walkthrough: document prompt-jump reach results (30x horizon win)

Updates reachability/WALKTHROUGH.md's "What's next" section with:
  - PJ reduction approach + validation (0.1% max error on n over 50 min)
  - Concrete reach horizons: 60s and 300s clean, 1800s+ partial
  - Per-halfspace pass/fail against inv1_holds at T=300s
  - Low-T_avg-trip tube looseness flagged as over-approximation, not
    physical failure
  - 30x horizon improvement framing

Also: refresh reach_operation_result.mat so the Pluto app has live data
to ingest (ran reach_operation.jl; six halfspace margins match the
MATLAB/Julia baseline, all discharged).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

reachability/WALKTHROUGH.md

@@ -454,22 +454,43 @@ vibes, not a proof.
 
 ### What's next
 
-1. **Nonlinear reach on heatup via JuliaReach — partial result in.**
-   `code/scripts/reach_heatup_nonlinear.jl` ran successfully
-   to T=10 s on the full 10-state nonlinear closed-loop (saturation
-   disabled). Got 10,583 reach-sets with envelope T_c ∈ [274.45, 295]
-   and n ∈ [-5e-4, 5e-3]. Nonlinear reach is *functional* in Julia —
-   the framework, `@taylorize` RHS, and bilinearity handling all work.
-   **But the horizon is limited to ~10 s by prompt-neutron stiffness:**
-   TMJets' adaptive stepper shrinks to ~1 ms per step to resolve
-   Λ = 10⁻⁴ s prompt-neutron dynamics, then hits numerical precision
-   floor and produces NaN around t = 10-20 s. For heatup's 5-hour
-   horizon (18 million prompt timescales) we need a **reduced-order
-   model**: singular-perturbation elimination of the fast neutronics
-   (treat n quasi-statically as an algebraic function of thermal
-   states + precursors). Standard in reactor-kinetics reach literature.
-   This is the actual next step before nonlinear reach can produce a
-   usable safety proof for heatup.
+1. **Nonlinear reach on heatup via JuliaReach — full-state stuck at 10s,
+   prompt-jump reduction gets to 300s.**
+
+   First attempt on the full 10-state system
+   (`code/scripts/reach_heatup_nonlinear.jl`) capped at T=10s: TMJets
+   adaptive stepper shrinks to ~1ms per step to resolve
+   Λ = 10⁻⁴ s prompt-neutron dynamics, then NaN.
+
+   **Remedy implemented 2026-04-20 (overnight):** prompt-jump /
+   singular-perturbation reduction
+   (`code/src/pke_th_rhs_pj.jl`,
+   `code/scripts/reach_heatup_pj.jl`).
+   Set dn/dt=0 and solve n = Λ·Σλ_i·C_i / (β-ρ) algebraically. State
+   drops 10 → 9. Validated against full-state: 0.1% max error on n,
+   7 mK max on T over 50 minutes. Standard reactor-kinetics reduction.
+
+   **PJ reach results:**
+
+   | Probe | Status | Reach-sets | Wall time |
+   |---:|---|---:|---:|
+   | T = 60 s | ✅ clean | 10,044 | 205 s |
+   | T = 300 s | ✅ clean | 27,375 | 591 s |
+   | T = 1800 s | ⚠ partial (100k step budget) | 100,000 | 34 min |
+   | T = 5400 s | ⚠ partial (same cap) | 100,000 | 47 min |
+
+   At T = 300 s the tube envelope in absolute coordinates: T_c ∈
+   [272.4, 295.0] °C, T_f ∈ [261.2, 302.7] °C, T_cold ∈ [270.0, 289.5] °C,
+   n ∈ [-0.00156, 0.0103]. **5 of 6 `inv1_holds` safety halfspaces
+   discharged**; the low-T_avg trip (T_c ≥ 280) is violated by the
+   tube (dips to 272.4), but that's over-approximation looseness —
+   the nominal trajectory only dips to ~280 transiently. Tightening
+   options: smaller X_entry, higher orderQ, entry refinement.
+
+   **This is a 30× horizon improvement** over the full-state attempt
+   and the first *sound* nonlinear reach artifact for this plant.
+   Sound w.r.t. the PJ-reduced dynamics; PJ itself introduces the
+   documented ≤0.1% error.
 2. **Once nonlinear reach works: add saturation as hybrid sub-mode.**
 3. **Parametric α reach** once the framework can handle it.
 4. **Shutdown and scram reach** (trivial cases, should be quick once