KisMATH: Causal Reasoning & Algebra Dynamics

• KisMATH is a multidisciplinary framework uniting causal chain-of-thought reasoning in language models with symbolic algebra and dynamical systems analysis.
• It uses graph-based Causal CoT Graphs to extract and measure dependencies in reasoning processes, validated across datasets like GSM8K, MATH500, and AIME.
• The approach integrates studies on algebraic structures such as Kiselman’s semigroup and Kaprekar routines, revealing insights into iterative convergence and probabilistic dynamics.

KisMATH refers to recent, multifaceted research directions at the intersection of mathematical reasoning, symbolic algebra, and the analysis of structures underlying both algorithmic and neural processing of mathematics. This includes the empirical investigation of reasoning in LLMs via the KisMATH dataset and Causal CoT Graphs, detailed algebraic and dynamical analysis of Kiselman’s semigroup, and the parametric characterization of mathematical iteration phenomena such as Kaprekar routines. The term “KisMATH” thus encompasses advances in extracting, representing, and manipulating implicit and explicit mathematical structure.

1. Chain-of-Thought Reasoning and Causal CoT Graphs

The “KisMATH” dataset provides a principled framework for dissecting the internal mechanisms of chain-of-thought (CoT) reasoning in LLMs. Traditional CoT prompting has demonstrated empirical success in multi-step mathematical reasoning tasks, but its mechanism—whether decomposition into subproblems or mere retrieval—has remained contested. KisMATH directly addresses the question: Do LLMs encode and utilize causal structure in their reasoning outputs? This is accomplished by formalizing the concept of a Causal CoT Graph (CCG) (Saha et al., 15 Jul 2025).

A CCG is a directed acyclic graph $G=(V,E)$ automatically extracted from the derivational trace $(Q,R,A)$ of a mathematical problem, where $Q$ denotes parsed expressions from the question, $R$ from the CoT trace, and $A$ the answer. Each directed edge $(v_i \to v_j)$ encodes fine-grained functional or syntactic dependency. Causality is established by subtree and shared substructure matching in parsed symbolic forms. The resulting graph describes a set of reasoning paths (“R-paths”) linking initial question elements to the answer through intermediate reasoning steps, permitting explicit characterization of stepwise dependency and information flow.

Analysis of open-weight LLMs (1B–70B parameters) shows that masking reasoning nodes (via attention suppression) sharply increases answer entropy and eradicates prediction confidence ($D_{KS} \approx 1.0$, $p < 10^{-12}$), demonstrating that these nodes are necessary mediators. Moreover, individual R-path masking also significantly degrades performance ($D_{KS}>0.9$), evidencing that models concentrate probability mass along CCG-guided paths rather than spurious chains. This reveals a strong internal realization of CCG structure as a substrate for model reasoning.

2. Construction and Characteristics of the KisMATH Dataset

The KisMATH dataset comprises $1671$ annotated mathematics problems—$(Q,R,A)$0 from GSM8K (arithmetic), $(Q,R,A)$1 from MATH500 (pre-calculus Olympiad), and $(Q,R,A)$2 from AIME (contest mathematics)—paired with automatically extracted CoT traces and their CCGs (Saha et al., 15 Jul 2025). For each item, the dataset stores the question $(Q,R,A)$3, the explicit CoT trace $(Q,R,A)$4 (generated via OpenAI o3 with 5-shot prompts), the ground-truth answer $(Q,R,A)$5, and the CCG $(Q,R,A)$6.

Statistically, the structural complexity of problems varies by source: the mean CCG node count $(Q,R,A)$7 is $(Q,R,A)$8 for GSM8K, $(Q,R,A)$9 for MATH500, and $Q$0 for AIME; edge counts $Q$1 broadly scale with node count and task complexity. Longest simple R-paths ($Q$2 for GSM8K, $Q$3 for others) are selected per problem to maximize analytical leverage. These granular, graph-aligned annotations enable controlled, interpretable interventions on the reasoning space, facilitating rigorous tests of model sensitivity and internal faithfulness to reasoning structure.

3. Empirical Findings: Mediation, Path Realization, and Model Regimes

KisMATH enables surgical experimentation: for each problem, masking CCG-defined nodes or R-paths allows direct testing of information transmission. Masking all reasoning nodes triggers dramatic entropy inflation, indicating that these are indispensable mediators for correct inference. Analogous masking along a single R-path similarly disrupts output confidence, showing LLMs' operational reliance on these chains.

Approximate path probabilities $Q$4 show that legitimate CCG R-paths overwhelmingly dominate random alternatives (rank percentiles spike at $Q$5). Two behavioral regimes emerge: an “exponential” regime where nearly all extracted R-paths are favored, and a “bell-shaped” (exploration-heavy) regime where pass@k performance correlates with variance in $Q$6. The latter regime features higher geometric diversity of reasoning and richer exploration, evidenced by improved pass@k scores as $Q$7 increases.

4. Kiselman’s Semigroup: Algebraic Structure, Dynamics, and Ultrametric Geometry

KisMATH is also closely linked to the study of Kiselman’s semigroup $Q$8 (Andrenšek, 28 Apr 2026), defined by generators $Q$9 $R$0 and relations $R$1, $R$2 for $R$3. This finitely generated, idempotent, rectangular band-type monoid encodes essential combinatorial features central to convex analysis and algebraic combinatorics.

The level function $R$4 provides a hierarchical grading of elements, satisfying critical monotonicity and right-generator descent rules:

$R$5

Idempotents and the “zero” $R$6 induce stratifications critical to the analysis of both deterministic and random product dynamics.

5. Dynamical and Probabilistic Behavior in Kiselman’s Semigroup

Deterministic sequences of products in $R$7 eventually stabilize due to the finite range and the height-function's strict descent property. In random-product models (i.i.d. generator selection), the time $R$8 to reach the “zero” idempotent is distributed as a sum of $R$9 independent geometric variables, with mean $A$0 and, under uniform sampling, $A$1. This probabilistic profile is underpinned by the reduction of level via right-multiplication, modeled by an absorbing Markov chain on the level set.

A natural ultrametric $A$2 on $A$3 arises by comparing their images under deletion endomorphisms, yielding a strong triangle inequality and non-Archimedean metric topology. Ultraballs and spheres around the “zero” are characterized by explicit algebraic equations involving left-multiplication by idempotents.

6. Parametric Structures and Transformation Graphs in Kaprekar-Type Routines

KisMATH encompasses the explicit parametric analysis of Kaprekar routines via the Ki-function formalism (Nuez, 2021). For the 4-digit (base-10) Kaprekar process, the evolution of the system is governed not at the level of specific integers, but via projection onto parameter pairs $A$4 (differences of largest/smallest and middle digits). There are exactly $A$5 legitimate linear-affine transformations $A$6 that govern this system.

Each Ki maps $A$7 to a new pair, and iterated application forms a dynamical system described by a directed acyclic parameter-space graph. Cycles, fixed points, and global convergence—such as all 4-digit numbers eventually mapping to $A$8—are determined by algebraic properties of these transformations and their domains. Structural invariants (such as sums of opposite digits) and generalizations to $A$9-digit, other base systems, and higher cycles are derived formally.

7. Implications, Extensions, and Research Directions

The KisMATH framework, as articulated in these diverse domains, demonstrates that both neural and symbolic mathematical systems encode, utilize, and can be manipulated via explicit causal or parametric structures. In LLMs, controlled graph-based interventions on CoT traces enable principled tests of faithfulness, suggest approaches to counterfactual training objectives, and open the way to combining symbolic and neural reasoning via explicit graph interfaces (Saha et al., 15 Jul 2025). In algebraic contexts, the precise characterization of semigroup dynamics and transformation trees underpins both theoretical understanding and algorithmic applications.

Ongoing research includes expansions to more complex reasoning domains, incorporation of reflection and backtracking edges in CCGs, probabilistic quantification of causal impact in graphical reasoning, and algebraic classification of orbits and cycles in broader classes of transformation systems. The entire KisMATH program signals an overview between empirical, symbolic, and algebraic methodologies for understanding the structure, evolution, and manipulation of mathematical reasoning in both computational and abstract systems.