RH PROJECT · SPECIALIST FEASIBILITY BRIEF · 2026-09-29 · V1

The actual residual needs an arithmetic estimate

Three agents examined the exact target, relevant literature and research priorities. This assessment produced no new unconditional partial lemma with a credible proof input. It does produce a precise question for specialist review.

Wall check: Same open gap — NB-GAIN, NS53/61/87. The full optimized residual is already the subject of NS87. Returning to it changes our priority, not the mathematical input. NS74 and NS83 remain binding; no frozen group is reopened.

1. Fix the original approximation problem

Use the fixed q=2 space H=L²((0,∞), dt/t²), with target τ(t)=1_t≥1 log t and integer atoms

a_n(t)=A(t/n), A(z)=∫₀^z {u} du/u.

Let Π_N project onto the span of a₁,…,a_N. Define the actual residual r_N=(I−Π_N)τ and squared error E_N=‖r_N‖². Keep the target, integer span, norm and smoothing order fixed. The existing criterion gives E_N→0 if and only if RH; no optimal convergence rate is required.

For N<n≤2N, set φ_n=(I−Π_N)a_n, H_N=(⟨φ_i,φ_j⟩), and h_N=(⟨r_N,φ_n⟩). The complete block identity is

Δ_N=E_N−E_2N=h_N^TH_N⁻¹h_N.

The full inverse and old-space projection matter. This target permits every new-block direction; it is not restricted to the prescribed logarithmic Möbius direction. Estimates for that particular direction remain sufficient possibilities with their separate numerator and cost obligations.

2. What is controlled, and what is missing

NS87 represents the correlations by finite signed arithmetic observations. For a physical cutoff T, put P_T(t)=∫_t^T log(u/t)r_N(u) du/u² and

q_n(T)=(1/n)∫₀^TP_T(t)dt − ∑_1≤k≤T/nP_T(kn).

Write G_N for the complete old Gram and C_N for its old/new cross block. The corrected observation vector is

ĥ_N,T=q_new(T)−C_N^TG_N⁻¹q_old(T).

Truncation destroys old-space orthogonality, so q_old cannot be discarded. The residual itself also contains the optimized arithmetic coefficients. This is the first unestimated signed vector.

At T=N⁶, define 𝒜_N=ĥ_N,T^TH_N⁻¹ĥ_N,T. The existing complete-tail bound gives

|𝒜_N/E_N−Δ_N/E_N| = O(N^−1/2 log²N).

This error is summable on N=2^j. Divergence of the sum of normalized observations is therefore equivalent to convergence of the actual errors to zero, through the exact error product. It remains an RH-equivalent target, not a weaker intermediate lemma. The familiar positive lower limit of j𝒜_2^j/E_2^j is one stronger sufficient condition. Neither arithmetic assertion is proved here. The asymptotic tail rate has no effective onset in the inherited argument. This is bookkeeping from existing bounds, not a new convergence theorem.

NS83 excludes eventual fixed-fraction whole-error contraction and eventual Δ_2^j/E_2^j≥a/j with a>1. NS74 shows that identical Gram geometry and impressive finite gain capture can coexist with a positive error floor in an altered family. A successful argument must use the original target correlations, beyond generic projection geometry.

3. What the checked literature supplies

Source	Transfer assessment
Bettin–Conrey–Farmer, Theorem 1	Full-line approximation, but assumes RH and a reciprocal-derivative moment hypothesis that also requires simple zeros. Existing smoothing does not remove these assumptions.
Short Mollifiers, Theorem 1	The stated theorem gives an unconditional positive zero proportion for sufficiently small fixed θ. Its differentiated target and moving-height moment do not supply the original full-line residual bound.
Bettin–Gonek, Theorems 1–2	Even coarse uniform long-mollifier bounds imply zero-free regions. A warning about the strength of the proposed input, not an impossibility theorem.

This was a bounded three-source applicability assessment. It supplies no proof that an adaptation is impossible and no global ranking of RH approaches. Physical cutoff T=N⁶ above must not be confused with the height parameter in these moment theorems.

4. The question for a specialist

Can an established arithmetic method estimate a precisely identified signed part of ĥ_N,T for the actual optimized coefficients, with uniform errors still useful after applying H_N⁻¹? If additional coefficient regularity is required, what exact unconditional statement would supply it?

A useful answer would name the expression, coefficient hypotheses, parameter ranges, estimating method and remaining shortfall. A smaller theorem may have standalone value without proving RH. An unspecified request for “more cancellation,” a generic solver improvement or another equivalent identity does not yet constitute such a lemma.

The complete dependency is present. A new estimating mechanism is absent from this assessment. That is the point at which the next research decision should be made.

5. Decision and review limits

Use this brief for a specialist feasibility assessment before another proof-oriented scan. No external message has been sent. If no concrete arithmetic statement emerges, leave the research groups frozen and stop automatic generation of adjacent variants.

The preference for full NB is organizational: its objects and missing estimate are explicit. It does not establish superiority over the signed Weil-floor problem. Arithmetic geometry remains a separate construction program requiring an actual pairing or operator and a proved transfer; the present team supplies neither.

Three agents provided separate reconstruction, literature and skeptical-strategy assessments, with communication between them. This is not external expert review, full independent verification of the repository, or numerical replication. No candidate screen was represented as passed; a later concrete candidate must meet the existing proposal and control gate.

Reviewed main: a992456a6305b0b01b0d6efe2b022c73d1c0cd3b. All 104 conclusion records and 14 continuation demands reviewed for scope. No new row, theorem node, thaw or manuscript version. RH, G2 and both missing-original recovery groups remain open.

Exact target · Source assessment · Skeptical review · Checks and limits