What does this research mean for the field?

Prenary Mathematics can be formalized by retaining the classical complex field as the object layer while introducing an independent action layer governed by an action-residual axiom, where every action leaves an irreducible positive residual. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

To establish a minimal axiomatic foundation for Prenary Mathematics by separating the classical object layer from an independent action layer characterized by an irreducible positive residual.

September 14, 2026Open Access

NEW(S)偏元数学地基定稿:动作留差公理与对象层/动作层分离(Prenary Mathematics Foundation: The Action-Residual Axiom and Object–Action Layer Separation)

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To establish a minimal axiomatic foundation for Prenary Mathematics by separating the classical object layer from an independent action layer characterized by an irreducible positive residual.
Formulated the action-residual axiom stating that actual outcomes retain an irreducible positive residual ε ∈ (0, δ₀) relative to classical mathematical outcomes.
Maintained the classical complex field (ℂ) as the unaltered object layer while defining degeneration conditions when the upper bound δ₀ = 0.
Developed and verified three formal proof files (Day22-01/02/03) using the Lean 4 interactive theorem prover and secondary comparator checks across two independent runs.
Successfully verified within the Lean 4 kernel (achieving 'No goals') the existence, positivity, open-interval bounds, and non-triviality of the action-residual function.
Formally proved the complete closure of the degeneration chain, recovering standard classical mathematics when δ₀ equals 0.
Established reproducible, machine-checked axiomatic consistency alongside explicitly stated falsification conditions without relying on empirical experimental validation.

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Implication

Mathematical analysis introduces an action-residual axiom separating object and action layers, suggesting a framework that formally degenerates to classical complex analysis.

Cite This Study

Song Chen (2026) studied this question.

synapsesocial.com/papers/6aa7b3aa0926e14a848b2c08 https://doi.org/10.5281/zenodo.22719724

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Implication

Mathematical analysis introduces an action-residual axiom separating object and action layers, suggesting a framework that formally degenerates to classical complex analysis.

NEW(S)偏元数学地基定稿:动作留差公理与对象层/动作层分离(Prenary Mathematics Foundation: The Action-Residual Axiom and Object–Action Layer Separation)

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Discussion

Implication

Cite This Study

NEW(S)偏元数学地基定稿:动作留差公理与对象层/动作层分离(Prenary Mathematics Foundation: The Action-Residual Axiom and Object–Action Layer Separation)

Key Points

Discussion

Implication

Cite This Study

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