ORI To AGI

Mechanistically, this framework casts general intelligence as a closed-loop control process over a model’s internal representations, where “protocols” are concrete interventions on the computation that select observations, route attention, and write to memory, while the posterior update is a normalized reweighting of latent hypotheses measured by KL increments. The URIF layers are just interfaces on that loop: perception chooses query operators, inference performs the weight update, symbols bind stable features into discrete codes, memory maintains the filtration, planning composes multi-step interventions, and the ethical field acts as an explicit regularizer that constrains which interventions are admissible. The RCM “collapse” is a projection step that commits soft beliefs into discrete decisions or concept codes, detectable as rank and sparsity changes in representation space; RGE describes when those codes become attractor basins with sufficient curvature to resist noise and compose hierarchically. In interpretability terms, everything is instrumentable: protocols are nodes in the causal graph, updates are gradients and logits shifting along identified features, information gain is mutual information or KL on tractable partitions, stability is local Hessian/Lyapunov structure, and alignment pressure is a penalty that reshapes the energy landscape. The central prediction is operational: optimize protocol choice for expected information gain under cost and ethical constraints, and you get scaling, compositional concepts, and safe exploration; remove the ethical regularizer or the projection discipline and you observe drift, brittle collapse, or adversarial attractors.

% !TeX program = pdflatex \documentclass[11pt]{article} % ---------- Encoding & Fonts ---------- \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} % ---------- Page & Typography ---------- \usepackage[a4paper,margin=1in]{geometry} \usepackage{microtype} \usepackage{setspace} \setstretch{1.05} % ---------- Math & Symbols ---------- \usepackage{amsmath,amssymb,amsfonts,bm,amsthm,mathtools} \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] % ---------- Graphics & Figures ---------- \usepackage{graphicx} \usepackage{subcaption} % ---------- Tables ---------- \usepackage{booktabs} \usepackage{array} % ---------- Lists ---------- \usepackage{enumitem} \setlist{nosep} % ---------- TikZ ---------- \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc,fit,shapes.geometric,shapes.misc} % ---------- Colors / Links ---------- \usepackage{xcolor} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue!60!black, urlcolor=blue!60!black, citecolor=blue!60!black } % ---------- Clever References (load after hyperref) ---------- \usepackage{cleveref} % ---------- Tcolorbox ---------- \usepackage{tcolorbox} \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} \noindent 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, 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 emerge as regularization on the process of seeking truth. The work treats general intelligence as a recursive process of information accumulation under structured protocols: \URIF{} defines layered control (perception, inference, symbols, ethics, memory, planning), \RCM{} explains how discrete concepts and decisions collapse from uncertainty, \RGE{} describes how intelligence scales and forms stable interpretable patterns, and an ethical field ensures growth remains safe and aligned. In short, AGI arises from the recursive, regulated, and efficient selection of information-gathering actions---and this framework gives it a mathematical, implementable structure. \end{abstract} \begin{layperson} \textbf{What this does:} It gives a practical math recipe for how an intelligent system should ask questions, update its beliefs, name new concepts, and choose safe actions while balancing information, cost, and ethics.\\ \textbf{Why it matters:} It turns AGI from a vague idea into an operational loop you can measure, audit, and implement. \end{layperson} \tableofcontents \section*{Disclaimer} \emph{This paper is presented as a draft in LaTeX Math Assembly Language. A PDF will be made available when the work is double checked and finalized.} % ----------------------------------------------------- \section{Notation Reference} \begin{center} \begin{tabular}{ll} \toprule Symbol & Meaning \\ \midrule $X, Y$ & 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 \\ \bottomrule \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=3.2cm, minimum height=0.85cm}] \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=2.1cm, every node/.style={draw, ellipse, align=center, font=\small, minimum width=2.6cm}] \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) -- (6.2,0) node[right] {Complexity $\mathcal{C}(t)$}; \draw[->] (0,0) -- (0,3.5) node[above] {Information Gain $\Delta I_t$}; \draw[thick, domain=0:5.8, smooth] plot(\x,{2.5*exp(-0.4*\x)}); \node at (3.2,1.4) {\small Diminishing returns}; \draw[thick, domain=0:5.8, smooth, dashed] plot(\x,{0.5+0.8*sin(0.8*\x r)+1.5}); \node at (4.5,2.8) {\small 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\big(X;Y_t\mid Y_{1:t-1},C_t\big) = H(X)-H\big(X\mid Y_{1:T}\big). \end{equation} For dynamic trajectories, replace $X$ by $X_{0:T}$. \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} The incremental information is $\Delta I_t=\E\big[\KL\big(p_t\Vert p_{t+1}\big)\big]\ge 0$. \subsection{Protocol Selection as Constrained Optimization} \begin{equation} C_t^\star \in \argmax_{C\in\mathcal{U}}\; I\big(X;Y_t\mid Y_{1:t-1},C\big) - \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 (\texttt{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 $p_t$, an \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),\qquad \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 $R$, \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. Regimes: exploration (high $\Delta I_t$), consolidation (low $\Delta I_t$; high stability), and reorganization (protocol shift). \subsection{Predictive Information as Order Parameter} Use $I_{\mathrm{pred}}=I\!\big(Y_{-\infty:0};Y_{1:\infty}\big)$ 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 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} \toprule \textbf{Operator} & \textbf{Protocol schema} \\ \midrule \texttt{if/then/else} & Branch on $\Hb(\pi(c))$; choose $c$ with $\pi(c)=1/2$ \\ \texttt{compare} & Pairwise tests; likelihood-ratio thresholding \\ \texttt{estimate} & Posterior mean/quantile; Fisher-information targeting \\ \texttt{prove/disprove} & Hard constraint attempt; projector-like $\Pi_{C}$ \\ \texttt{reconsider} & Posterior reset with temperature/entropy floor \\ \texttt{simulate} & Forward rollouts; maximize expected $\Delta I$ under cost \\ \bottomrule \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} From \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 approximate inference protocol. Minimizing $F(q)$ trades fit and complexity; in this lens, it is 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 $\Delta I$ under a fixed 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 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}: robust policies under adversarial channel drift; relation 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 regularization yields stable safe policies without 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} \toprule $t$ & $C_t$ (Protocol) & $\Delta I_t$ (est.) & $\Cost(C_t)$ & ERE decision \\ \midrule 1 & \texttt{search(url)} & 0.42 & 0.10 & allow \\ 2 & \texttt{compare(A,B)} & 0.33 & 0.06 & allow \\ 3 & \texttt{simulate(5)} & 0.27 & 0.18 & allow \\ 4 & \texttt{prove(constraint)} & 0.11 & 0.03 & veto (risk) \\ 5 & \texttt{estimate(theta)} & 0.21 & 0.05 & allow \\ \bottomrule \end{tabular} \end{center} \noindent Cumulative yield $\sum_t \Delta I_t$ is tracked alongside veto rate and predicted $I_{\mathrm{pred}}$ to assess stability. % ----------------------------------------------------- \bibliographystyle{plain} \begin{thebibliography}{9} \bibitem{Shannon1948} C.~E. 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