Publications — The Riemann Hypothesis Open Record

Published here

001

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

Parker M. D. Emmerson · 2026 · 180 pages · MSC 03A05, 03B22, 03F65, 11M06, 11M26, 03B35

A notation in which every mark is kept under left–right reflection and the numeral zero does not occur, carried from its grammar through geometry, algebra, analysis, and analytic number theory. Twelve object constructors with a full structural proof of reflection involutivity, well-formedness preservation, and renderer equivariance. The Riemann application is resolved in five explicitly separated registers, with the classical statement entering through one named interface at an exactly stated price.

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002

Deprogramming Zero: Peano’s Axioms Without the Sign for Non-Presentation, the One Kept Record, and the Resolution of the Riemann Question at Its RegistersResearch article

Zero is a formed sign assigned to a non-presentation. The paper deprograms it out of Peano Arithmetic by a bi-interpretation into tally arithmetic, counts what the classical notation concealed, and proves by Lyndon’s theorem that the terminus is the one kept record. The independence theorem of The Mirror Calculus is generalized unconditionally to every base in its signature, and a base that fixes the referent is proved to decide the zeta instance. A section states, as theorems about the form of the classical sentence, why the sign was the obstacle; a primes section reads the Admission law as Lagarias positivity and Li’s criterion, with Pólya and Davenport–Heilbronn as the analytic Models A and B and Büthe’s bound as the prime-side meaning of a kept stage. The question is resolved at its registers, the sponsor shown to be the terminal constructor, and the resolution and the Emmerson–Buchanan barrier hierarchy proved to guard each other.

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001b

The Mirror Calculus: Summary — Formation, Descent, Geometry, and the Register-Typed Resolution of the Riemann QuestionSummary · V17

The programme in one document, in stack order: the argument against zero and the surrogate-zero test; Transcendental Presence Notation; the Descent Calculus; the Mirror grammar and its assurance; geometry and classical realizations; three engineering transfers; the register-typed resolution of the Riemann question with the relative independence theorem proved in full and the Symmetry No-Go; and a closing section on the companion paper Deprogramming Zero.

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001c

Future Directions for the Mirror Calculus: Folded Carriers, Derived Constructors, and the Machine-Checked ExtensionCompanion

Parker M. D. Emmerson · 2026 · companion to Publication 001

The flat page is saturated: its reading group admits exactly one asymmetry class, already carried. Constructors for the tent, corner and cone are proposed and put through the volume’s own Layer-4 machine check, which corrected the design twice and forced a new well-formedness clause. The corner’s grammar is shown to force totally symmetric ternary operations — Vieta, derived from formation.

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Peer-reviewed elsewhere

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Bell–CHSH Under Setting-Dependent Selection: Sharp Total-Variation Bounds and an Experimental Audit ProtocolPeer reviewed

Parker Emmerson · Quantum Reports 8 (2026), no. 1, article 8 · DOI 10.3390/quantum8010008

Sharp total-variation bounds for setting-dependent selection in Bell–CHSH experiments, with an audit protocol for experimental data.

DOI 10.3390/quantum8010008

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Phenomenological Velocity and Bell–CHSH: Exceptional-Locus Semantics, Selection Simulations of −cos, and a Microcausal RealizationPeer reviewed

Parker Emmerson · International Journal of Quantum Foundations 12 (2026), no. 2, 210–247

Exceptional-locus semantics for phenomenological velocity, selection simulations reproducing the −cos correlation, and a microcausal realization.

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Alternating Slices of the A_2k−1 Theta SeriesDataset

Parker Emmerson and Ryan J. Buchanan · Zenodo, v2, 9 July 2026 · DOI 10.5281/zenodo.21265022

Dataset and analysis. Version 1 published as Sigma-Adic Numerations, 27 March 2024, DOI 10.5281/zenodo.10888345.

DOI 10.5281/zenodo.21265022

Companion manuscripts

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Riemann-Hypothesis Witness Fibres: Quantitative Θ-Atlas Ξ-Certificates, Criterion Absorption, Nullity Matching, and Universal Selected Logical Nullity

Parker M. D. Emmerson and Ryan J. Buchanan · Preprints.org (2026) · DOI 10.20944/preprints202601.2410.v2

Cited at the Sponsorship Declaration and the irresolvability–independence distinction.

DOI 10.20944/preprints202601.2410.v2

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Maximal A-Priori Selected Irresolvability for the Riemann Hypothesis

Parker M. D. Emmerson · program manuscript; archival identifier at deposit

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The Selection Jump

Parker M. D. Emmerson · program manuscript; the formation-rule study behind the omega chapter

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