21 SEPTEMBER 2026 · NS-32 · REVIEW v3

PR #14: revised, still held

Reviewed head f3611ef2c3464fa30e50408c62e339697def6c38, including the new concentration-function bound. Main at 06746f2 is also read. Reviews v1/v2 of the original head are preserved; this report supersedes their merge decision for the revised head. Three MAJOR findings remain. Do not merge or tag v1.48.

Actual revised-head build: 239 pages; 0 undefined references; 0 duplicate references; 0 overfull boxes. All four v1.48 manifest artifacts verify. The previous head built to 238 pages. No numerical experiment was run.

1. Proof and constants

NOTE — gap count, Borel domination and uniform constant pass. The forced level −D and at most K negative values give at most K gaps; QC is nonincreasing in K. The two defined measures are finite positive Borel measures, so setwise domination integrates nonnegative Borel functions. State that explicitly for clarity. Under an additional finite density cap R, packing gives QC_K ≥ φ²/(2KR). Absolute continuity from v1.47 alone does not supply that cap. Uniform density gives QC_K = φD/(2K) exactly, hence c=c′φ.

NOTE — original issues partly resolved. The revision drops the false “exactly when” wording and recognizes that trough critical values make the exact density unbounded. Its exact weighted-cost implication remains correct. QC₁ growth plus a separate lower shape bound K QC_K ≥ κQC₁ is a valid sufficient cofinal input; it is not yet a theorem for this symbol.

MINOR m1 — small-mass gap in the block proof. At lines 103–105, n=⌊m/M⌋ does not imply n(n−1) ≥ (m/M−1)(m/M−2) when 0<m/M<1. The claimed final inequality is nevertheless true (its right side is then negative). Repair by a case split, or retain the last partially filled block: for m/M=n+θ, the exact relaxed cost is M[n(n−1)/2+nθ] ≥ m²/(2M)−m/2. This even yields the stronger correction −hφ/2 in place of −3hφ/2. Also state φ>0 where M(h) is divided by, and handle zero denominators separately.

2. Current MAJOR findings

M1 — both new displayed K bounds have the wrong inference. At lines 62–65, division by sup_j j QC_j/QC₁ cannot provide a lower bound for QC_K. The needed quantity is a lower shape constant: κ=inf_j j QC_j/QC₁, and the conclusion is K ≥ c₀κQC₁/(C+Q_*). The v1.49 full-support countermodel has QC₁∼D/2, QC₂→0, but that supremum is at most (φD)/QC₁→1; the proposed bound would incorrectly force K→∞ from one-level growth.

For the concentration bound write A=hφ²/(2M(h)), B=3hφ/2, E=C+Q_*. The correct consequence of c₀(A/K−B)≤E is K ≥ c₀A/(E+c₀B), not c₀[A−B]₊/E. The two sufficient divergence conditions stated afterward are still valid when proved using the correct denominator. Remove the erroneous displayed claims.

M2 — the λ=4 weighted optimizer still fails feasibility. The saved c′=1 vector has minimum bin ratio 0.636<1, so its tiny energy is not an upper bound on the constrained minimum. The revised text still calls it a “minimum energy” and says the actual states satisfy the weighted hypothesis. This is contradicted by its archived output, not merely an interval-certification caveat. Even feasible twenty-bin states are not all-Borel witnesses and were not source-projected. The raw dual value alone establishes no feasible cheap state. Correct the manuscript status, diagnostic paragraph, evidence README, NS-30, map and both logs accordingly.

M3 — a largest aligned bin is not M(h). The new table and diagnostic paragraph identify the largest of twenty fixed-bin masses with the supremum over all translated intervals of length h. That maximum is only a lower bound for M(h), whereas the quantization lower bound requires an upper bound for M(h). For an exact aligned histogram on [−D,0], every interval of length h meets at most two bins, giving the safe bound M(h)≤2 max-bin-mass; midpoint frequency sampling would still need an error bound for a certificate. Relabel the saved numbers as aligned-bin maxima and do not substitute them as the true concentration function.

3. Remaining qualifications and sweep

MINOR m2 — critical order. Inverse-square-root blowup requires a nondegenerate extremum. At order r≥2, the correct exponent is 1/r−1. State the presence of a negative critical point in the selected range; arbitrary X need not contain a trough. A complete negative component inside the range suffices. This is a local analytic result, not a new certified trough-location computation.

MINOR m3 — diagnostic quantities. X=800 is independent of the prescribed X=2πN/L; D is sampled; the dynamic program optimizes a 200-bin midpoint measure with reconstruction levels on bin edges. These are not exact continuum costs. Values were computed only for K=1,2,4,8,16,32; hundreds/thousands are extrapolations. Four windows do not imply cofinal Q→0. The new density correction in main is preserved, but these remaining claims still need correction.

MINOR m4 — version and PR description. The PR body still describes the obsolete sup-density scalar. Its root README still presents v1.47 at the top. Rewrite the body around the final valid result and update the version only when ready. The status, evidence README, board, map and both logs now use the better exact-cost/concentration framing, but retain the infeasible-state and extrapolation overstatements.

The original scalar conceals an upper bound on every level band. ZLD gives lower band bounds; neither implies the proposed scalar divergence in general. Measure countermodels are not logical-independence results about zeta. The corrected zero field’s exact-cost growth and admissible packet coverage remain separate open inputs.

4. Source projection: no factor guarantee

For a raw unit LP vector f, set r = |⟨f,u_a⟩|, ε = ‖W u_a‖ and g = P_af/√(1−r²). With E = q[f] and δ = (2r+r²)ε,

(E−δ)/(1−r²) ≤ q[g] ≤ (E+δ)/(1−r²).

Neither a certified source nor small ε bounds 1/(1−r²) for this LP vector. A factor-two conclusion would follow, for example, from r² ≤ ¼ and δ ≤ E/2 with E certified positive. Those checks are absent. Fourier mass also changes by the pointwise subtraction in eq:v147-projected-density, so projection need not preserve WLH. The λ=3 source certificate concerns a different vector and cutoff; it does not certify this overlap.

Named missing computation, not run: recover/export the λ=3, c′=1 LP minimizer, match its centered N=256 coefficients to the exact normalized source, enclose r and its source-tail error, certify its complete q[f] and the cross/source energies (or ε), and check the projected weighted constraints. A two-precision interval replay is needed for a certified claim. Passing 20 bins would still leave the all-Borel requirement open. No new optimization, window or projection computation was run.

Disposition: leave PR #14 open and unmerged; no v1.48 tag. This is failure of stated deductions and diagnostic interpretations, not a refutation of the valid weighted-cost inequality or the physical form. Task B continues in v1.49 with exact cost growth, an explicit additional shape hypothesis where needed, and the source constraint retained. G2 and RH remain open.