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{"type": "markdown", "id": "cell_8d878ff95905", "created_at": "2026-07-27T15:30:57+00:00", "title": "Claim 2 \u2014 union bound 24% narrower under asymmetric arrival", "pinned": true, "pinned_at": "2026-07-27T15:30:57+00:00"}
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## Claim 2 — union bound tighter than variance upper bound under asymmetric arrival

> "Combining two arm-level anytime-valid confidence sequences via a union bound produces tighter intervals than the standard design-based variance upper bound when treatment induces asymmetric outcome arrival rates between arms (Section 4)."

This is Proposition 4.12: the union-bound width b(V₀;α/2)+b(V₁;α/2) is compared against the classical cumulative variance-upper-bound benchmark b(V₀/π+V₁/(1−π);α) (paper's Eq. 28), both using the exact stitching boundary b(V;α)=√[(Vη²+1)/η² · log((Vη²+1)/α²)] (paper's Eq. 21, α=0.05, η=0.5). We SIMULATE two DGP scenarios (N=500, R=250 replications, evaluated at t=25) and MEASURE V₀=Vₜ(0), V₁=Vₜ(1) from the real simulated potential-outcome trajectories (Eq. 18: the predictable quadratic variation, summed over units whose event has occurred) — the boundary formulas are then evaluated on these measured numbers, not fit to them.

**Symmetric scenario** (θ=(0.15,0.15), matched hazards and reward magnitudes): measured V̄₀=607.1, V̄₁=605.7 (ρ=V₀/(V₀+V₁)=0.501). Union width **183.51**, variance-UB width **183.03**, ratio **1.0026** — essentially 1.0, matching Proposition 4.12's ρ=0.5 theoretical value of exactly 1.0.

**Asymmetric scenario** (θ=(0.05,0.60), fast/large-magnitude control vs. rare/small-magnitude treatment): measured V̄₀=6110.7, V̄₁=23.6 (ρ=0.996 — one arm dominates the variance clock). Union width **330.91**, variance-UB width **434.83**, ratio **0.7610** — the union-bound CS is **23.9% narrower**. The theoretical Proposition-4.12 asymptote at this measured ρ is 0.7496 — our finite-sample measured ratio (0.7610) is close to but not identical to the asymptote, exactly as expected since V₀+V₁ is finite (not literally →∞); the extreme limit √(1−π)=0.7071 is approached but not yet reached.

**Verdict: VERIFIED** — the union bound is measurably tighter under simulated asymmetric arrival, and essentially tied under simulated symmetric arrival, matching the direction and rough magnitude of Proposition 4.12 in both regimes.

````raw
{
 "paper_id": "2603.25971",
 "alpha": 0.05,
 "eta": 0.5,
 "pi": 0.5,
 "scenarios": [
  {
   "scenario": "symmetric",
   "N": 500,
   "R": 250,
   "t": 25.0,
   "V0_mean": 607.1195136065204,
   "V1_mean": 605.7394213021958,
   "V0_se": 0.8029683801490114,
   "V1_se": 0.7532569423866434,
   "rho": 0.5005689418054329,
   "union_width": 183.5101380523057,
   "varub_width": 183.0259764590659,
   "width_ratio": 1.0026453162693443,
   "theory_asymptote_ratio": 0.9999998381525455,
   "union_narrower": false
  },
  {
   "scenario": "asymmetric",
   "N": 500,
   "R": 250,
   "t": 25.0,
   "V0_mean": 6110.737274044333,
   "V1_mean": 23.601861976934067,
   "V0_se": 5.141942398713432,
   "V1_se": 0.0866074159429563,
   "rho": 0.9961525012795034,
   "union_width": 330.9081620625933,
   "varub_width": 434.8335174109777,
   "width_ratio": 0.760999667258491,
   "theory_asymptote_ratio": 0.7496057412695782,
   "union_narrower": true
  }
 ],
 "elapsed_sec": 0.02797389030456543
}
````