riemann

v1.49 — PR #14 audit and the exact weighted zero input

Exact identities, proved implications and countermodels, plus named open inputs. No new numerical experiment or certificate. G2 and RH remain open.

Task A is NS-32 after a concurrent task-ID reconciliation. PR #14 was re-reviewed at f3611ef after its density correction. Three MAJOR findings remain: two invalid K rearrangements; a lambda=4 c’=1 state failing its bin constraint (0.636<1); aligned-bin maxima identified with the full concentration function. Its core weighted-cost implication, monotonicity, Borel domination and uniform-density constant pass. Revised head builds to 239 pages, 0 undefined/duplicate references; all four artifacts verify. The current audit is audits/2026-09-21-v1.48-pr14-v3.html; prior versions are preserved. PR #14 remains unmerged and untagged.

Task B is NS-33. The v1.49 manuscript is based on v1.47 main and is self-contained: it does not merge the rejected v1.48 wording. Its findings:

• The exact normalized level density is X^-1 times the sum of inverse slopes of the complete corrected zero field at every level preimage. A negative critical value produces an integrable but unbounded density; the continuous peak is infinite whenever the stated geometric condition holds. The finite histogram’s maximum near zero is not that peak.
• If the peak R is finite, phi^2/R diverges exactly when R=o(phi^2). Since phi<=1, R must tend to zero. Bounded above alone does not prevent divergence; uniform density phi/D is an example. The critical-level singularity, not bounded phi by itself, makes the crude bound vacuous.
• ZLD’s lower band bound and a peak-density upper bound do not imply each other for general level measures. No arithmetic independence is claimed.
• The exact weighted bound eta >= [c’ QC_K - Q]+ requires no bounded density. The density packing estimate is retained conditionally, and uniform density recovers c=c’phi exactly. A density-free concentration bound is proved with correction -hphi/2 and the proper K denominator.
• QC_1 = phi*D - mean(beta^-) is the averaged gap above the deepest negative value. Its growth alone is insufficient: an absolutely continuous, full-support countermodel has QC_1~D/2 but QC_2->0.
• The named input Exact Level Quantization Growth (QG) requires QC_K to diverge for every fixed K on a prescribed cofinal head-size family. It also requires separate admissible bounded-energy packets satisfying all-Borel weighted domination. A QC_1/K bound is a further shape hypothesis, not a consequence of one-level growth.

Neither a positive limit nor decay of phi follows from the cited depth corollary. No cofinal input is proved. The route remains blocked; the physical form and the valid weighted concentration criterion are not refuted.

No source-projection calculation was run. The audit names the missing lambda=3 overlap, complete-energy/cross-term and projected-constraint checks; a small source residual alone gives no constant-factor guarantee. NS-28 raw scripts and outputs are unchanged; its report gets a correction notice, preserving the current main correction and text below it.

Incremental record

• weighted_zero_input.tex: self-contained manuscript insert.
• scope_review.md: internal mathematical and claim audit.
• provenance.json: reviewed PR head, archived source/input hashes.
• build_report.json: actual final v1.49 build and reference counts.
• pr14_build_report.json: actual independent PR #14 build.
• research-report-2026-09-21-v1.html: user-facing report for both tasks.

The two NS-1 evidence groups remain OPEN. No new window, head run, tail metric, optimization or zero enclosure. No v1.48 merge/tag.

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