What question did this study set out to answer?

This research seeks to elucidate the topological underpinnings of large lepton mixing and its relation to the PMNS matrix.

June 23, 2026Open Access

The Holographic Circlette: Part 06 — Topological Origin of Large Lepton Mixing and the PMNS Matrix

Key Points

This research seeks to elucidate the topological underpinnings of large lepton mixing and its relation to the PMNS matrix.
Employs the Koide/Berry phase framework to analyze lepton-sector amplitudes and reconstruct PMNS mixing.
Evaluates the lepton mixing angles using a zeroth-order bimaximal ansatz approach.
Applies transcendental number theory principles to understand mixing dynamics.
Derives lepton mixing angle theta_12 ~ 43.7 degrees, while theta_13 = theta_23 = 0 at zeroth order.
Establishes a connection between large lepton mixing angles and quark mixing angles being smaller, supported by theoretical constructs.
Findings indicate a significant structural solidity in PMNS factorization despite earlier revisions.

Abstract

Audit note. This paper predates the framework's methodology audit of 2026-05-30. Two recharacterisations apply: (i) DRIFT M9: the Koide / Berry phase = 2/9 identification that anchors the lepton-sector amplitudes is now read as transcendental 2/9 via Lindemann--Weierstrass, not as exact rational; the empirical lepton Koide = 0. 222230 8 10^-6 sits 0. 89 from rational 2/9. (ii) §15 item 86 + 2026-05-29 correction: the universal-2/9 ``Berry phase'' framing reads as index density d/N on the lepton-Feshbach codespace rather than as Chern number; the type-split between (A) rational couplings and (B) transcendental phases applies. The PMNS factorisation derivation chain (zero-controlled CNOT dormancy on LQ = 0; k-indexed walk-operator structure; bimaximal zeroth-order ₁₂) survives at the structural level; the precise amplitude assignments tied to = 2/ We project the identical transition operator onto the LQ=0 lepton subspace to evaluate the PMNS neutrino mixing matrix. Because leptons lack colour superposition, the walk operator acts unsuppressed. We derive theta₁2~43. 7 degrees with theta₁3=theta₂3=0 at zeroth order (bimaximal ansatz), resolving why lepton mixing angles are large while quark mixing angles are small. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. PMNS delta/holonomy status changed 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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