A Proposed Hereditary Explicit-Formula Proof of the Riemann Hypothesis
Year: 2026
Type: preprint
Abstract: The argument is organized around a same-scalar completed explicit formula: before adaptive geometric choices are made, the arithmetic cells, residue conventions and normalizations for a detector bank are frozen, so the zero-side and prime-side estimates always refer to one identical arithmetic quantity. The zero geometry is treated at the resolution actually available to the detector. Separated zero packets are controlled by nonharmonic Fourier–Vandermonde frames. Sub-Rayleigh critical-line clusters are replaced by the multiplicity-weighted, carrier-weighted aggregate functionals that occur in the explicit formula, with carrier-weighted barycentres removing first-order cluster displacement. Extended packets are reduced by unit-slab tomography and pivotal refinement. Historical recurrence is recorded by physical zero identities and aggregate functionals, not by detector-dependent singular vectors; exact non-recycling reinstates every locally active historical zero in each newly built protected span. A hypothetical off-critical zero is converted into a one-sign physical debt. A shifted completed-neutral collar construction compresses the inner historical population to two stable aggregate coordinates, while a localized target-null rational payer controls historical outer orbits at polynomial cost. The final contradiction separates exposure from realization: once a nonzero physical debt is exposed, an independent homogeneous realization episode freezes its own arithmetic scalar and prescribes that same physical datum while assigning zero to the remaining active nuisance coordinates. Stable realization has analytic error strictly smaller than the locked debt. The realization step has two distinct height regimes, both of which are part of the proof. Write T0 = 3·1012. When the independently refrozen realization height satisfies H ≥T0, the all-height carrier, frame and matched-AFE certificates apply, and collective pivotal shadows spend the finite same-target Gevrey active occurrence arena of size O(LsH), 1 <s<2. When debt localization instead selects T0/2 ≤H <T0, no extrapolation of those all-height certificates is made: the whole refrozen Gaussian guard lies below the Platt–Trudgian verified height HPT, so every zero in that guard is already known to be critical, and a separate fixed-parameter factor-two continuation supplies the frame, AFE, completed-background and halo estimates on the entire interval [T0/2,T0). In this lower regime carrier-weighted slabs drive the pivotal geometry, and every continuing atomic successor spends a distinct physical zero value in a complete-guard occurrence space of dimension <96L2H + 3. These two regimes exhaust the possible refrozen heights because the middle-half debt localization gives H ∈[T/2,3T/2] from an exposure height T ≥T0. Target changes in either regime are controlled by a separate well-founded genealogy. Thus every noncontradictory branch either terminates, changes a permanent target coordinate, or consumes a finite bank-independent physical resource. The article contains the analytic derivations, finite-dimensional realization and routing theorems, directed numerical certificates, controlling output records and reproducibility hashes. Disclaimer: generative AI has been used extensively during the research. The author is an amateur who undertook this project as an experiment on learning and the possibilities of human-AI collaboration (the manuscript contains a more detailed disclaimer about authorship and AI). The research seemed to have reached a level where the only way forward would be independent verification by specialists. Any criticism and identification of potential errors are welcome.
Author: Kofrán, Gergely
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203890433
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