Elementary estimates for the smooth-successor extraction

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block_sieve: The weighted-dilation manuscript's block sieve (its Lemma 2.9) derived with explicit constants for z >= 1: pointwise upper and lower polynomials with coefficients of modulus at most one and level z^(4h+2), relative error O(e^(-h)) uniformly in growing h, and the fixed h=2 upper bound, with the five instances the successor extraction consumes checked against its hypotheses.

candidate_mass: The weighted-dilation manuscript's Proposition 7.1 derived for the successor candidate: pointwise bounds, prime support in [L^20,x^delta], harmonic masses, the P-part Euler product and tail, and the comparison X_A of order (x/L)H_Q H_P with constants independent of the later exceptional band j_** at least j_0; the micro-band mass comparison used by the balanced box is included.

classical_inputs: The exact Section 2 consumption of the successor extraction and the signed Type II estimate: the classical theorems used in standard form (prime number theorem, Mertens, Siegel-Walfisz, the multiplicative large sieve), the derived divisor moments and Fourier separation, and the Type I large-sieve application written out with its parameter order.

evidence/: Frozen subjects and the independently graded elementary examination, preserving the cutoff counterexample and exact surviving classical-premise scopes.

proxy_densities: The weighted-dilation manuscript's Lemma 7.4 derived at the ranges the successor extraction consumes: an absolute bound for the balanced prime proxy, short-cell upper densities for x^gamma-rough integers on compact sets with w-gamma bounded below, the cofactor bound (7.17) with w-gamma at least 2kappa, the minimum-coordinate identity (7.18), the bound (7.19) from a lower count and the block sieve, and the mesh step that sums the bins.

proxy_discrepancy: The weighted-dilation manuscript's Lemma 7.3 derived at the ranges the successor extraction consumes: full-cell rough counts, proxies bounded by one independently of the cell precision S, and character and Mellin discrepancy for every interval restriction inside the ambient range, all moduli and twists up to L^B, principal and imprimitive characters included, once S is at least 2B+2.

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This folder's results that are not drawn from a cited source are Plasma AI's, with the exact scope warranted by their independently graded examination. Its PASS grade accepts the literal E1 refutation and the surviving $z\ge1$ and E2–E5 deductions relative to the stated classical premises. The estimates themselves are OpenAI's, from Sections 2 and 7 of the weighted-dilation manuscript, read in the retained library transcription; result and equation numbers refer to that manuscript. Classical theorems are cited by their standard statements.

Target

The smooth-successor route to Problem 824 derives Li and Zhuang's Hypothesis $\mathrm A^+$, and from it $h(x)>x^{2-\varepsilon}$ for every $\varepsilon>0$ and all large real $x$, from two source premises: the uniform shift-cancellation estimate (P) and the elementary estimates (E). This folder is the editable proof sketch for (E). Its target is (E) exactly as the signed Type II estimate and the extraction of primes 2u-1 consume it, with their factor ranges, uniformity and order of choices:

• E1, the block sieve (Lemma 2.9) for $z\ge1$, at its five consumed scales;
• E2, proxy discrepancy (Lemma 7.3): the restricted differences $(I-B_I)\mathbf 1_K$ satisfy $(\mathrm D_B)$ for every interval restriction $K$, every character of modulus at most $L^B$ and every twist up to $L^B$, once the cell precision $S$ is large in terms of $B$, with the coefficient bound independent of $S$;
• E3, the densities (Lemma 7.4): the balanced prime proxy bound (7.16) with an absolute constant, the cofactor bound (7.17) on the range $w-\gamma\ge2\kappa+o(1)$, the identity (7.18) and the bound (7.19), and the mesh step that sums the bins;
• E4, candidate mass (Proposition 7.1) with comparison constants independent of the exceptional band $j_{**}\ge j_0$, and the micro-band comparison (7.25);
• E5, the other Section 2 inputs: harmonic character sums (Corollary 2.4), divisor moments (Lemma 2.5), Fourier separation (Lemma 2.8), and the large-sieve application in Type I.

The sketch

Every item below is a written derivation by Plasma AI. The independent grade accepts the operative E1 and exact E2–E5 scopes relative to the quoted classical premises. These elementary proofs do not use P; the weighted E3 consumer consequence additionally assumes the Type II/P replacement. No elementary proof involves the sign of $2u\pm1$ beyond the size relation $x\le2u-1\le5x$.

1. E1. The block sieve is derived for $z\ge1$ with coefficients in $\{-1,0,1\}$, level $z^{4h+2}$, relative error $c_1e^{-h}$ uniformly in growing even $h$, and the fixed $h=2$ upper bound, with $c_1$ and the threshold for $h$ depending only on the density constants. All five consumed instances are checked: cells and residue classes at $z=W$ with $h=2\lceil T^2\rceil$, the per-prime sieve of the composite subtraction, the preliminary sieve and ordinary rough integers at $z=x^{b_1}$ with $h=2\lfloor1/(40b_1)\rfloor$, and the balanced box at $h=2$. Every consumed cutoff tends to infinity. The unrestricted cutoff statement fails for $0<z<1$, as the block-sieve page shows; that range supplies no premise here.
2. E2. The proxy discrepancy is proved for every interval $K$ inside the ambient range $[x^{b_1/4},x^{1-b_1/4}]$, which contains every restriction the extraction makes, and for every $S\ge2B+2$; the extraction's choice $S>2B+3$ is included. Principal characters cancel on full cells, with at most two cut cells; nonprincipal characters are handled for the actual tests by nondecreasing prime lists and Siegel–Walfisz on one last-prime interval, and for the proxy by residue-class counts on every subinterval, as absolute-error statements. The losses $L^{S+1}$ from the number of cells meet savings $e^{-2T^2}$ and $x^{-b_1/4+o(1)}$ that beat every fixed power of $L$.
3. E3. The densities are derived from short-cell prime counts with prime-number-theorem accuracy $L^{-S-2}$ and a cube comparison of reciprocal-prime measures with $dt/t$. The bound (7.17) is proved only where it is used, with $(\gamma,w)$ in the compact set $\gamma\ge b_1$, $w-\gamma\ge\kappa$, $w\le1$; no density at $w=\gamma$ is used. The identity (7.18) is the minimum-coordinate decomposition with factor $l/l!$, and (7.19) follows from an ordinary lower count by ordered prime lists and the block-sieve upper bound at $z=x^{b_1}$. The balanced constant in (7.16) is $3$, independent of $\kappa$ and $S$.
4. E4. The candidate mass is derived in a sampling model. The upper comparison uses one pure slot of band $0$; the lower comparison restricts a fixed prefix and the P-part and uses only the deterministic range of the tail, which contains the exceptional slot because $j_{**}\ge j_0$. Both comparison constants and the exponent $C_A$ are independent of $j_{**}$; only the threshold for $x$ depends on it.
5. E5. The Section 2 inventory lists every consumed input with its parameters, derives the divisor moments and Fourier separation, and writes out the Type I large-sieve application with the order $D$, then the conductor split $C'$, then the Siegel–Walfisz exponent.

Classical boundary. The prime number theorem with error $O_A(y(\log y)^{-A})$, Mertens' estimates, Chebyshev's bound, the Siegel–Walfisz theorem and the multiplicative large sieve are used in their standard forms, with moduli and twists explicitly bounded by fixed powers of $L$. These are the only inputs not derived in this folder or on the pages it links.

Order of choices. The reconstruction respects the extraction's order with no feedback: the candidate geometry, the prefix, $(j',\theta)$ and $j_0$; then $\kappa$, using the balanced constant, which depends only on the absolute bound in (7.16), the comparison constants of E4 and absolute sieve constants; then $b_1$, $b_*=b_1/3$, $j_{**}$ and the bin mesh; then the Type II saving, $K_0$, $D_0$ and the discrepancy exponent $B$ supplied by (P); then the extraction's choice $S>2B+3$ and the prime-number-theorem accuracy $A=S+2$; then $x\to\infty$. The proxy coefficient bound is $1$ and does not depend on $S$, so the choice of $S$ cannot change $B$.

Examination scope. The examination of 2026-10-15 records refuted-as-stated for E1's frozen unrestricted cutoff domain, and refutation-failed for its operative $z\ge1$ range and E2–E5 relative to the stated classical premises. The current E1 statement adds $z\ge1$, and the block-sieve proof adds the empty-prime boundary case; these statement and proof additions differ from the examined wording. The frozen subjects and exact counterexample remain with the report. Its attacks include E3's cube comparison, E4's uniformity in the later fixed exceptional slot and E5's parameter order. The distinct whole-contract grade, examined on 2026-10-16 UTC, is PASS for these separate scopes. Neither the examination nor its grade establishes P or an unconditional Problem 824 conclusion.

Consequence for Problem 824

The graded interface comparison matches the accepted E scopes to the signed Type II, extraction and counting arguments. Conditional on the exact P endpoint hypothesis and the quoted classical premises, their independently graded composition gives $h(x)\ge h_{\rm sf}(x)>x^{2-\varepsilon}$ for each fixed $\varepsilon>0$ and every sufficiently large real $x$, with strict $1\le a<b<x$, coprimality and squarefreeness. P's separate author reconstruction has no accepted independent weight: its grade remains VOID. The OpenAI manuscript remains unrefereed, and the original classical proofs and PDF fidelity were outside these examinations. No unconditional acceptance of Problem 824 or conclusion about Problem 821 follows.