Persistent Memory and Identity Anchoring in Recursive Agents:
Identity, in this framework, is understood as a continuous process of maintaining coherence rather than as a fixed or separate self. Continuity of the “self” arises from measurable feedback among memory, perception, and adaptation systems that work to preserve stability. This stability depends on the system’s capacity to detect and correct small deviations, restoring internal consistency as conditions change. In this sense, persistence of identity is not a metaphysical notion but an observable property of self-regulating systems—a description of how order is maintained within structures that adapt over time.
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\title{\vspace{-0.5em}\textbf{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
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\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}
\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}
\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}
\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}
\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}
\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}
\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}
\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
]
\addplot+[thick] expression[domain=0:2000,samples=400]{0.002*sqrt(x)}; \addlegendentry{benign}
\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
]
\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}
\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}
% =========================================================
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% =========================================================
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\end{document}
