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\title{The Stability Boundary of Recursive Cognition:\\
A Control-Theoretic Program for Education, Culture, and Governance\\[0.35em]
\ Final Integrated Edition}
\author{C.\,L. Vaillant\\[2pt]
\small RGIM / OnToLogic Project \textbar{} Toronto, Canada\\
\small \texttt{author.email@domain.example}}
\date{October 05 2025}
\begin{document}
\maketitle
%========================================================
\begin{abstract}
This work presents a unified control-theoretic framework for modeling recursive cognition across education, culture, and governance. Individual and collective competencies are formalized as evolving state vectors regulated by systemic and personal feedback inputs. Using Lyapunov stability analysis, the model defines conditions under which cognitive and institutional systems maintain coherence while permitting controlled exploration through mechanisms such as hysteresis and thresholds of doubt. Robustness is established via explicit Lipschitz conditions, stochastic-process priors, and hierarchical measurement-invariance procedures. The framework is operationalized through an agent-based model (ABM) and a planned cluster randomized controlled trial (RCT) in education to empirically test superlinear competence propagation and adaptive stability. Ethical pluralism is integrated through a normative-authority design that balances justice, cooperation, and balance within bounded disagreement. Together, these elements form a tractable yet philosophically grounded approach to designing adaptive, legitimate, and ethically aligned social systems.
\end{abstract}
%========================================================
\section*{Public Summary}
This project explores how societies can stay stable while still learning, questioning, and evolving. It treats education, governance, and culture as living control systems—networks that constantly adjust themselves in response to feedback. The model describes how “common sense’’ and “critical thinking’’ interact dynamically: too little questioning leads to stagnation, while too much doubt causes chaos. The key idea is a mathematical “stability boundary’’ that lets systems explore new ideas without collapsing.
To make this real, the framework defines how to measure skills such as reasoning and emotional regulation, and how these skills spread through teaching and mentoring. It uses advanced statistical tools to track how change happens over time and agent-based computer simulations to see how learning spreads across social networks. A planned education trial will test the predictions in real classrooms.
The model also takes ethics seriously. It proposes ways for plural societies to manage deep value differences through fair deliberation, shared core principles, and transparent decision-making. In essence, this research is about building smarter, fairer systems—whether in schools or governments—that can adapt without losing their balance.
%========================================================
\section{Assumptions and Lipschitz Conditions}
$f\!:\R^n\!\times\!\R^m\!\times\!\Lambda\!\to\!\R^n$ satisfies
$\|f(x,I,\lambda)-f(y,I,\lambda)\|\le L_f\|x-y\|$ with $L_f<1$ and $\|\nabla_{\!\lambda} f(x,I,\lambda)\|\le L_\lambda$.
%========================================================
\section{Framework Recap}
$S_{t+1}=f(S_t,I_t,\lambda)$ with
$V(S,I,\lambda)=\mathcal{H}(S)+\alpha\|\nabla_{\!\lambda}f(S,I,\lambda)\|^2$.
Common sense minimizes $\mathcal{H}$; critical thinking is $\mathcal{C}=\nabla_{\!\lambda}f$.
%========================================================
\section{Threshold of Doubt and Stability}
Trigger when $\Delta_t=\|I_t-\hat I_t\|>\varepsilon_\lambda$, set $\eta_t=\beta \Delta_t/\varepsilon_\lambda$ and update
$S_{t+1}=f(S_t,I_t,\lambda)+\eta_t\,\mathcal{C}(S_t,I_t,\lambda)$.
If $\Delta V\le0$ and $f$ is Lipschitz, $S_t$ converges to an attractor in $\{\Delta V=0\}$.
%========================================================
\section{Education as a Control System}
$X_t=[\text{crit.think.},\text{common sense},\text{emo.reg.},\text{interpers.},\text{self-manage.}]^\top$,
\begin{equation}
X_{t+1}=A_tX_t+B_tU_t+\xi_t,\quad U_t=U_{\mathrm{sys}}+U_{\mathrm{pers}}.
\label{eq:edu}
\end{equation}
If $\rho(J_F(X^*))<1$ and $\sum_t\|\xi_t\|^2<\infty$, then $X_t\to X^*$.
%========================================================
\section{Discounted Transition Cost and Policy Saturation}
$C_{\mathrm{transition}}=\int_0^\infty e^{-\rho t}(\|A_t-A^*\|_F^2+\|B_t-B^*\|_F^2+\|U_t-U^*\|^2)dt$.
PID control: $U_{\mathrm{policy}}=K_p e+K_i\!\int e\,dt+K_d\dot e$, $|U_{\mathrm{policy}}|\le U_{\max}$.
%========================================================
\section{Measurement and Observability with Invariance}
\textbf{Workflow:} configural $\rightarrow$ metric $\rightarrow$ scalar invariance \cite{millsap2011}, accept if $\Delta$CFI ≤ 0.01. Partial invariance permitted.
Hierarchical priors: $\Lambda^{(q)}\!\sim\!\mathcal{N}(\bar\Lambda,\Sigma_\Lambda)$.
Ethical field $\mathbf{E}=(J,C,B)$ normalized to [0,1]; $\lambda=g(J,C,B)$ fitted via monotone splines.
Estimate $\nabla_\lambda f$ through DiD or IV comparisons.
%========================================================
\section{Identifiability and Priors for $A_t,B_t$}
Random-walk dynamics: $A_t=A_{t-1}+\epsilon_{A,t}$, $B_t=B_{t-1}+\epsilon_{B,t}$.
$\epsilon_{A,t}\!\sim\!\mathcal{N}(0,Q_A)$, $Q_A\!\sim\!\mathcal{IW}(\nu_0,\Psi_0)$ with $\nu_0=n+2$; analogous for $B_t$.
Tune by EM and forecast error \cite{durbin2001}; instrument $U_t$ via RD/IV.
%========================================================
\section{Endogenous Hybrid Switching with Hysteresis}
Regimes $S_{t+1}=f_{\sigma(t)}(S_t,I_t,\lambda)$, $\sigma(t)\!\in\!\{1,\dots,M\}$.
\[
\Pr(\sigma\!\ne\!\sigma_{t-1})=\mathrm{logit}^{-1}(\theta_0+\theta_1\Delta_t+\theta_2\dot V_t-\theta_3h_t),
\quad h_t=\max(0,h_{t-1}-\delta)+\mathbf{1}(\text{stress}_t>\tau).
\]
%========================================================
\section{Superlinear Propagation Mechanisms}
(a) Density-dependent $K_t=K_0\sigma_d(d_t)$, $\sigma_d(d)=d/(1+d)$.
(b) Peer interaction: $X_{t+1}=A_tX_t+B_tU_t+\gamma X_t\odot(\bar X_t-X_t)+\xi_t$.
(c) Aggregate logistic: $\tfrac{dM}{dt}=\beta M(1-M/M_{\max})-\mu M$.
%========================================================
\section{Scheduling Exploration $\alpha(t)$}
Annealed $\alpha(t)=\alpha_0e^{-t/\tau}$; Episodic (low vs high during crisis $\Delta_t>\varepsilon_{\text{crisis}}$); Adaptive $\alpha(t)=\alpha_0+\beta\,\text{sigmoid}(\mathcal{H}_t-H_{\text{target}})$.
%========================================================
\section{Ethical Pluralism and Authority Design}
Small-gain stability for $\gamma<1$ \cite{vidyasagar1993}.
Meta-norms: peaceful contestation, due process.
Authority design \cite{rawls1993}: shared core + parallel systems, supermajority rules, independent review board with appeal.
%========================================================
\section{Cross-Domain Coupling, Lags \& Saturation}
\[
U^{edu}_t=U^{base}_t+\gamma_{med}\tanh(\gamma Q^{info}_{t-\ell_1})+\gamma_{eco}\tanh(\gamma Budg_{t-\ell_2})+\eta_t^{edu}.
\]
Lag orders $\ell_k$ via AIC/BIC and Granger tests; asymmetric $\tanh$ for loss aversion optional.
%========================================================
\section{Goodhart/Campbell-Resilient Measurement}
Combine multi-indicators, audits, lagged incentives, and entropy band $[H_{\min},H_{\max}]$ with time-varying $\alpha(t)$ \cite{goodhart1975,campbell1976}.
%========================================================
\section{Falsifiable Predictions}
(1) Recovery time $\downarrow$ with Balance $B$.
(2) If $\rho(K)>1$, mentoring growth convex.
(3) Media over-stimulation $\uparrow \mathcal{H}_{social}$ $\Rightarrow$ longer $T_{rec}$.
(4) Anti-windup reduces overshoot vs controls.
%========================================================
\section{Education Trial Implementation}
Cluster RCT (2×2 factors: systemic PD/curriculum × personal self-management).
Power $d=0.2$–0.3, 40–60 clusters, 3–4 waves.
Measures baseline→follow-up; analysis via DiD, IV/RD, state-space priors. IRB approval and fairness audits.
%========================================================
\section{Agent-Based Model Specification and Calibration}
Agents $a=1…N$ with $X^a_t\in\R^n$ on small-world/scale-free network (exponent 2–3 \cite{barabasi1999}).
\[
X^a_{t+1}=A_{loc}X^a_t+\frac{\lambda_p}{|\mathcal{N}(a)|}\sum_{b\in\mathcal{N}(a)}\phi(X^b_t-X^a_t)+B_{loc}U^a_t+\zeta^a_t,
\]
$\phi(\Delta)=\Delta \mathbf{1}(\|\Delta\|\le\tau)$.
Mentoring prob $p_{ab}\propto e^{\kappa \text{sim}(X^a,X^b)}$.
Calibration targets: PISA SD ≈ 100, annual gain 10–15, teacher attrition 15–20 \%, network exponent 2–3.
Falsify if $\rho(K)>1$ yet decay, or anti-windup benefits absent.
%========================================================
\section{Related Work and Positioning}
Cybernetics \cite{wiener1948,ashby1956,forrester1968,ogata2010,khalil2002}, state-space \cite{durbin2001}, robust/MPC \cite{policyRobust,lqdiscount}, switched systems \cite{hespanha2004}, small-gain \cite{vidyasagar1993}, education \cite{cbebook}, AI alignment \cite{amodei2016,hadfield-menell2017,russell2019,alignmentOverview}, invariance \cite{millsap2011}, Goodhart/Campbell \cite{goodhart1975,campbell1976}, networks \cite{barabasi1999}.
Position: (A) normative design, (B) positive predictions, (C) heuristic interpretation.
%========================================================
\section{Normative Foundations and Political Economy}
$X^*$ and $w_k$ via deliberative processes (citizens’ juries, participatory budgeting, sunset clauses). Public reason \cite{rawls1993}.
Parallel systems with shared core, exit rights, minority protections, independent adjudication. Legitimacy tests: legality, transparency, proportionality, redress.
%========================================================
\appendix
\section{Parameter Dictionary}
\begin{longtable}{lllll}
\toprule
Symbol & Meaning & Domain & Typical Range & Estimation Method \\
\midrule
$L_f$ & Lipschitz constant & $\R^+$ & 0.7–0.95 & Theory/fit \\
$\alpha$ & Lyapunov weight & $\R^+$ & 0.1–10 & Pareto/values \\
$\beta$ & Doubt gain & $\R^+$ & 0.1–5 & Calibrated \\
$\rho$ & Discount rate & $\R^+$ & 0.01–0.10 & Policy norm \\
$Q_A,Q_B$ & Process covariance & PSD & small & IW prior/EM \\
$\mu_g,\kappa_g$ & Gov. gain/damping & $\R^+$ & context & VARX \\
$\ell_k$ & Coupling lags & $\mathbb{N}$ & 1–4 yrs & AIC/BIC \\
$\gamma,\delta$ & Saturation gains & $\R^+$ & 0.5–5 & Fit \\
$\alpha(t)$ & Schedule & $\R^+$ & varies & Rule \\
$K$ & Next-gen matrix & $\R^{m\times m}$ & $\rho(K)>1$ & Survey/fit \\
\bottomrule
\end{longtable}
\section{Historical Case-Study Protocol}
1. Choose window (e.g., Finland 1985–2015).
2. Identify reforms (shocks).
3. Estimate $(A_t,B_t)$ via state-space/ITS.
4. Generate \emph{ex ante} predictions for held-out years.
5. Compare with data using Bayes factors.
\section{ABM Implementation Plan}
Phase 1 (6 mo): Implement in Mesa/NetLogo \cite{wilensky1999}, match calibration targets.
Phase 2 (12 mo): Validate $X_t$ battery via multi-group CFA.
Phase 3 (6–12 mo): Partner identification and IRB.
Phase 4 (36–48 mo): RCT execution.
Phase 5 (ongoing): Model fit and publication.
%========================================================
\bibliographystyle{IEEEtran}
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\end{document}
