Smooth successor primes from the weighted-dilation argument

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evidence/: Frozen subjects, conditional whole-conclusion examination and passing grade, preserving the balanced-range qualification, P and classical premises.

li_zhuang_reduction: Li and Zhuang's counting reduction from smooth successor primes to h(x)>x^(2-epsilon), with the overlap bound, multiplicity of gcd removal, strict real endpoint, and squarefree conclusion written out.

shift_sign_analytic_core: The signed shift in the weighted-dilation transference and cancellation argument: the memory identity, norm bounds, equality-rank expansion, ideal kernels and local Fourier-energy estimates, with their parameter order and the precise source estimate still needed by the successor route. Author examination, without independent acceptance.

signed_type_ii: The independently graded conditional Type II derivation for mn=2u-1, retaining both signed determinant directions, the diagonal and exact unaccepted shift-cancellation endpoint hypothesis.

successor_prime_extraction: The independently graded conditional extraction of primes p=2u-1 with smooth p+1, using the proved balanced dyadic scope and exact P and classical premises; the unrestricted dyadic claim is refuted.

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This folder's results that are not drawn from a cited source are Plasma AI's, at the exact conditional scope accepted by the whole-conclusion examination and distinct passing grade. The route is Shisheng Li and Zhong Zhuang's, recorded on their claim page: their reduction follows Pollack and Pomerance after an idea of Erdős, and the counting method is OpenAI's weighted-dilation manuscript. This combined sketch derives the successor count from precise source estimates and carries their provisional standing into the final implication. The compact sign audit lists the substitutions relative to the predecessor proof; the derivations below supply their full conditional interfaces.

Target

Problem 824 asks about the count

$$h(x)=\#\{(a,b)\in\mathbb N^2: 1\le a<b<x,\ (a,b)=1,\ \sigma(a)=\sigma(b)\}.$$

The question is whether, for every $\varepsilon>0$ and every sufficiently large real $x$, $h(x)>x^{2-\varepsilon}$. Its recorded status is open. Pollack and Pomerance proved $h(x)>x^{1.4}$ (their Section 6 result). Write $L=\log x$ and $P^+(m)$ for the largest prime factor of $m$. The route derives three statements under the source premises below.

1. Successor count. For every fixed $0<\delta<1/4$ there are $c,C_A,C_w>0$ such that, for every sufficiently large real $x$, at least $c xL^{-C_A-1}\exp(-C_w\sqrt L)$ primes $p$ satisfy $2x-1\le p\le4x-1$ and $P^+(p+1)\le x^\delta$.
2. Hypothesis $\mathrm A^+$. For every $0<\delta<1/4$ and $\eta>0$, every sufficiently large real $y$ has at least $y^{1-\eta}$ primes $p\le5y$ with $P^+(p+1)\le y^\delta$.
3. Coprime pairs. For every $\varepsilon>0$ and every sufficiently large real $x$, $h(x)>x^{2-\varepsilon}$, even when both entries must be squarefree.

Exact source premises

(P) is the manuscript's Theorem 5.1, its lifted shift-cancellation estimate, restated with all endpoint hypotheses on the Type II page. It bounds

$$\sum_{z>0,\ z+k>0}\frac{\Psi(z/Z)}z\, z^{ia_1}(z+k)^{ia_2}\overline{F_0(z)}G_0(z+k)W_\ell(z)W_\ell(z+k) \ll L^{-D_0}$$

for arbitrary fixed $D_0$, uniformly for nonzero integer shifts $|k|\le L^C$, $Z=xL^{O(1)}$, and the prescribed endpoint class, once the original coefficient sequence $\alpha$ has sufficiently strong character and Mellin discrepancy. That endpoint class includes a compulsory integer factor in a fixed positive power range, rough above $\exp(\sqrt L)$. The source proposes a proof in Sections 3--5. It is an unrefereed premise here. The manuscript's predecessor-count statement concerns $p-1$ and does not itself imply $\mathrm A^+$.

The [[research/erdos_824_uniformity/shift_cancellation_reconstruction|local reconstruction of P]] supplies an author-recorded argument for the endpoint class used here. Its displayed pair (S) is the pair in the Type II page: the original restricted $\alpha$, the common interval $I_{**}$, the common residual coefficient $\overline{c_r}$, and characters modulo $2|k|$ are unchanged. Substitution uses $a_1=\varrho+\xi_3$, $a_2=-\varrho$ and $\Psi(t)=t\psi_Z(t)$; multiplication by $Z$ recovers the unweighted correlation. No discrepancy of a different sequence is inferred. The reconstruction's internal mark parameter $J$ is distinct from the proxy precision $S$ of the extraction: every choice inside P, including $B$, is fixed before $S>2B+3$. This is an interface identification, not independent acceptance of P or a proof of proxy discrepancy.

(E) consists of the manuscript's Section 2 estimates, Proposition 7.1 (candidate mass, uniformly in the exceptional-slot position), Lemma 7.3 (proxy discrepancy, uniformly under interval restrictions), and Lemma 7.4 (proxy densities and the Buchstab identity). Their needed statements, parameters and roles are given on the extraction page. Only the bound (7.17) mentions $2u+1$, and its proof uses that expression only through its size $x$ up to fixed factors. It gives the same bound for $2u-1$. The elementary reconstruction of E supplies written author proofs at these consumed ranges. Its block sieve has coefficients of modulus at most one and the required level, for both growing even $h$ and the fixed upper-bound choice $h=2$. Its proxy discrepancy uses full-cell averages followed by arbitrary interval restrictions inside $[x^{b_1/4},x^{1-b_1/4}]$. It handles principal and imprimitive characters and all twists of size at most $L^B$ once $S\ge2B+2$, with coefficient bound one independent of $S$.

The density proof supplies the balanced proxy bound with absolute constant three, the cofactor estimate on $\gamma\ge b_1$, $w-\gamma\ge\kappa$, $w\le1$, the minimum-coordinate identity and the finite-mesh subtraction. The actual cofactors have $w-\gamma\ge2\kappa+o(1)$, so no density at $w=\gamma$ is assumed. The candidate-mass proof makes its comparison constants independent of the later fixed $j_{**}\ge j_0$; only the threshold for $x$ may depend on that slot. The classical-input inventory derives the consumed moments and smooth separation and gives the complete Type I large-sieve application. Its external boundary is PNT with arbitrary logarithmic accuracy, Mertens, Chebyshev, Siegel--Walfisz and the multiplicative large sieve, in the precise versions stated there.

No E proof uses P. The candidate geometry, finite prefix and $j_0$ precede $\kappa$; then come $b_1$, the exceptional slot and the bin mesh; then the Type II saving and P's choices, including $B$; finally $S>2B+3$ and the prime-number-theorem accuracy $S+2$. The numerical coefficient bound does not feed the last choice back into P. The E deductions have an independent passing grade at the quoted classical boundaries. P remains author-recorded with zero accepted independent weight; the source manuscript remains unrefereed.

The combined argument

1. Type II for $2u-1$. Assuming (P) and the Section 2 estimates, the signed Type II derivation gives $\sum_{mn=2u-1}A(u)\Psi(u/x)\alpha_m\beta_n\ll xL^{-D_*}$ for bounded $\beta$ and an appropriately restricted $W$-rough $\alpha$ with discrepancy. The partition $u=eh$ retains all group factors and the compulsory rough integer in $h$. The off-diagonal identities are $m'-m=-2ek$, $h'-h=-kn$, and $mh'-m'h=k$. Their converse, using $(m,k)=1$, recovers a positive integral $n$. The resulting cross-products $z=m'h$ and $z+k=mh'$ have exactly the endpoint form of (P), with the original $\alpha$ still carrying discrepancy.
2. Prime extraction. Under (E), the signed sieve argument proves Type I distribution in the unit class $u\equiv\bar2$ for odd moduli, uses the Type II estimate to replace roughness and prime tests by their cell proxies, and subtracts composites. The balanced range uses $mn\equiv-1\pmod{D_1}$ and the same residue counts as the predecessor argument. It leaves weighted prime mass $\gg X_A/L$. Candidate mass $X_A\gg xL^{-C_A}$ and the pointwise bound $A(u)\Psi(u/x)\le\exp(C_w\sqrt L)$ give the successor count. Its ratio to $x^{1-\eta}$ tends to infinity for every fixed $\eta>0$, so taking $x=y$ gives the exact $\mathrm A^+$ above for all large real $y$.
3. Counting pairs. The full counting reduction gives Li and Zhuang's Theorem 1.3, including the squarefree conclusion and the strict range $1\le a<b<x$. For a fixed $0<\varepsilon<1$, choose $B\ge8/\varepsilon$, $\eta=\varepsilon/16$ and $\delta=1/(2B)$. It uses $\mathrm A^+$ at $y=(\log x)^B$. Products of $k=\lfloor\log x/\log(5y+1)\rfloor$ distinct available primes give $N\ge x^{1-\eta-1/B-o(1)}$ squarefree integers below $x$, with at most $U=x^{o(1)}$ divisor-sum values. Removing pairs with many common primes and dividing by the gcd costs a multiplicity of at most $2x^\eta$. Therefore $h(x)\ge N^2/(4Ux^\eta)>x^{2-\varepsilon}$. This implication uses neither (P) nor (E) except through $\mathrm A^+$.

The graded composition uses operative E1 and the exact E2–E5 scopes to deduce $\mathrm A^+$ under the complete P endpoint hypothesis and quoted classical premises, then the squarefree-pair bound. Its independent passing grade accepts that conditional argument. P's author reconstruction still has zero accepted independent weight; the source manuscript remains unrefereed and the original-source reading limits persist. No unconditional accepted resolution of Problem 824 follows.

Sign, ranges and uniformity

The change from $p-1$ to $p+1$ occurs inside the Type II and sieve proofs. Its essential checks are:

• Divisibility and positivity. The signed identities and their converse retain $UV\asymp x$, $x^{b_*}\le U,V\le x^{1-b_*}$, $E=UL^{-K_0}$, $\tfrac12EL^{-C_s}\le e\le2E$ and $0<|k|\ll L^{K_0+C_s}<W$. The cutoffs impose $e>0$ and $n\asymp V$ after reconstruction; no unsigned ordering of $m,m'$ is imposed.
• Units and local counts. The rough integers $m,m'$ are units modulo $2|k|$. Type I uses odd squarefree $d$ and its unit class $\bar2$. The balanced sieve's odd $D_1$ may have repeated prime factors and includes the exceptional integer. The congruence $mn\equiv-1\pmod{D_1}$ has $\varphi(D_1)$ unit pairs. Modulo each odd sieve prime $l$, the two disjoint bad sets $mn\equiv0$ and $mn\equiv-1$ have $2l-1$ and $l-1$ pairs.
• Moments and partition. Fixed divisor moments control $\sum_{u\asymp x}\tau(u)\tau(2u-1)$ and $\sum_{v\ll x}\tau(v)\tau(v+1)^3$. Only pure band primes enter $e$; all group factors and the exceptional rough integer stay in $h$. The costs before choosing $K_0$ do not depend on $K_0$.
• Analytic estimates. The analytic-core examination follows the memory identity, rare transfers, clean norm, equality-rank expansion, ideal comparison kernels and local Fourier-energy argument. The shift enters through $|k|$, translation, or invertibility modulo group primes larger than $|k|$. Positive shifted positions remain of size $x^{1+o(1)}$. Theorem 5.1 already asserts both signs.

No sign-specific obstruction was found in these deductions. The two Type II arguments use a common uniform class of instances of (P): endpoint forms, character moduli $2|k|$, polynomial logarithmic frequency bounds, scales $Z=xL^{O(1)}$, and both signs of $k$. Their pulled-back cutoffs and Fourier coefficients differ. This is a statement about a class of permitted data, not an assertion that the arguments make identical individual calls to (P).

A proof uniform over that endpoint class would support both transfers. Checking particular endpoints used by one argument, or accepting the predecessor-count statement alone, does not supply that uniform proof. Failure of (P) outside the consumed class would not by itself refute this route. The sufficient restriction is stated at the end of the analytic-core page.

Standing and the remaining dependence

The signed Type II and sieve implications are derived by Plasma AI under (P) and the operative E estimates. The final counting implication is Li and Zhuang's, written out here with its parameters checked. Their complete conditional examination has a distinct whole-contract PASS grade, examined on 2026-10-16 UTC, retaining every antecedent and reading limit. The reported external Lean proof covers the lower-bound implication at natural $x$, without the squarefree addendum; it has not been built or audited here.

The first deep analytic input is (P). Its author reconstruction depends on transference Theorem 3.5 and Corollary 3.11, ideal-kernel Theorem 4.1, and the comparison analysis of Section 5. It records derivations of these steps, with the classical source boundaries stated on its page; the manuscript remains unrefereed and the reconstruction has no independent acceptance. The separate E reconstruction supplies the elementary estimates at the intervals and choice order consumed above, with a separate passing grade relative to the quoted classical premises. The graded interface comparison finds no missing clause at the actual consumed scopes: the original restricted $\alpha$, common residual coefficient, rough interval, both shift signs, and modulus $2|k|$ in P agree with the Type II input; E supplies that original sequence's discrepancy and every density and mass bound used by the sieve.

The P examination and its VOID grade retain zero accepted independent weight. The separate exposure ruling resolves the eligibility question only; it supplies no whole-contract PASS. The E examination and the conditional-chain examination have distinct PASS grades at their separate stated scopes. The latter accepts refutation-failed for the complete P-and-E implication at every sufficiently large real $x$ with strict $1\le a<b<x$, coprimality and squarefreeness. It does not establish its antecedents. The frozen unrestricted E1 cutoff and isolated dyadic formulations have separate refuted-as-stated findings; only the consumed $z\ge1$ and balanced ranges survive. Their current explicit restrictions are disclosed on the owning pages. The complete proposal has no accepted unconditional conclusion here. Nothing here assesses the manuscript's further totient-fiber argument or changes the standing of Problem 821 or Problem 824.