Probability has four addresses – sharing the axioms does not mean sharing the answers.
The single word probability names at least four different things. This paper calls them by name and writes their addresses. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Kolmogorov’s axioms, the poor coverage of the Wald interval, credible intervals, the local polytope with CHSH, and Gleason’s theorem are all standard. Nor does this paper judge which position is right: what can be written is what each of them claims.
First, Kolmogorov’s axioms are a syntax, not an answer. On the same data, say zero successes in ten trials, the frequentist estimate is zero, the Bayesian posterior mean under a uniform prior is 0.083333, and under a Beta(10,1) prior it is 0.476190. All three satisfy the axioms in full – non-negativity, total mass one, additivity on disjoint sets (computed here). The axioms constrain the shape of an assignment, not its values: agreeing on Kolmogorov settles nothing about what to write down. So Kolmogorov probability is not an interpretation. The frequentist and the Bayesian are two interpretations standing on one syntax, and syntax and interpretation live on different layers, the same shape as the separation of convention from fact in Paper 95.
Second, where the data is thin the two part company. With no trials at all the frequentist estimate does not exist, being zero over zero, while the Bayesian one returns one half. After one trial with no success the two read zero against 0.333333, after five zero against 0.142857, after ten zero against 0.083333, and even after a hundred zero against 0.009804. Yet with thick data, five successes in ten, the two agree exactly. After ten trials with nothing observed one of them says the probability is exactly zero, closing off the next occurrence on the strength of the data alone. This paper does not judge which is right: what differs is what the answer is an answer about.
Third, and this is the core. A ninety-five per cent confidence interval does not cover ninety-five per cent of the time. Summing the binomial mass exactly, with no approximation and no random numbers, the Wald interval covers 0.6389 of the time at a sample of twenty with a true probability of 0.05; at 0.5 it covers 0.9586 and at 0.2 it covers 0.9208. Raising the sample to fifty lifts the 0.05 case to 0.9199, but a hundred brings it back down to 0.8775 – it is not even monotone in the sample size, and the intuition that more data always helps does not hold. What breaks here is not the axioms: it is that the number ninety-five has been taken to be about the wrong thing, being a property of a procedure under repetition rather than of the interval one is holding. The Wald interval is well known to behave badly and better ones exist; it is used here as a representative, and none of this is a criticism of frequentist statistics as such.
Fourth, a credible interval answers a different question. In the same situation, a sample of twenty with a probability of 0.05, the credible interval covers 0.9245 against the Wald interval’s 0.6389; at a sample of fifty the two read 0.9622 against 0.9199. But the credible interval was never asked to cover ninety-five per cent of the time. It was asked to contain ninety-five per cent of the posterior mass for this data. That it covers well is an outcome, not a claim. The two intervals are written with the same symbols, both announce ninety-five per cent, and answer different questions – the same shape of confusion that Paper 16 described for the equals sign.
Fifth, quantum probability does not fit that syntax globally. With two settings on each side and outcomes plus or minus one, a Kolmogorov model is a distribution over the sixteen deterministic assignments. Minimising over the simplex to approach the singlet’s four correlations, the closest classical mixture is about minus one half, minus one half, minus one half and plus one half, at a distance of 0.414214 – matching exactly the analytic distance to the CHSH facet, 0.414214 (computed here). No distribution whatever over the sixteen assignments reproduces those four correlations. This is not quantum probability violating Kolmogorov: inside any set of commuting observables, inside one context, the axioms hold in full. What fails is the existence of one joint distribution across all contexts. What must be moved is the context, not the axioms.
The census. Frequency means a long-run relative frequency, is moved by the design of repetitions, and is undefined at zero trials. Bayes means a degree of belief, is moved by the prior, and the prior changes the answer. Kolmogorov means axioms for a measure, is moved by nothing since it is a syntax, and fixes no value at all. Quantum probability, in Gleason’s sense, means a measure on closed subspaces, is moved by the context, and admits no global joint distribution. Four uses of one word: three of them interpretations of one syntax, and the fourth not fitting that syntax globally. The sentence the probability is 0.05 can be read four ways, and until one says which, it has not yet become checkable.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / which position is right (not judged) / a criticism of frequentist statistics (none). What is added: summing the coverage exactly rather than approximating or simulating it, setting it beside the credible interval to show the two answer different questions, minimising the distance to the local polytope independently and matching the analytic value, and writing for each of the four addresses what must be moved for it to move. What remains: sharing the axioms does not mean sharing the answers. The axioms settle only what may not be written; they say nothing about what to write. So reading a probability calls not for a check of the axioms but for a check of what the number is a number about.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Probability Has Four Addresses [Paper 164]. Zenodo.
DOI: 10.5281/zenodo.22092815
ORCID: 0009-0006-8587-3450