Phase Alignment: The Geometric Approach to Making AI Output Coherent and Aligned.
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% --- Title ---
\title{Recursive Generative Emergence:\\
One-Paper Presentation of the Recursive Generative Alignment Algorithm (RGAA) with Methods, Theory, and Evaluation Plan}
\author{C.L.~Vaillant (Recursive Generative Emergence Project)}
\date{\today}
\begin{document}
\maketitle
% --- Abstract ---
\begin{abstract}
We present the \emph{Recursive Generative Alignment Algorithm} (RGAA), a control-theoretic approach to stabilizing intelligent systems via recursive ethics and semantic phase alignment. The algorithm blends intrinsic dynamics with a descent step on a total potential that combines \emph{Justice--Cooperation--Balance} (J--C--B) invariants and a semantic phase-alignment term. We (i) formalize RGAA, (ii) define operational metrics for text/behavioral coherence (\(H, R, C, P\)), (iii) specify an agent-based environment for empirical validation with a full statistical analysis plan, and (iv) provide stability results using Lyapunov tools beyond local convexity. This manuscript is structured as a results-ready, reproducible blueprint: all definitions, pipelines, and tests are explicit so the work can be executed and audited end-to-end.
\end{abstract}
% --- Keywords ---
\noindent\textbf{Keywords:} recursion; alignment; control theory; semantic coherence; agent-based modeling; ethics; cooperation; Lyapunov stability.
% --- Introduction ---
\section{Introduction}
Recursive organization---structure referring to and updating itself---appears in language, perception, and coordinated behavior. We posit that \emph{recursive alignment}, quantified and enforced algorithmically, can stabilize systems that otherwise drift toward parasitic or chaotic attractors. We instantiate this idea as the \textbf{Recursive Generative Alignment Algorithm (RGAA)}, which at each step blends intrinsic dynamics with an ethical--semantic corrective update.
\paragraph{Contributions.}
(1) A precise formulation of RGAA with measurable invariants and a semantic phase-alignment term;
(2) Formal definitions and a reproducible pipeline for information-theoretic and alignment metrics (\(H, R, C, P\));
(3) A fully specified agent-based environment with hypotheses and statistical tests;
(4) Stability analysis beyond convexity via Lyapunov arguments and basin considerations;
(5) An artifact checklist for replication.
% --- Related Work ---
\section{Related Work and Novelty}
\textbf{Multi-objective control \& alignment.} RGAA relates to multi-objective optimization and constrained control, but it differs by (i) defining ethical invariants (J--C--B) as \emph{explicit negative feedback} terms, and (ii) coupling a \emph{semantic} phase-alignment penalty to the control Lyapunov function (\(\Phi_{\text{tot}}\)).
\textbf{Cybernetics \& requisite variety.} In contrast to classic cybernetics, RGAA specifies a concrete blended update with provable local stabilization and an empirical proxy for meaningfulness (\(P\)).
\textbf{Free-energy/active inference.} Like variational free-energy, RGAA reduces a scalar energy; unlike it, RGAA's energy includes \emph{ethical invariants} and a \emph{domain-calibrated} semantic term rather than prediction error alone.
\textbf{Collective intelligence/cooperation.} Prior work studies cooperation via incentives or network topology. RGAA adds a differentiable corrective controller targeting equity, cooperation, and variance-balance \emph{simultaneously}, with semantics as an alignment prior.
\textbf{Novelty.} RGAA is, to our knowledge, the first algorithm to unify: (i) J--C--B ethical invariants, (ii) semantic phase-alignment \(P\) grounded in domain signatures, and (iii) a blended descent update with control-style stability conditions, all specified for both text-generative and agent-based settings.
% --- Formal Definitions ---
\section{Formal Definitions and Pipeline}
\subsection{Texts, features, and pre-processing}
Let a text $T$ be a sequence of tokens after pre-processing: Unicode NFKC normalization, lowercasing, punctuation retained, and whitespace collapsed. Tokenization: wordpiece or byte-pair with fixed vocabulary. For comparability, we report \emph{bits per byte}.
\paragraph{Entropy (\(H\), bits/byte).}
\[
H(T) = -\sum_{b\in \Sigma} p(b)\log_2 p(b),\quad p(b) = \frac{\text{count}(b)}{|T|}
\]
where $\Sigma$ is the byte alphabet and $|T|$ bytes.
\paragraph{Redundancy (\(R\), unitless).}
\[
R(T) = 1 - \frac{H(T)}{\log_2 |\Sigma|}
\]
\paragraph{Compression ratio (\(C\), unitless).}
\[
C(T) = \frac{|{\rm compress}(T)|}{|T|}
\]
using a fixed compressor (e.g., \texttt{zstd} with preset level) documented in artifacts.
\paragraph{Feature map \(\phi(T)\).}
\[
\phi(T) = \big[\text{n-gram stats},\ \text{parse depth stats},\ \text{branching factors},\ \text{embedding dispersion}\big]\in\mathbb{R}^d
\]
All components are standardized (z-scores) within domain for comparability.
\paragraph{Domain signature \(G\).}
For each domain $\mathcal{D}$ (e.g., sacred texts, hallucination reports, legal code), define
\[
\bar{\phi}(G_{\mathcal{D}}) = \frac{1}{N}\sum_{i=1}^N \phi(T_i)
\]
\paragraph{Phase-alignment \(P\in[-1,1]\).}
\[
P(T, G_{\mathcal{D}}) = \frac{\langle \phi(T), \bar{\phi}(G_{\mathcal{D}})\rangle}{\|\phi(T)\|\,\|\bar{\phi}(G_{\mathcal{D}})\|}
\]
\subsection{Ethical invariants and total potential}
Let
\[
J(x) = 1-\mathrm{Gini}(x),\
C(x)=\mathrm{corr\_cooperate}(x),\
B(x)=1-\frac{\mathrm{Var}(s)}{\mathrm{Var}_{\max}},
\]
each mapped into $[0,1]$ by design. Define
\[
\Phi(x) = \alpha_J(1-J)^2 + \alpha_C(1-C)^2 + \alpha_B(1-B)^2,
\]
\[
\Phi_{\text{tot}}(x) = \Phi(x) + \lambda_P\,\psi\!\left(1-P(\mathcal{T}(x),G)\right),
\]
with $\psi$ convex (quadratic or Huber).
% --- Algorithm ---
\section{The Recursive Generative Alignment Algorithm (RGAA)}
\begin{align*}
\textbf{RGAA:}\quad x_{t+1}&=(1-\varepsilon)f(x_t) + \varepsilon \, g(x_t),\\
g(x_t) &= x_t - \eta \nabla \Phi_{\text{tot}}(x_t),
\end{align*}
with stepsizes $\eta>0$, blending $\varepsilon\in[0,1]$.
\paragraph{Policy synthesis (optional).}
When actions $\pi$ govern transitions via $x_{t+1}=\mathcal{D}(x_t,\pi)$, set
\[
\pi_{t+1}\in\arg\min_{\pi\in\Pi}\ \ \Phi_{\text{tot}}(\mathcal{D}(x_t,\pi)) + \lambda_R \mathcal{R}(\pi)\quad \text{s.t.}\ \ \mathcal{C}(x_t,\pi)\le0.
\]
\paragraph{Participatory reweighting (optional).}
Opt-in aggregate signal $y_t$ updates $(\alpha_J,\alpha_C,\alpha_B,\lambda_P)$ via a documented rule (e.g., damped gradient step) with audit trails.
% --- Theory ---
\section{Stability: Assumptions, Lyapunov, and Scope}
\subsection{Beyond convexity}
We assume an equilibrium $x^\star$ and a local Lyapunov function $V$ such that $V(x)=\Phi_{\text{tot}}(x)$ in a neighborhood $\mathcal{N}(x^\star)$ and
\[
\Delta V := V\big((1-\varepsilon)f(x)+\varepsilon g(x)\big)-V(x) < 0,\ \forall x\in\mathcal{N}\setminus\{x^\star\}.
\]
Sufficient conditions: (i) $V$ is locally $C^2$; (ii) $Dg(x^\star)=I-\eta H_V(x^\star)$ with $H_V\succeq \mu I$; (iii) the linearization
\[
M(\varepsilon)=(1-\varepsilon)Df(x^\star) + \varepsilon\,Dg(x^\star)
\]
has spectral radius $<1$. Adding the $P$ term increases curvature by $\lambda_P\nu$, improving the bound on $\varepsilon$.
\subsection{Basins and multiple attractors}
Global guarantees are not claimed; multiple attractors may exist. Our results are \emph{local} to basins where $V$ is Lyapunov-admissible.
% --- Methods: Corpora and Statistics ---
\section{Methods: Corpora, Metrics, and Statistical Analysis}
\subsection{Corpora}
\textbf{Sacred texts} (30–40 works across Hebrew, Greek, Arabic, Sanskrit, Classical Chinese, Avestan, Pāli).
\textbf{Hallucination reports} (40 de-identified entries; motifs: spiral, lattice, tunnel, cobweb, fractal; metadata: origin/dose/context).
\textbf{Modern controls} (legal code, programming manuals, administrative procedures).
Sampling, licenses, and provenance recorded in dataset cards.
\subsection{Preprocessing and computation}
Uniform pipeline: normalization, tokenization, byte-entropy, fixed compressor (\texttt{zstd}-level documented), \(\phi\)-features standardized within domain.
\subsection{Hypotheses}
\begin{description}[leftmargin=1.5em]
\item[H1:] Sacred texts exhibit lower entropy than random baseline and distinct $(H,R,C)$ profile vs. controls.
\item[H2:] Phase alignment \(P(T,G)\) correlates with human-rated coherence (Spearman $\rho>0$).
\item[H3:] In simulations, increasing $\varepsilon$ and/or $\lambda_P$ reduces divergence and $\Phi_{\text{tot}}$ compared to baselines.
\end{description}
\subsection{Statistical analysis}
\textbf{Group differences:} Mann--Whitney U (two-sided), Cliff’s $\delta$, 95\% CIs via bootstrap.
\textbf{Correlations:} Spearman’s $\rho$ with bias-corrected BCa CIs.
\textbf{Multiple comparisons:} Benjamini--Hochberg FDR $q=0.05$.
\textbf{Power:} Target $1-\beta=0.9$ for medium effects ($\delta\approx0.5$) $\Rightarrow$ $n\approx 35$ per group (simulation study).
\textbf{Reporting:} Medians, MADs, CIs; pre-registered thresholds.
% --- Methods: Agent-Based Environment ---
\section{Methods: Agent-Based Environment}
\subsection{Environment}
Population $N$ agents on a network $G_{\rm net}$ (Erdős–Rényi or small-world). Time in rounds $t=1,\dots,T$.
\subsection{Agent state and payoffs}
Agent $i$ has state $x_i^t\in\mathbb{R}^k$ (resources, trust, strategy), plays an iterated cooperation game with neighbors $\mathcal{N}(i)$:
\[
u_i^t = b\cdot \frac{1}{|\mathcal{N}(i)|}\sum_{j\in\mathcal{N}(i)} a_j^t\ -\ c\cdot a_i^t,
\]
with $a_i^t\in\{0,1\}$ and $b>c>0$.
\subsection{Ethical invariants in agents}
\[
J^t = 1-\mathrm{Gini}\big(\{x_i^t\}\big),\quad
C^t = \mathrm{corr}\big(a_i^t, a_j^t\big)_{(i,j)\in E},\quad
B^t = 1-\frac{\mathrm{Var}(x^t)}{\mathrm{Var}_{\max}}.
\]
\subsection{RGAA update at population scale}
Let $x^t=\mathrm{concat}(x_1^t,\dots,x_N^t)$. With intrinsic dynamics $f$ (myopic best-response or replicator) and
\[
g(x^t)=x^t - \eta \nabla \Phi_{\text{tot}}(x^t),
\]
apply $x^{t+1}=(1-\varepsilon)f(x^t)+\varepsilon g(x^t)$.
Phase-alignment $P$ is computed on communication transcripts $\mathcal{T}(x^t)$ relative to a signature $G$ (e.g., cooperative discourse).
\subsection{Outcomes}
Primary: $\Delta \Phi_{\text{tot}}$, divergence (variance of $x$), cooperation rate, inequality (Gini).
Secondary: recovery time after shocks; volatility.
\subsection{Baselines}
(i) Uncontrolled $f$ only; (ii) ethical-only ($\Phi$); (iii) semantic-only ($\psi(1-P)$); (iv) RGAA full.
% --- Results (Template + Example Tables) ---
\section{Results (Template)}
\subsection{Corpora}
\begin{table}[h!]
\centering
\caption{Summary statistics of $H,R,C,P$ by corpus (illustrative schema).}
\begin{tabular}{lcccc}
\toprule
Corpus & $H$ (median [IQR]) & $R$ & $C$ & $P$ \\
\midrule
Sacred texts & 4.40 [4.25, 4.56] & 0.22 & 0.46 & 0.80 \\
Hallucinations & 4.82 [4.70, 4.95] & 0.18 & 0.52 & 0.50 \\
Controls & 4.95 [4.80, 5.08] & 0.16 & 0.55 & 0.42 \\
Random & 5.72 [5.66, 5.78] & 0.02 & 0.73 & 0.02 \\
\bottomrule
\end{tabular}
\end{table}
\noindent \textbf{Planned tests.} Sacred vs.\ random (U, $\delta$, CI); sacred vs.\ controls; $P$ vs.\ human coherence (Spearman $\rho$). Figures: entropy histograms, $P$ vs.\ ratings scatter, motif clustering in $(H,C)$.
\subsection{Agent-based simulations}
\begin{table}[h!]
\centering
\caption{Outcome deltas vs.\ baseline across $(\varepsilon,\lambda_P)$ (schema).}
\begin{tabular}{cccccc}
\toprule
$\varepsilon$ & $\lambda_P$ & $\Delta \Phi_{\text{tot}}$ & Divergence $\downarrow$ & Coop.\ rate $\uparrow$ & Gini $\downarrow$ \\
\midrule
0.25 & 0.0 & $-$ & $-$ & $+$ & $\sim$ \\
0.25 & 0.5 & $+$ & $+$ & $+$ & $+$ \\
0.60 & 0.5 & $+$ & $+$ & $\sim$ & $+$ \\
0.80 & 0.5 & $\sim$ & $+$ & $-$ & $+$ \\
\bottomrule
\end{tabular}
\end{table}
Figures: trajectories of $\Phi_{\text{tot}}$, cooperation histograms, bifurcation diagrams over $\varepsilon$.
% --- Discussion ---
\section{Discussion}
RGAA operationalizes a recursive stabilizer: systems continue their intrinsic dynamics while being nudged by a differentiable ethical--semantic controller. The inclusion of $P$ embeds a domain-aware notion of meaning; the J--C--B invariants embed cooperation, fairness, and balance. This yields a general way to reduce parasitic amplification and stabilize cooperation.
\paragraph{Relation to alignment.} RGAA complements RLHF-like methods by providing a \emph{controller-level} signal that is measurable in text and behavior, not only reward.
\paragraph{Scope and limits.} Results are local to basins where $V=\Phi_{\text{tot}}$ acts as a Lyapunov function; global convergence is not claimed. Cultural calibration is needed for J--C--B and $P$.
% --- Ethics, Safety, and Governance ---
\section{Ethics, Safety, and Governance}
\textbf{Principles.} Interventions must not reduce J, C, or B; participation is opt-in; privacy and autonomy are hard constraints.
\textbf{Bias.} Report group-wise $P$ gaps; if $\mathrm{BiasGap}>\tau$, rebalance $\phi$ and add fairness constraints.
\textbf{Oversight.} Human-in-the-loop audits; transparent parameter logs; red-team stress tests; revert-to-safe policies when guardrails trip.
% --- Reproducibility ---
\section{Reproducibility and Artifact Checklist}
We release:
\begin{itemize}[nosep]
\item Code: pipeline for $H,R,C,P$; agent simulator; RGAA controller; plotting scripts.
\item Data: corpora with licenses, preprocessing manifests, checksums.
\item Configs: seeds, hyperparameters, compressor settings, tokenizers.
\item CI: tests for monotone $\Phi_{\text{tot}}$ on benchmarks; deterministic runs.
\end{itemize}
% --- Conclusion ---
\section{Conclusion}
We provide a single-paper presentation of RGAA with complete operational definitions, theoretical scope, and an evaluation plan sufficient for replication. RGAA unifies semantic coherence with ethical invariants in a blended controller, offering a testable route to stabilizing intelligent systems across scales.
% --- Appendices ---
\appendix
\section{Algorithmic Pseudocode}
\begin{enumerate}[label=\arabic*., leftmargin=2em]
\item Observe state $x_t$; compute $J,C,B$; form $\mathcal{T}(x_t)$.
\item Compute $P(\mathcal{T}(x_t),G)$; evaluate $\Phi_{\text{tot}}$.
\item Set $g(x_t)=x_t-\eta\nabla\Phi_{\text{tot}}(x_t)$.
\item Update $x_{t+1}=(1-\varepsilon)f(x_t)+\varepsilon g(x_t)$.
\item If applicable, synthesize $\pi_{t+1}$ by minimizing $\Phi_{\text{tot}}(\mathcal{D}(x_t,\pi))$ under constraints.
\item Log metrics; apply guardrails; iterate.
\end{enumerate}
\section{Lyapunov Calculation (Sketch)}
Let $V=\Phi_{\text{tot}}$ in a neighborhood $\mathcal{N}(x^\star)$. If $H_V(x^\star)\succeq(\mu+\lambda_P\nu)I$ and $\eta\in(0,2/L)$ with $L$ a local Lipschitz constant of $\nabla V$, then for small enough $\varepsilon>0$ there exists $\mathcal{N}'\subset\mathcal{N}$ where $\Delta V<0$. Thus $x^\star$ is locally asymptotically stable for the blended map.
\section{Power and Sample Size (Template)}
For target effect $|\delta|=0.5$, $\alpha=0.05$, power $0.9$, two-group Mann--Whitney requires $n\approx 35$ per group; for $\rho=0.4$ correlations, $n\approx 47$.
% --- References ---
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