MIND GAIN Governed Amplification via Inertial Momentum Transfer ,The Law of Inertial Amplification by Momentum Transfer

This paper formalizes the Law of Inertial Amplification by Momentum Transfer — designated GAIN (Governed Amplification via INertial Momentum Transfer) — as the third foundational pillar of Momentonics, alongside TIME (Theorem of Instantaneous Momentum Emergence) and MIND (Momentum in Inertial/Non-Inertial Dynamics). GAIN establishes that in any dynamic system satisfying four conditions simultaneously — solidary frame, closed mass cycle, constant generation cost, and controlled frame transition — the ratio of kinetic energy gained by the accumulator to energy invested by the generator per event grows proportionally with the velocity of the accumulator. The pure law is expressed as A(v) = 2v/v₀, containing no recovery term — GAIN exists without any energy recovery mechanism, as a direct consequence of dE = v·dp. When enhanced by regenerative recovery (Double Harvest), the amplified form GAIN⁺ = 2v/[v₀(1−η)] multiplies the pure effect by 1/(1−η), yielding amplification factors exceeding 89× at 3,000 RPM and 299× at 10,000 RPM for representative parameters. A four-level hierarchy of amplification is established: Level 0 (conventional, no GAIN), Level 1 (pure GAIN), Level 2 (GAIN⁺ with Double Harvest), and Level 3 (GAIN⁺ at extended radius). The amplification factor is shown to be the ratio of two integrals — one quadratic (Hamilton) and one linear (Noether). Both the ABÏON Drive and ABÏON Prime satisfy all four conditions with no intrinsic theoretical ceiling imposed by the law itself; practical limits arise from engineering constraints. A zero-free-parameter experimental prediction is provided. No new physics is proposed or required. The mathematical derivation follows from established classical mechanics. Experimental confirmation is in preparation. Keywords:GAIN, governed amplification, inertial amplification, momentum transfer, law of inertial amplification, Momentonics, TIME theorem, MIND framework, Dynamic State Principle, Noether's theorem, Hamilton's principle, frame asymmetry, ABÏON Drive, ABÏON Prime, closed mass cycle, frame transition, linear energy regime, quadratic barrier, energy storage, momentum engineering, Inertial Amplification Factor, Dynamic Recovery Architecture, falsifiability