Internal working note, 2026-09-30. This is bounded paper work, not a new candidate screen, theorem row, thaw, manuscript edit, publication, or numerical certificate.
Verdict: no new arithmetic lower estimate obtained. Wall check: Same open gap — NB-GAIN, closest NS53/61/87. The concrete attempted mechanism was to combine the original arithmetic coefficient recovery with forced Möbius-size new coefficients and the actual full refit. Two precise deficiencies emerge: recovery bounds the projected Gram in the direction that gives an upper, rather than lower, inverse-Gram bound; even an idealized Möbius new-block contribution to the recovery certificate has only a dyadically summable scale. Neither deficiency is an upper bound on the actual gain or a closure of original NB-GAIN.
Read the assignment brief first. The coordinator refreshed the public source at commit 769359298fada57711b6895c4bde3fabe2cc5168; this reviewer confirmed that exact HEAD in read-only checkout /private/tmp/riemann-prime-folding-review-20260930. Read AGENTS, the conclusion register/status inventory and continuation scopes, relevant current README/checkpoint and NEXT_STEPS entries, and the missing-evidence ledger. This is a conclusions/dependency review, not a replay of every historical proof. Both original-evidence recovery groups stay open.
Direct proof dependencies checked:
evidence/ns53_nb_blocks/proof.tex: exact Schur gain, optimizer update and scope of RBC; distinguish its unsmoothed target from the q=2 target below.evidence/v159/ns61/proof.tex: fixed q=2 criterion, complete average/trace bound, retained old-new correction, missing lower correlation.evidence/v163/ns78/proof.tex: complete norm/sample comparison and exact divisor inverse, including exterior eta and all cells.evidence/v161/ns74/proof.tex: same-Gram unitary control, altered loads and finite-statistics scope.evidence/v166/ns83/proof.tex: logarithmic lower floor and cumulative relative-gain restriction; imported historical dual-vector theory was not independently reconstructed here.evidence/ns86_optimized_arithmetic/argument.tex: original divisor/potential identities and hypothetical off-line-zero annihilator.evidence/ns87_joint_tail/argument.tex: complete corrected finite observations, inverse metric, safe step and polynomial tail cutoff.evidence/v162/ns75/proof.tex: exact next-block derivative-jump cancellation and its finite failure as unit-step descent.evidence/ns98_projected_cost/argument.tex: actual projected signed cross cost remains unestimated.Local prior-audit dependencies read in the primary workspace: audits/nb-six-review-2026-09-30-v1/reviewer-1-coefficients.md, coordinator-proof-check.md, reviewer-2-observations.md and reviewer-6-adversarial.md. The weighted coefficient lemma below is that prior checked corollary, not a result first proved by this note. Some broad combined reading output was truncated; no claim is made of revalidating every registry proof or line. No external theorem is newly imported; the elementary squarefree count used below is proved here.
| Fix q=2 once and for all. Let H=L2((0,infinity),dt/t²), a_n(t)=A(t/n), A(z)=integral_0^z {u}du/u, and tau(t)=1_(t>=1)log t. Real coefficients are unrestricted. Let f_N=Pi_N tau, r_N=tau-f_N, E_N= | r_N | ², c_N=G^(-1)b. Norms and every Gram entry include the complete domain. |
For arbitrary integers M>N>=1, let G be the old Gram, C the old/new block and D the new Gram. Set
h=b_new-CᵀG⁻¹b_old,
S=D-CᵀG⁻¹C >0,
y=S⁻¹h,
u=-G⁻¹Cy.
The actual optimizer increment has coefficient vector (u,y): old coefficients c_N+u and new coefficients y. Thus
f_M-f_N = sum_(m<=N) u_m a_m + sum_(N<n<=M) y_n a_n,
Delta_(N,M)=E_N-E_M = ||f_M-f_N||² = yᵀSy = hᵀS⁻¹h. (1)
This is the complete NS53/61 block formula. In particular y is not a prescribed Möbius vector, and u is not optional. Tails, small eigenvalues and all cross terms remain in (1).
The prior local weighted coefficient corollary says, for every finite real v,
sum_n |v_n|²/n⁴ <= B ||sum_n v_n a_n||²,
B=1536 zeta(2)². (2)
Its proof uses NS78 divisor inversion and a bounded Dirichlet convolution operator; it takes absolute values of Möbius coefficients. It is regularity/conditioning, not signed cancellation.
Define positive diagonal matrices
W_old=diag_(1<=m<=N)(m⁻⁴),
W_new=diag_(N<n<=M)(n⁻⁴),
L=W_new + CᵀG⁻¹W_old G⁻¹C.
For every real new coefficient vector v, the residualized synthesis has full coefficient vector (-G⁻¹Cv,v). Applying (2), with no optimization assumption on v, gives
vᵀLv <= B vᵀSv, hence S >= B⁻¹ L. (3)
Here L is positive definite because W_new is. This calculation retains the exact old refit and complete true energy. Inverting the positive matrices reverses their order:
S⁻¹ <= B L⁻¹,
Delta_(N,M) <= B hᵀL⁻¹h. (4)
Therefore the tempting implication “better coefficient coercivity supplies the desired lower inverse-Gram gain” has the wrong direction. It supplies (4), an upper estimate in the residual-correlation variable h. A lower gain from a metric comparison needs an upper bound on S (as NS61 already supplies), followed by an independent lower estimate for the corrected h. The latter is unchanged.
Substitution of the actual, still unknown y into (3) also gives the valid lower bound
Delta_(N,M) >= B⁻¹ [sum_(m<=N)|u_m|²/m⁴
+sum_(N<n<=M)|y_n|²/n⁴]. (5)
This is not contradictory to (4): (5) is in optimized coefficient coordinates y=S⁻¹h and (4) is in correlation coordinates h. Obtaining a lower bound on the right side of (5) remains an independent estimating task. Relabeling y as an arithmetic object does not estimate it.
Take M=2N. Even suppose, counterfactually as an unproved model hypothesis on the ACTUAL optimizer, that the new coefficients were exactly
y_n=-mu(n), N<n<=2N.
The new-block contribution in (5) would be exactly
b_N = (1/B) sum_(N<n<=2N) mu(n)²/n⁴
= [7/(24 B zeta(2))] N⁻³ + O(N^(-7/2)). (6)
| Elementary arithmetic justification: mu(n)²=sum_(d² | n)mu(d). Summing through x gives |
Q(x)=sum_(n<=x)mu(n)²
=sum_(d<=sqrt(x))mu(d) floor(x/d²)
=x/zeta(2)+O(sqrt(x)).
The floor errors total O(sqrt(x)), and the omitted absolutely convergent d⁻² tail contributes O(sqrt(x)). Partial summation against x⁻⁴ on (N,2N] gives (6), since integral_N^(2N) x⁻⁴ dx=7/(24N³). This calculation uses only squarefree density, not cancellation in the signs of mu.
The same order lower contribution follows if the new-block weighted distance to -mu is at most theta<1 times its weighted norm. The triangle inequality gives a factor (1-theta)². That closeness is NOT established for y. Fixed-index convergence is not a growing-block estimate, and even identification of every actual optimizer coefficient limit with -mu is already RH-equivalent in the prior audit.
The scale issue is independent of proving that strong coefficient hypothesis. NS83 gives E_N>=c/log N eventually for some c>0. Consequently the normalized certificate in (6), by itself, satisfies
0<=b_N/E_N=O((log N)/N³),
sum_j b_(2^j)/E_(2^j) < infinity. (7)
Thus even this idealized new-block piece cannot establish the needed nonsummable relative lower bound. Equation (7) bounds the size of this particular certificate, not the actual relative gain. The real gain could be much larger, through old refit or small-eigenvalue amplification. No upper bound Delta=O(N⁻³) is claimed.
To make the new-block part of (5) alone supply a/j relative gain for N=2^j, one would need, as a sufficient condition,
sum_(N<n<=2N)|y_n|²/n⁴ >= B a E_N/j. (8)
| Using n>N, (8) forces sum_new | y_n | ² >= B a N⁴ E_N/j. With the NS83 floor, this is of order at least N⁴/j², or root-mean-square coefficient size at least a constant times N^(3/2)/j. This is a necessary size for USING (8), not a necessary size for actual NB convergence. The established global O(N²) coefficient l2 upper bound does not force or forbid this distribution. The point is that an O(1)-size Möbius coefficient heuristic is far below what this particular coercivity route needs. |
For original atoms the current divisor defect at a prime p>N is exactly c_N(1), since its only old divisor is 1. More generally NS75 permits cancellation of every next-block defect by appending -delta_n while keeping old coefficients fixed. This is a concrete use of original arithmetic, unlike a generic Gram argument.
| However a local derivative jump is not the residual’s pairing with a complete atom or a residualized direction. The exact step gains 2<r_N,v>- | v | ²; jump cancellation controls neither signed pairing nor complete cost. NS75 already certifies held-old unit-step failures, including a negative pairing at N=256 for its stated direction. After old-space refitting the cancelled jumps need not remain cancelled. These are matched failures of this simple proposed implication, not a theorem excluding all divisor-feedback estimates. |
Accordingly no implication from “many forced arithmetic defects” to (8), to a nontrivial old-refit lower term in (5), or to a lower corrected correlation was obtained. The missing step cannot be replaced by assuming local energy removal adds independently across cells: a_n has a complete global tail and the prescribed refit couples the cells.
For the actual dyadic optimizer the desired edge remains
new proved estimates => beta_j <= Delta_(2^j,2^(j+1))/E_(2^j),
beta_j>=0, sum_j beta_j=infinity => E_N ->0 => RH.
No beta_j with divergent sum is constructed here. Equations (3)-(7) explain why the attempted coefficient-recovery mechanism does not supply one. The missing quantities are actual arithmetic coefficients/correlations, not a discarded numerical tail.
NS87 retains the full old-row correction in its finite observations and the complete inverse S⁻¹. At T=N⁶ its coupled tail error is dyadically summable on the normalized gain scale. That allows a lower arithmetic estimate to transfer if one is proved, but no positivity is created by localization. This note uses complete norms throughout and takes no physical or Mellin cutoff; there is no additional unbounded tail error hidden in (3)-(7).
NS74 matches the Gram/coercivity-only part: unitarily changed atoms preserve the full Gram, refit and (2) for pure combinations, while altering the target loads and leaving a positive approximation floor. It does NOT preserve the original alpha=1 divisor identity, so it is not a universal refutation of an original target-sensitive arithmetic estimate. The original positive-floor possibility is also visible conditionally in NS86’s bounded off-line-zero evaluator; no such zero is asserted.
NS83 excludes eventual constant whole-error contraction and eventual a/j with a>1. The attempted scale a/j with 0<a<=1 would be compatible, but remains unproved. No invented rate or effective onset is used. The hypothesis “new coefficients equal -mu” was an analytic stress test of an existing inequality, not an admitted candidate or a new numerical screen; no proposal gate, control replay, row or thaw is claimed.
Retain the explicit full-refit matrix-order check (3)-(4) and the scoped certificate-scale calculation (6)-(7) as audit bookkeeping. They prevent two invalid inferences from the previously established weighted estimate. They are not a new lower arithmetic estimate or an RH advance. No numerical search is justified by them. An attempt using the old refit must provide a separate arithmetic lower estimate of that term; an attempt using corrected correlations must estimate the full signed pairing with a compatible complete cost. Naming either obligation is insufficient.
Final classification: Same open gap — NB-GAIN / NS53/61/87. RH, G2, the actual cofinal signed-arithmetic lower bound and both historical original-evidence recovery groups remain open. No files outside this reviewer’s owned directory were edited.