Paper 10a: inflation from branch nucleation, withdrawn

Paper 10a is archived, and its corpse is the most educational in the whole collection, because the error is not arithmetic, not fitting, not dependency. The paper invented a number out of nothing, and the invention was invisible until you asked where the number came from.

What it tried

The paper extended the branch-interference picture from the microwave background, Paper 10’s territory, to inflation itself: the claim was that the inflationary epoch carries the same branch-ensemble structure, and that the ensemble’s phase structure fixes the spectral index and its running. If it worked, it would connect the framework to the two most measured numbers in early-universe cosmology.

How it died

The audit asked the paper’s mechanism to produce its exponent. The framework’s branch structure, its categories, levels and gradings, contains a fixed inventory of discrete data: levels, dimensions, fusion multiplicities, congruence classes. Inflation’s spectral index is a continuous number close to one, specifically about 0.965, and the running is a small derivative. To get from a discrete inventory to a continuous exponent, the mechanism needs an exponential of the inventory data, and the paper supplied one: an exponent built from the category’s data that landed, when evaluated, near the observed value.

The audit’s question was not whether the exponent evaluated correctly. It was what fixed the combination. The answer: nothing. The specific product of level data, the sign conventions, and the prefactor were all free, and among the free choices, exactly the combination landing on 0.965 was used. Other combinations in the same family gave 0.87, 1.08, values already excluded by data. The mechanism had no selection principle distinguishing the working combination from the dead ones, which means the match was selected from a menu by the answer, which is fitting, and the paper had presented it as derivation.

The technical verdict, in the language of the selection-principles document: the category constrains discrete data, and a continuous prediction requires either a dynamics that fixes continuous parameters or an admission that nothing is predicted. Paper 10a had neither, so it had nothing.

The rule this corpse bought

The archive entry’s residue is the cleanest rule in the programme’s method series: before accepting a numerical match, enumerate the menu. If the framework’s discrete data can be combined into a continuous number in more than one way, every combination must be listed, and the prediction is only the menu’s consensus, not its best member. A menu of one is a derivation. A menu of five where one matches is a fit with extra steps.

That rule is why Paper 10’s phase, the pi over 6, survived its own post-hoc scandal in a way this paper did not: the phase has exactly two preregistered routes, a menu small enough to audit. Paper 10a’s exponent had a menu nobody had counted. The counting is the physics, and this paper is the archive’s reminder that the counting comes before the match, not after.

The next paper is not physics at all. It is the anti-fit machinery that keeps the wager enforceable: Paper 10b.