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\title{\vspace{-0.5em}\textbf{RISE-01: Mechanistic Specification for Persistent Memory and Identity Anchoring in Recursive Agents}\\[0.25em]
\large System Definition, Stability Guarantees, Neural Implementations, Resource Models, and Verification Protocols}
\author{RGEmergence}
\date{\today}
\begin{document}
\maketitle
% =========================================================
% Abstract
% =========================================================
\begin{abstract}
This document specifies a mechanistic architecture for recursive agents that maintain persistent memory and a stable latent identity across updates, resets, and failures. The system integrates: (i) multiscale memory with cross-level consistency operators and redundancy management; and (ii) a re-entrant identity update that is contractive around a learned anchor manifold. We define operators, state variables, and invariants; provide stability bounds for identity drift; give repair guarantees under shard corruption; and specify recovery by checkpoint and log replay. Neural implementations, resource and complexity models, validation targets, and failure mode handling are included. Cross-references and parameter thresholds are explicit; all algorithms specify inputs/outputs and data shapes. All figures are generated inline (TikZ/PGFPlots) without external files.
\end{abstract}
% =========================================================
% 1. System Definition
% =========================================================
\section{System Definition}\label{sec:system}
% Purpose: define state, operators, invariants, and observables in a minimal way.
\paragraph{Time index.} Discrete recursion steps \(n \in \mathbb{N}\).
\paragraph{Agent state.}
\begin{itemize}[leftmargin=1.25em]
\item Latent identity \(I_n \in \R^{d}\).
\item Multiscale memory \(\mathcal{M}_n = \{M_n^{(0)}, M_n^{(1)}, \dots, M_n^{(L)}\}\), where level \(\ell\) stores representations at different time scales (working \(\to\) long-term).
\item Episodic buffer \(E_n\) (short horizon) and semantic store \(K_n\) (long horizon).
\item Autobiographical log \(\AL_n\): sequence of entries used for replay; entry schema in Sec.~\ref{sec:replay}.
\end{itemize}
\paragraph{Operators.}
\begin{itemize}[leftmargin=1.25em]
\item Coarsen \(S_{\down}^{(\ell)}: \mathcal{M}^{(\ell)} \rightarrow \mathcal{M}^{(\ell+1)}\) and Expand \(S_{\up}^{(\ell)}: \mathcal{M}^{(\ell+1)} \rightarrow \mathcal{M}^{(\ell)}\), assumed Lipschitz.
\item Episodic compressor \(\Gamma(E_n)\) and autobiographical summarizer \(\Upsilon(\AL_n)\).
\item Identity projector \(\Pi(I_n, \Gamma(E_n), \Upsilon(\AL_n))\).
\item Anchor projector \(\Pi_{\A}: \R^{d} \rightarrow \A\) (nonexpansive onto a learned anchor manifold; Sec.~\ref{sec:anchor}).
\item Redundancy graph \(\G=(\mathcal{V},\mathcal{E})\) with cross-check \(C^{(\ell)}\) and repair \(R^{(\ell)}\); Byzantine assumption \(|\mathcal{V}|\ge 3f+1\) (Sec.~\ref{sec:redundancy}).
\end{itemize}
\paragraph{Observables.} Identity drift \(\Delta_n = \norm{I_{n+1}-I_n}\), cumulative drift \(D_T=\sum_{n<T} w_n \Delta_n\); inconsistency scores \(\chiL_n\) for memory shards; symbolic self-model \(\Psi(I)\) for audit (Sec.~\ref{sec:psi}).
% =========================================================
% 2. Memory Architecture
% =========================================================
\section{Memory Architecture}\label{sec:memory}
% Goal: multiscale memory with explicit cross-level consistency and reconstruction.
\subsection{Cross-Level Consistency}
For each level \(\ell \in \{0,\dots,L-1\}\),
\begin{align}
\mathcal{L}^{(\ell)}_{\mathrm{rec}} &= \norm{M^{(\ell)} - S_{\up}^{(\ell)} S_{\down}^{(\ell)} (M^{(\ell)})}^2, \\
\mathcal{L}^{(\ell)}_{\mathrm{cross}} &= \norm{S_{\down}^{(\ell)}(M^{(\ell)}) - M^{(\ell+1)}}^2, \\
\mathcal{L}_{\mathrm{multiscale}} &= \sum_{\ell=0}^{L-1} \alpha_{\ell} \mathcal{L}^{(\ell)}_{\mathrm{rec}} + \sum_{\ell=0}^{L-1} \beta_{\ell} \mathcal{L}^{(\ell)}_{\mathrm{cross}}.
\end{align}
% Comment: These enforce cross-level agreement and reconstructability.
\subsection{Redundancy and Repair}\label{sec:redundancy}
Assume a \(k\)-vertex-connected graph \(\G\) and Byzantine tolerance with at most \(f\) adversarial nodes, \(|\mathcal{V}|\\ge 3f+1\). Store shards \(\{ M^{(\ell)}_{n,v} \}_{v\in\mathcal{V}}\). Define repaired estimate
\begin{equation}
\widetilde{M}^{(\ell)}_n := R^{(\ell)} \Big( \{ M^{(\ell)}_{n,v} \}_{v\in \mathcal{V}},\, C^{(\ell)} \Big).
\end{equation}
Assume at most \(\rho |\mathcal{V}|\) corrupted nodes with perturbation magnitude \(\delta\), and expand–coarsen reconstruction error \(\eta_\ell\).
With robust aggregation (median/trimmed mean) of breakdown \(>\rho\):
\begin{equation}
\norm{\widetilde{M}^{(\ell)}_n - M^{(\ell)}_n} \le c_\ell \delta + \eta_\ell.
\end{equation}
Define inconsistency
\begin{equation}
\chiL_n = \frac{1}{|\mathcal{V}|} \sum_{v} \norm{M^{(\ell)}_{n,v}-\widetilde{M}^{(\ell)}_n}.
\end{equation}
\paragraph{CrossCheck output format and thresholds.}
\textbf{CrossCheck} returns a tuple \((D, t, F)\) where: \(D\in\R^{|\mathcal{V}|\times|\mathcal{V}|}\) is the pairwise distance matrix across shards; \(t\in\R^{|\mathcal{V}|}\) is a per-node trust score computed from row statistics of \(D\); and \(F\in\{0,1\}^{|\mathcal{V}|\times|\mathcal{V}|}\) flags pairwise inconsistencies exceeding a level-specific tolerance. The repair operator may weight shards by \(t\). Repair is triggered when \(\chiL_n > \tau_\ell\).
\subsection{Threshold Calibration}\label{sec:thresholds}
Calibrate \(\tau_\ell\) using clean runs: estimate \(\mu_\ell=\E[\chiL]\), \(\sigma_\ell=\mathrm{Std}[\chiL]\) on validation traces without injected faults; set \(\tau_\ell=\mu_\ell + z_p \sigma_\ell\) for quantile \(p\) (e.g., \(p=0.995\)). Re-estimate after major updates.
% =========================================================
% 3. Identity Dynamics
% =========================================================
\section{Identity Dynamics}\label{sec:identity}
% Goal: re-entrant update, contraction, drift metrics, manifold control.
\subsection{Re-Entrant Update and Anchoring}\label{sec:update}
Let \(Z_{E,n}=\Gamma(E_n)\), \(Z_{A,n}=\Upsilon(\AL_n)\), and \(\norm{\xi_n}\le \varepsilon\).
\begin{equation}\label{eq:update}
I_{n+1} = (1-\alpha) I_n + \alpha\, \Pi(I_n, Z_{E,n}, Z_{A,n}) + \xi_n.
\end{equation}
Optional event gate \(g_n \in (0,1)\) multiplies the update delta for high-salience changes. Apply nonexpansive anchor mixing
\begin{equation}
\tilde I_{n+1}=(1-\alpha) I_n + \alpha\, \Pi(\cdot), \quad
I_{n+1} \leftarrow (1-\gamma)\tilde I_{n+1} + \gamma\,\Pi_{\A}(\tilde I_{n+1}), \;\; \gamma\in[0,1].
\end{equation}
\subsection{Stability and Drift}
Assume local Lipschitz constants \(L_I<1/\alpha\), \(L_Z<\infty\):
\[
\norm{\Pi(I',Z)-\Pi(I,Z)}\le L_I \norm{I'-I},\quad
\norm{\Pi(I,Z')-\Pi(I,Z)}\le L_Z \norm{Z'-Z}.
\]
\begin{theorem}[Contractive convergence]\label{thm:contract}
Let \(\kappa = (1-\alpha)+\alpha L_I < 1\). For fixed \(Z^\star\), there exists a unique fixed point \(I^\star\) s.t.
\[
\norm{I_{n+1}-I^\star} \le \kappa \norm{I_n - I^\star} + \alpha L_Z \norm{Z_n - Z^\star} + \varepsilon.
\]
If \(Z_n \rightarrow Z^\star\) (or has bounded variation), then \(I_n \rightarrow I^\star\) with steady error \(\le \frac{\alpha L_Z \sup_n \norm{Z_n-Z^\star} + \varepsilon}{1-\kappa}\).
\end{theorem}
Define \(\Delta_n=\norm{I_{n+1}-I_n}\) and \(D_T=\sum_{n<T} w_n \Delta_n\).
\begin{proposition}[Bounded drift]\label{prop:drift}
If \(\norm{Z_{n+1}-Z_n}\le \zeta\) and \(\norm{\xi_n}\le \varepsilon\), then
\[
\Delta_n \le \kappa \Delta_{n-1} + \alpha L_Z \zeta + 2\varepsilon,\qquad
\sup_n \Delta_n \le \frac{\alpha L_Z \zeta + 2\varepsilon}{1-\kappa}.
\]
\end{proposition}
\subsection{Information-Theoretic Persistence}\label{sec:info}
Treat the identity recursion as a channel. Define
\[
C_{\mathrm{id}} := \sup_{p(I_0)} \lim_{T\to\infty} \frac{1}{T}\, \mathcal{I}(I_0; I_T).
\]
\begin{proposition}[Capacity bound via SDPI]\label{prop:sdpi}
If the update channel from \(I_n\) to \(I_{n+1}\) is \(\eta\)-contractive in total variation (or satisfies a strong data-processing inequality with coefficient \(\eta<1\)), then \(\mathcal{I}(I_0; I_T) \le \eta^T \, \mathcal{I}(I_0; I_0)\) and thus \(C_{\mathrm{id}}\le -\log \eta\) bits/step. In the Lipschitz model above, \(\eta\) can be upper-bounded by \(\kappa\).
\end{proposition}
\noindent \emph{Proof sketch.} Apply strong data-processing inequalities for Markov chains (e.g., \cite{polyanskiy2017sdpi}) and the contraction mapping bound; relate the contraction of the map to a contraction coefficient on information.
\subsection{Anchor Manifold \texorpdfstring{$\A$}{}: Role, Learning, Stability}\label{sec:anchor}
\textbf{Role.} \(\A\) constrains \(I_n\) to a stable set where symbolic self-model \(\Psi(I)\) is well-defined and drift is bounded.\\
\textbf{Learning.} Learn \(\A\) as a low-dimensional submanifold or subspace using identity trajectories during a stabilization phase (e.g., contrastive clustering with smoothness regularization).\\
\textbf{Stability.} Nonexpansive projection \(\Pi_{\A}\) and small \(\gamma\) ensure geometric decay of distance to \(\A\) up to a noise floor.
\subsection{Parameter Guidance and Ablations}\label{sec:ablations}
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Parameter & Guidance & Rationale \\
\midrule
\(\alpha\) & \(\in (0, \frac{1}{1+L_I})\) & Ensure \(\kappa<1\) (Theorem~\ref{thm:contract}) \\
\(\gamma\) & 0.1--0.3 & Sufficient manifold pull without oscillation \\
Levels \(L\) & 2--4 & Tradeoff depth vs. consolidation delay \\
Reduction \(r_\ell\) & 2--8 & Coarsen factor per level (data-dependent) \\
Trim proportion & \(> \rho\) & Robust aggregation breakdown point \\
\bottomrule
\end{tabular}
\end{center}
% =========================================================
% 4. Neural Implementations
% =========================================================
\section{Neural Implementations}\label{sec:neural}
% Goal: translate operators into concrete modules with shapes and failure behavior.
\subsection{Identity Projector \texorpdfstring{$\Pi$}{Pi}}
\noindent Expected inputs (batch-first):
\begin{itemize}[leftmargin=1.25em]
\item \(I \in \R^{B \times 1 \times d}\): identity token (single query position).
\item \(Z_E \in \R^{B \times N_E \times d}\), \(Z_A \in \R^{B \times N_A \times d}\): memory tokens.
\end{itemize}
\begin{lstlisting}[style=py,caption={Identity projector with self- and cross-attention, plus gated fusion.}]
import torch, torch.nn as nn
class IdentityProjector(nn.Module):
def __init__(self, d_model=512, n_heads=8):
super().__init__()
self.self_attn = nn.MultiheadAttention(d_model, n_heads, batch_first=True)
self.mem_attn = nn.MultiheadAttention(d_model, n_heads, batch_first=True)
self.fuse = nn.Sequential(
nn.Linear(3*d_model, d_model),
nn.Sigmoid()
)
self.out = nn.Linear(d_model, d_model)
def forward(self, I, Z_E, Z_A):
"""
I: [B, 1, d_model]
Z_E: [B, N_E, d_model]
Z_A: [B, N_A, d_model]
"""
I_self, _ = self.self_attn(I, I, I) # [B,1,d]
mem = torch.cat([Z_E, Z_A], dim=1) # [B,N_E+N_A,d]
I_mem, _ = self.mem_attn(I, mem, mem) # [B,1,d]
fused = torch.cat([I, I_self, I_mem], dim=-1) # [B,1,3d]
gate = self.fuse(fused) # [B,1,d] in (0,1)
I_next = gate * I_self + (1.0 - gate) * I_mem # [B,1,d]
return self.out(I_next) # [B,1,d]
\end{lstlisting}
\subsection{Multiscale Operators \(S_{\downarrow}\) and \(S_{\uparrow}\)}
\begin{lstlisting}[style=py,caption={Linear compress/expand operators for one level.}]
class LinearCoarsen(nn.Module):
def __init__(self, d_model=512, r=4): # r: reduction factor in sequence length
super().__init__()
self.proj = nn.Linear(d_model, d_model)
self.pool = nn.AvgPool1d(kernel_size=r, stride=r)
def forward(self, X):
# X: [B, N, d] -> [B, N/r, d]
Xp = self.proj(X)
Xt = Xp.transpose(1,2) # [B,d,N]
Yt = self.pool(Xt) # [B,d,N/r]
return Yt.transpose(1,2) # [B,N/r,d]
class LinearExpand(nn.Module):
def __init__(self, d_model=512, r=4):
super().__init__()
self.proj = nn.Linear(d_model, d_model)
self.upsample = nn.Upsample(scale_factor=r, mode="nearest")
def forward(self, X):
# X: [B, N', d] -> [B, r*N', d]
Xp = self.proj(X)
Xt = Xp.transpose(1,2) # [B,d,N']
Yt = self.upsample(Xt) # [B,d,r*N']
return Yt.transpose(1,2) # [B,r*N',d]
\end{lstlisting}
\subsection{Reference Shapes (illustrative)}
\begin{itemize}[leftmargin=1.25em]
\item Level 0 (working): \([B, 1024, d]\)
\item Level 1 (episodic): \([B, 256, d]\)
\item Level 2 (semantic): \([B, 64, d]\)
\item Level 3 (thematic): \([B, 16, d]\)
\end{itemize}
% =========================================================
% 5. Resource and Complexity Model
% =========================================================
\section{Resource and Complexity Model}\label{sec:resources}
% Goal: symbolic formulas + worked examples; two profiles (full-tensor vs digest).
\subsection{Storage per Sample}
Let levels have lengths \(n_\ell\) and width \(d\), float32. \textbf{Per-replica} per-sample storage (no redundancy factor other than replication):
\begin{equation}
S_{\mathrm{sample}} = 4\, d \sum_{\ell=0}^{L} n_\ell \;\; \text{bytes}. \quad \text{(per replica)}
\end{equation}
With replication over \(|\mathcal{V}|\) nodes (simple replication model), \textbf{cluster total} per-sample storage:
\begin{equation}
S_{\mathrm{cluster}} = |\mathcal{V}| \cdot S_{\mathrm{sample}}.
\end{equation}
\textbf{Example:} \((n_0,n_1,n_2,n_3)=(1024,256,64,16), d=512\) \(\Rightarrow\) \(S_{\mathrm{sample}}\approx 2.66\) MB; with \(|\mathcal{V}|=16\), \(S_{\mathrm{cluster}}\approx 42.6\) MB (per sample).
\subsection{Cross-Check Bandwidth}
Two modes:
\begin{itemize}[leftmargin=1.25em]
\item \textbf{Full-tensor mode:} transmit \(4\, d \sum_\ell n_\ell\) bytes per update and per participating replica.
\item \textbf{Digest mode:} transmit robust sketches/checksums \(O\!\left(\sum_\ell n_\ell\right)\) bytes per update; on mismatch, fetch tensors for repair.
\end{itemize}
\subsection{Identity Projector Cost}
Let identity length \(Q\) (typically \(Q=1\)), memory length \(N=N_E+N_A\), heads \(H\), head width \(d_h=d/H\). Cross-attention cost:
\begin{equation}
\mathrm{FLOPs} \approx O\!\left(H \, Q \, N \, d_h\right) + O(H\, d\, d_h)
\end{equation}
(projections plus attention). For \(Q=1\), cost is linear in \(N\). If \(Q>1\), use \(O(H Q N d_h)\).
\subsection{Memory Consolidation Cost}
Assuming linear compress/expand with sequence pooling:
\begin{equation}
\mathrm{FLOPs} \approx \sum_{\ell=0}^{L-1} \big( a_\ell \, n_\ell d^2 + b_\ell \, n_{\ell+1} d^2 \big),
\end{equation}
where \(a_\ell,b_\ell\) depend on chosen modules (linear, conv, etc.).
\subsection{Two Resource Profiles (report both)}
\begin{itemize}[leftmargin=1.25em]
\item \textbf{Digest profile:} \(Q{=}1\), digest cross-checks; report FLOPs as functions and one numeric example \((N,d,H)\).
\item \textbf{Full-tensor profile:} \(Q\ge 1\), full-tensor cross-checks; report increased bandwidth/latency.
\end{itemize}
% =========================================================
% 6. Evaluation and Verification
% =========================================================
\section{Evaluation and Verification}\label{sec:evaluation}
% Goal: falsifiable tests with metrics, units, thresholds; tie to theorems and resource model.
\subsection{Identity Drift Tests}
\begin{itemize}[leftmargin=1.25em]
\item \textbf{Benign updates:} track \(D_T\) and \(\cos(I_0,I_T)\) over \(T\) steps with/without anchor mixing. Accept if \(D_T \le \delta\) and \(\cos(I_0,I_T) \ge s_{\min}\).
\item \textbf{Noise robustness:} inject bounded noise in \(Z_{E,n}, Z_{A,n}\); verify observed \(\sup_n \Delta_n\) matches Proposition~\ref{prop:drift} bound.
\end{itemize}
% --- Inline Figure: Drift over time (benign vs noisy) ---
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.8\textwidth,
height=6cm,
xlabel={steps},
ylabel={cumulative drift $D_t$ (arb.)},
legend style={at={(0.98,0.02)},anchor=south east},
ymin=0, xmin=0, xmax=2000
]
% benign: ~ a * sqrt(x)
\addplot+[thick] expression[domain=0:2000,samples=400]{0.002*sqrt(x)}; \addlegendentry{benign}
% noisy: benign + bump (1-exp(-kx)) + small sinusoidal term
\addplot+[thick,dashed] expression[domain=0:2000,samples=400]{0.002*sqrt(x) + 0.02*(1 - exp(-0.01*x)) + 0.002*sin(deg(0.02*x))};
\addlegendentry{noisy}
\end{axis}
\end{tikzpicture}
\caption{Cumulative identity drift $D_t$ over steps for benign vs.\ noisy updates. Thresholds map to Sec.~\ref{sec:evaluation}.}
\label{fig:drift_over_time}
\end{figure}
\subsection{Memory Retention and Self-Repair}
\begin{itemize}[leftmargin=1.25em]
\item \textbf{Retention:} after training on new items, probe recall of old items; report accuracy \(\ge r_{\min}\) and per-level reconstruction errors \(\eta_\ell \le \epsilon_\ell\).
\item \textbf{Shard failures:} randomly corrupt up to \(f\) nodes (Byzantine and non-Byzantine); measure accuracy drop \(\le \Delta_{\max}\) and repair latency distribution (median/p95).
\end{itemize}
% --- Inline Figure: Repair latency CDF for different corruption rates ---
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.8\textwidth,
height=6cm,
xlabel={repair latency (s)},
ylabel={CDF},
legend style={at={(0.98,0.3)},anchor=east},
ymin=0,ymax=1,xmin=0,xmax=2.5
]
% model CDFs with logistic-like curves
\addplot+[thick] expression[domain=0:2.5,samples=300]{1/(1 + exp(-8*(x-0.25)))}; \addlegendentry{10\% corruption}
\addplot+[thick,dashed] expression[domain=0:2.5,samples=300]{1/(1 + exp(-6*(x-0.35)))}; \addlegendentry{25\% corruption}
\addplot+[thick,dashdotted] expression[domain=0:2.5,samples=300]{1/(1 + exp(-4*(x-0.55)))}; \addlegendentry{40\% corruption}
\end{axis}
\end{tikzpicture}
\caption{Repair latency CDF at 10\%, 25\%, and 40\% corruption rates. Use this to set $p95$ latency targets in the validation suite.}
\label{fig:repair_latency_cdf}
\end{figure}
% --- Inline Figure: Retention accuracy over new training batches ---
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.8\textwidth,
height=6cm,
xlabel={new training batches},
ylabel={old-memory retention accuracy},
legend style={at={(0.98,0.98)},anchor=north east},
ymin=0.7,ymax=1.0,xmin=0,xmax=100
]
\addplot+[thick] expression[domain=0:100,samples=200]{0.92 - 0.0005*x + 0.005*exp(-x/12)};
\addlegendentry{baseline}
\addplot+[thick,dashed] expression[domain=0:100,samples=200]{min(1.0, 0.92 - 0.0005*x + 0.005*exp(-x/12) + 0.03*exp(-x/20))};
\addlegendentry{with anchor manifold}
\end{axis}
\end{tikzpicture}
\caption{Old-memory retention accuracy across new training batches. The anchor manifold variant slows retention decay.}
\label{fig:retention_over_time}
\end{figure}
\subsection{Checkpoint and Log Replay}\label{sec:replay}
\paragraph{Log entry schema.} Each entry \(e_k \in \AL\) is a tuple
\[
e_k = (\mathrm{ts}_k,\, \Delta I_k,\, \Delta M_k,\, \mathrm{etype}_k,\, \mathrm{meta}_k),
\]
where \(\mathrm{ts}\) is a timestamp or step index, \(\Delta I\) is an identity delta (or hash), \(\Delta M\) is a set of memory deltas or digests per level, \(\mathrm{etype}\) indicates event class, and \(\mathrm{meta}\) contains auxiliary data (e.g., salience).
\paragraph{Replay pseudocode.}
\begin{algorithm}[h]
\caption{Replay(\(\AL\)) for Recovery}
\begin{algorithmic}[1]
\Function{Replay}{$I_{t_j}, \mathcal{M}_{t_j}, \AL_{(t_j,t_{j+1}]}}$
\State $(I, \mathcal{M}) \gets (I_{t_j}, \mathcal{M}_{t_j})$
\For{each $e_k$ in chronological order}
\State apply $\Delta M_k$ to $\mathcal{M}$ (verify via digest; fetch full blocks on mismatch)
\State $I \gets (1-\alpha)I + \alpha\,\Pi(I, \Gamma(E_k), \Upsilon(\AL_{0:k}))$ \Comment{contractive step}
\State $I \gets (1-\gamma)I + \gamma\,\Pi_{\A}(I)$
\EndFor
\State \Return $(I, \mathcal{M})$
\EndFunction
\end{algorithmic}
\end{algorithm}
% --- Inline Figure: Restoration error vs checkpoint interval ---
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.8\textwidth,
height=6cm,
xlabel={checkpoint interval (minutes)},
ylabel={restoration error (norm difference)},
ymin=0,xmin=0,xmax=60
]
\addplot+[mark=*] coordinates {(1,0.055) (2,0.062) (5,0.075) (10,0.095) (20,0.135) (30,0.175) (60,0.29)};
\end{axis}
\end{tikzpicture}
\caption{Restoration error versus checkpoint interval. Choose intervals to satisfy the bound in Sec.~\ref{sec:replay}.}
\label{fig:restoration_error_vs_interval}
\end{figure}
\paragraph{Accumulated error.} If per-step log perturbations are bounded by \(\omega\) and the identity update is contractive with factor \(\kappa<1\), then the replay identity error after \(|\AL|\) steps is \(\le \frac{\omega+\varepsilon}{1-\kappa}\).
\subsection{Symbolic-Latent Audit and \texorpdfstring{$\Psi(I)$}{Psi(I)}}\label{sec:psi}
\textbf{Definition.} \(\Psi:\R^d\to\mathcal{S}\) is a learned decoder from latent identity to an interpretable self-model (e.g., trait vector, goal logits, or textual summary embedding).\\
\textbf{Training.} Fit \(\Psi\) jointly with identity dynamics using a bi-Lipschitz regularizer on \(\A\) so changes in \(I\) correspond to proportional changes in \(\Psi(I)\).\\
\textbf{Audit metric and acceptance.} Compute Spearman rank correlation over a held-out trajectory:
\[
\rho = \mathrm{Spearman}\Big(\cos(I_n,I_{n'}),\; \mathrm{sim}\big(\Psi(I_n),\Psi(I_{n'})\big)\Big).
\]
Accept if \(\rho \ge 0.85\) at \(\alpha=0.05\) (two-sided) with \(N\ge 200\) pairs.
% =========================================================
% 7. Limitations and Failure Modes
% =========================================================
\section{Limitations and Failure Modes}\label{sec:limits}
% Goal: enumerate failure signatures, triggers, mitigations, and escalation rules.
\subsection{Failure Catalog (structured)}
\begin{lstlisting}[style=py,caption={Failure modes and mitigations (structured).}]
FAILURE_MODES = {
"identity_collapse": {
"symptoms": ["cos(I_t, I_0) < s_min", "Delta_n spikes", "dist(I_t, A) increase"],
"triggers": ["unbounded salience", "distribution shift"],
"mitigations": ["reduce alpha", "increase gamma", "anchor projection", "checkpoint restore"],
"escalation": {"after_retries": 3, "action": "hard_restore"}
},
"memory_cascade_failure": {
"symptoms": ["chi_l > tau_l for multiple levels"],
"triggers": ["correlated node failures", "operator mismatch in S_up/S_down"],
"mitigations": ["degradation mode", "selective shard sacrifice", "operator retraining"],
"escalation": {"after_retries": 2, "action": "isolate_nodes_and_restore"}
}
}
\end{lstlisting}
\subsection{Adversarial Robustness}
If at most \(f\) of \(|\mathcal{V}|\) nodes submit arbitrary values and \(|\mathcal{V}|\ge 3f+1\), trimmed-mean with trim proportion \(>f/|\mathcal{V}|\) ensures
\[
\norm{R^{(\ell)}(\{M_v\}) - M^{(\ell)}} \le O(\delta).
\]
This extends repair guarantees in Sec.~\ref{sec:memory} to Byzantine submissions \cite{lamport1982,huber2004}.
% =========================================================
% 8. Relation to Modern Architectures (expanded)
% =========================================================
\section{Relation to Modern Architectures}\label{sec:relation}
\paragraph{Transformer integration example.} Implement \(I_n\) as a persistent learned token prepended to the sequence at each step. Replace a monolithic KV cache with multiscale memory shards \(\{M^{(\ell)}\}\). Use \(\Pi\) as a cross-attention block where the identity token queries \([Z_E; Z_A]\). Consolidation runs intermittently to update higher levels. Redundancy is implemented across model-parallel shards with digest cross-checks and on-demand full-tensor repair.
% =========================================================
% 9. Notation Summary
% =========================================================
\section{Notation Summary}\label{sec:notation}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Symbol & Meaning \\
\midrule
\(I_n \in \R^d\) & Latent identity at step \(n\) \\
\(\mathcal{M}_n = \{M^{(0)}_n,\dots,M^{(L)}_n\}\) & Multiscale memory levels \\
\(S_{\down}^{(\ell)}, S_{\up}^{(\ell)}\) & Coarsen/expand operators \\
\(\G=(\mathcal{V},\mathcal{E})\) & Redundancy graph \\
\(C^{(\ell)}, R^{(\ell)}\) & Cross-check and repair operators \\
\(\Gamma, \Upsilon\) & Episodic compressor and autobiographical summarizer \\
\(\Pi, \Pi_{\A}\) & Identity projector and anchor projector \\
\(\A\) & Anchor manifold \\
\(\alpha,\gamma\) & Identity step size and anchor mixing weight \\
\(\kappa\) & Contraction factor \((1-\alpha)+\alpha L_I\) \\
\(\eta_\ell\) & Expand–coarsen reconstruction error at level \(\ell\) \\
\(\rho,\delta\) & Fraction and magnitude of corrupted shards \\
\(\chi^{(\ell)}\) & Inconsistency score at level \(\ell\) \\
\(D_T\) & Cumulative identity drift over horizon \(T\) \\
\bottomrule
\end{tabular}
\end{center}
% =========================================================
% 10. Architecture Overview Figure (inline TikZ)
% =========================================================
\section{Architecture Overview (Inline Figure)}
\begin{figure}[h]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
% Multiscale rectangles
\def\xo{0.5}
\def\yA{6.0} \def\wA{10.0} \def\h{0.8}
\def\yB{4.5} \def\wB{8.0}
\def\yC{3.0} \def\wC{6.0}
\def\yD{1.5} \def\wD{4.5}
\draw (\xo,\yA) rectangle ++(\wA,\h);
\node[anchor=west] at (\xo+0.2,\yA+0.4) {Level 0: Working ($N{=}1024, d$)};
\draw (\xo,\yB) rectangle ++(\wB,\h);
\node[anchor=west] at (\xo+0.2,\yB+0.4) {Level 1: Episodic ($N{=}256, d$)};
\draw (\xo,\yC) rectangle ++(\wC,\h);
\node[anchor=west] at (\xo+0.2,\yC+0.4) {Level 2: Semantic ($N{=}64, d$)};
\draw (\xo,\yD) rectangle ++(\wD,\h);
\node[anchor=west] at (\xo+0.2,\yD+0.4) {Level 3: Thematic ($N{=}16, d$)};
% Coarsen arrows (right side, downward)
\draw[-{Stealth[length=3mm]}] (\xo+\wA+0.2,\yA+\h) -- ++(0,-1.3) node[midway,right,rotate=90] {$S_{\downarrow}$};
\draw[-{Stealth[length=3mm]}] (\xo+\wB+0.2,\yB+\h) -- ++(0,-1.3) node[midway,right,rotate=90] {$S_{\downarrow}$};
\draw[-{Stealth[length=3mm]}] (\xo+\wC+0.2,\yC+\h) -- ++(0,-1.3) node[midway,right,rotate=90] {$S_{\downarrow}$};
% Expand arrows (left side, upward)
\draw[-{Stealth[length=3mm]}] (\xo-0.2,\yB) -- ++(0,1.3) node[midway,left,rotate=90] {$S_{\uparrow}$};
\draw[-{Stealth[length=3mm]}] (\xo-0.2,\yC) -- ++(0,1.3) node[midway,left,rotate=90] {$S_{\uparrow}$};
\draw[-{Stealth[length=3mm]}] (\xo-0.2,\yD) -- ++(0,1.3) node[midway,left,rotate=90] {$S_{\uparrow}$};
% Redundancy graph overlay on Level 1
\pgfmathsetmacro{\nodes}{8}
\foreach \i in {0,...,7}{
\pgfmathsetmacro{\xi}{\xo+0.6 + \i*(\wB-1.2)/7}
\draw (\xi,\yB+0.4) circle (0.12);
}
% ring connections
\foreach \i in {0,...,7}{
\pgfmathtruncatemacro{\j}{mod(\i+1,8)}
\pgfmathsetmacro{\xii}{\xo+0.6 + \i*(\wB-1.2)/7}
\pgfmathsetmacro{\xjj}{\xo+0.6 + \j*(\wB-1.2)/7}
\draw (\xii,\yB+0.4) -- (\xjj,\yB+0.4);
}
% chords
\foreach \i in {0,...,7}{
\pgfmathtruncatemacro{\k}{mod(\i+2,8)}
\pgfmathsetmacro{\xii}{\xo+0.6 + \i*(\wB-1.2)/7}
\pgfmathsetmacro{\xkk}{\xo+0.6 + \k*(\wB-1.2)/7}
\draw[line width=0.6pt] (\xii,\yB+0.4) -- (\xkk,\yB+0.4);
}
\node at (\xo+\wB/2,\yB-0.4) {Redundancy graph $\mathcal{G}$ over shards};
% Identity update loop (right side)
\def\cx{11.8} \def\cy{5.5}
\draw (\cx,\cy) circle (0.4);
\node at (\cx,\cy) {$I_n$};
\draw (\cx-0.6,4.7) rectangle ++(1.2,0.5);
\node at (\cx,4.95) {$\Pi_{\mathcal{A}}$};
\draw[-{Stealth[length=3mm]}] (\cx,\cy-0.4) -- (\cx,5.2);
\draw[-{Stealth[length=3mm]}] (\cx,5.2) -- (\cx,\cy+0.4);
\node[anchor=west] at (\cx+0.2,5.2) {anchor mix};
% Cross-attention arrow from memory to identity
\draw[-{Stealth[length=3mm]}] (\xo+\wA, \yA+0.4) -- (\cx-0.5,\cy) node[midway,above] {$\Pi$ (cross-attn)};
\end{tikzpicture}
\caption{Architecture overview with multiscale levels, redundancy graph, and identity update loop with anchor projection.}
\label{fig:memory_pyramid}
\end{figure}
% =========================================================
% Bibliography
% =========================================================
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\end{document}
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