The Mirror Calculus: A Presence-Only Mathematical Language; Formation, Descent, Geometry, and the Register-Typed Resolution of the Riemann Question

This volume develops a mathematical notation in which every markis kept under left--right reflection and the numeral zero does not occur, and carries that notation from its grammar through geometry,algebra, analysis, and analytic number theory. Formation precedestruth. The Mirror Calculus enactspresence-only formation: an object-language inscription forms onlythrough a positive witness, trace, relation, transformation, roster,pairing, certificate, or explicit sponsorship event.Non-presentation forms no substitute object-language term. Numericalzero, empty collections, null returns, default falsity, vacuousjudgments, and negative records inferred from failed search areanalyzed as surrogate-zero devices: they make non-presentationparticipate as an object, value, or verdict. The strict Mirror grammar has exactly twelve object constructors:\[Bal,\ Row,\ Jux,\ Stk,,\ Dn,\ Lk,\ Ovl,,\ Box,\ OBox,\ Frac.\]Strict object atoms are fixed by horizontal reflection.Reflection reverses the arguments of\(Bal\), \(Row\), and \(Jux\), and actspointwise on the remaining constructors. The resulting involution,well-formedness preservation, and renderer equivariance have a fullstructural proof. A smaller implementation fragment\(\) is represented by the accompanying Lean source; the presentedition does not identify that fragment with the whole language.In particular, the current implementation datatype contains avariant named blank. The kernel artifact therefore concernsimplementation control syntax and has not yet formalized the stricter\(/\) separation adopted here. Blank is not a term denoting emptiness. It is a metasyntacticrenderer control for withheld emission:\((t)↓ e\) records successful emission and\((t)↑\) is a metalanguage judgment. A one-sided balanceor an enclosure with withheld content is consequently a diagnosticdisplay about evaluation, not a strict object-language sentence. The Riemann application is resolved only in explicitly separatedregisters. The literal zero-locus sentence\(\) is not in \(()\); the native axis sentence\(\) is. The explicitly sponsored extension\(=+\) derives the native sentence. The unextendedbase theory derives neither class-wide Admission nor itscounter-inscription. The witnessed semantics separately sponsors acoherent omega-kept family as a completion postulate, without therebysupplying its components or an analytic proof. Under the namedclassical interface,\[I().\]Symmetry alone is insufficient. Thus the work claims neither anunpriced classical proof nor a machine verification of the wholelanguage. It supplies a presence foundation, a strict mirrorgrammar, a typed root algebra with its dagger reading, scoped formaland computational assurance, a research program recorded at exactstrength, an external blind-reading audit of the object language, anative axiomatic derivation, a relative class-wide model theorem,and an exactly priced analytic interface.