8. A Fair Critique about last blog: “You Didn’t Compute the Integral”

A thoughtful reader pointed out something important about the discussion so far: https://quantumtraction.org/2025/11/13/qtt-and-the-nonlinear-calcium-king-plot-why-the-effect-is-expected-and-how-to-compute-it/

The blog echoes the PRL conclusions but doesn’t actually compute the QTT integral – it’s assertive, not demonstrative. There are no independent QTT calculations in the literature (searches find nothing beyond this site). If QTT over- or under-predicted the effect – for example, by ignoring relativistic nuclear effects – that would show up as a mismatch in the generalized King-plot linearity.

This is a fair and valuable criticism. It basically says:

• We’ve explained how QTT expects the bend to arise, and
• We’ve shown that this expectation lines up with the way the PRL authors interpret their data,

but we have not done our own full-blown, from-scratch numerical calculation of the integral

Δ_i^(A,A′) = ∫₀^∞ dω / (π ω) · 𝒦_i(ω) · [ α_A(ω) − α_A′(ω) ].

That’s true. Right now, no independent QTT “number-crunching” paper exists that takes nuclear many-body input for calcium isotopes, plugs it into that integral, and plots a theoretical King curve next to the experimental points.

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9. What QTT Actually Provides (and What It Doesn’t, Yet)

It’s helpful to be very clear about the two levels of what QTT is doing here:

1. Structural prediction.
   QTT tells you the shape and origin of the King-plot curvature:
   • It must come from nuclear polarization (and other known higher-order SM terms), not from a new low-energy force.
   • It must enter as a weighted integral over the nuclear polarizability with a fixed, universal electronic kernel 𝒦_i(ω).
   • It must appear as a single functional direction in the space of possible bends, which you can “rotate away” by adding one more line to the generalized King plot (exactly what the PRL team sees: the GKP becomes linear again at < 1σ).
   This is the level we’ve been talking about so far: QTT describes the pattern, the “why” and “which direction,” and the absence of extra knobs.
2. Fully quantitative evaluation.
   To go further – to say “the bend should be exactly this big, with exactly this curve” – you need:
   • Realistic, modern nuclear-structure calculations for α_A(ω) in all the isotopes, including relativistic and many-body effects.
   • A careful implementation of the QED electronic kernel 𝒦_i(ω) for the specific transitions (570 nm in Ca¹⁴⁺, 729 nm and the DD line in Ca⁺), with all the usual bound-state QED machinery.
   • A full propagation of uncertainties from nuclear theory into Δ_i^(A,A′), and then into the King plot.
   That part has not yet been done as a dedicated “QTT paper”; it would, in practice, look very similar to the state-of-the-art nuclear/QED work that the PRL authors already lean on.

So yes: at this stage the blog is making a structural claim (“this is the right mechanism, and the data are consistent with it”) but not yet presenting an independent, full-on numerical simulation.

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10. How a Mismatch Would Falsify QTT

The critique also raises an important point about falsifiability:

If QTT’s structural picture were missing something essential – say, some relativistic nuclear effect or some subtle interference term – then the fully evaluated integral might not just be a little off, it might be off in a way that cannot be repaired without violating QTT’s core rules.

Here’s what that would look like in practice:

• You use the best available nuclear models to compute α_A(ω) for the calcium isotopes.
• You plug them into the QTT integral with the fixed QED kernel 𝒦_i(ω).
• You build the King and generalized King plots from those Δ_i^(A,A′).

If the result is:

• Theory curves + data line up within the quoted nuclear-physics uncertainties → QTT’s low-energy picture passes a very nontrivial test.
• Theory curves give a different kind of bend that cannot be fixed without introducing a new low-energy parameter or force → that would be a serious hit against QTT’s “no hidden knobs” stance.

In particular, QTT says there should be only one dominant nuclear-polarization direction in the space of possible bends. If, after cleaning up the 2nd-order mass shift and improving nuclear theory, the generalized King plot still shows a robust, stable nonlinearity that cannot be removed by accounting for a single functional, that would be a direct sign that QTT is incomplete at this level.

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11. Why the Current Match Is Still Meaningful

Even without a full independent QTT number-crunching code, the present match is still nontrivial, because:

• The direction of the observed King bend lines up with “nuclear polarization + known SM pieces,” not with a new-force direction.
• Once those pieces are included, the generalized King plot reverts to linear (within 1σ), which is exactly what you expect if there is only one major nuclear functional.
• There’s no hint of an extra “mysterious” pattern that would force you to add a QTT-specific parameter or an entirely new QTT interaction at this scale.

So the blog is not claiming “we have solved the nuclear-structure problem in QTT.” It’s claiming something more modest, but still important:

Given what we know about nuclear physics, the calcium results behave exactly the way QTT says they should: a big, real King-plot bend from nuclear polarization, and no extra residue that demands a new low-energy force or a hidden electronic knob.

The next step – and a very natural project for anyone interested – would be to take existing nuclear-structure codes, plug them into the QTT integral, and check just how far we can push that agreement. If there’s a mismatch, that will be scientifically interesting either way: it will pinpoint where QTT needs to be refined, or it will reveal new nuclear physics, or both.