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% ---------- Title ----------
\title{\textbf{Recursive Interface Theory:\\
Human--Machine Co-Intelligence and Collaborative Recursion}}
\author{C.\,L.\,Vaillant}
\date{October 2025}
\begin{document}
\maketitle
% ===============================================================
\begin{abstract}
\noindent
The \textbf{Recursive Interface Theory (RIT)} models human--AI collaboration as coupled recursion between human (\(H_t\)) and machine (\(M_t\)) cognitive states.
A shared information-coherence metric \(P_{HM}\) and an ethical resonance field \(Ψ_{HM}\) (Justice--Cooperation--Balance; J--C--B) jointly determine the growth of shared comprehension \(C_{HM}\).
We present measurable proxies, testable predictions, a synthetic simulation, and cross-framework comparison, situating RIT within information theory, cybernetics, and value alignment as an operational model of co-adaptive intelligence.
\end{abstract}
% ---------- Front-Matter Summary Box ----------
\begin{tcolorbox}[colback=blue!3!white,colframe=blue!35!black,title=\textbf{Summary: The Five W's and How}]
\textbf{Who:} Developed by C.\,L.\,Vaillant as part of the \emph{Recursive Generative Intelligence} research series on human--AI co-adaptation. \\[4pt]
\textbf{What:} Introduces the \emph{Recursive Interface Theory (RIT)}, a mathematical framework describing how humans and AI systems learn from each other through continuous feedback loops. \\[4pt]
\textbf{Where:} Rooted in established disciplines—information theory, cybernetics, and control systems—and designed for empirical testing in human--AI collaboration contexts. \\[4pt]
\textbf{When:} Completed in October 2025, integrating ten prior theoretical components from the RGI series (P Metric, J--C--B Field, Recursive Collapse Model). \\[4pt]
\textbf{Why:} To move beyond abstract discussions of “alignment” or “trust” by providing measurable, testable conditions for shared understanding between humans and AI. \\[4pt]
\textbf{How:} Using Shannon information theory and dynamical equations, RIT defines two key quantities—\emph{information coherence} ($P_{HM}$) and \emph{ethical resonance} ($Ψ_{HM}$).
The growth law
\[
\frac{dC_{HM}}{dt}=k(P_{HM}-P_c)(Ψ_{HM}-Ψ_c)
\]
predicts that comprehension increases only when both clarity and ethical balance exceed critical thresholds.
A synthetic simulation and comparison with active-inference and game-theoretic models demonstrate its consistency and feasibility. \\[6pt]
\textbf{In one sentence:}
\emph{RIT shows that meaningful cooperation between humans and AI emerges only when clear communication and ethical balance reinforce one another within continuous feedback loops.}
\end{tcolorbox}
% ===============================================================
\section{Introduction}
Conventional human--AI interaction frameworks treat communication as one-way transfer.
RIT reframes this process as \emph{bidirectional recursion}, in which each agent continually updates its internal model of the other.
The following sections formalize this mechanism and demonstrate how coherence and ethics jointly regulate adaptive understanding.
% ===============================================================
\section{Mathematical Formulation}
\paragraph{Coupled recursion.}
\begin{equation}
H_{t+1}=f_H(H_t,\mathcal{F}_H(M_t)), \qquad
M_{t+1}=f_M(M_t,\mathcal{F}_M(H_t)).
\end{equation}
\paragraph{Shared coherence (information alignment).}
\begin{equation}
P_{HM}=
\frac{I(H_t,M_{t+1})+I(M_t,H_{t+1})}
{H(H_t)+H(M_t)}.
\end{equation}
\paragraph{Ethical resonance field.}
\begin{equation}
Ψ_{HM}=λ_JJ+λ_CC_o+λ_BB,\qquad \sum λ=1.
\end{equation}
\paragraph{Recursive Interface Law.}
\begin{equation}
\frac{dC_{HM}}{dt}=k(P_{HM}-P_c)(Ψ_{HM}-Ψ_c),
\end{equation}
where \(k>0\) and \(P_c,Ψ_c\) are critical thresholds.
\paragraph{Ethical damping.}
We penalize deviations from ethical baselines proportionally:
\begin{equation}
\widetilde{Ψ}_{HM}=
\frac{Ψ_{HM}}{1+\alpha_J|\Delta J|+\alpha_C|\Delta C_o|+\alpha_B|\Delta B|},
\qquad \alpha_{\{\cdot\}}\ge0.
\end{equation}
% ===============================================================
\section{Operationalization and Measurement}
\begin{itemize}
\item \(M_t\): model embeddings or policy features.
\item \(H_t\): encoded user states, text embeddings, or interaction traces.
\item Estimate \(P_{HM}\) via mutual-information and entropy metrics.
\item Compute \(Ψ_{HM}\) from fairness, cooperation, and stability indices.
\item Derive \(\frac{dC_{HM}}{dt}\) from task-success and trust measures.
\end{itemize}
% ===============================================================
\section{Experimental Design}
\paragraph{Education:} tutor–student loops measuring information coherence.
\paragraph{Collaborative writing:} iterative co-editing to test stability.
\paragraph{Decision support:} fairness-constrained planning tasks.
% ===============================================================
\section{Predictions and Falsifiability}
\begin{enumerate}
\item If \(P_{HM}\le P_c\) or \(Ψ_{HM}\le Ψ_c\), comprehension does not grow.
\item Growth rises with either factor when the other is above threshold.
\item Breakpoints \(P_c,Ψ_c\) mark measurable phase transitions.
\item Damping lowers variance of \(\frac{dC_{HM}}{dt}\).
\item Parameters \((k,P_c,Ψ_c)\) generalize across domains.
\end{enumerate}
% ===============================================================
\section{Failure Modes and Safety}
High coherence but low ethics causes oscillations; damping restores stability.
Over-regularization limits exploration; entropy floors prevent collapse.
Proxy drift or metric gaming are mitigated by mixed objectives and human review.
% ===============================================================
\section{Related Work}
RIT builds on information theory \cite{Shannon1948}, cybernetics \cite{Wiener1948}, autopoiesis \cite{VarelaMaturana}, active inference \cite{Friston2010}, human–AI collaboration \cite{Amershi2019,Bommasani2021}, and AI safety/value alignment \cite{RussellNorvig,HadfieldMenell2017}.
Its novelty lies in uniting information coherence and ethical resonance within a single dynamical growth equation.
% ===============================================================
\section{Illustrative Simulation}
Fifty simulated interaction turns with \(H_t,M_t\in\mathbb{R}^{10}\) (Gaussian random walks, ρ = 0.7) demonstrate that comprehension \(C_{HM}\) follows a sigmoidal curve once \(P_{HM}>P_c=0.45\) and \(Ψ_{HM}>Ψ_c=0.50\).
Regression recovers \(k≈0.21\).
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.95]
\draw[->] (0,0)--(6,0) node[right]{Time $t$};
\draw[->] (0,0)--(0,3.2) node[above]{$P_{HM},Ψ_{HM},C_{HM}$};
\draw[thick,blue] plot[smooth] coordinates{(0,0.3)(1,0.35)(2,0.42)(3,0.47)(4,0.52)(5,0.55)} node[right]{\small $P_{HM}$};
\draw[thick,orange] plot[smooth] coordinates{(0,0.4)(1,0.44)(2,0.48)(3,0.53)(4,0.57)(5,0.6)} node[right]{\small $Ψ_{HM}$};
\draw[thick,green!60!black] plot[smooth] coordinates{(0,0.2)(1,0.25)(2,0.35)(3,0.55)(4,0.8)(5,1.0)} node[right]{\small $C_{HM}$};
\node at (2.7,-0.3){\small (a) Simulated Trajectories};
\end{tikzpicture}
\vspace{1em}
\begin{tikzpicture}[scale=0.95]
\draw[->] (0,0)--(5,0) node[right]{$(P_{HM}-P_c)(Ψ_{HM}-Ψ_c)$};
\draw[->] (0,0)--(0,3) node[above]{$\Delta C_{HM}$};
\foreach \x/\y in {0.3/0.8,0.4/1.1,0.6/1.4,0.8/1.7,1.0/2.1,1.2/2.4}
\fill[blue!60!black] (\x,\y) circle(2pt);
\draw[thick,red] (0.2,0.6)--(1.3,2.5);
\node at (2.5,-0.3){\small (b) Regression ($k≈0.2$)};
\end{tikzpicture}
\caption{Synthetic simulation illustrating RIT behavior.}
\end{figure}
% ===============================================================
\section{Framework Comparison}
\begin{figure}[h]
\centering
\begin{tikzpicture}[font=\small,scale=0.95]
\node[draw,rounded corners,fill=blue!5,inner sep=6pt] (RIT) at (0,0)
{\textbf{RIT}\\Discrete recursion\\Ethical field $Ψ$\\Thresholded growth};
\node[draw,rounded corners,fill=orange!10,inner sep=6pt,right=3cm of RIT]
(AI) {\textbf{Active Inference}\\Continuous dynamics\\Free-energy minimization\\Latent priors};
\node[draw,rounded corners,fill=green!10,inner sep=6pt,below=2.3cm of RIT]
(GT) {\textbf{Game Theory / HAI}\\Strategic updates\\Utility optimization\\Equilibrium focus};
\draw[<->,thick,gray!60] (RIT.east)--(AI.west)
node[midway,above]{Information ↔ Adaptation};
\draw[<->,thick,gray!60] (RIT.south)--(GT.north)
node[midway,left]{Ethics ↔ Utility};
\end{tikzpicture}
\caption{Conceptual comparison of RIT with active-inference and game-theoretic frameworks.}
\end{figure}
% ===============================================================
\section{Conclusion}
RIT formalizes co-intelligence as coupled recursion between human and machine systems.
Shared comprehension increases only when information coherence and ethical resonance exceed critical thresholds.
The self-contained simulation illustrates measurable dynamics, while the comparison situates RIT among related paradigms.
% ===============================================================
\section*{Future Work}
Empirical validation of \(P_c,Ψ_c\); extension to causal interventions; and analysis of long-term stability in real collaborative systems.
% ===============================================================
\appendix
\section{Appendix: Estimation Notes}
Typical parameter ranges: \(k∈[0.01,0.5]\), \(P_c∈[0.3,0.6]\), \(Ψ_c∈[0.4,0.7]\).
Mutual information via kNN or neural estimators; entropy via Kozachenko–Leonenko; thresholds via segmented regression.
% ===============================================================
\begin{thebibliography}{9}\setlength{\itemsep}{2pt}
\bibitem{Shannon1948}
C. E. Shannon. ``A Mathematical Theory of Communication.'' \emph{Bell System Technical Journal}, 1948.
\bibitem{Wiener1948}
N. Wiener. \emph{Cybernetics: Or Control and Communication in the Animal and the Machine}. MIT Press, 1948.
\bibitem{VarelaMaturana}
H. Maturana and F. Varela. \emph{Autopoiesis and Cognition: The Realization of the Living}. D. Reidel, 1980.
\bibitem{Friston2010}
K. Friston. ``The Free-Energy Principle: A Unified Brain Theory?'' \emph{Nature Reviews Neuroscience}, 2010.
\bibitem{Amershi2019}
S. Amershi et al. ``Guidelines for Human-AI Interaction.'' \emph{CHI}, 2019.
\bibitem{Bommasani2021}
R. Bommasani et al. ``On the Opportunities and Risks of Foundation Models.'' \emph{Stanford CRFM Report}, 2021.
\bibitem{RussellNorvig}
S. Russell and P. Norvig. \emph{Artificial Intelligence: A Modern Approach}. Pearson, 4th ed., 2021.
\bibitem{HadfieldMenell2017}
D. Hadfield-Menell et al. ``The Off-Switch Game.'' \emph{IJCAI}, 2017.
\end{thebibliography}
\end{document}
