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The global classical zeta zero-free denominator improves to 4.824

The corrected fixed detector and retained parameters certify $4.824$ but reject $4.82$ at a source-valid Yang endpoint. This locates a boundary of that packet, not the best possible classical zero-free denominator.

Certified de Bruijn-Newman upper bound Lambda <= 0.175

This proves one unconditional upper bound. It does not prove RH, $\Lambda=0$, optimality of the parameters, or the continuing target $\Lambda<0.01$. The original C-0068 finite-canopy witness remains invalid because of the terminal-cutoff defect recorded in O-0177; C-0108 independently regenerated every finite cell with the cutoff-uniform activation-state repair.

Two-row-hook half-shifted cup positivity in every area

This theorem proves coefficientwise half-shifted positivity only for the two-row hooks $(n,1)$. It does not cover general two-row partitions, other hook shapes, the complete all-rank cup kernel, reciprocal-heat transport, or RH; the classical determinant and incidence machinery is not claimed as new.

Third-strip two-row half-shifted cup positivity in every area

The classical generalized-Vandermonde, divided-difference, Newton-tail, Temperley-Lieb, and inverse-incidence machinery is not claimed as new. The theorem covers only the two-row family $(n,3)$; it does not establish $(n,k)$ for $k\ge4$, all two-row partitions in one theorem, the complete all-rank cup kernel, reciprocal-heat transport, or RH.

The sharp full-range Rosser--Schoenfeld product and totient coefficients

Exact optima only for the published product and totient formula shapes on their unchanged ranges. They are not new asymptotic error terms or stronger conclusions outside those forms.

The reciprocal-xi orthogonal ensemble

Final submission-neutral manuscript package with frozen source and reproducibility materials.

The optimal nonnegative CHJ-II fixed-range Perron weight is constant

The unique optimum is for nonnegative normalized $C^1$ weights in the fixed $k=1$, $[T,2T]$ source construction. The pointwise $T^*$ corollary requires a real-valued remainder; only the averaged theorem retains arbitrary complex coefficients.

Second-strip two-row half-shifted cup positivity in every area

The classical generalized-Vandermonde, divided-difference, Newton-tail, Temperley-Lieb, and inverse-incidence machinery is not claimed as new. The theorem covers only the two-row family $(n,2)$; it does not establish the families $(n,k)$ for $k\ge3$, general hooks, the complete all-rank cup kernel, reciprocal-heat transport, or RH.

One-row and one-column half-shifted cup positivity in every area

This proves the two boundary partition families $(n)$ and $(1^n)$ in every admissible area and dimension. It does not cover arbitrary partitions, the complete all-rank cup kernel, reciprocal-heat transport, or RH; the classical determinant, divided-difference, Riordan, and inversion ingredients are not claimed as new.

Johnston--Yang's surviving global VK psi decay improves to 0.2043325

The proof uses only the surviving denominator $51.34$, closed inward to $51.3401$. The decay $0.2043325$ is certified with margin but is not an optimality claim for this or any broader method.

Interior barriers for robust positivity programs toward the Riemann hypothesis

Draft snapshot. The reproducibility archive contains the exact TeX source and build instruction for this snapshot.

Fiori--Kadiri--Swidinsky's global absolute psi decay improves to 0.8817882

The low-range splice at $\log x=2008$ is active. At amplitude $9.2202181$, its exact decay cap is $0.8817882015416\ldots$, only about $1.5416\times10^{-9}$ above the displayed decay. No larger same-amplitude decay follows merely from another asymptotic zero-free improvement.

Every admissible two-row half-shifted cup coordinate is coefficientwise nonnegative

The theorem covers every admissible two-row partition, but not partitions with three or more positive parts, the complete all-rank cup kernel, reciprocal-heat transport, or RH. The generalized-Vandermonde product, arithmetic divided differences, Newton-tail determinant, standard Temperley--Lieb immanants, and inverse noncrossing-incidence machinery are classical and are not claimed as new.

Degree seven is the sharp all-rank Karlin polynomial threshold

The sharp threshold is for positive-leading polynomials and the signed centered consecutive-derivative family; it is not a theorem about arbitrary Toeplitz minors, full PF_m, or every rank failing in degree seven.

Complete area-at-most-six half-shifted cup positivity in every rank

The theorem is complete only for partitions of area at most six; it proves neither area-seven positivity nor the complete all-rank cup kernel, and its cup-stability extension uses the explicit written outer-arc and area-order proof.

Bellotti--Wong's six-parameter zero-count pair sharpens exactly

The source's six-parameter certification is assumed rather than reconstructed, the Artin branch requires Artin holomorphy, and the pair is only a new Pareto point, not a global optimum.

Axler's global fourth- and fifth-order prime intervals improve sharply

The order-four coefficient is the actual least full-range value because of the gap $1327<1361$. The order-five coefficient is exact only for the corrected FKS step-envelope splice and is not asserted globally least. No order-six or higher result is included.

All half-shifted flagged cup coordinates are positive through rank eight

The complete-coordinate positivity theorem stops at rank eight; only the explicit negative leading minor is all-rank, and no arbitrary-rank entrywise positivity or standard Temperley-Lieb immanant identification is proved.

A smaller explicit threshold for decimal digit sums of primes

The theorem imports the analytic lemmas in Lehmann arXiv:2606.04677v1 and certifies their specialization and threshold closure. It improves one explicit sufficient threshold by about 22.38%, but makes no least-threshold, global-optimality, or full first-principles rederivation claim.

The reciprocal-xi Fourier transform is strictly PF7

This is a fixed-order computer-assisted theorem for orders $1$ through $7$. Strictness requires strictly ordered distinct nodes. The default audit validates retained certificates and reruns C-0017 but does not replay every producer; inherited quadrature enclosures remain pinned inputs. No PF8, order-uniform, PF-infinity, zero-location, or RH conclusion is asserted.

Strict reflected Lorentz cones for Rankin coefficient packets

The theorem concerns exact Dirichlet coefficients only in the absolute-convergence Euler-product normalization. It does not carry the cone through analytic continuation, prove a completion theorem, control critical-strip curvature, locate zeros, or imply RH.

Strict TP4 for the folded reciprocal heat kernel on arithmetic cells

The result is strict TP4 only for four ordered arithmetic square cells and their stated positive row integrations; it is not unrestricted total positivity, an all-rank theorem, or an RH implication.

Pi/6 is the sharp zero-sector threshold for centered factorial D3 positivity

The sharp threshold applies to centered factorial order-three determinants only; it does not imply the full Polya-frequency hierarchy, exclude nonreal zeros, or imply RH.

Completed Weil positivity through support log 7

Continuum positivity is proved only through autocorrelation support $\log7$. Uniform eventual finite-matrix positivity is proved only through $\log6$, with a non-effective threshold, and the result is not an RH implication.

All-order positive kernels for curves over finite fields

The all-order kernel theorem is confined to curves over finite fields. The spectral Gram statement holds at every positive node, while the closed-point Euler series is used only for $\operatorname{Re}z>1/2$. No analogous number-field polarization or Riemann-zeta consequence is asserted.

A certified Polya-frequency order of 9,425,328,785,007

This is a finite-order corollary of the verified height $3,000,175,332,801$. The stated order is the largest certified by the displayed worst-case sector inequality at that endpoint, not a global optimality claim. It gives no order-uniform estimate, no passage to $PF_\infty$, and no implication for RH.

The sharp Rosser--Schoenfeld weighted-prime-sum upper coefficient

The upper value is sharp only for x >= 319 in the stated 1/log(x) formula. The lower value is not asserted optimal, and neither statement changes the asymptotic error order.

The sharp Rosser--Schoenfeld reciprocal-prime coefficient is 0.487203269718814...

Exact for the fixed range x >= 286 and the stated log^-2 formula shape. It is not a new asymptotic error term, an optimum for another starting point, or a lower-bound result.

The global Rosser--Schoenfeld reciprocal-prime lower coefficient is below 1/306

A same-shape splice of published estimates, not a new asymptotic error term and not the optimal coefficient for the actual reciprocal-prime Mertens error.

The displayed sector proves PF43, not PF44

Katkova · CMFT 7 (2007)

The paper's displayed 43π/44 sector, read through Schoenberg's criterion, certifies PF₄₃. PF₄₄ requires 44π/45. Later verified zero heights independently make PF₄₄ true.

Authors notified.

The coefficient-1.149 row is false at its published endpoint

Axler · Integers 24 (2024), Table 9

The exact same-range threshold is 1.1490003091852194519..., so 1.14900031 repairs the coefficient while preserving the published start. Retaining 1.149 moves the least integer start to 42,575,222,505.

The coefficient-1.071 row is false and has a sharp repair

Axler · Integers 24 (2024), Table 8

The stated inequality fails at p = 22,078,033. The exact same-range threshold is 1.071000358265763676..., so 1.07100036 repairs the coefficient without changing the range; retaining 1.071 moves the least integer start to 22,078,034.

Sharp fixed-template bounds for the pole-cancelled zeta derivative

Sharpness is proved only for the two displayed fixed denominator templates. The result does not optimize over varying denominators or arbitrary rational envelopes, and the exact corollary is an immediate algebraic extraction from a stronger pinned source theorem.

Lemma 5.2's printed product interval is inconsistent

Ramaré–Zuniga-Alterman · arXiv:2603.25961v3

For W = ∏_p(1 - 2/p² + 1/p³), the printed interval (0.4028, 0.4029) is incompatible with the formula. A certified lower bound is W > 0.421; direct evaluation gives approximately 0.4282495637.

Authors notified.

Broadbent et al.'s global theta coefficient improves from 0.024334 to 0.0143406585

Exact for the splice of published rounded-up upper step envelopes. It is not the optimal coefficient for the actual theta function, does not improve the lower signed error, and does not by itself propagate numerical improvements to downstream theorems.

Axler's compact prime-counting denominator coefficient improves from 70.935 to 44.053

The decimal $44.053$ is safe for the fixed published-envelope proof and is not claimed least for the actual $\pi(x)$. The conclusion preserves Axler's exact denominator shape and range and has no all-height RH implication.

Axler's Table 9 coefficient-1.149 row fails at its endpoint and has a sharp repair

This corrects only the coefficient-1.149 row of Axler's Proposition 11 and Table 9. The coefficient classification is restricted to the positive-denominator domain. The full finite interior has one exhaustive prime-enumeration implementation, with independent reconstruction of partition counts and decisive margins.

Axler's Table 8 coefficient-1.071 row fails and has a sharp repair

This repairs only Axler's coefficient-$1.071$ row. The exact optimum $a_*$ is for the published half-line $x\ge22{,}078{,}017$ and the one-parameter denominator $x/(\log x-a)$, not for arbitrary prime-counting bounds.

An exact phase diagram for the JustZeta comparisons

This is a qualitative range sharpening of the comparisons following the Garcia--Grenie--Molteni envelope. It does not improve that analytic envelope itself, prove optimality among all bounds for $(s-1)\zeta(s)$, or make an exhaustive novelty claim.

All 42 rows are valid, but every coefficient can be lowered on its published range

Axler · Integers 24 (2024), Table 7

The published starts are minimal. For each row, the exact same-range threshold is log(x₀) - x₀/π(x₀), with a certified smaller strict decimal repair. These are fixed-range optima, not global denominator optima.

Zeta zeros verified critical-line and simple through height 3,000,175,332,801

The theorem stops at the inclusive finite height $3,000,175,332,801$. Nothing is asserted above that endpoint, and the one-unit executable extension is pinned to its recorded CPython 3.14 macOS ARM64 python-flint 0.9.0 and FLINT 3.6.0 environment.