Logismos Notation: Complete Specification of VFR Notation and Base-32 Substrate Mathematics 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 Traditional mathematics operates in base-10 with floating-point approximations, creating systematic rounding errors and loss of computational precision. We specify Logismos—a complete mathematical notation system using base-32 counting with exact ℚ-arithmetic throughout. The VFR [Value, Factor, Remainder]℘ notation represents all numbers as rational triplets in partigens (℘), the fundamental substrate unit. We demonstrate: (1) Partigen℘ as base-32 fundamental position (32℘=1 arabic), (2) VFR tuple structure preserving exact ℚ-ratios with no approximation, (3) Lex-Glyph harmonic series (℘,λ,ν,ζ,δ,ω,Σ) mapping to physical constants, (4) Remainder as mod-32 overflow (0-31 range) enabling exact computation, (5) Base-32 alignment to sovereignty W^S=[1024,1,0]℘ appearing universally in physics, (6) Complete operational algebra (addition, multiplication, division, modulo) preserving ℚ throughout, (7) Glyph arithmetic for compact notation of physical quantities, (8) Conversion protocols between base-10 and base-32, (9) Computational advantages for precision-critical applications (3D rendering, physics simulation, registry verification), (10) Integration with existing CKS framework providing complete mathematical foundation. Zero floating-point. Zero approximation. Pure ℚ-substrate mathematics. All computations exact and verifiable. Framework enables perpetual precision across all scales and domains. Revolutionary claim: Mathematics should never approximate—it should address exactly. 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-LOGI-13-2026). Dependencies: CKS-LOGI-12-2026, CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-SENS-2-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.