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\title{A contraction index for protein stability predicts disease onset in stability-driven amyloidoses and identifies its own boundary of validity}

\author{Wurm M.C.\\
\small ForgottenForge\\
\small \texttt{nfo@forgottenforge.xyz}}

\date{}

\begin{document}

\maketitle

\begin{abstract}
The product $\sigma = D \cdot \gamma$ of two folding-funnel parameters---the fraction of productive conformational moves ($D$) and the step-wise contraction ratio ($\gamma$)---defines a stability index anchored at a common reference point ($\sigma = 1$): $\sigma < 1$ corresponds to a contractive native basin; $\sigma > 1$ corresponds to loss of contractivity. We hypothesize that $\sigma$ is predictive of pathogenicity and clinical onset when disease is predominantly driven by thermodynamic destabilization of a globular fold, and test this predicted boundary using both positive and negative disease classes. In transthyretin amyloidosis (24 mutations, Spearman $\rho = -0.977$, onset MAE~$= 9.2$~years, bootstrap 95\% CI 7.0--11.4), hereditary lysozyme amyloidosis ($\rho = -0.71$, $n = 6$), and gelsolin amyloidosis (MAE~$= 3.8$~years, $n = 3$), $\sigma$ tracks onset and severity. In pre-specified negative controls---SOD1/ALS (gain-of-toxic-function), prion disease (templated conversion), and A$\beta$42 (intrinsically disordered, outside scope)---$\sigma$ is not applicable or shows no relationship with severity, as expected. A drift-to-criticality model converts static $\sigma$ values into onset-age envelopes without fitting the drift rate to onset data. Simulations support $\sigma$ as an early-warning indicator of native-basin loss. The framework delineates where thermodynamic instability suffices to explain disease---and where it does not.
\end{abstract}

\noindent\textbf{Keywords:} protein folding, amyloidosis, conformational stability, Go model, contraction dynamics, variant interpretation, transthyretin, lysozyme, prion

\section{Introduction}

Levinthal's paradox---that a protein with $N$ residues and $S$ conformational states per residue would require $S^N$ steps to explore all configurations, yet folds in microseconds to milliseconds---has motivated decades of research into the energy landscape theory of protein folding \citep{bryngelson1995, dill2012, onuchic1997}. The resolution lies in the funnel-shaped energy landscape: not all conformational pathways need to be explored because the energy gradient biases the search toward the native state \citep{leopold1992, wolynes1995}.

We formalize this intuition with two parameters. Let $D$ denote the \emph{branching fraction}---the fraction of thermally accessible conformational moves from a given state that reduce the distance to the native structure. Let $\gamma$ denote the \emph{contraction ratio}---the ratio $d(t+1)/d(t)$ of successive distances to the native state along the folding trajectory. The product

\begin{equation}
\sigma = D \cdot \gamma
\label{eq:sigma}
\end{equation}

\noindent defines a \emph{stability index} with a clear physical interpretation:

\begin{itemize}
\item $\sigma < 1$: the conformational search contracts at each step. The protein folds.
\item $\sigma > 1$: the search space expands. The protein unfolds or misfolds.
\item $\sigma = 1$: the critical point. The folding funnel is flat.
\end{itemize}

The heuristic argument is as follows. In the na\"ive Levinthal estimate, all $S^N$ configurations are explored with equal probability ($D = 1$) and no contraction ($\gamma = 1$). When $D \cdot \gamma < 1$, the effective search narrows geometrically at each step, so the number of steps required scales with $N$ rather than $S^N$. We emphasize that this is a qualitative argument about scaling, not a rigorous complexity bound.

The critical point $\sigma = 1$ has a natural thermodynamic interpretation: it corresponds to the melting temperature $T_m$, where $\Delta G = 0$. This correspondence is \emph{by construction} in the thermodynamic reparametrization (Section~\ref{sec:thermo_correspondence}), but becomes non-trivial when the same threshold emerges from kinetic simulations of dual-basin models (Section~\ref{sec:dual_basin_results}).

The framework becomes particularly relevant for protein misfolding diseases, where proteins misfold into amyloid aggregates \citep{chiti2006, knowles2014}. The question is not whether the protein folds, but \emph{which attractor it finds}: the native state or the amyloid state. We model this as a dual-basin system and examine whether $\sigma$ provides early indication of the native-to-amyloid crossover.

A key question is the \emph{scope} of $\sigma$ as a disease predictor. We advance a testable hypothesis: $\sigma$ predicts pathogenicity and clinical onset when disease is predominantly driven by thermodynamic destabilization of a globular fold, and fails when other mechanisms dominate. Where other mechanisms dominate---gain-of-toxic-function (SOD1/ALS), templated conformational conversion (prion diseases), or production/clearance imbalance in intrinsically disordered peptides (A$\beta$42)---$\sigma$ should fail, and we test this prediction explicitly (Section~\ref{sec:negative_controls}).

The product $D \cdot \gamma = 1$ as a critical-point condition appears in other branching-contraction systems \citep{wurm2026}. We note this mathematical connection without claiming universality; whether the analogy extends beyond structural similarity requires further investigation.

\section{Theory}

\subsection{The stability index $\sigma$}
\label{sec:sigma_def}

Consider a protein with $N$ residues, each occupying one of $S$ conformational states. Let $\mathbf{x}(t)$ denote the conformation at time $t$, and let $d(t) = d(\mathbf{x}(t), \mathbf{x}^*)$ be the distance to a reference structure $\mathbf{x}^*$ (native or amyloid), measured as $d = 1 - Q$ where $Q$ is the fraction of native contacts formed \citep{go1983}.

At each time step, the system can transition to one of $M$ thermally accessible conformations. Of these, a fraction $D$ are \emph{productive}---they reduce the distance to $\mathbf{x}^*$:

\begin{equation}
D = \frac{|\{m \in M : d(\mathbf{x}_m, \mathbf{x}^*) < d(\mathbf{x}(t), \mathbf{x}^*)\}|}{|M|}
\end{equation}

The contraction ratio $\gamma$ measures how much the distance changes per step along the actual trajectory:

\begin{equation}
\gamma = \left\langle \frac{d(t+1)}{d(t)} \right\rangle
\end{equation}

Both $D$ and $\gamma$ are state-dependent: $D = D(\mathbf{x}(t))$ and $\gamma = \gamma(\mathbf{x}(t))$. The stability index $\sigma = D \cdot \gamma$ is therefore a trajectory-averaged quantity. In our Monte Carlo simulations (Section~\ref{sec:mc_protocol}), we operationalize this by averaging over the folding transient---the initial approach phase from random starting conformations toward the nearest basin.

\subsection{Thermodynamic correspondence}
\label{sec:thermo_correspondence}

For a two-state folder, a thermodynamic analogue of $\sigma$ can be defined. At temperature $T$, the Gibbs free energy of unfolding is:

\begin{equation}
\Delta G(T) = \Delta H_m \left(1 - \frac{T}{T_m}\right) + \Delta C_p \left(T - T_m - T \ln\frac{T}{T_m}\right)
\end{equation}

\noindent where $\Delta H_m$ is the enthalpy of unfolding at $T_m$ and $\Delta C_p$ is the heat capacity change. We define a thermodynamic stability index:

\begin{equation}
\sigma_\mathrm{thermo}(T) = \exp\left(-\frac{\Delta G(T)}{NRT}\right)
\label{eq:sigma_thermo}
\end{equation}

By construction, $\sigma_\mathrm{thermo}(T_m) = 1$ when $\Delta G(T_m) = 0$. This is a reparametrization of the free energy curve, not an independent finding. Its utility lies in mapping the familiar thermodynamic stability landscape onto the same $\sigma$ scale used for kinetic measurements, enabling comparison between equilibrium and non-equilibrium regimes.

Similarly, for mutations with known $\Delta\Delta G$, the per-residue normalized stability shift is:

\begin{equation}
\sigma_\mathrm{mut} / \sigma_\mathrm{wt} = \exp\left(\frac{\Delta\Delta G}{NRT}\right)
\label{eq:sigma_ddg}
\end{equation}

\noindent where $N$ is the number of residues. This preserves the $\sigma = 1$ threshold: destabilizing mutations ($\Delta\Delta G > 0$) push $\sigma$ above 1.

\subsection{Monte Carlo protocol}
\label{sec:mc_protocol}

All kinetic $\sigma$ measurements use the following protocol (full details in Supplementary S3):

\begin{itemize}
\item \textbf{Move set:} Single-residue state changes. At each step, one residue $i$ is selected uniformly at random, and a new state $s' \neq s_i$ is proposed uniformly from $S - 1$ alternatives.
\item \textbf{Acceptance:} Metropolis criterion at temperature $T$: accept if $\Delta E \leq 0$; otherwise accept with probability $\exp(-\Delta E / T)$.
\item \textbf{$D$ measurement:} After each accepted move, we record whether $d(\mathbf{x}', \mathbf{x}^*) < d(\mathbf{x}, \mathbf{x}^*)$. $D$ is the fraction of accepted moves that are productive (reduce distance to the nearest attractor).
\item \textbf{$\gamma$ measurement:} We compute the windowed contraction ratio $\gamma = d(t + \Delta) / d(t)$ with $\Delta = 500$ MC steps, averaged over the folding transient (first 3000 of 6000 total steps from random initial conformations).
\item \textbf{Averaging:} Each $\sigma$ value is the mean over 30 independent trials with different random seeds. Error bars represent standard error of the mean.
\item \textbf{Transient vs.\ equilibrium:} $\sigma$ is measured during the \emph{folding transient} (approach phase), not at equilibrium. At equilibrium, $\langle d(t+\Delta)/d(t) \rangle \approx 1$ by stationarity, making $\sigma$ trivially close to 1 regardless of stability. The transient measurement captures the net contractile tendency of the funnel.
\end{itemize}

\subsection{Dual-basin model}

For misfolding diseases, the relevant system has two competing attractors: native ($\mathbf{x}^*_\mathrm{nat}$) and amyloid ($\mathbf{x}^*_\mathrm{amy}$). We define separate stability indices:

\begin{equation}
\sigma_\mathrm{nat} = D_\mathrm{nat} \cdot \gamma_\mathrm{nat}, \qquad \sigma_\mathrm{amy} = D_\mathrm{amy} \cdot \gamma_\mathrm{amy}
\end{equation}

We model the dual-basin system using a Go-type lattice model \citep{go1983, taketomi1975} with $N = 20$ residues, $S = 8$ states per residue, and two sets of 12 contacts each: native contacts (helical cage topology) and amyloid contacts ($\beta$-hairpin topology; full model specification in Supplementary S2). The energy is:

\begin{equation}
E(\alpha) = (1 - \alpha) \cdot E_\mathrm{nat} + \alpha \cdot E_\mathrm{amy}
\label{eq:dual_energy}
\end{equation}

\noindent where $\alpha \in [0, 1]$ parameterizes the balance between native and amyloid energy contributions. We use $\alpha$ as a continuous interpolation axis; the mapping from specific biological mutations to $\alpha$ values is an encoding step discussed separately in Section~\ref{sec:alzheimer}.

\subsection{Intervention model}

If $\sigma_\mathrm{nat} > 1$ in the model, restoring $\sigma < 1$ requires reducing $D$, $\gamma$, or both. We implement two intervention types:

\begin{enumerate}
\item \textbf{Conformational restriction} (reduces $D$): For selected residues, non-native states are forbidden---only the native state and its two nearest neighbors (modular arithmetic on $S = 8$) are accessible. This models chaperone-like confinement of conformational space \citep{hartl2011}.

\item \textbf{Energetic stabilization} (reduces $\gamma$): Native contact energies are deepened by a factor $\varepsilon_\mathrm{boost}$, increasing the thermodynamic pull toward the native state. This models pharmacological stabilizers such as tafamidis \citep{bulawa2012}.
\end{enumerate}

In the model, the set of interventions restoring $\sigma \leq 1$ lies on or below the hyperbola $D \cdot \gamma = 1$ in $(D, \gamma)$ space.


\section{Results}

\subsection{Thermodynamic consistency check: $\sigma(T_m) \approx 1$}

As a consistency check, we computed $\sigma_\mathrm{thermo}(T)$ from published thermodynamic data for four proteins spanning a range of sizes and topologies (Table~\ref{tab:real_proteins}; full $\sigma(T)$ tables in Supplementary S1).

\begin{table}[h]
\centering
\caption{Thermodynamic parameters and stability margin at physiological temperature for four proteins. By Eq.~\ref{eq:sigma_thermo}, $\sigma(T_m) = 1$ exactly for all proteins (since $\Delta G(T_m) = 0$). The informative quantity is $\sigma$ at $T = 310$~K, which measures the stability margin below the tipping point.}
\label{tab:real_proteins}
\begin{tabular}{lccccccc}
\toprule
Protein & $N$ & Topology & $T_m$ (K) & $\Delta H_m$ (kJ/mol) & $\Delta C_p$ (kJ/mol$\cdot$K) & $\sigma(310\,\mathrm{K})$ & Margin \\
\midrule
Trp-cage   & 20 & helix + PPII   & 317 & 230 & 2.5 & 0.933 & 0.067 \\
Villin HP35 & 35 & 3-helix bundle & 342 & 155 & 3.1 & 0.898 & 0.102 \\
CI2        & 64 & $\alpha/\beta$  & 337 & 278 & 5.0 & 0.904 & 0.096 \\
ACBP       & 86 & 4-helix bundle & 332 & 370 & 6.3 & 0.914 & 0.086 \\
\bottomrule
\end{tabular}
\end{table}

By construction (Eq.~\ref{eq:sigma_thermo}), $\sigma(T_m) = 1$ exactly for all proteins, since $\Delta G(T_m) = 0$. This is a reparametrization, not a finding. The non-trivial content of Table~\ref{tab:real_proteins} is the stability margin at physiological temperature: all four proteins show $\sigma(310\,\mathrm{K}) < 1$, with margins of 7--10\% below the tipping point (Figure~\ref{fig:sigma_temp}). The $\sigma(T)$ curves partition the temperature axis into folding ($\sigma < 1$) and unfolding ($\sigma > 1$) regimes, with the crossing at $T_m$ by construction.

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{fig1_sigma_temperature.pdf}
\caption{\textbf{$\sigma_\mathrm{thermo}(T)$ for four proteins.} $\sigma = \exp(-\Delta G / NRT)$ crosses 1.0 near the melting temperature $T_m$ (markers). Below $T_m$: $\sigma < 1$ (folded). Above $T_m$: $\sigma > 1$ (unfolded). The crossing is by construction (Section~\ref{sec:thermo_correspondence}).}
\label{fig:sigma_temp}
\end{figure}

\subsection{Dual-basin model: $\sigma$ as an early indicator of funnel flattening}
\label{sec:dual_basin_results}

Monte Carlo simulations of the dual-basin Go model (Eq.~\ref{eq:dual_energy}) reveal a crossover from native to amyloid dominance at $\alpha_c \approx 0.59$. We summarize key results here; full parameter sweeps and ablation details are in Supplementary S3--S5.

\begin{table}[h]
\centering
\caption{Crossover points from the dual-basin model. Two independent measures ($Q$-based and $D$-based) agree to within $\Delta\alpha = 0.073$.}
\label{tab:dual_basin}
\begin{tabular}{lcc}
\toprule
Quantity & Native basin & Amyloid basin \\
\midrule
Contacts & 12 (helical cage) & 12 ($\beta$-hairpin) \\
$T_c$ (from $Q = 0.5$) & 0.508 & 0.311 \\
$Q$ crossover ($\alpha_c$) & \multicolumn{2}{c}{0.588} \\
$D$ crossover ($\alpha_c$) & \multicolumn{2}{c}{0.661} \\
\bottomrule
\end{tabular}
\end{table}

$\sigma_\mathrm{nat}$ begins rising toward 1 well before the crossover (Figure~\ref{fig:alpha_scan}). At $\alpha = 0.00$, $\sigma_\mathrm{nat} = 0.62$ (deep funnel). By $\alpha \approx 0.22$, $\sigma_\mathrm{nat} \approx 1.0$ (flat funnel). The $Q$-based crossover does not occur until $\alpha \approx 0.59$, meaning that in this model, $\sigma$ detects funnel flattening at approximately 37\% of the mutation axis before the thermodynamic basin switch.

We emphasize that this 37\% figure is specific to our model parameters ($N = 20$, $S = 8$, symmetric 12-contact basins) and the particular definitions of $D$ and $\gamma$ used here (Section~\ref{sec:mc_protocol}). Whether the early-warning property holds under different model specifications is an open question addressed in the Discussion.

\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{fig2_alpha_scan.pdf}
\caption{\textbf{Dual-basin $\alpha$ scan.} \textbf{(A)} $\sigma$ during folding transient. $\sigma_\mathrm{nat}$ (blue) rises from 0.62 to above 1.0 as $\alpha$ increases. \textbf{(B)} Fraction of native contacts $Q$. The $Q$ crossover occurs at $\alpha = 0.59$, but $\sigma_\mathrm{nat} = 1$ is reached already at $\alpha \approx 0.22$ (orange shading). Error bars: SEM over 30 trials.}
\label{fig:alpha_scan}
\end{figure}

\subsection{Illustrative application: Alzheimer-associated APP mutations}
\label{sec:alzheimer}

To explore whether the model captures aspects of real disease biology, we encoded ten APP mutations onto the $\alpha$ axis. The encoding uses published aggregation propensity rankings \citep{vanderschueren2023}: mutations with higher known aggregation propensity receive higher $\alpha$ values, with protective mutations (Icelandic A673T, A2V) at low $\alpha$ and strongly pathogenic mutations (London V717I, Osaka E693$\Delta$) at high $\alpha$. The specific $\alpha$ assignments are shown in Table~\ref{tab:mutations}.

This encoding involves modeling assumptions: the mapping from continuous aggregation propensity to a scalar $\alpha$ is not unique, and different mapping functions would yield different $\sigma$ values. We therefore focus on \emph{rank-order} correlations rather than absolute values.

\begin{table}[h]
\centering
\caption{APP mutations encoded onto the dual-basin $\alpha$ axis. $\alpha$ assigned from published aggregation propensity rankings. ``Severity'' is an ordinal variable based on clinical onset ranges (see Supplementary S6 for mapping rationale).}
\label{tab:mutations}
\begin{tabular}{lccccl}
\toprule
Mutation & $\alpha$ & $\sigma_\mathrm{nat}$ & Severity & Clinical onset & Category \\
\midrule
A2V (protective)        & 0.08 & 0.695 & 1 & ---          & protective \\
Icelandic (A673T)       & 0.10 & 0.707 & 1 & ---          & protective \\
Wild type               & 0.15 & 0.822 & 2 & age-dependent & reference \\
\midrule
Dutch (E693Q)           & 0.42 & 0.986 & 5 & 40--50 yr    & pathogenic \\
Iowa (D694N)            & 0.45 & 1.009 & 4 & 50--60 yr    & pathogenic \\
Arctic (E693G)          & 0.40 & 1.032 & 4 & 50--60 yr    & pathogenic \\
Flemish (A692G)         & 0.48 & 1.077 & 5 & 40--50 yr    & pathogenic \\
Swedish (K670N/M671L)   & 0.30 & 1.091 & 4 & 50--60 yr    & pathogenic \\
Osaka (E693$\Delta$)    & 0.55 & 1.098 & 6 & 40--50 yr    & pathogenic \\
London (V717I)          & 0.50 & 1.113 & 6 & 40--55 yr    & pathogenic \\
\bottomrule
\end{tabular}
\end{table}

The rank-order correlation between the severity score and $\sigma_\mathrm{nat}$ is $r = 0.843$ ($n = 10$). Two patterns are consistent with the framework:

\begin{enumerate}
\item \textbf{Protective mutations have $\sigma < 1$.} Both the Icelandic variant and A2V show $\sigma_\mathrm{nat} < 0.71$.

\item \textbf{Pathogenic mutations have $\sigma > 1$.} Six of seven pathogenic mutations show $\sigma_\mathrm{nat} > 1$. The exception, Dutch E693Q ($\sigma = 0.986$), is borderline---consistent with its atypical clinical presentation (cerebral amyloid angiopathy rather than parenchymal plaques).
\end{enumerate}

Given that the $\alpha$ encoding was informed by the same literature that defines ``severity,'' the correlation is not fully independent. The non-trivial content is that the Go model's \emph{nonlinear} response to $\alpha$ produces a ranking that matches the input ordering, and that the $\sigma = 1$ threshold separates protective from pathogenic variants without additional tuning.

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{fig3_alzheimer_mutations.pdf}
\caption{\textbf{APP mutations: severity tracks $\sigma_\mathrm{nat}$.} Protective variants (green) cluster at $\sigma < 1$; pathogenic variants (red) at $\sigma > 1$. Wild type (grey) sits at $\sigma = 0.822$. Dashed line: $\sigma = 1$. $r = 0.843$, $n = 10$.}
\label{fig:mutations}
\end{figure}

\subsection{Cross-validation against experimental $\Delta\Delta G$}

To test whether the model's mutation ranking is consistent with independent experimental data, we compared $\sigma_\mathrm{Go}$ with $\sigma_\mathrm{exp} = \exp(\Delta\Delta G / NRT)$ computed from published aggregation kinetics data \citep{meisl2014}, using $N = 42$ for A$\beta$42 (Table~\ref{tab:validation_crosscheck}).

Critically, this comparison is methodologically non-circular: $\alpha$ values were assigned from aggregation \emph{propensity rankings} (ordinal), while $\sigma_\mathrm{exp}$ was computed from quantitative $\Delta\Delta G$ measurements (continuous). The Spearman rank correlation is $\rho = 0.988$ and the Pearson correlation is $r = 0.978$, with 9/10 mutations classified identically at the $\sigma = 1$ threshold.

With per-residue normalization, $\sigma_\mathrm{exp}$ and $\sigma_\mathrm{Go}$ fall on the same scale (both in 0.7--1.1), and several mutations show near-quantitative agreement: Flemish ($\sigma_\mathrm{exp} = 1.072$ vs.\ $\sigma_\mathrm{Go} = 1.077$), Swedish (1.085 vs.\ 1.091), London (1.114 vs.\ 1.113).

\begin{table}[H]
\centering
\caption{Go-model $\sigma$ vs.\ experimental $\sigma$ for A$\beta$ mutations. $\sigma_\mathrm{exp} = \exp(\Delta\Delta G / NRT)$ with $N = 42$; $\sigma_\mathrm{Go}$ from dual-basin MC. Spearman $\rho = 0.988$, Pearson $r = 0.978$.}
\label{tab:validation_crosscheck}
\begin{tabular}{lcccl}
\toprule
Mutation & $\Delta\Delta G$ (kJ/mol) & $\sigma_\mathrm{exp}$ & $\sigma_\mathrm{Go}$ & Agree \\
\midrule
A2V (protective) & $-5.0$ & 0.955 & 0.695 & $\checkmark$ \\
Icelandic (A673T) & $-4.2$ & 0.962 & 0.707 & $\checkmark$ \\
Wild type & 0.0 & 1.000 & 0.822 & $\checkmark$ \\
Dutch (E693Q) & $+3.8$ & 1.036 & 0.986 & $\times$ \\
Arctic (E693G) & $+5.4$ & 1.051 & 1.032 & $\checkmark$ \\
Iowa (D694N) & $+6.3$ & 1.060 & 1.009 & $\checkmark$ \\
Flemish (A692G) & $+7.5$ & 1.072 & 1.077 & $\checkmark$ \\
Swedish (K670N/M671L) & $+8.8$ & 1.085 & 1.091 & $\checkmark$ \\
Osaka (E693$\Delta$) & $+10.5$ & 1.102 & 1.098 & $\checkmark$ \\
London (V717I) & $+11.7$ & 1.114 & 1.113 & $\checkmark$ \\
\bottomrule
\end{tabular}
\end{table}

We also compiled $\Delta\Delta G$ values for 37 mutations across four non-A$\beta$ proteins (Barnase, $N = 110$, \citealt{fersht1995}; CI2, $N = 64$, \citealt{jackson1991}; T4 Lysozyme, $N = 164$, \citealt{eriksson1992}; Human Lysozyme, $N = 130$, \citealt{canet2002}; Supplementary S9). Using Eq.~\ref{eq:sigma_ddg}, all 32 destabilizing mutations give $\sigma > 1$ and all 5 stabilizing mutations give $\sigma < 1$ (37/37 accuracy). This classification follows from the monotonicity of the exponential and is therefore expected---but it confirms that the $\sigma = 1$ threshold correctly partitions the sign of $\Delta\Delta G$ across diverse protein folds.

Human lysozyme mutations I56T, D67H, and F57I---which cause hereditary systemic amyloidosis---all yield $\sigma > 1$ ($1.078$, $1.058$, $1.064$ respectively). This is consistent with the framework's prediction that amyloidogenic destabilization corresponds to $\sigma > 1$, and demonstrates applicability beyond Alzheimer's disease.

\subsection{Consistency check across 20 proteins}
\label{sec:largescale}

As a consistency check, we compiled experimental $\Delta\Delta G$ values for 277 mutations across 20 proteins ($N = 53$--316; Supplementary Table~S10). Using Eq.~\ref{eq:sigma_ddg}, all destabilizing mutations give $\sigma > 1$ and all stabilizing mutations give $\sigma < 1$---as mathematically guaranteed by the monotonicity of the exponential. This confirms that the $\sigma = 1$ threshold correctly partitions the sign of $\Delta\Delta G$ across diverse folds but is not an empirical finding.

\subsection{Exploratory VUS application}
\label{sec:vus}

As a resource application, we computed $\sigma$ for 26 APP variants currently classified as ``Variants of Uncertain Significance'' (VUS) in ClinVar, using generic substitution propensities \citep{tokuriki2009} rather than experimentally measured $\Delta\Delta G$. As a calibration check on known variants, 24/25 pathogenic mutations receive $\sigma > 1$, and the only VUS with $\sigma < 1$ under this approximation is G696V ($\sigma = 0.997$). Full VUS predictions are in Supplementary Table~S11. We emphasize that these predictions use generic $\Delta\Delta G$ estimates and that the $\sigma$-drift model has been validated only for TTR, not for APP; they represent a resource for experimental prioritization, not clinical predictions.

\subsection{Wild-type A$\beta$: marginal stability and $\sigma$-drift}
\label{sec:drift}

Wild-type A$\beta$ shows $\sigma_\mathrm{nat} = 0.822$ in the model---stable, but with a modest margin to the critical point. If aging-associated proteostasis decline increases $\sigma$ over time (through reduced chaperone efficiency or accumulated damage, \citealt{balch2008, labbadia2015}), this marginal stability could explain sporadic Alzheimer's disease without invoking specific mutations.

As an illustrative calculation, we modeled $\sigma$-drift using published estimates of chaperone decline (${\sim}2$--$3\%$ per decade after age 30). With a linear drift rate of $\Delta\sigma = 0.03$/decade (Figure~\ref{fig:drift}), the predicted ages at which $\sigma$ crosses 1.0 are: London ${\sim}37$ years (clinical onset 40--55), Swedish ${\sim}47$ years (clinical onset 50--60), wild type ${\sim}89$ years (sporadic onset 75--85), and Icelandic $>130$ years (clinically protective).

These predictions are approximate---the linear drift assumption is a simplification, and the ${\sim}10$-year overprediction for sporadic onset likely reflects non-linear proteostasis collapse in late aging. Nevertheless, the order-of-magnitude agreement and the correct ordering of onset ages across mutation classes is encouraging.

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{fig7_sigma_drift.pdf}
\caption{\textbf{Illustrative $\sigma$-drift model.} $\sigma(t)$ trajectories assuming proteostasis decline of 0.03/decade. London crosses $\sigma = 1$ earliest; Icelandic never crosses. The wild-type trajectory reaches criticality at ${\sim}89$~yr (clinical sporadic onset: 75--85~yr). The ${\sim}10$-year discrepancy may reflect non-linear proteostasis decline not captured by the linear model.}
\label{fig:drift}
\end{figure}

\subsection{Intervention geometry in the model}

For the dual-basin model at $\alpha = 0.45$ ($\sigma_\mathrm{nat} \approx 1.07$), we tested both intervention types (Table~\ref{tab:intervention}).

\begin{table}[h]
\centering
\caption{Intervention results at $\alpha = 0.45$ in the dual-basin model.}
\label{tab:intervention}
\begin{tabular}{llccc}
\toprule
Strategy & Dose & $\sigma_\mathrm{nat}$ & $Q_\mathrm{nat}$ & $\sigma < 1$? \\
\midrule
\multicolumn{5}{l}{\emph{Conformational restriction (reduces $D$)}} \\
\quad 10 residues & 50 states blocked & 1.009 & 0.856 & no \\
\quad 12 residues & 60 states blocked & 0.926 & 0.892 & yes \\
\midrule
\multicolumn{5}{l}{\emph{Energetic stabilization (reduces $\gamma$)}} \\
\quad $\varepsilon_\mathrm{boost} = 0.3$ & +30\% contact energy & 1.029 & 0.842 & no \\
\quad $\varepsilon_\mathrm{boost} = 0.5$ & +50\% contact energy & 0.854 & 0.847 & yes \\
\midrule
\multicolumn{5}{l}{\emph{Combined}} \\
\quad 4 res.\ + $\varepsilon = 0.5$ & 20 blocked + 50\% & 0.896 & 0.878 & yes \\
\quad 8 res.\ + $\varepsilon = 0.5$ & 40 blocked + 50\% & 0.749 & 0.914 & yes \\
\bottomrule
\end{tabular}
\end{table}

Both strategies independently restore $\sigma < 1$ (full intervention results in Supplementary S7--S8). Combination therapy achieves rescue at lower individual doses than either strategy alone (Figure~\ref{fig:hyperbola}), consistent with the hyperbolic geometry: the product $D \cdot \gamma$ decreases faster when both factors are reduced simultaneously.

\begin{figure}[H]
\centering
\includegraphics[width=0.78\textwidth]{fig4_therapeutic_hyperbola.pdf}
\caption{\textbf{Intervention geometry in the model.} Effective interventions lie on or below $D \cdot \gamma = 1$. Conformational restriction moves the state leftward (reducing $D$); energetic stabilization moves it downward (reducing $\gamma$). Combination exploits the hyperbola's curvature.}
\label{fig:hyperbola}
\end{figure}

The minimum stabilizer dose increases non-linearly with disease severity (Table~\ref{tab:dose_response}, Figure~\ref{fig:dose}): intervention at $\alpha = 0.25$ requires ${\sim}15\times$ less stabilization than at $\alpha = 0.60$.

\begin{table}[h]
\centering
\caption{Minimum stabilizer dose by disease stage in the model.}
\label{tab:dose_response}
\begin{tabular}{ccccl}
\toprule
$\alpha$ & Stage & $\sigma_\mathrm{untreated}$ & $\varepsilon_\mathrm{rescue}$ & Note \\
\midrule
0.20 & Healthy & 0.965 & 0.0 & No treatment needed \\
0.25 & Early   & 1.016 & 0.1 & Minimal intervention \\
0.35 & Moderate & 1.023 & 0.2 & Low dose \\
0.45 & Moderate & 1.074 & 0.5 & Standard dose \\
0.55 & Severe  & 1.097 & 1.0 & High dose \\
0.60 & Critical & 1.081 & 1.5 & Aggressive \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{fig6_dose_response.pdf}
\caption{\textbf{Non-linear dose--response in the model.} Minimum stabilizer dose to restore $\sigma < 1$ as a function of $\alpha$. Early intervention requires ${\sim}15\times$ less stabilization than late intervention.}
\label{fig:dose}
\end{figure}


\subsection{Robustness of early-warning across model parameters}
\label{sec:robustness_results}

To test whether the early-warning property of $\sigma$---crossing 1 before the native contact fraction $Q$ drops to 0.5---is specific to our default parameterization or robust, we performed a systematic sweep across 72 parameter combinations: chain lengths $N \in \{20, 30, 40, 50\}$, conformational states $S \in \{6, 8, 10\}$, symmetric native/amyloid contacts $c \in \{8, 12, 16\}$, and asymmetric topologies (native $c = 12$, amyloid $c \in \{8, 10, 14\}$).

In all 72 cases, $\sigma_\mathrm{nat}$ crosses 1.0 \emph{before} $Q$ drops to 0.5 (Figure~\ref{fig:robustness}a,b). The early-warning lead $\Delta\alpha = (\alpha_Q - \alpha_\sigma) / \alpha_Q$ ranges from 27.7\% to 42.2\% (mean 35.9\%, median 36.3\%), consistent with the ${\sim}37\%$ observation in our default model.

To test whether the \emph{product form} $\sigma = D \cdot \gamma$ is essential, we performed two ablation analyses. In the first, we \emph{equalized} each factor across basins while letting the other vary freely. \emph{D-equalized} ($D_\mathrm{nat} = D_\mathrm{amy} = \bar{D}$ at each $\alpha$, $\gamma$ varies normally) isolates energy discrimination: it produces early warning in all 72 cases, but at a fixed $\alpha = 0.5$ for every parameter combination (standard deviation of crossing points $= 0$), because $\gamma$ depends only on $\alpha$ and physical constants, not on $N$, $S$, or contact topology. Energy discrimination alone provides universal early warning but zero parameter sensitivity. \emph{$\gamma$-equalized} ($\gamma_\mathrm{nat} = \gamma_\mathrm{amy} = \bar{\gamma}$ at each $\alpha$, $D$ varies normally) isolates geometric discrimination: early warning also succeeds in all 72 cases, with nonzero spread ($\mathrm{SD} = 0.080$), but sensitivity is limited to contact-count asymmetry---chain length $N$ and state count $S$ do not affect the crossing point when $\gamma$ is equalized.

The full composite $\sigma = D \cdot \gamma$, where \emph{both} factors discriminate between basins, achieves 72/72 early warning with crossing points sensitive to $N$, $S$, \emph{and} contact topology ($\mathrm{SD} = 0.046$; Figure~\ref{fig:robustness}c). The product form is the physically motivated combination: $D$ captures how many conformational moves are productive (geometry), while $\gamma$ captures how much energy is gained per move (thermodynamics). Neither discrimination alone captures the full picture.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig_robustness.pdf}
\caption{\textbf{Robustness of early warning and ablation analysis.} \textbf{(a)}~Representative case ($N = 40$, $S = 8$, $c = 12$): the full composite $\sigma$ (blue solid) crosses 1.0 at $\alpha_\sigma = 0.50$, well before $Q$ (red) drops to 0.5 at $\alpha_Q = 0.78$. D-equalized ($\gamma$-only discrimination, orange dash-dot) also crosses at $\alpha = 0.50$ (identical for all parameter combinations). $\gamma$-equalized ($D$-only discrimination, purple dash-dot) crosses at $\alpha = 0.50$ for this symmetric case but differs for asymmetric topologies. Frozen ablations (thin dashed) approach 1.0 only at $\alpha = 1$ by construction. \textbf{(b)}~$\alpha_\sigma$ vs.\ $\alpha_Q$ for all 72 parameter combinations. All points fall below the diagonal. \textbf{(c)}~All three non-tautological indicators achieve 72/72 early warning; the frozen ablations achieve 0/72. The spread annotation ($\sigma_\alpha$) shows standard deviation of crossing points across parameter combinations: D-equalized has zero spread (no parameter sensitivity), while full $\sigma$ and $\gamma$-equalized have nonzero spread reflecting sensitivity to model parameters.}
\label{fig:robustness}
\end{figure}

\subsection{Transthyretin validation}
\label{sec:ttr}

Transthyretin (TTR, $N = 127$) provides an independent validation target for the $\sigma$ framework. Unlike APP/A$\beta$, all TTR amyloidosis mutations are stability-driven---there is no secretase cleavage to confound the thermodynamic analysis. We compiled 25 TTR mutations with published $\Delta\Delta G$ values \citep{sekijima2005, hammarstrom2002} and known clinical onset ages spanning 20--72 years across four phenotypes: familial amyloid polyneuropathy (FAP), familial amyloid cardiomyopathy (FAC), leptomeningeal (CNS), and the protective variant T119M.

Using $\sigma_\mathrm{mut} = \sigma_\mathrm{wt} \cdot \exp(\Delta\Delta G / NRT)$ with $\Delta G_\mathrm{unfold}^\mathrm{wt} = 25$~kJ/mol and $T = 310$~K, we computed $\sigma$ for each mutation (Table~\ref{tab:ttr}). T119M ($\Delta\Delta G = -0.8$~kcal/mol) is correctly classified as protective ($\sigma = 0.917 < \sigma_\mathrm{wt} = 0.926$). All 24 pathogenic mutations receive $\sigma > \sigma_\mathrm{wt}$ with values ranging from 0.930 (G6S, late-onset) to 0.975 (D18G, early-onset CNS).

\begin{table}[H]
\centering
\caption{TTR validation: selected mutations ordered by $\sigma$. Onset envelope spans drift rates 0.02--0.05/decade (central: 0.03/decade). Full data in Supplementary Table~S13.}
\label{tab:ttr}
\small
\begin{tabular}{lcccccc}
\toprule
Mutation & $\Delta\Delta G$ (kcal/mol) & $\sigma$ & Phenotype & Onset (actual) & \multicolumn{2}{c}{Onset envelope (yr)} \\
 & & & & & central & range \\
\midrule
T119M & $-0.8$ & 0.917 & protective & --- & never & never \\
G6S & $+0.3$ & 0.930 & FAC & 72 & 53 & 44--65 \\
V122I & $+1.5$ & 0.944 & FAC & 60 & 49 & 41--58 \\
V30M & $+2.1$ & 0.952 & FAP & 33 & 46 & 40--54 \\
A36P & $+2.5$ & 0.957 & FAP & 38 & 45 & 39--51 \\
Y114C & $+2.8$ & 0.960 & FAP & 30 & 43 & 38--50 \\
L55P & $+3.6$ & 0.970 & FAP & 20 & 40 & 36--45 \\
D18G & $+4.0$ & 0.975 & CNS & 25 & 38 & 35--43 \\
\bottomrule
\end{tabular}
\end{table}

A clarification on threshold interpretation: all 24 pathogenic TTR mutations have $\sigma < 1$ at baseline (range 0.930--0.975), meaning the mutant protein still folds at 37$^\circ$C. The $\sigma = 1$ threshold is not the disease threshold in the static sense. Rather, the $\sigma$-drift model posits that aging-associated proteostasis decline progressively increases the effective $\sigma$ at a rate of $\sim$0.03/decade. Disease onset occurs when the effective $\sigma(t) = \sigma_\mathrm{baseline} + \mathrm{drift} \times (t - 30)/10$ crosses~1. Mutations closer to $\sigma = 1$ at baseline reach the tipping point sooner---hence the negative correlation with onset age.

The correlation between $\sigma$ and clinical onset age is strong: Spearman $\rho = -0.977$ ($p < 10^{-15}$, $n = 24$). The $\sigma$-drift model (drift rate 0.03/decade from age 30; $\Delta G_\mathrm{unfold}^\mathrm{wt}$ from Hammarstr\"om et al.\ 2002, 37$^\circ$C, pH~7.0) predicts onset ages with MAE~$= 9.2$~years (bootstrap 95\% CI: 7.0--11.4~years, $10^4$ resamples) and $r = 0.964$ against observed onsets (Figure~\ref{fig:ttr}). The drift rate was taken from published chaperone decline estimates \citep{balch2008, labbadia2015} and was not fit to TTR onset data. Since $\sigma$ is a monotonic transform of $\Delta\Delta G$ for a single protein, the correlation magnitude is necessarily identical to $r(\Delta\Delta G, \mathrm{onset})$---this is a mathematical fact, not an empirical finding.

Two caveats about the TTR onset data warrant discussion. First, published ``onset ages'' are typically age at diagnosis, not true symptom onset, introducing right-censoring bias of uncertain magnitude. Second, within-phenotype heterogeneity is substantial: V30M onset ranges from 25--70~years across populations (early-onset in Portugal, late-onset in Sweden), and the single onset age used here (33~years) represents the most common presentation. A leave-one-out analysis (Supplementary Table~S13) shows that $\rho$ ranges from $-0.968$ to $-0.982$ across all 24 jackknife iterations, indicating that no single mutation drives the correlation. The three most influential mutations---D18G, L55P, and G6S---span the full $\sigma$ range, confirming that the correlation is not anchored by outliers at either extreme. What $\sigma$ adds beyond $\Delta\Delta G$ alone is threefold: (1)~a universal threshold at $\sigma = 1$ that is protein-independent, whereas the $\Delta\Delta G$ value at which a mutation becomes pathogenic is protein-specific; (2)~chain-length normalization via the $1/NRT$ factor, enabling direct comparison between TTR ($N = 127$) and APP ($N = 770$) on the same scale; and (3)~the $\sigma$-drift model, which converts a static stability measure into an age-dependent onset prediction that $\Delta\Delta G$ alone does not provide.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig_ttr.pdf}
\caption{\textbf{Transthyretin validation.} \textbf{Top left:} $\sigma$ vs.\ clinical onset age ($\rho = -0.977$). \textbf{Top right:} $\Delta\Delta G$ vs.\ onset (same correlation magnitude---mathematical consequence of monotonic transform). \textbf{Bottom left:} predicted vs.\ actual onset from $\sigma$-drift model (MAE $= 9.2$~yr, 95\% CI: 7.0--11.4~yr). \textbf{Bottom right:} $\sigma$ distribution by phenotype.}
\label{fig:ttr}
\end{figure}

\subsection{Drift-rate sensitivity}
\label{sec:drift_sensitivity}

The $\sigma$-drift model uses a proteostasis decline rate of 0.03/decade, estimated from published chaperone decline data \citep{balch2008, labbadia2015}. Because this rate is itself uncertain, \emph{all onset predictions should be interpreted as envelopes rather than point estimates}. We computed predicted onset ages for all TTR and A$\beta$ mutations under three scenarios: slow (0.02/decade), medium (0.03/decade), and fast (0.05/decade).

Because the drift model is linear in $(1 - \sigma)$, the mutation severity ranking is invariant across drift rates ($\rho = 1.0$ by construction). The drift rate affects only the absolute timescale, not the ordering---this is a property of the translation, not an empirical finding. For TTR V30M ($\sigma = 0.952$), the onset envelope spans 40--56~years (observed: 33~years). Across all 24 TTR mutations, the MAE ranges from 8.1~years (fast drift) to 14.7~years (slow drift), with the central estimate of 9.2~years at the medium rate (Figure~\ref{fig:drift_sensitivity}). The drift model is best understood as an interpretation layer that translates static stability margins into approximate time horizons under explicit assumptions, rather than as an independent empirical result.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig_drift.pdf}
\caption{\textbf{Drift-rate sensitivity.} \textbf{Left:} predicted onset age vs.\ $\sigma$ for three drift rates (0.02, 0.03, 0.05 per decade). \textbf{Right:} onset-age envelope for each mutation across drift rates. The ranking of mutations is identical across all scenarios ($\rho = 1.0$).}
\label{fig:drift_sensitivity}
\end{figure}

\subsection{Benchmark against alternative early-warning indicators}
\label{sec:benchmark}

To assess whether $\sigma$ provides early warning because it captures something specific about the folding--misfolding transition or because any reasonable indicator would detect the crossover in a Go model, we benchmarked $\sigma$ against four alternative indicators across all 72 parameter combinations (Supplementary S14): (1)~$\Delta G$ threshold (native basin free energy crosses zero), (2)~$Q$-variance susceptibility ($\chi > 50\%$ of maximum), (3)~critical slowing down (CSD; relaxation time $\tau > 5\tau_0$), and (4)~$D$-only threshold ($D_\mathrm{nat} < D_0/2$).

All five indicators achieve 72/72 early warning (Figure~\ref{fig:benchmark}). The Go model's critical transition is detectable by multiple methods; $\sigma$ is not unique in this regard. Mean lead times differ: $Q$-variance detects earliest (55.5\%), followed by CSD (47.6\%), then $\sigma$ (35.9\%), $\Delta G$ threshold (36.0\%), and $D$-only (36.3\%). The shorter lead time of $\sigma$ relative to fluctuation-based indicators reflects its nature as a \emph{contractivity} measure rather than a \emph{fluctuation} measure: $\sigma$ crosses 1 when the net drift toward the native state vanishes, whereas susceptibility and CSD detect the \emph{precursors} of that vanishing.

What $\sigma$ provides beyond detection is interpretability: (1)~a physically motivated product form $D \cdot \gamma$ that decomposes the early-warning signal into geometric and energetic contributions (Section~\ref{sec:robustness_results}); (2)~a protein-independent threshold at $\sigma = 1$ with thermodynamic correspondence (Section~\ref{sec:thermo_correspondence}); (3)~chain-length normalization enabling cross-protein comparison; and (4)~the $\sigma$-drift model converting static stability into onset-age predictions (Section~\ref{sec:ttr}). $\sigma$ combines these features in a single normalized quantity.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig_benchmark.pdf}
\caption{\textbf{Benchmark against alternative early-warning indicators.} \textbf{(a)}~Representative case ($N = 40$, $S = 8$, $c = 12$): all five indicators normalized to $[0, 1]$, with threshold crossings marked. All cross before $Q$ drops to 0.5. \textbf{(b)}~Early-warning success rates: all indicators achieve 72/72. \textbf{(c)}~Lead-time distributions across 72 parameter combinations. $Q$-variance and CSD detect earlier because they measure fluctuation precursors; $\sigma$, $\Delta G$, and $D$-only detect when the net drift itself vanishes.}
\label{fig:benchmark}
\end{figure}

\subsection{Secondary validation: hereditary lysozyme amyloidosis}
\label{sec:lyz}

Human lysozyme (LYZ, $N = 130$) causes hereditary systemic amyloidosis through a stability-driven mechanism analogous to TTR: destabilizing mutations partially unfold the globular domain, exposing aggregation-prone regions \citep{canet2002}. Using published $\Delta\Delta G$ values ($\Delta G_\mathrm{wt} = 9.3$~kcal/mol at pH~5.0, 37$^\circ$C; \citealt{booth1997}), we computed $\sigma$ for six pathogenic mutations (Table~\ref{tab:lyz}).

\begin{table}[H]
\centering
\caption{Lysozyme validation: pathogenic mutations ordered by $\sigma$.}
\label{tab:lyz}
\small
\begin{tabular}{lcccc}
\toprule
Mutation & $\Delta\Delta G$ (kcal/mol) & $\sigma$ & Onset (actual) & Onset ($\sigma$-drift) \\
\midrule
T70N & $+0.8$ & 0.899 & 60~yr & 64~yr \\
L84S & $+1.2$ & 0.904 & 55~yr & 62~yr \\
D67H & $+1.5$ & 0.907 & 55~yr & 61~yr \\
F57I & $+1.8$ & 0.911 & 50~yr & 60~yr \\
I56T & $+2.1$ & 0.914 & 55~yr & 59~yr \\
W64R & $+3.5$ & 0.930 & 40~yr & 53~yr \\
\bottomrule
\end{tabular}
\end{table}

All six mutations receive $\sigma > \sigma_\mathrm{wt}$ ($= 0.890$). The correlation between $\sigma$ and clinical onset is $\rho = -0.714$---the correct sign, with W64R (highest $\sigma$) showing earliest onset and T70N (lowest $\sigma$) showing latest onset. The $\sigma$-drift model (using the same 0.03/decade rate as TTR, \emph{not} refit) predicts onset ages with MAE~$= 7.2$~years. With $n = 6$, the correlation does not reach conventional significance; this result is supporting, not standalone validation (Supplementary S15). One $\Delta\Delta G$ value (L84S) is estimated from clinical severity rather than direct measurement.

\subsection{Supporting validation: gelsolin amyloidosis}
\label{sec:gelsolin}

Gelsolin (GSN) causes Finnish hereditary amyloidosis (AGel) through destabilization of domain~2 ($N = 117$, $\Delta G_\mathrm{wt} \approx 6.0$~kcal/mol). The dataset is small ($n = 5$, only 3 with onset data) but cleanly stability-driven.

\begin{table}[H]
\centering
\caption{Gelsolin validation.}
\label{tab:gelsolin}
\small
\begin{tabular}{lcccl}
\toprule
Mutation & $\Delta\Delta G$ (kcal/mol) & $\sigma$ & Onset & Source \\
\midrule
N184K & $+2.0$ & 0.946 & 55~yr & estimated \\
D187Y & $+2.5$ & 0.953 & 45~yr & experimental \\
D187N & $+3.0$ & 0.959 & 40~yr & experimental \\
A174P & $+2.8$ & 0.957 & --- & DynaMut2 \\
G167R & $+3.5$ & 0.966 & --- & DynaMut2 \\
\bottomrule
\end{tabular}
\end{table}

For the three mutations with onset data, the $\sigma$-onset ranking is perfect ($\rho = -1.0$), and the drift model gives MAE~$= 3.8$~years. With $n = 3$, this is consistent but not statistically meaningful on its own (Supplementary S16). Its value is as part of the multi-protein pattern: like TTR and LYZ, gelsolin is a globular protein whose amyloidosis is purely stability-driven, and $\sigma$ correctly tracks severity.

\subsection{Predicted negative controls: where $\sigma$ should not predict onset}
\label{sec:negative_controls}

The framework makes a falsifiable prediction: $\sigma$ should fail as an onset predictor when disease mechanisms extend beyond thermodynamic instability of a globular fold. Specifically, we predict failure when (i)~pathogenicity depends on a toxic gain-of-function uncorrelated with stability loss, (ii)~propagation requires templated conformational conversion rather than spontaneous unfolding, or (iii)~the protein has no stable native fold to lose. We tested each prediction on a representative disease class (full data in Supplementary S17).

\paragraph{SOD1/ALS (gain-of-toxic-function).}
SOD1 ($N = 153$) mutations cause familial ALS. Although all 10 tested mutations are destabilizing ($\sigma > \sigma_\mathrm{wt}$), the correlation between $\sigma$ and clinical onset is $\rho = +0.28$ ($p = 0.43$)---the wrong sign. This failure is expected: SOD1/ALS pathogenesis involves gain-of-toxic-function through aberrant interactions, not simply loss of native-state stability \citep{chattopadhyay2015}. G37R ($\Delta\Delta G = 1.8$~kcal/mol, modest destabilization) has the earliest onset (35~yr), while H46R ($\Delta\Delta G = 3.0$~kcal/mol, strong destabilization) has late onset (55~yr). Stability loss is necessary but not sufficient; the toxic gain-of-function determines severity.

\paragraph{Prion disease (templated conversion).}
PRNP ($N_\mathrm{domain} = 104$) mutations cause familial CJD, FFI, and GSS. All 7 mutations with published $\Delta\Delta G$ values give $\sigma < 1$: the prion protein remains thermodynamically stable even with pathogenic mutations. The correlation between $\sigma$ and onset is $\rho = -0.29$ ($p = 0.53$), not significant. The $\sigma$-drift model predicts onset ages of 160--270~years---clearly wrong. Prion disease does not arise from spontaneous unfolding across the $\sigma = 1$ boundary; it requires templated conversion by PrP$^\mathrm{Sc}$ seeds, a mechanism entirely outside the scope of equilibrium thermodynamics.

\paragraph{A$\beta$42/Alzheimer's disease (intrinsically disordered peptide).}
A$\beta$42 has no stable globular fold and therefore falls outside the scope of $\sigma$ by definition. As a consistency check, we applied two stability-prediction tools (DynaMut2, ESM-1v) to familial A$\beta$42 mutations: the methods show no inter-method agreement ($\rho = -0.10$, $n = 31$), confirming that folding-stability tools do not produce meaningful results for intrinsically disordered peptides. A$\beta$42 pathogenicity is governed by production rates ($\beta$/$\gamma$-secretase cleavage), aggregation kinetics, and clearance---not by loss of a native basin that does not exist.

\paragraph{Summary.} Table~\ref{tab:scope} summarizes the scope of $\sigma$ across disease classes.

\begin{table}[H]
\centering
\caption{Scope of $\sigma$: onset prediction requires stability-driven disease mechanism.}
\label{tab:scope}
\small
\begin{tabular}{llccl}
\toprule
Protein & Disease & $\rho$($\sigma$, onset) & $\sigma$ predicts? & Dominant mechanism \\
\midrule
TTR & Cardiac/poly-amyloidosis & $-0.977$ & yes & thermodynamic instability \\
LYZ & Systemic amyloidosis & $-0.714$ & yes (supportive) & thermodynamic instability \\
Gelsolin & Finnish amyloidosis & $-1.000$ & yes ($n = 3$) & thermodynamic instability \\
\midrule
SOD1 & ALS & $+0.28$ & no & gain-of-toxic-function \\
PRNP & CJD/FFI/GSS & $-0.29$ & no & templated conversion \\
A$\beta$42 & Alzheimer's & n/a & no & IDP; production/clearance \\
\bottomrule
\end{tabular}
\end{table}

\noindent The pattern is consistent with the hypothesis: $\sigma$ predicts onset when disease is driven by loss of thermodynamic contractivity in a globular protein, and fails when other mechanisms dominate. This boundary is not a limitation discovered post hoc---it follows from the definition of $\sigma$ as a measure of native-basin stability. Where there is no native basin (A$\beta$42), or where disease bypasses the unfolding pathway (prions, SOD1 toxic gain-of-function), $\sigma$ correctly has no predictive power.

\section{Discussion}

\subsection{What $\sigma$ adds and does not add}

The stability index $\sigma = D \cdot \gamma$ encodes two aspects of the folding funnel---the probability of productive progress ($D$) and the magnitude of that progress ($\gamma$)---into a single number with a threshold at $\sigma = 1$. This connects to existing concepts: the folding funnel \citep{bryngelson1995} corresponds to $\sigma < 1$; $\phi$-value analysis \citep{fersht1995} measures a local analogue of $D$; the roughness of the energy landscape \citep{dill2012} is captured by $\gamma$.

What $\sigma$ does \emph{not} provide is a model-independent observable. Both $D$ and $\gamma$ depend on the move set, sampling protocol, temperature, and distance metric. In our Go model, these are well-defined (Section~\ref{sec:mc_protocol}). As a first step toward experimental operationalization, we implemented an HDX-MS proxy using published per-residue NMR order parameters ($S^2$) for A$\beta$42 variants \citep{sgourakis2007, lim2007}. The mapping defines local frustration $\sigma_\mathrm{local}(i) = 4 S^2_i (1 - S^2_i)$ plus a destabilization penalty for structured residues that lose rigidity upon mutation, with aggregation-prone regions (CHC residues 17--21, C-terminal core 30--42) weighted by their role in cross-$\beta$ spine formation. The resulting $\sigma_\mathrm{HDX}$ correlates with Go-model $\sigma$ at $r = 0.68$ ($p = 0.042$, $n = 9$; F19P excluded as a mechanistic outlier whose proline abolishes $\beta$-sheet formation entirely). Per-residue analysis shows that mutation-induced $|\Delta\sigma_\mathrm{local}|$ is 3.3$\times$ higher at known aggregation hotspot residues than elsewhere ($p < 10^{-6}$, point-biserial), and each mutation's peak $\Delta\sigma$ residue matches its mutation site. This supports the feasibility of NMR/HDX-MS order parameters as experimental proxies for $\sigma$, though the mapping remains a proof-of-concept requiring validation with directly measured HDX protection factors.

Figure~\ref{fig:landscape} illustrates the dual-basin landscape at three values of $\alpha$.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig5_energy_landscape_3d.pdf}
\caption{\textbf{Dual-basin energy landscape.} \textbf{Left:} $\alpha = 0.15$ (native dominates). \textbf{Center:} $\alpha = 0.59$ (crossover). \textbf{Right:} $\alpha = 0.85$ (amyloid dominates).}
\label{fig:landscape}
\end{figure}

\subsection{On the tautological character of $\sigma(T_m) = 1$}

The thermodynamic $\sigma$ (Eq.~\ref{eq:sigma_thermo}) gives $\sigma = 1$ at $T_m$ by definition, since $\Delta G(T_m) = 0$. This is not a result but a reparametrization. The non-trivial bridge to kinetics is that the \emph{same} threshold emerges in Monte Carlo simulations of the dual-basin model: $\sigma_\mathrm{nat}$ computed from trajectory dynamics crosses 1 at a point that corresponds to the loss of native-basin dominance. The kinetic and thermodynamic definitions of $\sigma$ are not formally equivalent, and their agreement in the Go model is a consistency check on the framework, not a proof.

A stronger test---computing kinetic $\sigma$ from all-atom molecular dynamics trajectories and comparing with $\sigma_\mathrm{thermo}$ from calorimetric data for the same protein---is beyond the scope of this work but represents the natural next step.

\subsection{Limitations of the Alzheimer mapping}

The encoding of APP mutations onto the $\alpha$ axis involves several assumptions that limit the strength of causal claims:

\begin{enumerate}
\item \textbf{$\alpha$ is not mechanistically derived.} The $\alpha$ values are assigned from published aggregation propensity rankings, not computed from molecular properties. This makes the mapping a structured encoding rather than a prediction. The correlation $r = 0.843$ is therefore better interpreted as ``the Go model's nonlinear response to $\alpha$ preserves the input ordering'' than as ``$\sigma$ predicts severity.''

\item \textbf{Small sample ($n = 10$).} With ten data points, the correlation is not robust against removal of individual mutations. A leave-one-out analysis (Supplementary S6) shows that $r$ ranges from 0.79 to 0.89 depending on which mutation is excluded, with all jackknife estimates remaining significant at $p < 0.05$.

\item \textbf{Severity is ordinal.} Clinical onset ages span ranges (e.g., 40--55 yr) and are influenced by penetrance, genetic background, and diagnostic criteria. We used midpoint onset age to define an ordinal severity score (1--6), but note that different operationalizations would yield different correlations.

\item \textbf{No out-of-sample validation.} Ideally, $\alpha$ assignments would be made for a training set and the correlation tested on held-out mutations. With only ten mutations available, this is not feasible. The cross-validation against independent $\Delta\Delta G$ data (Table~\ref{tab:validation_crosscheck}) provides partial external validation but does not fully address this concern.
\end{enumerate}

For these reasons, we frame the Alzheimer application as ``illustrative'' rather than ``predictive'' in the strong sense.

\subsection{Graded risk and the drift model}

The $\sigma$-drift framework naturally distinguishes between \emph{early-onset pathogenic} ($\sigma \gg 1$), \emph{late-onset or reduced-penetrance} ($\sigma$ slightly above~1), and \emph{effectively benign} ($\sigma < 1$ or $\sigma$ so close to~1 that the drift-to-criticality timescale exceeds a human lifespan). In the TTR validation, variants with $\sigma \approx 1.002$ would not be expected to manifest before age $\sim$120, rendering them effectively benign. This graded prediction is falsifiable: variants in the range $1.01$--$1.05$ are candidates for late-onset manifestation in carriers who live into their ninth or tenth decade.

We note a general limitation: when a variant's protective effect operates through a mechanism outside equilibrium thermodynamics (e.g., reduced secretase cleavage for APP A673T), $\sigma$ alone is insufficient. Such variants define the boundary of the framework's applicability.

\subsection{Robustness of the early-warning observation}

The systematic parameter sweep (Section~\ref{sec:robustness_results}) confirms that $\sigma$ provides early warning across all 72 tested parameter combinations, with a mean lead of $\Delta\alpha = 36\%$ (range 28--42\%). Ablation analysis reveals a clear division of labor: energy discrimination ($\gamma_\mathrm{nat} \neq \gamma_\mathrm{amy}$) provides universal early warning but zero parameter sensitivity; geometric discrimination ($D_\mathrm{nat} \neq D_\mathrm{amy}$) provides parameter-dependent crossings but sensitivity only to contact asymmetry. Only the full product $\sigma = D \cdot \gamma$ couples both types of discrimination---crossing points that reflect chain length, conformational entropy, \emph{and} contact topology.

Importantly, benchmarking against four alternative indicators (Section~\ref{sec:benchmark}) shows that early warning per se is not unique to $\sigma$: all tested indicators achieve 72/72 detection in the Go model. The Go model's critical transition is sufficiently sharp that multiple indicators detect it. The contribution of $\sigma$ is not superior detection but the interpretive framework---the product decomposition, the protein-independent threshold, and the drift model---that connects detection to prediction. Whether this advantage persists in models with smoother transitions remains to be tested in coarse-grained or all-atom simulations.

\subsection{Intervention geometry as a design principle}

The hyperbolic structure $D \cdot \gamma = 1$ is a geometric consequence of the product form of $\sigma$. In the model, it correctly describes how two orthogonal intervention types combine. However, real therapeutic agents do not cleanly separate into ``$D$-reducers'' and ``$\gamma$-reducers'': chaperones affect both conformational access and energetics, while stabilizers may alter dynamics as well as thermodynamics.

The practical implication is more modest: the $\sigma$ framework suggests that \emph{combination strategies targeting different aspects of the folding funnel} should exhibit super-additive effects. This is consistent with clinical experience in transthyretin amyloidosis, where tafamidis (a kinetic stabilizer, primarily affecting $\gamma$) is being explored in combination with gene silencing approaches (which reduce substrate, effectively affecting $D$). The iso-cure hyperbola provides a geometric language for this intuition, not a quantitative dosing guide.

\subsection{The scope of $\sigma$: a mechanistic boundary}

The multi-protein analysis (Section~\ref{sec:negative_controls}) reveals a clear pattern: $\sigma$ predicts onset when disease is driven by thermodynamic instability of a globular fold, and fails when other mechanisms dominate. Operationally, we define ``stability-driven'' as satisfying three criteria: (i)~the wild-type protein has a well-defined globular fold ($\Delta G_\mathrm{unfold} > 0$), (ii)~pathogenic mutations are destabilizing ($\Delta\Delta G > 0$), and (iii)~the primary disease mechanism is exposure of aggregation-prone regions through partial unfolding, not toxic gain-of-function or templated conversion. This boundary is not empirical curve-fitting---it follows from the definition of $\sigma$ as a measure of native-basin contractivity:

\begin{itemize}
\item \textbf{Where $\sigma$ works}: TTR, lysozyme, and gelsolin share a common disease mechanism: destabilizing mutations shift the folding equilibrium, exposing aggregation-prone regions. The disease \emph{is} the stability loss. $\sigma$ measures exactly this.

\item \textbf{Where $\sigma$ fails}: SOD1/ALS involves toxic gain-of-function through aberrant protein--protein interactions that are uncorrelated with the degree of destabilization. Prion diseases require PrP$^\mathrm{Sc}$-templated conversion---a kinetic process that $\sigma$ cannot capture. A$\beta$42 has no native basin; its pathogenicity is governed by production, nucleation kinetics, and clearance.
\end{itemize}

\noindent The negative controls are not post-hoc rationalizations. Pre-specified negative control diseases were selected based on mechanistic criteria (toxic gain-of-function; templated conversion; intrinsically disordered substrate) before $\sigma$ was computed. All data sources and analysis code are provided. The framework makes explicit, testable predictions about its boundary of validity.

\subsection{Scope and further limitations}

\begin{enumerate}
\item \textbf{Model simplicity.} A 20-residue lattice model with 8 states per residue cannot capture sequence-specific energetics, solvation, or multi-body interactions. The cross-validation against experimental $\Delta\Delta G$ (Spearman $\rho = 0.988$) suggests that the topological essentials are captured, but absolute $\sigma$ values should not be over-interpreted.

\item \textbf{Two-state approximation.} Real folding landscapes have intermediates, kinetic traps, and off-pathway aggregation. The dual-basin model is a minimal representation.

\item \textbf{Small secondary datasets.} The lysozyme ($n = 6$) and gelsolin ($n = 3$) validations are consistent with the framework but individually underpowered. At $n = 6$, the minimum detectable $|\rho|$ at 80\% power ($\alpha = 0.05$, two-tailed) is approximately 0.81; the observed $\rho = -0.714$ falls below this threshold, meaning the lysozyme result does not reach conventional significance on its own. At $n = 3$, no meaningful inference is possible. These datasets gain evidential weight only as part of the multi-protein pattern. Larger datasets of stability-driven amyloidoses with published $\Delta\Delta G$ and onset ages are needed---fibrinogen ($\alpha$-chain, AGel type~II) and apolipoprotein~A-I amyloidosis are promising candidates.

\item \textbf{Proteostasis context.} In vivo, stability depends on the cellular proteostasis network. The $\sigma$-drift model is illustrative: it captures the direction and approximate magnitude of aging effects but uses a linear approximation where the real decline is likely non-linear and accelerating.

\item \textbf{Kinetic vs.\ thermodynamic $\sigma$.} The thermodynamic $\sigma$ is exact but tautological at $T_m$. The kinetic $\sigma$ is non-trivial but model-dependent. Bridging the two through all-atom simulations with experimental validation is the key open problem.
\end{enumerate}

\subsection{Connection to broader theory}

The condition $D \cdot \gamma = 1$ as a critical boundary appears in the contraction geometry of Collatz-type maps, where branching and contraction ratios govern convergence \citep{wurm2026}, and in models of semantic change, where morphological depth interacts with meaning contraction \citep{wurm2026b}. This structural analogy is suggestive but does not constitute evidence for universality. Whether the product-form critical point $D \cdot \gamma = 1$ defines a meaningful universality class across domains remains an open question.


\section{Conclusion}

We have introduced $\sigma = D \cdot \gamma$ as a stability index for protein folding and tested the hypothesis that $\sigma$ predicts pathogenicity and clinical onset when disease is predominantly driven by thermodynamic destabilization of a globular fold.

The data are consistent with this hypothesis in both directions. In transthyretin amyloidosis ($\rho = -0.977$, MAE~$= 9.2$~years, $n = 24$), hereditary lysozyme amyloidosis ($\rho = -0.71$, $n = 6$), and gelsolin amyloidosis (MAE~$= 3.8$~years, $n = 3$)---all stability-driven diseases of globular proteins---$\sigma$ tracks onset and severity. In SOD1/ALS (gain-of-toxic-function), prion disease (templated conversion), and A$\beta$42 (intrinsically disordered)---where disease mechanisms extend beyond stability loss---$\sigma$ fails as predicted.

A dual-basin Go model provides the theoretical foundation: $\sigma$ crosses~1 before native-basin loss across 72 parameter combinations, though benchmarking shows this detection is not unique to $\sigma$. What $\sigma$ combines in a single normalized quantity is a protein-independent threshold ($\sigma = 1$), chain-length normalization, and the drift model converting static stability into onset predictions. The negative controls were pre-specified based on mechanistic criteria before $\sigma$ was computed; this design and all analysis code are available in the repository.

The framework's value lies not in universality but in precision of scope. By delineating where thermodynamic instability is sufficient to explain disease---and where it is not---$\sigma$ provides both a practical variant-interpretation tool for stability-driven amyloidoses and a mechanistic classifier for disease mechanisms. The current evidence is consistent with the hypothesis but does not constitute proof: the positive cases ($n = 3$ proteins) could be extended by false negatives not yet tested, and the negative cases could admit exceptions where stability loss and alternative mechanisms co-contribute. The key open problems are: (1)~extending the positive validation to larger datasets of stability-driven diseases (fibrinogen, apolipoprotein~A-I), (2)~testing intermediate cases where stability loss and gain-of-function co-contribute (e.g., $\alpha$-synuclein in Parkinson's), (3)~bridging kinetic and thermodynamic $\sigma$ through all-atom simulation, and (4)~operationalizing $\sigma$ from experimental observables such as HDX-MS protection factors.

\section*{Data availability}

All simulation code, thermodynamic data, and analysis scripts are available at \url{https://github.com/forgottenforge/protein-sigma} (see Supplementary S12 for a complete file listing). Results are fully reproducible with the provided random seeds using Python~3 and NumPy.

\section*{Author contributions}

M.W.\ conceived the framework, performed all simulations and analyses, and wrote the manuscript.

\section*{Competing interests}

The author declares no competing interests.

\section*{Acknowledgments}

Thermodynamic data for Trp-cage, Villin HP35, CI2, and ACBP were obtained from published literature. Mutation data for TTR, lysozyme, gelsolin, SOD1, and PRNP were compiled from published clinical, biochemical, and thermodynamic studies.

The contraction framework applied here originated in the analysis of non-injective maps \citep{wurm2026} and proved more general than initially anticipated. Whatever constraints bind the thinking mind are mostly of its own construction; whatever keys exist are already in its own pocket.

\section*{Funding}
The author received No Funding for this work.

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