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V1. E = (x^2 + y^5, 2), characteristic 2, ambient A^2
f = y^5 + x^2   ord_0 f = 2   in_0 f = x^2
Sing(E) ideal (f, df): [y, x]  -> the origin only
d_x^(2) f = 1   (a unit: so Diff^(2)(p(E,2)) = O, hence p^nega(E,-a)_0 = O for all a>0)
step 1: chart y: f' = y^3 + x^2;  ord_0 f' = 2;  points of order>=2 over GF(4) on this chart: [(0, 0)]
           chart x: f' = x^3*y^5 + 1;  points of order>=2 over GF(4): []
step 2: chart y: f' = x^2 + y;  ord_0 f' = 1;  points of order>=2 over GF(4) on this chart: []
           chart x: f' = x*y^3 + 1;  points of order>=2 over GF(4): []
   -> resolved (order < 2).
Observation: every point of order 2 of every transform lies on x'=0, the strict
transform of {x=0}; the edge data (3-tuple (n, n-r, q)) of the p-algebra at the
singular point is the same at every step (the only order-1 element of degree 1 is x),
while the residual order ord(eps) drops 5 -> 3 -> 1.  The manuscript's Thm 16.6(2)
asserts a strict lexicographic drop of the Inv-string at every point of pi^{-1}(nabla)\nabla'.
J#|_{x=0} (generators mod x): [y^5, y^10, y^9, y^9, y^8]  -> F(Y) = (y^5, 2) on Y = A^1
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V2. Narasimhan's example, characteristic 2, ambient A^4
f = Y^7*W + Y*Z^3 + Z*W^3 + X^2
curve C: (x,y,z,w) = (t^32, t^7, t^19, t^15)
f(C) = 0
  d_x f = 0;   restricted to C: 0;   ord_0 = +Infinity
  d_y f = Y^6*W + Z^3;   restricted to C: 0;   ord_0 = 3
  d_z f = Y*Z^2 + W^3;   restricted to C: 0;   ord_0 = 3
  d_w f = Y^7 + Z*W^2;   restricted to C: 0;   ord_0 = 3
dimension of the critical scheme {f = df = 0}: 1
I(C) has 13 generators; minimal order of a generator at 0: 2  (no order-1 element: C lies in no smooth hypersurface at 0)
semigroup <32,7,19,15>: minimal generators: [32, 7, 19, 15]  (all four => embedding dimension 4 at 0)
radical(crit) == I(C): True
point eta on C over GF(4): (a + 1, a, a, 1)  ord_eta f = 2  in_eta f = X^2 + (a + 1)*Y^2 + (a + 1)*Y*Z + (a + 1)*Z^2 + Y*W + Z*W + a*W^2
ord_eta(d_z f) = 1  -> a degree-1 element of p(E,1) of order 1 at eta; so the edge data at eta is (4, <=2, ...) < (4,3,2) = edge data at 0.
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V3. Hauser's kangaroo sequence (arXiv:0811.4151, sect. G), characteristic 2, ambient A^3
f0 = y^7 + y*z^4 + x^2   Sing(f0,2) ideal radical: [x, y^3 + z^2]
f1 = f0(xy,y,zy)/y^2 = y^3*z^4 + y^5 + x^2
f2 = f1(xz,yz,z)/z^2 = y^5*z^3 + y^3*z^5 + x^2
f3 = f2(xz,yz+z,z)/z^2 = y^5*z^6 + y^4*z^6 + y^3*z^6 + y^2*z^6 + x^2
  ord at a0: 2, initial form: x^2
  ord at a1: 2, initial form: x^2
  ord at a2: 2, initial form: x^2
  ord at a3: 2, initial form: x^2
  a0: coefficient part w.r.t. x=0 (exceptional monomial removed) = y^7 + y*z^4, residual order = 5
  a1: coefficient part w.r.t. x=0 (exceptional monomial removed) = z^4 + y^2, residual order = 2
  a2: coefficient part w.r.t. x=0 (exceptional monomial removed) = y^2 + z^2, residual order = 2
  a3: coefficient part w.r.t. x=0 (exceptional monomial removed) = y^5 + y^4 + y^3 + y^2, residual order = 2
  a3: square part of eps = y^4*z^6 + y^2*z^6  = ( y^2*z^3 + y*z^3 )^2 ;  non-square part h = y^5*z^6 + y^3*z^6
  manuscript-style tail at a3: x + phi = y^2*z^3 + y*z^3 + x  ; hypersurface {x + phi = 0}
  f3 in coordinates x -> x + phi: y^5*z^6 + y^3*z^6 + x^2
  residual order w.r.t. the new hypersurface: 3  (was 2 at a2 w.r.t. {x=0}: INCREASE 2 -> 3)
  the transform of the old tail {x=0} still contains Sing(f3,2) but has residual order 2 which is NOT the residual order realised by the correct tail.
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V4. Ridge (edge data of the tangent cone) at near points, random hypersurfaces, char 2
  near points tested: 375; lexicographic increases of (n-r, q) found: 0