Auditable Mathematics_ A New Foundation

Abstract: Auditable Mathematics (AM) Auditable Mathematics (AM) is a meta-theoretical framework designed to dismantle the "logical inflation" caused by the unconditional acceptance of infinity and continuity in traditional mathematics. It seeks to re-anchor mathematical truth in the finitude of the physical world through resource-based constraints and rigorous computational verification. 1. Fundamental Axioms Rejection of Actual Infinity: AM treats mathematics as a process rather than a collection of pre-existing entities. Numbers are defined as "verifiable records" of computation rather than abstract points. Universal Computational Bound (UCB): A hard ceiling is established based on the physical limits of the observable universe (approximately atoms). Values exceeding the universe's total information-carrying capacity are classified as "Hyper-audited Numbers" and lose deterministic truth. Pixelated Space: The system rejects infinite subdivision, defining a minimum spatial unit (anchored to the Planck scale). This removes the "sites" required for infinite recursive folding. 2. Meta-Computational Logic Constructive Existence: The existence of a mathematical object is strictly bound to a constructive algorithm that terminates within the UCB. AM rejects pure reductio ad absurdum, asserting that "non-non-existence" does not automatically grant existence. The Entropy Tax (): Incorporating Landauer’s Principle, AM acknowledges that every logical operation has an energy cost. Theorems are evaluated based on their "Computational Work," and those requiring excessive resources are flagged as "expensive hallucinations". Complexity Collapse: When a proof's complexity exceeds a civilization's audit capacity, the conclusion transitions from an "absolute truth" to a "statistical truth". 3. Deconstruction of Paradoxes and Pathologies Fractal Stripping: AM resolves the "finite area, infinite perimeter" paradox by enforcing boundary thickness and recursive truncation at . Perimeters are proven to converge into finite constants at microscopic scales. Calculus Liquidation: The framework replaces limit language with a Finite Difference Protocol. Derivatives are redefined as interval slopes with a mandatory minimum window . Smoothing Singularities: In physical modeling, singularities (like those in black holes) are audited as mathematical overflows and auto-corrected into smooth distributions of finite density. 4. Impact on AI and Society Algorithmic Justice: AM provides a "survival protocol" for AI, establishing auditable logic chains for every prediction to eliminate "black box" hallucinations. Financial Integrity: By introducing discrete time-scale auditing, AM exposes system risks in models (like Black-Scholes) that ignore physical latency and execution delays. Keywords:Auditable Mathematics (AM);Universal Computational Bound (UCB);Pixelated Space Axiom;Potential Infinity;Constructive Existence;Entropy Tax;Finite Difference Protocol;Complexity Collapse;Singularity Smoothing;Algorithmic Traceability;