The Machine with Two Levers: Expanded Halving and Doubling

The Machine with Two Levers is a standalone WidiVision exploration of Expanded Halving and Doubling (EHD). A museum-story Vignette introduces a machine with two principal motions—MEDIATE and DUPLICATE—and a small REMEMBER tray. The following Illumination reorganizes addition, subtraction, multiplication, and division as different journeys through a common architecture of descent, ascent, parity memory, comparison, selection, and reconstruction. Version 2.0 extends that architecture in two directions. The nearest-square journey reads square-root extraction through base-four packets, doubled sides, and small signed misses. The factor-door journey grows paired square sides until their difference matches the number field, exposing a rectangle whose sides are a factor pair. The closing section brings this view together with WidiVision’s earlier gnomonic rearrangement and local digit-halving perspectives. The purpose is conceptual rather than computational speed. The publication makes no claim that the classical procedures, digit-recurrence square root, difference-of-squares factoring, binary tree search, or their underlying algorithms are new. Its contribution is a unified WidiVision interpretation that makes the connections among mediation, duplication, remembered discrepancy, square-root choice, paired-square growth, and factor recovery visible to a general reader. The deposit includes a separate technical companion proving the nearest-square recurrence, paired-square factor correspondence, finite-horizon completeness result, and safe pruning lemma. It also includes Computational Explorations EHD.1–EHD.3 as offline browser laboratories for the four basic operations, nearest-square extraction, and factor-door journeys. Bill Widi conceived EHD and proposed the square-root and paired-square factoring extensions. ChatGPT (OpenAI) assisted with mathematical formalization, proofs, computational testing, drafting, software design, prior-art framing, and editorial development. Bill Widi directed the project, reviewed the completed work, and takes responsibility for its final form.