= Solution
The <Cox proportional-hazards model> specifies
$$
\boxed{h_i(t)=h_0(t)\exp(\beta^Tz_i),}
$$
where $h_0(t)$ is an unspecified common baseline <hazard> and $z_i$ is the individual's <covariate> vector. The <hazard ratio> for two fixed <covariate> vectors is $\exp[\beta^T(z_i-z_k)]$, constant in time. Assume independent subjects and <independent censoring> conditional on their <covariates>, and use the question's absence of tied events.
Let $t_j$ be an event time, $i_j$ its failing subject, and $R_j$ the <risk set> immediately before that time. In a short interval of length $dt$, the conditional probability that subject $i$ fails, given one failure among those at risk, is
$$
\frac{h_i(t_j)dt+o(dt)}{\sum_{k\in R_j}h_k(t_j)dt+o(dt)}
\longrightarrow\frac{e^{\beta^Tz_i}}{\sum_{k\in R_j}e^{\beta^Tz_k}}.
$$
The baseline <hazard> cancels. Multiplying the successive conditional event-identity factors gives the <Cox partial likelihood> for $\beta$:
$$
\boxed{L_p(\beta)=\prod_j
\frac{e^{\beta^Tz_{i_j}}}{\sum_{k\in R_j}e^{\beta^Tz_k}},\qquad
\ell_p(\beta)=\sum_j\left[\beta^Tz_{i_j}-\log\sum_{k\in R_j}e^{\beta^Tz_k}\right].}
$$
This is a partial <likelihood>, not the full survival <likelihood> for the unspecified $h_0$. <Risk sets> evolve with the observed history; the derivation uses their successive conditional <hazards>, rather than assuming the <risk sets> themselves are independent.
Define $w_{jk}(\beta)=e^{\beta^Tz_k}/\sum_{\ell\in R_j}e^{\beta^Tz_\ell}$ and $\bar z_j(\beta)=\sum_{k\in R_j}w_{jk}(\beta)z_k$. For component $r$,
$$
\boxed{\frac{\partial\ell_p}{\partial\beta_r}
=\sum_j\left[z_{i_j,r}-\sum_{k\in R_j}w_{jk}(\beta)z_{k,r}\right].}
$$
Differentiating again gives
$$
\frac{\partial^2\ell_p}{\partial\beta_r\partial\beta_s}
=-\sum_j\left[\sum_{k\in R_j}w_{jk}z_{k,r}z_{k,s}
-\bar z_{j,r}\bar z_{j,s}\right].
$$
Thus the <score and information of Cox partial likelihood> are the sum of event-minus-risk-mean contrasts and the sum of risk-weighted covariance matrices, respectively. The Hessian is negative semidefinite; the <likelihood> is concave, with strict identifiability requiring variation in the relevant risk-set <covariates>.
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