Spectral Distance √3
Spectral Distance √3 (IV.D43) is the τ-categorical fixed spectral separation arising from the three-fold internal-fiber structure. It is a dimensionless invariant equal to √3, fixed by the Three-Fold Distance Squared theorem (IV.T11/IV.T115); it appears as a structural constant in spectral / coupling formulas.
τ-Definition
Spectral Distance √3 (IV.D43) is the τ-categorical fixed spectral separation arising from the three-fold internal-fiber structure. It is a dimensionless invariant equal to √3, fixed by the Three-Fold Distance Squared theorem (IV.T11/IV.T115); it appears as a structural constant in spectral / coupling formulas.
Categorical invariant. SpectralDistance(internal-fiber) := √3, fixed by three-fold internal symmetry; an E2 invariant of the τ-spectral lattice.
Primary registry anchor:
IV.D43
τ-Derivation Chain
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I.K0— Universe Postulate establishes τ -
IV.T11— Three-Fold Distance Squared theorem fixes the squared spectral distance to 3 -
IV.D43— Spectral Distance √3 — the resulting structural constant -
IV.T115— Three-fold distance squared (alternate proof / restatement)
Lean modules referenced:
TauLib.BookIV.Calibration.DimensionlessCouplings
SI Translation
Numerical value: 1.732050808 ± 0 (dimensionless)
Calibration anchor: PG-P01-neutron
Calibration chain:
- m_n (anchor)
- Spectral Distance √3 enters as a dimensionless factor in coupling formulas anchored to m_n via IV.D300
Manuscript reference: manuscript-sources/book-04/part02-spectral/ch-three-fold-distance.tex
Lean Coverage
See Also
Cross-domain bridges
This glossary term sits on the boundary between domains. The τ-framework's cross-domain pivots are the structural junctions where physics, life, and metaphysics readouts meet.
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PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D06-truth4-logicTruth4 Logic -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D09-calibrated-split-complexCalibrated Split-Complex Codomain -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D10-split-complex-scalarsSplit-Complex Scalars -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-D15-boundary-ringBoundary Ring and Scalars -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T06-prime-polarityPrime Polarity Theorem -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T07-split-complex-forcedSplit-Complex Forced -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T08-crt-coherenceCRT Coherence Constraint -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T09-algebraic-lemniscateAlgebraic Lemniscate -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T11-mutual-determinationMutual Determination (5-Way Equivalence) -
PG-Q12-spectral-distance-sqrt3Spectral Distance √3 →MathG-T13-spectral-trichotomySpectral Trichotomy Lemma