A leakage probability that exceeded one

The fourth correction is the shortest arithmetic in the archive and the longest-lived error: a probability formula that was not a probability, p proportional to e to the Psi, unbounded above, sitting at the centre of the boundary-permeability model for several paper generations. Its failure mode deserves respect precisely because it never misfired in any calculation anyone ran.

The defect

The boundary field Psi measures stress between branches, and the model wanted leakage to grow with stress. The natural first move: p equals e to the Psi. The function grows, it is smooth, it is positive, and in the regime where Psi is small it is essentially linear, so every perturbation calculation is comfortable. The defect is visible in one glance at the range: e to the Psi has no ceiling. Psi is a field value with no a priori bound, and a moderate input, sixty-two in the model’s units, drives the formula past ten to the 27. A probability of ten to the 27 is not an extreme probability. It is a proof that the formula is not a probability.

Why did this survive so long? Because probability densities and probabilities get confused in a way that hides the defect. If p is a density, it may exceed one, and the observable quantity is its integral. The draft’s language drifted between density and probability across drafts, and each usage was locally defensible. But the leakage quantity fed into decision rules and detection thresholds that needed a genuine probability, and there the unbounded form was simply wrong: the interval is zero to one, and any formula that leaves it is refuted by arithmetic before any experiment speaks.

The fix and its cost

The repair is one line: p equals e to the Psi, over one plus e to the Psi. The logistic function maps any input into the unit interval, is monotone, and agrees with the naive form for small Psi to corrections of order e to the Psi, which is why every existing small-stress calculation survived the repair unchanged. For large Psi, the new formula saturates at one: maximum leakage, never more. The wall can become perfectly permeable, which is the physical ceiling, and the formula now respects it.

The cost of the repair is a parameter: the saturation needs a scale, the critical stress Psi c at which the crossover happens. The audited papers carry that scale as an input, bounded by consistency, not derived. That is the honest residue of the error, and it is cheaper than the disease: one bounded input parameter, stated as such, versus a formula whose outputs left physics entirely.

The general lesson, which is about ranges

The reason to publish this as a correction rather than a footnote is that the failure class is enormous. Every model that maps a driver to a response via an exponential, and physics is full of them, owes a check that the response lands in its physically allowed range. Arrival probabilities, branching ratios, detection efficiencies: all are trapped in the unit interval, and their driver formulas usually are not. The check costs one line. The archive’s version of the lesson: before asking whether a formula is accurate, ask what values it can produce, because a formula that can produce impossible values is already known to be wrong in some regime, and the only question is whether that regime is one you visit. In this case it was not, for years, and the visit, when it finally came, was arranged by an audit rather than by nature.

The last post in the series is the argument for publishing all of this at all: why I publish my own corrections.