Verdict: both notes are admissible as scoped audits after one correction to gain02. Neither supplies an independently estimated nonsummable signed gain. No new candidate, computation, thaw or research claim is admitted.
Wall check: Same open gap — NB-GAIN. Closest full-gain results: NS53/61/87; selected-cost/margin results: NS90–93/98; controls: NS74/83. The coefficient-recovery and structured-phase inputs are mathematically different, but neither estimates the missing actual-target quantity at the required scale. This is not a general obstruction to either arithmetic family.
Reviewed both complete notes: gain/01-derivation/derivation-2026-09-30-v1.md and the corrected gain/02-independent/assessment.md. Reviewed the current public source at commit 769359298fada57711b6895c4bde3fabe2cc5168 in the isolated checkout specified by the coordinator. AGENTS, conclusion/continuation register, current status, freezes and evidence ledger were read in this continuing review session. Direct proof checks covered NS53, NS61, NS74, NS75, NS78, NS83, NS87, NS90, NS93 and NS98, plus the prior local weighted-coefficient lemma and coordinator check. Imported historical dual-vector theory and historical interval certificates were not independently replayed. No numerical candidate calculation was performed.
The setting is fixed q=2 with the original target and complete space L2((0,infinity),dt/t²). With old Gram G, cross block C, new Gram D, corrected load h and Schur complement S, the optimizer increment is exactly
y=S^-1 h, u=-G^-1 C y,
Delta=E_N-E_M=||(sum_old u_m a_m)+(sum_new y_n a_n)||²=hᵀS^-1h.
The prior coefficient lemma is applicable to every finite pure combination, hence to every residualized new vector, not merely the optimizer. I independently checked its conservative constant: the complete NS78 sample inequality contributes 16; the weighted cumulative-divisor estimate contributes 24; differencing contributes 4; convolution by mu contributes at most zeta(2)². Their product is B=1536 zeta(2)². The convolution series is absolutely convergent in operator norm; no Möbius sign cancellation was used.
For W_old=diag(m^-4), W_new=diag(n^-4),
L=W_new+CᵀG^-1 W_old G^-1 C>0,
L<=B S, therefore S^-1<=B L^-1.
The order reversal is correct. Thus Delta<=B hᵀL^-1h, an upper bound in load coordinates. At the actual y, Delta>=yᵀLy/B is also valid; it is a lower bound in still-unestimated optimizer coordinates. There is no contradiction and no lower inverse-Gram estimate. The old coefficient correction is fully present in L. All matrices and synthesis norms are complete-space objects, so no tail is discarded.
Under the explicitly unproved stress-test hypothesis y_n=-mu(n) on N<n<=2N, the new-coordinate part is
b_N=(1/B) sum_new mu(n)²/n⁴
=7/[24 B zeta(2)] N^-3+O(N^-7/2).
| The density proof is sound: mu(n)²=sum_(d² | n)mu(d), floor errors and the omitted d^-2 tail are O(sqrt(x)), and partial summation integrates x^-4 from N to 2N to 7/(24N³). Weighted closeness with relative error theta<1 gives the stated lower factor (1-theta)². No optimizer closeness is established. |
| Using the inherited eventual NS83 bound E_N>=c/log N yields b_N/E_N=O(log N/N³), hence a summable dyadic certificate. This is not an upper bound on Delta/E_N. The old-refit contribution and small-eigenvalue amplification may be much larger. The stated coefficient-size requirement for this particular certificate to provide a/j also checks: sum_new | y_n | ² must be at least a constant times N⁴/j², hence root-mean-square size at least order N^(3/2)/j. This is not a requirement for NB convergence itself. |
NS75 supports the limited jump-feedback warning: raw jump cancellation retains other cells and tails; refitting can undo it; a negative finite pairing exists at the stated N=256. None is a cofinal no-go. No substantive defect found in gain01.
I independently checked Green–Tao, Annals 175 (2012), equation (1.1) and Theorem 1.1, pages 542–543. The uniform linear-phase Möbius estimate and the nilsequence theorem’s complexity dependence support the source description. I did not audit their proof. A fixed exponent A cannot be promoted to A(N) with unchanged implied constants.
Assume the note’s additional kernel representation for the complete projected B_N=N H_N, with Fourier coefficient mass L_N and uniform remainder epsilon_N. Partial summation against w(t)=log(2N/t) is legitimate: w has bounded supremum and total variation; interval partial sums are differences of uniform Davenport prefix estimates. Factoring each separated phase produces two tapered Möbius sums. Consequently
K_N <= C_A L_N N/log^(2A)N +(log2)² N epsilon_N.
| The factor N is correct: two sums have scale N each, and the kernel normalization divides by N. The remainder uses | d | _1<=(log2)N. Neither small Fourier mass nor a small approximation remainder is proved for the changing Schur complement. Even hypothetical polylogarithmic mass yields only a growing allowance for every fixed A; it does not imply growth of the actual cost. A decaying mass, a sharper bilinear estimate or another representation is not excluded. |
The separate averaged numerator requirement is correctly retained. Put r_j=(ell_(2^j)/E_(2^j))+. If sum K_j<=CJ and sum r_j>=epsilon J on every sufficiently large dyadic index block J<=j<2J, then exact line search, E(2^j)>=c/j and weighted Cauchy–Schwarz give a positive lower total gain on each such block. This is the existing NS93 sufficient mechanism. No one of these arithmetic hypotheses is supplied by the phase theorem. Kernel information alone cannot determine the original-target loads; NS74 remains an appropriately scoped control.
G03-01 — minor scope error, resolved. The first gain02 version said that a model observation error “must” be little-o of sqrt(E_N/log N). This unnecessarily strengthened a sufficient hypothesis into a necessity. The author corrected it after review. Fixed constants 0<=a<b already suffice if
||h_*||_(H^-1)>=b sqrt(E_N/log N),
||h-h_*||_(H^-1)<=a sqrt(E_N/log N).
The reverse triangle inequality gives Delta>=(b-a)² E_N/log N. Little-o is one convenient sufficient condition. Neither a model signal lower bound nor its useful inverse-metric error is established here. The correction does not admit the attempted route.
The full desired edge is unchanged: for the actual original q=2 dyadic residuals, prove 0<=beta_j<=Delta_j/E_j eventually with sum beta_j=infinity; telescoping then gives E_N→0 and the established RH implication. No such independently estimated beta_j is obtained.
Additional missing inputs are explicit: gain01 needs an arithmetic lower estimate for actual new coefficients or the old refit, strong enough in the weighted norm; gain02 needs a usable complete-kernel representation/complexity-and-remainder bound and a compatible actual signed numerator. A direct model approach instead needs both an actual model-signal lower bound and a sufficiently separated error in the complete inverse metric. These are alternatives, not assumptions silently supplied by one another.
NS87 already makes its normalized localization loss dyadically summable at T=N⁶; that is a transfer theorem, not positivity of the interior observations. NS83 still excludes eventual fixed-fraction whole-error gain and eventual a/j with a>1, while compatible slower nonsummable gains remain open. No whole-space or same-Gram control was misapplied to exclude an original target-sensitive arithmetic hypothesis.
Retain the matrix-order and certificate-scale calculations as audit findings, and the structured-phase source check as a bounded failed literal import. Stop these attempts without scans or stronger route-closure language. Final classification: Same open gap. RH, G2, actual cofinal signed gain and both historical-original evidence groups remain open.