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% 2026-04-20 — overnight session: prompt-jump reduction + nonlinear reach
% A-style invention-log entry, written in real-time during the session.
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\session{2026-04-20 (overnight)}{open-ended autonomous session}{Implement
the singular-perturbation (prompt-jump) reduction of the PKE+T/H model.
Validate it against the full 10-state.  Re-run TMJets nonlinear reach
on heatup and find the new horizon wall.  Extend the Pluto app to read
reach results live.  Document everything for review in the morning.}

\section{2026-04-20 (overnight) — Prompt-jump nonlinear reach}
\label{sec:20260420-overnight}

\subsection*{Origin}

The 2026-04-20 evening session ended with TMJets working on the full
10-state heatup at $T = 10$~\unit{\second} but exhausting its 50{,}000-step
budget by $T = 60$~\unit{\second}.  Diagnosis: prompt-neutron timescale
$\Lambda = 10^{-4}$~\unit{\second} forces $\sim$1~\unit{\milli\second}
adaptive steps to bound Taylor remainder.  Over hours, infeasible.

The known remedy: \emph{singular-perturbation reduction} — set
$\dot n = 0$ and solve algebraically for $n$, removing the prompt
timescale from the dynamic state.  Standard reactor-kinetics move,
documented in textbooks (Hetrick \emph{Dynamics of Nuclear Reactors},
ch.\ 4; Ott \& Neuhold).  Auto mode active; Dane's instruction at
session start: ``take a big fat overnight rip as far as you can on the
prompt jump assumption and doing the reachability and app buildout.
Document things in the journal and we'll review in the morning.''

\subsection*{Part 1: The prompt-jump derivation}

\begin{derivation}
Starting from the 10-state PKE+T/H system, focus on the neutron-balance
equation:
$$\dot n = \frac{\rho - \beta}{\Lambda} n + \sum_{i=1}^{6} \lambda_i C_i.$$

The prompt-neutron generation time $\Lambda \sim 10^{-4}$~\unit{\second}
makes the first term \emph{very fast} relative to the precursor and
thermal dynamics (precursor timescales 0.3 to 80~\unit{\second}; thermal
$\sim$10--100~\unit{\second}).  A standard regular-perturbation argument
(Hetrick, ch.~4) shows that on timescales $\gg \Lambda$, the prompt
term equilibrates rapidly and we can set
$$\dot n \approx 0 \quad \Longrightarrow \quad
\frac{\rho - \beta}{\Lambda} n + \sum_i \lambda_i C_i = 0.$$
Solving for $n$:
$$\boxed{\;n_{\mathrm{PJ}}(C, \rho) = \frac{\Lambda \sum_i \lambda_i C_i}{\beta - \rho}\;}$$
valid when $\beta - \rho > 0$, i.e.\ sub-prompt-critical.  For our
heatup controller $\rho = K_p \cdot e$ with $K_p e \ll \beta$, so the
denominator is well bounded away from zero.

Substituting back into the precursor and fuel equations:
\begin{align*}
\dot C_i &= \frac{\beta_i}{\Lambda} n_{\mathrm{PJ}} - \lambda_i C_i
        = \frac{\beta_i \sum_j \lambda_j C_j}{\beta - \rho} - \lambda_i C_i \\
\dot T_f &= \frac{P_0 \, n_{\mathrm{PJ}} - hA(T_f - T_c)}{M_f c_f}
        = \frac{P_0 \Lambda \sum_j \lambda_j C_j / (\beta - \rho) - hA(T_f - T_c)}{M_f c_f}.
\end{align*}

The state vector drops from 10 to 9: $x = [C_1, \ldots, C_6, T_f, T_c, T_{\mathrm{cold}}]^\top$.
The dynamics gain a rational nonlinearity ($1/(\beta - \rho)$).  The
fastest dynamic timescale becomes $1/\lambda_6 = 0.33$~\unit{\second}
— still fast, but \emph{three orders of magnitude} slower than $\Lambda$.

\textbf{Soundness cost:} the prompt transient (the $\sim$50~\unit{\micro\second}
adjustment of $n$ after a step in $\rho$) is no longer captured.  For
hours-long heatup reach, that transient is irrelevant to safety claims.
For prompt-supercritical regimes ($\rho \to \beta$) the algebraic
formula diverges and the reduction is invalid — but those regimes are
themselves accident-class, outside the scope of normal-operation reach.
\end{derivation}

\subsection*{Part 2: Implementation}

Two new files in \texttt{code/}:

\begin{itemize}
  \item \texttt{src/pke\_th\_rhs\_pj.jl} — sim version of the reduced
        RHS, with allocating + helper functions for IC and $n$-reconstruction.
  \item \texttt{scripts/validate\_pj.jl} — side-by-side sim of full
        vs.\ reduced PKE on the heatup scenario.
\end{itemize}

The reduced RHS is structurally identical to the full one with two
differences: (a) no $\dot n$ equation; (b) $n$ inside the precursor and
fuel-temperature equations is replaced by $n_{\mathrm{PJ}}(C, \rho)$,
introducing the rational denominator.

\subsection*{Part 3: Validation against full-state}

\texttt{validate\_pj.jl} runs both models from the same heatup IC
($n_0 = 10^{-3}$, $T = T_{\mathrm{standby}}$ everywhere) for 50 minutes
and tabulates pointwise error.

\begin{lstlisting}[style=terminal]
=== PJ vs full-state, heatup scenario ===
  t [s]    n_full    n_pj      |Δn|/n_full   T_c err   T_f err   T_cold err
     1.0   1.000e-03 1.000e-03   8.32e-07     4.839e-09   1.718e-08   6.642e-10
     5.0   1.000e-03 1.000e-03   3.08e-06     3.970e-08   9.392e-08   1.921e-08
    10.0   1.001e-03 1.001e-03   5.59e-06     1.295e-07   2.320e-07   7.945e-08
    60.0   1.017e-03 1.018e-03   3.70e-05     3.826e-06   4.534e-06   3.446e-06
   300.0   1.310e-03 1.311e-03   3.77e-04     1.867e-04   1.960e-04   1.816e-04
  1200.0   3.414e-03 3.410e-03   1.02e-03     2.177e-03   2.111e-03   2.213e-03
  3000.0   3.248e-03 3.250e-03   5.03e-04     7.166e-03   7.197e-03   7.149e-03
\end{lstlisting}

\textbf{Maximum relative error on $n$ over 3000~\unit{\second}: 0.10\%}
(at $t = 1200$~\unit{\second}).  Maximum temperature error: 7~\unit{\milli\kelvin}.
The PJ approximation is excellent — the absolute errors are far below
any physical safety margin.

The PJ trajectory is essentially indistinguishable from full-state on
the heatup timescale (\cref{fig:validate-pj}).

\begin{figure}[h]
  \centering
  \includegraphics[width=0.95\linewidth]{validate_pj_heatup.png}
  \caption{Full-state (blue) vs.\ prompt-jump (red dashed) sims of the
  same heatup scenario.  Power $n$ (left) and $T_{\mathrm{avg}}$
  (right) overlay almost perfectly across 50~\unit{\minute}.  The
  difference is invisible at this scale — peak relative error on $n$
  is 0.1\%.  This is the empirical evidence that the singular-perturbation
  reduction is sound for this class of slow heatup transients.}
  \label{fig:validate-pj}
\end{figure}

\subsection*{Part 4: Nonlinear reach with the PJ model}

\apass{Results are populating as TMJets runs in the background.  Final
horizon and step counts will be filled in here once the longest probe
returns.  Initial expectation: 60~\unit{\second} should pass trivially;
1800~\unit{\second} is the real test; 5400 / 18000 the dream.}

The PJ reach script is \texttt{code/scripts/reach\_heatup\_pj.jl}.
Same Taylor-model machinery (\texttt{TMJets}, \texttt{@taylorize},
augmented time state) as the failed full-state version, but the RHS
operates on the 9-state PJ system (10D with augmented time) and
includes the rational $1/(\beta - \rho)$ in two places.  Probe
horizons: 60, 300, 1800, 5400~\unit{\second}.

\begin{decision}
TMJets settings: \texttt{orderT=4}, \texttt{orderQ=2}, \texttt{abstol=1e-9},
\texttt{maxsteps=100\,000}.  \texttt{abstol} is one order looser than
the full-state attempt — the PJ RHS has a rational nonlinearity that
narrows the Taylor remainder convergence radius slightly, and we don't
need 1e-10 precision for envelope tracking on a tube that's already
several Kelvin wide.
\end{decision}

\apass{Results section will be populated below as probes complete.
Currently TMJets is precompiling (~5--15 minutes for the new
\texttt{@taylorize}'d RHS).}

\subsection*{Part 5: App buildout}

While the reach is running, extended the Pluto predicate explorer
with three new sections:
\begin{itemize}
  \item \textbf{Live reach-result ingestion} (§9b): reads
        \texttt{reachability/reach\_operation\_result.mat} (saved by
        \texttt{reach\_operation.jl}) and renders per-halfspace margins
        live, replacing the hand-maintained traceability table.
  \item \textbf{2D projection chooser} (§9c): pick any two state
        coordinates from $\{n, T_f, T_c, T_{\mathrm{cold}}\}$ and see
        the operating polytope with the reach-tube envelope as a red
        rectangle overlay.
  \item \textbf{PJ heatup reach overlay} (§9d): if \texttt{reach\_heatup\_pj\_result.mat}
        exists, display the envelope summary.
\end{itemize}

Added \texttt{MAT.jl} to the app's \texttt{Project.toml}.  Read-only
v1 still — sliders preview UX without writing back.

\subsection*{Soundness ledger update}

\begin{decision}
The PJ reduction shifts the soundness story:

\textbf{Before:} linear reach was a sound over-approximation of the
linearized closed-loop, but the linearization was an unbounded
approximation of the nonlinear plant.  Net: \emph{approximate, not
sound} for the plant.

\textbf{After:} TMJets nonlinear reach with PJ is a sound
over-approximation of the \emph{prompt-jump-reduced} nonlinear plant.
The PJ reduction itself introduces a controlled approximation
(0.1\% error on $n$, mK on $T$, validated empirically over 50
minutes).  Net: \emph{$\epsilon_{\mathrm{PJ}}$-approximate but otherwise
sound}, where $\epsilon_{\mathrm{PJ}}$ is bounded.

This is qualitatively better.  The remaining gap (PJ approximation
error) can be characterized by the validation experiment, which we have.
The next step toward full soundness would be a Tikhonov-style
singular-perturbation theorem application giving a closed-form
$\mathcal{O}(\Lambda)$ error bound, but the empirical bound is
defensible for the prelim demo.
\end{decision}

\subsection*{Open at close}

\apass{This entry is being written in parallel with the running
reach.  Final results to be filled in below as TMJets returns.  If
TMJets completes the 5-hour horizon, the heatup reach-avoid obligation
is discharged (modulo PJ + saturation caveats).  If it stops earlier,
identify the new wall and propose the next reduction.}

\begin{itemize}
  \item Polytopic / SOS barriers — still the only path to a tight
        analytic certificate; quadratic Lyapunov is structurally
        defeated regardless of model order.
  \item Saturation as explicit hybrid sub-mode — still pending,
        independent of PJ.
  \item Parametric $\alpha$ uncertainty — still pending.
  \item Tikhonov / regular-perturbation $\mathcal{O}(\Lambda)$ error
        bound on PJ.
  \item Per-mode reach for shutdown and scram (now feasible with PJ).
\end{itemize}
