Clade-Metaproductivity (CMP) Overview

• CMP is a framework that decomposes matrices into core (non-nilpotent) structures using explicit algebraic constructions.
• It extends to weighted and weak inverses with determinantal formulas, facilitating applications in signal processing, control, and related fields.
• In machine learning, CMP metrics aggregate descendant performances to evaluate agents’ long-term self-improvement potential.

Clade-Metaproductivity (CMP) encompasses a family of theoretical and practical constructs spanning matrix theory, operator algebra, and machine learning. In matrix theory, CMP refers to a class of generalized inverses—specifically, the CMP and its weighted and weak variants—designed to extract the "core" productive part of a matrix, isolating it from nilpotent components through explicit algebraic constructions. In the context of self-improving machine learning agents, CMP is a metric evaluating the long-term self-improvement potential of an agent by systematically aggregating the results achieved by all its descendants in the expansion tree, thus prioritizing agent lineages with high evolutionary promise.

1. Algebraic Foundations of CMP and CMP Inverse

The algebraic notion of Clade-Metaproductivity arose to formalize matrix decompositions where a given matrix $A$ is split into a "core" (non-nilpotent, information-carrying) part and a nilpotent part. For matrices $A$ in $\mathbb{H}^{n \times n}$ or $\mathbb{C}^{n \times n}$ admitting a core–nilpotent decomposition $A = A_1 + A_2$, with $A_2$ nilpotent and $A_1A_2 = A_2A_1 = 0$, the (unweighted) CMP inverse is defined by

$A_{c,\tau} = A^\dagger A_1 A^\dagger$

where $A^\dagger$ is the Moore–Penrose inverse of $A$. The CMP inverse is the unique $X$ that satisfies the system: $$\begin{aligned} & XAX = X \ & AXA = A_1 \ & AX = A_1 A^\dagger \ & XA = A^\dagger A_1 \end{aligned}$$ This construction "filters out" the nilpotent part, isolating the productive core. The CMP framework extends to weighted cases and even more general scenarios via minimal rank weak Drazin inverses, resulting in several closely related generalized inverse definitions.

2. Weighted and Weak CMP Inverses

The weighted CMP inverse broadens applicability to the setting $A \in \mathbb{H}^{m \times n}$ with a weighting matrix $W \in \mathbb{H}^{n \times m}$, replacing appearance of the Drazin (or core) inverse with the $W$-weighted Drazin inverse $A_{d,W}$:

$A_{c,\tau,W} = A^{\dagger} W A_{d,W} A^{\dagger}$

This variant allows for additional flexibility in contexts where weighting reflects application priorities or modeling constraints. The inverse remains characterized by a system of equations analogously filtering core versus nilpotent components.

The weak CMP inverse, introduced by employing any minimal rank weak Drazin inverse $X$ in place of the Drazin inverse, is defined for $A \in \mathbb{C}^{n \times n}$ as

$A^{(w,c,\dagger)} = A^\dagger A X A A^\dagger$

This definition encompasses the standard CMP and MPCEP inverses as special cases, providing a unified, strictly broader class. As demonstrated in (Xu et al., 10 Sep 2025), the weak CMP inverse admits explicit block expressions using the Hartwig–Spindelböck decomposition and is deeply interrelated with weak MPD and Bott–Duffin $(e,f)$-inverses.

3. Determinantal Representations and Computational Aspects

A major advance for practical computation of the CMP, especially over the quaternion skew field $\mathbb{H}$, is the derivation of explicit determinantal (Cramer-rule type) formulas for (weighted) CMP inverses. For $A \in \mathbb{H}^{m \times n}$ and $W \in \mathbb{H}^{n \times m}$, if $k = \max\{\mathrm{Ind}(WA), \mathrm{Ind}(AW)\}$ and $\mathrm{rk}((WA)^k)=r_1$, the $(i,j)$ entry is given in terms of noncommutative row and column determinants (rdet$_i$, cdet$_j$): $$[a_{ij}]_{c,\tau,W} = \frac{\sum_{s\in \mathcal{I}} Q_{is} \cdot \mathrm{rdet}_i((AA^*)_j(\omega_i))}{\text{denominator}}$$ (see Theorems 5.3 and 5.4 in (Kyrchei, 2020) for details). For complex matrices, noncommutative determinants reduce to standard determinants, yielding more familiar formulas.

Algorithmic workflows involve:

• Forming intermediate products ($U=WA$, $A^*A$, $AA^*$, $V=AW$),
• Computing principal powers and ranks,
• Constructing auxiliary matrices using the column–row determinant scheme,
• Composing the generalized inverse explicitly by the determinantal formulas.

A detailed example in (Kyrchei, 2020) walks through such computations for quaternionic data, confirming feasibility for explicit, exact algebraic solutions.

4. CMP in Lineage-Based Metaproductivity Metrics for Self-Improvements

In machine learning, CMP has been adopted to formalize and address the "Metaproductivity–Performance Mismatch" in the adaptive evolution of self-modifying agents (Wang et al., 24 Oct 2025). There, CMP serves as a metric for estimating an agent’s long-range self-improvement potential:

$$\mathrm{CMP}_{\pi}(\mathcal{T}, a) = \mathbb{E}_{\mathcal{T}_B \sim p_{\pi}(\cdot|\mathcal{T},a)} \left[ \max_{a^{\prime}\in C(\mathcal{T}_B, a)} U(a^{\prime}) \right]$$

with $C(\mathcal{T}_B, a)$ denoting the clade (all descendants) rooted at $a$ and $U$ a utility function (e.g., coding benchmark accuracy).

Empirically, CMP is estimated via the cumulative ratio of successful to attempted tasks over an agent’s entire clade: $$\widehat{\mathrm{CMP}}(a) = \frac{n_{\mathrm{success}}^{C}(a)}{n_{\mathrm{success}}^{C}(a) + n_{\mathrm{failure}}^{C}(a)}$$ where $$n_{\mathrm{success}}^{C}(a) = \sum_{a'\in C(a)} n_{\mathrm{success}}(a')$$.

Within the Huxley–Gödel Machine (HGM), CMP is used for:

• Expansion decisions via Thompson sampling on success/failure Beta posteriors,
• Steering self-improvement away from greedy, locally optimal, but ultimately unproductive trajectories.

Experiments on SWE-bench Verified and Polyglot datasets confirm that agents guided by CMP attain higher long-term benchmark performance, require fewer computational resources, and reach human-level coding proficiency when compared with agents guided only by immediate evaluation performance (Wang et al., 24 Oct 2025).

5. Relationships to Other Generalized Inverses and Operator Theory

Theoretical work demonstrates that the (weak) CMP inverse generalizes, extends, and unifies various established classes of generalized inverses:

• Reduces to the Drazin or MPCEP inverse in specific limits,
• Coincides with the Moore–Penrose inverse under explicit criteria (e.g., $\mathrm{ind}(A) \leq 1$, $A = AA^D A$, and $A = XA^2$ for some $\chi$-inverse $X$),
• Is characterized as a strong Bott–Duffin $(e, f)$-inverse for suitable idempotents $e, f$:

$YAY = Y,\quad YA = e,\quad AY = f$

This demonstrates robust algebraic structure and illuminates deep connections between structural matrix decompositions and projective algebra.

6. Applications and Significance

Both CMP inverses and CMP metrics have found extensive applications:

• In matrix and operator theory: symbolic and exact solutions to singular or rank-deficient linear and matrix equations, leveraging Cramer-type determinantal expressions (Kyrchei, 2020).
• In signal processing, robotics, and control: essential where quaternionic matrices model three-dimensional transformations.
• In self-improving AI agents: as a principled metric for navigated search through lineage trees, correcting for short-term benchmark-optimization bias and focusing on evolutionary potential (Wang et al., 24 Oct 2025).
• In image and signal reconstruction: enabling recovery via "core" structure isolation.
• In the theory of generalized inverses and algebraic system theory: providing new perspectives on projectors, decomposable systems, and interconnected subsystems.

A plausible implication is that as the spectrum of matrix, operator, and agent designs becomes more intricate, CMP-based approaches—be they algebraic or probabilistic—offer principled mechanisms to extract, measure, or optimize for productive structure or potential in both static and evolving systems.

7. Summary Table of CMP Concepts

Domain	CMP Object	Defining Feature/Formula
Linear algebra	CMP inverse	$A_{c,\tau} = A^\dagger A_1 A^\dagger$ using core–nilpotent decomposition
Linear algebra	Weighted CMP inverse	$A_{c,\tau,W} = A^\dagger W A_{d,W} A^\dagger$
Linear algebra	Weak CMP inverse	$A^{(w,c,\dagger)} = A^\dagger A X A A^\dagger$ with $X$ minimal rank weak Drazin inverse
AI agent optimization	CMP metaproductivity metric	$\mathrm{CMP}_\pi(\mathcal{T}, a)$, aggregating descendant utilities

In all settings, CMP formalizes the extraction or measurement of productive core—with respect to invertibility, structure, or evolutionary potential—out of complex, possibly degenerate systems. Its continued development unifies core algebraic structures and modern adaptive search, providing both explicit algorithms and guidance for optimization in high-dimensional, lineage-based explorations.