The Proton, Sharper

What AP30 predicted. At first order, 1836.152663699, five parts per billion from the measurement. With the carried sixteen at second order, 1836.152673444 — the value the corpus froze on 2 August 2026 as its registered prediction D, at 0.57σ from CODATA 2022 — and a next term of unknown size, sign and coefficient, which the pre-registration bound the corpus not to choose against the residual.

What this paper predicts. With the sixteen decomposed and the series closed:

m_p/mₑ = 1836 + 21α·(1 − 1/(84π)) + 21α·r/(1 − r), r = 16α/1836, = 1836.152673445.

Measured, CODATA 2022: 1836.152673426(32). Residual: −1.9 × 10⁻⁸, which is −10 parts per trillion, −0.59σ. Against the independent H₂⁺ value of 2025, 1836.152673414(47): −0.66σ. The prediction’s own uncertainty, from α’s, is 2 × 10⁻¹¹, thirteen parts in a quadrillion.

What this supersedes, and what changed. D, the frozen value of 2 August, is superseded by D_closed = 1836.152673444951, which references the pre-registration’s digest and stands beside D, never in its place. What changed: the tail, +6.2 × 10⁻¹⁰, the third order and everything after it. The third-order coefficient is c₃ = 21·16²/1836² = +1.595 × 10⁻³, sign positive, stated here in advance as the pre-registration requires; the measurement’s own constraint on c₃, −0.047 ± 0.082, holds it at 0.6σ. The pre-registration noted that D lies above the measurement, so any term that closed the gap would have to be negative. The chain does not close the gap and does not try to: every holder adds, so the closed form sits 6 × 10⁻¹⁰ further from today’s central value than D did, 0.57σ to 0.59σ. A negative term would be a coefficient chosen against the residual, which the corpus bound itself not to offer; the chain has none, and if the residual grows negative as the measurement sharpens, the chain has nothing with which to repair itself. That is what makes KS-HOLD.1 a switch and not a promise.

What became sharper. Not the digit — at the measurement’s precision the two values are the same. What sharpened is the formula. The sixteen is a count with a reason, one rule with the proton’s 84 and the neutron’s 8. The series has a ratio and a sum; there is no truncation and no unknown next term. And the next term is known, with its sign: +6.2 × 10⁻¹⁰, with everything beyond it summed inside the closed form. AP30’s open switch closes. The author said the proton would have to become sharper; this is what that means in the formula’s own terms.

The chain against the measurement. Two things move as the holders are added, and they are different things. The residual — the measurement minus the prediction — is what the sky says. The omitted structure — everything the truncated formula leaves out, which the closed form supplies — is what the structure says is still missing. The measurement’s one sigma is 3.2 × 10⁻⁸, and the table reads both against it:

This is how a prediction sharpens as structural omissions are accounted for, and why it is structural and not a fit. Each holder added removes what the previous truncation left out, by the same factor r, six parts in a hundred thousand: the static structure alone misses by nearly five million sigma; the repair brings it to three hundred; the second holder brings it inside the noise, to 0.57σ, where the measurement can no longer tell the formula from the truth. From there the residual does not move, because it cannot: the third holder shifts the prediction by 0.019σ and the fourth by a millionth of a sigma. The measurement’s resolving power is fixed; the structure’s omission keeps shrinking under it. The line between the second and third holder is where the chain passes beneath what the sky can see, and that is why the third holder is the registered prediction: it is the first term the structure gives that the measurement has not yet been able to test. When the measurement improves about fifty-fold, the third holder is on the table; at nearly a million-fold, the fourth. The formula has no further adjustment to make in the meantime, because the closed form has already spent them all.

In plain words. The proton is a record that costs something to keep. AP30 counted the record standing still and the cost of keeping it, and the cost of keeping the cost, and stopped there with a number it couldn’t explain. This paper explains the number — the four things a record needs, each held through the three directions of space and through time, sixteen — and notices that the keeping never ends: each keeping is kept by the next, at the same small share, forever. Add that up and you get the proton’s weight to the digit the best measurement has, with the next digit predicted, and its sign.

What it would take to see the third order, and the fourth — and to kill the closed form. The third-order term is three parts in ten thousand of a part per billion. The best measurement is good to seventeen parts per trillion. To resolve the term at one sigma the measurement must improve about fifty-fold; the fourth order would need nearly a million-fold. The kill comes first: with today’s central value unchanged, the closed form is 3σ from the measurement once the uncertainty reaches 3.4 parts per trillion — about five times today’s precision — and 5σ at 2.1; the molecular-ion and Penning-trap programmes project that range within a few years. The prediction is registered now, before any such measurement exists; that is its whole point, and it is why the closed form, and not any single order, is the claim.