Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter
For the power mean $P_t$, $t<1$, let \[ \Lambda_N(t)=\sup_{x_k>0} \frac{\sum_{n=1}^N P_t(x_1,\ldots,x_n)}{\sum_{n=1}^N x_n} \] be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to $t<0$, $t=0$, and $0<t<1$. With the analytic parameter $q=t/(1-t)\in(-1,\infty)$, and writing $\lambda_N(q)=\Lambda_N(q/(1+q))$, we derive one exact scalar shooting map for all three regimes; its apparent singularity at $q=0$ is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at \[ y_*(q)=1+q,\qquad \Lambda_*(q)=(1+q)^{(1+q)/q}. \] We prove that there is a unique real-analytic phase $\kappa(q)$ for which the finite-defect expansion has zero cubic coefficient and, uniformly for $q$ in compact subsets of $(-1,\infty)$, for every fixed $L\ge2$, \[ \Lambda_*(q)-\lambda_N(q) =\sum_{j=2}^{L}\frac{A_j(q)}{(\log N+\kappa(q))^j} +O\!\left((\log N)^{-L-1}\right). \] In this phase-normalized scale, \[ A_2=2\pi^2(1+q)\Lambda_*,\qquad A_3=0,\qquad A_4=-\frac{\pi^2}{3}(2q^2+5q+5)A_2. \] Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain $A_2(0)=2e\pi^2$ and $A_4(0)=-(10/3)e\pi^4$; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is $A_5$, at order $(\log N+\kappa(q))^{-5}$.