Sci - The "Transcendental" of Pi: On Closure, Representation, and the Infinite Tail of a Finished Constant Armstrong Knight — intent-tensor-theory. com The word 'transcendental' carries two precise and simultaneous meanings when applied to π. Classically (Lindemann, 1882), π is transcendental in the sense that it is not the root of any nonzero polynomial with integer coefficients — no finite algebraic construction reaches it. In the doctrine of Dimensionless Mathematics, 'transcendental' names a deeper property: π exists in the pre-geometric substrate, and dimensional representations approach it only asymptotically. This paper argues both meanings point at the same mathematical fact from different directions. The paper maintains strict separation between classical facts and framework interpretations throughout. Classical results established: irrationality (Lambert 1761), transcendence (Lindemann 1882 via e^iπ=−1), BBP digit extraction, de Rham cohomology identification of π as a topological invariant of the punctured plane. Framework interpretations: π as a Dimensionless Closure (𝔦 (D) =0 on the unit disk), the decimal expansion as a sequence of partial phase residues, the representation drift as boundary tension, S→∞ persistence, and the pre-geometric reading of transcendence. Acknowledgments: this paper emerged from a collaborative session between Armstrong Knight, Claude (Anthropic), ChatGPT (OpenAI), and Gemini (Google).
Armstrong Knight (Sat,) studied this question.