PWR-HYBRID-3/claude_memory/2026-04-28-path1-sos-pj-sketch.md

2026-04-28 — Path 1 sketch: SOS on PJ polynomial dynamics

Status: Sketch only. Do not run yet. Dane wants this saved for an overnight session. Closing the soundness gap on operation/hot-standby barriers — moving from linearized SOS to nonlinear-SOS-with-PJ.

The obstacle

Dynamics in pke_th_rhs_pj.jl are rational, not polynomial, because of the prompt-jump algebraic relation:

n(x) = Λ · Σ λᵢ Cᵢ / (β - ρ(x))

with ρ(x) = u + α_f(T_f - T_f0) + α_c(T_c - T_c0) affine. SumOfSquares.jl only handles polynomials.

The trick: multiply through by D(x) = β - ρ(x)

D(x) is affine in (T_f, T_c) and strictly positive on the operating envelope (this is exactly the prompt_critical_margin_* predicate — already in predicates.json).

For each PJ state-equation dxᵢ/dt = fᵢ(x)/D(x) + polynomial, multiply:

D(x) · dxᵢ/dt = fᵢ(x) + D(x) · polynomial

The right-hand side is now polynomial. The Lyapunov/barrier condition dB/dt ≤ 0 becomes (multiplying by D > 0):

D(x) · ∇B(x) · f(x) ≤ 0      ← polynomial inequality

which SumOfSquares can handle. Soundness preserved because D > 0 on the operating envelope (and we discharge that as a separate predicate, already done in current SOS work via the prompt_critical_margin_* halfspace).

Polynomial RHS spelled out

PJ state vector: x = [C₁..C₆, T_f, T_c, T_cold], u constant per mode. Define S = Σⱼ λⱼ Cⱼ (linear in C). Then:

D · dCᵢ/dt   = βᵢ S - λᵢ Cᵢ D                              ← deg 2
D · dT_f/dt  = (P₀ Λ S - hA (T_f - T_c) D) / (M_f c_f)      ← deg 2
D · dT_c/dt  = ((hA (T_f - T_c) - 2 W c_c (T_c - T_cold)) D) / (M_c c_c)   ← deg 2
D · dT_cold/dt = ((2 W c_c (T_c - T_cold) - Q_sg) D) / (M_sg c_c)           ← deg 2

All right-hand sides are degree ≤ 2 in (C, T_f, T_c, T_cold). With B a degree-4 polynomial, the SOS Lie condition D ∇B · f is degree ≤ 4 + 1 + 2 = 7. Multipliers degree 2. SDP size scales with monomial count of the relevant degrees in the relevant variables.

Dimensionality / tractability

Full 9-state SOS, degree 4: C(9+4, 4) = 715 monomials. Big SDP but tractable on modern CSDP/MOSEK with chordal sparsity. May need MOSEK rather than CSDP for solver robustness at this scale.

Realistic first attempt: 3D projection on (T_f, T_c, T_cold) with the precursor dynamics treated as bounded uncertainty. Degree 4 in 3 vars = 35 monomials — same scale as the existing 2D scripts. The precursors' contribution to thermal dynamics enters only through S = Σ λⱼ Cⱼ in the T_f equation, and S is bounded on the reach envelope (we have intervals from prior reach runs). Treat S as [S_lo, S_hi] parametric uncertainty in the SOS — clean Putinar multiplier on [S - S_lo, S_hi - S].

This is the pragmatic path: 3D polynomial SOS with bounded S. Sound for the nonlinear PJ plant on the operating envelope where D > 0 and S ∈ [S_lo, S_hi].

Session plan for the overnight run

1. Add D(x) and S(x) symbolic primitives to sos_barrier.jl.
2. Extend solve_sos_barrier_2d → solve_sos_barrier_nd with parametric-uncertainty multipliers (Putinar on S ∈ [S_lo, S_hi]).
3. Compute S envelope from reach_operation_result.mat (already exists for operation) and from a long shutdown sim for hot-standby.
4. Build barrier_sos_3d_pj_operation.jl:
   • 3D state (δT_f, δT_c, δT_cold) around operation equilibrium
   • Polynomial RHS via multiply-through
   • D > 0 and S ∈ [S_lo, S_hi] as Putinar safety multipliers
   • Same X_entry, X_unsafe as current 2D operation barrier
   • Run, compare to linearized result.
5. Build barrier_sos_3d_pj_shutdown.jl — analogous.
6. If both succeed, write a journal entry making the soundness claim explicit. The new barrier is sound for the PJ-reduced nonlinear plant on {D > 0} ∩ {S ∈ [S_lo, S_hi]}. The PJ reduction itself has the Tikhonov error bound (already worked out 2026-04-21). Composing these gives a sound certificate for the full nonlinear plant up to the (small, characterized) PJ error.

Subtleties / things that might bite

• CSDP may not converge at degree 4 in 3D with multiple Putinar multipliers. If so, switch to MOSEK (license setup needed) or drop to degree 2 with iterative refinement.
• The multiply-through is only valid where D > 0. If the SOS finds a B that's only valid on a region where the SOS solver's certificate space includes D ≤ 0 points, the result is bogus. Mitigate: use D as a Putinar multiplier on the entry/safe sets (so SOS only reasons over {D > 0}).
• Precursor coupling not certified. The 3D projection drops the 6 precursor states. They're bounded in S but their individual dynamics aren't certified. If the SOS proves invariance of (T_f, T_c, T_cold) while precursors drift outside [S_lo, S_hi], the certificate is invalid. Need to either (a) certify precursor bounds separately as a forward invariant, or (b) include precursor states in the SOS (back to 9D).
• Connection to Tikhonov. The PJ reduction has error O(Λ) = O(10⁻⁴). Composing PJ-validity-guaranteed barrier + PJ Tikhonov bound = sound certificate for the full plant, modulo the (small) PJ error. Worth working out the full chain rigorously.

Files that would change

• code/src/sos_barrier.jl — generalize to ND + parametric uncertainty
• code/scripts/barrier/barrier_sos_3d_pj_operation.jl — new
• code/scripts/barrier/barrier_sos_3d_pj_shutdown.jl — new
• journal/entries/YYYY-MM-DD-pj-sos-soundness.tex — new
• code/CLAUDE.md — update soundness claim section

Estimated time

8–12 hours focused work. Realistic overnight run if the SDP is well-conditioned. If MOSEK setup eats time, push to a second night.