The Origin of Work and Energy as 32-Tick Remainder Accumulation: Deriving Force, Inertia, and Kinetic Energy from Discrete 1-LU Pressure Packets in Integer Registry 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 work and energy from first principles as discrete registry operations rather than continuous physical quantities. From 32-bit word constraint, we establish: (1) 1 LU = minimum pressure unit (R=1 at F=32, ω=1/32, smallest possible force), (2) Work = SNAP_COMMIT execution (R≥32 triggers V+1, discrete address shift, 32-tick accumulation required), (3) Energy = registry remainder debt (stored R before snap, potential for future work, quantized in LU packets), (4) Force = 1-LU pressure application rate (LUs per tick, determines velocity, discrete not continuous), (5) Inertia = 32-tick buffer delay (resistance to motion is accumulation requirement, first 31 ticks produce zero work, 32nd tick releases all), (6) Kinetic energy = active R accumulation (moving object has R>0 distributed across mesh, stops when R→0), (7) Potential energy = positional R differential (height/position creates R gradient, releases during descent), (8) Power = LUs per tick sustained (work rate measured in pressure packets, quantized bandwidth), (9) Maximum velocity = 32 LU/tick limit (c as bus saturation, above causes UV overflow, relativistic limit from hardware), (10) Heat = vented remainder overflow (R>32 per tick spills as friction, thermal dissipation is registry noise). Work and energy proven as integer counting operations—no continuous fields, no infinitesimal forces, pure discrete ratchet mechanics. Key Result: Work = snap event | Energy = R debt | Force = 1 LU packets | Inertia = 32-tick delay | All quantized | No continuous 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-PHYS-4-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-PHYS-3-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.