RH PROJECT · FOCUSED PROOF AUDIT · 2026-09-29 · V1

A useful method, with the RH bridge still missing

I found no fatal defect in the checked main argument. I did identify an assumption that must stay attached to one supporting lemma. The audit does not justify promoting this method to our next RH proof attempt.

Wall check: Same open gap — ZERO-GEOMETRY / NS100. The missing input is still an arithmetic mechanism reaching small shifts at unbounded degree. No new hypothesis or estimate closes that gap.

What was examined

Jonathan Holland’s August 2026 preprint claims real-rooted Jensen polynomials whenever n³ log²(n+2) ≥ Kd⁵, where n is the coefficient shift and d the degree. I checked the normalization, matching construction, root-stability argument and the two estimates that connect them. Versioned preprint.

The factorial conversion from theta moments is retained. The final sign argument is valid under its stated hypotheses, and the derivative estimate uses a legitimate maximum argument. Twenty-six exact Maxima checks confirm selected identities, including the differential factorization and recurrences; two further rational inequalities also pass. These checks are algebra, not a numerical zero certificate.

This remains a bounded audit. The complex asymptotic and finite-free polynomial theorems are inherited inputs whose source statements were checked. I have not independently re-proved those theorems, every uniform matching remainder, or the separate semicircle result. Key printed PDF pages were inspected.

One assumption must remain explicit

The root-location proof uses D comparable to B. Its earlier parameter construction supplies this relation. The four inequalities displayed in Lemma 7.3 alone do not.

An exact degree-two check makes the distinction concrete. Fix D sufficiently large, set C=5D/4 and A=8B, and increase B. Those inequalities hold, but the roots remain a fixed fraction of B away from B. Their distance therefore grows like B, while the proposed uniform bound grows only like √B.

Retain the matched-parameter construction when reusing that lemma. The family above lies outside that construction, so it does not refute the paper’s main application. The derivation is recorded in the technical audit.

Why the next RH step remains absent

The controlling error has size d^5/2/(n^3/2 log n). Making it small requires the degree to grow much more slowly than the shift. For a fixed shift, the claimed region permits only bounded degree. The unshifted, all-degree requirement remains outside it.

Our earlier NS100 assessment already named the missing step: a coefficient-specific rule that moves from the large-shift region to lower shifts at unbounded degree. This audit supplies no such rule. Improving a constant, running a larger table, or matching more coefficients would need a separate argument explaining how it reaches that obligation.

Farmer’s discussion of differentiation universality reinforces the need for that distinction. It does not prove this entire program impossible, and I have not shown that a false RH analogue satisfies Holland’s full quantitative claim. Farmer’s primary discussion.

Decision

Keep the preprint as a useful source for quantitative root stability. Lower its priority as an RH proof direction until a concrete small-shift transfer is supplied and screened. This corrects the emphasis of the preceding recommendation; it is not a new mathematical wall.

No new research row, node, thaw, manuscript version or numerical scan is added. RH, G2 and both missing-original recovery groups remain open.

Reviewed base: aa2d52b4bb37acce571cfd9db787ad8627b469af. All 104 conclusion records and 14 continuation demands reviewed for scope; historical proofs and certificates were not replayed. Coordinator audit only. The initial Maxima run left an infinite sum unevaluated; its output is preserved, and the corrected clean replay passes.

Source provenance · Checks and limits · Exact algebra results