The logistic permeability function

This is the shortest correction in the whole programme, and the one that still embarrasses me most, because the error survived in a central formula across several papers, and the fix is one line. It took an outside audit to prompt it. Both halves of that sentence are lessons.

The broken formula

The boundary field Psi measures stress on the wall between branches. The leakage probability was written as proportional to e to the Psi. The intent: stress opens the boundary exponentially.

The defect: e to the Psi is not a probability. It is unbounded. Psi is a field value with no a priori ceiling, and a modest input, around 62 in the units the papers use, drives e to the Psi past 10 to the 27. A probability of 10 to the 27 is not a large probability. It is a category error. Probabilities live in the interval from zero to one, and anything outside it is not an extreme value of your theory, it is a refutation of your formula.

Nobody noticed for a long time for a boring reason. In every regime the papers examined, Psi was small enough that e to the Psi was a small, well-behaved number, and all downstream conclusions stayed finite. The formula was wrong in a regime nobody visited, which is the most expensive place to be wrong, because nothing in your daily practice ever forces you there.

The one-line repair

Wrap the exponential in a logistic function: p equals e to the Psi, over one plus e to the Psi, times the baseline. That is the whole fix. For small Psi the new formula agrees with the old one to within corrections of order e to the Psi, so every small-stress result in the existing papers stands. For large Psi, instead of exploding, p saturates at one. The wall never becomes more than perfectly permeable, which is the least a sane formula can promise.

The logistic function is the standard way to turn an unbounded driver into a bounded response, and it is the same function that turns any real number into a value in the unit interval in statistics, in neural networks, in population models. There was no cleverness available here to use. The repair is the obvious one, applied late.

What the fix buys, and what it does not

With the repair, the permeability function has honest limits everywhere: it is monotone in stress, it vanishes as the baseline vanishes, it saturates at one under extreme stress, and it never leaves the unit interval. The derivative structure is smooth, so the perturbation calculations around small Psi are unaffected. In later papers the same saturating form appears as p leak equals delta naught e to the Psi over one plus delta naught e to the Psi, with the baseline delta naught itself exponentially suppressed. That expression is bounded twice over, once by the logistic and once by the tiny baseline, and the papers’ claims about patterned leakage all live in the small-p regime where the two forms agree.

What the fix does not buy is any new physics. A formula that never contradicts itself is a floor requirement, not a result. The physical content of the leakage hypothesis, that the suppression is structured rather than thermal, and that it scales with the code distance, is untouched by all of this. Which is exactly why the error was worth confessing at length: it damaged nothing but credibility, and credibility is the thing a speculative programme spends.

The meta-lesson, which I will keep returning to in the method series: audits are not admissions of failure, they are the only mechanism by which a wrong formula in an unvisited regime gets found before it costs you a prediction. Someone had to read the formula and ask the childish question. What happens if Psi is big. The answer was 10 to the 27, and then it was one line.