What question did this study set out to answer?

The aim is to refine the Triptych-GUT model by incorporating new algebraic and spectral components to enhance grand unification predictions.

June 13, 2026Open Access

Triptych-GUT V9 — An Algebraic, Spectral and Fixed-Point Dynamical Framework for Grand Unification with a Single Dimensional Calibration

Key Points

The aim is to refine the Triptych-GUT model by incorporating new algebraic and spectral components to enhance grand unification predictions.
Introduced dimensional calibration at v_PDG = 246.220 GeV with precision derived from G_F.
Applied NNLO spectral corrections using the Parker-Toms recursion on S^4. Generated spectral invariants in the CKM sector.
Developed the conditional Galois raccord to minimize discrepancies between versions in the framework.
Observed a significant reduction in the V1/V9 discrepancy, from 442 ppm to 1.33 ppm, post-correction.
Achieved a final residual uncertainty of 0.04 ppm against the G_N uncertainty of approximately 22 ppm.
Validated CKM sector parameters with PDG agreement below 0.001°.

Abstract

DESCRIPTION EN Ninth version of the Triptych-GUT programme. The unique dimensional calibration remains vPDG = 246. 220 GeV (from GF, precision 0. 26 ppm). V9 contributions relative to V8: (i) Conformal coupling H and BPS-Pell chain. The C2 closure derives the conformal coupling xiH* = Vₘin/6 from the standard Pati-Salam NCG structure and the Pell duality gY*/g₂₋* = Vₘin. The BPS-Pell chain establishes deltaC/Cbare = -Vₘin/Nc³ (conditionally composed). (ii) NNLO spectral correction. The Parker-Toms recursion da₂₊/dₓi = -12 a₂₊-₂ on S⁴ gives a₄ (xi*) = -1/90 and da₆/dₓi|xi* = 2/15 (unprotected). The stationary point xiₑff = (30 - sqrt (30) ) /180 is the minimum of a₆. Universality of etaPT = 1 - 28*sqrt (30) /705 in H₇ follows from TBL, G = 0 (7 verified commutators). NNLO spectral correction: epsilon = (7/6) etaPT epsilon₀ = 1. 3676 ppm. (iii) CKM sector. The spectral invariants of STC establish ACKM = g₃³ - 1, gamma = arccos (Vₘin (1-lambda⁴) ) = 65. 6° (PDG agreement < 0. 001°), alpha = pi - beta - gamma = 91. 9°, and J = J₀ sin (pi/8). (iv) Conditional Galois raccord (GUT-568/569). The conjecture DeltaIₜotal = phi, with IFₑff = 180*sqrt (5) - 37 in Q (sqrt (5) ) interpreted as a zeta-regularised fermionic spectral weight, reduces the V1/V9 gap from 442 ppm to 1. 33 ppm. The subsequent correction -epsilonNNLO gives a final residual of 0. 04 ppm, operationally zero against the current GN uncertainty (approximately 22 ppm). DESCRIPTION FR Neuvième version du programme Triptyque-GUT. La calibration dimensionnelle unique reste vPDG = 246. 220 GeV (obtenue via GF avec une précision de 0. 26 ppm). Apports V9 par rapport à V8: (i) Couplage conforme H et chaîne BPS-Pell. La fermeture C2 dérive le couplage conforme xiH* = Vₘin/6 depuis la structure NCG standard de Pati-Salam et la dualité de Pell gY*/g₂₋* = Vₘin. La chaîne BPS-Pell établit deltaC/Cbare = -Vₘin/Nc³ de manière conditionnelle composée. (ii) Correction spectrale NNLO. La récursion Parker-Toms da₂₊/dₓi = -12 a₂₊-₂ sur S⁴ donne a₄ (xi*) = -1/90 et da₆/dₓi|xi* = 2/15 (non protégé). Le point stationnaire xiₑff = (30 - sqrt (30) ) /180 est le minimum de a₆. L’universalité du facteur etaPT = 1 - 28*sqrt (30) /705 dans H₇ découle de TBL, G = 0 (7 commutateurs vérifiés). Correction spectrale NNLO: epsilon = (7/6) etaPT epsilon₀ = 1. 3676 ppm. (iii) Secteur CKM. Les invariants spectraux de STC établissent ACKM = g₃³ - 1, gamma = arccos (Vₘin (1-lambda⁴) ) = 65. 6° (accord PDG à 0. 001°), alpha = pi - beta - gamma = 91. 9°, et J = J₀ sin (pi/8). (iv) Raccord galoisien conditionnel (GUT-568/569). La conjecture DeltaIₜotal = phi, avec IFₑff = 180*sqrt (5) - 37 appartenant à Q (sqrt (5) ) interprété comme poids spectral fermionique zêta-régularisé, réduit l’écart V1/V9 de 442 ppm à 1. 33 ppm. La correction -epsilonNNLO donne ensuite un résidu final de 0. 04 ppm, opérationnellement nul devant l’incertitude sur GN (environ 22 ppm).

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