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(*==========================================================*)(*RecursiveGeometricDerivationofInverseFineStructureConstant*)(*==========================================================*)(*1.DefinetheRecursiveGeometricOperator(TheFractalLimit)*)(*Mathematicallydefinedasthepositiverootof:x^2+(N-f0)x-f0(N+1)=0*)GeometricOp[f0_,Nval_]:=Root[#^2+(Nval-f0)#-f0(Nval+1)&,2](*2.DefineParametersbasedonArchimedeanBasis*)(*GeometricSeed(f_vac):NormalizedAreaDifferencebetweenSquareandInscribedCircle*)fvac=(4-Pi)/24;(*EffectiveSymmetryBase(N_eff):IntegerBase154withSelf-InteractionFeedback*)(*Thefeedbacktermcorrespondsto-alpha/2*)Neff[alphaInv_]:=154-(1/alphaInv)/2;(*3.SolvetheSelf-ConsistentEquation*)(*Thevalueisderivedas:alpha^-1=137+GeometricOp(f_vac,N_eff)*)solLowEnergy=FindRoot[x==137+GeometricOp[fvac,Neff[x]],{x,137.036},WorkingPrecision->20];alphaInvCalc=x/.solLowEnergy;(*4.OutputtheResults*)Print[Style["=== Geometric Derivation Results ===",16,Bold]];(*CalculatedValue*)Print[Style[Row[{"Calculated \!\(\*SuperscriptBox[\(α\), \(-1\)]\): ",NumberForm[alphaInvCalc,12]}],14,Blue]];(*CODATA2022Standard*)Print[Style["CODATA 2022 Value: 137.035999177(21)",12]];(*PrecisionCheck*)Print["----------------------------------------"];Print[Style[Row[{"Difference: ",NumberForm[alphaInvCalc-137.035999177,10]}],12,Red]];(*==========================================*)(*GeometricVisualizationofAlphaInverse*)(*Structure:Circle+2ActiveCorners*)(*==========================================*)Graphics[{(*----------------------------------------*)(*1.TheExcludedVacuum(Residue)*)(*----------------------------------------*)(*Representsthe'virtual'or'subtracted'geometry*){FaceForm[Opacity[0.15,Red]],EdgeForm[Directive[Thin,Dashed,Red]],(*Top-LeftCorner*)Rectangle[{-1,0},{0,1}],(*Bottom-RightCorner*)Rectangle[{0,-1},{1,0}]},(*----------------------------------------*)(*2.The137Node(ActiveEntity)*)(*----------------------------------------*)(*RepresentsthefusionoftheCircleand2Spikes*){FaceForm[RGBColor[0.2,0.5,0.9]],(*SolidBlue*)EdgeForm[None],(*Nointernaledgestoshowfusion*)(*TheActiveCorners(Spikes)*)Rectangle[{0,0},{1,1}],(*Top-Right*)Rectangle[{-1,-1},{0,0}],(*Bottom-Left*)(*TheCoreCircle*)Disk[{0,0},1]},(*----------------------------------------*)(*3.Outlineofthe137Node*)(*----------------------------------------*)(*Optional:Addsacleanborderaroundthefusedblueshape*){FaceForm[None],EdgeForm[Directive[Thickness[0.005],RGBColor[0.1,0.3,0.7]]],Disk[{0,0},1],Rectangle[{0,0},{1,1}],Rectangle[{-1,-1},{0,0}]},(*----------------------------------------*)(*4.TheSymmetryBase(Frame)*)(*----------------------------------------*){FaceForm[None],EdgeForm[Directive[Thin,Black]],Rectangle[{-1,-1},{1,1}]}},(*PlotSettings*)PlotRange->{{-1.2,1.2},{-1.2,1.2}},ImageSize->500,PlotLabel->Style["Geometric-Arithmetic Schema of \!\(\*SuperscriptBox[\(α\), \(-1\)]\)",18,Bold,FontFamily->"Arial"]]
=== Geometric Derivation Results ===
Calculated : 137.035999177
-1
α
CODATA 2022 Value: 137.035999177(21)
----------------------------------------
Difference: 5.826450433×
-11
10
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