== R1  exact identities (Fraction jets, random rational points)
PASS a = x d_x log|f| = 1 + 2lu for f = x(x^{2l}+z^2), l = 1..8, 200 points
PASS x^2 V_eff = 3/4 + 2l(1-l)u + 3l^2u^2, l = 1..8, 200 points
PASS f = c x(x^{2l}+rho): a = 1 + 2l x^{2l}/(x^{2l}+rho) >= 1, 200 cases
PASS l=1: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = 3/4 at u*=(l-1)/(3l)
PASS l=2: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = 5/12 at u*=(l-1)/(3l)
PASS l=3: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -7/12 at u*=(l-1)/(3l)
PASS l=4: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -9/4 at u*=(l-1)/(3l)
PASS l=5: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -55/12 at u*=(l-1)/(3l)
PASS l=6: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -91/12 at u*=(l-1)/(3l)
PASS l=7: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -45/4 at u*=(l-1)/(3l)
PASS l=8: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -187/12 at u*=(l-1)/(3l)
PASS l=9: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -247/12 at u*=(l-1)/(3l)
PASS l=10: min_[0,1] x^2 V_eff = 3/4-(l-1)^2/3 = -105/4 at u*=(l-1)/(3l)
PASS ridge: a(x, x+x^2) = 3 - 1/x and x^2 V_eff(x, x) = 35/4 - 1/x^2 (39 points)
PASS Lemma 2.1 pointwise multiplier identity for random polynomial m, b, w (60 cases)
PASS b=(1+a)/2 gives 1/4 + x^2 V_eff and b=1 gives a (model family, l=1..5)
PASS PRS Ex. 4.1 case 2 (m<=-1, k<0, any l): a = -m - k l u >= 1 (300 cases)

== R2  exact constants in the proof of Proposition 7.3
PASS x^2 f <= (110125/16384) z^7 on {|x-z|<=z^2}
PASS D >= d0 z0^-4 with d0 = 3724541952/165256328125 = 0.022538 > 1/45
PASS A <= (182/9)/c1 z0^-3 <= 21/c1 z0^-3
PASS B <= 12096 z0^3

== R3  ridge example: numerical A, B, D, N versus the bounds of Proposition 7.3 (TESTED)
  sigma=y^2 (c1 = 1.000000)
   z0       A z0^3     B/z0^3     D z0^4     N/(D z0^2)   (A+B)/(D z0)
   0.12500  4.7246     33.9616    0.49765    1.26413      9.4940    
   0.06250  4.8186     23.0268    0.43389    1.51474      11.1055   
   0.02000  4.8608     19.7551    0.38786    1.75849      12.5322   
   0.01000  4.8649     19.4752    0.37671    1.82711      12.9143   
   0.00250  4.8662     19.3877    0.36825    1.88184      13.2144   
   0.00100  4.8663     19.3828    0.36655    1.89314      13.2759   
PASS sigma=y^2: A <= 21 z0^-3/c1, B <= 12096 z0^3, D >= z0^-4/45, N <= 16 z0^2 D at all z0
  sigma=sin^2 y (c1 = 0.405285)
   z0       A z0^3     B/z0^3     D z0^4     N/(D z0^2)   (A+B)/(D z0)
   0.12500  4.7434     33.8358    0.49812    1.26359      9.5229    
   0.06250  4.8235     22.9953    0.43396    1.51463      11.1151   
   0.02000  4.8613     19.7519    0.38787    1.75849      12.5335   
   0.01000  4.8650     19.4744    0.37671    1.82711      12.9146   
   0.00250  4.8662     19.3877    0.36825    1.88184      13.2144   
   0.00100  4.8663     19.3828    0.36655    1.89314      13.2759   
PASS sigma=sin^2 y: A <= 21 z0^-3/c1, B <= 12096 z0^3, D >= z0^-4/45, N <= 16 z0^2 D at all z0

== R4  fibre Hardy constants by P1 FEM on (xmin, xmax), log-graded mesh (upper bounds; TESTED)
  l=1  rho=1: 1.03563  rho=1e-3: 1.05670  rho=1e-6: 1.10386  rho=0: 4.01870  (exact for rho=0: (l+1)^2 = 4)
  l=2  rho=1: 1.03431  rho=1e-3: 1.04254  rho=1e-6: 1.05414  rho=0: 9.01933  (exact for rho=0: (l+1)^2 = 9)
  l=3  rho=1: 1.03401  rho=1e-3: 1.03913  rho=1e-6: 1.04549  rho=0: 16.02145  (exact for rho=0: (l+1)^2 = 16)
  l=5  rho=1: 1.03384  rho=1e-3: 1.03675  rho=1e-6: 1.04006  rho=0: 36.03628  (exact for rho=0: (l+1)^2 = 36)
PASS all FEM fibre constants >= 1 (consistent with Lemma 2.1 + a >= 1)
  control m = x^(-1/2) (a = 1/2, exact best constant 9/16 = 0.5625): FEM 0.58112
PASS control weight with a = 1/2 gives a constant < 1

== R5  PRS Example 7.8: det(frame) = [t(t^{2l}+f)]^{n-1}
PASS det(X0, X_1..X_{n-1}, Y_1..Y_{n-1}) = F^{n-1} for n = 2,3,4, l = 1,2,3

== R6  literal model on R^2: <Delta U, W> - <U, Delta W> (TESTED)
  l=1  difference = 4.101562500000e-01   int x phi^2 = 4.101562500000e-01
  l=2  difference = 4.101562500000e-01   int x phi^2 = 4.101562500000e-01
PASS boundary form equals int x phi^2 dx (l = 1, 2)

ALL REFEREE CHECKS PASSED
