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s>3 and the Newton–Toeplitz average — residue

Hundertmark–Pattakos–Schulz, arXiv:2504.18487v1. Opened this session: the abs and the HTML, Theorem 2.2, Remark 2.3, Lemma 4.3, Proposition 4.5, Lemma 5.9, and the proof of Theorem 2.2 in §7. Replay: python3 toeplitz_probe.py (writes toeplitz_probe.json). No face enumeration, no verify_beta3.c.

Status: residue. No path to b(4) in Theorem 2.2. No global γ > fmin on every radial probability. The compact-aspect number 0.901924 (leading 1.1087 on aspect ≤ 4) is not used and is not unrestricted.

A. s>3: the two-shell sign, and nothing else feeds b(4)

q2 already had the closed quadratic. Independently:

Q_s(α, β; t, ℓ) = α² t^{s−1} + β² + αβ (t^{s+ℓ} + t^ℓ).

At s=4, ℓ=1, t=1/8, α=16, β=−1 this is the dyadic rational

16² · (1/8)³ + 1 − 16 · ((1/8)⁵ + 1/8)
  = 256/512 + 1 − 16/32768 − 2
  = −1/2 − 1/2048
  = −1025/2048 < 0.

The same opposite, inner-heavy dipole is negative at s=7/2 (1/√2 − 1 − 1/(65536√2) < 0) and at s=3.1. At s=3 the discriminant is the identity

(t⁴ + t)² − 4 t² = t² (t³ − 1)(t³ + 3) ≤ 0    on (0, 1],

so the dipole form stays nonnegative. One-shell, same-sign, and quadrupole (ℓ=2, Hardy threshold 5) stay positive on (3, 4]. Interval enclosures sit in compute/q2/certs/s_gt_3.json; this folder only replays the algebra.

That kills Lemma 4.3 for every s>3: there exist measures orthogonal to radial functions with I_s(ν) < 0, so I_s(ρ) need not dominate I_s(ρ̄).

Does any other HPS path justify b(4) ≈ 1.083 in Theorem 2.2?

No. The printed chain is short, and the missing piece is only β_4 ≥ b(4)^{-1}.

Theorem 2.2 is N_c < b(s) Z + c(s) Z^{1/3} for s ∈ (1, 3]. The §7 proof solves the Lemma 5.9 comparison (5.50) for N and substitutes the Proposition 4.5 floor β_s ≥ b(s)^{-1}. Lemma 5.9 already runs for 2 ≤ s ≤ 4. It compares α_{N,s} to β_s. It does not produce a lower bound on β_s. On (3, 4] it has nothing to feed into unless someone else supplies β_s ≥ b(s)^{-1}.

Proposition 4.5 without Lemma 4.3 still gives β_s^{rad} ≥ b(s)^{-1} for every s>1, by Newton and the elementary minimum of (1+t^s)/(1+t^{s−1}). Theorem 2.2 consumes the full β_s (infimum over all probabilities in D_s), not the radial infimum. Lemma 5.9 is written with that full β_s. Substituting β_4^{rad} would be a different lemma.

The other candidates in the paper, or next to it, do not close the gap:

No other comparison in HPS (Newton for non-radial measures, multipole remainder, Lemma 5.5, the kinetic remainder in §6) replaces β_4 ≥ b(4)^{-1}. A new pair estimate that never passes through β_4 would be a different paper. None is certified here. b(4) ≈ 1.0830249 is smaller than 1.1185 and cannot be used in Theorem 2.2.

B. After Newton, Q is an average of f — no cheaper floor

For a radial probability, Newton gives

g(r,u) = (r³ + u³) / (2 max(r,u)) = f(t) (r² + u²) / 2,
t = min(r,u) / max(r,u),    f(t) = (1 + t³) / (1 + t²).

So Q = I/D is a weighted average of f. In particular Q ≥ fmin = 1/b(3) ≈ 0.89410745697, which is Proposition 4.5 at s=3. Figure 2 already says this is not sharp: the HPS power-law trial sits near 0.9207. A global γ > fmin would move the leading coefficient below b(3). Three cheap attempts do not produce one.

Two-atom exact minimum

Two radii, ratio t ∈ (0,1], D-mass w on the inner atom. The quotient is

Q(w,t) = [w² + (1−w)² t² + w(1−w)(1+t³)] / [w + (1−w)t²].

For fixed t the unique interior critical point is w = t/(1+t), and

Q2(t) = (1 + 2t + t³) / (1 + 2t + t²).

Q2'(t) = 0 on (0,1) at the positive root of t² + 3t − 2 = 0,

t* = (−3 + √17) / 2 ≈ 0.5615528128,
Q2* = (17√17 − 55) / 16 ≈ 0.9432997272,
1/Q2* = (55 + 17√17) / 118 ≈ 1.0601084376.

That coefficient is below 1.1185. It is a lower bound only for 2-atomic measures. q2's 3-atomic search already has I/D ≈ 0.931 < Q2*. The two-atom floor is not global.

At the HPS ratio t0 one has the identity t0³ = 2 − 3 t0, so

Q2(t0) = (3 − t0) / (1 + t0)² ≈ 0.94366,

slightly above Q2*. The pairs that saturate f = fmin are not the two-atom minimizer.

Infinite t0-chain

Equal m-mass on r_k = t0^{-k}, k = 0, …, n−1. With u = t0^{-2} the energy is a finite geometric sum,

D = (1/n) (u^n − 1)/(u − 1),
I = (1/n²) Σ_d f(t0^{|d|}) · ((1 + u^d)/2) · (u^{n−|d|} − 1)/(u − 1).

The n=3 line collapses to

Q3 = (1/3) [ 1 + fmin (1+u)²/(1+u+u²) + f(t0²) (1+u²)/(1+u+u²) ]
   ≈ 0.9379263180.

That is the minimum of the family: Q1 = 1, Q2 ≈ 0.94705, Q3 ≈ 0.93793, then Q_n increases (Q24 ≈ 0.98823). Equal D-mass is the inversion of equal m-mass on a geometric set and gives the same Q_n.

The n → ∞ limit is 1, not fmin. Write Q = 1 − (1/n) Δ / Σ u^k with Δ = Σ_{i,j} (1 − f_{|i−j|}) (u^i + u^j)/2. Both Δ and Σ u^k are O(u^n), so the deficit is O(1/n). A long t0-chain does not put almost all pair-weight at t0: either n is small and the t=1 self-pairs stay large, or n is large and the 1/n masses kill the deficit.

Q3 ≈ 0.9379 is a trial, hence an upper bound on β_3^{rad}. It sits above the HPS power-law 0.9207. It is not a lower bound, and it does not beat 1.1185 as a global coefficient.

1D convolution on log-radius

Put x = log r. Then

I = ∬ f(e^{−|x−y|}) (e^{2x} + e^{2y})/2 ρ(dx) ρ(dy),
D = ∫ e^{2x} ρ(dx),

for a probability ρ on the line. The translation-invariant kernel K(h) = f(e^{−|h|}) satisfies K ≥ fmin, with equality only at |h| = −log t0. Any average of K is therefore ≥ fmin. That is again Proposition 4.5. There is no Bochner gap: K(h) − γ is not a positive-definite bump for any γ > fmin, because K(∞) = 1 and K attains fmin at two finite points.

Dropping positivity makes it worse. The truncated Toeplitz symbol of K on a 16-point t0-step grid, evaluated at θ = π, is ≈ 0.11187 < fmin. Signed log-densities are not a path to a higher floor.

Positive trials on at most 16 log-nodes (the cap here):

None of these is a lower bound. None of them raises the global floor above fmin. A number that does not beat 1.1185 globally is not a dent of the printed leading coefficient.

The leftover handle is unchanged: a proof that a minimizer has bounded aspect, or any other argument that gives Q ≥ γ > fmin for every radial probability. The compact-class certificate Q ≥ 0.901924 on aspect ≤ 4 is not that argument.

What is proved vs still open

Proved here: the q2 two-shell sign at s=4 is the exact rational −1025/2048; Lemma 4.3 does not extend; Lemma 5.9, Newton-on-radial, pointwise |x−y| ≤ 2 max, and signed convolution do not justify feeding b(4) into Theorem 2.2; the two-atom min is (17√17 − 55)/16 and is not global; a t0-chain has closed form Q_n → 1 with minimum Q3 ≈ 0.9379 and is a trial; the log-radius kernel recovers fmin as a global floor and nothing strictly above it.

Not proved: β_s ≥ b(s)^{-1} for any s>3; β_s < b(s)^{-1}; a global γ > fmin for β_3^{rad}. Not a dent of 1.1185. Remark 2.3 remains a conjecture. The compact-class 1.1087 is not unrestricted.