Peer-Inexpressible Residual (PIER)

• PIER is a measure that quantifies an AI model's intrinsic behavioral uniqueness by isolating its irreducible response component, which cannot be replicated by any convex combination of peer models.
• It employs in-silico interventions and active auditing under strict ISQED protocols to ensure identifiability and guide effective model substitution and consolidation in heterogeneous AI ecosystems.
• Empirical findings across vision, language, and forecasting demonstrate PIER’s capacity to detect nuanced shifts—including adversarial stresses and domain changes—supporting robust AI governance.

The Peer-Inexpressible Residual (PIER) quantifies the intrinsic behavioral uniqueness of a target model within a heterogeneous AI ecosystem. PIER formalizes the maximal irreducible component of a model's response that cannot be replicated by any stochastic convex combination of its peer models, evaluated under precisely matched in-silico interventions. Vanishing PIER characterizes scenarios where a target’s outputs are functionally substitutable via routing among its peers, while nonzero PIER certifies strict behavioral novelty. The concept is central to model ecosystem auditing, underpinning formal approaches to redundancy detection, actionable substitution, and system consolidation in evolving AI infrastructures (You, 30 Jan 2026).

1. Formal Construction of PIER

Let $X$ denote the input space, $\Theta$ the space of intervention strengths, and $T : \Theta \times X \to X$ a “Type-B” intervention map (e.g., token masking at level $\theta$). Each model $f_j : X \to Y$, $j=1,\dots,N$, produces outputs mapped by a scalarizer $g : Y \to \mathbb{R}$, yielding scalarized responses $Y_j(x,\theta) := g(f_j(T(\theta, x)))$. Fixing a target model $t \in \{1,\dots,N\}$ and denoting its peer set by $J = \{1,\dots,N\}\setminus\{t\}$, the peer response vector is $$Y_{-t}(x,\theta) = (Y_j(x,\theta))_{j\in J} \in \mathbb{R}^{N-1}$$.

The convex peer-expressivity class $\mathcal{C}$ comprises all convex combinations of peers:

$$\mathcal{C} = \left\{ h_w(x,\theta) = w^\top Y_{-t}(x,\theta) : w \in \Delta^{N-2} \right\}$$

with $$\Delta^{N-2} = \{ w \in \mathbb{R}^{N-1} : w_j \ge 0, \sum_j w_j = 1 \}$$. The population projection weight is

$$w^* := \underset{w \in \Delta}{\arg\min}\ \mathbb{E}_P \left[(Y_t(X,\theta) - w^\top Y_{-t}(X,\theta))^2 \right]$$

given design distribution $P$ over $X \times \Theta$. The PIER is the orthogonal projection residual

$$R_t(x,\theta) := Y_t(x,\theta) - {w^*}^\top Y_{-t}(x,\theta).$$

A canonical scalar uniqueness score is obtained via the expected absolute residual under an independent evaluation design $P_\text{eval}$:

$$U := \mathbb{E}_{(X,\theta)\sim P_\text{eval}}\left[ |R_t(X,\theta)| \right].$$

U is strictly non-negative; $U \approx 0$ implies the target is functionally substitutable by convex routing over peers, while $U > 0$ witnesses irreducible uniqueness.

2. In-Silico Quasi-Experimental Design (ISQED) and Identifiability

PIER is well-defined and identifiable only under ISQED, wherein auditors enforce matched queries across all models: fitting data $$(X_i^\text{fit}, \theta_i^\text{fit}) \sim P_\text{fit}$$ for calibrating the convex surrogate, and evaluation data $$(X_i^\text{eval}, \theta_i^\text{eval}) \sim P_\text{eval}$$ for estimating PIER multiplicity. Uniformity and independence between fitting and evaluation samples (“honesty”) are required.

Key assumptions:

• Boundedness and continuity of $(Y_j)$ on $X \times \Theta$.
• $P_\text{fit}$ and $P_\text{eval}$ absolutely continuous with respect to a shared reference $P$.
• Loss $L(w) = \mathbb{E}_P[(Y_t-w^\top Y_{-t})^2]$ strongly convex near $w^*$, with unique minimizer in the simplex interior.

Crucially, with only observational logs (each model accessed under its own process $Q_j$), there exist ecosystems that cannot be distinguished by any function of the logs, even though they possess distinct uniqueness scores $U$. Thus, PIER is generally non-identifiable from passive records; matched interventions across the ecosystem restore identifiability by resolving ambiguities in the joint response law.

3. Sample Complexity and Minimax Lower Bound

In a local linear regime, $Y_j(x,\theta) = \varphi(x,\theta)^\top \beta_j$ with known feature map $\varphi$ and coefficients $\beta_j$, uniqueness reduces to geometric separation: is $\beta_t$ outside the convex hull of $\{\beta_j:j\in J\}$? Define the uniqueness margin $$\gamma := \mathrm{dist}(\beta_t,\,\mathrm{conv}\{\beta_j:j\in J\}) > 0$$ and let noise have sub-Gaussian parameter $\sigma^2$.

Active auditing entails selecting $d$ linearly independent design points and, at each, querying models $r = O((\sigma^2/\gamma^2) \log(Nd/\delta))$ times to achieve error $\le \gamma$, with failure probability at most $\delta$. Thus, total queries per model scale as

$$O\left(d \sigma^2 \gamma^{-2} \log(Nd/\delta)\right).$$

A tight minimax lower bound holds: no procedure can have better worst-case dependence on $(d,\sigma^2,\gamma,N,\delta)$.

4. PIER Estimation via the DISCO Estimator

Given fitting data $$\{(X_i^\text{fit}, \theta_i^\text{fit}), Y_{j,i}^\text{fit}\}$$, construct the peer matrix $X_{-t}^\text{fit} \in \mathbb{R}^{m \times (N-1)}$ and target vector $y_t^\text{fit} \in \mathbb{R}^m$. Estimate $\hat w$ via regularized quadratic programming:

$$\hat w = \underset{w \in \Delta}{\arg\min} \left( \frac{1}{m} \|y_t^\text{fit} - X_{-t}^\text{fit} w\|_2^2 + \lambda_m \|w\|_2^2 \right).$$

Given independent evaluation data $$\{(X_i^\text{eval}, \theta_i^\text{eval}), Y_{j,i}^\text{eval}\}$$, residuals are

$$R_i = Y_{t,i}^\text{eval} - \hat w^\top Y_{-t,i}^\text{eval},$$

and the empirical uniqueness score is

$\hat U = \frac{1}{n} \sum_{i=1}^n |R_i|.$

Convergence and inferential guarantees:

• $\hat w \to_p w^*$ as $m \to \infty$.
• $\hat U \to_p U$ as $m,n \to \infty$.
• Asymptotic normality: $\sqrt{n}(\hat U-U)$ converges in law to Normal.
• Finite-sample design error is controlled by $\|\hat w-w^*\|_2$ plus discretization effects.

5. Empirical Findings Across Modalities

Audits employing PIER and the DISCO estimator have been conducted across diverse architectures and modalities:

• Computer vision under adversarial stress (ImageNet, $\ell_\infty$ attacks): ResNet-50 shows PIER collapse under moderate perturbations ($\downarrow\,0.55\times$), indicating substitutability by peers when compromised. Conversely, ConvNeXt exhibits amplified PIER ($\uparrow1.2\times$), revealing novel irreducible behavior under attack.
• Dataset shift (texture vs. shape bias): Relative PIERs differ strongly across context, with shape-biased ResNets manifesting large amplifications, exposing context-dependent uniqueness.
• LLMs (SST-2, token masking): At low masking ($\theta$), ALBERT exhibits high PIER aligning with unique sensitivity to negation and complex syntax; at moderate $\theta$, DistilBERT’s vanishing PIER shows full behavioral expressibility by peer routing.
• Traffic forecasting (31-city ensemble): Scatter analysis of PIER versus MAE-penalty upon substitution reveals decoupling, with “unique but unhealthy” and “irreplaceable” islands. Consolidation via PIER pruning outperforms utility-based ranking for robust MAE budgeting.

Significantly, raw pairwise disagreement and PIER are decorrelated: PIER enables actionable routing, while disagreement solely measures geometric distance without policy implications.

6. Key Theoretical Properties of PIER

Properties of PIER and its estimation are as follows:

• Monotonicity: Enlarging the peer set can only decrease PIER; the convex hull grows, so the residual norm shrinks.
• Conservatism: PIER upper bounds uniqueness measured by more expressive classes such as linear spans or RKHS.
• Actionable routing: If $U\approx 0$, routing via $w^*$ exactly replicates the target’s behavior in expectation.
• Passive consistency and asymptotics: Estimators $\hat w$, $\hat U$ converge at standard parametric rates, and a central limit theorem applies.
• Finite-sample design error: Controlled by the deviation $\|\hat w-w^*\|_2$ plus discretization terms.
• Sampling optimality: Active auditing achieves minimax-optimal query complexity up to constants.
• Detection vs. estimation cost: Detecting nonzero uniqueness is as hard as estimation to $O(\gamma)$ accuracy in the 1D case.
• Shapley paradox: Cooperative game-theoretic methods such as Shapley values can assign nonzero credit to fully redundant models, underscoring the fundamental distinction between credit assignment and redundancy auditing.
• Necessity of interventions: Matched in-silico interventions are essential; observational logs cannot identify PIER.

7. Significance for AI Ecosystem Governance

PIER and its active identification framework fundamentally extend trustworthy AI beyond single-model explainability, enabling rigorous quantification and evidencing of behavioral novelty and redundancy. This paradigm supports principled model consolidation, reliable substitution, and targeted governance across evolving, heterogeneous AI ecosystems, as demonstrated in vision, language, and multi-agent spatiotemporal forecasting scenarios. The formal apparatus decouples utility from uniqueness, exposes actionable redundancies overlooked by attribution metrics, and establishes intervention-based auditing as the foundation for future governance protocols in large-scale AI deployments (You, 30 Jan 2026).