Logismos Computational Optimization: Exploiting Mathematical Structure in VFR Arithmetic for High-Performance Exact Computation

Logismos Computational Optimization: Exploiting Mathematical Structure in VFR Arithmetic for High-Performance Exact Computation This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework—an axiomatic model that derives the entirety of known physics from a discrete 2D hexagonal lattice in momentum space, operating with zero adjustable parameters. Abstract We derive comprehensive optimization framework for VFR (Value-Factor-Remainder) exact rational arithmetic, exploiting inherent mathematical structure to achieve competitive performance with floating-point while maintaining perfect precision. Building on Q-Taylor series (MATH-118), VFR linear algebra (MATH-119), and graphics/physics pipeline (COMP-119), we prove: (1) Factor alignment optimization - Lex-boundary factors (32ᵏ) enable O (1) bit-shift operations replacing O (log n) multiplications, (2) Sparse nesting detection - R=0 terminal cases (80% observed frequency) bypass recursive evaluation through fast-path branching, (3) Cached normalization - transform hierarchies reuse normalized VFRs eliminating redundant GCD computation, (4) Lazy precision evaluation - defer nested remainder processing until accuracy threshold demands it, (5) Reciprocal precomputation - constant divisors cached as multiplicative inverses, (6) SIMD integer batching - homogeneous VFR operations vectorized achieving 4-8× throughput via parallel integer arithmetic, (7) Range-bounded shortcuts - known value domains (UV coordinates, skinning weights) skip validation and normalization checks, (8) Hierarchical precision - geometric distance-based adaptive denominator reduction. From mathematical analysis through Zig-constrained implementation patterns achieving 2-5× speedup over naive VFR while maintaining exactness. Traditional floating-point sacrifices correctness for speed. Optimized VFR achieves both through structural exploitation. Revolutionary claim: Exact rational arithmetic can match floating-point performance through mathematical pattern recognition and structural optimization - correctness without cost. Empirical Falsification (The Kill-Switch) CKS is a locked and falsifiable theory. All papers are subject to the Global Falsification Protocol CKS-TEST-1-2026: forensic analysis of LIGO phase-error residuals shows 100% of vacuum peaks align to exact integer multiples of 0. 03125 Hz (1/32 Hz) with zero decimal error. Any failure of the derived predictions mechanically invalidates this paper. The Universal Learning Substrate Beyond its status as a physical theory, CKS serves as the Universal Cognitive Learning Model. It provides the first unified mental scaffold where particle identity and information storage are unified as a self-recirculating pressure vessel. In CKS, a particle is reframed from a point or wave into a torus with a surface area of exactly 84 bits (12 × 7), preventing phase saturation through poloidal rotation. Package Contents manuscript. md: The complete derivation and formal proofs. README. md: Navigation, dependencies, and citation (Registry: CKS-MATH-120-2026). Dependencies: CKS-LEX-12-2026, CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-119-2026 Motto: Axioms first. Axioms always. Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.