RESEARCH AUDIT · NS99 · SEPTEMBER 23, 2026 · VERSION 1

Which approaches share the gap?

Several do—but there are two different shared-gap families, plus separate unresolved routes. NS98 did not prove that the projected cost is unbounded. It found that keeping the exact projection still leaves an unestimated signed arithmetic cross term.

The direct answer

• Exactly the same selected cost: NS73’s p1 Mobius/log direction, carried through NS90–93 and NS98. The compatible numerator remains independently unproved. NS92’s larger signed suffix budget is sufficient, not equal to the cost.
• The broader missing arithmetic decrease: the optimized-correlation, repaired-kernel, curvature, preconditioner, joint-observation and square-product lines return to an actual cofinal gain estimate. They use different directions and sometimes different smoothing orders or block sizes. They do not all share the same denominator.
• Another exact repetition, in earlier geometry: CAE, unrestricted adaptive concentration, critical Clifford contraction (NS47) and critical localization absorption (NS50) express the same complete signed Weil floor under their stated domain constraints.
• Different gaps: full-kernel injectivity, CCM convergence, canonical growth, signed sample-error decay, continuous-to-integer transfer, level/packet complexity, heat transport and zeta-specific geometric rigidity. Their relationship to RH does not make their proof obligations identical.

Wall check: Same open gap. Closest results: NS92–98. What changes is the dependency map, not the arithmetic estimate. All 104 statuses remain intact; 63 nodes now name their unfinished continuation. A proved finite result does not become conditional because its further use needs an open bound.

What this changes next time

Before trying a new formula, identify its input below. A new name, coordinate system, normalized trace or finite-size table is insufficient. A useful proposed test must add a specific estimate or change a relevant hypothesis, and state what success or failure would establish.

Dad’s original count-to-infinity inference is closed by an off-line symmetric control. That does not close geometry. A revival needs a property of the actual zeta function which the control lacks. A separate agent has reassessed this possibility in the NS100 report; its tests and evidence status are separate from this map audit.

MOBIUS-COST

For d_n=-mu(n) log(2N/n), N<n<=2N, retain K_N=||(I-Pi_N) sum d_n a_n||². The NS93 sufficient averaged route asks for bounded means of K_(2^j) over J<=j<2J, together with MOBIUS-MARGIN on the same ranges. Pointwise bounded K_N is stronger. NS98 does not supply either bound.

Scope: This is the same direction and exact projected cost in NS73 p1 and NS90–93/98. It is not the full NS73 three-template span or the curvature direction. NS92 gives K_N<=U_N<=kappa log(2) S_N; bounding the signed suffix energy S_N is sufficient but stronger. NS93 excludes only the fully absolute product majorant. NS98 bounds the projected diagonal but leaves its signed cross term unestimated. A new decomposition, finite table or free trace rescaling alone does not change the input.

MOBIUS-MARGIN

With ell_N=<r_N,sum d_n a_n>, obtain a lower bound for its gain relative to E_N and the same projected cost K_N. The NS91 lower bound b_N retains the joint interior allowance and complete tail. NS93 gives a sufficient averaged criterion: positive lower means of (b_(2^j)/E_(2^j))_+ on the same ranges as the bounded cost means.

Scope: A cost upper bound alone does not prove descent. The averaged sufficient criterion is not claimed necessary; actual ell_N may admit bounds that b_N cannot detect. The N^6 grouped tail is controlled, but its interior margin remains open. The divisor-inversion quadratic rewrite under the old normal equations gives the same numerator.

NB-GAIN

For a fixed smoothing order and actual optimized residual, prove a nonsummable sequence of relative gains. In q=2, the full block target is Delta_N/E_N, Delta_N=h_N^T H_N^(-1)h_N. NS87 localizes the actual arithmetic observations with complete tails at T=N^6; a positive lower liminf of j A_(2^j)/E_(2^j) is sufficient. This remains unproved.

Scope: Related selected-direction bounds are sufficient routes to this objective, not the same scalar cost. NS71 needs its model-inverted numerator and true-energy ratio; NS72 needs actual curvature and selected cost; NS95/96 use a restricted N-to-N² span and complementary gain. NS53/59 use the unsmoothed analogue, so their formulas are not identified with q=2. NS80–82 guarantee a fraction of available gain, not a fraction of total error. NS83 excludes eventual fixed-fraction whole-error contraction, not all nonsummable slower gains.

NB-SAMPLES

Construct integer approximants whose complete reciprocal-sample or zero-exterior error tends to zero. NS78 compares the complete sample sum with the full physical norm; NS89 preserves every decay rate under its optimal normalization. Neither result bounds the arithmetic sum cofinally.

Scope: This is an upper-error route to the same convergence objective, not the MOBIUS-COST scalar or a new lower-gain identity. A fixed-template failure does not exclude freely chosen coefficients; coefficientwise convergence is not norm convergence.

NB-GROWTH

Prove subpolynomial growth of the canonical smoothed norm, or separately establish a useful bounded cofinal subsequence with its required implication. NS66 identifies the power-growth exponent; NS67 supplies complete cells and tail; NS68 transfers the forcing power through the positive renewal operator.

Scope: Different object from the optimized projected cost. Strong full-sequence convergence is already excluded; weaker growth targets are not. Cancellation merely between the two complete centered Abel terms cannot change the exponent. Renewal positivity cannot manufacture the missing forcing estimate.

NB-DILATION

Force divergence of the cumulative normalized absolute dilation defect appearing in NS88’s whole-error upper envelope.

Scope: A different scalar and sufficient route to convergence; no identity with the fixed Mobius cost. The eventual energy/j^p lower bound for p<1 is excluded; p=1 remains open within the stated constant restriction. Finite sign changes do not settle cumulative growth.

NB-CONTINUOUS

Supply a valid additional bridge from the continuous dilation closure to integer approximation, or prove actual zero continuous defect by new arithmetic input.

Scope: The continuous closure is strictly larger. Its small certified upper bound is not an upper bound for integer error and is not zero. This does not share the projected-cost estimate.

WEIL-FLOOR

Prove q_a[f]>=-C||f||² with a common finite C on complete admissible domains: the actual even source complement and the odd sector, with the established source transfer to the full form. All prime, pole, exterior and cross terms must remain.

Scope: CAE, unrestricted exact adaptive concentration, critical Clifford domination and coefficient-one localization absorption are versions of this same floor under their stated domains. Their stronger fixed-slack or channel-deletion failures do not refute the floor. This is a second genuine shared-gap family, distinct from the NB selected numerator/cost. RH implications alone do not identify their estimates.

KERNEL-API

Establish arithmetic affine-potential injectivity (API), respecting parity, endpoints, all prime shifts and poles. The homogeneous translated-radical boundary test is exactly the null equation, not an independent rigidity estimate.

Scope: This is a different unresolved input. Local open-set UCP and ordinary-cone positivity are false for the actual operator; those controls are not full-window nullvectors. No generic Sobolev bootstrap or whole-space compact inverse supplies API.

CCM-LIMIT

Control the complete cutoff residual relative to a complete separator (or the stated Rayleigh-excess/gap variant) cofinally, plus entire-transform evaluator/tightness bounds. Global-ground claims also require the specified odd ordering.

Scope: Fixed-window real-zero transforms and proximity to the first zeta zero do not supply convergence to Xi. Absolute residual/resolvent data and a polynomial positive separator do not replace the named relative inputs. Distinct from the NB cost and API.

LEVEL-PACKETS

Verify the relevant ZLD/QG level-distribution hypothesis and separate bounded-energy source-admissible packet coverage needed for the stated minorant-complexity implication.

Scope: These hypotheses concern an obstruction to bounded minorant complexity, not a proof of the signed Weil floor. ZLD implies only the stated QG growth clause; packet feasibility does not follow. Do not merge this gap with WEIL-FLOOR.

ZERO-GEOMETRY

A geometric revival must add a property of the actual xi/theta/Euler-product object that symmetric off-line controls do not share. The NS94 power-product detector still requires an all-height vanishing proof.

Scope: Equal inversion counts and an infinite exterior similarity point do not imply RH: the point depends on circle parameters. That inference is closed, not every geometric approach. A classical RH-equivalent radial-growth or disk-contraction criterion is not a new arithmetic estimate; any new proposed sufficient mechanism needs its own test and scope. NS100 tests a concrete stronger condition using the actual theta kernel: its p=0 slice cosine transform at frequency 40 is strictly negative, and continuity extends failure to small p>0. This closes only per-slice positivity; the full integrated kernel remains unestimated. Its classical associated-kernel criterion is not a novel mechanism.

HEAT-TRANSPORT

Bound the external zero-cluster interactions and support dependence that survive symmetry-paired heat transport.

Scope: Internal inverse-gap cancellation alone is insufficient. The unmatched-real-zero obstruction is conditional and its hypothesis was not verified. This is not the NB projected-cost question.

OPERATOR-BRIDGES

These are separate unfinished constructions, grouped only to mark their independence from NS98: evaluator estimates and closed realization (fixed-space); actual structural/asymptotic transfer (circle/Toeplitz); a valid unshifted-sign/extension bridge (de Branges); number-field intersection construction and positivity transfer (Hodge/F_1).

Scope: No equivalence between these missing inputs is asserted. The old circle/Toeplitz measurement lane is not Dad’s tangent-circle zero-count proposal. Shifted finite-window data do not determine an unshifted sign; analogy with function fields does not construct a number-field intersection theory.

Review base: 918dc6c. This is a conclusions/dependency review, not a new proof audit or numerical replay of all historical work. Both original-evidence recovery groups stay open. Manuscript v1.66 is unchanged; RH and G2 remain open. Generated map and source links · Audit records.