What question did this study set out to answer?

The aim is to provide a rigorous proof that the observed Lorentz anisotropy in the Truncated Cubic Honeycomb vanishes logarithmically in the infrared limit.

June 14, 2026Open Access

Velocity Unification on the Truncated Cubic Honeycomb: A Wilsonian Irrelevant-Operator Proof That the 41% Bare Lorentz Anisotropy Vanishes Logarithmically in the Infrared

Key Points

The aim is to provide a rigorous proof that the observed Lorentz anisotropy in the Truncated Cubic Honeycomb vanishes logarithmically in the infrared limit.
Analysis of the operator governing Lorentz anisotropy involving higher derivatives.
Utilization of Wilsonian renormalisation-group techniques to examine the behavior of the anisotropy at different energy scales.
Application of dimensional analysis to establish the consequences for Lorentz invariance.
The coefficient of the anisotropic operator flows to zero as energy approaches zero, demonstrating vanishing anisotropy.
Wilsonian classification confirms that this anisotropy is effectively irrelevant in the renormalisation group sense.
The identification of the Eg dynamical mass scale aligns with predictions indicating a relationship to the Planck mass.

Abstract

ThecanonicalHolographicCirclette3DsubstrateofZ3 ⊗Q3 (theTruncatedCubicHoneycomb t4, 3, 4) exhibits a first-order O (k) Lorentz anisotropy at the Planck scale, with direction- dependent bare group velocities satisfying v100/v111= √2 (a 41% maximum-to-minimum ratio) governed by the cubic harmonic invariant I4 (ˆ n) = n2 xn2 y + n2 y n2 z + n2 z n2 x (ANCHOR §15 item 102). Naively, this represents a catastrophic substrate-level violation of macroscopic Lorentz invariance. We demonstrate that the anisotropy is, in fact, governed by an operator of the form Oaniso ∼a2 (∂2ϕ) 2, whose coefficient carries mass dimension−2 in d= 4 spacetime — making it a strictly irrelevant operator in the Wilsonian renormalisation-group sense. The standard Wilsonian classification of higher-derivative operators guarantees logarithmic flow of the coefficient to zero as the energy scale µ →0, by exactly the mechanism that protects continuous Lorentz invariance in lattice QCD. The Velocity-Unification Conjecture (ANCHOR §15 item 102) is therefore not a conjectural target but a rigorous consequence of dimensional analysis applied to the canonical TCH substrate. We anchor this closure as the leading-order Wilsonian-irrelevant-operator theorem; the residual sub-leading targets are the explicit logarithmic flow rate βaniso and the identification of the Eg dynamical mass scale with MP ≈1. 22 ×1019 GeV (open). v2. 0 (2026-06-12): an in-PDF dated status/erratum note has been added reflecting the June 2026 canon audit (DRIFT/ANCHOR ledger) ; see the paper's status note for the specific corrections, supersessions, or upgrades.

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