P_max error replacements (q4 work)

Context: compact certificate at R=12, n=22 gives φ_target=0.9055, P=(q−1)/(q+1)≈0.0564, err=P(1−fmin)≈0.00597, γ=0.899526, 1/γ≈1.1118. Reweighting step: I/D = Σ_i s_i w_i λ_i / Σ_i s_i w_i with w_i=c_i μ_i, s_i∈[1/q,q], λ_i=Σ_j F_{ij}μ_j∈[fmin,1], φ(μ) the s≡1 value.

Not a dent. No h(D_L,D_R) tail.

1. Replace (1−fmin) by λ_max or a pair-aware amplitude?

1a. Uniform P·(λ_max − fmin) with λ_max<1

No as a global replacement for P(1−fmin).

• For any bin i, F_{ii}=1 (same-bin pair hits t=1), so a vertex μ=e_i has λ_i=1. Hence max_i λ_i(μ)=1 on a set of measures that still matters for the certified φ inf.
• At those vertices the reweighting drop is zero (only one active w_i), so the error term is slack, but a uniform certificate must use amplitude 1−fmin, not λ_max−fmin<1−fmin, if it is to hold for every μ without a state-dependent bookkeeping.

1b. Per-μ spread P·(max_i λ_i − min_i λ_i)

Yes (conditional) — plausible valid tightening; no counterexample found.

Lemma (target). Fix μ. With λ_i=(Fμ)_i, w_i=c_iμ_i, s_i∈[1/q,q]:

Proof idea. Q(s) is a weighted mean of the numbers λ_i with weights t_i=s_i w_i; varying s only scales each weight by a factor in [1/q,q]. For equal w_i, the sharp two-level adversary gives drop =P·(\max λ−\min λ) (the classical (κ−1)/(κ+1) bound with κ=q). Unequal w_i can decrease the drop (e.g. two-bin equal w: drop =0 while spread >0), never increase it above P·spread in exhaustive n≤8 vertex searches (max ratio ≈0.9996 at n=3, still ≤1).

Numerics at R=12,n=22 (pmax_try.py, reweight_gap.py):

μ* is the SLSQP minimizer of φ; spread bound beats λ_max−fmin because min_i λ_i > fmin there.

Pair-aware without adversarial independence of s and λ: Yes, this is exactly the spread bound. Correlation helps: large w_i/w_j kills reweighting freedom (drop →0), while large spread requires spread μ where the effective weights on competing bins are moderate. The adversarial two-valued s construction is sharp for the amplitude but pessimistic for the compact grid (positive slack in reweight_gap.json: ≈0.0067 at R=12).

Counterexample to spread bound: not found. Saturating configs exist (n=3, ratio drop/(P·spread)≈1), so spread is nearly sharp, not loose.

Caveat for certification: faces certify φ(μ)≥φ_target uniformly; replacing the error requires a uniform proof of the spread lemma with varying w_i=c_iμ_i, not only grid search. Until that is written, treat P·spread(μ*) as numerical residue, not a new certificate.

2. F_hi on diagonal / nearby, F_lo on far bins?

No — cannot lower-bound I with F_hi on any pair.

• Valid kernel entries must satisfy K_{ij} ≤ min_{(r,u)\in bin_i×bin_j} f(r/u). F_hi[i,j] is an upper endpoint of the interval enclosure; using it when F_hi>F_lo would inflate the quadratic lower bound and is unsound.
• On the certified mesh R=12,n=22 (and R=4,n=18): F_hi=F_lo pointwise on every pair (max(F_hi−F_lo)=0). A near/far split collapses to the same matrix already used in A_lo.
• Conceptually, near pairs already carry the tight per-pair minima in F_lo; the loss in P(1−fmin) is from reweighting s, not from slack in diagonal F_{ii} (already 1).

Scripts

• pmax_try.py — spread / λ_max savings at μ*, mixed-F sanity, random counterexample scan (no violation with ratio >1 in tested regimes).

Bottom line