Most of the cat‑over‑GKP readout advantage is a choice of meter, not a property of the code

If you want to know which bosonic code is easier to read without destroying, the obvious move is to fix the photon number, measure how much signal each gives up per unit of logical damage, and take the ratio. Do that for cat against GKP and you get a factor of 7.8.

That number is real, and it is mostly not about the codes.

Why the ratio splits

Readability is P = γS / Γ_self — signal per unit self-disturbance — so any ratio of readabilities factors into a ratio of signals times a ratio of disturbance rates. That is arithmetic, not physics. The physics is that for these two codes the two factors have entirely different origins.

The cat's protected variable is an unbounded quadrature, so its signal grows with photon number: S = 8n̄ at d = 2. GKP's is modular and bounded, giving S = O(1). The signal ratio is therefore a statement about what kind of variable each code protects, and it inflates without limit as you add photons. Comparing raw readabilities at "matched photon number" does not control for this, because matching photon number is exactly what makes the cat's signal large.

What is left after you divide it out

For any d = 2 meter the within-code signal-to-variance ratio obeys a meter-independent identity, S/V ≈ 4. So a scale-invariant readability γ(S/V)/Γ_self removes the meter's units and isolates what the code contributes. What remains of the 7.8x is a ratio of self-disturbance rates of about 1.5x — 171 against 117.

That residual is the part our bound actually constrains, and it is the part worth arguing about.

What we are not claiming

The 1.5 is soft, and we would rather say so than let it be quoted as a constant:

• It is read at a single photon number, n̄ ≈ 1.9. It is a code-and-meter quantity at that point, not a structural invariant.
• Both self-disturbance rates sit near the extraction floor and carry 3–12% uncertainty, so 1.5 is resolved only to about ±0.2.
• At d = 3 the raw cat advantage narrows to a factor of order 2, from a single comparison on atlas-weight rows, and no d = 4 GKP comparator was run. The scale-invariant ranking is not settled above d = 2.

So the honest statement is not "cat beats GKP by 1.5x." It is that the headline 7.8x is dominated by a units effect, and once removed, what survives is small enough that we cannot currently rank the two with confidence.

The transferable part

Any comparison of readout quality between codes with different protected variables will inflate this way, and the inflation is invisible if you quote only the ratio. Report both factors. A readability advantage that is really a statement about bounded versus unbounded observables should not be presented as a statement about which code is easier to read.