Recursive Knowledge Synthesis (RKS)

• Recursive Knowledge Synthesis (RKS) is a framework that recursively refines knowledge states in multi-LLM systems using a tri-agent architecture.
• It employs semantic generation, analytical consistency checking, and transparency auditing to iteratively enforce coherence and stability.
• Grounded in contraction mappings and fixed-point theory, RKS guarantees convergence, transparency, and explainability in practical deployments.

Recursive Knowledge Synthesis (RKS) is a formal framework for recursive, stability-guaranteed knowledge refinement in LLM ensembles. RKS is defined in terms of a tri-agent architecture: semantic generation, analytical consistency checking, and transparency auditing are delegated to separate, heterogeneous LLM modules, which interact through an explicitly recursive cycle. Each module imposes distinct, mutually constraining transformations on the shared “knowledge state,” yielding synthesized outputs that are irreducible to any single-model process. The central innovation in RKS is its mathematical foundation in contraction mappings and fixed-point theory, providing theoretically and empirically validated guarantees of convergence, transparency, and explainability in multi-LLM systems under realistic, public-access settings (Shigemura, 17 Dec 2025).

1. Mathematical Foundations and Formal RKS Model

Let $\mathcal{K}$ denote the Banach space of knowledge states, with metric $d(x, y) = \|x - y\|_2$. Recursive Knowledge Synthesis decomposes the transformation of knowledge as sequential application of three operators:

• $M_S: \mathcal{K} \to \mathcal{K}$ — Semantic generation
• $M_A: \mathcal{K} \to \mathcal{K}$ — Analytical consistency
• $M_T: \mathcal{K} \to \mathcal{K}$ — Transparency auditing

The composite validation operator is defined as $V_{\mathrm{Op}} = M_T \circ M_A \circ M_S$. Recursive updates proceed according to

$$\mathrm{Knowledge}_{t+1} = V_{\mathrm{Op}}(\mathrm{Knowledge}_t)$$

A core requirement is that $V_{\mathrm{Op}}$ be a contraction: $\exists\, \gamma \in [0, 1)$ such that

$$\|V_{\mathrm{Op}}(x) - V_{\mathrm{Op}}(y)\|_2 \leq \gamma \|x - y\|_2 \quad \forall x, y \in \mathcal{K}$$

This is practically enforced by the $d(x, y) = \|x - y\|_2$0 module, which acts as a projection onto the space of transparency-compliant knowledge states:

$d(x, y) = \|x - y\|_2$1

With $d(x, y) = \|x - y\|_2$2 and $d(x, y) = \|x - y\|_2$3 at most non-expansive ($d(x, y) = \|x - y\|_2$4 and analogous for $d(x, y) = \|x - y\|_2$5), Banach’s Fixed-Point Theorem ensures a unique fixed point $d(x, y) = \|x - y\|_2$6 and geometric-rate convergence.

2. Tri-Agent Architecture and Recursive Cycle

RKS implements a structured tri-agent system, with each functional role encapsulated by a separate LLM module:

• Semantic Generator ($d(x, y) = \|x - y\|_2$7): Receives the knowledge draft in natural-language–encoded vector form, outputs a semantically coherent revision.
• Analytical Consistency Checker ($d(x, y) = \|x - y\|_2$8): Processes $d(x, y) = \|x - y\|_2$9’s output, identifies and corrects logical inconsistencies.
• Transparency Auditor ($M_S: \mathcal{K} \to \mathcal{K}$0): Ingests the previous stage’s result and evaluates compliance with ethical and explainability constraints, producing an audited knowledge state.

The recursive operation cycle is as follows: $V_{\mathrm{Op}} = M_T \circ M_A \circ M_S$2

3. Evaluation Metrics and Stability Measures

RKS system performance is quantitatively evaluated using four primary, human-annotated metrics:

• Transparency Score ($M_S: \mathcal{K} \to \mathcal{K}$1): Averaged from Explainability Coefficient ($M_S: \mathcal{K} \to \mathcal{K}$2) and Traceability Parameter ($M_S: \mathcal{K} \to \mathcal{K}$3),

$M_S: \mathcal{K} \to \mathcal{K}$4

High compliance: $M_S: \mathcal{K} \to \mathcal{K}$5; acceptable: $M_S: \mathcal{K} \to \mathcal{K}$6; forced re-audit if $M_S: \mathcal{K} \to \mathcal{K}$7.

• Deviation Detection Rate ($M_S: \mathcal{K} \to \mathcal{K}$8):

$M_S: \mathcal{K} \to \mathcal{K}$9

where $M_A: \mathcal{K} \to \mathcal{K}$0 is the seeded set of contradictions.

• Correction Success Rate ($M_A: \mathcal{K} \to \mathcal{K}$1):

$M_A: \mathcal{K} \to \mathcal{K}$2

• Reflex Reliability Score ($M_A: \mathcal{K} \to \mathcal{K}$3): Weighted composite,

$M_A: \mathcal{K} \to \mathcal{K}$4

The weighting reflects the empirically observed dominance of missed deviations.

4. Empirical Outcomes and Experimental Configuration

Empirical investigations involved 47 independent trials (maximum 25 recursive iterations or 120 minutes per trial) using public-access, free-tier deployments of commodity LLMs:

• $M_A: \mathcal{K} \to \mathcal{K}$5: ChatGPT (OpenAI GPT-5.0 free tier)
• $M_A: \mathcal{K} \to \mathcal{K}$6: Gemini Pro (Google free tier)
• $M_A: \mathcal{K} \to \mathcal{K}$7: Copilot (Microsoft M365 free tier)

Standardized prompt templates orchestrated the agent cycle. The convergence criterion enforced $M_A: \mathcal{K} \to \mathcal{K}$8.

Empirical findings include:

Metric	Value/Result	Description
RRS (mean)	$M_A: \mathcal{K} \to \mathcal{K}$9	Composite system reliability
$M_T: \mathcal{K} \to \mathcal{K}$0	68% of trials	High transparency compliance
$M_T: \mathcal{K} \to \mathcal{K}$1	92% of trials	Acceptable transparency compliance
DDR (mean, $M_T: \mathcal{K} \to \mathcal{K}$2)	$M_T: \mathcal{K} \to \mathcal{K}$3	Consistency error detection
CSR (mean)	$M_T: \mathcal{K} \to \mathcal{K}$4	Correction following detected deviations
Convergence rate	89% of trials	Achieved fixed point
Mean iteration to converge	$M_T: \mathcal{K} \to \mathcal{K}$5	Recursive cycles to solution

Enforcement of $M_T: \mathcal{K} \to \mathcal{K}$6 thresholds led to forced re-audits, preventing degradation in transparency. DDR was observed as the major contributor to composite reliability, justifying its preeminent weight in $M_T: \mathcal{K} \to \mathcal{K}$7.

5. Theoretical and Practical Implications

Observed convergence rates and transparency compliance support the interpretation that the $M_T: \mathcal{K} \to \mathcal{K}$8 module effectively induces a contraction constant $M_T: \mathcal{K} \to \mathcal{K}$9, aligning empirical outcomes with Banach Fixed-Point Theorem predictions. In situations where $V_{\mathrm{Op}} = M_T \circ M_A \circ M_S$0 dipped below compliance thresholds, the recursive correction protocol prevented drift, reinforcing the system’s stability. The empirical convergence rate (89%) is consistent with an inferred contraction constant $V_{\mathrm{Op}} = M_T \circ M_A \circ M_S$1.

A plausible implication is that the logical structure imposed by recursive tri-agent validation, with transparency auditing as the contraction operator, is necessary for ensuring unique, explainable, and compliant knowledge representations in multi-LLM deployments.

6. Broader Impact and Applicability

The RKS paradigm demonstrates that safety-preserving, explainable, and convergent multi-LLM systems are feasible using only publicly accessible LLM APIs, bridging resource disparities in research environments. The human-supervised, session-level role decomposition architecture prioritizes transparency and auditability, trading off some efficiency for strict oversight.

RKS provides a foundational recipe for future safety-first, human-in-the-loop multi-LLM deployments. It exemplifies a scalable, contractive approach to composite LLM knowledge synthesis and offers reproducible metrics for cross-model stability and explainability assessments (Shigemura, 17 Dec 2025).