Feynman anticheckers:
auxiliary computations
M.Skopenkov, A.Ustinov

Here we check auxiliary computations in the proofs and provide a source code for the plots illustrating the main results.
We denote e:=epsilon, d:=delta epsilon. 
In the earlier notation of arXiv:2208.14247, we have d:=delta.

1.Checking of Proposition 22

u=Exp[I w e]{{1,-d},{-d,1}}.{{Exp[I p e],-I m e},{-I m e,Exp[-I p e]}}/(Sqrt[1-d^2]Sqrt[1+m^2e^2]);
u//MatrixForm
FullSimplify[Det[IdentityMatrix[2]-u]]
a=(({{(Sqrt[1-(d^2)]) (Sqrt[1+(e^2) (m^2)])(E^(-I e w)) -(E^(-I e p))-I d e m, -d (E^(-I e p))-I e m}, {-d (E^(I e p))-I e m, (Sqrt[1-(d^2)]) (Sqrt[1+(e^2) (m^2)])(E^(-I e w)) -(E^(I e p))-I d e m}}))/(2 (-I d e m-Cos[e p]+(Sqrt[1-(d^2)]) (Sqrt[1+(e^2) (m^2)]) Cos[e w]));
FullSimplify[a.(IdentityMatrix[2]-u),1>d>0&&m>0&&e>0]//MatrixForm

2.Checking of Example 4

tt=2;xx=2;
p[k_]:=2 Pi k/(xx e);
w[l_]:=2 Pi l/(tt e);
ae1[x_,t_]:=1/(2tt xx)Sum[
  (-I m e Sqrt[1-d^2]-d Sqrt[1+m^2e^2]Exp[I w[l] e])Exp[I p[k] x-I w[l] t]/
(Sqrt[1-d^2]Sqrt[1+m^2e^2]Cos[w [l] e]-Cos[p[k]e]-I m e d),
{k,1,xx},{l,1,tt}];

ae2[x_,t_]:=1/(2tt xx)Sum[
( Sqrt[1-d^2]Exp[-I p[k] e]-Sqrt[1+m^2e^2]Exp[I w[l] e])Exp[I p[k] x-I w[l] t]/(Sqrt[1-d^2]Sqrt[1+m^2e^2]Cos[w [l] e]-Cos[p[k]e]-I m e d),
{k,1,xx},{l,1,tt}];
ae1conjugate[x_,t_]:=1/(2tt xx)Sum[
  (I m e Sqrt[1-d^2]-d Sqrt[1+m^2e^2]Exp[-I w[l] e])Exp[-I p[k] x+I w[l] t]/
(Sqrt[1-d^2]Sqrt[1+m^2e^2]Cos[w [l] e]-Cos[p[k]e]+I m e d),
{k,1,xx},{l,1,tt}];
ae2conjugate[x_,t_]:=1/(2tt xx)Sum[
(Sqrt[1-d^2]Exp[I p[k] e]-Sqrt[1+m^2e^2]Exp[-I w[l] e])Exp[-I p[k] x+I w[l] t]/(Sqrt[1-d^2]Sqrt[1+m^2e^2]Cos[w [l] e]-Cos[p[k]e]+I m e d),
{k,1,xx},{l,1,tt}];
Table[FullSimplify[ae1[k e,l e]],{l,tt-1,0,-1},{k,0,xx-1}]//MatrixForm
Table[FullSimplify[ae2[k e,l e]],{l,tt-1,0,-1},{k,0,xx-1}]//MatrixForm
pe[x_,t_]:=FullSimplify[ae1[x,t]ae1conjugate[x,t]+ae2[x,t]ae2conjugate[x,t],m>0&&e>0&&0<d<1];
psume[t_]:=Sum[pe[k e,t],{k,0,xx-1}]
FullSimplify[psume[0],m>0&&e>0&&0<d<1]
FullSimplify[psume[e],m>0&&e>0&&0<d<1]

3.Checking of Lemma 6

p=.
FullSimplify[D[((- e^2Sin[e p]Cos[e p])/((((e^2) (m^2)+((Sin[e p])^2)))^(3/2))),e]2((((e^2) (m^2)+((Sin[e p])^2)))^(5/2))- e(2Sin [p e]^2 (p e(1-Cos [p e])^2+2(p e-Sin [p e])Cos [p e])+m^2 e^2 (Sin [2 p e]-2p e Cos [2 p e]))]

4.Plots in Figures 1 and 3

m=4;ε=3/100;y=200ε;
(*m=1;ε=1;y=100ε;*)
(*m=8;ε=1/8;y=100ε;*)
w[p_,m_,e_]:=(1/e)ArcCos[Cos[p e]/Sqrt[1+m^2e^2]];
(* In the following formula, (t-x)/e must be even *)
a1[x_,y_,m_,e_]:=If[Abs[x]>Abs[y],0,(1/(1+m^2e^2)^((y/e-1)/2))If[x==y,0,Sum[Binomial[(x+y-2e)/(2e),r]Binomial[(y-x-2e)/(2e),r](m e)^(2r+1)(-1)^r,{r,0,y/(2e)}]]];
a2[x_,y_,m_,e_]:=If[Abs[x]>Abs[y],0,(1/(1+m^2e^2)^((y/e-1)/2)) If[x==y>0,1,
Sum[Binomial[(x+y-2e)/(2e),r]Binomial[(y-x-2e)/(2e),r-1](m e)^(2r)(-1)^r,{r,1,y/(2e)}]]];
(* In the following formula, (t-x)/e must be odd *)
b1[x_,t_,m_,e_]:=Re[m e^2/(Pi Sqrt[1+m^2e^2]) NIntegrate[Cos[w[p,m,e](t-e)]Exp[-I p x]/Sin[w[p,m,e]e],{p,0,Pi/e}]];
b2[x_,t_,m_,e_]:= Re[e/Pi NIntegrate[(-Sin[w[p,m,e](t-e)]+I Cos[w[p,m,e](t-e)]Sin[p e]/( Sqrt[1+m^2e^2]Sin[w[p,m,e]e]))Exp[-I p (x-e)],{p,0,Pi/e}]];chargedensity[x_,t_,m_,e_]:=If[EvenQ[(t-x)/e],Abs[b1[x,t+e,m,e]]^2/2+Abs[a2[x+e,t+e,m,e]]^2/2,Abs[a1[x,t+e,m,e]]^2/2+Abs[b2[x+e,t+e,m,e]]^2/2];
altchargedensity[x_,t_,m_,e_]:=If[EvenQ[(t-x)/e],Abs[b1[x,t+e,m,e]]^2/2+Abs[b2[x,t+e,m,e]]^2/2,Abs[a1[x,t+e,m,e]]^2/2+Abs[a2[x,t+e,m,e]]^2/2];
(*charge[x_,t_,m_,e_]:=Sum[chargedensity[xx,t,m,e],{xx,-2t,x,e}];*)
averageddensity[x_,t_,m_,e_]:=chargedensity[x,t,m,e]+chargedensity[x+e,t,m,e];
theta[x_,y_,m_,ε_]:=(y/ε) ArcSin[((m ε)/(Sqrt[(1+(m^2)(ε^2))(1-((((x/y)))^2))]))]-(x/ε) ArcSin[((m ε(x/y))/(Sqrt[1-((((x/y)))^2)]))];
thetain[x_,t_,m_,ε_]:=(t/ε) ArcTan[Sqrt[t^2-(1+(m^2)(ε^2)) x^2]/(m ε t)]-(x/ε) ArcTan[Sqrt[t^2-(1+(m^2)(ε^2)) x^2]/(m ε x)];
thetaout[x_,y_,m_,ε_]:=-(y/ε) ArcCosh[((m ε)/(Sqrt[(1+(m^2)(ε^2))(1-((((x/y)))^2))]))]+(x/ε) ArcCosh[((m ε(x/y))/(Sqrt[1-((((x/y)))^2)]))];
largeb1[x_,y_,m_,ε_]:=ε(Sqrt[((2m)/(π y))])(((1-(1+(m^2)(ε^2))((((x/y)))^2)))^(-1/4))Cos[theta[x,y,m,ε]+(π/4)];
largeb2[x_,y_,m_,ε_]:=-ε(Sqrt[((2m)/(π y))])(((1-(1+(m^2)(ε^2))((((x/y)))^2)))^(-1/4))(Sqrt[((y+x)/( y-x))])Sin[theta[x,y,m,ε]+(π/4)];
b1bessel[x_,t_,m_,ε_]:=(I)^((Abs[x]-t+ε)/ε)  ε Sqrt[m]thetain[Abs[x],t,m ,ε]^(1/2)/(Sqrt[3 ] (t^2-(1+(m^2)(ε^2)) x^2)^(1/4))(BesselJ[1/3,thetain[Abs[x],t,m ,ε]]+BesselJ[-1/3,thetain[Abs[x],t,m ,ε]]);
b2bessel[x_,t_,m_,ε_]:=(I)^((Abs[x]-t)/ε)  ε Sqrt[m]Sqrt[(t+x)/(t-x)]thetain[Abs[x],t,m ,ε]^(1/2)/(Sqrt[3 ] (t^2-(1+(m^2)(ε^2)) x^2)^(1/4))(BesselJ[1/3,thetain[Abs[x],t,m ,ε]]+BesselJ[-1/3,thetain[Abs[x],t,m ,ε]]);
outsideb1[x_,y_,m_,ε_]:=-ε(Sqrt[m])(-1)^((x-y)/(2ε))(((y^2-(1+(m^2)(ε^2))(x^2)))^(-1/4))((thetaout[x,y,m,ε])^(1/2))BesselK[1/3,thetaout[x,y,m,ε]];
feynman1[x_,t_,m_]:= -(m /4) BesselY[0,m Sqrt[t^2-x^2]];
feynman2[x_,t_,m_]:= (m /4) (x+t)/Sqrt[t^2-x^2]BesselY[1,m Sqrt[t^2-x^2]];
feynman3[x_,t_,m_]:= m /(2Pi) BesselK[0,m Sqrt[x^2-t^2]];
feynman4[x_,t_,m_]:= m /(2Pi) (x+t)/Sqrt[x^2-t^2]BesselK[1,m Sqrt[x^2-t^2]];
feynman5[x_,t_,m_]:= -(m /4) BesselJ[0,m Sqrt[t^2-x^2]];
feynman6[x_,t_,m_]:= (m /4) (x+t)/Sqrt[t^2-x^2]BesselJ[1,m Sqrt[t^2-x^2]];
feynmanchargedensity[x_,t_,m_]:=If[Abs[x]<Abs[t],feynman1[x,t,m]^2+feynman2[x,t,m]^2+feynman5[x,t,m]^2+feynman6[x,t,m]^2,feynman3[x,t,m]^2+feynman4[x,t,m]^2];
hyperbola[x_,t_,m_]:=Sqrt[m]/(2Sqrt[2 Pi Sqrt[x^2-t^2]])Exp[-m Sqrt[x^2-t^2]];
ListPlot[{Table[feynman1[x,y,m],{x,-y+2ε,y-2ε,2ε}],Table[Im[b1bessel[x,y,m,ε]]/(4ε),{x,-y+2ε,y-2ε,2ε}],Table[b1[x,y+ε,m,ε]/(4ε),{x,-y+2ε,y-2ε,2ε}]}, Joined->{True,True,False},PlotStyle->{Black,Default,Black},DataRange->{-y,y},AxesLabel->{x,Im[(Subscript[A, 1])]/(4"ε")}]
ListPlot[{Table[feynman2[x,y,m],{x,-y+ε,y-3ε,2ε}],Table[Im[b2bessel[x,y,m,ε]]/(4ε),{x,-y+ε,y-3ε,2ε}],Table[b2[x+ε,y+ε,m,ε]/(4ε),{x,-y+ε,y-3ε,2ε}]}, Joined->{True,True,False},PlotStyle->{Black,Default,Black},DataRange->{-y,y},AxesLabel->{x,Im[(Subscript[A, 2])]/(4"ε")}]
ListPlot[{Table[Abs[outsideb1[x,y,m,ε]],{x,y-2ε,y/Sqrt[1+m^2ε^2],-2ε}],Table[Abs[b1[x,y+ε,m,ε]],{x,y-2ε,y/Sqrt[1+m^2ε^2],-2ε}]}, Joined->{True,False},PlotStyle->{Black,Black},DataRange->{y,y/Sqrt[1+m^2ε^2]},AxesLabel->{x,Im[(Subscript[A, 1])]},ScalingFunctions->"Log"]
(*feynmancharge[x_,t_,m_]:=NIntegrate[feynmanchargedensity[xx,t,m],{xx,-2t,x}];
ListPlot[{Table[feynmancharge[x,y,m],{x,-y+ε,y-ε,ε}],Table[charge[x,y,m,ε]/(16ε^2),{x,-y+ε,y-ε,ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{-y,y},AxesLabel->{x,Q/(16ε^2)}]*)
(*ListPlot[{Table[feynmanchargedensity[x,y,m],{x,-y+ε,y-ε,ε}],Table[( m y/(4Pi))(y^2-x^2)^(-1/2)/(y-x),{x,-y+ε,y-ε,ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{-y,y},AxesLabel->{x,(((Abs[(Subscript[A, 1])])^2)+((Abs[(Subscript[A, 2])])^2))/(32"ε"^2)}]
ListPlot[{Table[feynmanchargedensity[x,y,m]-( m y/(4Pi))(y^2-x^2)^(-1/2)/(y-x),{x,-y+ε,y-ε,ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{-y,y},AxesLabel->{x,(((Abs[(Subscript[A, 1])])^2)+((Abs[(Subscript[A, 2])])^2))/(32"ε"^2)}]
ListPlot[{Table[feynmanchargedensity[x,y,m],{x,-y+ε,y-ε,ε}],Table[altchargedensity[x,y,m,ε]/(16ε^2),{x,-y+ε,y-ε,ε}],Table[averageddensity[x,y,m,ε]/(8ε^2),{x,-y+ε,y-ε,ε}]}, Joined->{True,False,False},PlotStyle->{Orange,Blue,Black},DataRange->{-y,y},AxesLabel->{x,(((Abs[(Subscript[A, 1])])^2)+((Abs[(Subscript[A, 2])])^2))/(32"ε"^2)}]
ListPlot[{Table[feynmanchargedensity[x,y,m],{x,-y+ε,y-ε,ε}],Table[chargedensity[x,y,m,ε]/(16ε^2),{x,-y+ε,y-ε,ε}],Table[averageddensity[x,y,m,ε]/(8ε^2),{x,-y+ε,y-ε,ε}]}, Joined->{True,False,False},PlotStyle->{Orange,Blue,Black},DataRange->{-y,y},AxesLabel->{x,(((Abs[(Subscript[A, 1])])^2)+((Abs[(Subscript[A, 2])])^2))/(32"ε"^2)}]
*)
ListPlot[{Table[feynmanchargedensity[x,y,m],{x,-y+2ε,y-2ε,2ε}],Table[averageddensity[x,y,m,ε]/(8ε^2),{x,-y+2ε,y-2ε,2ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{-y,y},AxesLabel->{x,Q}]
(*AxesLabel->{x,(Q[x,t,"m","ε"]+Q[x+"ε",t,"m","ε"])/(8"ε"^2)}]*)
ListPlot[{Table[feynmanchargedensity[x,y,m],{x,y+2ε,3y/2,2ε}],Table[averageddensity[x,y,m,ε]/(8ε^2),{x,y+2ε,3y/2,2ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{y,1.5y},AxesLabel->{x,Q},ScalingFunctions->"Log"]
(*ListPlot[{Table[feynmanchargedensity[x,y,m],{x,y+ε,1.5y,ε}],Table[averageddensity[x,y,m,ε]/(8ε^2),{x,y+ε,1.5y,ε}]}, Joined->{True,False},PlotStyle->{Orange,Black},DataRange->{y,1.5y},AxesLabel->{x,(((Abs[(Subscript[A, 1])])^2)+((Abs[(Subscript[A, 2])])^2))/(32"ε"^2)},ScalingFunctions->"Log"]
ListPlot[{Table[a1[x,y,m,ε],{x,-y+2ε,y-2ε,2ε}],Table[largea1[x,y-ε,m,ε],{x,-y+2ε,y-2ε,2ε}]}, Joined->{False,True},DataRange->{-y,y},AxesLabel->{x,(Subscript[a, 1])},LabelStyle->Directive[FontSize->Tiny]]
ListPlot[{Table[a2[x,y,m,ε],{x,-y+4ε,y-4ε,2ε}],Table[largea2[x-ε,y-ε,m,ε],{x,-y+4ε,y-4ε,2ε}]}, Joined->True,DataRange->{-y,y}]
ListPlot[{Table[feynman1[x,y,m],{x,-y+2ε,y-2ε,2ε}],Table[largeb1[x,y,m,ε]/(4ε),{x,-y+2ε,y-2ε,2ε}],Table[b1[x,y+ε,m,ε]/(4ε),{x,-y+2ε,y-2ε,2ε}]}, Joined->{True,True,False},PlotStyle->{Black,Default,Black},DataRange->{-y,y},AxesLabel->{x,Im[(Subscript[A, 1])]/(4"ε")}]
ListPlot[{Table[feynman2[x,y,m],{x,-y+ε,y-3ε,2ε}],Table[largeb2[x,y,m,ε]/(4ε),{x,-y+ε,y-3ε,2ε}],Table[b2[x+ε,y+ε,m,ε]/(4ε),{x,-y+ε,y-3ε,2ε}]}, Joined->{True,True,False},PlotStyle->{Black,Default,Black},DataRange->{-y,y},AxesLabel->{x,Im[(Subscript[A, 2])]/(4"ε")}]*)
ListPlot[{Table[feynman3[x,y,m],{x,y+2ε,1.5y,2ε}],Table[b1[x,y+ε,m,ε]/(4ε),{x,y+2ε,1.5y,2ε}]}, Joined->{True,False},PlotStyle->{Black,Black},DataRange->{y,1.5y},AxesLabel->{x,Im[(Subscript[A, 1])]/(4"ε")},ScalingFunctions->"Log"]
ListPlot[{Table[feynman4[x,y,m],{x,y+ε,1.5y,2ε}],Table[b2[x+ε,y+ε,m,ε]/(4ε),{x,y+ε,1.5y,2ε}]}, Joined->{True,False},PlotStyle->{Black,Black},DataRange->{y,1.5y},AxesLabel->{x,Im[(Subscript[A, 2])]/(4"ε")},ScalingFunctions->"Log"]
ListPlot[{Table[hyperbola[x,y,m],{x,y+2ε,1.5y,2ε}],Table[b1[x,y+ε,m,ε]/(4ε),{x,y+2ε,1.5y,2ε}]}, Joined->{True,False},PlotStyle->{Black,Black},DataRange->{y,1.5y},AxesLabel->{x,Im[(Subscript[A, 1])]/(4"ε")},ScalingFunctions->"Log"]