# Skeptical strategy and control review Date: 2026-09-29. Reviewed base: `a992456a6305b0b01b0d6efe2b022c73d1c0cd3b`. Owner: `nb_strategy_challenge`. This is an internal route-selection review, not a new theorem, candidate screen, research row, thaw or independent replay of historical certificates. The coordinator performed the complete register review; this review read current instructions and continuation scopes and checked the relevant NS74/83 proofs and current NB/Weil/operator conclusions. The two original-evidence groups in `evidence/MISSING.md` remain open. **Wall check: Same open gap — NB-GAIN, closest NS87, with NS53/61 as its criterion and budget predecessors.** No arithmetic hypothesis has changed. Success of this review means a better specified feasibility decision, not a lower bound. Failure to find an admissible partial lemma means no new research candidate is authorized by this review. ## Recommendation and challenge to the previous recommendation Prioritize one expert-readable feasibility brief about the full optimized fixed-order NB residual. Do not describe this as the mathematically most promising route, a new project mechanism, or evidence that NB has become more likely to succeed. The defensible preference is organizational: the relevant objects and their exact dependency are already specified, and the complete observation tail is controlled at the needed scale. That makes the unresolved question relatively easy to present accurately to a specialist. It does not make the arithmetic inequality easy. Returning from a prescribed Möbius direction to the full block removes a self-imposed restriction. It does not introduce a new estimate: NS87 already concerns that full block and the actual target. In particular, an optimum over all directions is a more permissive sufficient target, but permissiveness by itself supplies no constructive lower bound. The residual coefficients depend on the full inverse Gram matrix, so familiar theorems for fixed Möbius or divisor weights cannot be imported without a uniformity argument for these changing coefficients. The exact-bottleneck agent identifies the first unestimated vector as `q_new(T) - G_old,new^T G_N^(-1) q_old(T)`: the finite integral-minus-lattice observations of the actual residual, with the old-space correction retained. The arithmetic enters twice, in that residual and in the refit. No useful standalone summand with an independent estimating mechanism has been identified in this review. The literature agent's bounded search likewise finds no unconditional theorem supplying this input. Therefore I recommend **no newly proposed partial lemma and no computation now**. That is a valid outcome of the feasibility stage. ## Comparison of three directions | Direction | What is already concrete | What remains missing | Priority consequence | | --- | --- | --- | --- | | NB-GAIN | Full block gain, actual target, finite joint arithmetic observations and complete tails | A lower bound on actual projected signed observations relative to the full current error, at a compatible nonsummable cofinal scale | First choice for a bounded feasibility assessment; unchanged NS87 input, not a thaw | | WEIL-FLOOR | Complete signed form and bounded-floor-to-RH transfer | One common cofinal lower constant on the actual admissible domains, keeping every prime, pole, exterior and cross term | Keep available, but no reason to reopen coordinate/localization variants without a new signed estimate | | Arithmetic geometry or operator construction | Explicit inventories of fixed-space, de Branges and number-field geometric obligations | A route-specific construction and a proved transfer to the actual arithmetic object; these missing constructions are not equivalent to one another | Longer-term specialist-led formulation only; no supplied construction to develop now | WEIL-FLOOR is not refuted by tiny positive energies or by the failure of a uniform positive gap. Its target allows a common absolute lower error. NS34, NS43, NS47 and NS50 instead delimit particular ways of losing signed energy or restating the same floor. They do not establish NB's superiority. Likewise, a missing operator construction is not an impossibility theorem. ## Controls that must precede any later proposed partial lemma 1. **NS74: identify what knows the target.** The altered atoms preserve all complete Gram entries and geometric preconditioners while changing the loads and retaining an explicit positive approximation floor. Any argument using only the Gram, condition numbers, or a finite fraction of available block gain cannot establish original convergence. A supporting generic solver lemma may survive this control; its separate target-sensitive arithmetic premise must then be stated and proved. NS74 is not a counterexample to an estimate that genuinely uses the unchanged original target loads or original Mellin atoms. 2. **NS83: retain fixed order and an admissible rate.** For every fixed order, the critical-zero lower bound rules out eventual constant relative contraction at every doubling and eventual `a/j` contraction with `a>1`. It leaves weaker nonsummable gains open; it does not give an individual upper bound on every block. Changing smoothing order changes the problem and invokes the existing changing-norm caveat, not an automatic escape. 3. **Retain the actual span and terms.** NS76/77 prevent importing the relaxed continuous-space error as an integer-space bound. NS80–82 give efficiency within available gain, not available gain relative to the whole error. NS95/96 distinguish restricted product/LCM spans, refitting and the full block; their controls must be matched if those restrictions return. 4. **Match any new arithmetic claim to its own control.** If a fixed Möbius direction returns, NS90–93/98's numerator, cost and signed-cross-term obligations return too. For a functional-equation or symmetry-only positivity proposal, use the applicable false analogue, including the Davenport–Heilbronn screen, with actual hypotheses. Do not present an unrelated successful example screen as validation of an NB lemma. These are scope checks, not a demand that every useful supporting lemma individually imply RH or distinguish every false analogue. It is the complete claimed implication that must survive the applicable controls. ## What would count as a worthwhile partial result? An unconditional partial result could have standalone value without yielding RH if it supplies a previously absent signed estimate for a fully specified original arithmetic expression, with a useful uniform range and an honest remaining shortfall. It must improve a stated bound, expose a genuine arithmetic obstruction, or resolve a precise feasibility question. Its proof must name the step that fails for generic coefficients; putting arithmetic symbols into a Hilbert-space projection identity is not that step. No such new statement is proposed here. In particular, it would be premature to call an unspecified estimate on “one signed term” a partial lemma: until the full expression, range and remaining transfer are identified, it is only a request to find one. The two other agents' checks do not presently repair that omission. A faster exact inverse or a better constant for capturing available gain may be useful numerical-analysis work, but it is solver work if its guarantee holds for arbitrary loads. Publish it on that basis if desired; do not score it as arithmetic progress. A larger finite table, another coordinate system, or a positive fraction of a possibly vanishing available gain does not meet the partial-arithmetic milestone. ## Bounded next action and accept/stop criteria Finish one short brief containing the exact actual-target expression, the complete tail allowance, the required cofinal scale, NS74/83, and the exact parameter mismatch for the closest literature results. Its immediate use is a **specialist feasibility assessment**, not a request to endorse an RH proof. No outreach is authorized or performed by this review. Ask the specialist to choose one of three concrete outcomes: - Identify an existing unconditional theorem and show how its coefficient hypotheses and uniformity cover a stated piece of the actual expression. - Propose one smaller unconditional arithmetic statement, with its exact range, a plausible method, and the gap that would remain after proving it. - Explain that no such statement is currently justified and identify what new arithmetic structure would be needed before investing further. Accept a new research task only when the dependency edge and arithmetic use are written down, all companion estimates are named, and success/failure changes an existing proof obligation or route choice. Apply the repository proposal gate and the actual candidate's matched screen then, before proof or computation. Stop the automatic research cycle if this bounded assessment returns no statement. Do not respond by generating adjacent variants until one happens to look numerically positive. The present evidence favors this decision discipline over further unsupervised candidate proliferation; it does not rank mathematical success probabilities. ## Dad's geometry interest The meaningful geometric question is not whether a familiar surface resembles the zeta phase. It is whether one can construct an arithmetic pairing, a number-field intersection form, or a closed operator whose proved positivity or spectral statement transfers to the original zeta object with the required domain and boundary terms. The corresponding transfer and construction are absent. Circle-count symmetry is already controlled by NS94; the separate fixed-space and de Branges obligations in OPERATOR-BRIDGES remain open under their own hypotheses. This gives geometry a precise role without equating surface analogy with arithmetic content or claiming a general geometric wall. Final classification: **Same open gap — NB-GAIN / NS87.** The team has improved the specification and selection decision, not the lower bound. RH, G2 and the actual cofinal arithmetic estimate remain open; all freeze conditions remain.