The equal-tail/likelihood-ratio edge for a single observation: six affine branches of V=(am²+bm+c)^(3/2), two of them new, and a boundary-atom test that closes both

the Bar-Lev–Hoessly ET–LR operator factors exactly: writing V = u³ (u = V^(1/3) > 0) and W = u² = V^(2/3), one has 9V²V"' - 9VV'V" + 4(V')³ = (27/2) u⁷ W"', so the equation (11) of arXiv:2608.28221v1 is a polynomial identity in (u, u', u", u"') with no fractional power anywhere

for u > 0, equation (11) holds at every point iff the third derivative of W = u² vanishes at every point; and a real function whose third derivative vanishes identically is the quadratic W(x) = W"(0)/2 x² + W'(0) x + W(0)

under the reduction, the variance function is the 3/2 power of a quadratic that is positive everywhere: there exist a, b, c with a m² + b m + c > 0 and V(m) = (a m² + b m + c) * sqrt(a m² + b m + c)

the pair (sign of a, sign of the discriminant b² - 4ac) of the ET–LR quadratic W = a m² + b m + c is an affine invariant of the NEF: an affine change of the observation x -> alpha x + beta with the forced rescaling W -> lambda W (lambda > 0) multiplies a by lambda/alpha² > 0 and the discriminant by (lambda/alpha)² > 0

a quadratic that is positive at some point falls into exactly SIX pairwise disjoint affine cells: (a=0,b=0), (a=0,b!=0), (a>0,disc<0), (a>0,disc=0), (a>0,disc>0), (a<0); in particular a < 0 forces disc > 0, so there are six cells and not eight

completed-square normal forms for the four non-degenerate cells, and: on the cell a < 0 the set m: W(m) > 0 is bounded, of half-length sqrt(disc/(4a²))

the cumulant k(theta) = -sqrt(c² - theta²) on |theta| < c (c > 0) is strictly convex, has mean function k'(theta) = theta/sqrt(c²-theta²) onto R, and has variance function V(m) = (1+m²)^(3/2)/c: the identity k"(theta) = V(k'(theta)) holds identically

the OUTER hyperbolic branch, cell a > 0 and disc > 0: for every c > 0 the function k(theta) = -sqrt(theta² - c²) on theta < -c is a strictly convex candidate cumulant whose mean function takes values in (1, infinity) and whose variance function is V(m) = (m²-1)^(3/2)/c, i.e. V^(2/3) = (m²-1)/c^(2/3), a quadratic with a > 0 and disc > 0

the INNER hyperbolic branch, cell a < 0: for every c > 0 the function k(theta) = +sqrt(theta² + c²) on all of R is a strictly convex candidate cumulant whose mean function takes values in (-1, 1) – a BOUNDED mean domain – and whose variance function is V(m) = (1-m²)^(3/2)/c, i.e. V^(2/3) = (1-m²)/c^(2/3), a quadratic with a < 0

each of the three sqrt-quadratic cells is realised by its cumulant: (c*V)^(2/3) equals 1+m² (disc = -4 < 0), m²-1 (disc = 4 > 0) and 1-m² (a = -1 < 0) respectively; and the ET–LR operator is homogeneous of degree three in V, so the modulus c is irrelevant to equation (11)

dominated convergence for a strictly negative exponent, and its two absolute-continuity corollaries: if mu << Lebesgue is carried by (-infinity, beta] (resp. [beta, infinity)) and exp(x-beta) (resp. exp(beta-x)) is mu-integrable, then int exp((n+1)(x-beta)) dmu -> 0

the inner hyperbolic exponent limit: sqrt(theta² + c²) - theta -> 0 as theta -> +infinity, so exp(k(theta) - theta) -> 1

the outer hyperbolic exponent limit: -sqrt(theta² - c²) - theta -> 0 as theta -> -infinity, so exp(k(theta) - theta) -> 1

the Tweedie p = 3/2 exponent limit: -4c²/theta -> 0 as theta -> -infinity, so exp(k(theta)) -> 1 and the generating measure of that branch puts positive mass on the boundary point 0 of its support

controls: the inverse Gaussian branch has k(theta) = -sqrt(-2 c theta) -> -infinity as theta -> -infinity (so exp(k) -> 0 and there is NO boundary atom), and the Gaussian branch has sigma² theta²/2 - beta theta -> +infinity for every beta (so the test does not apply at all)

NEITHER new cell admits a Lebesgue-dominated generating measure: if mu << Lebesgue is carried by (-infinity,1] (resp. [1,infinity)), has exp(x-1) (resp. exp(1-x)) integrable, and satisfies int exp(theta(x-1)) dmu = exp(k(theta) - theta) along theta = n+1 (resp. theta = -(n+1)) for k = kIn c (resp. kOut c), then False – the left side tends to 0 by absolute continuity and the right side tends to 1 by the exponent limit

controls on A1-A6: without absolute continuity the zero-limit conclusion is FALSE (for the Dirac mass at the boundary the integral is identically 1), and all three non-Gaussian solutions of the quadratic-root condition are non-constant variance functions (vNig at 0 vs 3/4, vOut at 5/4 vs 5/3, vIn at 0 vs 4/5)

the unrestricted one-observation ET–LR classification: if a continuous one-parameter NEF satisfies Assumption 2.5 of arXiv:2608.28221v1 and p_ET,theta = p_LR,theta for every theta in Theta with a single observation, then the family is Gaussian up to affine transformation – no power-variance restriction and no n -> infinity limit

the local expansion at the median is EXHAUSTED by the source's two conditions: once equation (11) and the density identity (19) hold, every higher odd-order condition Gₘ^((2k+1))(1/2) = 0 is automatically satisfied – verified identically to series order 16 in exact rational arithmetic on the full solution class V = (a m² + b m + c)^(3/2)

the printed p-values of Section 5.1 of arXiv:2608.28221v1 (Levy / inverse-Gaussian NEF, mu = 1, lambda = 1/2) reproduce to every printed digit: p_ET(0.5) = 0.980277 (printed 0.9803), p_LR(0.5) = p_UMPU(0.5) = 0.617075 (printed 0.6171), p_ET(1) = 0.572416 (printed 0.5724), p_LR(1) = 1