Decision: no currently supported arithmetic lemma or estimating method meets a thaw condition. Stop automatic theta variations; do not automatically switch to NB, Möbius, Weil or operator variants. This is a bounded feasibility assessment, not a claim that these routes are impossible or equally difficult.
Reviewed remote base: 18d324703969b5bcc9aa94043cfd90a6e8211f34 (PR76), refreshed by the coordinator; worktree HEAD 32f599ced1abb2bf2376a715042e6e4101591639 additionally claims the interpretation and present audit. Read AGENTS section 0/gates, all 104 conclusion records and 14 continuation demands, current README/checkpoints, freeze conditions and evidence ledger; inspected the NB exact-target/strategy notes, September 30 gain audits, PR76 derivation and two-prime operator audit. This is a conclusions/dependency assessment, not a historical proof or certificate replay. Both missing-original groups remain OPEN. No new external theorem is imported.
Wall check: Same open gap. Closest: ZERO-GEOMETRY/NS100–101/PR76; alternatives NB-GAIN/NS53/61/87, MOBIUS-COST/MARGIN/NS90–98, WEIL-FLOOR/v1.36, and the separately scoped OPERATOR-BRIDGES. What changes: nothing in their arithmetic hypotheses. This review tests whether continued theta preference has a defensible estimating mechanism; it finds none.
PR76 fixes the exact original trajectory
Y(r) = -Lambda(1/2+ir), p(r)=r²+1/4, X(r)=p(r)Y(r)/4,
J(r)=p(r)²(Y'(r)²-Y(r)Y''(r))+2(r²-1/4)Y(r)².
The first target is J(r)>=0 for every real r, including zeros of Y, where the undivided formula stays valid. It equals 16 L1[X]; first-Laguerre positivity alone is not RH. Any proposed proof must identify a separate subsequent bridge before being described as an RH route completion.
This is narrower than proving every associated-kernel inequality, but narrower does not mean presently estimable. The positive decaying primitive, its envelope and unit tail are shared by NS101 controls that violate J’s sign. Their original lattice coefficients/Jacobi IVP differ; that mismatch leaves the original method open, without supplying a lower estimate. The known exact IVP has already been used to derive the mixed form. Repeating its identity is not a new thaw input.
The estimating obligation remains a complete signed comparison using those original coefficients or that IVP at a step the controls cannot share. Fixed reflected prefixes eventually fail; smooth-completion error bounds have no matching lower margin; the Bessel reference margin has no original-theta comparison. PR76 resolves legitimate transforms, not any of these inequalities. The new four-coordinate picture carries no trajectory estimate. Recency, a pleasing formula and fewer displayed variables provide no reason to prioritize more theta work.
| Family | Exact remaining input and present assessment |
|---|---|
| Full NB-GAIN | At fixed q=2 and actual target, lower-bound a nonsummable aggregate of complete relative gains. The tail is already sufficiently controlled; the signed corrected observations are not. This is the best specified existing handoff, an operational advantage only. |
| MOBIUS-COST/MARGIN | For d_n=-mu(n)log(2N/n), control the actual projected cost and a compatible signed numerator on the same ranges. Neither is supplied; choosing fewer coefficients narrows the available gain and creates two explicit obligations. |
| WEIL-FLOOR | Prove one common finite C in q_a[f]>=-C||f||², cofinally on the actual even source complement and odd domain, with source transfer and every prime, pole, exterior and cross term. Tiny eigenvalues do not refute this target: it requires no positive gap. No independent complete comparison is currently available. |
| Operator bridges | Construct the particular missing realization and transfer: Sonine evaluators/closed operator; actual circle/Toeplitz asymptotics; unshifted de Branges sign/extension bridge; or number-field intersection and positivity transfer. These are separate constructions, not one interchangeable arithmetic estimate. None is supplied. |
For NB the normalization must remain
H=D-C^T G^-1 C, h=b_new-C^T G^-1 b_old,
Delta_N=h^T H^-1 h=E_N-E_(2N), c_N=G^-1 b_old.
The old optimizer is refitted: for new coordinates y=H^-1h, its change is -G^-1Cy. At T=N^6, the finite vector is q_new(T)-C^TG^-1q_old(T), in the same complete inverse metric. Truncated old correlations do not vanish. NS87’s normalized tail error is dyadically summable, with no effective asymptotic onset asserted; improving it again does not estimate the interior. Divergence of the localized relative observation energies is equivalent to convergence under this error control, hence is not a newly weaker partial lemma.
NS74 preserves the full Gram while changing the target loads and leaving an approximation floor: the missing arithmetic step must know the original target. NS83 excludes eventual fixed-fraction whole-error contraction and eventual a/j contraction with a>1; irregular slower nonsummable gains remain admissible. Neither control excludes every original signed estimate.
September 30 gain01’s coefficient coercivity gives an upper inverse-Gram comparison in corrected-load coordinates. Its hypothetical Möbius-size new-block lower certificate is summable; this is not an upper bound on actual gain. Gain02 leaves kernel complexity, a factor-N allowance and the numerator unestimated. Fixed strict signal/error separation could suffice; little-o error is not necessary. These findings provide no ready replacement candidate.
For the fixed Möbius direction, the fully absolute product majorant is already excluded by NS93. The signed projected cross term and companion margin remain open; free normalization, another finite table or bounded cost alone cannot supply gain. For operators, the connected two-prime model admits arbitrary assigned positive periods and its natural scalar realization has translation spectrum, not discrete eigenvalues. Changing the realization could escape that scoped failure, but a proposed domain, arithmetic completion and sign transfer must actually be constructed. A larger prime list does not provide them.
Use one final synthesis of this already commissioned twelve-review round, capped at 30 minutes of paper comparison and zero computations, to check whether another reviewer supplies one concrete lemma missing here. Require its exact expression/domain/rate, original-arithmetic estimating step, complete conditional transfer, every unproved companion estimate and matched control. A partial result need not prove RH; it must improve a named bound or settle an independently testable feasibility question. Naming an unchanged target is insufficient.
If one proposal meets that standard, assess only that proposal under its own thaw condition and proposal gate. Success would authorize consideration of a specific research task; this review itself authorizes no scan, row or thaw. If none does, finish this audit and stop the automatic research cycle. Do not commission another generic next-step round, enlarge a finite table, or draft another NB brief: the existing brief is already prepared. No outreach is performed or recommended as an automatic action. Reopening requires a concrete new input, not a scheduled retry.
Final classification: Same open gap. No mathematical ranking of the surviving routes is justified by this assessment. RH, G2 and the actual cofinal signed-arithmetic lower bound remain open; no manuscript version, node, computation or thaw follows.