Run the couplings up. Watch sin²θW hit exactly 3/8.
Cube geometry supplies one weak-mixing number with nothing fitted: the infrared anchor sin²θW(0) = 3/(4π). The ultraviolet anchor sin²θW = 3/8 is the standard SU(5) tree-level value, reached by construction wherever the GUT-normalized g₁ equals g₂; it is not forced by the cube. This script takes the PDG couplings at MZ, runs the standard-model renormalization-group equations at one and two loops, and finds the scale where g₁ = g₂, which is where sin²θW = 3/8 identically. Then it repeats the whole test with MSSM beta functions. The crossing value is exact algebra; the crossing scale turns out to be spectrum-dependent, and the page says so.
3/(4π) and 3/8
Low-energy cube reading 3/(4π) = 0.23873. A Lean THEOREM
(Masses/WeakMixingFromCube.lean, weakMixingFromCube_consistent) certifies it lies
inside the measured 1σ band [0.23851, 0.23883] of sin²θW(0), the Q=0
Thomson-limit value, about 0.4σ from center. Identifying the cube ratio with
sin²θW(0) at all is HYPOTHESIS tier; the module names its falsifier
(weakMixingFromCube_falsifier). The MZ value 0.23121 lives at a different scale;
comparing 0.23873 to it is a different-scale comparison with no Lean certificate. The high-energy reading
3/8 = 0.375 is the standard SU(5)/GUT tree-level value, a definition in Lean, not a cube derivation.
Separately, the primary Lean Weinberg-angle prediction line is (3 − φ)/6 ≈ 0.2306
(T20 / WeinbergAngleScoreCard), a distinct object from both cube anchors. The preregistered question here:
does standard-model running put the 3/8 point inside the conventional GUT window 10¹⁵ to
10¹⁷ GeV?
SM fails the window; MSSM passes
The verdict is printed, not softened: with the standard-model spectrum alone the crossing lands two orders below the window. Add superpartners and it lands inside. The scale belongs to the spectrum, not the cube.
sin² at μ* = 0.37500
At the g₁ = g₂ point the script prints sin²θW = 0.37500 with the note "should be 0.375". If the RGE solver drifted, that line would catch it. It is the page's null test on its own numerics.
The running, drawn
One-loop running of sin²θW from MZ to 10¹⁷ GeV (points from the reference output). The gold marker is the SM crossing at 1.03×10¹³ GeV; the green marker is the MSSM crossing at 2.01×10¹⁶ GeV, inside the shaded GUT window.
Run it yourself
$ python3 weak_mixing_crossing.py == two-loop g1=g2 crossing (sin^2 = 3/8) == mu* = 1.089e+13 GeV sin^2 at mu* = 0.37500 (should be 0.375) VERDICT (panel criterion: mu* in GUT window 1e15 - 1e17 GeV) 2-loop: sin^2=3/8 reached at 1.089e+13 GeV -> OUTSIDE [1e15,1e17] == same test, MSSM beta functions == g1=g2 at mu* = 2.013e+16 GeV -> INSIDE [1e15,1e17] => the crossing scale is dominated by the (unknown) BSM content
Mixed tiers, stated exactly. The IR anchor: a Lean THEOREM
(weakMixingFromCube_consistent) certifies 3/(4π) lies inside the measured 1σ band of
sin²θW(0) at Q=0; identifying the cube ratio with that observable is HYPOTHESIS, with
the falsifier named in the module. The UV anchor 3/8 is an imported standard GUT definition (DEF) in Lean
(sin2ThetaW_GUT := 3/8), not cube-forced; the only cube-forced weak-mixing number in Lean is
3/(4π). The RGE crossing scale is a standard SM/MSSM computation (MEASURED, computational) and is
spectrum-dependent: it moved three orders of magnitude when superpartners were added, so it is set by particle
content, not geometry. Do not cite this page as a derivation of the GUT scale. What survives: sin² = 3/8
is reached exactly when g₁ = g₂ (an identity of the GUT normalization), and the comparison of
3/(4π) to the MZ value 0.23121 is a different-scale curiosity with no certificate behind
it.