Assumptions and source interfaces

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This is part of the restatement and premise account of the examination report, dated 15 October 2026. The census comes from the supplied statements, proof sites and consequence sentences. It is not inferred from the metadata's premise list. All references below resolve within the retained frozen subject.

P: the complete consumed shift-cancellation hypothesis

Use L=log x, T=log L, W=exp(sqrt L), with all fixed parameters chosen before x tends to infinity. Prime factors are positive-prime factors of positive integers; P^-(1)=infinity and P^+(1)=1. Falling factorials count ordered distinct marks. A dyad means [Y,2Y), or a fixed constant enlargement.

The group data are fixed constants 0<a<b<c<d<0.47, c_0,C_0,v_-,v_+>0, q in (0,1), and integer ell>=1. For each large x there are disjoint small and big group index sets, each of cardinality between c_0 T and C_0 T. All prime groups are mutually disjoint. Small primes belong to [exp(L^a),exp(L^b)]. Each big group consists of all primes in an interval inside [exp(L^c),exp(L^d)]. Its harmonic mass V_g is between v_- and v_+. For omega_g(n) the number of distinct divisors from group g, define n_* as the largest group-free divisor of n and

$$W_\ell(n)=\prod_g q^{\omega_g(n)-\ell} \frac{(\omega_g(n))_\ell}{V_g^\ell}.$$

For B fixed, D_B means that for every interval I contained in [1,x^2], every Dirichlet character chi of modulus at most L^B, including principal and imprimitive characters, and every real |t|<=L^B,

$$\left|\sum_{m\in I}\frac{\alpha_m\chi(m)m^{it}}m\right|\le L^{-B}.$$

An allowed endpoint has the form

$$F_0(n)=\sum_{mpr=n_*}\alpha_m\chi_0(m)m^{i\sigma_0} \mathbf1_{p\in I_p,\ P^-(p)>W}p^{i\sigma_1}c_r.$$

Here p is an integer, not necessarily prime; alpha is supported on integers rough above W; |alpha_m|,|c_r|<=L^C; I_p is an interval in [x^tau,x^eta] for fixed 0<tau<eta<1/4; chi_0 has modulus at most L^C; and |sigma_0|,|sigma_1|<=L^C. There is no roughness condition on r. The endpoint depends on n only through n_*.

P asserts the following entire uniform implication, as supplied in signed_type_ii.md, lines 80–108. For fixed group data, endpoint bounds C,tau,eta, target D_0>0, and a fixed positive compact support interval [c_Psi,C_Psi], there exists a fixed B such that, for all large x, all nonzero integers k with |k|<=L^C, both signs included, all xL^(-C)<=Z<=xL^C, all |a_1|,|a_2|<=L^C, all smooth Psi on that support with derivative bounds norm(Psi^(j))_infinity<<_j L^(C(j+1)), and all allowed F_0,G_0 whose alpha in F_0 satisfies D_B,

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

The hypothesis also includes conjugation of the discrepancy-bearing endpoint or its coefficients. This review assumes the full statement, including its coefficient classes and uniformity. It supplies no proof of P and no audit of the omitted Sections 3–5. The supplied statement is the interface being tested; its fidelity to an original Section 5 PDF is not separately established.

E and the band weight in the deductions

The general Type II bands have fixed 0<c_1<c_2, s'>0 and lambda>1, with c_2s'<1/4 and s_j=s'L lambda^(-j), for 0<=j<=j_x. The band Q_j consists of all primes with logarithms in [c_1s_j,c_2s_j]. These bands are pairwise disjoint and disjoint from the groups. The last scale lies between c_- T and c_+ T for fixed 0<c_-<=c_+. Each band has a fixed integer number r_0 of labelled slots, with repetitions allowed and (r_0-1)c_1/lambda>=c_2+1. All slots contain band primes except a designated slot of fixed band j_, containing an integer rough above W in [x^tau_,x^eta_], where tau_*=c_1s'lambda^(-j_) and eta_=c_2s'lambda^(-j_**). The normalized list count is a_Q, with factor product_j (1/r_0!), and A(u)=a_Q(u_)W_ell(u).

The signed Type II statement quantifies over every fixed D_,b_,C>0 with eta_<b_/4, fixed smooth Psi supported in a positive compact interval, UV comparable to x with x^b_<=U,V<=x^(1-b_), and complex sequences alpha,beta bounded by L^C. Alpha is rough-supported and its actual dyadic and any further interval restriction satisfies the required D_B. Some B depending only on the fixed data suffices; beta has no discrepancy hypothesis. The conclusion is the uniform bound O(x L^(-D_*)) for the weighted sum over mn=2u-1. This is a deduction from P and E, not an extra premise.

For extraction, fix 0<delta<1/4 and a nonnegative smooth Psi supported on [1,2], positive on a closed interior interval of positive length. Take q=1/2, ell=1, lambda=2, (c_1,c_2,c_3,c_4)=(1,6/5,3/2,9/5), s'=1/[r_0(c_1+c_2)], with r_0 fixed so c_2s'<delta and (r_0-1)c_1/2>=c_2+1. Take j_x maximal with c_1s_(j_x)>=20T. The group logarithms [c_3s_j,c_4s_j] lie wholly in either [L^(1/10),L^(1/5)] or [L^(3/10),L^(2/5)]. Fix the exceptional band j_**>=j_0 after the early geometry. Put X_A=sum A(u)Psi(u/x), H_Q=sum a_Q(v)/v and H_P=sum_(r_*=1) W_1(r)/r.

The sieve normalizations are V(y)=product_(p<=y)(1-1/p), V_1(y)=product_(3<=p<=y)(1-1/(p-1)), and mathfrak S=2 product_(p>=3)(1-1/(p-1)^2)>0. Thus V_1(y)/V(y) tends to mathfrak S; the factor 2 accounts for the omitted prime 2 when the sifted integer is 2u-1.

E, as specified in extraction lines 3–17 together with signed lines 3–7, contains the elementary Section 2 estimates and Proposition 7.1 and Lemmas 7.3,7.4 at their stated strength. The exact consumed clauses are:

The tests are I_gamma(m)=1 when P^-(m)>x^gamma and I_pr(m)=1 when m is prime. Cells are (Y,exp(Delta_L)Y] with Delta_L=L^(-S); their defining counts remain full-cell counts even when K cuts a cell. The rough-number density is the finite sum

$$D_\gamma(w)=\frac1w+ \sum_{l\ge2}\frac1{l!} \int_{t_i\ge\gamma,\ \sum_{i<l}t_i\le w-\gamma} \frac1{w-\sum_{i<l}t_i}\prod_{i<l}\frac{dt_i}{t_i}.$$

At fixed small b_1 it satisfies

$$\int_{b_1}^{1/2}D_\alpha(1-\alpha)\frac{d\alpha}{\alpha} =D_{b_1}(1)-1, \qquad D_{b_1}(1)\le\frac{e^{-\gamma_E}}{b_1}(1+O(e^{-c/b_1})).$$

The endpoint alpha=1/2 is read by a left limit and has no effect on the integral. The density application uses N_u comparable to x, a fact verified for the changed sign in the extraction keeper.

The PNT, Mertens, Siegel–Walfisz and large-sieve statements explicitly supplied in Section 2 are also required. They are not omitted from E's external interface merely because the word elementary precedes that label. Their exact forms and reading limits are recorded next. No assertion of the truth of the complete antecedent E is part of this examination's verdict.

Classical external interfaces and reading depth

The versions below are identified by the supplied manuscript bibliography. The original sources were not supplied or read. “Claims checked” here means the supplied statement was checked against its actual specialization, not verified against an unavailable original. These statements remain explicit assumptions of the conditional argument.

The full Section 2 text was read. The proofs of the consumed local fixed-moment, Fourier and block-sieve forms were examined at their displayed scope, with their classical inputs exposed. Corollary 2.4's harmonic character estimate is a derived interface of Siegel–Walfisz, not a new external premise. Section 2's mean-square lemma, derivative tests and rough-polynomial corollary were read but are not load-bearing inputs to the deductions outside the assumed P/E results; no independent certification of their unused scopes is claimed.

Kevin Ford and Heini Halberstam, The Brun–Hooley sieve, 2000, Section 1 Lemma 1, is credited behind the block construction. A. Buchstab, 1937, is credited behind the density identity. Their original papers are unread. The full local block construction and finite-simplex identity are supplied and examined; the deductions do not import an unspecified stronger theorem from those citations. The remaining bibliography entries are not premises merely because they occur in the source list.

Other source ownership and boundaries

OpenAI's Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release, is identified as an unrefereed manuscript. Sections 2, 6 and 7 and the bibliography were read in the supplied transcription in full. The signed proof and all essential source-proof reconstructions consumed in the extraction were examined, including Type I and the whole balanced sieve. The keepers record the actual derivations. There is no original classical-proof reading or PDF-image comparison, and no claim of manuscript-wide acceptance.

The successor application and final reduction are attributed to Shisheng Li and Zhong Zhuang, Coprime pairs with equal sum of divisors, assuming smooth shifted primes, 11 October 2026 draft, Theorem 1.3 and Section 3, pages 1–3. The provided form is the full project exposition, whose counting proof has been examined in full. The original PDF is unread, so original-source statement and proof fidelity are not certified. Pollack and Pomerance after Erdos is method provenance, not a separate imported premise of the explicit counting proof. The comparison to Li–Zhuang Theorem 1.4 in the extraction is checked for interval containment only; its original wording is unavailable.

The catalog excerpt supplies only the formulation of Problem 824. No live catalog standing or external acceptance evidence was read or inferred.

Native premise census

No native L-claim is consumed as an established premise by the supplied statements, proofs or consequence sentences. The mathematical census on 15 October 2026 found none; no live ledger read was performed or authorized. There is no prospective native grade being used as if already accepted.

The internal logical order is P/E, then signed Type II, Type I and the successor sieve, then A+, then the squarefree-pair reduction. Each intermediate deduction is examined in this package; no circular native promotion or unreported established premise supplies a missing link. The assumption truth, original-source fidelity, and eventual independent grading remain separate from this conditional deduction review.