== C1  Laplace-Beltrami: Riemannian formula == frame formula, on test functions, l=1..8
PASS l=1: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=2: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=3: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=4: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=5: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=6: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=7: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)
PASS l=8: Delta u = u_xx - (F_x/F) u_x + F d_z(F u_z)

== C2  a := -delta*Delta_w(delta) = x F_x/F = 1 + 2 l u,  u = x^{2l}/(x^{2l}+z^2)
PASS l=1: -x*Delta_w(delta) == 1 + 2 l u
PASS l=2: -x*Delta_w(delta) == 1 + 2 l u
PASS l=3: -x*Delta_w(delta) == 1 + 2 l u
PASS l=4: -x*Delta_w(delta) == 1 + 2 l u
PASS l=5: -x*Delta_w(delta) == 1 + 2 l u
PASS l=6: -x*Delta_w(delta) == 1 + 2 l u
PASS l=7: -x*Delta_w(delta) == 1 + 2 l u
PASS l=8: -x*Delta_w(delta) == 1 + 2 l u

== C3  x^2 V_eff = 3/4 + 2l(1-l)u + 3 l^2 u^2   (V_eff = (Delta delta)^2/4 + (Delta delta)'/2)
PASS l=1: x^2 V_eff formula
PASS l=1: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=2: x^2 V_eff formula
PASS l=2: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=3: x^2 V_eff formula
PASS l=3: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=4: x^2 V_eff formula
PASS l=4: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=5: x^2 V_eff formula
PASS l=5: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=6: x^2 V_eff formula
PASS l=6: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=7: x^2 V_eff formula
PASS l=7: V_eff == (3F_x^2-2FF_xx)/(4F^2)
PASS l=8: x^2 V_eff formula
PASS l=8: V_eff == (3F_x^2-2FF_xx)/(4F^2)

== C4  cross-check with Prandi-Rizzi-Seri (JST 2018) Example 7.7, eqs (241)-(242), n=2..5
       density rho = [t(t^{2l}+f)]^{-(n-1)}  (slot x=t, slot z=f)
PASS n=2 l=1: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=2 l=2: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=2 l=3: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=2 l=4: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=2 l=5: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=2 l=6: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=1: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=2: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=3: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=4: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=5: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=3 l=6: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=1: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=2: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=3: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=4: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=5: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=4 l=6: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=1: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=2: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=3: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=4: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=5: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1
PASS n=5 l=6: V_eff == PRS (241)+(242);  -t Delta delta == (n-1)(1+2lu) >= 1

== C5  unitary-map form identities (jets: v, v' in slots 2,3; v'' in slot 4)
PASS l=1 x-part: F^-1|(F^1/2 v)_x|^2 = |v_x|^2 + V_x|v|^2 + D_x(...)
PASS l=1 z-part: F|(F^1/2 v)_z|^2 = F^2|v_z|^2 + W|v|^2 + D_z(...)
PASS l=1  W == -x^2 (x^{2l} + 2 z^2)
PASS l=2 x-part: F^-1|(F^1/2 v)_x|^2 = |v_x|^2 + V_x|v|^2 + D_x(...)
PASS l=2 z-part: F|(F^1/2 v)_z|^2 = F^2|v_z|^2 + W|v|^2 + D_z(...)
PASS l=2  W == -x^2 (x^{2l} + 2 z^2)
PASS l=3 x-part: F^-1|(F^1/2 v)_x|^2 = |v_x|^2 + V_x|v|^2 + D_x(...)
PASS l=3 z-part: F|(F^1/2 v)_z|^2 = F^2|v_z|^2 + W|v|^2 + D_z(...)
PASS l=3  W == -x^2 (x^{2l} + 2 z^2)
PASS l=4 x-part: F^-1|(F^1/2 v)_x|^2 = |v_x|^2 + V_x|v|^2 + D_x(...)
PASS l=4 z-part: F|(F^1/2 v)_z|^2 = F^2|v_z|^2 + W|v|^2 + D_z(...)
PASS l=4  W == -x^2 (x^{2l} + 2 z^2)

== C6  Hardy pointwise identity (m = 1/F, w in slot 2, w_x in slot 3):
       m w_x^2 = m (w_x - w/x)^2 + (F_x/(xF)) m w^2 + D_x( m w^2 / x )  and  F_x/(xF) = (1+2lu)/x^2
PASS l=1 Hardy identity
PASS l=2 Hardy identity
PASS l=3 Hardy identity
PASS l=4 Hardy identity
PASS l=5 Hardy identity
PASS l=6 Hardy identity
PASS l=7 Hardy identity
PASS l=8 Hardy identity

== C7  minimum of q(u) = 3/4 + 2l(1-l)u + 3l^2u^2 on [0,1] (the pointwise V_eff criterion)
l= 1  u*=0       min x^2 V_eff = 3/4       1/4+min = 1         (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 2  u*=1/6     min x^2 V_eff = 5/12      1/4+min = 2/3       (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 3  u*=2/9     min x^2 V_eff = -7/12     1/4+min = -1/3      (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 4  u*=1/4     min x^2 V_eff = -9/4      1/4+min = -2        (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 5  u*=4/15    min x^2 V_eff = -55/12    1/4+min = -13/3     (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 6  u*=5/18    min x^2 V_eff = -91/12    1/4+min = -22/3     (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 7  u*=2/7     min x^2 V_eff = -45/4     1/4+min = -11       (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 8  u*=7/24    min x^2 V_eff = -187/12   1/4+min = -46/3     (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l= 9  u*=8/27    min x^2 V_eff = -247/12   1/4+min = -61/3     (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
l=10  u*=3/10    min x^2 V_eff = -105/4    1/4+min = -26       (V_eff criterion needs >= 3/4; first-order weight 1+2lu >= 1)
PASS C7 closed form min = 3/4 - (l-1)^2/3 at u*=(l-1)/(3l), l=1..10

== C8  fibre operator in t = log(x/|z|^{1/l}): V(t) = 1 + 2l(1-l)u + 3l^2u^2, u = e^{2lt}/(1+e^{2lt})
       with psi = (1-u)^{1/2} = (1+e^{2lt})^{-1/2}:  V - 1 - psi''/psi = 2 l u >= 0   (d/dt = 2l u(1-u) d/du)
PASS l=1  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=2  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=3  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=4  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=5  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=6  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=7  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu
PASS l=8  V formula, V-1-psi''/psi = 2lu, dlog psi/dt = -lu

== C9  literal model on R^2: z = +-infinity is at finite distance / finite volume
PASS l=1  int dz/F = pi/x^{l+1};  F/z^2 = x + x^{2l+1}/z^2;  F d_z(F d_z z^-1) = 2x^{2l+2}(x^{2l}+z^2)/z^3
PASS l=2  int dz/F = pi/x^{l+1};  F/z^2 = x + x^{2l+1}/z^2;  F d_z(F d_z z^-1) = 2x^{2l+2}(x^{2l}+z^2)/z^3
PASS l=3  int dz/F = pi/x^{l+1};  F/z^2 = x + x^{2l+1}/z^2;  F d_z(F d_z z^-1) = 2x^{2l+2}(x^{2l}+z^2)/z^3
PASS l=4  int dz/F = pi/x^{l+1};  F/z^2 = x + x^{2l+1}/z^2;  F d_z(F d_z z^-1) = 2x^{2l+2}(x^{2l}+z^2)/z^3
PASS l=5  int dz/F = pi/x^{l+1};  F/z^2 = x + x^{2l+1}/z^2;  F d_z(F d_z z^-1) = 2x^{2l+2}(x^{2l}+z^2)/z^3

== C10 fibre scaling: F(c y, c^l) = c^{2l+1} F(y, 1)  (slot x=y, slot z=c)
PASS l=1 scaling
PASS l=2 scaling
PASS l=3 scaling
PASS l=4 scaling
PASS l=5 scaling
PASS l=6 scaling
PASS l=7 scaling
PASS l=8 scaling

== C11 generalisations of the first-order weight a = -t d_t log(density)
PASS l=1  f = x(x^{2l}+rho):  a = 1 + 2l x^{2l}/(x^{2l}+rho)  (>= 1 since rho >= 0)
PASS l=2  f = x(x^{2l}+rho):  a = 1 + 2l x^{2l}/(x^{2l}+rho)  (>= 1 since rho >= 0)
PASS l=3  f = x(x^{2l}+rho):  a = 1 + 2l x^{2l}/(x^{2l}+rho)  (>= 1 since rho >= 0)
PASS l=4  f = x(x^{2l}+rho):  a = 1 + 2l x^{2l}/(x^{2l}+rho)  (>= 1 since rho >= 0)
PASS l=5  f = x(x^{2l}+rho):  a = 1 + 2l x^{2l}/(x^{2l}+rho)  (>= 1 since rho >= 0)
PASS PRS Ex.4.1 (m,k,ell)=(-1,-1,4): a = -m - k ell u >= -m >= 1
PASS PRS Ex.4.1 (m,k,ell)=(-1,-3,9): a = -m - k ell u >= -m >= 1
PASS PRS Ex.4.1 (m,k,ell)=(-2,-1,7): a = -m - k ell u >= -m >= 1
PASS PRS Ex.4.1 (m,k,ell)=(-5,-2,11): a = -m - k ell u >= -m >= 1

== C12 ridge example f = x((z-x)^2 + x^4) (real-analytic, Z={x=0}, no tangency): both pointwise criteria fail
  x=1e-1  a(x, x+x^2) = -7   x^2 V_eff(x, x) = -91.25
  x=1e-2  a(x, x+x^2) = -97   x^2 V_eff(x, x) = -9991.25
  x=1e-3  a(x, x+x^2) = -997   x^2 V_eff(x, x) = -999991
  x=1e-4  a(x, x+x^2) = -9997   x^2 V_eff(x, x) = -1e+08
  x=1e-5  a(x, x+x^2) = -99997   x^2 V_eff(x, x) = -1e+10
  x=1e-6  a(x, x+x^2) = -999997   x^2 V_eff(x, x) = -1e+12
PASS a(x,x+x^2) == 3 - 1/x  (first-order criterion a >= 1 - k x fails)
PASS x^2 V_eff(x,x) == 35/4 - 1/x^2  (PRS/FPR criterion x^2 V_eff >= 3/4 - k x fails)

== C13 multiplier family A = -beta grad(delta)/delta: Hardy weight delta^2 W_beta = beta(1+a-beta) - delta d_delta beta
       beta = (1+a)/2 reproduces 1/4 + x^2 V_eff (the PRS/FPR route); beta = 1 gives W = a (the route used here)
PASS l=1  beta=(1+a)/2: W == 1/4 + x^2 V_eff (formula and direct);  beta=1: W == a
PASS l=2  beta=(1+a)/2: W == 1/4 + x^2 V_eff (formula and direct);  beta=1: W == a
PASS l=3  beta=(1+a)/2: W == 1/4 + x^2 V_eff (formula and direct);  beta=1: W == a
PASS l=4  beta=(1+a)/2: W == 1/4 + x^2 V_eff (formula and direct);  beta=1: W == a
PASS l=5  beta=(1+a)/2: W == 1/4 + x^2 V_eff (formula and direct);  beta=1: W == a

ALL CHECKS PASSED
