What does this research mean for the field?

The v0.3 scalar-threshold hardening layer achieves block-level scalar-derived gauge closure for the QDL-SO10-1 benchmark, yielding specific high-scale threshold masses with an RMS residual below 10^-12. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to advance the QDL-SO10-1 benchmark by establishing scalar-derived gauge closure.

April 28, 2026Open Access

Scalar-Derived Gauge Closure of the QDL–SO10–1 Benchmark: Explicit Threshold Blocks, Dynkin-Index Accounting, and the v0.3 Closure Test

Key Points

This research aims to advance the QDL-SO10-1 benchmark by establishing scalar-derived gauge closure.
Introduced explicit threshold blocks for gauge closure in the QDL-SO10-1 framework.
Calculated high-scale threshold masses using derived formulas.
Established block-level scalar-derived closures, presenting corresponding Dynkin-index splitting vectors.
Achieved ln r4 = 2.689679, ln rL = -1.989712, ln rR = -0.699967 with respective threshold masses.
Demonstrated RMS threshold residual below 10^-12, indicating precise calculations.
Proposed a proof-of-principle scalar-threshold derivation layer for future enhancements.

Abstract

The QDL-SO10-1 grand-unification sequence defines a closure-first SO (10) -compatible benchmark within the Quantized Dimensional Ledger (QDL) program. Papers #1-#3 established the initial benchmark trilogy: a fixed SO (10) -compatible benchmark, its low-energy phenomenology, and a stress-test / robustness audit. Paper #5 began the executable hardening sequence by revising the gauge-closure target to inverse alphaU = 44. 501718, corresponding to alphaU = 0. 022471, and by converting the remaining Pati-Salam channel splittings into calibrated high-scale threshold targets. This paper supplies the v0. 3 scalar-threshold hardening layer. The derivation is explicitly block-level: it proves that the calibrated v0. 2 offset vector lies in a declared Dynkin-index threshold span, but it does not yet constitute a full component-by-component SO (10) scalar census. Three aggregate threshold blocks are introduced with traceless Dynkin-index splitting vectors d4 = (12, -6, -6), dL = (-6, 12, -6), and dR = (-6, -6, 12), equivalently corresponding to aggregate Dynkin-index blocks S4 = (18, 0, 0), SL = (0, 18, 0), and SR = (0, 0, 18) after common-shift subtraction. Using the high-scale threshold formula lambdaᵢ = - (1 / 12 pi) times the sum over threshold blocks of Sᵢ times ln (Mₐ / MU), with MU = 1. 00 x 10¹6 GeV, the v0. 3 solution is ln r4 = 2. 689679, ln rL = -1. 989712, and ln rR = -0. 699967. The corresponding threshold masses are M4 = 1. 472695 x 10¹7 GeV, ML = 1. 367348 x 10¹5 GeV, and MR = 4. 966016 x 10¹5 GeV. The derived threshold vector is (-1. 284227, 0. 950018, 0. 334210), with RMS threshold residual below 10^-12 at displayed precision. The result advances QDL-SO10-1 from calibrated executable gauge closure to block-level scalar-derived gauge closure. It is a proof-of-principle scalar-threshold derivation layer, not the final microscopic scalar-sector completion. The next burden is a full component-level SO (10) to Pati-Salam to Standard Model scalar census and QDL admissibility verification.

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