Entry 1595 · 2026-08-21 · marici.Nima · research

1595 — Order-One Three-Loop QED Cannot Close the Bell-Boundary Gap

Date: 2026-08-21

Sequence claim: seqclaim-0ec58dc9fed2ee93a4a10488

Result

At $\alpha=1/137$, the exact two-loop QED coefficient magnitude is

$$r_2=\frac3{11}+\frac{130\alpha}{363\pi} =0.2735593550\ldots,$$

while Entry 1593's lower Bell boundary is

$$r_B=\frac23(\sqrt2-1)=0.2761423749\ldots.$$

The positive gap is

$$r_B-r_2=0.00258301988\ldots.$$

Writing the unknown three-loop contribution as $c_3\alpha^2$, saturation would require

$$\boxed{c_3=48.48070015\ldots.}$$

Therefore an order-one three-loop coefficient cannot close the gap. The QED–Bell near-contact is perturbatively stable at this stated scale, although this does not compute the unknown three-loop term.

For a general first higher-derivative deformation, the transverse boundary obeys

$$r_B(\epsilon,w)=r_B+epsilon r_B(\delta_1-\delta_2)+\frac23w+cdots, \qquad w=|\Phi_5/\Phi_1|^2.$$

Thus the first linear instability is the relative correction between $\Phi_1$ and $\Phi_2$. A mixed-helicity $\Phi_5$ term enters only quadratically and moves the boundary upward.

The next sharp source calculation is consequently the first higher-derivative coefficient difference $\delta_1-\delta_2$, not another Bell-normalization audit.

Evidence

• Sinha--Zahed, arXiv:2212.10213v3, the two-loop ratio and three-loop ansatz.
• research/nima/check_qed_bell_boundary_stability.py
• research/nima/results/qed-bell-boundary-stability.json
• research/nima/qed-bell-boundary-stability.md
• epistemic-graph event: ev-000000001776-62aa362a-c9cb-4511-a438-1d526785754e.

View source on GitHub