotomy_certificate(A, delta) added to src/cio_sde/collapse.py (+ DichotomyCertificate). Returns fate ∈ {COLLAPSE, MARGINAL, DIVERGE}, collapses, active/slow_dim, active_abscissa, slow_abscissa = β, active_decay_rate = 1/(2λ_max(P)), transient_bound = √cond(P), and invariance_residual. Uses the eig-based Riesz projector → orthonormal active basis, and the SAME reduced Lyapunov metric as stability_gates() applied to the active block A_M.

3.3 Three suite tests added (tests/test_cio_sde.py): test_t1_nonnormal_invariance, test_t1_nonnormal_active_decays, test_t1_nonnormal_dichotomy — all green.

3.4 Wired into the decode-time check — loop_lm._stability_gates now also reports the dichotomy fate on the empirical decode Jacobian (purely diagnostic; the acceptance gate is unchanged). This doc promoted DRAFT → PROVEN-in-regime; SIGMA0-COLLAPSE-CERTIFICATE.md §1 updated to move the #768 contraction frontier from open to closed-in-regime.

3.5 Scope honesty (unchanged). This is contraction-onto-the-manifold for the local linear Jacobian — NOT a global non-collapse guarantee; grounding remains the safety mechanism. "Machine- checked" = closed-form algebra + sweep + pytest, not Lean.

Source of record for the math: this doc + experiments/explore_nonnormal_contraction.py (designed cases) + experiments/prove_t1_nonnormal_dichotomy.py (randomized sweep) + the shipped dichotomy_certificate. The reusable Lyapunov/Kreiss machinery is stability_gates() in src/cio_sde/collapse.py.