Instanton actions, from 8 pi squared over g squared to 4 pi squared c

Some tunnellings are forbidden classically but allowed quantum mechanically, and the rate at which they happen is suppressed by an exponential of a quantity called the instanton action. This post explains that standard piece of physics, because the framework has two different instanton actions in it: one that was retracted, and one that survived and turned out to explain a coincidence.

The standard formula

In Yang-Mills theory, the amplitude for a tunnelling event between topologically distinct vacua is suppressed by e to the minus S, where S is the instanton action, equal to 8 pi squared over g squared. The coupling g measures how strongly the gauge field interacts, and the whole suppression lives in that division: weak coupling means an astronomically large action and an astronomically small rate.

The formula is textbook. It is also where the framework’s best-surviving physics claim comes from, and the one before it went wrong.

The one that was retracted

The interface papers needed a small number: the baseline probability that anything crosses a branch boundary. They built it as e to the minus 147, with the action 147 obtained as the code distance 64 times a logarithmic ratio. I have already told that story in the post on the code distance: the ratio inside the logarithm was reverse-engineered from the target, so the 147 was an echo of the answer it was meant to produce. That construction is archived. It is not physics, it is typesetting that looks like physics.

The one that survived

Separately, the framework carries a number called rho N, about 1.1 times 10 to the minus 120, which is the vacuum-energy scale, and its logarithm is about 276. Where does an instanton action of 276 come from? The candidate that survives auditing is the standard formula with the coupling pinned by topology: if the coupling is set by the boundary category’s central charge, g squared equals 2 over 7, then 8 pi squared over g squared equals 4 pi squared times 7, which is 276.35. Compare the observed log, 276.19. The gap is five hundredths of one per cent.

With a subleading correction from the Z6 quotient structure, minus one sixth, the prediction becomes 276.19 and the gap drops to about four thousandths of a per cent. That is the same precision at which the framework’s better numerical coincidences live, and it is much better than numerology deserves.

The difference between the two actions is instructive. The retracted 147 was assembled from a sector count and a fitted ratio to hit a target. The surviving 276 is the standard Yang-Mills exponent with one input, the central charge, that is itself a theorem of the category fixed by independent arguments. The target was known before the 147 existed. The central charge was known before the coincidence was noticed. That ordering is what separates an explanation from an echo.

What would kill it

A coupling pinned by topology can be computed wrong. The kill criterion is stated in the papers: an honest boundary calculation of the instanton coupling for this category, done by established methods, either returns 2 over 7 and the coincidence becomes a derivation, or it does not and the match goes back to being a coincidence, and I will say so in the corrections series. Until that calculation exists, the claim carries the benchmark label: mechanism plausible, arithmetic exact, authority pending. Nothing in this programme has a detection attached to it, and this is the result most likely to be famous if it ever earns one.