Self-Harmony Framework in AI & Social Models

• Self-Harmony Framework is a set of methodologies that harmonizes diverse signals to produce stable, invariant results across AI, logic, and social modeling.
• It employs a dual-view mechanism with harmonic mean aggregation to robustly select pseudo-labels, mitigating biases and enhancing adaptation.
• The framework extends its principles to social systems and proof-theoretic semantics, uniting diverse rules through mutual containment and consensus-driven dynamics.

The Self-Harmony Framework encompasses a set of methodologies and theoretical foundations for achieving robust, stable, and reliable adaptation and consensus in AI, social systems, and proof-theoretic semantics. Across domains, the framework is characterized by the systematic balancing or harmonization of diverse, often conflicting, internal or external perspectives, signals, or rules, relying on principles of consistency, mutual containment, and information-theoretic invariance. The following presents a comprehensive synthesis of the key instantiations of the Self-Harmony Framework, as articulated in machine learning, social modeling, and logic.

1. Motivation and Conceptual Underpinnings

The Self-Harmony Framework arises from the need to induce robust generalization and adaptation without external supervision or explicit human labeling. In the context of test-time reinforcement learning (TTRL), a central challenge is the construction of trustworthy pseudo-labels when adapting LLMs using only synthetic, model-generated signals. Standard aggregation methods, such as majority voting, often suffer from systemic biases, reinforcing spurious but popular answers (the "echo chamber" effect). Self-Harmony is motivated by the observation that correct answers should demonstrate stability across different semantically equivalent phrasings or representations, while spurious answers lack such invariance.

Analogously, in proof-theoretic semantics and computational ludics, self-harmony encapsulates the harmony condition, where logical connectives are harmonious if their introduction and elimination rules coincide, ensuring a balance between syntactic formation and semantic admissibility.

In social modeling, the "harmony with diversity" phase in agent-based attraction-repulsion models represents a state where systems sustain diversity in opinions or behaviors while maintaining global coherence by equilibrating attractive and repulsive social forces.

2. Methodological Framework in Test-Time RL

In the TTRL setting, Self-Harmony operationalizes its principles by assigning a single model to two cooperative roles:

• Solver ($\mathcal{T}_\theta$): Generates answers for a given query.
• Reframer ($\mathcal{P}_\theta$): Paraphrases the query, inducing a semantically equivalent but distinct version.

The algorithm involves:

1. Generating multiple answers for both the original and reframed questions using the Solver role.
2. Aggregating answer frequencies from both views via the harmonic mean, defined as

$S(a) = \frac{2\, p_0(a)\, p_1(a)}{p_0(a) + p_1(a)}$

where $p_0(a)$ and $p_1(a)$ are the empirical frequencies of candidate answer $a$ for the original and reframed views, respectively.

1. Selecting as pseudo-label the answer $y^*$ with maximum $S(a)$:

$y^* = \underset{a}{\mathrm{arg\,max}}\; S(a)$

2. Updating model parameters via reinforcement learning using a reward based on pseudo-label agreement in both views and penalizing lack of diversity or format infidelity in paraphrasing.

This harmonic mean mechanism forms a view-invariant filter, strongly penalizing view-dependent artifacts and preferentially selecting answers robust to rephrasing.

3. Theoretical Guarantees and Formal Properties

The pseudo-label selection procedure is grounded in an information-theoretic objective that seeks to maximize mutual information between the model’s answers across views while penalizing view-dependent mutual information. The harmonic mean emerges as the optimal solution to this penalized mutual information criterion under mild regularity conditions. This produces pseudo-labels that are consistent and stable under question reframing, which is theoretically justified to reduce the likelihood of reinforcing spurious or brittle answers during test-time adaptation.

Moreover, the Self-Harmony framework in proof theory, as articulated in computational ludics, specifies the harmony condition for logical connectives. Harmony obtains if and only if the set of introduction rules matches exactly the set of elimination rules, formalized as

$\alpha^I = \alpha^E$

where $\alpha^I$ and $\alpha^E$ denote the sets of introduction and elimination actions, respectively. This mutual containment guarantees both invertibility and recoverability of connectives, unifying the inversion and recovery principles into a single structural criterion.

4. Comparative Analysis with Alternative Approaches

A key distinction of Self-Harmony arises in its divergence from majority voting and related aggregation methods:

Aggregation Method	Description	Vulnerabilities
Majority Voting	Selects answer with highest frequency in original	Amplifies systemic bias
Dual-View Majority Voting	Sums counts across both original and reframed views	Can favor inconsistent answers
Harmonic Mean (Self-Harmony)	Selects answer frequent in both views; penalizes view-dependent artifacts	Suppresses spurious/unbalanced answers

In contrast, harmonic mean consensus yields stable and reliable pseudo-labels, enables rapid test-time adaptation without human-generated labels, and outperforms classical baselines on a suite of reasoning benchmarks. Quantitative results demonstrate superior performance: Self-Harmony ranked first in 28 of 30 settings, with substantial accuracy gains (e.g., GSM8K Llama-3.1-8B: 91.6% vs. 60.5% for majority voting).

A similar logic underpins self-harmony in social agent-based models: only with intermediate interaction strength, sufficient susceptibility, and high tolerance does the system stabilize in the "harmony with diversity" phase, as shown via phase diagrams and order parameter analysis.

5. Empirical Validation and Stability

Extensive empirical testing demonstrates Self-Harmony’s practical benefits:

• State-of-the-art accuracy in test-time RL, with top ranking on nearly all measured configurations.
• Zero training failures across all experiments, indicating unprecedented empirical robustness.
• Consistency and reliability even in early adaptation stages, where standard aggregation methods fail.
• Ablation studies verify that each framework component (Solver, Reframer, harmonic mean aggregation, diversity reward) is essential to optimal performance.
• Human and LLM evaluation confirm successful semantic equivalence in paraphrasing, validating the Reframer’s efficacy.

The ablation and comparative studies corroborate the necessity of view-invariant selection and diversity in paraphrases, directly linking stability to the harmonic mean aggregation.

6. Extensions: Social Systems and Proof-Theoretic Semantics

In social modeling (Cui, 2023), the Self-Harmony Framework is instantiated as an "attraction-repulsion" model, revealing conditions for stability in collective opinion dynamics. The necessary regime for stable "harmony with diversity" requires moderate interaction strength, sufficient susceptibility, and high tolerance across agents and interactions. The framework provides order parameters—entropy, standard deviation, and susceptibility of entropy—to empirically demarcate the phase space and identify critical "triple points" where harmony, consensus, or polarization may emerge.

In proof theory (Naibo et al., 2021), the Self-Harmony Framework is realized through the harmony condition for logical connectives: harmony obtains precisely when introduction and elimination rules coincide, yielding syntactic, model-theoretic, and dynamic equivalence via dual decomposability in computational ludics. This provides a structural and semantic foundation for the definition of logical constants and underlies generalizations to novel logics.

7. Synthesis and Theoretical Significance

The Self-Harmony Framework, as unified across learning, logic, and social modeling, emphasizes the construction of stable, invariant signals or rules by enforcing concordance across internally distinct yet semantically equivalent views, subsystems, or rulesets. In machine learning, this is operationalized by dual-view, harmonic mean consensus; in logic, by mutual containment of introduction and elimination rules; in social systems, by balancing attractive and repulsive influences to sustain coherent yet diverse group states.

The framework demonstrates that harmonizing self-generated signals (self-supervision, self-play), or balancing rules of formation and admissibility, promotes stability and generalization in high-dimensional, label-free, or weakly supervised settings. Its principles are theoretically motivated, empirically validated, and foundational for the design of robust AI adaptation, logical deduction systems, and social consensus models. The Self-Harmony Framework thus stands as a paradigm for constructing invariance, reliability, and coherence without recourse to external supervision or rigid control.