Riemann Integration from Three Primitives

The Riemann integral a f (x) dx is derived entirely from the Tree of Continua C andthe three primitives — same, different, opposite — without invoking limits, ε-δ arguments,or the classical construction of the real line.The interval a, b is partitioned into cylinder sets by the base-k expansion. At finitedepth d, there are (k + 1)d cylinder sets of equal width ∆xd = (b − a)/(k + 1)d . A functionf : a, b → R is a labeling of these cylinder sets by values in Q(ωN ) ⊂ Per(C). TheRiemann sum at depth d: is a finite sum of periodic orbits — itself a periodic orbit in Per(C), exact requiring no analysis.The Riemann sums form a compatible family (Rd (f ))d≥1 via the partition identityFVd (x) = s Vd+1 (xs ), which forces the averaging restriction map. The Riemann integralis the IPG reading at ∞ of this compatible family.The Fundamental Theorem of Calculus is a finite algebraic identity at each depth d —the finite difference operator applied to the depth-d Riemann sum returns f at depth d —whose IPG reading at ∞ gives the classical statement.The signed area — the reversal a f = − b f — is the chiral involution on theorientation of the interval: the opposite primitive.Riemann integrability is restated as a cylinder set condition: f is Riemann integrableif and only if the compatible family (Rd (f ))d≥1 has a well-defined IPG reading at ∞,which holds if and only if the set of discontinuities of f has measure zero — a statementabout which cylinder sets fail to converge.Lebesgue integration partitions the range cylinder sets rather than the domain cylindersets. It is the TolFilt morphism theory of f viewed as a map from domain cylinder setsto range cylinder sets. Measure theory is TolFilt morphism theory.No limits in the classical sense. No ε-δ. No measure axioms. Three primitives. Oneintegral.