Section 2 inputs consumed by the successor extraction

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This page records which estimates of Section 2 of the weighted-dilation manuscript the signed Type II estimate and the extraction of primes 2u-1 consume, at which parameters, and how each is supplied. It belongs to the elementary estimates of the smooth-successor route. The estimates are the manuscript's or classical; the inventory and the derivations are Plasma AI's. The E5 examination records refutation-failed relative to the stated classical premises, whose original proofs were unread. Its distinct whole-contract grade, examined on 2026-10-16 UTC, is PASS at those exact scopes and reading boundaries. The reconstruction of premise P checked the Section 2 inputs used inside P's own proof; that coverage is not relied on here, and the inputs of the extraction are listed separately.

Classical theorems used in standard form

These are classical theorems, used as stated and not re-derived. Their displayed statements in the cited sources supply the classical boundary; the derived inputs and applications below do not assume the rest of the weighted-dilation manuscript.

• The prime number theorem, in the form $\pi(y)=\mathrm{Li}(y)+O_A(y(\log y)^{-A})$ for every fixed $A>0$; see Terence Tao, 254A, Notes 2, Corollary 39 and Exercise 40. It is used for prime counts in cells of relative length $L^{-S}$ with $A=S+2$, chosen after $S$; for last-prime counts in the rough-integer densities; for the free slot of the candidate; and for the harmonic masses of bands and groups.
• Mertens' estimates, $\sum_{p\le y}1/p=\log\log y+M+O(1/\log y)$ and $\prod_{p\le y}(1-1/p)\sim e^{-\gamma_E}/\log y$; see Tao, 254A, Notes 1, Theorems 15 and 26. The Chebyshev bound $\sum_{y<p\le2y}1/p\ll1/\log y$, for $y\ge2$, follows already from $\pi(2y)/y\ll1/\log y$.
• The Siegel–Walfisz theorem (the manuscript's Theorem 2.2): for fixed $A,C$, uniformly for $(a,r)=1$ and $r\le(\log y)^C$, $\pi(y;r,a)=\mathrm{Li}(y)/\varphi(r)+O_{A,C}(y(\log y)^{-A})$, with ineffective constants. Tao's Notes 2, Exercise 64, states its von Mangoldt form; removing prime powers and partial summation give this form, with arbitrarily large fixed logarithmic accuracy.
• The multiplicative large sieve (the manuscript's Theorem 2.3): for coefficients on an interval of length $H$, $\sum_{f\le R}\frac f{\varphi(f)}\sum^*_{\chi\bmod f}|\sum a_n\chi(n)|^2 \ll(H+1+R^2)\sum|a_n|^2$. This is Montgomery and Vaughan, Multiplicative Number Theory II, Theorem 19.16, enlarging a real interval to consecutive integers, at a cost of at most one in its length.
• Decay of Fourier transforms of smooth compactly supported functions, by integration by parts.

Derived inputs

Harmonic character sums (the manuscript's Corollary 2.4) are derived from the Siegel–Walfisz theorem on the proxy-discrepancy page, uniformly for every interval in $[\exp(L^c),x^{C_0}]$, every nonprincipal character of modulus at most $L^B$, primitive or not, and every twist up to $L^B$.

The block sieve (the manuscript's Lemma 2.9) is derived on the block-sieve page, with its five consumed instances checked.

Divisor moments (the manuscript's Lemma 2.5). For fixed $k\ge0$,

$$\sum_{n\le Y}\frac{\tau(n)^k}n\le\prod_{p\le Y}\sum_{a\ge0}\frac{(a+1)^k}{p^a} \le\prod_{p\le Y}\Bigl(1+\frac{2^k}p+\frac{c_k}{p^2}\Bigr)\ll_k(\log2Y)^{2^k},$$

by unique factorization and Mertens, and $\sum_{n\le Y}\tau(n)^k\le Y\sum_{n\le Y}\tau(n)^k/n$. The consumed forms follow by Cauchy's inequality: $\sum_{u\le2x}\tau(u)\tau(2u-1)\ll xL^{O(1)}$ in the Type II absolute bound, $\sum_{v\ll x}\tau(v)\tau(v+1)^3\ll xL^{O(1)}$ in its diagonal, the endpoint square sums with $\tau(z_*)^4$ in its separation step, and $\sum_{r\asymp U}b(r)^2\ll UL^{O(1)}$ for the Type I coefficients $0\le b(r)\le L^{O(1)}\tau(r)$. The exponents depend only on $k$ and the fixed coefficient bounds, never on later precisions.

Fourier separation (the manuscript's Lemma 2.8). Let $F$ be smooth on $\mathbb R^b$, $b$ fixed, supported in a fixed box, with $j$-th derivatives $O_j(L^{C(j+1)})$, with $C\ge1$ after enlargement if necessary. Integrating by parts $j$ times, $|\widehat F(\xi)|\ll_jL^{C(j+1)}(1+|\xi|)^{-j}$. With $j=b+1$ this gives $\int|\widehat F|\ll L^{C(b+2)}$, and for $C'=2C$,

$$\int_{|\xi|>L^{C'}}|\widehat F(\xi)|\,d\xi\ll_jL^{C(j+1)-C'(j-b)} =L^{-Cj+C(2b+1)},$$

which is at most $L^{-A}$ once $j\ge2b+1+A/C$. So $C'$ is fixed before the desired saving $A$, as the Type II page requires. For a fixed smooth $\Psi$ on $[1,2]$, Mellin inversion $\Psi(y)=\frac1{2\pi}\int\widetilde\Psi(\tau) y^{i\tau}d\tau$ with $\widetilde\Psi$ the Fourier transform of $\Psi\circ\exp$ has $\int|\widetilde\Psi|<\infty$; this is the bounded integrated cost used in Type I.

Elementary sums. Positivity and the squarefree Euler product give $\sum_{d\le y}\mu^2(d)/\varphi(d)\le\prod_{p\le y}(1-1/p)^{-1} \ll\log(2y)$, used for the principal character in Type I and for the squarefree $h$-sum below. For the balanced-box remainders let $d_3(n)=\#\{(a,b,c)\in\mathbb N^3:abc=n\}$. At each prime power $d_3(p^a)=\binom{a+2}{2}\ge3$ for $a\ge1$, hence $3^{\omega(n)}\le d_3(n)$ and

$$\sum_{n\le D}3^{\omega(n)}\le\sum_{ab\le D}\left\lfloor\frac D{ab}\right\rfloor \le D\left(\sum_{a\le D}\frac1a\right)^2 \le D(1+\log D)^2.$$

In particular both $\sum_{d\le D}\nu(d)/d$ and $D^{-1}\sum_{d\le D}\nu(d)$ are $O(D(1+\log D)^2)$ whenever $0\le\nu(d)\le d\,3^{\omega(d)}$, as in the balanced box.

The large sieve in Type I

The Type I proposition on the extraction page is the only consumer of the large sieve, which P does not use. Its application is checked here at the exact parameters. Fix $\vartheta<1/2$ and $D>0$. After extracting the first slot $p\in\mathcal Q_0$ of band $0$, $A(u)=\sum_{pr=u}b(r)$ with $0\le b(r)\le L^{c_b}\tau(r)$, $c_b$ depending only on the candidate. As $\mathcal Q_0\subset[x^{c_1s'},x^{c_2s'}]$ and $c_2s'<1/4$, every contributing dyad pair $p\asymp P$, $r\asymp U$ has $PU\asymp x$ and $x^{a_0}\ll P,U\ll x^{1-a_0}$ with $a_0=c_1s'$. A nonprincipal character modulo squarefree odd $d=hf$ is the primitive $\chi^*\bmod f>1$ together with coprimality to $h$, and $\varphi(d)=\varphi(h)\varphi(f)$.

Fix $c'$ large enough, in terms of the early candidate choices, that the divisor moments give both $\sum_{r\asymp U}b(r)\ll UL^{c'}$ and $\sum_{r\asymp U}b(r)^2\ll UL^{c'}$. The parameter $c'$ precedes $D,C'$ and all proxy choices.

Large conductors. For $R/2<f\le R$ with $R>L^{C'}$, separate $\Psi(pr/x)$ by Mellin inversion. For each $\tau$ put $a_p=\mathbf 1_{(p,h)=1}p^{i\tau}$ on the dyad and $b'_r=b(r)\mathbf 1_{(r,h)=1}r^{i\tau}$. Since $1/\varphi(f)\le(2/R)f/\varphi(f)$, Cauchy's inequality and the large sieve give

$$\sum_{R/2<f\le R}\frac1{\varphi(f)}\sum^*_{\chi\bmod f} \Bigl|\sum_pa_p\chi(p)\Bigr|\Bigl|\sum_rb'_r\chi(r)\Bigr| \ll\frac1R\bigl((P+1+R^2)P\cdot(U+1+R^2)UL^{c'}\bigr)^{1/2},$$

with $c'$ from the divisor moments. With $PU\asymp x$ this is $\ll xL^{c'/2}(R^{-1}+P^{-1/2}+U^{-1/2}+R/\sqrt x)$. Summing over the $O(L)$ dyadic $R\in(L^{C'},x^\vartheta]$, over squarefree $h\le x^\vartheta$ with weight $1/\varphi(h)$ (total $O(L)$), over the $O(L^2)$ dyad pairs and over $\tau$ against $|\widetilde\Psi|$ gives $xL^{c'/2+4}(L^{-C'}+x^{-a_0/2}+x^{\vartheta-1/2})$. Choose $C'\ge D+c'/2+4$; the restrictions $(f,h)=1$ and squarefreeness of $f$ were dropped only to enlarge a sum of absolute values. Keeping $h$ squarefree makes its $O(L)$ mass precisely the elementary sum proved above.

Small conductors. For $1<f\le L^{C'}$, fix $h$, $\chi^*$ and $r$. The primes $p\mid h$ in $\mathcal Q_0$ are at most $\vartheta/a_0+1$ in number and cost $\sum_rb(r)\ll x^{1-a_0}L^{c'}$ per dyad pair. For the rest, partial summation against $y\mapsto\Psi(yr/x)$ on the dyad, whose total variation is $O(1)$ including the two cutoffs, and the bound $\sum_{p\le y}\chi^*(p)\ll_KyL^{-K}$ give $O_K(PL^{-K})$. For completeness, $\log y\asymp L$ at these endpoints, so $f\le L^{C'}\le(\log y)^{2C'+1}$ eventually. Apply Siegel–Walfisz with accuracy $K+C'+2$ and sum over at most $f\le L^{C'}$ reduced classes; the nonprincipal main terms cancel and the resulting bound is stronger than $O_K(yL^{-K})$. Thus the loss from residue classes is included. Summing over $r$, the $\varphi(f)$ characters with weight $1/\varphi(f)$, $f\le L^{C'}$, $h$ and dyads gives $\ll xL^{c'+C'+3-K}+x^{1-a_0}L^{c'+C'+3}$. Choose $K\ge D+c'+C'+3$ after $C'$.

For the principal character, replacing its coprimality-restricted mass by $X_A$ costs at most

$$X_A\max_{u\le2x,\ A(u)>0} \sum_{l\mid u}\frac1{l-1} \sum_{d'\le x}\frac{\mu^2(d')}{\varphi(d')} \ll X_A\frac{L}{\log L}L^{-20}L \ll X_AL^{-18}.$$

Here $l$ is prime, every such $l$ is at least $L^{20}$, and the divisor sum over $d'$ bounds the original sum with $(d',l)=1$ dropped. With the nonprincipal estimates this proves the Type I bound $\sum_{d\le x^\vartheta}|r(d)|\ll xL^{-D}+X_AL^{-18}$ for odd squarefree $d$. The order is $D$, then $C'$, then the Siegel–Walfisz exponent $K$; none depends on $\kappa$ beyond $\vartheta=1/2-\kappa/2<1/2$, and none on $S$.

Consumption by step

No step consumes the mean-square lemma (Lemma 2.6), the derivative tests (Lemma 2.7) or the rough-integer polynomial (Corollary 2.10); those enter only P's own proof.