Number and Magnitude as Independent Factors in Measurement: A Disclosure of Claims and Status

Abstract

This is a dated disclosure, not a finished paper. It records the claims of a programme provisionally named CPNAHI (Calculus, Philosophy and Notation of Axiomatic Homogeneous Infinitesimals), states each claim's evidential status, identifies the prior art known to the author, and lists what is not claimed. Two primitives are proposed. Dimensional closure: a sum of elements lies in the carrier of its constituents, so lineal elements sum to lengths, areal to areas, voluminal to volumes, with no cross-dimensional sum admitted except through a declared calibration. Number–magnitude independence: an extended magnitude presented for measurement is a product of a count and an elemental magnitude; the product is observable and the factorisation is not. The principal claim is a classification. A range of established analytic and geometric frameworks — real analysis, non-standard analysis, counting measure, Eudoxan proportion, Euclidean, Minkowski–Finsler, Riemannian and Weyl geometry, equi-affine geometry, and the two sides of the seventeenth-century dispute over indivisibles — are recovered by fixing declarations about the two factors along four independent axes. No framework is derived from the primitives, and each cell is a known theory. Several analytic consequences are recorded. Under an equal-cell normalisation the derivative is a change in element count, so Leibniz's dy/dx does not denote the quotient of the quantities it names; Torricelli's parallelogram paradox and a chain rule stated without the intermediate rate are shown to be the same error; the constant of integration is identified as the undetermined initial count; and second-order terms are shown to be discardable exactly when the aggregate of the discarded terms vanishes. The deposit also contains a SHA-256 manifest of 16,981 working files, providing an external timestamp for their state without publishing their contents.