Interface Theory:
Mechanistically speaking, the Recursive Interface Theory describes human and AI interaction as two coupled learning systems whose internal representations evolve together through feedback. Each updates its parameters based on the other’s outputs, like linked gradient flows, while information coherence measures how aligned their feature spaces become. Ethical resonance functions as a stabilizing constraint, similar to regularization in optimization, ensuring that shared growth remains balanced and value-consistent. The main growth law expresses comprehension as increasing only when both informational clarity and ethical balance exceed threshold values, and damping prevents instability or overfitting. In essence, RIT models alignment as a co-training process where interpretability and ethical stability emerge from the continuous synchronization of human and machine representations.
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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 $\Psi_{HM}$ (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} ($\Psi_{HM}$).
The growth law
\[
\frac{dC_{HM}}{dt}=k\big(P_{HM}-P_c\big)\big(\Psi_{HM}-\Psi_c\big)
\]
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\big(H_t,\mathcal{F}_H(M_t)\big), \qquad
M_{t+1}=f_M\big(M_t,\mathcal{F}_M(H_t)\big).
\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}
\Psi_{HM}=\lambda_J J+\lambda_C C_o+\lambda_B B, \qquad \lambda_J+\lambda_C+\lambda_B=1.
\end{equation}
\paragraph{Recursive Interface Law.}
\begin{equation}
\frac{dC_{HM}}{dt}=k\big(P_{HM}-P_c\big)\big(\Psi_{HM}-\Psi_c\big),
\end{equation}
where $k>0$ and $P_c,\Psi_c$ are critical thresholds.
\paragraph{Ethical damping.}
We penalize deviations from ethical baselines proportionally:
\begin{equation}
\widetilde{\Psi}_{HM}=
\frac{\Psi_{HM}}{1+\alpha_J|\Delta J|+\alpha_C|\Delta C_o|+\alpha_B|\Delta B|},
\qquad \alpha_J,\alpha_C,\alpha_B\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 $\Psi_{HM}$ from fairness, cooperation, and stability indices.
\item Derive ${dC_{HM}}/{dt}$ 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 $\Psi_{HM}\le \Psi_c$, comprehension does not grow.
\item Growth rises with either factor when the other is above threshold.
\item Breakpoints $P_c,\Psi_c$ mark measurable phase transitions.
\item Damping lowers variance of ${dC_{HM}}/{dt}$.
\item Parameters $(k,P_c,\Psi_c)$ 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, $\rho=0.7$) demonstrate that comprehension $C_{HM}$ follows a sigmoidal curve once $P_{HM}>P_c=0.45$ and $\Psi_{HM}>\Psi_c=0.50$.
Regression recovers $k\approx0.21$.
\begin{figure}[h]
\centering
% --- (a) Sim trajectories ---
\begin{tikzpicture}[scale=0.95]
\draw[->] (0,0)--(6,0) node[right]{Time $t$};
\draw[->] (0,0)--(0,3.2) node[above]{$P_{HM},\ \Psi_{HM},\ C_{HM}$};
\draw[thick,blue] plot[smooth] coordinates{(0,0.30)(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{(0,0.40)(1,0.44)(2,0.48)(3,0.53)(4,0.57)(5,0.60)} node[right]{\small $\Psi_{HM}$};
\draw[thick,green!60!black] plot[smooth] coordinates{(0,0.20)(1,0.25)(2,0.35)(3,0.55)(4,0.80)(5,1.00)} node[right]{\small $C_{HM}$};
\node at (2.7,-0.3){\small (a) Simulated Trajectories};
\end{tikzpicture}
\vspace{1em}
% --- (b) Regression ---
\begin{tikzpicture}[scale=0.95]
\draw[->] (0,0)--(5,0) node[right]{$(P_{HM}-P_c)(\Psi_{HM}-\Psi_c)$};
\draw[->] (0,0)--(0,3) node[above]{$\Delta C_{HM}$};
\foreach \x/\y in {0.30/0.80,0.40/1.10,0.60/1.40,0.80/1.70,1.00/2.10,1.20/2.40}
\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 ($k\approx0.2$)};
\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 $\Psi$\\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 $\leftrightarrow$ Adaptation};
\draw[<->,thick,gray!60] (RIT.south)--(GT.north)
node[midway,left]{Ethics $\leftrightarrow$ 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 $P_c,\Psi_c$; extension to causal interventions; and analysis of long-term stability in real collaborative systems.
% ===============================================================
\appendix
\section{Appendix: Estimation Notes}
Typical parameter ranges: $k\in[0.01,0.5]$, $P_c\in[0.3,0.6]$, $\Psi_c\in[0.4,0.7]$.
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}
