What question did this study set out to answer?

This work aims to provide a theoretical basis for the mass ratios of fermions and their interactions using holographic codes.

June 23, 2026Open Access

The Holographic Circlette: Part I: The Encoding and Its Dynamics

Key Points

This work aims to provide a theoretical basis for the mass ratios of fermions and their interactions using holographic codes.
Defined an 8-bit quantum error-correcting code on a holographic lattice.
Derived key physical parameters from a single geometric parameter δ = 2/9.
Identified the CNOT gate as the governing dynamic rule related to weak interactions.
Charged lepton mass ratios were derived to 0.007% precision.
Weak mixing angle was calculated as sin²θ_W = 2/9 with a 0.5% error.
Dynamic dark energy model aligned with DESI DR2 observations within 1.5% accuracy.

Abstract

We propose a unified physical framework in which the Standard Model fermionspectrum corresponds to the set of valid codewords of an 8-bit quantumerror-correcting code defined on a holographic lattice. Four local constraintsselect exactly 45 valid matter states from 256 possibilities. The dynamicsare governed by a unique update rule - a CNOT gate at the bridge-isospinboundary - identified as the weak interaction. From this information-theoretic foundation, we derive: the charged lepton massratios to 0. 007\% precision from a single geometric parameter = 2/9;the weak mixing angle ²W = 2/9 (0. 5\% error) ; the W/Z bosonmass ratio MW/MZ = 7/9 (0. 06\% error) ; and the PMNS neutrinomixing angles under a bimaximal lattice ansatz. Gravity emerges as thecurvature of the rank-2 Fisher information tensor; special relativity as abandwidth constraint; and the cosmological constant as the vacuum informationfloor. A dynamic dark energy model matches DESI DR2 observations to within1. 5\%. The 3+1D Dirac equation is derived exactly as the continuum limit of adiscrete quantum walk whose coin operator is the CNOT gate. Rest mass is theCNOT execution frequency. The complex unit i is forced by the unitarity of areversible Boolean swap. The central result of this paper is that the integer ratio = 2/9 - the ratio of a 2-bit topological defect to a 9-bitplaquette - determines the charged lepton mass spectrum, the electroweakmixing angle, the Cabibbo angle, and the PMNS mixing matrix, with onecontinuous free parameter (the overall mass scale). A companion paper (Part~II) extends the framework to composite particles, proving a zero-sumconservation identity and deriving the W-boson quantum numbers and theMajorana nature of the neutrino from the code algebra. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Early part-series foundation; superseded by forced-byte/R14 framing The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.

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