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

Abstract

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:\[\mathsf{Bal},\ \mathsf{Row},\ \mathsf{Jux},\ \mathsf{Stk},\\mathsf{Up},\ \mathsf{Dn},\ \mathsf{Lk},\ \mathsf{Ovl},\\mathsf{Adh},\ \mathsf{Box},\ \mathsf{OBox},\ \mathsf{Frac}.\]Strict object atoms are fixed by horizontal reflection.Reflection reverses the arguments of\(\mathsf{Bal}\), \(\mathsf{Row}\), and \(\mathsf{Jux}\), and actspointwise on the remaining constructors. The resulting involution,well-formedness preservation, and renderer equivariance have a fullstructural proof. A smaller implementation fragment\(\GK\) 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 \texttt{blank}. The kernel artifact therefore concernsimplementation control syntax and has not yet formalized the stricter\(\PreTerm/\Term\) separation adopted here. Blank is not a term denoting emptiness. It is a metasyntacticrenderer control for withheld emission:\(\Eval(t)\downarrow e\) records successful emission and\(\Eval(t)\uparrow\) 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\(\RHClassZero\) is not in \(\Sent(\GP)\); the native axis sentence\(\RHMirror\) is. The explicitly sponsored extension\(\Pdag=\Pbase+\AdmZeta\) 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,\[I(\AdmZeta)\quad\Longleftrightarrow\quad\RHClass.\]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.