Binary reactions and the physical-time Bell limit

We give the tracking and low-population steps separately. The same pathwise inventory and variation estimates hold after conditioning on the initial census; expectations below average that census as well as reaction clocks. Fix a cutoff $\delta>0$ and, on each edge, run a companion signed queue from the same empty state and the same exports with service function

All divided-difference slopes lie between $a_0=\kappa_-\delta$ and $a_1=\kappa_+$. The companion is driven by adapted coefficients; they are not asserted independent of either process. Since each physical reaction uses one of the $NL+O(1)$ packets, the sum of absolute population jumps is at most $2L+O(N^{-1})$. Hence the variations of $a_\pm$ are bounded independently of $N$ at fixed $\delta$.

For $m^*=\mu_NZ^*/N$, compensated reaction counting yields

Let $y$ solve the same adapted finite-variation equation without $M$, and let $y_J$ omit both $M$ and $de^N$. Scalar monotonicity and $\|e^N\|_\infty\le c_0/N$ give

For the first inequality, write the difference equation with a measurable divided-difference coefficient $a(t)\in[a_0,a_1]$. Its solution at $t$ is $\mu_N\int_0^t\exp[-\mu_N\int_s^t a(u)du]\,de^N(s)$. This is a pathwise Stieltjes identity, not an anticipative stochastic integral. Integration by parts bounds its absolute value by $2\mu_N\|e^N\|_\infty$. The second inequality follows because the drift points toward $[-J_*/a_0,J_*/a_0]$.

The instantaneous root $f(t)=[J(t)]_+/a_+(t)-[-J(t)]_+/a_-(t)$ satisfies

Contraction between jumps of $f$ and summation of its jumps imply

For $\xi=m^*-y$, production jumps cancel. The square-jump identity and monotonicity, with $V=\mathbb E\xi^2$ and $\alpha=\mu_N/N$, give

Young's inequality absorbs the square-root term into $\mu_Na_0V$ and gives $\sup_tV\le C_\delta(\alpha+\alpha^2)$. Thus

This controls directional flux because a signed queue exposes only one orientation, and

It remains to remove $\delta$ without assuming the desired population closeness. At time $t$ let $K=\{r:x_r<\delta\}$, and let $I_K,O_K$ be normalized queued charge directed into and out of $K$. Summing (4.4) over $K$ gives

An incoming queue has origin outside $K$, so its origin population is at least $\delta$. Its expected service count therefore bounds

Consequently

The target directional current whose physical origin lies in $K$ is therefore bounded, using (3.2) and Cauchy–Schwarz, by

Couple physical and companion queues with identical exports and minimum-rate baseline services. View companion service as baseline service with physical coefficients plus extra service where $x<\delta$. Every unmatched baseline service contracts their absolute signed queue difference by $1/N$; an extra companion service can enlarge it by at most $1/N$. Starting from equality, the expected number of baseline mismatches is at most the expected number of extra services. The latter normalized count is at most $R_{\delta,N}+D_{\delta,N}$, since it is supported on low-population origins. Comparing the two directional flux vectors thus costs at most twice this count. Adding the direct companion error gives

Take $N\to\infty$ at fixed $\delta$, then $\delta\downarrow0$, to prove flux convergence. Finally the physical population process has drift $B(\Phi^{N,+}-\Phi^{N,-})$ and an $O(N^{-1})$ quadratic-variation budget, because it has at most $NL+O(1)$ jumps of size $1/N$. The martingale maximal inequality, initial calibration and integrated flux convergence prove (4.6) tends to zero.

Each open set $\{t:w_r(t)>0\}$ is a countable union of intervals. No finiteness of the nodal set is inferred from piecewise $C^1$ regularity. On compact subintervals of these positive-weight components, standard integrated-hazard first-jump construction is unique until explosion or a nodal boundary; the increasing compact localization defines the minimal process on their union. Killing at such boundaries gives the minimal forward solution. The nonnegative vector $w$ solves its forward balance equation, so successive first-jump iteration, or positive Volterra iteration, bounds each partial-transition sum by $w$; with initial $\nu$ the bound is $Cw$. If a holding path remains at $r$ while $w_r$ tends to zero, write incoming and outgoing positive currents as $I_r,O_r$. Then $\dot w_r=I_r-O_r$ and $\lambda^B_{\rm out}(r)=O_r/w_r\ge-\dot w_r/w_r$. Integrating shows that the holding survival to that zero is zero. The dominated killed law also gives

Thus neither explosion nor nodal killing loses mass. For initial $w(0)$, normalization and domination imply equality with $w(t)$. For general $\nu$, normalization gives the asserted dominated process. The first-jump construction determines its law uniquely. This is the finite-graph existence argument underlying the standard Bell process [DGGTZ]; it is not a selection of that process among all event laws.

Let $N_\varepsilon$ count crossings of the deterministic weights through a regular level $\varepsilon$. One-dimensional coarea gives $\int_0^1N_\varepsilon d\varepsilon\le\sum_r\kinVar(w_r)$. There is a sequence $\varepsilon_k\downarrow0$ such that $\varepsilon_kN_{\varepsilon_k}\to0$: otherwise $N_\varepsilon$ would have a nonintegrable $c/\varepsilon$ lower bound near zero. The probability that the Bell path visits a sector while its weight is at most $\varepsilon$ is bounded by initial small-weight mass, jump influx into those sectors, and deterministic downcrossings at which the path is already in that sector. By Lemma 4.2 and (3.2), a valid bound is

Jump influx uses the integrated current bound at a small destination; each downcrossing contributes at most $C\varepsilon$.

Couple the target and tagged jumps at minimum conditional intensities until they disagree, the target reaches the small-weight region, or $\sup_t\|x^N-w\|_1>\varepsilon/2$. In the remaining states,

The full microscopic marginal is preserved by this coupling although its queues depend on the tag's past. The target marginal retains its own Markov rates. A union and compensator bound gives

Take $N\to\infty$ at each fixed $\varepsilon_k$, then $k\to\infty$. The queue proof's cutoff $\delta$ was already removed; the two localizations are not interchanged. This proves the whole-path claim.