Multi-Computation from Exponential Race Events: Extracting Simultaneous Observables from Stochastic Barrier Arrays (v3)

An exponential race over N elements - the mechanism by which an array of stochastic magnetic tunnel junctions produces a softmax sample in O(1) time - yields more than the winner index. This paper catalogues what can be read from one race, prices each observable in bits per event, and separates what is exact from what is only correlational. A single race event yields three mathematically exact observables: the softmax sample (winner index), a maximum-likelihood estimate of the partition function, and pairwise log-probability ratios. A fourth single-race quantity, an entropy estimate from the inter-spike intervals, is correlational: no bound is proved, and it is validated as a rank correlation only. Two parallel arrays add attention-head similarity by coincidence detection. Four further operations - rank information, full softmax estimation, attention-weighted value computation and Fisher information - require repeated races and are priced separately. Version 3 corrects version 2 downward in two places. (1) The complete rank ordering is not a single-race observable: at N = 1024 with N(0,4) logits, one race gives the exact top-5 set with probability 3.5e-5 and the exact top-5 order with probability 4.7e-7, because one race carries 12.8 bits about the top-5 while naming a 5-subset of 1024 costs 43.1 bits. Recovering the set by voting needs about 3.6e3 races. The single-race count therefore drops from five to four and the throughput multiplier from 5-6x to 4-5x. (2) Information utilization is bounded by the mutual information between logits and permutation, not by the permutation entropy: 1030 of 8769 bits at N = 1024, a ceiling of 11.7 percent falling to 7.5 percent at N = 32768, rather than the near-100 percent claimed in version 2. Version 3 further relabels Operation 5 from an entropy bound to an entropy estimate, since no bound is proved; adopts the known deterministic time-to-threshold race for selection and prices it against the stochastic race in a common unit (0.0915 nat of pairwise device mismatch against 1.81 nat of thermal spread); adds a biological cross-domain section with two explicit import prohibitions; places the partition-function estimator in the Gumbel-trick literature; and corrects several references and cross-references. A full list of changes is in CHANGELOG_v3.md. Files: the paper (PDF and LaTeX source), the changelog, and a code archive containing the validation suite and its recorded results, with a README mapping each table of the paper to the script and result file behind it.