The Conjunctive Fraction of Embodied Benchmarks: Composition Cannot Be Measured Where the Write Path Has Already Destroyed It

The Capability Convergence Hypothesis (CCH) holds that capability is purchased by access structure: the joint possession of a compressive bounded-state channel and a scalable verbatim-index channel. It proves composition strictly super-additive on a witness family that obstructs each channel separately, and names as its largest open risk a measurement it reports "has not been run by anyone, including us": how often natural workloads instantiate that joint load. We run it for embodied multimodal streams. An arithmetic criterion, computable before any run, first decides whether a benchmark can exhibit the effect at all: a bounded score cannot host an interaction whose components already sum past the bound. A census of public video question-answering corpora then puts the conjunctive fraction in the low single digits, so solving that subset perfectly would move the richest benchmark by less than the run-to-run floor at which these benchmarks are compared. We supply the witness the hypothesis implies (735 judge-free items over head-mounted kitchen streams), and in the registered grid every prediction fails and the channel dichotomy reverses. A diagnostic locates the failure upstream of every arm, in the shared single-frame write path, which decides the locator verb only 11-23 the ledger from annotations instead, composition by address beats a same-cost different-occurrence control by 14-26$ points on every backbone and is null on both single-axis families (the corpus is a valid witness and the loss is on the write), yet even then it answers less than half.