WP-79, The Ratio and the Scale, was written in August against a neutral-atom quantum simulator result and asked a good question: of the numbers this corpus publishes, which could an experiment actually refute? Its answer was that a spectrum measurement reports level ratios, En/Em, and that in a ratio every overall scale cancels. So a scale cannot be tested by a ratio, and most of the corpus’s numbers are scales.
That much is right, and block [1] confirms it on its own terms: under a rescaling E → aE the level ratio is unchanged, and under t → bt the product μ T* is unchanged. COMPUTED
Ratios are falsifiable. Scales are not.
What is wrong is not the finding but its scope. The sentence is stated as a property of measurement. It is a property of spectrum instruments, which was the only class WP-79 had in front of it. The qualifier was never written down, and without it the conclusion classifies as unfalsifiable a set of quantities that a different apparatus refutes routinely.
Here is where the reasoning slipped, and it is a distinction worth having in general.
A phase is dimensionless and is not scale-invariant. WP-79’s filter tests the second property; the eye reads the first; and because the two coincide for the quantities a spectrometer reports, nothing forced them apart. COMPUTED
Block [3] makes the same point in SI base units on the free-fall phase, m g h T ⁄ ℏ, which reduces to 1 — dimensionless — while every factor inside it carries a scale. The quotient remains observable. That is the whole of the objection in one expression.
The reason the second instrument behaves differently is not subtle, and it is structural rather than technical.
A spectrum instrument reports En/Em. Multiply every energy in the system by a and the report is identical: the apparatus is blind to the overall scale by construction, which is exactly why WP-79’s quotient is the right thing to take.
An action instrument — an atom interferometer — reports S⁄ℏ. And ℏ is fixed by nature. It is not a parameter of the model that a rescaling may carry along; it is a constant the apparatus measures against. So the rescaling symmetry WP-79 quotients by is broken by the apparatus, and a scale that no spectroscopy can touch becomes something an interferometer can refute.
Ratios are falsifiable by any instrument. Scales are unfalsifiable by spectrum instruments, and falsifiable by instruments whose report is an action in units of ℏ.
WP-79’s sentence is recovered as the first half.
| quantity | spectrum instrument | action instrument |
|---|---|---|
| k-nacci roots ηk | — | — |
| level ratios (CFT data) | exposed | exposed |
| μmax × T* (Floquet) | exposed | exposed |
| T* — the period alone | — | exposed |
| μmax — the rate alone | — | exposed |
| light-cone velocity v | — | exposed |
Three quantities move. T*, μmax and v were classified as beyond the reach of experiment on the strength of a filter that only ever described one apparatus. They are not. COMPUTED
It is worth being explicit about which direction this correction runs. It does not retract a claim or narrow one. It hands three of this corpus’s own numbers to an experiment that could kill them. A correction that increases the surface exposed to refutation is the useful kind, and it is the second in this volume — WP-97 replaced “the only field admitting both is trivial”, which named nothing observable, with “it would show thirty sides”, which a telescope can refuse.
The k-nacci roots ηk stay where WP-79 put them, and no third column would move them either. They are theorems about the roots of polynomials. There is no apparatus whose report bears on whether a polynomial has the root it has, and no instrument class — existing, planned or imagined — changes that.
WP-79 said so, and on this the earlier verdict stands without amendment.
That an action instrument determines a scale absolutely is the operating principle of atom interferometry as a metrology field, not an observation made here. Cold-atom gravimeters measure g absolutely. Recoil interferometry measures h⁄m and through it the fine structure constant — Parker et al. (2018) with caesium, Morel et al. (2020) with rubidium. Those are absolute determinations of scales, by exactly the mechanism described in §3. CHECKED
Nothing in this note is offered to that community as news. The finding is internal. WP-79 ran a falsifiability audit against one instrument class and stated its conclusion without the qualifier; this note supplies the qualifier and redoes the sort. The physics is the standard physics and is load-bearing only as the reason the correction holds.
The physics is standard in metrology, and § 6 says so without qualification. The place it is not standard is the literature WP-79’s sentence actually came from: the foundations-of-constants argument about how many fundamental constants there are, and which quantities have physical meaning.
The canonical text is Duff, Okun and Veneziano, Trialogue on the number of fundamental constants (JHEP 03 (2002) 023). Okun’s position is that “only dimensionless variables, functions and constants have physical meaning in a theory”; Veneziano’s framework is built on quantities “typically representing ratios of quantities having the same dimensions”, with α and me⁄mp as the examples. CHECKED A dimensionless observable that is not scale-invariant — an action or an interferometric phase, S⁄ℏ — is not treated in that exchange.
This does not make § 2 new physics, and does not upgrade any row of the inventory below. The metrology community operates on the distinction daily; a foundations paper not discussing an observable is not the same as the observable being unknown. What it does explain is how a correct-sounding sentence gets written: “only dimensionless quantities are measurable” is the form the argument takes in the place one goes to read about constants, and in that form it slides into “only scale-invariant quantities are measurable” without anyone noticing the step. WP-79 took that step. The two-column sort of § 5 is what catches it. OPEN whether the foundations literature engages it somewhere this note has not looked.
Under the rule of 2026-09-01: this paper’s finding is a domain result — which physical quantities an instrument class can refute — occasioned by an external experiment. Machine assistance found it. Assistance used is not the test; assistance as the subject is. Same class as WP-90, which reports defects in an external preprint and carries no tag either.
| Claim | Status | What it rests on |
|---|---|---|
| level ratio is invariant under E → aE | Computed | block [1] |
| μ T* is invariant under t → bt | Computed | block [1] — WP-79 is correct for its instrument |
| a phase is dimensionless and not scale-invariant | Computed | block [2]; φ → a φ |
| [m g h T ⁄ ℏ] = 1 in SI base units | Computed | block [3] |
| an action instrument breaks the rescaling symmetry | Follows | §3; ℏ is a constant of nature, not a model parameter |
| T*, μmax and v are exposed to an action instrument | Computed | block [5]; three quantities move |
| the k-nacci roots are exposed to no instrument | Follows | they are theorems about polynomial roots |
| absolute determination of scales by interferometry | Classical | Parker 2018, Morel 2020 — cited, not claimed |
| this note contains new physics | FALSE | §6; the finding is internal to this corpus |
| a third instrument class moves the k-nacci roots | OPEN | none is proposed here, and none is expected |
Every row marked Computed is regenerated by wp96-verify.py, which sits beside this note and exits non-zero on any failure. The row marked FALSE is there because a reader skimming §3 could reasonably take the interferometry argument for a result of this corpus, and it is not one.
wp96-verify.py was committed on 2026-09-06 ahead of the page it belongs to, and CLAUDE.md recorded that the paper was held in a session on another account and that no session should draft a replacement. This page was written on 2026-09-07 at the author’s instruction, rendered from the verification script rather than from that draft. If the held version arrives, the two are renderings of the same checked claim set and differ in prose alone; reconcile them by the script, which is the authority for every number above.