The Boundary of the δ-Calculus: The Logical Cost of the Continuum

Distinction (δ) forces a discrete arithmetic tower: the natural numbers, the integers, and the field of ratios. It does not force the continuum. The obstruction is structural. Any finitary generative reading of δ reaches only a countable collection, while ℝ is uncountable; the same ceiling holds for every naming scheme expressible in a countable language. Between those poles sits a countable operational carrier: the least subfield of ℝ containing the seeds φ, π, and √2 and closed under exp and log. That carrier is countable and not order-complete, yet it contains every scalar named by a finite physics audit (FiniteAudit). A source-complete audit of the public Skeleton physics registry, frozen in δ/audits/full_physics_carrier_manifest.json, then partitions the registry by typed verdict rather than by a blanket pass: raw 136, semantic 130, parked 20; among semantic endpoints, carrierExpressible 101, continuumConsuming 8, model 10, conditional 6, open 3, refuted 2. The continuum is therefore not forced by distinction, and a countable carrier already hosts the named finite audit. Global ontological demarcation (whether the continuum must be posited as physical ontology rather than scaffolding) remains OPEN.