Recommendation: do not allocate another prime-prefix, damping, or contour-identity task. No original-arithmetic estimating lemma is admitted by this review. One unconditional smoothing mechanism was assessed below. It controls remote residues, but leaves precisely the local signed quantity that needs new arithmetic information.
Wall check: Same open gap — ZERO-GEOMETRY. Closest result: PR76, review 08; NS100/101 for structural controls. What changes: an explicit unconditional smoothed-prime formula is tested against its local and remote residue budgets. No new hypothesis estimates the local signed part. NS47 concerns a different all-test comparison and is not a no-go theorem for this pointwise question.
Reviewed public baseline: 18d324703969b5bcc9aa94043cfd90a6e8211f34 (PR76), in the refreshed isolated worktree. Worktree HEAD when inspected was 32f599ced1abb2bf2376a715042e6e4101591639, whose additional commits claim the explanation and this assessment. Read AGENTS, the full registered-node status/note/wall-input projection, the shared-input scopes, current README/checkpoints, NEXT_STEPS frozen groups, MISSING, PR76 derivation and review 08. This is a dependency and feasibility review, not a replay of historical proofs or certificates. Both original-evidence recovery groups remain OPEN.
The relevant prior art is Gonek–Hughes–Keating, arXiv:math/0511182v1, Section 2, Lemma 1. Its exact smoothed logarithmic-derivative formula is unconditional and allows any nonnegative smooth mass-one cutoff supported in [1,e]. It excludes evaluation at zeros and poles. Its proof retains nontrivial zeros, the pole at 1, and trivial zeros; smoothing supplies absolute convergence. The paper’s subsequent moment models and temporary RH interpretation are not imported. In particular, I do not differentiate the error term in its approximate product theorem.
Fix, independently of the cutoff parameter, a nonnegative function
u in C_c^infinity((sqrt(e),e)), integral u(y)dy = 1.
x >= e, L = log x,
v(y) = integral_y^infinity u(z)dz,
A_x(z) = integral u(y) exp(-L*z*log y)dy,
R_x(z) = A_x(z)/z,
S_x(s) = sum_(2<=n<=x) Lambda(n) log(n) n^(-s) v(n^(1/L)).
The arithmetic coefficients are exactly the original von Mangoldt coefficients, with their prime-power support. The cutoff equals one through sqrt(x) and zero from x onward. Locally differentiating the exact formula, rather than an asymptotic remainder, gives
F(s) := (zeta'/zeta)'(s)
= S_x(s) + sum_rho R_x'(s-rho)
- R_x'(s-1) + sum_(m>=1) R_x'(s+2m). (1)
Zeros are counted with multiplicity; all nontrivial zeros are retained. The formula is used at s=1/2+i*r, r real, zeta(s) != 0. Smooth-cutoff decay justifies locally uniform differentiation away from zeros/poles. The pole term has the displayed minus sign: near s=1, it contributes +1/(s-1)^2. Each zero term instead contributes -1/(s-rho)^2. No zero-free strip is assumed.
This is a legitimate representation to which an estimate could be applied. Its existence is already prior art, and is not the proposed advance.
There is a useful unconditional remote-zero bound. Put U>=1, and remove from the zero sum all terms with abs(Im rho-r)<=U. For any fixed integer K>=1, the remaining sum satisfies
sum_(abs(Im rho-r)>U) abs(R_x'(s-rho))
<= C_(u,K) * sqrt(x) * L^(1-K)
* log(2+abs(r)+U) / U^K. (2)
This is an analytic bound with a cutoff-dependent constant determined by fixed derivative norms of u, not a numerical certificate or an optimized constant. To see the scale, change variables q=log y and write g(q)=e^q u(e^q), supported inside (1/2,1). Integrating g and q*g by parts K times yields, for z=s-rho,
abs(A_x(z)) + abs(integral q*g(q)*exp(-L*z*q)dq)
<= C_(u,K) * sqrt(x)/(L*abs(z))^K,
R_x'(z) = -A_x(z)/z^2
- (L/z)*integral q*g(q)*exp(-L*z*q)dq.
Only 0<Re rho<1 is used for sqrt(x). Summing over unit ordinate intervals using the unconditional O(log(2+abs(t))) local zero-count bound gives (2). The trivial-zero terms converge rapidly because the support of u stays above 1. The pole term and gamma completion remain explicit.
Thus remote residues can be made smaller than a previously established positive margin by enlarging U. This is real tail control, but it is not control of the local residues. Increasing x also incurs the horizontal sqrt(x) allowance; it does not remove all residues for free.
There are two precise limitations, neither an impossibility theorem for the original arithmetic problem.
A. An all-height distance-free absolute residue error is already unavailable near known critical-line zeros. If rho=1/2+i*gamma, then as r->gamma, its term in (1) is
R_x'(i*(r-gamma)) = 1/(r-gamma)^2 + O_u(L^2).
Consequently the complete zero contribution cannot be put into a uniformly bounded logarithmic-quotient error across all nonzero points of such a neighborhood. It must retain the pole, or work with the undivided expression and its exact vanishing factors. The positive sign of this particular principal part is helpful, but an absolute-value estimate loses that help.
B. Keeping nearby residues exactly exposes an unestimated signed local term. Set mu_j=integral u(y)(log y)^j dy. Direct Taylor expansion gives
R_x'(z) = -1/z^2 + L^2*mu_2/2 - L^3*mu_3*z/3
+ L^4*mu_4*z^2/8 + ... .
For a hypothetical symmetric pair rho_+=1/2+a+i*gamma, rho_-=1/2-a+i*gamma, at s=1/2+i*gamma, their contribution is therefore
R_x'(-a)+R_x'(a) = -2/a^2 + L^2*mu_2 + O_u(L^4*a^2),
for 0<a*L<=1/2. (3)
It is negative and of inverse-square size when a*L is small. Ordinate counting, multiplicity bounds and smoothness provide no lower horizontal separation a and no compensating positive local arithmetic term. Zero-density estimates that allow even one such exceptional pair do not repair that pointwise defect by counting it sparsely. Taking L comparable to or larger than 1/a is not a supplied schedule: the location is unknown, the signed terms change, and the remote/error budgets also change.
Equation (3) is an algebraic stress test of a residue estimate using only zero geometry/counting. It is not an assertion that zeta has this pair, a matched counterexample to the original Euler product, or a claim that the other terms cannot compensate it. Precisely that compensation is missing. No new prime-specific cancellation argument was found that bounds the coupled local zero sum and S_x from below.
Retain the completion
C(s) = -1/s^2 - 1/(s-1)^2 + psi1(s/2)/4,
E(r) = Re(C(1/2+i*r)+F(1/2+i*r)).
PR76 gives, away from zeros,
J(r) = p(r)^2 * Y(r)^2 * E(r), p(r)=r^2+1/4.
If E_local denotes (1) with the remote nontrivial-zero terms removed, then (2) controls abs(E-E_local). A sign proof would require a signed lower bound for this actual local expression exceeding that error at every real nonzero point, using the exact prime coefficients. Neither the original Euler product nor (2) supplies it. There is no established positive margin to choose U against. A bare o(1) remainder would not suffice in the absence of such a margin; even an O((1+abs(r))^-2) remainder needs a compatible signed lower estimate.
At zeros use PR76’s undivided J=p^2*Y'^2>=0, without assuming simplicity. Multiplying a quotient estimate by p^2*Y^2 cancels its poles but does not manufacture the missing sign between zeros. No additional implication from the first-Laguerre target alone to RH is claimed.
Budget and falsifiable outcome: a 1–2 hour paper-only feasibility slot is sufficient for this proposed smoothing mechanism; it should stop at the local-residue decision above. Success would require an independent, original-coefficient lower estimate controlling the signed local prime-plus-residue combination with a compatible error. Failure means do not commission larger prime tables, a damping scan, or another identity audit in this lane. It does not close all Euler-product methods, and this note does not propose an adjacent regularization as the next task.
Final classification: Same open gap. The available mechanism estimates the distant error but leaves the local signed arithmetic input untouched. No research row, thaw, theorem node, certificate, manuscript version, or RH/G2 advance follows.