Collision-free distributed tangent realization from residual ledgers

Let $V$ be finite, with an injective prime labeling $p:V\to\mathbb N$, and let $F\ge0$ be a flow whose positive edges have distinct endpoints. Suppose its divergence is a residual vector $r$, its total traffic is at most the distributed residual/cut ledger, and the incident flow at $p_v$ is at most $|r_v|+2b_v$, where $b$ is the port-load vector. Assume $p_v|r_v|\le R_*$ and $p_vb_v\le B_*$, and $p_v\le H$ with $H\le n$, $H^2\le n$, $n>0$. Fix a paper bank, a compatible guarded central-anchor certificate, a finite exceptional set, and natural list parameters $K,h,P_{\rm head},X_0$.

Take $L,\sigma,N,d>0$ with $LN=n$. Suppose the total ledger is at most $C_T C_\tau Nw+e_TN$, and $R_*+2B_*\le C_I C_\tau Nw+e_IN$. The source's distributed main, error and ceiling budgets for these constants are required to be at most $d^2/48,d^2/96,d^2/96$, respectively. Every split request must have a positive lower-cardinality bound $m_q$ no greater than its clean multiplier-list size, and must satisfy $dn\le m_q a$ at either endpoint label $a$.

Then one can choose a multiplier $z_q$ from each clean list so that all selected endpoint products are distinct. Moreover, divergence is $r$, and for every natural prime-index $\ell$,

Here $q$ ranges over the split positive-flow requests, $w_q$ is its split weight, and $s_q,t_q$ its labels; $v_\ell$ denotes the natural factorization coordinate, defined also for nonprime $\ell$. This turns explicit residual, traffic, density and parameter budgets into a collision-free realization.