β_3^{rad} — residue, not a dent of 1.1185

Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1, Proposition 4.5: β_3 ≥ min_t f(t) = 1/b(3) ≈ 0.894107, with f(t)=(1+t^3)/(1+t^2) and b(3)<1.1185. Figure 2 says this is not sharp: power-law trials sit about 3% below b(3). A certified γ > 1/b(3) that holds for every radial probability in (4.1) would be a new leading coefficient in the same §5–§7 chain.

Status: residue. The printed leading coefficient 1.1185 is unchanged.

Numerical target (upper bounds on β_3)

Replay: explore_beta3.py. Wrong direction for an ionization upper bound.

• Power law m(dr) ∝ r^α dr on [1,n], HPS 3D density A|x|^{-p} with α=2-p. Best at α≈−2, n≈3.50, p≈4: I/D ≈ 0.920655, so β_3^{-1} ≈ 1.08618 (2.88% below b(3)=1.11843). Matches Figure 2.
• k-atomic: 1.000, 0.9433, 0.9310, 0.9265, 0.9244, 0.9232 for k=1…6, decreasing toward the power law.
• Piecewise-constant log-grid: I/D ≈ 0.921.

Apparent inf of β_3 ≈ 0.921. Apparent inf of β_3^{-1} ≈ 1.086.

Withdrawn 1.1168

An earlier certificate claimed β_3 ≥ 0.895396, hence β_3^{-1} ≤ 1.11682 < 1.1185, by lifting a middle-window bound with the tail polynomial h(D_L, D_R). That lift is false:

• h(0,1) ≈ 0.991, but the HPS power-law trial has I/D ≈ 0.921 when placed entirely in that “tail”.
• I_CC ≥ β D_C is false; the correct sub-measure bound is I_CC ≥ β D_C M_C.

certs/beta3_rad.json is marked withdrawn. Do not cite 1.1168.

Compact class (correct, not a Theorem 2.2 dent)

On D-measures of aspect ≤ R, after scaling inf(supp z)=1 so supp z ⊆ [1,R], the mid-radius Rayleigh with F_ij = min f on each bin-pair, minus the reweighting error P_max(1−fmin) with P_max = (q−1)/(q+1), q=R^{1/n}, is a valid lower bound.

Face enumeration (verify_beta3.c) at R=4, n=18, φ_target=0.906, 0 singular faces, gives

Q ≥ 0.901924    on aspect ≤ 4,
1/Q ≤ 1.108741.

That would beat 1.1185 if it held for every radial probability. It does not: HPS β_3 is an inf over all μ in D_s. Replay: certs/beta3_compact.json. Independent n=16 rebuild: verify_beta3.rs.

A two-window mixture Q ≥ γ_R − p12(γ_R−fmin) with cross terms at fmin (lift_global.py) has grid max p12 ≈ 0.995, so the lift collapses to fmin. Adjacent-window pairs are the t≈t0 pairs.

Aspect / first variation (search, not a bound)

aspect_try.py: geometric t0-chains stay at Q ≥ 0.9379; 200 random atomic trials stayed above 0.930; the power-law n=3.5 trial is Q ≈ 0.9207. No explicit measure in that scan dips below the compact γ. Truncation of those trials to aspect 4 did not increase Q. That is evidence, not a proof that every measure has aspect ≤ 4, and not a lift.

The leftover handle is a proof that a minimizer has bounded aspect, or any other global lower bound strictly above min f.

s>3

b(4)≈1.083 would be a real jump if Lemma 4.3 extended. Two-shell dipoles make I_s(ν) negative for every tested s>3. Closed form at s=4: Q=−1025/2048. See work/s_gt_3_notes.md.