The Boundary of the δ-Calculus: The logical cost of the continuum

Iterated distinction generates a canonical arithmetic tower N ⊂ Z ⊂ Q with finite codes and computable equality and order. Does it also determine the real line on which the theory’s laws are written? Countable finitary generation cannot produce every real, and the ordered-field laws together with the recognition cost J(x) = ½(x + x⁻¹) − 1 admit incomplete models, so order-completeness is an additional axiom with no countable model. Separately, the exact decisions that are free on the tower have strengths on Cauchy reals measured by principles of omniscience: deciding x = 0 is the weak limited principle, trichotomy is the limited principle, weak dichotomy is the lesser limited principle. The theory’s displayed scalar formulas, including its indexed mass families, lie in the least subfield of R containing π and closed under exp and log, a countable field that also contains φ, √2, and log 2; the masses themselves lie in Q(φ). Structural laws determine J on the nonzero rationals; continuity then determines the extension agreeing there on R>0, and rational agreement alone does not. Finally, a finite protocol, a decoder and a publisher of records with a round-trip identity, exists for a set classically exactly when the set injects into the finite records, while an effective protocol is a stronger requirement; N, Z, and Q have effective protocols and R has no protocol at all. Under one stated physical premise, every physically real value space has an effective protocol, and none has all of R as its exact-output range.