/* Exact finite algebra only. These checks do not certify any Weil window. */
display2d:false$
checks:[]$
check(label,expr):=block([answer:is(ratsimp(expr)=0)],
  checks:endcons([label,answer],checks),
  if answer # true then error("Exact check failed",label,expr))$

/* ConformalSubs F(m), in signature (4,1), using scalar coordinates. */
lorentz(u,v):=sum(u[i]*v[i],i,1,4)-u[5]*v[5]$
embed(v):=block([r2:sum(v[i]^2,i,1,3)],
  append(v,[(r2-1)/2,(r2+1)/2]))$
xx:[x1,x2,x3]$ yy:[y1,y2,y3]$
fx:embed(xx)$ fy:embed(yy)$ np:[0,0,0,1,1]$
check("Conformal point is null",lorentz(fx,fx))$
check("Conformal point normalization",lorentz(fx,np)+1)$
check("Conformal pairing is minus half squared distance",
  lorentz(fx,fy)+sum((xx[i]-yy[i])^2,i,1,3)/2)$

/* Bounded multiplier identity: keep the null-equation term explicit. */
kk:matrix([k11,k12,k13],[k12,k22,k23],[k13,k23,k33])$
ff:[f1,f2,f3]$ mm:[m1,m2,m3]$
qmf:sum(sum(kk[i,j]*mm[i]*ff[i]*mm[j]*ff[j],j,1,3),i,1,3)$
afm2f:sum(sum(kk[i,j]*ff[j]*mm[i]^2*ff[i],j,1,3),i,1,3)$
edge:sum(sum(kk[i,j]*ff[i]*ff[j]*(mm[i]-mm[j])^2,j,1,3),i,1,3)/2$
check("Signed multiplier identity",qmf-afm2f+edge)$

/* A positive semidefinite signed matrix can have positive off-diagonal entries. */
aa:matrix([3/4,-1,1/4],[-1,2,-1],[1/4,-1,3/4])$
ones:matrix([1],[1],[1])$
check("Signed example nullvector",sum((aa.ones)[i,1]^2,i,1,3))$
check("Signed example eigenvalues 0, 1/2, 3",
  determinant(aa-z*ident(3))+z*(z-1/2)*(z-3))$

/* Representative prime-power lengths log 2 and log 4 = 2 log 2. */
bb:-2*d2*cos(t)-2*d4*cos(2*t)$
check("Perturbation symbol mean zero",integrate(bb,t,-%pi,%pi))$
check("Perturbation symbol positive mean square",
  integrate(trigreduce(expand(bb^2)),t,-%pi,%pi)/(2*%pi)-2*(d2^2+d4^2))$
check("Single-shift symbol at zero",subst(t=0,-2*d*cos(t))+2*d)$
check("Single-shift symbol at pi",subst(t=%pi,-2*d*cos(t))-2*d)$

/* Explicit real parity witnesses used only to check correlation signs. */
assume(a>0)$
even_corr:integrate(cos(%pi*(x+a/2)/(2*a))*cos(%pi*x/(2*a)),x,-a,a/2)$
odd_corr:integrate(sin(%pi*(x+a/2)/a)*sin(%pi*x/a),x,-a,a/2)$
check("Even cosine shift correlation",even_corr-a*sqrt(2)*(3/8+1/(2*%pi)))$
check("Odd sine shift correlation",odd_corr-a/(2*%pi))$
print(checks)$
print("PASS:",length(checks),"exact algebra checks; no interval or RH certificate.")$
quit();