This file exists because the manuscript’s window parameter lambda is not
the parameter used in the published literature, and a referee cannot tell
from either side whether the W_4 >= 0 certificate is beyond or inside
the published state of the art. The translation below settles it.
Only published (arXiv) prior art is compared here. Self-published and repository-only work is deliberately excluded.
The semilocal window lambda > 1 corresponds to test functions supported
multiplicatively in [lambda^{-1}, lambda], i.e. in the logarithmic
coordinate to support half-width
a = log lambda
The published literature parameterizes directly by that half-width (Zhu
writes it L; the classical Yoshida / Connes–Consani bound is stated as
supp f ⊂ [-(log 2)/2, (log 2)/2]). So every result can be placed on one
axis:
| result | half-width a |
lambda = e^a |
|---|---|---|
| classical Yoshida / Connes–Consani | 0.3466 | 1.414 |
Zhu, arXiv:2608.24827, certified L = 0.8 |
0.8000 | 2.226 |
this manuscript, lambda = 3 (prop:v116-window-positive) |
1.0986 | 3.000 |
this manuscript, lambda = 4 (W_4 >= 0, prop:v126-full-window) |
1.3863 | 4.000 |
The lambda = 4 certificate is at 1.73× the half-width of the
published Zhu window and 4× the classical range. It is not subsumed on
reach.
The comparison is legitimate because the semilocal form QW_lambda and
the global Weil form agree on every test supported inside the window
(prop:v121-cofinal-rh relies on exactly this), so the two sides certify
the same object on the same functions.
Zhu (a = 0.8) |
this manuscript (a = 1.386) |
|
|---|---|---|
| statement | two-sided quantitative enclosure 8.9e-18 <= lambda_min <= 2.27e-17 |
nonnegativity of the complete form W_4 >= 0; v1.40 also certifies a simple ground and ordinary gap > 9.7546e-74 |
| parity | both sectors; ground state simple and even | both sectors; complete ground simple and even at lambda=4 (prop:v140-ground4), with 0<mu0<2.454e-75, mu1>1e-73 |
| method | one-stroke reduction to a single finite PSD matrix | structured inverse, simultaneous 17-column certificate, direction–complement gluing |
| arithmetic | interval | Arb ball arithmetic, 768–1280 bits, two-precision replay |
Zhu supplies an explicit positive lower bound for the ground value on a smaller window. This manuscript is stronger in reach. Neither subsumes the other, and the manuscript should say so rather than leave it to the reader.
The LDL* pivots and the invariant margin 1 - lambda_max(U, K) recorded in
evidence/v126/g2_simultaneous/ are not an ordinary-norm eigenvalue
bound and must not be quoted as one; see the scope string inside each
certificate JSON.
The former roughly-20-times comparison used unconverged finite compressions. It is not a semantic lock and does not determine the complete ground value. For lambda=4, the quoted empirical prediction 2.07e-73 is more than 84 times the certified complete-ground upper bound 2.454e-75. For lambda=6 and 8 the available unconverged compressions cannot test the law. The numerical comparison does not determine an asymptotic constant.
A separate external numerical benchmark is now available: at the existing lambda=3, N=120 benchmark, the first eight local transform-zero discrepancies agree with CCM section 6 Figure 1 at its displayed precision. The first is approximately 1.582329697193127e-34, matching 1.6e-34. The interval gates replay at 768 and 1024 bits; see v1.41. This is a finite-compression reproduction, not a complete-ground or cofinal convergence theorem.
prop:v125-cutoff-cost proves the scalar diagonal far majorant needs a
remote cutoff N+1 > L exp(M_phi/(1-c)) with M_phi / lambda -> 1.
This is a necessary lower barrier with doubly exponential growth in the
half-width a, not an asymptotic formula for an optimal cutoff.
Zhu’s abstract states that any one-stroke certificate must resolve
frequencies up to 2 pi e^{A_L} with A_L ~ 4 e^L — the same doubly
exponential shape, drawn as the same conclusion (“why the positivity route
cannot reach RH unassisted”). Groskin, arXiv:2607.02828, gives the
corresponding brute-cutoff cost (T ~ 10^{63} at c = 100) for
Connes–van Suijlekom truncations, using cutoff-free interval LDL^T —
the same factorization technique used here.
The objects differ: Zhu bounds frequency resolution for a one-stroke certificate; this manuscript bounds the remote cutoff for one specific scalar majorant, and says explicitly that this is a cost of that majorant rather than of every possible argument. But the result belongs to a published family and should be cited as such, not presented as new.
No published prior art was found for: norm contraction
(prop:v120-norm-counterexample), Schatten / Hilbert–Schmidt summation
(prop:v132-schatten), reciprocal-band approximate commutation (v1.33),
or the unconditional divergence of the scalar signed-primitive budget
(cor:v137-scalar-no-go).
The certified-Weil-positivity literature now shares a house style (certificate-first, “not a proof of RH”, scoped no-gos, checksummed witnesses), so tone does not distinguish this project. Two things would:
evidence/diag_ns2_semantic_lock/ is retained as a
diagnostic; the eight interval root gates in evidence/v141/ are the
certified finite-compression statement.evidence/MISSING.md and the
certificate ledger. Self-review cannot reach a REVIEWED state.(log 2)/2 positivity range.The CCM operator framework, de Branges / Hermite–Biehler constructions and matrix decompositions describe related parts of the problem. Their established identities and missing transfer hypotheses differ. A shared RH objective does not identify those hypotheses or their proof difficulty.
CCM define μ_λ = inf spec A_λ for exactly the form QW_λ this manuscript
certifies, prove μ_λ is decreasing in λ, write “we cannot assert that
μ_λ ≥ 0”, and prove lim μ_λ = 0 ⇒ RH. Hence, by monotonicity:
W_4 >= 0 ⟺ μ_λ >= 0 for every λ <= 4,
replacing “cannot assert” across a range 1.73× Zhu’s certified window and
beyond CCM’s own λ = 3 numerics. The floors W_λ >= −8·I are lower bounds
on μ_5, μ_6, μ_8. This is the cleanest positioning statement available and
it is a target statement, not a shortcut: cofinal μ_λ >= 0 is RH.
NS-5 is now completed in v1.40. The complete ground of W_4 is
simple, isolated and even, with 0<mu0<2.454e-75, mu1>1e-73 and
ordinary gap greater than 9.7546e-74 (prop:v140-ground4). The
1024/1280-bit shifted-inertia certificates retain the full tail inverse
change and reuse the archived complete lambda4 Schur bounds. These
figures replace no finite eigenvalue diagnostic: they are bounds for
the complete operator. The earlier eight-order figure concerns the
finite even-sector separation, not the complete global gap.
The precise real-distribution and trigonometric-core hypotheses of
Connes–van Suijlekom Theorem6.1 (arXiv:2511.23257) are verified in
cor:v140-real-zeros. Thus xi-hat_4, the entire Fourier transform
of the complete ground eigenfunction, has only real zeros. This does
not assert simplicity of those zeros or convergence to Riemann Xi.
See the proof and certificate.
Suzuki’s fixed-window construction uses a shift strictly below the spectral
bottom. The existence of that shifted positive space does not decide the
unshifted bottom’s sign. In particular, W_lambda >= -8 I safely permits a
shift 8+epsilon; the nonnegative endpoint shift may have a nullspace and
needs separate quotient/domain analysis. This is a limitation of the stated
shift construction, not a theorem excluding all de Branges methods.
Positive local screw data admit positive extensions under the cited extension theorems. The required arithmetic conclusion is that the prescribed global Weil screw function is positive. Uniqueness of arbitrary extensions is not established and is not substituted for that identification. NS-51 below instead uses the published nondegeneracy/continuity formulation and states its missing arithmetic injectivity input explicitly.
The prior Li assessment concerns the direct identification of rational Li
transforms with finite-window entire Paley–Wiener tests. It does not prove
that every indirect transfer is impossible. The historical diagnostics are
retained in evidence/diag_routes/li/.
Nyman–Beurling is reopened by NS-53. The established criterion is d_N -> 0;
any proved upper rate tending to zero suffices. Burnol’s lower bound of order
1/log N for the squared distance does not require a matching upper bound.
The archived finite distances in evidence/diag_routes/nb/ establish no
asymptotic decay theorem. NS-53 gives the exact residualized block gain
h^T S^(-1) h, with the full Schur complement, and the unconditional lower
bound ||h||^2 / (kappa log(M/N)), kappa=log(2pi)-gamma. A divergent
cumulative relative gain would prove convergence; the needed arithmetic
residual block-correlation bound is explicitly OPEN, not a consequence of
the finite identities. See evidence/ns53_nb_blocks/.
evidence/diag_circle_split/)The archived split contains Toeplitz-structured commutator and reflection
terms, including divided differences. It has not been identified with the
standard single-symbol parity pencils T(phi) +/- H(phi) under hypotheses
that would license the corresponding asymptotic theorems. The previous
unqualified identification is withdrawn, as required by the NS-48
conclusions audit, MAJOR A1.
The finite lambda=3 diagnostic records substantial cancellation between
positive- and negative-symbol contributions and concentration at low
frequencies. Its displayed tiny final energies are not complete-space
interval certificates. A nonzero compactly supported state has nonzero
Fourier tails: the previous claim of all mass below about 10 and no contact
with high-frequency negative regions is false. Quantified complete tail
control and a uniform signed estimate remain necessary. See the NS-31
circle/symbol audit and evidence/diag_true_symbol/ for the diagnostic
scope. The earlier KMS interpretation also remains withdrawn.
RH is equivalent to Lambda=0 after Rodgers–Tao’s unconditional lower bound.
The historical Polymath15 barrier reproduction does not exclude all bridges
from Weil forms. NS-52 fixes the normalization Z_t(w)=8 H_t(2w), so the
heat coefficient in the ordinary zeta coordinate is 1/4. Finite zero-cluster
transport has an exact contour formula; inverse-gap singularities cancel
inside the paired contribution, but the exterior-cluster logarithmic
derivative remains. The smooth-core complete zero sum exists; differentiating
it and controlling it uniformly in support are separate demands.
A particular same-test comparison q_0 >= q_t0 - C ||f||^2 is impossible
if the real-zero deformed function has even one zero at which the original
radical transform is nonzero. The result is conditional: this checkpoint
does not verify an unmatched zero at a specified time. It closes neither
the heat approach nor a comparison with a different justified test transport.
See evidence/ns52_heat_pairs/. No physical negative Weil direction follows.
For f in the complete logarithmic form domain, A_a f=0 exactly when
its screw potential (g * E_a f)|(-a,a) is affine. This local continuous
integral equation avoids an unproved H_0^1 assumption. Even parity allows
a constant potential; odd parity allows b x, and its derivative constant
must be retained. Ordinary L2 screw-operator injectivity cannot be silently
substituted for injectivity on the derivative completion.
A bounded self-adjoint rank-one perturbation of the same logarithmic
principal operator has a zero eigenfunction with logarithmic boundary decay
outside H_0^1. Thus the proposed generic bootstrap fails; an arithmetic
bootstrap is neither proved nor refuted. Affine-potential injectivity (API)
on the complete form domain remains OPEN. With continuity of the lowest
eigenvalue and a positive small-window starting point it would prevent a
first sign crossing, without a uniform positive gap. This is the established
nondegeneracy route in an explicit integral formulation, not a proved new
positivity mechanism. See evidence/ns51_kernel/.
The assessed finite-window data do not supply the missing cofinal assertions. They do not prove that every alternative mechanism fails, or that logically equivalent criteria have equally difficult proofs. Literal Toeplitz–Hankel structure and applicable asymptotics remain separate obligations; a truncated moment analogy is not a proved arithmetic determinacy equivalence. NS-51–53 record concrete first lemmas and their remaining inputs without promoting an unexcluded route to a proved estimate.
The sinc-lattice conclusion in the original diag_ns14_zeros report is
withdrawn. Its reconstruction omitted the (-1)^n change from unshifted
to centered cosine coefficients. The corrected exact formula has removable
poles, and its value at a retained lattice point is sqrt(L/2)(-1)^k v_k,
not zero merely because v_k is small. The new lambda=3, N=120 certificate
reproduces CCM’s local zero discrepancies. It does not rerun NS-14’s N=256
experiment or transfer zero locations to the complete ground. Thus no
negative conclusion about CCM step (b) follows from the old diagnostic.
See the preserved report’s correction.
G2 and RH remain open.
The complete lambda=4 ground transform now has a certified unique simple
zero in (gamma_1-0.1, gamma_1+0.1); see v1.42.
The even-sector separator 1e-67 and the exact trial’s complete energy
bound its orthogonal ground-projection error. Endpoint signs and a
positive derivative pass at 1024/1280 bits, with all old infinite-tail
bounds retained. This is a coarse local enclosure. It does not determine
the sign of the discrepancy or transfer the minute finite-compression
values, and it does not order earlier zeros. Cofinal convergence, G2
and RH remain open.
The complete first positive lambda=4 ground-transform zero is now certified
simple and within 8.752082e-33 of gamma_1. At 1024/1280 bits the evaluator
dual comparison gives sqrt(C_ell*rho)/0.0016 = 8.7520815739...e-33.
Its limit is the upper energy budget times the evaluator constant, not
the original sqrt(rho/b) ordinary projection error.
A hypothetical certified lower ground-energy bound l would improve the
separate trial-transfer estimate through b*(rho-l)/(b-l). Resolution
1e-71 would need a gap about 3.2032e-153, relative 1.3055e-78.
No such estimate is archived, and the frozen trial’s transform value
1.0641...e-40 at gamma_1 would still prevent that transfer alone from
giving the desired gamma-centered enclosure. The direct energy bound
instead requires an upper-energy improvement; a lower bound is not a
substitute for that improvement in the direct formula. It can improve
the separate transfer enclosure toward its frozen-trial floor
abs(fhat(gamma_1))/d = 1.064123340460...e-40/0.0016, about 6.65e-38.
This is a floor of that enclosure formula, not of the actual discrepancy;
attainment is not asserted.
The finite N=120 discrepancy (2.9250,2.9251)e-71 is separately certified.
Its scale lies below the resolution of the displayed archived estimates;
the actual complete discrepancy is not certified to have that scale.
No universal impossibility claim about every use of v125/v126 data is made.
This is a bound failure, not an object failure. CCM’s source-to-ground
approximation and cofinal convergence, G2 and RH remain open.
For an exact source-admissible pencil direction, the weighted criterion
requires L_a[v_k]/q_a^+[v_k] <= nu_k + eta_a ||v_k||^2/q_a^+[v_k], where
L_a is the loss between the exact symbol and its proposed lower minorant.
At zero allowed error, the relative loss budget is nu_k. For the weaker
bounded-error goal the eta term remains; tiny nu_k alone is not a no-go.
The section-7 diagnostic uses the uncompressed even head. Its twelve lambda=4 N=48 values below 1e-8 are diagnostic only, and do not establish source orthogonality. The exact class is restricted by the source constraint; mixtures require a full matrix inequality, not only directional checks. A fixed head supplies no complete-complement or cofinal estimate. The live criterion still requires uniform errors on a cofinal family, with the bounded-error weakening and both-parity/source hypotheses retained. The task stops at this distinction; no new concentration bound, G2 or RH claim.
The symbol is the sharply truncated explicit formula with the shifted Perron denominator. Its archimedean block cancels by the functional equation, without RH; the strict prime-power endpoint, trivial zeros and finite-prefix remainder remain. This is classical explicit-formula content, not a new lens or sign theorem. The lattice envelopes vary. The archived x=9,16,36,64 are integers; treating them as nonintegers loses a substantial endpoint term. Forty low sample locations and fourteen deep-trough/centroid locations were reproduced numerically with an explicit Gaussian/desmoothing convention, sufficient zero counts and residuals, all below 1e-6. This is not a cofinal trough-density or zero-spacing theorem.
The minorant task proves eta >= [(b+B)c-B]+ and eta >= [c*delta-Q]+, retaining positive spill and original packet energy. A conditional distinct- level bound eta >= [c D_a/(2K)-Q]+ gives the requested linear complexity consequence only with a uniform arithmetic level-distribution lower bound seen by a source-admissible bounded-energy packet. v1.37’s unconditional D_a->infinity does not establish these hypotheses. The J-trough lobe-floor family has the exact bound eta>=d_(J+1), but the required cofinal trough multiplicity is also not established. A two-value countermodel demonstrates why depth and concentration alone are insufficient; it is not a counterexample to the arithmetic criterion. Source projection and odd-sector scope are explicit. The route remains blocked, not closed. G2 and RH remain open.
The exact corrected zero field equals beta_a. Uniform lower measures of all its level bands form the named open input ZLD; a lower bound on each cumulative sublevel set is insufficient. The reduction separately requires a source-admissible packet whose Fourier density covers those same bands and whose original energy is bounded. On [-theta D,0] these hypotheses give eta >= [b alpha kappa theta^2 D/K - Q]+.
The cited zero-counting, zero-density, Landau-Gonek and pair-correlation results do not provide this signed value-distribution statement. No implication or equivalence with RH, pair correlation, the density hypothesis or Lindelof is established; logical independence is not claimed either. v1.37 gives global depth divergence, not uniform band measure. The explicit local slope estimate has a window-dependent cost. Broad Fourier pulses have energy log X + O(1), so broad spread alone does not give uniform Q. Both arithmetic distribution and bounded-energy coverage remain open. This is a gap in the proposed minorant obstruction, not an object failure; the route remains blocked and the weighted criterion valid. G2 and RH remain open. See evidence/v147/ and its primary-source bibliography.
The proposed peak-density scalar is not a bound justified by the finite histogram. A negative critical value makes the continuous coarea density essentially unbounded; v1.47’s absolute continuity never asserted a bounded peak. The exact weighted quantization implication survives without that assumption, with the uniform case giving c=c’*phi exactly. PR #14’s v1.48 revised candidate f3611ef recognizes the peak problem but remains held for incorrect K rearrangements, a bin maximum identified with the full concentration function, and an infeasible lambda=4 diagnostic vector (minimum bin ratio 0.636 at c’=1). The concentration-function bound is proved here with its correct rearrangement.
QC_1=phi*D-mean(beta^-) is an exact corrected-zero functional, but its growth alone is insufficient: even full-support absolutely continuous measures can have QC_1~D/2 and QC_2->0. The named cofinal input QG is growth of QC_K for every fixed K, plus separate admissible bounded-energy weighted packets. A QC_1/K law requires an additional quantization-shape hypothesis. Neither input is established; depth divergence gives no fraction limit. The failure is of the crude bound and diagnostic interpretation, not of the valid concentration criterion or physical form. G2 and RH remain open.
The positive Gaussian arithmetic radical h=E(H)(exp x) is an explicit comparison weight with pole mass 1/sqrt(3), independent of an unknown lowest eigenvector. Its full Picone identity gives q[hg]=J_h[g] minus (2/3)Var_nu(g) and the negative odd sinh square. The exact finite-window exterior contribution is retained; this is not a new tail metric.
For this weight, generic strict absorption fails even at the weaker bounded-error target. Discarding delta>0 of transformed energy forces eta >= [delta*kappa exp(gamma lambda^2)-O(lambda log lambda)]+. Deleting only prime-2 energy also fails. The proof uses compact far-translated bumps in two parity spaces; one even linear constraint removes the actual source without uncontrolled cross terms. It establishes no negative physical Weil direction and does not exclude a different weight.
The surviving Critical Arithmetic Energy comparison (CAE), with full coefficient one and the odd pole retained, is the existing uniform complement-floor requirement written in these coordinates. It is open; so is the stronger local capacity test that discards the positive potential. The calculation identifies the accuracy the weight demands but supplies no independent arithmetic sign bound. Both parities, source residual and cofinal uniformity are still needed. G2 and RH remain open.
The logarithmic-Laplacian theorem requires u = L_Delta u = 0 on the same open
set. Its hypothesis does not transfer to the arithmetic bounded nonlocal
remainder. The exact prime-2 construction in evidence/v155/proof.tex proves
local UCP false for the actual operator when 2a > log 2, in both parities for
a > log 2 and after finitely many source constraints. It never asserts
A_a f = 0 on the whole window. This is a failure of a proposed local property,
not a negative Weil direction or a failure of API.
The full real-space semigroup is not order preserving when 2a > log r, r^3-r-1=0, by an exact positive off-diagonal pairing of disjoint nonnegative tests. Bounded positive multiplication cannot repair this cone property; quadratic-form positivity remains a different question. A prime-free exterior-of-support observation cell yields a valid restricted uniqueness lemma. First-zero support saturation prevents obtaining such a cell inside the window from zero endpoint traces alone.
The complete-domain inward-dilation identity leaves precisely the signed
prime autocorrelation changes together with both pole changes to estimate.
No such arithmetic exclusion is supplied. The supporting primitive equation,
prime-entry norm obstruction and fixed-window Fredholm reduction are archived
in evidence/ns55_kernel/; the physical exceptional coupling remains unresolved.
Full build: 285 pages, zero undefined/duplicate references. API, G2 and RH remain open.
The q=2 criterion and its Gram formula are prior art in Werner Ehm, arXiv:2405.06349v1. This checkpoint independently states the tail-space obstruction to an off-line zero, adds a positive adjacent-average identity and an elementary N^(-2) trace budget, and keeps the optimized cross-correlations explicit. No priority claim is made for the criterion or the general smoothing idea. The certified finite quantities do not imply a cofinal lower bound.
The sharp-interpolant obstruction uses the classical dilation-invariant unitary in Baez-Duarte, math/0011254v1, Section 4. The localized Laguerre calculation extends that argument through Mellin integration; its scope is the full canonical sequence, not every subsequence or the optimal approximation. The changing-order one-atom control is unconditional and supplies no RH information. These NB results make no change to the certified Weil window or its comparison with published work.
NS-65 checks fixed iterations of ordinary arithmetic Cesaro averaging, not all summability methods. NS-66 records the exact fixed-order canonical norm-growth exponent using the classical zero-free/Mobius-series implication in Baez-Duarte, math/0202141v2, Lemma 2.1. This is a reformulation of zero location, not an independently improved Mertens estimate or zero-free region. No priority claim is made.
The complete q=2 Gram formula remains Ehm’s prior work. This checkpoint checks its elementary/remainder split at the quadratic-form level. The small entrywise correction cancels an explicit diagonal cusp; the repaired background is positive definite but cannot be uniformly comparable to the actual Gram on all finite integer coefficients. The lower exclusion uses known critical-line zeros; the upper uses unbounded critical-line zeta values, for example Soundararajan, Theorem 1. These are scoped comparisons, not negative Weil directions or failure of optimized approximation.
The canonical cell, renewal and norm-growth calculations are exact reformulations and bounds with their arithmetic input explicit; no priority claim or new zero-free region is asserted. The finite preconditioner uses the repaired kernel only to select coefficients, measuring every energy with the actual full Gram. Its efficiency does not establish the asymptotic numerator/cost estimate needed for convergence. Existing certified Weil windows and comparison claims are unchanged.
The rational repaired kernel yields a whole-line Sobolev energy, a cubic spline curvature comparison and two explicit coefficient rules. These are model identities and bounds; every finite gain retains the actual arithmetic Gram. No priority claim is made for the spline or Hardy-space mechanisms. The Mobius/log-taper test has prior motivation in Bettin–Conrey–Farmer, Theorem 1, whose q=1 asymptotic assumes RH and an additional inverse-derivative moment bound. Our q=2 projected block comparison neither uses nor disproves that theorem.
The inserted Blaschke factor is an explicit altered-family control: identical Grams and close finite statistics do not imply zero limiting error. The classical inner-factor role in NB approximation is explained by Burnol’s projection formula. No zero or positive limiting error is inferred for the original zeta family. The original cofinal arithmetic contraction remains unproved.
The q=2 distance extends the classical Hardy projection calculation in Burnol, Theorem 1.2 and Lemma 2.4. No novelty claim is made for inner-factor geometry or Beurling–Lax theory. The derivation preserves the complete exterior tail and proves a strict integer/continuous closure separation; the separation alone does not establish unequal distances for this particular target.
The new positive tail budget uses the full convergent zero sums. Its numeric calibration takes the published finite-height result of Platt–Trudgian and the explicit counting bound of Hasanalizade–Shen–Wong as external inputs. Only scalar evaluation is replayed here. No new zero verification, integer error upper bound, or RH conclusion is asserted.
The norm comparison uses elementary logarithmic interpolation on every reciprocal cell; its comparator retains the actual arithmetic samples. The projected condition bound follows by minimizing over old coordinates, and the gain guarantee uses the classical Kantorovich inequality, proved explicitly. No novelty claim is made for mass-matrix norm equivalence, Mobius inversion, Schur monotonicity or preconditioned steepest descent. No numerical implementation of the complete infinite sampling inverse is claimed. Its uniform guarantee does not apply to the smooth curvature rule.
The distinction between integer and continuous closure already appears in Baez-Duarte’s strengthening; NS76 gives an explicit q=2 separating witness, not a new general observation about these two kinds of closure. The related general strong criterion states smooth kernel hypotheses. Our logarithmic target is unbounded at zero, so we do not apply that theorem directly to it: the relevant fixed-order criterion is the separately justified bounded-integration argument in NS61.
The coefficient estimate uses elementary weighted Cauchy–Schwarz and divisor inversion on a finite sample prefix. No priority claim is made for these standard mechanisms or for polynomial conditioning of related NB systems. The statement is specific to the displayed q=2 physical-coordinate Gram. The explicit large-cutoff construction is an upper bound on sufficient work, not a necessary cutoff, complexity lower bound, or practical implementation. It does not resolve the original cofinal approximation input.
NS82 applies an elementary norm-perturbation estimate to the complete Stirling remainder already retained in NS78. It changes the comparator: finite physical integrals plus a rank-two logarithmic tail, instead of truncated reciprocal samples. The cutoff improvement is a sufficient construction bound; no priority, necessary complexity or convergence claim.
Burnol, Advances in Mathematics 170 (2002), 56–70, Sections 3–5, arXiv:math/0103058, supplies the phase-operator dual-vector construction and squared-multiplicity Hilbert blocks. NS83 extends that calculation by explicit additional Hardy operators; it does not claim a new lower-bound mechanism or priority. The small-endpoint and complete-tail estimates are carried through for each fixed order, and the target weight changes to |rho|^(-2q).
This is a lower liminf, not an asymptotic equality or a finite-N bound with known onset. The resulting restriction on total relative contraction does not conflict with the finite-block preconditioner guarantees. The missing unconditional upper estimate remains missing. No RH assumption is used in the critical-zero construction; no statement about off-line zeros is inferred.
The full conclusion register and archived audit dispositions were reviewed. This pass repairs the residual Toeplitz/Fourier-support overclaims above, clarifies the cutoff as a lower barrier, and removes map arrows that could misrepresent QG/ZLD plus packets as a positivity proof. These inputs obstruct specified low-complexity minorants; CAE is the distinct signed-floor target.
NS-85 tests the classical full logarithmic Mobius taper in the fixed q=2
norm, including every physical tail. At N=256 its squared error is about
0.352175; optimizing only the first coefficient reduces it to about
0.045293, still far above the unrestricted optimum about 0.0000141424.
Its exact interior residual is a logarithmically smoothed prime-counting
discrepancy. No growing-interval mean-square bound is obtained. Finite
underperformance neither disproves its possible asymptotic convergence nor
imports the RH and inverse-derivative hypotheses of Bettin–Conrey–Farmer.
See evidence/ns85_taper_test/ and the versioned NS-84/85 review. No new
manuscript version, cofinal upper estimate, G2 or RH claim is made.