Entropy Budgets and Weighted Concentration in Parallel Games

How do small individual losses accumulate when many tests are performed together and their outcomes may be strongly correlated? This paper proves a weighted concentration theorem from an explicit conditional entropy budget. The natural scale is the sum of the squared benchmark losses, and the bound holds at every finite length without assuming independence of the outcomes. The result For benchmark losses εᵢ, let S = ∑ᵢ εᵢ² and let Z = ∑ᵢ εᵢ(1 − Wᵢ) be the weighted loss. Under the stated budget and charge assumptions, the paper proves Pr[Z ≤ (1 − r)S] ≤ exp[−r⁵S/(2²¹K⁴(1 + C₀))] for 0 < r ≤ 1. A sharper all-wins estimate is also established. How the argument works Independent sparse auxiliary tests reveal selected successful coordinates. An exact posterior entropy identity and a differential bound then control the probability of unusually small total loss. The underlying success indicators themselves may be arbitrarily correlated. The paper also characterises the predicates admitting exact positive-factor mass bounds, proves exponential mass obstructions outside that class, and gives a robust common-information obstruction to uniformly low-mass accurate filters on all near-perfect tested instances with growing answer alphabets. Independent question-free nuisance registers nevertheless cancel exactly from entropy increments under the specified product assumptions. Scope The commuting-operator game applications, including disjoint-block and projection predicates and a charged extension, assume the explicitly stated finite-tracial reconstruction hypothesis FR(K). That hypothesis is not proved here; at its existential level it is equivalent to the corresponding finite-tracial scalar budget. The general quadratic parallel-repetition problem remains open. The factorisation obstructions constrain particular reconstruction methods and do not disprove general quadratic repetition.