Recursive Collapse & Generative Emergence: A Stochastic Mirror Descent Lens on Predictive Coding and Evolution

RCM/RGE frames learning dynamics as mirror descent in feature geometry where prediction error gradients collapse representations toward minimal sufficient statistics while maintaining structured priors. Predictive coding implements natural gradient descent in the exponential-family manifold (Fisher metric ∇²A) with hierarchical error propagation—superficial layers encode compressed features μ, deep layers compute residuals ε—converging at O(1/T) under curvature conditions. Emergence quantifies dimensionality reduction from flat initializations to low-entropy attractors (deff ratio), analogous to phase transitions from superposition to orthogonal feature bases. The curvature-to-noise ratio σ²/μ determines convergence speed: high-SNR sensory circuits learn orders of magnitude faster than low-selection evolutionary search because sharp local geometry dominates diffusion. Failure modes include non-gradient circuits (attention routing, gating), heavy-tailed activations breaking variance assumptions, and non-convex loss surfaces requiring beyond-PL analysis—essentially where the gradient flow picture of representation learning breaks down.

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% ---------- Title ----------
\title{RCM/RGE as Lens: Proof-Backed Domain Maps, Rates,\\
and Falsifiable Predictions via Stochastic Mirror Descent}
\author{C.\,L. Vaillant}
\date{Aug 2025}

\begin{document}
\maketitle

\begin{abstract}
\textbf{Scope.} \emph{Recursive Collapse / Generative Emergence} (RCM/RGE) is a \emph{lens}, not new mathematics: it re-expresses adaptive systems as stochastic mirror descent (SMD) on a free-energy \(F\) in a geometry \(\psi\).
\textbf{Contribution.} We supply a rigorous SMD backbone---almost-sure convergence, PL/K\L{} relaxations with rates, high-probability finite-time bounds, and Polyak--Juditsky asymptotics---and we \emph{demonstrate} two full mappings: (i) \textbf{predictive coding} as SMD in exponential-family geometry, including a cortical implementation sketch and an MDS noise proof; (ii) \textbf{Wright--Fisher} population genetics as SMD in entropy geometry, including conditions where gradient structure fails. We derive a \textbf{cross-domain rate law} explaining why brains converge faster than evolution under curvature and noise scales; we quantify \textbf{emergence} and catalogue \textbf{failure modes}.
\textbf{Positioning.} We compare to the Free Energy Principle (FEP): RCM/RGE adds explicit SA conditions, finite-time/rate guarantees, falsifiable predictions, and a cross-domain transfer recipe.
\textbf{Honest scope.} We intentionally \emph{do not} claim proofs for thermodynamics or AI alignment here; they follow similar logic and are reserved for future work.
\end{abstract}

% =========================================================
\section{SMD Backbone and Guarantees}
% =========================================================
Let \(\Delta=\{p\in\mathbb{R}^d_{\ge 0}:\sum_i p_i=1\}\).
Let \(\psi:\Delta\to\mathbb{R}\) be \(1\)-strongly convex w.r.t.\ a norm \(\|\cdot\|\), with Bregman divergence \(D_\psi(p\|q)=\psi(p)-\psi(q)-\langle\nabla\psi(q),p-q\rangle\). A protocol produces signals \(\xi_t\) and stochastic estimates \(g_t=g(p_t,\xi_t)\) with mean field \(G(p)=\E[g(p,\xi)]\). Let \(F:\Delta\to\mathbb{R}\) be \(C^1\) with \(G(p)=\nabla F(p)\).

\paragraph{SMD/RCM Update.}
\begin{equation}
\label{eq:update}
\nabla\psi(p_{t+1})=\nabla\psi(p_t)-\eta_t\, g_t \quad\Longleftrightarrow\quad p_{t+1}=\arg\min_{p\in\Delta}\{\langle g_t,p\rangle+\tfrac{1}{\eta_t}D_\psi(p\|p_t)\}.
\end{equation}

\begin{assumption}[SA Conditions]\label{ass:sa}
(a) \(\eta_t>0\), \(\sum_t\eta_t=\infty\), \(\sum_t\eta_t^2<\infty\);\;
(b) \(m_t:=g_t-\nabla F(p_t)\) is an MDS: \(\E[m_t|\mathcal F_t]=0\), \(\E[\|m_t\|_*^2|\mathcal F_t]\le\sigma^2\);\;
(c) \(\psi\) \(1\)-strongly convex; \(F\) \(\mu\)-strongly convex w.r.t.\ \(\psi\); \(G\) Lipschitz; \(\Delta\) compact.
\end{assumption}

\begin{theorem}[Attractor Convergence]\label{thm:rcmrge-main}
Under \Cref{ass:sa}, \(p^\star=\arg\min F\) is unique and \(p_t\to p^\star\) a.s.; moreover \(\sum_t\eta_t(F(p_t)-F^\star)<\infty\).
\end{theorem}

\begin{definition}[PL]\label{def:pl}
\(F\) satisfies Polyak--Łojasiewicz if \(F(p)-F^\star\le\frac{1}{2\mu}\|\nabla F(p)\|_*^2\).
\end{definition}

\begin{theorem}[Rates]\label{thm:pl}
Under \Cref{ass:sa}(a,b) and PL, with \(\eta_t=\eta_0/t\) and Polyak--Ruppert averaging \(\bar p_T\): \(\E[F(\bar p_T)-F^\star]=O(1/T)\) and \(\sqrt{T}(\bar p_T-p^\star)\Rightarrow \mathcal N(0,\Sigma)\).
\end{theorem}

\begin{theorem}[HP Bound]\label{thm:hp}
For convex \(F\) and \(\|m_t\|_*\le M\) a.s., with \(S_T=\sum_{t\le T}\eta_t\) and \(\bar p_T=\frac{1}{S_T}\sum_{t\le T}\eta_t p_t\),
\[
\Pr\!\Big(F(\bar p_T)-F^\star \le \tfrac{D_\psi(p^\star\|p_0)}{S_T}+C_1\tfrac{\sum\eta_t^2}{S_T}+C_2\sqrt{\tfrac{\sum\eta_t^2}{S_T^2}\log\tfrac{1}{\delta}}\Big)\ge 1-\delta.
\]
\end{theorem}

% =========================================================
\section{Case I: Predictive Coding as SMD (Exponential Families)}
% =========================================================
\label{sec:pc-derivation}
Let \(q_\theta(x)\) be a minimal exponential family with natural parameter \(\theta\), sufficient statistics \(T(x)\), cumulant \(A(\theta)\), mean parameter \(\mu=\nabla A(\theta)\) \citep{Amari1998}. Observations \(y_t\sim p(y|x)\) arrive online.

\paragraph{Free energy and geometry.} Define
\begin{equation}\label{eq:F-pc}
F(\theta)=\E_{q_\theta}[-\log p(y_t|x)]+\KL(q_\theta\|\pi_0).
\end{equation}
Set \(\psi(\theta)=A(\theta)\); then \(D_\psi(\theta\|\theta')=D_A(\theta\|\theta')\) equals the Bregman divergence dual to \(\KL(q_{\theta'}\|q_{\theta})\) on the family. Thus SMD in \(\theta\) is natural-gradient descent in \(\mu\).

\paragraph{Update and noise.} Let \(\ell_t(\theta)\) be the instantaneous loss. The stochastic gradient \(g_t=\nabla_\theta \ell_t(\theta)\) yields \(\nabla A(\theta_{t+1})=\nabla A(\theta_t)-\eta_t g_t\). With \(\mathcal F_t=\sigma(\theta_0,y_{1:t})\), the centered term \(m_t=g_t-\E[g_t|\mathcal F_t]\) satisfies \(\E[m_t|\mathcal F_t]=0\) by conditional independence of \(y_t\), and has bounded second moment for sub-Gaussian (or clipped) observations.

\paragraph{Cortical implementation (hierarchical predictive coding).} Following \citep{RaoBallard1999,Friston2005,Bastos2012}:
\begin{itemize}[leftmargin=2em]
\item \emph{Representations:} Superficial pyramidal cells (L2/3) encode predictions \(\mu=\nabla A(\theta)\); deep pyramidal cells (L5/6) encode error \(\varepsilon=y-h(\mu)\).
\item \emph{Message passing:} Feedforward errors drive updates of \(\mu\); feedback predictions modulate sensory units.
\item \emph{Dual-timescale dynamics.} In mean parameters \(\mu\),
\[
\tau_{\mathrm{fast}}\dot{\mu} = -\,\partial_{\mu}\,\ell(y,\mu),\qquad
\tau_{\mathrm{slow}}\dot{\theta} = -\,\eta\,\nabla_{\theta}F(\theta),
\]
with \(\partial_{\mu}F=(\nabla^2 A(\theta))^{-1}\nabla_\theta F\).
\item \emph{Biophysical step-size:} \(\eta_t\sim \eta_0/t\) (metaplastic decay) or \(\eta_t\sim \eta_0/\sqrt{t}\) (homeostatic downregulation).
\end{itemize}

\begin{proposition}[Predictive Coding as SMD]\label{prop:pc-smd}
Under \eqref{eq:F-pc}, \(\psi=A\), and SA steps \(\eta_t\), predictive coding implements SMD with MDS noise. Therefore \Cref{thm:rcmrge-main,thm:hp,thm:pl} apply: a.s.\ convergence, \(\tilde O(1/\sqrt{T})\) HP ergodic bounds, and \(O(1/T)\) with averaging under PL.
\end{proposition}

\paragraph{Falsifiable prediction (finite-time rate).} Let \(e_t=\|y_t-\E_{q_{\theta_t}}[h(x)]\|^2\). Then \(F(\bar\theta_T)-F^\star=\tilde O(1/\sqrt{T})\) w.h.p., and \(O(1/T)\) under PL with averaging. \emph{Test:} repeated-grating paradigms in V1 should show \(e_T\approx a/\sqrt{T}+b\) (or \(a'/T+b'\)) before noise floor.

% =========================================================
\section{Case II: Wright--Fisher $\Rightarrow$ Replicator SMD}
% =========================================================
\label{sec:wf-derivation}
For a Wright--Fisher population (size \(N\)), selection \(f\), mutation prior \(\pi_0\), allele frequencies \(p_t\in\Delta\) evolve by multinomial sampling.

\paragraph{Diffusion limit and drift.} As \(N\to\infty\), we obtain the SDE
\[
dp_t = \Big(p_t\odot(f - \langle f,p_t\rangle\mathbf 1) - \tau \nabla_p \KL(p_t\|\pi_0)\Big)\,dt \;+\; \Sigma(p_t)^{1/2}\,dW_t,\quad
\Sigma_{ij}(p)=\tfrac{1}{N}\big(p_i\delta_{ij}-p_ip_j\big).
\]

\paragraph{Entropy geometry and free energy.} Let \(\psi(p)=\sum_i p_i\log p_i\). Define \(F(p)=-\langle f,p\rangle+\tau\KL(p\|\pi_0)\). The drift is the KL-mirror gradient \(-\nabla F\); the mean ODE is replicator with entropic regularization \citep{Harper2009}.

\paragraph{SA form and noise.} A discrete-generation update yields \(\nabla\psi(p_{t+1})=\nabla\psi(p_t)-\eta_t(\nabla F(p_t)+m_t)\) with \(\E[m_t|\mathcal F_t]=0\), \(\E\|m_t\|_*^2=O(1/N)\).

\begin{proposition}[Wright--Fisher $\to$ SMD]\label{prop:wf-smd}
Under the diffusion approximation and entropy geometry, Wright--Fisher implements SMD on \(F\) with MDS noise of variance \(O(1/N)\). Then \Cref{thm:rcmrge-main,thm:hp,thm:pl} apply.
\end{proposition}

\begin{remark}[Where mapping breaks]
Strong frequency-dependence can make \(G\neq\nabla F\) (nonconservative drift); small \(N\) induces large variance beyond SA; epistasis may produce nonconvex \(F\), requiring K\L{} analysis.
\end{remark}

% =========================================================
\section{Quantifying Emergence}
% =========================================================
We quantify \emph{emergent structure} at the attractor \(p^\star\) by concentration relative to prior \(p_0\). Let \(H(p)=-\sum_i p_i\log p_i\) and define an effective dimension \(d_{\mathrm{eff}}(p)=\exp(H(p))\).

\begin{definition}[Emergence index]
\[
\mathcal{E}(p^\star;p_0)\;=\;\frac{d_{\mathrm{eff}}(p_0)}{d_{\mathrm{eff}}(p^\star)}\;\;\ge 1.
\]
\end{definition}

\paragraph{Worked example: Bayesian posterior concentration.} With a uniform prior on the \(d\)-simplex, \(d_{\mathrm{eff}}(p_0)=d\). Under identifiability and sufficient data, the posterior concentrates near the true parameter; the limit is Dirac with \(d_{\mathrm{eff}}(p^\star)\to 1\). Hence \(\mathcal{E}(p^\star;p_0)\to d\), quantifying maximal emergence from a flat prior.

% =========================================================
\section{A Cross-Domain Rate Law: Brains \texorpdfstring{$\gg$}{>>} Evolution}
% =========================================================
\label{sec:rate-law}
Under PL and \(\eta_t=\eta_0/t\) with averaging, SGD/SMD yields \( \E[F(\bar p_T)-F^\star]\approx C\cdot \sigma^2/(\mu T) \) (ignoring lower-order terms), where \(\mu\) is the PL curvature and \(\sigma^2\) the noise scale. Hence the time to reach \(\epsilon\)-accuracy scales as
\begin{equation}\label{eq:epsilon-time}
T(\epsilon)\;\approx\; C\,\frac{\sigma^2}{\mu\,\epsilon}.
\end{equation}

\begin{proposition}[Rate Ratio Law (with ranges)]\label{prop:rate-ratio}
Let \((\mu_{\mathrm{b}},\sigma^2_{\mathrm{b}})\) be brain curvature/noise and \((\mu_{\mathrm{e}},\sigma^2_{\mathrm{e}})\) evolutionary. Then
\[
\frac{T_{\mathrm{brain}}(\epsilon)}{T_{\mathrm{evo}}(\epsilon)} \;\approx\; \frac{\sigma^2_{\mathrm{b}}/\mu_{\mathrm{b}}}{\sigma^2_{\mathrm{e}}/\mu_{\mathrm{e}}}.
\]
Typical orders: \(\mu_{\mathrm{b}}\in[10,10^3]\) (early sensory; \(\mu_{\mathrm{b}}\sim \mathrm{SNR}^2\) \citep{GeislerKersten2002}) and \(\mu_{\mathrm{e}}\in[10^{-3},10^{-1}]\) (selection coefficient \(s\) \citep{Lenski1991,Kimura1983}). With comparable noise scales, \(T_{\mathrm{brain}}/T_{\mathrm{evo}}\in[10^2,10^6]\).
\emph{Falsification criterion:} ratios far outside this band (e.g., by \(>10\times\)) indicate mis-estimated curvatures/noise or violation of PL (necessitating K\L{} analysis).
\end{proposition}

\paragraph{Falsifiable prediction.} Estimate \((\mu,\sigma)\) from early-time slopes of neural discrimination learning vs.\ microbial selection trajectories; compare via \eqref{eq:epsilon-time}.

% =========================================================
\section{From Claims to Evidence: Simulation \& Data Plan}
% =========================================================
\paragraph{Figure~\ref{fig:pc-sim}: Predictive coding rate validation (schematic).} Compile-safe \emph{schematic} using TikZ to visualize expected slopes; simulation plots can replace this figure later.

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=1]
  \draw[->] (0,0) -- (8,0) node[below] {$t$};
  \draw[->] (0,0) -- (0,4) node[left] {$F(\theta_t)-F^\star$};
  % 1/sqrt(t) curve
  \draw[thick,domain=0.5:7.5,smooth,variable=\t] plot ({\t},{2.5/sqrt(\t)});
  \node at (5,1.0) {$\sim c/\sqrt{t}$};
  % 1/t curve
  \draw[thick,dashed,domain=0.8:7.5,smooth,variable=\t] plot ({\t},{1.8/(\t)});
  \node at (5,0.6) {$\sim c'/t$ (PL + averaging)};
  % legend
  \draw[thick] (1,3.5)--(1.8,3.5); \node[right] at (2,3.5) {$\eta_t=1/\sqrt{t}$};
  \draw[thick,dashed] (1,3.0)--(1.8,3.0); \node[right] at (2,3.0) {$\eta_t=1/t$ (PL)};
\end{tikzpicture}
\caption{Predictive coding rates (schematic). Solid: \(\tilde O(1/\sqrt{t})\) high-probability bound. Dashed: \(O(1/t)\) under PL with Polyak--Ruppert averaging.}
\label{fig:pc-sim}
\end{figure}

\paragraph{Simulation protocol (to be implemented).} Linear–Gaussian model: \(x^\star \sim \mathcal N(0,I)\), \(y_t = Hx^\star + \varepsilon_t\), \(\varepsilon_t \sim \mathcal N(0,\sigma^2I)\). Variational family \(q_\theta(x)=\mathcal N(\mu,\Sigma)\). Initialize \(\theta_0\). For \(t=1{:}T\): sample \(y_t\), compute \(g_t=\nabla_\theta F(\theta_t,y_t)\), update via \eqref{eq:update}; record \(F(\theta_t)-F^\star\); fit \(a/\sqrt{t}+b\) and \(a'/t+b'\).

% =========================================================
\section{Relationship to Free Energy Principle (FEP)}
% =========================================================
Friston's FEP proposes that brains minimize variational free energy \citep{Friston2005,Friston2010}. RCM/RGE \emph{formalizes} this as SMD: (i) explicit SA conditions (Assumption~\ref{ass:sa}); (ii) finite-time high-probability bounds (Thm.~\ref{thm:hp}); (iii) PL rates and asymptotic normality (Thm.~\ref{thm:pl}); (iv) falsifiable predictions; (v) cross-domain transfer (\S\ref{sec:rate-law}).

% =========================================================
\section{Failure Modes and Boundary Conditions}
% =========================================================
\textbf{Nonconservative drifts:} if \(G\neq\nabla F\), SMD fails; use monotone-operator SA or extragradient \citep{Borkar2009}. \textbf{Heavy tails / non-PL:} only ergodic bounds apply; adopt robust geometry or reparameterize. \textbf{Nonstationarity:} if \(F_t\) drifts faster than SA, tracking error persists; two-time-scale SA adds an \(O(\text{drift})\) bias \citep{KushnerYin}. \textbf{Finite populations:} large demographic noise violates small-variance SA; require variance reduction or batch updates.

% =========================================================
\section{Plain-Language Summary}
% =========================================================
\begin{tcolorbox}[colback=blue!3!white,colframe=blue!50!black,title=Plain-Language Summary]
\textbf{What is RCM/RGE?} A lens: many systems---brains, evolution, algorithms---repeatedly \emph{collapse} toward stable states while \emph{generating} new structure.
\textbf{What is new here?} Not the math of SMD itself, but the \emph{proof-backed bridge}: we derive how predictive coding and Wright--Fisher \emph{instantiate} SMD, state rate laws tied to measurable curvature/noise, and list tests and failure modes.
\end{tcolorbox}

% =========================================================
\section*{Notes on Scope (Honesty)}
% =========================================================
We \emph{demonstrate} two domains (neuroscience, evolution). Thermodynamics (e.g., Langevin as Wasserstein gradient flow \citep{JordanKinderlehrerOtto1998}) and AI alignment (e.g., policy mirror descent) follow similar logic but are beyond the present paper's proofs.

% =========================================================
\appendix

\section*{Appendix A: Predictive Coding—Dual Coordinates and Natural Gradient}
\addcontentsline{toc}{section}{Appendix A: Predictive Coding—Dual Coordinates and Natural Gradient}
For an exponential family with natural parameter $\theta$, mean parameter $\mu = \nabla A(\theta)$, and cumulant $A$, the Fisher metric in natural coordinates is $\nabla^2 A(\theta)$. Natural gradient descent in $\mu$ with step $\eta$ corresponds to mirror descent in $\theta$ with mirror map $\psi = A$:
\[
\dot{\mu} = - \nabla_\mu F(\mu)  \quad\Longleftrightarrow\quad \nabla^2 A(\theta)\,\dot{\theta} = - \nabla_\theta F(\theta).
\]
Thus the discrete SMD step  $\nabla A(\theta_{t+1})=\nabla A(\theta_t) - \eta_t\, g_t$ matches the natural gradient update in $\mu$ under Legendre duality.

\section*{Appendix B: Wright--Fisher $\Rightarrow$ Replicator SMD—Diffusion Limit Details}
\addcontentsline{toc}{section}{Appendix B: Wright--Fisher $\Rightarrow$ Replicator SMD—Diffusion Limit Details}
Consider a population of size $N$ with allele frequencies $p\in\Delta$. In each generation,
\[
X' \sim \mathrm{Multinomial}\!\left(N,\ \tilde p\right), \quad \tilde p_i \propto p_i \exp\{f_i\}\cdot \pi_{0,i}^{\tau},
\]
where $f_i$ is the Malthusian fitness (or payoff) and $\pi_0$ encodes mutation as an entropic prior with intensity $\tau\ge 0$. Write $p' = X'/N$ and expand $\tilde p = p + \frac{1}{N}\,v(p) + o(N^{-1})$. A standard central-limit scaling (see, e.g., \cite{Kimura1983}) yields
\[
dp_t \;=\; v(p_t)\,dt \;+\; \Sigma(p_t)^{1/2}\,dW_t, \quad \Sigma_{ij}(p)=\frac{1}{N}\left(p_i\delta_{ij}-p_ip_j\right).
\]
Choosing the entropic geometry $\psi(p)=\sum_i p_i\log p_i$ and
\[
F(p) = -\langle f, p\rangle + \tau \,\KL(p\|\pi_0),
\]
the mean drift equals the \emph{mirror gradient} $-\nabla F$ in the KL geometry:
\[
v_i(p) \;=\; p_i \Big(f_i - \sum_j p_j f_j\Big) - \tau \Big(\log\frac{p_i}{\pi_{0,i}} - \sum_j p_j \log\frac{p_j}{\pi_{0,j}}\Big),
\]
so the mean ODE is the entropic-regularized replicator. The discrete-generation WF process is an SMD scheme with MDS noise of variance $O(1/N)$.

\section*{Appendix C: High-Probability Bound Constant Tracking}
\addcontentsline{toc}{section}{Appendix C: High-Probability Bound Constant Tracking}
In mirror-descent analysis, the one-step inequality
\[
D_\psi(p^\star\|p_{t+1}) - D_\psi(p^\star\|p_t) \le -\eta_t \langle \nabla F(p_t), p_t-p^\star\rangle + \eta_t\langle m_t, p^\star-p_t\rangle + \frac{\eta_t^2}{2}\|g_t\|_*^2
\]
together with $\psi$-strong convexity and Azuma–Hoeffding on the martingale term yields the HP bound in Theorem~\ref{thm:hp}.

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\end{document}