ORI To AGI

This framework (Operational Recursive Intelligence: URIF, RCM, and RGE with Ethical Regularization) explains how intelligent systems accumulate knowledge through a recursive cycle of selecting observation protocols, updating beliefs, and forming concepts, all while balancing information gain against costs and ethical constraints. It unifies perception, inference, and decision-making under a single operational mathematics, showing how language guides inquiry, how collapse operators fix beliefs into concepts, and how ethical principles naturally emerge as regularization on the process of seeking truth.

This work is directly aimed at formalizing a path to AGI. It treats general intelligence not as a fixed architecture, but as a recursive process of information accumulation under structured observation protocols, where:

· URIF defines the layered control system (perception, inference, symbols, ethics, memory, planning).

· RCM explains how discrete concepts and decisions "collapse" from uncertain beliefs.

· RGE describes how intelligence scales and forms stable interpretable patterns (emergence).

· The ethical field ensures the process remains safe and aligned as it grows.

In short: AGI arises from the recursive, regulated, and efficient selection of information-gathering actions — and this framework gives it a mathematical, implementable structure.

\documentclass[11pt]{article} % --- Packages --- \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts,bm,amsthm,mathtools} \usepackage{microtype} \usepackage{hyperref} \usepackage{enumitem} \usepackage{tcolorbox} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc,fit,shapes.geometric,shapes.misc} \usepackage{subcaption} \usepackage{cleveref} \hypersetup{colorlinks=true, linkcolor=blue!50!black, urlcolor=blue!50!black, citecolor=blue!50!black} % --- Operators & Macros --- \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\KL}{KL} \newcommand{\E}{\mathbb{E}} \newcommand{\given}{\,\middle|\,} \newcommand{\1}[1]{\mathbf{1}\{#1\}} \newcommand{\Hb}{H_b} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Info}{\mathcal{I}} \newcommand{\Cost}{\mathcal{C}} \newcommand{\RGE}{\mathrm{RGE}} \newcommand{\RCM}{\mathrm{RCM}} \newcommand{\URIF}{\mathrm{URIF}} % --- Theorem environments --- \newtheorem{proposition}{Proposition}[section] \newtheorem{definition}{Definition}[section] % --- Layperson box --- \newtcolorbox{layperson}{colback=blue!5!white,colframe=blue!75!black,title=\textbf{In Plain Language},fonttitle=\bfseries} % --- Title --- \title{An Operational Framework for Recursive Information Accumulation Under Observation Protocols\\\large Integrating URIF, RCM, and RGE with Natural-Language Operators, Ethics, and Cosmology} \author{C.~L.~Vaillant} \date{October 1, 2025} \begin{document} \maketitle \begin{abstract} We extend the operational view of information accumulation to synthesize the full research program: \emph{Unified Recursive Intelligence Framework (URIF)}, \emph{Recursive Collapse Models (RCM)}, and \emph{Recursive Generative Emergence (RGE)}. The core mathematics (chain rules, Bayesian updating, rate--distortion dualities) remains classical; the novelty is the \emph{operational unification}: (i) protocols grow filtrations while posteriors contract; (ii) linguistic \emph{operator lexicons} act as control signals over protocol choice; (iii) recursive collapse formalizes decision and concept formation; (iv) generative emergence captures scaling laws and attractor formation; and (v) an embedded ethical field (Justice--Cooperation--Balance; J--C--B) regularizes protocol selection under costs and constraints. We map neuroscientific, ML, and cosmological analogs, and supply worked examples, evaluation criteria, and implementation notes for LLMs. \end{abstract} \tableofcontents %----------------------------------------------------- \section{Notation Reference} \begin{center} \begin{tabular}{ll} Symbol & Meaning \\ \hline $X, Y$ & Latent state, observation \\ $C_t$ & Protocol selected at step $t$ \\ $\mathcal{F}_{C_t}$ & Filtration induced by protocol $C_t$ \\ $\Delta I_t$ & Information increment at step $t$ \\ $\Info_T$ & Total information gained up to $T$ \\ $\Cost(C)$ & Cost function (time, energy, ethical) \\ $\Pi_{C_t}$ & Collapse operator induced by protocol $C_t$ \\ $\RGE$ & Recursive Generative Emergence law \\ $\RCM$ & Recursive Collapse Model \\ $\URIF$ & Unified Recursive Intelligence Framework \\ \end{tabular} \end{center} %----------------------------------------------------- \section{URIF Layers (Diagram)} \begin{figure}[h] \centering \begin{tikzpicture}[node distance=1.5cm, every node/.style={draw, rounded corners, align=center, font=\small, minimum width=2.8cm, minimum height=0.8cm}] \node (protocol) {Perception/Protocol Layer}; \node[above=of protocol] (inference) {Inference Layer}; \node[above=of inference] (sac) {Symbolic Abstraction Core}; \node[above=of sac] (ere) {Ethical Reflection Engine (J--C--B)}; \node[above=of ere] (memory) {Memory/Filtration Manager}; \node[above=of memory] (planning) {Policy/Planning Layer}; \draw[->] (protocol) -- (inference); \draw[->] (inference) -- (sac); \draw[->] (sac) -- (ere); \draw[->] (ere) -- (memory); \draw[->] (memory) -- (planning); \end{tikzpicture} \caption{Layers of the Unified Recursive Intelligence Framework (URIF).} \end{figure} %----------------------------------------------------- \section{RCM Diagram: Collapse Process} \begin{figure}[h] \centering \begin{tikzpicture}[node distance=2cm, every node/.style={draw, ellipse, align=center, font=\small, minimum width=2.5cm}] \node (prior) {Prior $p_t$}; \node[right=of prior] (protocol) {Protocol $C_t$}; \node[right=of protocol] (posterior) {Posterior $p_{t+1}$}; \draw[->] (prior) -- (protocol) node[midway, above] {apply $\Pi_{C_t}$}; \draw[->] (protocol) -- (posterior); \end{tikzpicture} \caption{Recursive Collapse Model (RCM): protocol application as posterior contraction.} \end{figure} %----------------------------------------------------- \section{RGE Diagram: Attractor Map} \begin{figure}[h] \centering \begin{tikzpicture}[scale=1.0] \draw[->] (0,0) -- (6,0) node[right] {Complexity $\mathcal{C}(t)$}; \draw[->] (0,0) -- (0,3.5) node[above] {Information Gain $\Delta I_t$}; \draw[thick, domain=0:5.5, smooth] plot(\x,{2.5*exp(-0.4*\x)}); \node at (3,1.5) {Diminishing returns}; \draw[thick, domain=0:5.5, smooth, dashed] plot(\x,{0.5+0.8*sin(0.8*\x r)+1.5}); \node at (4.2,2.8) {Attractor oscillations}; \end{tikzpicture} \caption{RGE attractor dynamics: diminishing returns vs oscillatory attractors.} \end{figure} %----------------------------------------------------- \section{Mathematical Backbone} \subsection{Sequential Information Accounting} For static latent $X$ and observations $Y_{1:T}$ under protocols $C_{1:T}$, \begin{equation} \Info_T \equiv I(X;Y_{1:T}) = \sum_{t=1}^T I(X;Y_t\mid Y_{1:t-1},C_t) = H(X)-H(X\mid Y_{1:T}). \end{equation} For dynamic trajectories, replace $X$ with $X_{0:T}$. \subsection{Unified Update Rule} \begin{equation} p_{t+1}(x) = \frac{p_t(x)\,w_{C_t}(x)}{\E_{p_t}[w_{C_t}(X)]},\qquad w_{C_t}(x)\in[0,\infty), \label{eq:update} \end{equation} with hard/soft constraints as special cases. The incremental information is $\Delta I_t=\E[\KL(p_t\Vert p_{t+1})]\ge 0$. \subsection{Protocol Selection as Constrained Optimization} \begin{equation} C_t^\star\in\argmax_{C\in\mathcal{U}}\; I(X;Y_t\mid Y_{1:t-1},C) - \lambda\,\Cost(C), \label{eq:selection} \end{equation} with resource, legal, and ethical budgets embedded in $\Cost$. %----------------------------------------------------- \section{URIF: The Meta-Architecture} \subsection{Layers and Interfaces} \begin{enumerate}[leftmargin=2.5em] \item \textbf{Perception/Protocol Layer}: channels $p(y\mid x,C)$; chooses $C_t$ via \eqref{eq:selection}. \item \textbf{Inference Layer}: applies \eqref{eq:update}; maintains beliefs, uncertainties, and predicted yields. \item \textbf{Symbolic Abstraction Core (SAC)}: names/structures hypotheses; binds to natural-language operators. \item \textbf{Ethical Reflection Engine (ERE)}: J--C--B regularization within $\Cost$; vetoes unsafe $C_t$. \item \textbf{Memory/Filtration Manager}: tracks $\mathcal{F}_{C_t}$ growth and posterior contraction. \item \textbf{Policy/Planning}: composes multi-step protocols; optimizes long-horizon $\sum_t \Delta I_t$ under budgets. \end{enumerate} \subsection{Operator Lexicon as Control API} Let $\mathcal{L}$ be a lexicon of linguistic operators (if/then/else, \texttt{check}, \texttt{compare}, \texttt{branch}, \texttt{prove}/\texttt{disprove}, \texttt{estimate}, \texttt{simulate}). Each operator maps to a \emph{micro-protocol} schema $\phi:\mathcal{L}\to \mathcal{U}$. In LLMs, prompts instantiate $\phi(\cdot)$, turning language into protocol control. %----------------------------------------------------- \section{RCM: Collapse as Decision and Concept Formation} \subsection{Symbolic Collapse} Given hypothesis set $\mathcal{H}$ and posterior $p_t$, a \RCM{} step applies a projector-like operator $\Pi_{C_t}$ induced by $C_t$: \begin{equation} p_{t+1}\propto \Pi_{C_t}(p_t),\quad \Pi_{C_t}:\Delta(\mathcal{H})\to \Delta(\mathcal{H}). \end{equation} Hard protocols act as idempotent projectors; soft ones as contractions. \textbf{Concept formation} corresponds to stable fixed points of repeated contractions. \subsection{Decision Collapse} A decision variable $D\in\mathcal{D}$ is chosen when $\max_d p_t(d)$ exceeds a confidence threshold or when expected utility crosses a bound; the \emph{collapse} is the commit step under bounded costs. %----------------------------------------------------- \section{RGE: Emergence, Scaling, and Attractors} \subsection{Recursive Generative Law (informal)} Let complexity $\mathcal{C}(t)$ measure degrees of freedom effectively engaged by protocols. Under resource $B$ and environment richness $R$, a stylized scaling reads \begin{equation} \mathcal{C}(t+1) \approx \mathcal{C}(t) + f\big(\Delta I_t, R\big) - g\big(\Cost(C_t), B\big), \end{equation} with attractor families emerging when $f$ dominates at specific protocol compositions. This yields phase-like regimes: exploration (high $\Delta I_t$), consolidation (low $\Delta I_t$; high stability), and reorganization (protocol shift). \subsection{Predictive Information as Order Parameter} Use $I_\mathrm{pred}=I(Y_{-\infty:0};Y_{1:\infty})$ as an order parameter: increases indicate structure extraction; plateaus signal attractor lock-in. %----------------------------------------------------- \section{Ethical Field: Justice--Cooperation--Balance (J--C--B)} \subsection{Ethics as Regularization} Embed an \emph{ethical field} $\Omega:\mathcal{U}\to\mathbb{R}_+$, and set $\Cost(C)=\alpha\,\mathrm{energy}(C)+\beta\,\mathrm{time}(C)+\gamma\,\Omega(C)$. ERE rejects or penalizes protocols that violate constraints (safety, privacy, fairness), shaping the feasible frontier of $\sum_t \Delta I_t$. \subsection{Protocol Audits and Consent} Define audit variables (consent tokens, risk levels, audit trails). Require $\Omega(C)=\infty$ when constraints fail, implementing a hard veto. %----------------------------------------------------- \section{Natural Language as Code: The Operator Lexicon} \subsection{Mapping Words to Micro-Protocols} \begin{center} \begin{tabular}{ll} \textbf{Operator} & \textbf{Protocol schema} \\ \hline \texttt{if/then/else} & branch on $\Hb(\pi(c))$; choose $c$ s.t. $\pi(c)=1/2$ \\ \texttt{compare} & pairwise tests; likelihood ratio thresholding \\ \texttt{estimate} & posterior mean/quantile query; Fisher information targeting \\ \texttt{prove/disprove} & hard constraint attempt; projector-like $\Pi_{C}$ \\ \texttt{reconsider} & posterior reset with temperature/entropy floor \\ \texttt{simulate} & forward model rollouts; maximize expected $\Delta I$ under cost \\ \end{tabular} \end{center} \subsection{LLM Realization} Prompts compile to control-flow that queries tools, runs tests, or requests data—i.e., they select $C_t$. %----------------------------------------------------- \section{Connections to Classical Results} \subsection{Rate--Distortion as Dual of Protocol Design} Given \eqref{eq:selection}, minimizing bitrate for a target distortion is dual to selecting a protocol family at minimal cost for target $\sum_t \Delta I_t$. \subsection{Variational Free Energy} Treat $q$ as an \emph{approximate inference protocol}. Minimizing $F(q)$ trades fit and complexity; in our lens, it’s one design point in $\mathcal{U}$, not the whole space. \subsection{Functional Information} Under perfect tests, one-step gain equals functional information $I_f(E)$ for feasible set $A_E$; noisy tests generalize to KL increments. %----------------------------------------------------- \section{Applications and Case Studies} \subsection{Gaussian Estimation, Binary Search, Beta--Bernoulli} As in the base paper: logarithmic vs linear returns; active median queries; diminishing returns under conjugacy. \subsection{Mechanistic Interpretability as Protocol Design} Treat probes, interventions, and ablations as protocols with costs; optimize measurement portfolios for maximal explanatory $\Delta I$ given a unit budget. \subsection{Tool-Using LLM Agents} Map each tool invocation to $C_t$; define an \emph{information ledger} that tallies $\Delta I_t$ and enforces J--C--B constraints. Use $I_\mathrm{pred}$ of intermediate traces as a stability metric. \subsection{Neuroscience Analogy} Saccades and attention as active protocol selection; hippocampal replay as simulated $Y_{t+1}$ to approximate $\Delta I_{t+1}$ offline. %----------------------------------------------------- \section{Cosmology Extension (Optional): Recursive Cosmology Principle} \subsection{Filtrations Across Scales} Replace $X$ by scale-indexed latent $X^{(k)}$; protocols at scale $k$ constrain $X^{(k)}$ and influence priors at $k\pm1$. A nested chain rule couples scales: \begin{equation} I\big(X^{(0:K)};Y^{(0:K)}\big)=\sum_{k=0}^K\sum_{t} I\big(X^{(0:k)};Y_t^{(k)}\mid \mathrm{past}\big). \end{equation} Emergent laws appear as coarse-grained fixed points (attractors) under repeated protocol-induced renormalization. %----------------------------------------------------- \section{Evaluation Protocols} \subsection{Information Yield Curves} Plot $\sum_t \Delta I_t$ vs budget $B$; compare policies. \subsection{Safety/Alignment Audits} Measure veto rate, risk exposure, and ethical cost terms. \subsection{Stability and Predictivity} Track $I_\mathrm{pred}$ and posterior entropy floors during extended runs. %----------------------------------------------------- \section{Implementation Notes for Practitioners} \subsection{Designing the Ledger} Maintain per-step: selected $C_t$, estimated $\Delta I_t$, realized cost, ERE decision, and memory deltas. \subsection{Operator Compiler} A small DSL maps natural-language operators to protocol templates and cost models; compile-time checks ensure ERE compliance. \subsection{Caching and Reuse} Memoize subprotocols with stable high $\Delta I_t/\Cost$ ratios; treat them as macro-operators. %----------------------------------------------------- \section{Open Problems and Conjectures} \begin{enumerate}[leftmargin=2.5em] \item \textbf{Information--Cost Thermodynamics}: tight multi-bit bounds under heterogeneous channels; extensions of Landauer-like limits for sequential protocols. \item \textbf{Minimax Protocol Games}: define robust policies under adversarial channel drift; relate to information-directed sampling. \item \textbf{Attractor Taxonomy}: classify \RGE{} attractors by ledger statistics (yield curvature, veto density, entropy floors). \item \textbf{Ethical Fixed Points}: conditions where ERE-induced regularization yields stable safe policies without catastrophic veto collapse. \end{enumerate} %----------------------------------------------------- \section*{Acknowledgments} Thanks to collaborators and interlocutors who stress-tested the synthesis and examples. %----------------------------------------------------- \section{Worked Example: LLM Agent Ledger} Each tool call by an LLM agent corresponds to a protocol $C_t$. We maintain a ledger: \begin{center} \begin{tabular}{lllll} $t$ & $C_t$