bibkey: srivastav2025sievevaughan authors: Priyamvad Srivastav year: 2025 title: Log-free bounds on exponential sums over primes doi: null url: https://arxiv.org/abs/2505.07803v2 claim: The sieve-weighted Vaughan identity retains exact von Mangoldt recovery for general normalized sieve weights by preserving extra convolution terms; the stated additive exponential-sum bound supplies no centered signed Robin-tail estimate at zero frequency. strata_touched: [] license: citation-only triage: anchor

Exact recovery with sieve weights and the remaining signed estimate

The inspected primary is arXiv:2505.07803v2, revised 27 January 2026; v1 was submitted 12 May 2025. The locators below refer to v2. This note attributes the source’s results and records their parameter correspondence; it supplies neither an independent proof audit nor Lean certification. The published identities and sieve arguments are reused, not new project mathematics.

The recovery operator includes additional convolutions

Write $*$ for Dirichlet convolution and $1 (n) = 1$ for $n \geq 1$ . Juxtaposition of arithmetic functions means pointwise multiplication. Section 2, Lemma 2.1 and its first Remark allow arbitrary weights with

$λ (1) = 1, θ + θ^{'} = μ .$

With the source’s least-common-multiple transform

$h (d) = [d_{1}, d_{2}] = d \sum λ (d_{1}) θ^{'} (d_{2}),$

and $V > 1$ , the cited identity is

$Λ = h * lo g - 1 * h * Λ_{\leq V} + [(1 * θ) (1 * λ)] * Λ_{> V} + Λ_{\leq V} . (SV1)$

Here $Λ_{\leq V}$ and $Λ_{> V}$ truncate the complete von Mangoldt function, including prime powers. The source also states the corresponding identity for $μ$ . Its standard weight choice has $1 < U < U_{1}$ , $R > 1$ , with $θ^{'}$ supported on $d \leq U_{1}$ and $λ$ on $d \leq R$ ; consequently $h$ is supported on $d \leq U_{1} R$ . The first Remark explicitly permits other normalized weight choices.

This recovery operator differs from requiring $\sum_{d ∣ m} a (d) = Λ (m)$ for a single coefficient vector $a$ . A restriction proved for that simple divisor response would not exclude (SV1). Optimizing sieve weights and retaining exact recovery are compatible in this published formulation, provided all additional terms are kept. Those terms have no favorable sign asserted by Lemma 2.1.

The divisor-to-frequency expansion is already supplied

The weights in equation (2.1) are

$λ (d) = \frac{d μ ( d )}{φ ( d )} \frac{G _{q d} ( R / d )}{G _{q} ( R )} 1_{(d, q) = 1}, G_{ℓ} (x) = r \leq x (r, ℓ) = 1 \sum \frac{μ ( r ) ^{2}}{φ ( r )} .$

Section 5.1, Lemma 5.1 gives, for every integer $n \geq 1$ ,

$G_{q} (R) d ∣ n \sum λ (d) = r \leq R (r, q) = 1 \sum \frac{μ ( r )}{φ ( r )} c_{r} (n), c_{r} (n) = a mod r (a, r) = 1 \sum e^{2 πian / r} . (SV2)$

The paper attributes this Selberg-weight/Ramanujan-sum connection to Kobayashi and Huxley. The cited predecessors are Kobayashi, A note on the Selberg sieve and the large sieve, Proc. Japan Acad. 49 (1973), 1–5, and Huxley, The distribution of prime numbers (1972). Their full texts were not independently inspected here.

Thus an existing source supplies this particular transition from divisor responses to additive frequency channels. It is not a new FIB spectral theorem. Neither (SV1) nor (SV2) identifies these additive frequencies with the nontrivial zeros of the Riemann zeta function or transports the five Zeckendorf occupancy classes to the needed arithmetic amplitudes.

The exponential-sum theorem keeps its original parameters

To distinguish the paper’s small parameter from the Robin excess exponent, write it as $κ$ . Theorem 1 assumes $0 < κ \leq 1/10$ , sufficiently large $x \geq x_{0} (κ)$ , and

$u = \frac{a}{q} + \frac{δ}{x}, (a, q) = 1, ∣ δ ∣ \leq \frac{x ^{1/5 + κ}}{q}, 1 \leq q \leq x^{2/5 - κ}, δ_{0} = max {1, ∣ δ ∣/4} .$

It states the uncentered bound

$n \leq x \sum Λ (n) e^{2 πin u} \leq \frac{q}{φ ( q )} F_{κ} (\frac{lo g ( δ _{0} q )}{lo g x}, \frac{lo g ^{+} ( δ _{0} / q )}{lo g x}) \frac{x}{δ _{0} q}, (SV3)$

with $F_{κ}$ specified in equation (1.4) and $lo g^{+} z = max {lo g z, 0}$ . The theorem also bounds the corresponding Möbius exponential sum. The Remarks explicitly include $q = 1$ and $q = 2$ . The source calls the result semi-explicit: $x_{0} (κ)$ is effectively computable but is not given as a numerical threshold.

For the Robin Chebyshev sum, the additive frequency is $u = 0$ , represented by $a = 0$ , $q = 1$ , $δ = 0$ . Here $δ_{0} = 1$ and the two arguments of $F_{κ}$ are zero. The cited theorem bounds the uncentered $ψ (x)$ ; it states no estimate for the centered difference $ψ (x) - x$ at this frequency. Its nonzero-frequency savings cannot be substituted for the original signed tail merely by changing coordinates.

The same-source transport obligation

Use the existing jointly selected source class, with its same actual integer $N$ , clock $A = lo g N$ , own CA price, proper GA1, regularity and comparison against every integer $m \geq N$ . That source application is not Lean certified. Its fixed excess exponent $0 < η < 1/2$ is independent of the parameter $κ$ in (SV3).

If (SV1) is used as the recovery supplier, write its entire right-hand side as

$C (n) = (h * lo g - 1 * h * Λ_{\leq V} + [(1 * θ) (1 * λ)] * Λ_{> V} + Λ_{\leq V}) (n) .$

For any chosen normalized weights, the unpaid quantitative input is a same-source lower bound of the form

$A lo g A Y \to \infty lim \int_{A}^{Y} (n \leq t \sum C (n) - t) \frac{1 + lo g t}{t ^{2} lo g ^{2} t} d t \geq - B_{new} (A), B_{new} (A) = o (A^{1/2 - η} lo g A) . (SV4)$

The weight choice may depend on $N$ , but all terms in $C$ must use that same choice. Equation (SV1) provides the exact full $Λ$ recovery in this expression, not an estimate for (SV4). The subtracted baseline is still $t$ and the outer integration still starts at the actual $A$ . The inner summatory functions retain terms below $A$ and all prime powers; the finite $Λ_{\leq V}$ contribution and every convolution crossing a cutoff remain present. The limit in (SV4) is taken for the joint centered expression, without asserting separate convergence of the components.

The original account

$- X_{N} = I_{ψ} (A) + K_{N} (A), X_{N} = lo g G (N) - γ,$

keeps its original $K_{N} (A)$ . Neither the norm estimates used for the type-II sums nor (SV3) supplies a sign or a rate for (SV4) on this selected class. A future estimate must also justify its connection to the actual CA competitors of this $N$ . The source contributes an exact alternative recovery mechanism and the existing additive-frequency expansion; it contributes no new signed Robin margin in this application.

A classical mean supplier for one fixed mixed response

For a specific permissible weight choice in (SV1), use the same actual integer source and clock $A = lo g N > 1$ , and put

$L = lo g A, r = L /8, R = e^{r}, U = e^{2 r}, U_{1} = e^{3 r}, V = e^{4 r},$

$ρ_{z} (d) = μ (d) lo g (z / d) 1_{d \leq z}, λ = ρ_{R} / r, θ^{'} = (ρ_{U_{1}} - ρ_{U}) / r, θ = μ - θ^{'} .$

These are logarithmic Barban–Vehov weights. They satisfy the general identity’s normalization; they are not the optimized weights in (2.1). The source’s quantitative exponential-sum theorems are not asserted for this replacement. For the same LCM transform $h$ , define

$H_{A} (y) = d \sum h (d) ⌊ y / d ⌋, S_{A} = d \sum \frac{h ( d )}{d}, H_{A}^{-} (y) = max {- H_{A} (y), 0} .$

The finite support lies in $d \leq R U_{1} = A$ . Use the bilinear form $⟨ f, g ⟩ = \sum_{d, e} f (d) g (e) / [d, e]$ , and put $Q (z) = ⟨ ρ_{z}, ρ_{z} ⟩$ and $E (a, b) = ⟨ ρ_{b} - ρ_{a}, ρ_{b} - ρ_{a} ⟩$ . The relevant existing supplier is Carneiro–Chirre–Helfgott–Mejía-Cordero, Optimality for the two-parameter quadratic sieve, arXiv:2005.03162v6, Theorem 1.1, equations (1.7)–(1.8). Its normalized weights are defined in (1.4)–(1.5). Multiplying the source’s $M (1, z; h_{0})$ by $(lo g z)^{2}$ gives $Q (z)$ ; multiplying $M (a, b; h_{0})$ by $(lo g (b / a))^{2}$ gives $E (a, b)$ . Thus the published expansions read

$Q (z) = lo g z - κ_{BV} + O (e^{- c_{0} l o g z}),$

$E (a, b) = lo g (b / a) - 2 κ_{BV} + O (e^{- c_{0} l o g (b / a)} + e^{- c_{0} l o g a}) .$

Here $κ_{BV}$ is the constant in the cited Theorem 1.1, distinct from the small parameter $κ$ in (SV3), and $c_{0} > 0$ . Polarization supplies the parameter correspondence

$2 r^{2} S_{A} = Q (U_{1}) - Q (U) - E (R, U_{1}) + E (R, U) .$

Both leading terms and both second-order constants cancel. The cited classical theorem therefore gives the two-sided envelope

$∣ S_{A} ∣ ≪ \frac{e ^{- c_{1} L}}{L ^{2}} .$

Here $c_{1} > 0$ is fixed.

This application is not a new mean theorem or Lean certification. It does not determine the sign of $S_{A}$ . Nor does it estimate all finite negative responses: the exact floor correction remains

$H_{A} (y) = y S_{A} - d \sum h (d) {y / d},$

with the full support retained even when $d > y$ .

For the unchanged actual integer $N$ , write $a_{p} = v_{p} (N)$ and $ϵ = 1/ (A lo g A)$ . The actual next-layer gains and losses are

$β_{j} (p) = lo g \frac{1 - p ^{- (j + 1)}}{1 - p ^{- j}}, δ_{N, p, k} = ϵ lo g p - β_{a_{p} + k} (p) .$

Own-price global CA optimality and decreasing layer gains supply $0 \leq δ_{N, p, k} \leq ϵ lo g p$ . The local negative-part candidate asks for an eventual same-source bound

$A \leq x \leq 2 A sup \frac{\sum _{p^{k} \leq x} δ _{N, p, k} H _{A}^{-} ( x / p ^{k} )}{ϵ x} \leq A^{- η} .$

This candidate remains unpaid. A two-sided estimate for $S_{A}$ , or positivity of that mean alone, does not supply it. A proof of this candidate would control only one local conversion defect; the Euler gain, other recovery convolutions, centered baseline and complete infinite tail in (SV4) would still need estimates with the same source and weight choice. No original Robin-budget improvement is claimed from the classical mean correspondence.

Fixed-truncation Mellin interface and remaining source coupling

The same source’s Theorem 1.2 also applies with the common parameters $(D_{1}, D_{2}) = (1, R^{3})$ . The two normalized profiles have derivative supports $(2/3, 1)$ and $(0, 1/3)$ , respectively. Their derivative inner product is zero, so this theorem supplies a two-sided error rather than a positive main term for the fixed mixed mean. The following interface uses the published Mellin kernel in section 2.1, equation (2.5). The finite-truncation and local-residue computations are mathematical applications of that kernel; no originality or Lean certification is asserted.

Write

$G (s, w) = p \prod (1 - \frac{p ^{- 2 - s - w} ( 1 - p ^{- s} ) ( 1 - p ^{- w} )}{( 1 - p ^{- 1 - s} ) ( 1 - p ^{- 1 - w} )}) .$

This is the source’s Euler factor in equivalent notation, with

$d, e \geq 1 \sum \frac{μ ( d ) μ ( e )}{[ d , e ] d ^{s} e ^{w}} = \frac{ζ ( 1 + s + w ) G ( s , w )}{ζ ( 1 + s ) ζ ( 1 + w )} (ℜ s, ℜ w > 0) .$

Keeping the actual finite $d$ -cutoff, define

$P_{R} (w) = d \leq R \sum \frac{μ ( d ) lo g ( R / d )}{d} p ∣ d \prod \frac{1 - p ^{- w}}{1 - p ^{- 1 - w}}, F_{R} (w) = \frac{( R ^{3 w} - R ^{2 w} ) P _{R} ( w )}{w ^{2} ζ ( 1 + w )} .$

Summing the $e$ -variable by its Euler product and applying Mellin inversion gives the exact interface

$r^{2} S_{A} = \frac{1}{2 πi} \int_{(c)} F_{R} (w) d w, c > 0. (SV5)$

Here $P_{R} (0) = r$ and $F_{R} (0) = r^{2}$ by removable continuation; zero is not an omitted pole. Choose $0 < b < 1$ and $Y > 0$ so that the boundary of the rectangle with vertical sides $c, - b$ and horizontal sides $\pm Y$ meets no zero of $ζ (1 + w)$ . Let $Γ$ run from $c - iY$ to $- b - iY$ , then to $- b + iY$ and $c + iY$ . The full finite-height identity is

$r^{2} S_{A} = ζ (ν) = 0 ℜ ν > 1 - b, ∣ℑ ν ∣ < Y \sum Res_{w = ν - 1} F_{R} (w) + E_{R},$

$E_{R} = \frac{1}{2 πi} [\int_{Γ} F_{R} (w) d w + \int_{c - i \infty}^{c - iY} F_{R} (w) d w + \int_{c + iY}^{c + i \infty} F_{R} (w) d w] .$

Zeros are summed by distinct position; pole orders retain their multiplicities. The finite Euler factors have possible poles on $ℜ w = - 1$ , which this rectangle does not cross. In particular, the auxiliary height $Y$ neither truncates away the remaining contributions nor replaces the original arithmetic scale $T = A lo g A$ . For

$M_{R} (c) = d \leq R \sum \frac{∣ μ ( d ) ∣ lo g ( R / d )}{d} p ∣ d \prod \frac{1 + p ^{- c}}{1 - p ^{- 1 - c}},$

the two right-line tails obey

$\int_{∣ t ∣ > Y} ∣ F_{R} (c + i t) ∣ d t \leq \frac{2 ( R ^{3 c} + R ^{2 c} ) ζ ( 1 + c ) M _{R} ( c )}{Y} .$

No required fixed-power bound for $E_{R}$ follows from this identity or its displayed tail estimate.

For a fixed zero $ρ = β + i τ$ with $β > 1/2$ and $τ > 0$ , put $α = ρ - 1$ and $δ = 1 - β \in (0, 1/2)$ . In a fixed sufficiently small neighborhood of $α$ , the truncated factor has the local expansion

$P_{R} (w) = R^{- w} \frac{G ( - w , w )}{w ^{2} ζ ( 1 - w )} + O_{ρ} (R^{δ /2}) . (SV6)$

Its contour derivation starts from

$P_{R} (w) = \frac{1}{2 πi} \int_{(a)} \frac{R ^{s}}{s ^{2}} \frac{ζ ( 1 + s + w )}{ζ ( 1 + s )} G (s, w) d s, a > δ,$

with the neighborhood chosen small enough for initial absolute convergence. Move this line to $ℜ s = δ /2 > 0$ . The only crossed pole is $s = - w$ . On the new line, $ζ (1 + s)$ has real-part parameter greater than one, and $ℜ w > - 1/2$ ensures locally uniform absolute convergence of this Euler product. The usual vertical growth bound for $ζ (1 + s + w)$ together with $s^{- 2}$ gives the stated error. This local argument does not extend the source’s narrower uniform region by assertion. Uniformity on the neighborhood also controls each fixed derivative of the error by Cauchy’s estimate.

The coefficient in (SV6) is nonzero at $w = α$ . Indeed,

$G (- w, w) = p \prod \frac{( p - 1 ) ( p + 1 - p ^{w} - p ^{- w} )}{( p ^{1 + w} - 1 ) ( p ^{1 - w} - 1 )} .$

For $∣ℜ w ∣ < 1$ this product is locally absolutely convergent. Its denominators are nonzero. If its numerator vanished and $z = p^{w}$ , then $z + z^{- 1} = p + 1$ , whereas $p^{- 1} < ∣ z ∣ < p$ implies $∣ z + z^{- 1} ∣ \leq ∣ z ∣ + ∣ z ∣^{- 1} < p + 1$ . Thus every factor is nonzero. Also $ζ (1 - α) = ζ (2 - ρ) \neq = 0$ since its real-part parameter exceeds one. The first term at $α$ has size a nonzero constant times $R^{δ}$ , so $P_{R} (ρ - 1) \neq = 0$ for all sufficiently large $R$ .

Let $m \geq 1$ be the multiplicity of $ρ$ and define

$C_{ρ} = \frac{m G ( 1 - ρ , ρ - 1 )}{( ρ - 1 ) ^{4} ζ ( 2 - ρ ) ζ ^{(m)} ( ρ )} \neq = 0.$

For this pair alone, the normalized residue contribution is

$Z_{ρ} (r) = \frac{1}{r ^{2}} (Res_{w = ρ - 1} F_{R} (w) + Res_{w = \overset{ρ}{ˉ} - 1} F_{R} (w)) = - 2 r^{m - 3} R^{- δ} ℜ (C_{ρ} e^{i τ r}) + O_{ρ} (r^{m - 3} R^{- δ} (r^{- 1} + R^{- δ /2})) . (SV7)$

To obtain the coefficient, insert (SV6) into $F_{R}$ . The main meromorphic part is $(R^{2w}-R^w)\mathfrak G(-w,w)/ (w^4\zeta(1-w)\zeta(1+w))$. The leading Laurent term of $1/ ζ (1 + w)$ at $α$ is $m! / (ζ^{(m)} (ρ) (w - α)^{m})$ . The $(m - 1)$ st derivative of $- R^{w}$ contributes $- C_{ρ} r^{m - 1} R^{α}$ before normalization. Lower derivatives, the $R^{2 w}$ term and the error in (SV6) give the displayed local error. Other zero contributions remain outside that local error.

For a rectangle containing this pair, let $B_{ρ, N}$ contain all other residues and $E_{R}$ , divided by $r^{2}$ . Then $S_{A} = Z_{ρ} (r) + B_{ρ, N}$ exactly. If a same-source estimate $(S_{A})_{-} \leq C_{η} A^{- η} / L$ were supplied, (SV7) would require

$2ℜ (C_{ρ} e^{i τL /8}) \leq \frac{B _{ρ, N}}{r ^{m - 3} A ^{- δ /8}} + K_{ρ} (r^{- 1} + A^{- δ /16}) + \frac{C _{η}}{8} r^{2 - m} A^{- η + δ /8} (SV8)$

for a constant $K_{ρ}$ independent of $N$ . When $β > 1 - 8 η$ , the last term tends to zero. Thus an individual zero package cannot be paid by its decay alone at this scale; the full other contributions or the actual source phases must be controlled. Neither an actual negative-phase source subsequence nor that compensating control is supplied here. Local oscillation on continuous clocks does not establish a sign change of the full $S_{A}$ or a failure on the selected integer family.

The phase in this interface is $τL /8$ . Identifying it with FIB composition rotation $C = M J$ would require an additional intertwining map preserving these weights and source conditions. This interface constructs no such map. The CA fixed point, all-integer right-tail maximality and source excess remain the joint hypotheses of the original $N$ ; their implication of (SV8) is the open coupling obligation. No required same-source one-sided mean bound, negative-part bound or complete original Robin estimate is obtained from this interface.

A classical higher-prime-power budget with the mean unpaid

Keep the same actual $N$ , fixed $η$ , weights and cutoffs. Write $σ_{A} = (S_{A})_{-}$ and, for each fixed integer $m \geq 2$ , define

$D_{N}^{(\geq m)} (x) = p^{k} \leq x k \geq m \sum δ_{N, p, k} H_{A}^{-} (x / p^{k}) .$

The finite-correlation supplier is Chen An, A Generalization of Graham’s Estimate on the Barban-Vehov Problem, arXiv:2206.10104v1, Theorem 1.3. Its rational-field case, explicitly attributed there to Graham (1978, p. 84), gives

$C (w, z; t) := n \leq t \sum d ∣ n \sum ρ_{w} (d) e ∣ n \sum ρ_{z} (e) = t lo g w + O (t), 1 \leq w < z \leq t .$

No condition $w z \leq t$ is imposed. For $y \geq U_{1}$ , apply this supplier with $(w, z, t) = (R, U_{1}, y)$ and $(R, U, y)$ . For $U < y < U_{1}$ , first use the exact truncation relation

$C (R, U_{1}; y) = C (R, y; y) + r lo g (U_{1} / y),$

then apply the supplier with $(R, y, y)$ and $(R, U, y)$ . The resulting transition term $lo g (U_{1} / y) / r$ in $H_{A} (y)$ is nonnegative. For $1 \leq y \leq U$ , the divisor sum of $θ^{'}$ is supported only at $n = 1$ , so $H_{A} (y) = 1$ exactly. For $0 \leq y < 1$ , $H_{A} (y) = 0$ . Consequently an absolute constant $C > 0$ supplies

$H_{A}^{-} (y) = 0 (0 \leq y \leq U), H_{A}^{-} (y) \leq C y / L^{2} (y > U) .$

The full-support floor correction supplies the second envelope

$H_{A}^{-} (y) \leq σ_{A} y + B_{A}, B_{A} = \frac{( R - 1 ) ( U _{1} - U )}{r ^{2}} \leq \frac{64 A}{L ^{2}} .$

$E_{A} (y) = (H_{A}^{-} (y) - σ_{A} y)_{+}, a = C / L^{2} .$

For all $y \geq 0$ the two envelopes imply $E_{A} (y) \leq min (a y, B_{A})$ and $H_{A}^{-} (y) \leq σ_{A} y + E_{A} (y)$ . No sign of $S_{A}$ is assumed.

Reuse the existing Dusart prime-power input: $Γ_{2} (t) = ψ (t) - ϑ (t) ≪ t$ and $ϑ (t) < 2 t$ for $t \geq 1$ . For every fixed $m \geq 2$ , classical prime-power counting therefore gives

$Γ_{m} (t) := p^{k} \leq t k \geq m \sum lo g p = k = m \sum ⌊ l o g_{2} t ⌋ ϑ (t^{1/ k}) \leq K_{m} t^{1/ m} .$

Here $K_{m}$ is a fixed constant; the tail after $k = m$ is bounded by $2 lo g_{2} t t^{1/ (m + 1)} = O_{m} (t^{1/ m})$ for $t \geq 2^{m}$ , and the count is zero below $2^{m}$ . This is an application of the classical counting bound, not a new prime-distribution theorem.

Put $b = B_{A} / x > 0$ and $z = a / b > 0$ . Enlarging the nonnegative residual sum to all $p, k \geq m$ and integrating against its counting measure gives the exact identity

$p, k \geq m \sum lo g p min (a / p^{k}, b) = b Γ_{m} (z) + a \int_{(z, \infty)} \frac{d Γ _{m} ( t )}{t} = a \int_{z}^{\infty} \frac{Γ _{m} ( t )}{t ^{2}} d t \leq \frac{m K _{m}}{m - 1} a^{1/ m} b^{1 - 1/ m} .$

The boundary terms cancel because $b = a / z$ . The formula includes an atom at $z$ in the first term, so it also holds when the real splitting point is a prime power. Since $0 \leq δ_{N, p, k} \leq ϵ lo g p$ , the uniform application is

$A \leq x \leq 2 A sup \frac{D _{N}^{(\geq m)} ( x )}{ϵ x} \leq c_{m} σ_{A} + C_{m} \frac{A ^{- γ_{m}}}{L ^{2}}, γ_{m} = \frac{m - 1}{2 m}, c_{m} = p \sum \frac{lo g p}{p ^{m - 1} ( p - 1 )} < \infty. (SV9)$

All $k \geq m$ are retained. In particular $m = 2$ gives the floor residual $O (A^{- 1/4} / L^{2})$ . If the separate same-source mean requirement $σ_{A} = O (A^{- η} / L)$ were supplied, this complete higher-power component would be $o (A^{- η})$ for the original $0 < η \leq 1/4$ , including the endpoint. That requirement is unpaid; the existing two-sided mean envelope does not have this fixed-power rate. For $η > 1/4$ , the $m = 2$ upper envelope alone is insufficient, which does not refute the negative-part candidate.

For any original fixed $0 < η < 1/2$ , take $m_{η} = ⌈(1 - 2 η)^{- 1} ⌉ \geq 2$ and define the finite low-layer residual on that same actual $N$ by

$R_{N, η} = A \leq x \leq 2 A sup \frac{1}{ϵ x} k = 2 \sum m_{η} - 1 p^{k} \leq x \sum δ_{N, p, k} E_{A} (x / p^{k}) .$

An empty sum is zero. Since $E_{A} \leq H_{A}^{-}$ and $γ_{m_{η}} \geq η$ , the same envelopes give

$R_{N, η} \leq A \leq x \leq 2 A sup \frac{D _{N}^{(\geq 2)} ( x )}{ϵ x} \leq R_{N, η} + c_{2} σ_{A} + C_{m_{η}} \frac{A ^{- γ_{m_{η}}}}{L ^{2}} . (SV10)$

The last term is unconditionally $o (A^{- η})$ . Conditional on the still-unpaid mean requirement, obtaining $sup_{A \leq x \leq 2 A} D_{N}^{(\geq 2)} (x) / (ϵ x) = o (A^{- η})$ is equivalent to $R_{N, η} = o (A^{- η})$ . This stronger target is sufficient for the higher-power component of the local candidate, whose original threshold is $A^{- η}$ . For $1/4 < η \leq 1/3$ this leaves the square layer only; all $k \geq 3$ residuals already have the required rate. No exponent is lowered.

The actual CA layer weights do not supply another power saving by themselves. The logarithmic series gives $β_{j + 1} (p) \leq p^{- 1} β_{j} (p)$ , and own-price optimality gives $β_{a_{p} + 1} (p) \leq ϵ lo g p$ . Thus for all $k \geq 2$ ,

$(1 - p^{1 - k}) ϵ lo g p \leq δ_{N, p, k} \leq ϵ lo g p .$

In particular, $D_{N}^{(\geq 2)} (x) / (ϵ x)$ is between one half and one times $x^{- 1} \sum_{p^{k} \leq x, k \geq 2} lo g p H_{A}^{-} (x / p^{k})$ . This comparison gives no lower bound for $H_{A}^{-}$ and leaves open improvement through its sampling on the selected sources.

These bounds are paper-level applications of existing suppliers, without Lean certification or a claim of original number theory. The mean requirement, finite low layers when needed, prime layer $k = 1$ , other recovery terms and full signed tail in (SV4) remain unpaid. No complete Robin-budget gain follows from (SV9) or (SV10).

A signed lowest-frequency band for these actual coefficients

Keep exactly the logarithmic weights, cutoffs and actual integer source above. The following application uses the existing classical Mertens input, not a new cancellation theorem. Write

$W (u) = \frac{u ^{3/5}}{( lo g u ) ^{1/5}}, M (t) = n \leq t \sum μ (n) ≪ t e^{- c W (l o g t)} .$

All estimates in this section are for sufficiently large $A$ , with fixed positive constants. For an interval $I \subset (V /2, V]$ , the actual LCM decomposition forces

$[d, e] \in I ⟹ (d, e) = 1, [d, e] = d e, d > R /2, e > U_{1} /2 > U .$

Indeed $d e / [d, e] < 2$ and this ratio is the positive integer $(d, e)$ . Consequently

$ℓ \in I \sum h (ℓ) = \frac{1}{r ^{2}} R /2 < d \leq R \sum μ (d) lo g (R / d) d e \in I, e \leq U_{1} (e, d) = 1 \sum μ (e) lo g (U_{1} / e) .$

The coprimality condition can be retained uniformly. The classical finite convolution identity and a split at $a = t$ give

$M_{d} (t) ∣ M_{d} (t) ∣ := n \leq t (n, d) = 1 \sum μ (n) = a \leq t a ∣ d^{\infty} \sum M (t / a), ≪ t e^{- c_{1} W (l o g t)} \frac{d}{φ ( d )} + t^{3/4} p ∣ d \prod (1 - p^{- 1/2})^{- 1} .$

Here $a ∣ d^{\infty}$ means that every prime factor of $a$ divides $d$ . The first term uses $W ((lo g t) /2) ≍ W (lo g t)$ ; the second is the usual Rankin bound on the complementary smooth reciprocal sum. The elementary bounds $d / φ (d) ≪ lo g (2 d)$ and $\prod_{p ∣ d} (1 - p^{- 1/2})^{- 1} ≪ d^{1/4}$ then yield

$M_{d} (t) ≪ U_{1} e^{- c_{2} W (L)} (d \leq R, U_{1} /2 \leq t \leq U_{1}) .$

The polynomial remainder is $O (U_{1}^{5/6})$ since $U_{1} = R^{3}$ . It and the logarithmic factor are absorbed by decreasing $c_{2}$ . Abel summation against $lo g (U_{1} / e)$ on the inner interval, followed by $lo g (R / d) \leq lo g 2$ , therefore gives the uniform application

$ℓ \in I \sum h (ℓ) ≪ \frac{V}{L ^{2}} e^{- κ W (L)}, I \subset (V /2, V], κ > 0. (SV11)$

To identify its actual finite frequency band, put $b (t) = {t} - 1/2$ , $c_{A} = \frac{1}{2} \sum_{ℓ} h (ℓ)$ and $g_{A} (y) = F_{A} (y) - c_{A} = \sum_{ℓ} h (ℓ) b (y / ℓ)$ . Reuse the classical Vaaler approximation, as stated in Baker, Banks, Brüdern, Shparlinski and Weingartner, Piatetski-Shapiro sequences, arXiv:1203.5884v1, §2, equation (2.1), PDF p.6, with attribution there to Vaaler (1985). The source statement was inspected; its underlying proof is not independently audited here. Use its polynomial $P_{J}$ and nonnegative majorant $D_{J}$ , with $∣ b - P_{J} ∣ \leq D_{J}$ and $∣ P_{J} (n) ∣ ≪ 1/∣ n ∣$ for $n \neq = 0$ . At integers the representatives are explicitly $b = - 1/2$ , $P_{J} = 0$ and $D_{J} = 1/2$ . Define

$g_{A, J} (y) = ℓ \sum h (ℓ) P_{J} (y / ℓ), r_{A, J} (y) = ℓ \sum ∣ h (ℓ) ∣ D_{J} (y / ℓ),$

$b_{A, J} (ξ) = ℓ \leq V, 1 \leq ∣ n ∣ \leq J n / ℓ = ξ \sum h (ℓ) P_{J} (n) .$

The finite positive frequency band $B_{+} = [1/ V, 2/ V)$ forces $n = 1$ and $V /2 < ℓ \leq V$ . Thus its merged coefficients are exactly $b_{A, J} (1/ ℓ) = h (ℓ) P_{J} (1)$ , without an infinite Fourier substitution or a missing harmonic cutoff. Abel summation using (SV11) gives, for every interval $I \subset B_{+}$ , $y \geq 0$ and integer $J \geq 1$ ,

$ξ \in I \sum b_{A, J} (ξ) e^{2 πi y ξ} ≪ \frac{V}{L ^{2}} e^{- κ W (L)} (1 + y / V) . (SV12)$

In particular the real contribution of this band and its conjugate,

$g_{top, J} (y) = 2ℜ P_{J} (1) V /2 < ℓ \leq V \sum h (ℓ) e^{2 πi y / ℓ},$

has the same bound. Put $g_{rest, J} = g_{A, J} - g_{top, J}$ and $Z_{rest, J} (y) = c_{A} + g_{rest, J} (y) - y (S_{A})_{+}$ . The exact identity $E_{A} (y) = [c_{A} + g_{A} (y) - y (S_{A})_{+}]_{+}$ and the positive part’s $1$ -Lipschitz property show where this bound is consumed. For a block $P$ of actual primes, $y_{p} = x / p^{k} \leq Y$ , and $w_{N, p, k} = δ_{N, p, k} / ϵ$ , let $M_{P} = \sum_{p \in P} w_{N, p, k}$ . Then

$\frac{1}{x} p \in P \sum w_{N, p, k} E_{A} (y_{p}) \leq \frac{1}{x} p \in P \sum w_{N, p, k} [Z_{rest, J} (y_{p})]_{+} + C \frac{M _{P}}{x} \frac{V}{L ^{2}} e^{- κ W (L)} (1 + Y / V) + \frac{1}{x} p \in P \sum w_{N, p, k} r_{A, J} (y_{p}) . (SV13)$

No sign of $S_{A}$ or replacement of the actual weights is assumed. For a dyadic block $P \subset {p : P < p \leq 2 P}$ with $Y = x / P^{k}$ , the existing $w_{N, p, k} \leq lo g p$ and $ϑ (2 P) < 4 P$ give $M_{P} ≪ (x / Y)^{1/ k}$ . At $Y ≍ V$ , fixed $k \geq 2$ and $A \leq x \leq 2 A$ , the displayed band term is therefore $O_{k} (A^{- γ_{k}} L^{- 2} e^{- κ W (L)})$ , $γ_{k} = (k - 1) / (2 k)$ . For each remaining layer $k < m_{η}$ , $η > γ_{k}$ and $W (L) = o (L)$ , so this upper bound alone does not reach $o (A^{- η})$ . Other bands, the actual positive part, and the pointwise remainder are still present. This is an application of existing classical inputs, without Lean certification or a claim of original number theory; it gives no complete Robin gain.

A small-truncation boundary for the absolute remainder envelope

Use the classical Fejér-normalized Vaaler majorant

$D_{J} (t) = \frac{1}{2 ( J + 1 ) ^{2}} (\frac{sin π ( J + 1 ) t}{sin π t})^{2} = \frac{1}{2 ( J + 1 )} ∣ n ∣ \leq J \sum (1 - \frac{∣ n ∣}{J + 1}) e^{2 πin t},$

with continuous value $1/2$ at integers. This is the standard Vaaler construction, attributed to Vaaler (1985); the original proof has not been independently inspected. The approximation statement used above is also supplied by the inspected Baker et al. equation (2.1). All estimates below retain the actual coefficients and hold for sufficiently large $A$ .

Choose primes $q, s$ in the fixed-ratio ranges

$3 R /4 < q \leq 7 R /8, 3 U_{1} /4 < s \leq 7 U_{1} /8.$

These ranges are disjoint and $V /2 < q s < V$ . The only permitted LCM decomposition at $ℓ = q s$ is $d = q, e = s$ : the preceding coprimality observation gives $d e = q s$ , while $s > R$ and $q s > U_{1}$ . Thus

$h (q s) = \frac{lo g ( R / q ) lo g ( U _{1} / s )}{r ^{2}} ≫ L^{- 2}, V /2 < ℓ \leq V \sum ∣ h (ℓ) ∣ ≫ \frac{V}{L ^{4}} . (SV14)$

The second bound directly applies the existing unconditional prime number theorem to the two fixed-ratio intervals; it is not a new prime-counting result. Each pair gives a different product, and their count is $≫ R U_{1} / (lo g R lo g U_{1}) ≍ V / L^{2}$ .

If $U < y < 2 U$ , $ℓ > V /2$ and $J + 1 \leq V / (8 U)$ , then $0 < (J + 1) y / ℓ < 1/2$ . The elementary bounds $sin π u \geq 2 u$ for $0 \leq u \leq 1/2$ and $sin π t \leq π t$ imply $D_{J} (y / ℓ) \geq 2/ π^{2}$ . Together with (SV14), this gives $r_{A, J} (y) ≫ V / L^{4}$ throughout $U < y < 2 U$ .

For a fixed integer $k \geq 2$ , define the full active-sample cost

$T_{N, k, J} (x) = \frac{1}{x} p^{k} < x / U \sum w_{N, p, k} r_{A, J} (x / p^{k}) .$

The actual own-price comparison already established above gives $(1 - p^{1 - k}) lo g p \leq w_{N, p, k} \leq lo g p$ . Consequently the primes with $x / (2 U) < p^{k} < x / U$ have weighted mass $≫_{k} (x / U)^{1/ k}$ , by the same prime number theorem. Uniformly for $A \leq x \leq 2 A$ ,

$T_{N, k, J} (x) ≫_{k} \frac{V}{x L ^{4}} (\frac{x}{U})^{1/ k} ≍_{k} \frac{A ^{- 1/2 + 3/ (4 k)}}{L ^{4}}, J \geq 1, J + 1 \leq \frac{V}{8 U} . (SV15)$

For squares this is $A^{- 1/8} / L^{4}$ , larger than the already available $A^{- 1/4} / L^{2}$ residual upper envelope for $E_{A}$ . This is a lower bound for the absolute approximation envelope, not for $E_{A}$ or the signed error.

Both terms of the Fourier upper budget must therefore be retained. Put $B_{1} = \sum_{ℓ} ∣ h (ℓ) ∣ \leq B_{A}$ , $M_{N, k} (x) = \sum_{p^{k} < x / U} w_{N, p, k}$ and $\mathcal S_{N,k,x}(\alpha)=\sum_{p^k<x/U} w_{N,p,k}e^{2\pi i\alpha x/p^k}$. The Fejér formula gives

$T_{N, k, J} (x) \leq \frac{B _{1} M _{N, k} ( x )}{2 x ( J + 1 )} + \frac{1}{x ( J + 1 )} ℓ \sum ∣ h (ℓ) ∣ n = 1 \sum J (1 - \frac{n}{J + 1}) ∣ S_{N, k, x} (n / ℓ) ∣. (SV16)$

At $J + 1 = ⌊ V / (16 U)⌋$ , the first term is $O_{k} (A^{- 3/4 + 3/ (4 k)} / L^{2})$ , using $M_{N, k} (x) ≪_{k} (x / U)^{1/ k}$ and $B_{A} ≪ V / L^{2}$ . Its ratio to the lower scale in (SV15) is $O_{k} (A^{- 1/4} L^{2}) \to 0$ . Hence the second, oscillatory upper-budget term in (SV16) is itself at least a fixed positive multiple of the scale in (SV15) eventually. Paying only the zero-frequency term cannot pay this absolute envelope.

This excludes only the small- $J$ sufficient condition that asks the whole absolute envelope to be small. It excludes neither larger $J$ nor cancellation in the signed error, and gives no counterexample to the actual-source target or Robin’s inequality. The estimates are attributed paper-level applications, not Lean-certified conclusions or claims of original number theory.

A weaker joint condition remains possible. Write $Δ_{J} = g_{A} - g_{A, J}$ . The elementary positive-part inequality

$[Z + Δ]_{+} \leq [Z]_{+} + [Δ_{+} - (- Z)_{+}]_{+}$

and the same band removal give

$E_{A} (y) \leq [Z_{rest, J} (y)]_{+} + [(Δ_{J} (y))_{+} - (- Z_{rest, J} (y))_{+}]_{+} + ∣ g_{top, J} (y) ∣.$

Only the positive signed error exceeding the actual negative margin is charged in this expression. No quantitative joint bound at the original $η$ rate has been supplied for it. The remaining bands, mean requirement, low prime-power layers, prime layer $k = 1$ , recovery terms and complete signed tail (SV4) remain unpaid; the original source and target are unchanged.

Exact-integer sample corrections are paid by a classical divisor bound

Keep the actual normalized weights, so $∣ λ (d) ∣, ∣ θ^{'} (d) ∣ \leq 1$ . For a positive integer $m$ , the LCM definition gives the jump coefficient

$J_{A} (m) := ℓ ∣ m \sum h (ℓ) = d ∣ m \sum λ (d) e ∣ m \sum θ^{'} (e), ∣ J_{A} (m) ∣ \leq τ (m)^{2} . (SV17)$

Here $τ$ is the ordinary divisor-count function. Put $J_{A} (y) = 0$ for positive noninteger $y$ . Use the symmetric sawtooth representative $b_{sym} (t) = b (t) + \frac{1}{2} 1_{Z} (t)$ , with value zero at integers, and define $g_{sym, A} (y) = \sum_{ℓ} h (ℓ) b_{sym} (y / ℓ)$ . For $y > 0$ , $g_{sym, A} (y) = g_{A} (y) + \frac{1}{2} J_{A} (y)$ . For $E_{sym, A} (y) = [c_{A} + g_{sym, A} (y) - y (S_{A})_{+}]_{+}$ , the positive part’s Lipschitz property gives

$∣ E_{A} (y) - E_{sym, A} (y) ∣ \leq \frac{1}{2} ∣ J_{A} (y) ∣. (SV18)$

The same finite Vaaler polynomial $P_{J}$ approximates $b_{sym}$ with nonnegative majorant $D_{J} - \frac{1}{2} 1_{Z}$ : off the integers the original estimate applies; at integers both $b_{sym}$ and $P_{J}$ are zero and $D_{J} = 1/2$ . Thus a midpoint Fourier convention can be used only with the explicit correction (SV18) when returning to the original response.

For every prime-power layer, including $k = 1$ , the global own-price comparison gives $0 \leq w_{N, p, k} \leq lo g p$ . Define its full endpoint cost

$C_{N} (x) := \frac{1}{2 x} p^{k} \leq x k \geq 1 \sum w_{N, p, k} ∣ J_{A} (x / p^{k}) ∣.$

If $x$ is noninteger this cost is zero. If $x = M$ is a positive integer, only $p^{k} ∣ M$ contribute. Since $τ (M / p^{k}) \leq τ (M)$ and $\sum_{p^{k} ∣ M} lo g p = lo g M$ , (SV17) gives

$C_{N} (M) \leq \frac{τ ( M ) ^{2} lo g M}{2 M} . (SV19)$

Directly reuse Nicolas’s divisor-count maximizing reference, printed p.117, equations (6)–(8), with fixed exponent $ν = 1/8$ . Its reference $Q_{ν}$ maximizes $τ (m) m^{- ν}$ over all positive integers, and therefore

$τ (m) \leq τ (Q_{ν}) Q_{ν}^{- ν} m^{ν}, A \leq x \leq 2 A sup C_{N} (x) ≪ A^{- 3/4} lo g A = o (A^{- η}) (0 < η < 1/2) . (SV20)$

The constant is independent of the selected $N$ and $A$ ; $Q_{ν}$ is a fixed divisor-count reference, distinct from the actual CA integer $N$ and its varying price $ϵ$ . This pays the exact-integer representative correction for all layers in the original local sampling budget. It is a classical application, not a new divisor estimate or a Lean-certified result. It controls neither near-jump errors nor the signed-error/negative-margin coupling, mean, low-layer residuals, general prime-layer cost, recovery terms or complete signed Robin tail.

A common large truncation pays the higher-power approximation error

Keep the original sawtooth representative and actual coefficients. Put $M_{J} = J + 1$ and retain every layer in

$T_{N, J}^{(\geq 2)} (x) = \frac{1}{x} p, k \geq 2 p^{k} < x / U \sum w_{N, p, k} ℓ \leq V \sum ∣ h (ℓ) ∣ D_{J} (x / (p^{k} ℓ)) .$

For each fixed auxiliary $0 < ν < 1$ , the existing Nicolas reference at exponent $ν /4$ gives $τ (n)^{2} lo g n ≪_{ν} n^{ν}$ . The actual weights and the LCM definition therefore give the aggregate mass at any positive integer jump $n$ :

$p, k \geq 2, ℓ p^{k} ℓ ∣ n \sum w_{N, p, k} ∣ h (ℓ) ∣ \leq τ (n)^{2} p, k \geq 2 p^{k} ∣ n \sum lo g p \leq τ (n)^{2} lo g n ≪_{ν} n^{ν} . (SV21)$

Indeed $\sum_{ℓ ∣ n} ∣ h (ℓ) ∣ \leq τ (n)^{2}$ follows by taking absolute values before the LCM sum, using $∣ λ ∣, ∣ θ^{'} ∣ \leq 1$ . The divisor-growth input is reused, rather than reproved.

The Fejér formula gives $D_{J} (t) \leq min (1/2, (8 M_{J}^{2} ∥ t ∥^{2})^{- 1})$ , with its original value $1/2$ at integers. For $q = p^{k} ℓ \leq 2 x$ , assign the summand to $n = q ⌊ x / q + 1/2 ⌋$ . Then $1 \leq n \leq 2 x$ , $q ∣ n$ and $∥ x / q ∥ = ∣ x - n ∣/ q$ . Consequently

$D_{J} (x / q) \leq \frac{1}{2} min (1, \frac{( x / M _{J} ) ^{2}}{∣ x - n ∣ ^{2}}) . (SV22)$

At $x = n$ the minimum means one. For every real $x$ and $a > 0$ , $\sum_{n \in Z} min (1, a^{2} /∣ x - n ∣^{2}) ≪ 1 + a$ : retain the nearest integers, count those within distance $a$ , and sum the square-reciprocal tail. Grouping (SV22) by $n$ and using (SV21) thus bounds all $q \leq 2 x$ contributions, uniformly for $A \leq x \leq 2 A$ , by $O_{ν} (A^{ν} (A^{- 1} + M_{J}^{- 1}))$ .

For $q > 2 x$ , use $D_{J} (x / q) \leq q^{2} / (8 M_{J}^{2} x^{2})$ . No part of the support $ℓ > x / p^{k}$ is discarded. The existing $Γ_{2} (z) ≪ z$ and $B_{1} \leq B_{A} ≪ V / L^{2}$ give

$T_{q > 2 x} \leq \frac{B _{A} V ^{2}}{8 M _{J}^{2} x ^{3}} (\frac{x}{U})^{2} Γ_{2} (x / U) ≪ \frac{A ^{3/8}}{M _{J}^{2} L ^{2}} . (SV23)$

Together these bounds yield the actual, all-layer sampling estimate

$A \leq x \leq 2 A sup T_{N, J}^{(\geq 2)} (x) ≪_{ν} A^{ν} (A^{- 1} + M_{J}^{- 1}) + \frac{A ^{3/8}}{M _{J}^{2} L ^{2}} . (SV24)$

Choose the same $J = ⌈ V ⌉$ for every sample and layer. With $ν = (1/2 - η) /2$ , the original fixed $0 < η < 1/2$ is unchanged and

$A \leq x \leq 2 A sup T_{N, J}^{(\geq 2)} (x) ≪_{ν} A^{- 1/2 + ν} + A^{- 5/8} L^{- 2} = o (A^{- η} / L) . (SV25)$

This estimates the full real-variable envelope directly at a larger $J$ . It leaves (SV15) intact and does not bound the two Fourier upper-budget terms in (SV16) separately.

To return to the original response, retain all frequency bands jointly: $Z_{all, J} (y) = c_{A} + g_{A, J} (y) - y (S_{A})_{+}$ . Then $∣ E_{A} (y) - [Z_{all, J} (y)]_{+} ∣ \leq r_{A, J} (y)$ . Let $R_{N, η, J}^{poly}$ be the residual (SV10) with $E_{A}$ replaced by $[Z_{all, J}]_{+}$ , restricted to the same active samples $p^{k} < x / U$ and $2 \leq k < m_{η}$ . Since $E_{A} (y) = 0$ for $y \leq U$ , (SV25) and the Lipschitz property give

$∣ R_{N, η} - R_{N, η, J}^{poly} ∣ = o (A^{- η} / L), J = ⌈ V ⌉ . (SV26)$

Likewise the positive signed error exceeding the negative margin is bounded by $r_{A, J}$ , so its higher-power sampled cost is paid by (SV25). The remaining polynomial positive part is not bounded by this argument. The constants, drift and lowest frequency band must still be estimated jointly. This is an application of existing Fejér, divisor-growth and prime-power inputs, without a claim of original number theory or Lean certification. The mean, general prime layer, recovery terms and complete signed Robin tail remain open.

A classical high-harmonic supplier on critical rectangular blocks

Directly reuse Liu–Wu–Yang, Proposition 3.1, equation (3.2), retaining all four terms of that published estimate. Write $e (t) = e^{2 πi t}$ and, for an actual rectangle $P < p \leq 2 P$ , $D < ℓ \leq 2 D$ , $H < j \leq 2 H$ , put $Y = x / P^{k}$ and

$B_{D, H, J} (y) = D < ℓ \leq 2 D \sum h (ℓ) H < j \leq 2 H j \leq J \sum P_{J} (j) e (j y / ℓ) .$

Here $P, D, H \geq 1$ , $k \geq 2$ is fixed, $A \leq x \leq 2 A$ and $H \leq J = ⌈ V ⌉$ . Set the source parameters to

$(α, β, γ, δ) = (1, k, 1, 0), (M_{src}, N_{src}, X_{src}) = (P, D, H Y / D) .$

Its remaining hypothesis is $H \leq P^{k - 1} D$ . The actual prime and active-sample indicators, $w_{N, p, k}$ , $H P_{J} (j)$ and any test values $∣ u_{p} ∣ \leq 1$ enter $a_{j, p}$ ; $h (ℓ)$ enters $b_{ℓ}$ . Normalize the coefficients using $w \leq lo g (2 P)$ , $∣ P_{J} (j) ∣ ≪ H^{- 1}$ and $∣ h (ℓ) ∣ \leq τ (ℓ)^{2}$ . The existing Nicolas input absorbs this divisor factor into an arbitrarily small fixed power of $A$ . The source theorem is uniform in these coefficients, so choosing $u_{p}$ as the complex sign of the actual block sum gives

$\frac{1}{x} P < p \leq 2 P p^{k} < x / U \sum w_{N, p, k} ∣ B_{D, H, J} (x / p^{k}) ∣ ≪_{k, ν} \frac{A ^{ν}}{x} [x P^{1 - k} + D P / H + P D + P D D / (H Y)] . (SV27)$

The source $X_{src}^{ε}$ , coefficient logarithms and dyadic summation losses are absorbed by choosing smaller auxiliary exponents before the prescribed $ν > 0$ . No independence of the test values from the actual coefficients is required. The same bound handles conjugate frequencies and actual positive-part indicators.

On $D ≍ Y ≍ V$ , $P ≍ A^{1/ (2 k)}$ , (SV27) reduces to

$≪_{k, ν} A^{ν} (A^{- 3/4 + 1/ (2 k)} + A^{- γ_{k}} H^{- 1/2}), γ_{k} = \frac{k - 1}{2 k} . (SV28)$

For an unpaid fixed layer $2 \leq k < m_{η}$ , choose a fixed $d > 0$ small enough that $2 (η - γ_{k}) + d < 1/2$ , and set $H_{0} = A^{2 (η - γ_{k}) + d}$ . Taking $0 < ν < min (d /2, 3/4 - 1/ (2 k) - η)$ pays the dyadic blocks $H \geq H_{0}$ in this critical region to $o (A^{- η})$ . The source condition holds throughout these blocks for large $A$ , since $J ≍ A^{1/2}$ and $P^{k - 1} D ≍ A^{1 - 1/ (2 k)}$ .

This supplies only the indicated high harmonics on critical rectangles. Low harmonics, other rectangles, the joint drifted positive part and the complete original budget remain unestimated. It is a direct application of an existing exponential-sum theorem, not a new prime-distribution result or an identification of its frequencies with FIB composition rotations.

The same approximation budget includes the prime layer at a larger cutoff

Let $T_{N, J}^{(\geq 1)}$ be the full-support sampling envelope defined above with $k \geq 1$ , retaining the active condition $p^{k} < x / U$ . The grouping in (SV21)–(SV22) applies unchanged: with all layers, $\sum_{p^{k} ∣ n, k \geq 1} lo g p = lo g n$ , so the mass at each positive integer jump is still at most $τ (n)^{2} lo g n ≪_{ν} n^{ν}$ . For $q > 2 x$ , directly reuse the existing classical bound $ψ (z) ≪ z$ in place of $Γ_{2} (z) ≪ z$ in (SV23). This bounds the full far support by $B_{A} V^{2} / (M_{J}^{2} U^{3}) ≪ A^{3/4} / (M_{J}^{2} L^{2})$ . Consequently

$A \leq x \leq 2 A sup T_{N, J}^{(\geq 1)} (x) A \leq x \leq 2 A sup T_{N, J_{1}}^{(\geq 1)} (x) J_{1} ≪_{ν} A^{ν} (A^{- 1} + M_{J}^{- 1}) + \frac{A ^{3/4}}{M _{J}^{2} L ^{2}}, ≪_{ν} A^{- 1 + ν} + A^{- 5/4} L^{- 2} = o (A^{- η} / L), = ⌈ A ⌉, ν = (1 - η) /2, 0 < η < 1/2. (SV29)$

This is a corollary of the same classical suppliers and jump grouping, with the original coefficients, representative, source and $η$ . It pays the prime-layer approximation error as well. The prime-layer polynomial positive part is not bounded by (SV29).

For the higher-power residual, keep $J_{0} = ⌈ V ⌉$ . At each actual sample the two polynomial positive parts differ by at most $r_{A, J_{0}} + r_{A, J_{1}}$ . Summing over the same active samples and using (SV25) and (SV29) gives $|\mathcal R^{\rm poly}{N,\eta,J_1} -\mathcal R^{\rm poly}{N,\eta,J_0}|=o(A^{-\eta}/L)$. Thus the existing $J_{0}$ high-harmonic supplier remains usable for this residual. No application of Liu–Wu–Yang to the extra $J_{1}$ harmonics is asserted. The mean, joint polynomial positive parts, other recovery terms and complete signed Robin tail remain unpaid. This application has no Lean certification and makes no claim of original number theory.

Exact local-support recombination pays high harmonics on all rectangles

Keep the original actual source, fixed $η$ , integer representative and full mean $σ_{A} = (S_{A})_{-}$ . The finite floor response gives $E_{A} (y) = [- H_{A} (y) - σ_{A} y]_{+}$ . Put

$c_{A} (y) = \frac{1}{2} ℓ \leq y \sum h (ℓ), s_{A} (y) = ℓ \leq y \sum \frac{h ( ℓ )}{ℓ}, g_{loc, J} (y) = ℓ \leq y \sum h (ℓ) P_{J} (y / ℓ),$

where $h$ still has its original support $ℓ \leq V$ . Then

$Z_{loc, J} (y) ∣ E_{A} (y) - [Z_{loc, J} (y)]_{+} ∣ = c_{A} (y) + g_{loc, J} (y) - y (s_{A} (y) + σ_{A}), \leq ℓ \leq y \sum ∣ h (ℓ) ∣ D_{J} (y / ℓ) \leq r_{A, J} (y) . (SV30)$

This is exact recombination of the original floor sum: its terms with $ℓ > y$ are zero. Their fractional-part contributions in the earlier full-support expression are combined with its full linear term, rather than discarded. The full $σ_{A}$ remains in (SV30); it is not replaced by the negative part of $s_{A} (y)$ . The common $J_{0} = ⌈ V ⌉$ approximation cost is already paid by (SV25).

Apply the existing four-term supplier (SV27) on dyadic blocks, now keeping $ℓ \leq y_{p} = x / p^{k}$ inside each block. This moving prefix can be handled by standard finite Fourier completion. For an integer dyadic $D \geq 1$ , write $Q = 2 D$ and $b_{p} = max (0, min (D, ⌊ y_{p} ⌋ - D))$ . On $v = ℓ - D \in {1, \dots, D}$ the exact identity is

$1_{v \leq b_{p}} ∣ β_{r} (p) ∣ = r = 0 \sum Q - 1 β_{r} (p) e (r v / Q), β_{r} (p) = \frac{1}{Q} u = 1 \sum b_{p} e (- r u / Q), \leq \frac{C}{1 + min ( r , Q - r )}, r = 0 \sum Q - 1 \frac{C}{1 + min ( r , Q - r )} ≪ lo g (2 D) . (SV31)$

Finite Fourier orthogonality supplies the identity and the finite geometric-sum bound supplies the displayed majorants. For each $r$ , put the normalized $β_{r} (p)$ in the allowed coefficient $a_{j, p}$ , together with the actual test sign, own-price weight and sample flags; put $h (ℓ) e (r (ℓ - D) / Q)$ in $b_{ℓ}$ . Thus (SV27) still applies with only a logarithmic completion cost. No unestimated separation of a joint $p, ℓ$ indicator is assumed.

For a nonempty block, $D \leq min (V, Y)$ with $Y = x / P^{k}$ . Under the original source condition $H \leq P^{k - 1} D$ , the four terms in (SV27) give the uniform block cost

$≪_{k, ν} A^{ν} (A^{- 1/2} + A^{- γ_{k}} H^{- 1/2}), γ_{k} = \frac{k - 1}{2 k} . (SV32)$

Indeed the first term is at most $A^{- 1/2}$ . Splitting at $Y = V$ gives $P D / x ≪ A^{- 3/4 + 1/ (2 k)} \leq A^{- 1/2}$ , $D P / x ≪ A^{- 1/2 + 1/ (4 k)} \leq A^{- γ_{k}}$ and $P D D / Y / x ≪ A^{- γ_{k}}$ . These are bounds for the same rectangle. If the source condition fails, use direct absolute values: the cost is $≪_{k, ν} A^{ν} P D / x$ , and $P D < H P^{2 - k} \leq J_{0}$ for $k \geq 2$ . This again costs $O_{k, ν} (A^{- 1/2 + ν})$ without using the source theorem outside its range. The term $ℓ = 1$ , outside the dyadic denominator blocks, has total cost $O_{k, ν} (A^{- 1 + 3/ (4 k) + ν})$ from $p^{k} < x / U$ and the harmonic coefficient bound, and fits the same budget.

For $1 \leq H_{0} \leq J_{0}$ , define the complete local high-frequency part

$G_{k, > H_{0}} (y) = ℓ \leq y \sum h (ℓ) H_{0} < ∣ j ∣ \leq J_{0} \sum P_{J_{0}} (j) e (j y / ℓ) .$

Dyadic decomposition includes the block crossing $H_{0}$ : take $H \geq H_{0} /2$ and retain both flags $j > H_{0}$ and $j \leq J_{0}$ in $a_{j, p}$ . Summing (SV32), its condition-failure bound and the $ℓ = 1$ contribution, and including conjugate frequencies, yields

$A \leq x \leq 2 A sup \frac{1}{x} p^{k} < x / U \sum w_{N, p, k} ∣ G_{k, > H_{0}} (x / p^{k}) ∣ ≪_{k, ν} A^{ν} (A^{- 1/2} + A^{- γ_{k}} H_{0}^{- 1/2}) . (SV33)$

Here $k \geq 2$ is fixed. Divisor normalization, completion and all dyadic losses are absorbed by choosing smaller auxiliary exponents before the prescribed $ν > 0$ . The bound is uniform in the same actual $x$ and sample-dependent test signs; no independent source phases are chosen.

For each original unpaid layer $2 \leq k < m_{η}$ , take a common fixed $0 < d < 1 - 2 η$ and set $H_{0, k} = A^{2 (η - γ_{k}) + d}$ . Its exponent lies between zero and $1/2$ . Choose $0 < ν < min (d /2, 1/2 - η)$ ; then (SV33) is $o (A^{- η})$ for every such layer. Define

$g_{loc, k}^{low} (y) = ℓ \leq y \sum h (ℓ) 1 \leq ∣ j ∣ \leq H_{0, k} \sum P_{J_{0}} (j) e (j y / ℓ),$

and let $R_{N, η}^{low}$ be (SV10)’s residual over the same active samples, replacing $E_{A} (y_{p})$ in each layer $k$ by $[c_A(y_p)+g_{{\rm loc},k}^{\rm low}(y_p) -y_p(s_A(y_p)+\sigma_A)]_+$. Equation (SV30), the positive part’s Lipschitz property and (SV33) give

$∣ R_{N, η} - R_{N, η}^{low} ∣ = o (A^{- η}) . (SV34)$

Empty layer sums remain zero. This applies the same published supplier to all rectangles after exact local-support recombination, extending the critical-rectangle scope of (SV28). The low frequencies, moving constant and drift remain jointly inside the same positive part; they have no bound here at the required rate. The separate full-mean requirement, prime layer, other recovery terms and complete signed Robin tail remain unpaid. This is an attributed application of classical inputs without Lean certification, original-number-theory priority or an identification with FIB composition rotation.

Actual deep-layer weights differ only below the residual target scale

Retain the same selected $N$ , its actual $A = lo g N$ , original $η$ , $E_{A}$ , active samples and finite layers in $R_{N, η}$ . Let $R_{A, η}$ be that same supremum with only $w_{N, p, k}$ replaced by $lo g p$ . The existing own-price layer comparison gives $0 \leq lo g p - w_{N, p, k} \leq p^{1 - k} lo g p$ for $k \geq 2$ . Using the original envelope $E_{A} \leq B_{A}$ , the difference between the two nonnegative sums at each $x \in [A, 2 A]$ is at most

$\frac{B _{A}}{x} p \leq x / U \sum \frac{lo g p}{p} + p \sum k \geq 3 \sum \frac{lo g p}{p ^{k - 1}} = \frac{B _{A}}{x} p \leq x / U \sum \frac{lo g p}{p} + c_{2} .$

The existing $ϑ (t) < 2 t$ input and Abel summation give $\sum_{p \leq z} (lo g p) / p ≪ lo g (2 z)$ ; $c_{2} < \infty$ is already defined in (SV9). Since $B_{A} ≪ A / L^{2}$ , the bound is uniformly $O (A^{- 1/2} / L)$ . Pointwise order and the supremum inequality therefore give, without requiring either supremum to be attained,

$0 \leq R_{A, η} - R_{N, η} ≪ \frac{A ^{- 1/2}}{L} = o (A^{- η} / L), 0 < η < 1/2. (SV35)$

Thus the original local requirement $R_{N, η} = o (A^{- η})$ is equivalent to the same requirement for $R_{A, η}$ along the same selected sources. The actual clock has not been replaced by an arbitrary continuous one. This application first uses the envelope for $E_{A}$ ; it does not assert that an arbitrary polynomial positive part is bounded by $B_{A}$ . Transport to a polynomial residual requires the already paid approximation comparison (SV26).

The depth-dependent lowering of these weights changes this residual only below its target scale. This leaves open compensation from the actual selected clock, the joint low response and other terms of the complete recovery identity. It provides neither the separate mean bound nor a complete signed Robin estimate, and has no Lean certification or claim of original number theory.

Rectangle-dependent cutoffs also pay some of the lowest harmonics

The four-term supplier retains more information than the uniform critical cutoff in (SV34). For the same local-support rectangle, put

$Q_{k} (P, D) = \frac{D P}{A} + \frac{P ^{1 + k /2} D ^{3/2}}{A ^{3/2}}, t_{k} (P, D) = A^{2 η + d} Q_{k} (P, D)^{2}, (SV36)$

where $d$ and the auxiliary exponents obey the choices in (SV34). Since $A \leq x \leq 2 A$ , (SV27) with the exact prefix completion (SV31) gives the block cost $≪_{k, ν} A^{ν} (A^{- 1/2} + Q_{k} (P, D) H^{- 1/2})$ under its size condition. The two terms of $Q_{k}$ are precisely the second and fourth terms of that supplier, before maximizing over rectangles.

Remove from the existing low part only the frequencies $t_{k} (P, D) < ∣ j ∣ \leq H_{0, k}$ . If $t_{k} \geq 1$ , include the crossing dyadic block $H \geq t_{k} /2$ and both exact frequency flags. If $t_{k} < 1$ , all positive frequencies in that rectangle are removed and $H \geq 1$ . In either case $Q_{k} / H ≪ A^{- η - d /2}$ on the removed blocks. The flags are allowed in $a_{j, p}$ for each fixed $P, D, H$ . When the source size condition fails, the already paid direct bound from (SV32) applies instead. Finite dyadic summation, with the same small-exponent slack as before, costs $o (A^{- η})$ uniformly in $x$ .

This deletion includes $j = 1$ when $t_{k} < 1$ . To handle that singleton within the source’s convention $H < h \leq 2 H$ , take source $H = 1$ , keep only $h = 2$ in $a_{h, p}$ , and set $X_{src} = x / (2 P^{k} D)$ . The phase is then exactly $x / (p^{k} ℓ)$ , the condition $1 \leq P^{k - 1} D$ holds, and the four cost terms change only by fixed factors. Negative frequencies are conjugates. The entire $ℓ = 1$ harmonic contribution is separately paid by the absolute bound already used in (SV32), also when its lowest frequencies are included.

Let $P (p)$ be the dyadic scale satisfying $P (p) < p \leq 2 P (p)$ for the actual prime $p$ . Define the remaining response, depending on $p$ and $x$ , by

$g_{ad, k} (p; x) = D dyadic D \leq V \sum D < ℓ \leq 2 D ℓ \leq m i n (V, y_{p}) \sum h (ℓ) 1 \leq ∣ j ∣ \leq m i n (J_{0}, H_{0, k}, t_{k} (P (p), D)) \sum P_{J_{0}} (j) e (j y_{p} / ℓ), y_{p} = x / p^{k} .$

The dyadic denominator sum begins at $D = 1$ ; it excludes the separately paid $ℓ = 1$ . In the original residual, replace $E_{A} (y_{p})$ by $[c_{A} (y_{p}) + g_{ad, k} (p; x) - y_{p} (s_{A} (y_{p}) + σ_{A})]_{+}$ and call the resulting supremum $R_{N, η}^{ad}$ . The constants, full mean, drift, active samples and own-price weights are unchanged. The positive-part Lipschitz inequality and (SV34) therefore give

$∣ R_{N, η} - R_{N, η}^{ad} ∣ = o (A^{- η}) . (SV37)$

For fixed scales $P ≍ A^{u}$ , $D ≍ A^{v}$ , an entire rectangle, including its first harmonic, is paid if

$u + 2 v < 2 - 2 η - d, (k + 2) u + 3 v < 3 - 2 η - d .$

These sufficient conditions describe a region, rather than a single example; they are not necessary conditions for a useful estimate. The retained rectangles and their joint constant/drift positive part still have no bound here at the required rate. This is another parameter-specific application of the same published four-term supplier, without a new exponential-sum theorem, Lean certification or complete signed Robin gain.

A published moment obstruction applies to the original linear factor

The remaining joint estimate cannot be supplied by assuming that linear smoothing automatically gives uniformly bounded high moments of each factor. Granville–Koukoulopoulos–Maynard, Sieve weights and their smoothings, arXiv:1606.06781v4, definition (1.9) and Theorem 1.3, directly apply to the original $λ (d) = μ (d) (1 - lo g d / lo g R)_{+}$ . With the source’s moment index $k = 3$ and smoothing exponent $1$ , their exponent is $(3 6) - 6 (1 + 1) = 8$ . Thus

$M_{6} (R) := d_{1}, \dots, d_{6} \leq R \sum \frac{λ ( d _{1} ) \dots λ ( d _{6} )}{[ d _{1} , \dots , d _{6} ]} = c_{3, 1} (lo g R)^{8} + O ((lo g R)^{7}), c_{3, 1} > 0. (SV38)$

At the unchanged actual clock, $R = A^{1/8}$ . Expanding the sixth power of $Λ_{R} (n) = \sum_{d ∣ n} λ (d)$ gives

$\frac{1}{A} n \leq A \sum Λ_{R} (n)^{6} = M_{6} (R) + O (\frac{R ^{6}}{A}) = c_{3, 1} (lo g R)^{8} + O ((lo g R)^{7} + A^{- 1/4}) .$

The expansion error uses only $∣ λ (d) ∣ \leq 1$ for the actual $d \geq 1$ and $|\lfloor A/[d_1,\ldots,d_6]\rfloor -A/[d_1,\ldots,d_6]|\le1$. This is an application of the published moment theorem, not a new high-moment result or Lean certification.

It rules out a uniformly bounded sixth moment for this single factor under ordinary integer averaging. It does not rule out compensation in the mixed LCM response $h$ , in its drifted positive part, or under the actual reciprocal-prime-power sampling: those are different quantities and measures. In particular, (SV38) is neither a lower bound for the unpaid residual nor a counterexample to its required estimate. Replacing the linear weight by a more smoothed one would change the recovery contract and does not inherit the estimates for the current $h$ .

The original response envelopes localize the unpaid prime samples

Choose a fixed $0 < ω < 1/2 - η$ and retain only the original finite unpaid layers $2 \leq k < m_{η}$ . Put

$P_{-} = A^{1/2 - η - ω}, P_{+, k} = A^{(η + ω) / (k - 1)} .$

Let $R_{N, η}^{win}$ be the original $E_{A}$ residual with the additional sample condition $P_{-} < p \leq P_{+, k}$ , intersected with the same active condition $p^{k} < x / U$ . Apply the envelopes to $E_{A}$ , before any polynomial replacement. At each $x \in [A, 2 A]$ , the discarded small-prime part in a layer is at most

$\frac{B _{A}}{x} p \leq P_{-} \sum lo g p ≪ \frac{A ^{- 1/2} P _{-}}{L ^{2}} = \frac{A ^{- η - ω}}{L ^{2}} .$

For the large-prime part, use $E_{A} (x / p^{k}) \leq C x / (p^{k} L^{2})$ and the existing $ϑ (t) < 2 t$ input. Abel summation gives, for $z \geq 1$ , $\sum_{p > z} (lo g p) / p^{k} \leq 2 k z^{1 - k} / (k - 1)$ . Thus this part is also $≪_{k} A^{- η - ω} / L^{2}$ . Enlarging each discarded nonnegative sum is legitimate, including primes outside the active range. Finite layer summation and the pointwise order therefore imply

$0 \leq R_{N, η} - R_{N, η}^{win} ≪_{η, ω} \frac{A ^{- η - ω}}{L ^{2}}, (SV39)$

without requiring an attained supremum. For each unpaid layer $γ_{k} < η$ , so $P_{-} < A^{1/ (2 k)} < P_{+, k}$ . This is an explicit prime window containing the critical scale; its exponent width need not be small. Empty layers and empty intersections still contribute zero.

In this same window, replace the response by the joint adaptive positive part of (SV36)–(SV37), and denote its residual by $R_{N, η}^{ad, win}$ . The absolute approximation errors only shrink on a sample subset, and the window flags are permitted in the arbitrary $a_{j, p}$ arrays used to delete harmonics. Consequently the original sampled $L^{1}$ argument applies to this subset with the same rate. Together with (SV39), it gives

$∣ R_{N, η} - R_{N, η}^{ad, win} ∣ = o (A^{- η}) . (SV40)$

This conclusion uses that absolute-error argument, not an inference from the difference of the two unrestricted suprema in (SV37). The constants and full drift stay jointly inside each retained positive part. No envelope for an arbitrary polynomial positive part has been assumed. The remaining task is now restricted in prime samples as well as in harmonics and denominators; the separate mean, prime layer, other recovery terms and complete signed tail still need their original estimates. Equations (SV39)–(SV40) are a localization application of the existing envelopes and prime-counting input, without a new prime-distribution theorem or Lean certification.

Shared sampling primes can be removed from the periodic sector

Retain the same actual source, original fixed $η$ , support $V = A$ , and full local slope $s_{A} (y)$ and mean $σ_{A}$ . For each sampling prime $p$ , define

$F_{A, p}^{cop} (y) = ℓ \leq m i n (V, y) p ∤ ℓ \sum h (ℓ) {y / ℓ}, E_{A, p}^{cop} (y) = [F_{A, p}^{cop} (y) - y (s_{A} (y) + σ_{A})]_{+} .$

The local floor identity in (SV30) gives the same expression for $E_{A}$ before the condition $p ∤ ℓ$ is inserted. Only its fractional-part sector is removed. In particular, the original $s_{A} (y)$ still contains the terms with $p ∣ ℓ$ , and $σ_{A}$ still uses the full $S_{A}$ .

Directly reuse $∣ h (ℓ) ∣ \leq τ (ℓ)^{2}$ , the Nicolas divisor-growth supplier already used in (SV27), and $0 \leq {t} < 1$ . For every prescribed $ν > 0$ , they give

$∣ E_{A} (y) - E_{A, p}^{cop} (y) ∣ \leq ℓ \leq m i n (V, y) p ∣ ℓ \sum ∣ h (ℓ) ∣ ≪_{ν} A^{ν} \frac{min ( V , y )}{p} . (SV41)$

The count is zero when $p > min (V, y)$ ; $ℓ = 1$ is never removed. This is an absolute coefficient-counting bound and requires no independence between the prime and the denominator.

For any fixed $k \geq 1$ , use the original $0 \leq w_{N, p, k} \leq lo g p$ where that weight is defined. The existing $ϑ (t) < 2 t$ input and Abel summation give, with $z = (x / V)^{1/ k} \geq 1$ ,

$\frac{1}{x} p \sum lo g p \frac{min ( V , x / p ^{k} )}{p} = \frac{V}{x} p \leq z \sum \frac{lo g p}{p} + p > z \sum \frac{lo g p}{p ^{k + 1}} ≪_{k} \frac{V}{x} lo g (2 z) + z^{- k} ≪_{k} A^{- 1/2} lo g (2 A), A \leq x \leq 2 A .$

An atom at $p = z$ belongs to the first sum. All summands in this majorant are nonnegative, so the same bound holds on the original active samples, their prime window, or any subset. Choosing a smaller auxiliary divisor exponent first absorbs the logarithm. Thus for every prescribed $ν > 0$ and fixed $k$ ,

$A \leq x \leq 2 A sup \frac{1}{x} p^{k} < x / U \sum w_{N, p, k} ∣ E_{A} (x / p^{k}) - E_{A, p}^{cop} (x / p^{k}) ∣ ≪_{k, ν} A^{- 1/2 + ν} . (SV42)$

For the original finite unpaid layers, choose $0 < ν < 1/2 - η$ . Their total is $o (A^{- η})$ . The fixed-layer estimate also permits $k = 1$ , but it is not an estimate for an unbounded sum of layers or a bound for the remaining prime-layer response.

The same deletion applies directly to the already retained adaptive polynomial. Define

$c_{A, p}^{cop} (y) = \frac{1}{2} ℓ \leq y p ∤ ℓ \sum h (ℓ),$

and define $g_{ad, k}^{cop} (p; x)$ by inserting $p ∤ ℓ$ in the denominator sum of $g_{ad, k} (p; x)$ , with all its original frequency cutoffs intact. Since $∣ P_{J_{0}} (j) ∣ ≪ 1/∣ j ∣$ , removing the common-prime terms from the constant and oscillatory sectors costs at most

$C lo g (2 J_{0}) ℓ \leq m i n (V, y_{p}) p ∣ ℓ \sum ∣ h (ℓ) ∣.$

The same sampling calculation pays this extra logarithm. In the original window, replace the retained adaptive response by

$[c_{A, p}^{cop} (y_{p}) + g_{ad, k}^{cop} (p; x) - y_{p} (s_{A} (y_{p}) + σ_{A})]_{+}, y_{p} = x / p^{k},$

and call its residual $R_{N, η}^{ad, win, cop}$ . The positive part’s Lipschitz bound, the sampled absolute bound above, and (SV40) give

$∣ R_{N, η} - R_{N, η}^{ad, win, cop} ∣ = o (A^{- η}) . (SV43)$

The supremum need not be attained. No coprimality indicator is assumed to factor into the independent coefficient arrays of (SV27): this comparison deletes the common-prime sector by absolute counting after the existing harmonic comparisons. Neither $E_{A, p}^{cop}$ nor an arbitrary new polynomial positive part is asserted to inherit the original $E_{A}$ envelopes or its vanishing below $U$ . The original active flags and the full local slope and mean remain explicit.

This is an application of existing divisor-growth and prime-counting inputs to the original response, not a new coprime exponential-sum theorem, original-number-theory claim, or Lean certification. It removes the shared sampling prime from the retained periodic and constant sectors. The coprime low response, its joint full drift, the separate mean estimate, prime layer, other recovery terms and complete signed Robin tail still have no bound here at the original target rate.

The actual response has a quadratic bound just above its cutoff

Keep the original weights, actual source and fixed exponent. Write

$a_{R} (n) = d ∣ n \sum λ (d), b_{A} (n) = e ∣ n \sum θ^{'} (e) .$

The defining finite LCM sum gives $H_{A} (y) = \sum_{n \leq y} a_{R} (n) b_{A} (n)$ . For $n \leq U$ , $b_{A} (n) = 1_{n = 1}$ . If $U < n \leq U (1 + δ) < min (2 U, U_{1})$ , every proper divisor of $n$ is at most $U$ . Only the divisor $e = n$ can differ from the untruncated Möbius response, and

$b_{A} (n) = θ^{'} (n) - μ (n) = - \frac{μ ( n )}{r} lo g (n / U) .$

Consequently, uniformly for real $U < y \leq U (1 + δ)$ ,

$H_{A} (y) = 1 - \frac{1}{r} U < n \leq y \sum μ (n) a_{R} (n) lo g (n / U) . (SV44)$

There is no rounding of the observation endpoint. For sufficiently large $A$ , every $0 < δ \leq 1/2$ satisfies the displayed cutoff conditions. The existing absolute weight bounds give

$U < n \leq U (1 + δ) \sum ∣ a_{R} (n) ∣ \leq d \leq R \sum ∣ λ (d) ∣ (⌊ \frac{U ( 1 + δ )}{d} ⌋ - ⌊ \frac{U}{d} ⌋) \leq δ U (1 + r) + \frac{R - 1}{r} .$

Here $\sum ∣ λ (d) ∣/ d \leq 1 + r$ and $\sum ∣ λ (d) ∣ \leq (R - 1) / r$ . The first follows from $∣ λ (d) ∣ \leq 1$ and the harmonic-sum bound; the second is the same decreasing-logarithm sum used in the original $B_{A}$ envelope. Using $lo g (n / U) \leq δ$ , $r \geq 1$ and the unchanged $σ_{A} \geq 0$ in $E_{A} (y) = [- H_{A} (y) - σ_{A} y]_{+}$ gives

$E_{A} (y) \leq 2 δ^{2} U + \frac{δ R}{r ^{2}}, U < y \leq U (1 + δ) . (SV45)$

This controls the complete original positive part. It does not estimate the mean by a power, remove its compensation, or bound the constant and oscillatory sectors separately.

For the square layer put $P = (x / U)^{1/2}$ , $A \leq x \leq 2 A$ , and restrict to any subset of the actual primes satisfying

$U < x / p^{2} \leq U (1 + δ) .$

They lie in $[P (1 + δ)^{- 1/2}, P)$ , containing at most $C δ P + 1$ integers. The endpoint term is retained. Moreover $p \geq A^{3/8} / 1 + δ > U (1 + δ)$ eventually, uniformly in $x$ and $δ$ . Thus every denominator in the moving local support is smaller than $p$ , and $E_{A, p}^{cop} (x / p^{2}) = E_{A} (x / p^{2})$ with the full drift and mean. Equation (SV45) and $w_{N, p, 2} \leq lo g p$ yield

$A \leq x \leq 2 A sup \frac{1}{x} U < x / p^{2} \leq U (1 + δ) \sum w_{N, p, 2} E_{A} (x / p^{2}) ≪ L δ^{3} A^{- 3/8} + L δ^{2} A^{- 3/4} + \frac{δ ^{2} A ^{- 1/2}}{L} + \frac{δ A ^{- 7/8}}{L} . (SV46)$

Only when the original fixed $3/8 < η < 1/2$ , choose $b = (η - 3/8) /3$ and $δ_{2, A} = A^{- b} / L$ . The first term is $A^{- η} / L^{2}$ and the other three have strictly larger power exponents. The whole bound is $O_{η} (A^{- η} / L^{2})$ . Restricting to the original prime window preserves this bound. Removing these square samples from the original nonnegative residual therefore changes its supremum by at most that amount; an attained supremum is unnecessary.

For comparison, the old $E_{A} (y) ≪ y / L^{2}$ envelope and the same short-interval integer count give only $O (δ A^{- 3/8} / L + A^{- 3/4} / L)$ here. With $δ = δ_{2, A}$ , its first term is $A^{- η + 2 b} / L^{2}$ , so that upper bound does not pay this wider strip. This is a comparison of sufficient bounds, not a lower bound on the actual response. When $η \leq 3/8$ , the old envelope and the existing $ϑ (P) ≪ P$ supplier already pay a fixed-width square strip at $O (A^{- 3/8} / L^{2})$ ; that case is not an additional payment.

The finite onset calculation is an application of the original weight definitions and counting bounds, not a new classical sieve theorem, historical originality claim or Lean certification. It pays the complete response only in the displayed shrinking strip. The critical region $y ≍ V$ , remaining low layers, mean rate, prime layer, other recovery terms and full signed Robin tail retain their original obligations.

A published short-sum bound also pays prime-layer onset samples

The floor-counting error in (SV45) can be avoided in a sufficiently long onset interval by directly reusing Henriot’s corrected 2014 New Theorem 5, erratum printed p.377. Use its single primitive polynomial $Q (t) = t$ and the fixed function $F (n) = τ (n)$ . Its parameters are

$g = k = r_{Q} = 1, α = 1/4, δ_{coef} = 1, ε = 1/800 < 1/400, A_{0} = 2,$

with a fixed divisor-growth constant $B_{0} = B_{0} (ε)$ . The class follows from multiplicativity, $τ (p^{j}) = j + 1 \leq 2^{j}$ and the same classical divisor-growth input already used in (SV27). These constants do not grow with the Robin source. Here $δ_{coef}$ is the theorem’s polynomial parameter, not the interval width $δ$ .

Set the source’s lower endpoint to $U$ and its interval length to $Δ = δ U$ . Its range is satisfied when $U^{1/4} < Δ \leq U$ and $U$ is sufficiently large. For this same single-factor polynomial, the corrected root density recorded in the cited note is

$\frac{ρ ˘ _{Q} ( a )}{aκ ( a )} = \frac{1}{a} p ∣ a \prod (1 - 1/ p) \leq \frac{1}{a} .$

The existing harmonic divisor sum is at most $(1 + lo g U)^{2}$ . Rosser–Schoenfeld Theorem 8, (3.28) gives $\prod_{p \leq U} (1 - 1/ p) ≪ 1/ lo g U$ . Thus the corrected theorem supplies

$U < n \leq U (1 + δ) \sum τ (n) ≪ δ U lo g U, U^{1/4} < δ U \leq U .$

This invokes the existing short-sum theorem; it does not reprove it or extend Shiu’s literal $0 < a < q$ residue statement to $a = q = 1$ . Since $∣ λ (d) ∣ \leq 1$ gives $∣ a_{R} (n) ∣ \leq τ (n)$ , substitution into the unchanged (SV44) yields

$E_{A} (y) ≪ δ^{2} U, U < y \leq U (1 + δ), 0 < δ \leq 1/2, δ U > U^{1/4} . (SV47)$

The positive terms $1$ and $σ_{A} y$ are retained in the exact response before this upper bound is taken. The theorem includes real endpoints; there is no omitted unit error in the inner short sum.

For $k = 1, 2$ , put $P_{k} = (x / U)^{1/ k}$ and $a_{k} = \frac{3}{4} (1 - 1/ k)$ , so $a_{1} = 0$ , $a_{2} = 3/8$ . The actual onset primes lie in $[P_{k} (1 + δ)^{- 1/ k}, P_{k})$ . Its integer count is $O_{k} (δ P_{k} + 1)$ , and each prime has $lo g p = O (L)$ for $A \leq x \leq 2 A$ . The unchanged weights $0 \leq w_{N, p, k} \leq lo g p$ and (SV47) give

$A \leq x \leq 2 A sup \frac{1}{x} U < x / p^{k} \leq U (1 + δ) \sum w_{N, p, k} E_{A} (x / p^{k}) ≪_{k} L δ^{3} A^{- a_{k}} + L δ^{2} A^{- 3/4} . (SV48)$

The outer endpoint term is explicit. For both $k = 1, 2$ , $p > y = x / p^{k}$ eventually throughout this strip. Hence the same estimate applies exactly to $E_{A, p}^{cop} (y)$ with the full local slope and mean. Any original sample or window subset is allowed.

For the original fixed $0 < η < 1/2$ , choose

$δ_{1, A} = A^{- η /3} / L . (SV49)$

Its inner length is $A^{1/4 - η /3} / L ≫ U^{1/4}$ ; the exponent gap is $3/16 - η /3 > 1/48$ . Equation (SV48) for $k = 1$ becomes $O_{η} (A^{- η} / L^{2})$ , since its second term has exponent $3/4 + 2 η /3 > η$ . Thus this shrinking part of the original prime layer, including its entire drifted positive part, is paid at $o (A^{- η})$ without assuming the missing mean rate. The old envelope and the same integer count would give $O (δ_{1, A} / L + A^{- 3/4} / L)$ , which does not pay this strip. Again this compares sufficient upper bounds, not actual lower bounds.

When the original $η > 3/8$ , the square width $δ_{2, A}$ in (SV46) also satisfies the short-sum condition. Equation (SV48) gives the same $O_{η} (A^{- η} / L^{2})$ payment with a simpler two-term bound; it is not a second payment of that square strip. For $η \leq 3/8$ the existing fixed-width square payment remains the applicable reuse.

The square strip can be deleted from the original nonnegative residual before applying (SV40)–(SV43). Restricting each of their existing sampled absolute-error bounds to the retained samples preserves the adaptive comparison at $o (A^{- η})$ . This uses the bounds on the original response; it gives no new polynomial positive part the original envelopes or vanishing property. The prime-layer calculation concerns its original response and does not import the higher-layer adaptive cutoffs into $k = 1$ .

These are attributed applications of the corrected short-sum and prime-product suppliers, without Lean certification or a historical originality claim. They control onset strips only. The remaining prime and square samples, critical region $y ≍ V$ , other finite low layers, same-source mean rate, remaining recovery convolutions and complete signed Robin tail still require estimates at the original fixed rate.

Reusing short prime intervals enlarges the paid onset range

The outer integer count in (SV48) can be replaced by the existing Nicolas comparison note’s (GS4). That application supplies the uniform short-increment bound from Guth–Maynard Corollary 1.3, including longer intervals and the prime-power correction. Combining the two existing inputs keeps the actual source, its full drifted positive part, and the original fixed exponent.

For $k = 1, 2$ , put $P_{k} = (x / U)^{1/ k}$ , $A \leq x \leq 2 A$ , and $c_{k} = P_{k} (1 + δ)^{- 1/ k}$ . In addition to (SV47)’s conditions, suppose $h_{k} = P_{k} - c_{k} + 1 \geq P_{k}^{2/3}$ throughout this window. Apply (GS4) with its dummy clock $Y = P_{k}$ , lower endpoint $u = c_{k} - 1$ and interval length $h_{k}$ . For sufficiently large $A$ , $Y /2 \leq u < Y$ and $u + h_{k} = Y \leq 3 Y /2$ ; also $h_{k} ≍_{k} δ P_{k}$ . The onset primes lie in $[c_{k}, P_{k})$ , contained in $(c_{k} - 1, P_{k}]$ . Hence (GS4) directly gives

$c_{k} \leq p < P_{k} \sum lo g p \leq ψ (P_{k}) - ψ (c_{k} - 1) ≪_{k} δ P_{k} .$

The enlarged interval includes the endpoint contribution. The supplier is uniform at these real endpoints, and any subset of the original prime samples is allowed. With the unchanged $w_{N, p, k} \leq lo g p$ , equation (SV47) now gives

$A \leq x \leq 2 A sup \frac{1}{x} U < x / p^{k} \leq U (1 + δ) \sum w_{N, p, k} E_{A} (x / p^{k}) ≪_{k} δ^{3} A^{- a_{k}}, a_{k} = \frac{3}{4} (1 - 1/ k) . (SV50)$

For $k = 1$ and every original fixed $0 < η < 1/2$ , and for $k = 2$ only when the original $3/8 < η < 1/2$ , choose

$b_{k} = \frac{η - a _{k}}{3} > 0, δ_{k, A} = \frac{A ^{- b_{k}}}{lo g L} . (SV51)$

These widths satisfy all hypotheses uniformly in $A \leq x \leq 2 A$ . Indeed $δ_{1, A} U = A^{1/4 - η /3} / lo g L ≫ U^{1/4}$ ; the square layer has a larger exponent. At the outer endpoints, $h_{1} ≍ A^{3/4 - η /3} / lo g L ≫ P_{1}^{2/3}$ because $3/4 - η /3 > 7/12 > 1/2$ , and $h_{2} ≍ A^{3/8 - (η - 3/8) /3} / lo g L ≫ P_{2}^{2/3}$ because $3/8 - (η - 3/8) /3 > 1/3 > 1/4$ . Both widths tend to zero. Equation (SV50) yields

$O_{η} (\frac{A ^{- η}}{( lo g L ) ^{3}}) = o (A^{- η})$

for the complete original response on these strips. Throughout them $p > y = x / p^{k}$ , so the coprime response equals the original response, including its mean and drift. Deleting the square samples before (SV40)–(SV43) again uses those equations’ existing sampled absolute-error bounds on the retained subset. No higher-layer adaptive cutoff or envelope for a new polynomial positive part is imported into the prime layer.

This is a logarithmic enlargement at the original $o (A^{- η})$ target. At (SV51)’s width, the existing integer-count bound (SV48) has first term $O (L A^{- η} / (lo g L)^{3})$ , which does not establish that target. This comparison concerns sufficient upper bounds, not a lower bound on the actual response. The explicit widths (SV51) are $L / lo g L$ times those in (SV46) and (SV49). When $η \leq 3/8$ , the previously paid fixed-width square strip is likewise reused without recounting it. The stronger $O (A^{- η} / L^{2})$ estimates on the narrower old strips remain available. The power exponent has not improved, the prime strip still shrinks, and the remaining samples, mean rate, critical region, other recovery terms and complete signed Robin tail remain unpaid. This is an attributed parameter application of the existing (SV47) and (GS4), without a historical originality or Lean-certification claim.

Signed short-interval inputs control the actual activation response

The actual coefficients permit a further application of published Möbius estimates. Matomäki–Shao–Tao–Teräväinen, Higher uniformity of arithmetic functions in short intervals I. All intervals, arXiv:2204.03754v4, Theorem 3.1(i), supplies arbitrary fixed logarithmic savings for the untwisted Möbius sum at the $7/12$ length threshold. Its starred estimate includes the ordinary interval sum. The theorem’s upper length cutoff is $t^{1 - ε}$ ; longer intervals up to $t$ are covered by equal pieces of length comparable to $t^{1 - 2 ε}$ , with all their lower endpoints in $[t, 2 t]$ . Choose $ε$ small enough to keep these pieces above the lower cutoff. This yields, for fixed $θ > 7/12$ and every fixed $B > 0$ ,

$∣ M (t + h) - M (t) ∣ ≪_{θ, B} \frac{h}{( lo g t ) ^{B}}, t^{θ} \leq h \leq t, M (t) = n \leq t \sum μ (n) .$

For fixed $θ > 19/35$ and $0 < B < 1/3$ , the same estimate is already supplied by Matomäki–Pandey–Pliego–Teräväinen–Wang, arXiv:2610.09567v1, Theorem 1.4. Choose its parameter $ε$ smaller than both $θ - 19/35$ and $1/3 - B$ . These inputs hold at every sufficiently large lower endpoint. No exceptional-set avoidance or unrestricted nilsequence testing weight is assumed. The following is a paper-level application, without Lean certification or a historical novelty claim.

Preserve coprimality and all short prefixes

Fix a supplied pair $1/2 < θ < 1$ , $B > 0$ , and then fix $0 < κ < 1 - θ$ . Suppose

$R^{θ - 1 + κ} \leq δ \leq 1/2, r = lo g R, U = R^{2} .$

For $0 \leq H \leq δ U$ , define the actual signed prefix $F_{R} (H) = \sum_{U < n \leq U + H} μ (n) a_{R} (n)$ . The exact identity $μ (d m) μ (d) = μ (d)^{2} μ (m) 1_{(d, m) = 1}$ gives

$F_{R} (H) = \frac{1}{r} d \leq R \sum μ (d)^{2} lo g (R / d) U / d < m \leq (U + H) / d \sum μ (m) 1_{(m, d) = 1} . (SV52)$

Reuse the finite coprime convolution preceding (SV11). Its inner sum is exactly

$a ∣ d^{\infty} a \leq (U + H) / d \sum [M (\frac{U + H}{d a}) - M (\frac{U}{d a})] .$

Here $a ∣ d^{\infty}$ retains all prime powers supported on $d$ . Split this same finite sum at $Z = R^{κ /4}$ . For $a \leq Z$ , $d \leq R$ and $t = U / (d a)$ ,

$R^{1 - κ /4} \leq t \leq R^{2}, δ t^{1 - θ} \geq R^{κ (3 + θ) /4} .$

If $0 \leq h \leq δ t$ is below $t^{θ}$ , the integer count $∣ M (t + h) - M (t) ∣ \leq h + 1 \leq t^{θ} + 1$ is absorbed by $δ t / r^{B}$ using the displayed positive power margin. For $h \geq t^{θ}$ , apply the supplied Möbius estimate. Thus every such prefix, including the arbitrarily short ones, satisfies $∣ M (t + h) - M (t) ∣ ≪ δ t / r^{B}$ .

Using $\sum_{a ∣ d^{\infty}} 1/ a = d / φ (d)$ , the contribution with $a \leq Z$ in (SV52) is at most

$\frac{C δ U}{r ^{B + 1}} d \leq R \sum \frac{μ ( d ) ^{2} lo g ( R / d )}{φ ( d )} ≪ δ U r^{1 - B} .$

The last bound uses $\sum_{d \leq R} 1/ φ (d) ≪ 1 + r$ . The elementary divisor identity $d / φ (d) = \sum_{q ∣ d} μ (q)^{2} / φ (q)$ and the convergent product $\sum_q\mu(q)^2/(q\varphi(q)) =\prod_p(1+1/(p(p-1)))$ directly supply it.

For the complementary $a > Z$ , put $P_{d} = \prod_{p ∣ d} (1 - p^{- 1/2})^{- 1}$ . The same Rankin estimates used for the coprime convolution give

$a > Z a ∣ d^{\infty} \sum \frac{1}{a} \leq Z^{- 1/2} P_{d}, # {a \leq 2 U / d : a ∣ d^{\infty}} \leq (2 U / d)^{1/2} P_{d} .$

Uniformly for $d \leq R$ , $P_{d} \leq exp (C r)$ : enumerate its distinct prime factors and use $p_{j} \geq j + 1$ and $ω (d) \leq r / lo g 2$ in the logarithm of the product. Retain the interval count $δ U / (d a) + 1$ for each large $a$ . Then this part of (SV52) is bounded by

$exp (C r) (δ U r Z^{- 1/2} + \frac{R ^{3/2}}{r}) .$

The second term includes every unit endpoint error. Here $\sum_{d \leq R} lo g (R / d) / d ≪ r^{2}$ and $\sum_{d \leq R} lo g (R / d) / d ≪ R$ . Relative to $δ U r^{1 - B}$ , the two terms are at most

$exp (C r) (r^{B} R^{- κ /8} + r^{B - 2} R^{1/2 - θ - κ}) = o (1) .$

All parameters here are fixed and $θ > 1/2$ . Consequently, with $T_{R} (Y) = \sum_{U < n \leq Y} μ (n) a_{R} (n) lo g (n / U)$ , Abel summation of the original logarithmic weight gives

$0 \leq H \leq δ U sup ∣ F_{R} (H) ∣ T_{R} (Y) U < Y \leq U (1 + δ) sup E_{A} (Y) ≪ δ U r^{1 - B}, = F_{R} (Y - U) lo g (Y / U) - \int_{U}^{Y} F_{R} (t - U) \frac{d t}{t}, = U < Y \leq U (1 + δ) sup [T_{R} (Y) / r - 1 - σ_{A} Y]_{+} ≪ \frac{δ ^{2} U}{r ^{B}} . (SV53)$

The complete constant and drift enter before the upper bound; only their already known nonnegativity is used. This is uniform control of the original positive part at every real endpoint, rather than an average taken before applying the positive part.

Wider actual prime and square strips

Return to the same $R = A^{1/8}$ , $r = L /8$ and actual source. The existing outer prime increment in (SV50), combined with (SV53), gives for its admissible interval lengths

$A \leq x \leq 2 A sup \frac{1}{x} U < x / p^{k} \leq U (1 + δ) \sum w_{N, p, k} E_{A} (x / p^{k}) ≪_{k, θ, B, κ} \frac{δ ^{3} A ^{- a_{k}}}{L ^{B}} . (SV54)$

For $k = 1$ with the original $0 < η < 5/32$ , and for $k = 2$ with the original $3/8 < η < 1/2$ , every fixed $M > 0$ permits

$δ = A^{- (η - a_{k}) /3} L^{M}, B = 3 M + 2. (SV55)$

Indeed $8 (η - a_{k}) /3 < 5/12$ ; choose $θ > 7/12$ and $κ > 0$ leaving a strict power margin in $δ \geq R^{θ - 1 + κ}$ . Equation (SV54) is then $O_{η, M} (A^{- η} / L^{2})$ . The outer lengths still exceed $P_{k}^{2/3}$ by a positive power, as in (SV51), and these strips still shrink because $M$ is fixed.

For the additional prime-layer range $5/32 \leq η < 6/35$ , the newer $19/35$ supplier applies because $8 η /3 < 16/35$ . Choose fixed $0 < ξ < 1/9$ and $3 ξ < B < 1/3$ , and use

$δ = A^{- η /3} L^{ξ} .$

Equation (SV54) gives $O (A^{- η} / L^{B - 3 ξ}) = o (A^{- η})$ , with the same strict length margin. The original fixed exponent has not been replaced. For $η \geq 6/35$ , this signed transfer supplies no additional prime strip; the previous unsigned (SV51) remains available. The fixed-width square payment for $η \leq 3/8$ is reused.

These enlarged strips retain $p > y = x / p^{k}$ , the full coprime response, and the original subset and square-deletion rules in (SV50)–(SV51). They improve the permitted logarithmic factor in the width; they do not improve its power exponent. Choosing $M = M (A)$ is not allowed. At a fixed width the stated bounds provide only fixed logarithmic savings, insufficient to certify the original power target when $η > a_{k}$ . No same-source mean rate, critical $y ≍ R^{4}$ response, remaining recovery convolution or complete signed Robin tail has been supplied by this application.

Almost-all short sums transfer to the actual dense inner mesh

The almost-all supplier is already cited in FIB §400: Matomäki–Radziwiłł–Shao–Tao–Teräväinen, Higher uniformity of arithmetic functions in short intervals II. Almost all intervals, arXiv:2411.05770v2, Theorem 1.1(i), with the starred norm defined in equation (1.4). For fixed $ε > 0$ and every fixed $K > 0$ , it bounds the untwisted starred Möbius sum by $H / (lo g T)^{K}$ for real starting points in $[T, 2 T]$ , outside a set of measure $O_{K, ε} (T / (lo g T)^{K})$ , whenever $T^{1/3 + ε} \leq H \leq T^{1 - ε}$ . Take the testing function identically one and fix the nilsequence complexity parameter. The starred supremum already includes every initial subinterval; no new maximal-prefix theorem is required.

The following application combines this cited supplier with the all-interval supplier used in (SV52)–(SV53). It does not assume that the actual sampled endpoints avoid the exceptional set. It has no Lean certification or historical originality claim.

Fix a supplied pair $1/2 < θ < 1$ , $B > 0$ , and fix

$0 < c < \frac{2 ( 1 - θ )}{2 - θ}, R^{- c} \leq δ \leq R^{- c /2} . (SV56)$

The all-interval input is $∣ M (t + h) - M (t) ∣ ≪ h / (lo g t)^{B}$ for $t^{θ} \leq h \leq t$ . Keep the actual $r = lo g R$ , $U = R^{2}$ , coefficients and signed prefix $F_{R} (H)$ of (SV52). The conclusion is

$0 \leq H \leq δ U sup ∣ F_{R} (H) ∣ ≪_{c, θ, B} δ U r^{1 - B}, U < Y \leq U (1 + δ) sup E_{A} (Y) ≪_{c, θ, B} \frac{δ ^{2} U}{r ^{B}} . (SV57)$

The smooth-source tail retains all endpoint errors

Choose fixed $0 < α < 1 - c$ and put

$P_{d} (α) = p ∣ d \prod (1 - p^{- α})^{- 1}, g_{α} (q) = p ∣ q \prod \frac{p ^{- α}}{1 - p ^{- α}} (q squarefree),$

with $g_{α} (1) = 1$ and $g_{α} (q) = 0$ otherwise. Expanding the product gives $P_{d} (α) = \sum_{q ∣ d} g_{α} (q)$ , and

$q \sum \frac{g _{α} ( q )}{q} = p \prod (1 + \frac{p ^{- α}}{p ( 1 - p ^{- α} )}) < \infty.$

The ordinary weighted harmonic sums therefore give

$d \leq R \sum \frac{lo g ( R / d )}{d} P_{d} (α) ≪_{α} r^{2}, d \leq R \sum \frac{lo g ( R / d )}{d ^{α}} P_{d} (α) ≪_{α} R^{1 - α} . (SV58)$

Indeed, after writing $d = q m$ , the inner sums are bounded by $O ((1 + lo g (R / q))^{2})$ and $O_{α} ((R / q)^{1 - α})$ , respectively; the remaining factor is $g_{α} (q) / q$ . Thus no uniform subexponential bound for $P_{d}$ is needed here.

Set $Z = r^{C}$ with fixed $C > 0$ . In the exact coprime convolution of (SV52), the terms with $a > Z$ , $a ∣ d^{\infty}$ , retain the integer interval bound $δ U / (d a) + 1$ . The Rankin bounds are

$a > Z a ∣ d^{\infty} \sum \frac{1}{a} \leq Z^{- (1 - α)} P_{d} (α), # {a \leq 2 U / d : a ∣ d^{\infty}} \leq (2 U / d)^{α} P_{d} (α) .$

Together with (SV58), their total contribution to $F_{R}$ is at most

$O_{α} (δ U r Z^{- (1 - α)} + \frac{R ^{1 + α}}{r}) . (SV59)$

The second term includes every unit endpoint error. Choose $C (1 - α) > B + 1$ . Both terms are then $o (δ U r^{1 - B})$ : the second has the strict power margin $1 - c - α > 0$ . These choices are fixed before $R$ grows.

Sparse starts use the uniform input; dense starts use an integral

Choose fixed $ω > 0$ sufficiently small that

$c + ω < 1, (2 - c - 2 ω) (1 - θ) > c,$

and put $D_{0} = R^{c + ω}$ . For $a \leq Z$ , $d \leq D_{0}$ and $t = U / (d a)$ , eventually $t \geq R^{2 - c - 2 ω}$ . Thus $δ t^{1 - θ}$ grows by a positive power of $R$ . The same short-prefix argument in (SV52) bounds every inner prefix by $O (δ t / r^{B})$ , including lengths below $t^{θ}$ . Reuse $\sum_{a ∣ d^{\infty}} 1/ a = d / φ (d)$ and the weighted harmonic bound there. This sparse part is $O (δ U r^{1 - B})$ .

For the remaining $d > D_{0}$ , fix $a \leq Z$ and a dyadic block $D < d \leq 2 D$ , padding boundary blocks only in the nonnegative envelope bound. The actual restored weights remain restricted to $d \leq R$ , so $lo g (R / d)$ is never used at a negative value. Put $T = U / (2 a D)$ and define the common-prefix envelope

$Q_{T} (t) = 0 \leq h \leq 2 δ T sup ∣ M (t + h) - M (t) ∣.$

The sampled starts $t_{d} = U / (a d)$ lie in $[T, 2 T]$ , with separation comparable to $T / D$ . Uniformly for these blocks, $T ≫ R / Z$ , $T ≪ R^{2}$ , and $lo g T ≍ r$ . Condition (SV56) implies $c < 2/3$ . Choose a fixed sufficiently small $ε > 0$ . The strict margins in $δ T^{2/3 - ε} ≫ 1$ and $δ T^{ε} ≪ 1$ ensure that the almost-all supplier applies to length $2 δ T$ , also on a fixed dyadic cover of $[T /2, 3 T]$ . On the exceptional set use $Q_{T} \leq 2 δ T + 1$ . For every fixed $K > 0$ it follows directly that

$\int_{T /2}^{3 T} Q_{T} (t) d t ≪_{c, C, K} \frac{δ T ^{2}}{r ^{K}} .$

Take disjoint cells of length comparable to $T / D$ around the starts $t_{d}$ . Since $∣ μ (n) ∣ \leq 1$ , moving the start by $v$ changes any fixed-length interval sum by at most $2∣ v ∣ + 2$ . Averaging over each cell therefore gives the deterministic bound

$D < d \leq 2 D \sum Q_{T} (t_{d}) ≪ \frac{δD T}{r ^{K}} + T + D . (SV60)$

This uses the exceptional-set integral itself, including any sampled points it contains. It is valid for the actual mesh at every sufficiently large $R$ .

For an upper bound, drop $a ∣ d^{\infty}$ and $μ (d)^{2}$ only in this nonnegative small- $a$ sum. Restore the actual factor $lo g (R / d) / r$ from (SV52). Summing (SV60) over the blocks and $a \leq Z$ gives

$O (δ U r^{1 - K} lo g Z + \frac{U}{D _{0}} lo g Z + \frac{RZ}{r}) .$

Here $\sum_{a \leq Z} 1/ a = O (lo g Z)$ , $\sum_{D} lo g (R / D) = O (r^{2})$ , $\sum_{D} lo g (R / D) / D = O (r / D_{0})$ , and $\sum_{D} D lo g (R / D) = O (R)$ . Choose $K > B + 2$ . The first term is $o (δ U r^{1 - B})$ ; the other two have the positive power margins $ω$ and $1 - c$ . Combining this with the sparse part and (SV59) proves the prefix estimate in (SV57). The existing Abel-summation and full-positive-part argument of (SV53) supplies its second estimate. The constant $1$ , full drift $σ_{A} Y$ and original coefficients are retained.

The mesh application enlarges the signed prime-onset range

Combining (SV57) with the existing prime increment in (SV50) gives exactly the cost bound (SV54), now with condition (SV56). The supplied source and its original fixed $η$ are unchanged.

For every original $0 < η < 15/68$ and every pre-fixed $M > 0$ , take

$δ = A^{- η /3} L^{M}, B = 3 M + 2. (SV61)$

Choose $θ > 7/12$ sufficiently close to $7/12$ . Since $c = 8 η /3 < 10/17 = 2 (1 - 7/12) / (2 - 7/12)$ , condition (SV56) holds eventually. The all-interval supplier allows this fixed $B$ , and the almost-all supplier allows any fixed $K > B + 2$ . Thus the actual prime-onset cost is $O_{η, M} (A^{- η} / L^{2})$ . This extends the arbitrary-fixed- $M$ range of (SV55), which previously required $η < 5/32$ .

For the additional original range $15/68 \leq η < 4/17$ , take fixed $0 < ξ < 1/9$ and $3 ξ < B < 1/3$ , and set

$δ = A^{- η /3} L^{ξ} . (SV62)$

Choose $θ > 19/35$ close enough to leave a strict margin, because $c = 8 η /3 < 32/51 = 2 (1 - 19/35) / (2 - 19/35)$ . The same cost is $O (A^{- η} / L^{B - 3 ξ}) = o (A^{- η})$ . This extends the small-positive-logarithmic-width range of (SV55), which previously stopped at $η < 6/35$ .

The outer prime lengths in both cases still exceed $P_{1}^{2/3}$ by a positive power, as required in (SV50), and $p > y$ remains true. The actual weights, coprime response, full positive part and existing sampled deletion rules therefore retain their original scope. The square-layer applications of (SV55) are already available and are directly reused.

These are still shrinking strips with the same power exponent in their width. No growing $M (A)$ , fixed-width prime payment, same-source mean estimate, critical $y ≍ R^{4}$ estimate, remaining recovery convolution or complete signed Robin-tail estimate follows from this application. For $η \geq 4/17$ , the unsigned strips remain available; no enlarged signed strip is supplied here.

The full complementary hyperbola bounds the actual finite response

The general coprime Mertens estimate preceding (SV11) also applies outside that equation’s highest-support band. Keep all original coefficients and define the continuous complementary ramp

$χ_{R} (e) = ⎩ ⎨ ⎧ 0, lo g (e / U) / r, 1, e \leq U, U < e \leq U_{1}, e > U_{1} .$

The actual difference weight satisfies $θ^{'} (e) = μ (e) (1 - χ_{R} (e))$ . Reuse $a_{R} (n) = \sum_{d ∣ n} λ (d)$ and the original divisor-product formula for $H_{A}$ . The classical identity $\mathbf1*\mu= \mathbf1_{{1}} $g i v es, f ore v eryre a l$ y\ge1$,

$H_{A} (y) = 1 - m e \leq y e > U \sum μ (e) χ_{R} (e) a_{R} (m e) . (SV63)$

This finite sum includes all $m \leq y / U$ and the saturated range $e > U_{1}$ . It is not restricted to the first activation or to the highest LCM band. The following is an application of the already cited classical Mertens input, without Lean certification or historical originality claim.

For squarefree $d \leq R$ , let $g = (d, m)$ and $q = d / g$ . Then $d ∣ m e$ is equivalent to $q ∣ e$ , and $(g, q) = 1$ . Writing $e = q b$ retains the exact sign identity

$μ (d) μ (q b) = μ (g) μ (q)^{2} μ (b) 1_{(b, q) = 1} .$

In particular the common-factor sign $μ (g)$ is not assumed positive. Use the existing coprime bound preceding (SV11), with $W (u) = u^{3/5} / (lo g u)^{1/5}$ ,

$∣ M_{q} (t) ∣ ≪ t e^{- c_{1} W (l o g t)} \frac{q}{φ ( q )} + t^{3/4} p ∣ q \prod (1 - p^{- 1/2})^{- 1}, M_{q} (t) = n \leq t (n, q) = 1 \sum μ (n) .$

The bounds already used in (SV52)–(SV53), $q / φ (q) ≪ r$ and $\prod_{p ∣ q} (1 - p^{- 1/2})^{- 1} \leq e^{C r}$ for $q \leq R$ , therefore give, uniformly for $t \geq R$ ,

$∣ M_{q} (t) ∣ ≪ t Ξ_{R}, Ξ_{R} = e^{- c_{2} W (r)} .$

The remainder divided by $t$ is at most $R^{- 1/4} e^{C r}$ , which is absorbed into $Ξ_{R}$ . For each nonempty inner interval in (SV63), put $T = y / (m q)$ . Its lower endpoint is $U / q \geq R$ . The weight $b \mapsto χ_{R} (q b)$ is continuous, nondecreasing, bounded by one, has total variation at most one, and vanishes at $U / q$ . Abel summation thus supplies

$U / q < b \leq T \sum μ (b) 1_{(b, q) = 1} χ_{R} (q b) ≪ T Ξ_{R} (y > U) .$

All real endpoints and the ramp’s saturation are retained. Expand the actual $a_{R}$ in (SV63) and only then take absolute values. The common-factor identity gives

$∣ H_{A} (y) - 1∣ ≪ \frac{y Ξ _{R}}{r} d \leq R \sum \frac{μ ( d ) ^{2} lo g ( R / d )}{d} m \leq y / U \sum \frac{( d , m )}{m} .$

For $M \geq 1$ , the classical gcd-divisor identity gives

$m \leq M \sum \frac{( d , m )}{m} = b ∣ d \sum \frac{φ ( b )}{b} j \leq M / b \sum \frac{1}{j} \leq (1 + lo g M) τ (d) .$

Reuse the ordinary divisor harmonic bound $\sum_{d \leq R} τ (d) lo g (R / d) / d ≪ r^{3}$ . For $U < y \leq 2 A = 2 R^{8}$ , $lo g (y / U) = O (r)$ , so the preceding estimate is $O (y r^{3} Ξ_{R})$ . Absorbing these fixed logarithmic factors and using $r = L /8$ yields a fixed $c_{0} > 0$ such that

$∣ H_{A} (y) - 1∣ ≪ y Ξ_{A}, E_{A} (y) ≪ y Ξ_{A}, Ξ_{A} = e^{- c_{0} W (L)}, U < y \leq 2 A . (SV64)$

The second bound concerns the complete original positive part: $E_{A} (y) = [(1 - H_{A} (y)) - 1 - σ_{A} y]_{+} \leq ∣1 - H_{A} (y) ∣$ because $σ_{A} \geq 0$ . It assumes no sign of $S_{A}$ and takes no average before the positive part. The existing two envelopes can be combined with this bound on its stated range by taking their minimum.

Together with the existing $E_{A} (y) = 0$ for $y \leq U$ , this bound covers every actual argument $x / p^{k}$ in the local response problem with $A \leq x \leq 2 A$ . The existing exact floor correction at $y = A$ , together with $\sum_{ℓ} ∣ h (ℓ) ∣ \leq B_{A} ≪ R^{4} / r^{2}$ , also gives

$∣ S_{A} ∣ \leq \frac{∣ H _{A} ( A ) - 1∣ + 1 + B _{A}}{A} ≪ Ξ_{A} + \frac{1 + B _{A}}{A} ≪ Ξ_{A} .$

This is a two-sided subexponential mean envelope, not the required fixed-power mean estimate. For every fixed $η > 0$ , $Ξ_{A} / (A^{- η} / L) = L exp (η L - c_{0} W (L)) \to \infty$ ; the stated error allowance does not pay that original rate. No estimate for the jointly compensated improper tail follows by splitting off this mean.

A critical prime strip is paid at every original fixed exponent

For the same actual source, $A \leq x \leq 2 A$ and $0 < δ \leq 1/16$ , retain only prime-layer samples with $V < x / p \leq V (1 + δ)$ . Put $P = x / V ≍ A^{1/2}$ . Their primes lie in $[P / (1 + δ), P)$ , which has at most $O (δ P + 1)$ integers. Keep the actual $0 \leq w_{N, p, 1} \leq lo g p ≪ L$ and use (SV64). This gives

$A \leq x \leq 2 A sup \frac{1}{x} V < x / p \leq V (1 + δ) \sum w_{N, p, 1} E_{A} (x / p) ≪ L Ξ_{A} (δ + A^{- 1/2}) . (SV65)$

The second term includes the outer integer endpoint error. For every original fixed $0 < η < 1/2$ , choose

$δ_{crit} = A^{- η} exp (\frac{c _{0}}{2} W (L)) . (SV66)$

This tends to zero. The first term in (SV65) is $L A^{-\eta}e^{-c_0\mathcal W(L)/2} =O(A^{-\eta}/L^2)$, and the endpoint term has the additional strict power margin $1/2 - η > 0$ . Thus this actual critical strip costs $O_{η} (A^{- η} / L^{2})$ at the original target scale.

Here $p ≍ R^{4} > U_{1}$ . Every nonzero original LCM coefficient uses $d \leq R$ , $e \leq U_{1}$ ; its support therefore contains no such $p$ . The original and coprime responses coincide, and no common sampling-prime contribution has been omitted. The previously paid sampled approximation and deletion rules can still be used on this subset.

The extra width factor in (SV66) exceeds every pre-fixed power of $L$ , but the width remains $A^{- η + o (1)}$ and shrinks. This does not improve its fixed power exponent. At fixed width, (SV65) gives only subexponential decay in $L$ , which does not certify the original $A^{- η}$ budget. The remaining critical samples, same-source mean rate, other recovery terms and complete signed Robin tail retain their original obligations; no RH conclusion or estimate for an arbitrary new polynomial positive part is asserted.

The actual type-II coefficients have an infinite negative-part cost

Fix the original actual $N$ , its clock $A$ , price $ϵ = 1/ (A lo g A)$ , and the weights and cutoffs above. The following is an application of the published recovery (SV1), not a new recovery theorem or a Lean-certified result. It tests a coefficientwise payment scheme; the original joint target (SV4) is unchanged.

To distinguish the type-II factor from the existing divisor response $b_{A} (n)$ , put

$b_{A}^{II} = (1 * θ) (1 * λ) = δ_{1} - 1 * h .$

The second equality is the divisor-product identity used in the proof of Lemma 2.1: $(1 * μ) (n) = δ_{1} (n)$ , $λ (1) = 1$ , and $(1 * θ^{'}) (1 * λ) = 1 * h$ . Reuse that identity directly. For every ordinary prime $p$ , let $a_{p} = v_{p} (N)$ and define the original increment and its complement by

$β_{j} (p) W_{N} (p^{k}) Γ_{N} (p^{k}) = lo g \frac{1 - p ^{- (j + 1)}}{1 - p ^{- j}}, j \geq 1, = w_{N, p, k} = lo g p - ϵ^{- 1} β_{a_{p} + k} (p), = ϵ^{- 1} β_{a_{p} + k} (p), k \geq 1.$

Both arithmetic functions vanish away from prime powers. They retain every actual multiplicity, and $W_{N} + Γ_{N} = Λ$ . The original CA price gives $0 \leq W_{N} (p^{k}) \leq lo g p$ . Consequently the complete type-II term splits exactly as

$b_{A}^{II} * Λ_{> V} = T_{N}^{w} + T_{N}^{g}, T_{N}^{w} = b_{A}^{II} * (W_{N})_{> V}, T_{N}^{g} = b_{A}^{II} * (Γ_{N})_{> V} . (SV67)$

Actual rough multiples retain the original gains

Call $n$ $V$ -rough when every prime factor of $n$ exceeds $V$ . Since $R < U_{1} < V$ , for such $n > 1$ the only divisors in the supports of $λ$ and $θ^{'}$ are one. Their values at one are both one. Hence

$b_{A}^{II} (1) = 0, b_{A}^{II} (n) = - 1 (n > 1 is V -rough) .$

If additionally $(n, N) = 1$ and $n$ has at least two distinct prime factors, every prime-power divisor is proper. All the convolution factors $b_{A}^{II} (n / p^{k})$ are therefore $- 1$ , and $a_{p} = 0$ on these primes. The complete layer telescoping gives

$T_{N}^{w} (n) = - p^{k} ∣ n \sum w_{N, p, k} = - lo g n + ϵ^{- 1} lo g Z (n) . (SV68)$

The same source’s all-integer right-tail comparison applies to the actual competitor $N n$ . Multiplicativity, using $(n, N) = 1$ , gives

$0 \leq lo g Z (n) \leq lo g \frac{lo g ( A + lo g n )}{lo g A} = o_{N} (lo g n) .$

Thus (SV68) is $- lo g n + o_{N} (lo g n)$ on these integers. The comparison is applicable but is not necessary for this asymptotic: the ordinary bound $Z (n) \leq \sum_{d \leq n} 1/ d \leq 1 + lo g n$ already suffices at fixed $N$ . No favorable actual source phase or additional prime-distribution hypothesis has been selected.

The full kernel detects divergence on one arithmetic progression

Set

$M_{N} = lcm rad (N), p \leq V \prod p .$

Choose two distinct primes $p_{0}, q_{0} > max {V, P^{+} (N)}$ and put $n_{j} = p_{0} q_{0} (1 + j M_{N})$ for $j \geq 0$ . Euclid’s theorem supplies these two primes; no enumeration or progression-prime theorem is needed. Each $n_{j}$ is $V$ -rough, coprime to $N$ , and has at least the two distinct prime factors $p_{0}, q_{0}$ . Also $n_{j} > A$ , since $p_{0} q_{0} > V^{2} = A$ . Every $N n_{j}$ is an actual integer covered by the original source comparison.

For any fixed real $a < 1$ and $b$ , (SV68) gives eventually

$[- T_{N}^{w} (n_{j}) - a lo g n_{j} - b]_{+} \geq \frac{1 - a}{2} lo g n_{j} .$

Retain the complete original kernel, including its logarithmic correction:

$k (t) = \frac{1 + lo g t}{t ^{2} lo g ^{2} t} = - \frac{d}{d t} \frac{1}{t lo g t}, \int_{n}^{\infty} k (t) d t = \frac{1}{n lo g n} (n > 1) .$

Tonelli’s theorem for the nonnegative integrand now gives

$\int_{A}^{\infty} n \leq t \sum [- T_{N}^{w} (n) - a lo g n - b]_{+} k (t) d t \geq \frac{1 - a}{2} j \geq j_{0} \sum \frac{1}{n _{j}} = \infty. (SV69)$

The last series is a fixed positive multiple of $\sum_{j} (1 + j M_{N})^{- 1}$ . Fixing the original $N$ precedes taking the infinite integration endpoint; a large fixed modulus cannot make this harmonic series converge.

In particular, no representation $T_{N}^{w} (n) + a lo g n + b = Q_{N} (n) + E_{N} (n)$ with $Q_{N} (n) \geq 0$ can satisfy $\int_{A}^{\infty} \sum_{n \leq t} ∣ E_{N} (n) ∣ k (t) d t < \infty$ when $a < 1$ : at each negative coefficient, $∣ E_{N} (n) ∣ \geq [- T_{N}^{w} (n) - a lo g n - b]_{+}$ . This excludes such coefficientwise nonnegative combinations of source comparison defects plus an absolutely integrable error. It asserts nothing for $a \geq 1$ or for a joint signed estimate.

Natural centering retains the coefficientwise obstruction

The divergence in (SV69) also applies after subtracting the actual type-II logarithmic drift and its full constant mean. Keep the same fixed $N$ and all the prime powers in (SV67), and put

$J_{A}^{II} D_{N}^{g} = ℓ \leq V \sum \frac{h ( ℓ ) lo g ℓ}{ℓ}, = p^{k} > V \sum \frac{Γ _{N} ( p ^{k} )}{p ^{k}}, M_{V} c_{N}^{II} = d \leq V \sum \frac{Λ ( d )}{d}, = 1 + J_{A}^{II} + S_{A} (M_{V} + D_{N}^{g}) . (SV70)$

These names distinguish the constants from the existing harmonic atom $J_{A} (m)$ and sampled correction $C_{N} (x)$ . Since $0 \leq β_{a_{p} + k} (p) \leq β_{k} (p) \leq p^{- k}$ , $D_{N}^{g}$ is absolutely convergent. More explicitly, for fixed $N$ ,

$p^{k} \leq x \sum Γ_{N} (p^{k}) = O_{N} (lo g x), x p^{k} > x \sum \frac{Γ _{N} ( p ^{k} )}{p ^{k}} = O_{N} (1) .$

The ordinary harmonic and square-reciprocal tails supply these bounds; different prime powers have different integer values. There is no cutoff of the infinite gain sum in (SV70).

Sum the complete recovery before centering

Reuse the published PNT consequence $ψ (x) = x + o (x)$ , with all prime powers included. A directly recorded supplier is Johnston–Yang, arXiv:2204.01980v2, Theorem 1.1, equation (1.3). Only this asymptotic consequence is used here; its explicit error is not promoted to the original fixed-power Robin allowance.

The finite support of $h$ and Stirling’s formula give

$n \leq x \sum (h * lo g) (n) = S_{A} x lo g x - (S_{A} + J_{A}^{II}) x + O_{A} (lo g x) .$

The existing full-support floor correction is $H_{A} (y) = S_{A} y + O_{A} (1)$ . Therefore the fixed low convolution and the gain term have the respective sums

$n \leq x \sum (1 * h * Λ_{\leq V}) (n) n \leq x \sum T_{N}^{g} (n) = S_{A} M_{V} x + O_{A} (1), = - S_{A} D_{N}^{g} x + O_{N} (lo g x) .$

For the second line, sum the exact factor $\sum_{m \leq y} b_{A}^{II} (m) = 1 - H_{A} (y)$ for $y \geq 1$ . The preceding harmonic and reciprocal-tail estimates pay both the finite convolution error and the omitted part of $D_{N}^{g}$ .

Now solve (SV1), with (SV67), for the sum of $T_{N}^{w}$ . The finite $Λ_{\leq V}$ term is retained and contributes $O_{A} (1)$ as $x \to \infty$ . This gives

$n \leq x \sum T_{N}^{w} (n) = - S_{A} x lo g x + (c_{N}^{II} + S_{A}) x + o_{N} (x) .$

Consequently the actual centered coefficients

$T_{N}^{w} (n) = T_{N}^{w} (n) + S_{A} lo g n - c_{N}^{II}$

satisfy $\sum_{n \leq x} T_{N}^{w} (n) = o_{N} (x)$ . This is a fixed- $N$ asymptotic, with no claimed uniform rate as $N$ grows and no claim about its own signed improper integral.

The accepted finite mean envelope after (SV64) already gives $∣ S_{A} ∣ \to 0$ along the original large sources. In particular, $S_{A} < 1/2$ eventually. Use this existing consequence without reproving that envelope or claiming a fixed-power improvement. Taking $a = S_{A}$ and $b = - c_{N}^{II}$ in (SV69) yields, for every sufficiently large original fixed source,

$n \leq x \sum T_{N}^{w} (n) = o_{N} (x), \int_{A}^{\infty} n \leq t \sum [- T_{N}^{w} (n)]_{+} k (t) d t = \infty. (SV71)$

Thus subtraction of the logarithmic drift, constant mean and Euler-gain contribution does not make coefficientwise negative mass integrable under the original kernel.

The complete recovery still cancels on the same integers

On each actual $n_{j}$ in (SV69), $h$ has no divisor in its support other than one, so $(h * lo g) (n_{j}) = lo g n_{j}$ . No prime power at most $V$ divides $n_{j}$ . Thus the other low terms vanish, while all the high gains satisfy

$T_{N}^{w} (n_{j}) = - lo g n_{j} + ϵ^{- 1} lo g Z (n_{j}), T_{N}^{g} (n_{j}) = - ϵ^{- 1} lo g Z (n_{j}) .$

The original complete recovery therefore gives exactly

$C (n_{j}) = lo g n_{j} + T_{N}^{w} (n_{j}) + T_{N}^{g} (n_{j}) = 0 = Λ (n_{j}) . (SV72)$

Prime powers also retain their full compensation: for $p > V$ and $j \geq 1$ , $(b_{A}^{II} * Λ_{> V}) (p^{j}) = - (j - 1) lo g p$ and $(h * lo g) (p^{j}) = j lo g p$ , leaving $Λ (p^{j}) = lo g p$ .

The three operations $\sum_{n \leq t} [- T_{N}^{w} (n)]_{+}$ , $[- \sum_{n \leq t} T_{N}^{w} (n)]_{+}$ and $[- \sum_{n \leq t} C (n) + t]_{+}$ are different. Equations (SV69)–(SV71) concern only the first and its stated centering. They supply no lower bound for the original sampled $H_{A}$ or $E_{A}$ losses, and do not assert divergence of either of the latter two operations. This rules out the specified coefficientwise payment scheme, while preserving joint arithmetic compensation as the route required by (SV4). The full original signed bound and RH remain unproved; no historical originality or Lean certification is claimed for this application.

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