Examination of the elementary premise E

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Subject and independence

Subject. The complete E1–E5 assertions and every essential deduction in wiki/research/smooth_successor_estimates/_index.md, block_sieve.md, proxy_discrepancy.md, proxy_densities.md, candidate_mass.md and classical_inputs.md in that directory, in the neutral extraction frozen on 2026-10-15 at 20:36:13.943931 UTC. The three consumer pages under wiki/research/smooth_successor_primes/ and the supplied manuscript excerpts are supporting inputs, with their exact native paths, retained copies, line locators and reading depths in the subject map. Examination date: 2026-10-15. No assertion here covers later text.

The reviewer is a fresh, isolated adversarial reviewer, disclosed model gpt-6-astra, who neither authored nor collaborated on the subject and had not previously built on it. The input consisted of the commission, its frozen mathematical extraction, named operating instructions and neutral metadata. Continuation used only this examination's own notes. No author advice, private author plans, previous report, sibling verdict or unrelated research was supplied or read. The reviewer does not grade this report or decide its own eligibility.

The mathematical read set was the six estimate pages, all three consumer pages, the source excerpt containing Section 2, Section 6 definitions and statement, full Section 7 and bibliography, and the extraction's README and read map. All were read in full. All substantive mathematical reading used that extraction. Named operating sources were compared with the delivered manifest and frozen operating copies before use; those comparisons matched. The complete neutral verification standard and its campaign adapter were read, as were the required corpus, evidence, approach-vetting and authoring rules. The frozen subject stayed unchanged. Observed subscription tables at the first boundary and after startup were empty; no inbox or feed was opened. No excluded communication or search exposure was observed. No outside search, other checkout, live claim ledger, synchronization or additional mathematical subject was read. These are supporting facts for a distinct grader to assess, not a self-awarded independence certificate.

Reasoning and reconstruction. The independent written keepers are weak steps and proof coverage. The subject map retains the exact ten mathematical excerpts and explains how to rebuild the assessment from ordinary files. No mathematical program or numerical check was run, and no cached output is mathematical evidence. Administrative byte comparisons establish identity only. This is one commissioned examination; no claim of a disjoint second review cycle is made.

The source is OpenAI, Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release, an unrefereed manuscript. Its mathematics retains that authorship. Reconstruction, uniformity accounting and original project reasoning are Plasma AI's. The supplied manuscript is a transcription. No original classical proofs or PDF-fidelity evidence were supplied or inspected.

Verdict and its limits

E1 is refuted-as-stated on the literal written domain. Its statement places no lower bound on the sieve cutoff $z$ and allows the prime set to be empty. At $z=1/2$, one object of weight 1, $X=0$ and $r(1)=1$ satisfy every written hypothesis, but the remainder level is below 1 and omits $r(1)$. Both sieve inequalities and the polynomial-support clause fail. The exact witness and every hypothesis check appear below and in the first section of proof coverage.

This is a cutoff-domain defect. If $z\ge2$ is treated as an implicit sieve convention, the witness is outside that convention. Neither the supplied statement nor the supplied source conventions state it. All five consumed cutoffs tend to infinity, so this witness does not refute any consumed instance. The complete proof for $z\ge1$, including growing $h$, coefficients, levels and fixed $h=2$, survived the attacks recorded here. This distinction is necessary for the grader to assess the finding's weight.

Thus the conjunction of the unrestricted written assertions is refuted-as-stated, on E1's literal domain. No unsupported essential inference or counterexample was found in the operative E1 fragment or in E2–E5 under the stated classical premises. This is not an unqualified whole-E acceptance and awards no tier. A distinct grader must assess the complete report, the interpretation of the cutoff domain and the external reading limitations. If that grader finds the conventional cutoff already implicit, the recorded operative reasoning supports the narrower conclusion refutation-failed, not the literal counterexample.

P remains an explicit unverified antecedent of the consumer implications. It is not a premise of the reconstructed E proofs. This report establishes no unconditional conclusion about Problem 824 and does not inspect the manuscript's further totient-fiber argument.

Restatement and conventions

All factor variables are positive integers. Prime-indexed sums and products run over primes; multiplicities do not change a distinct-prime count. $P^+(1)=1$, $P^-(1)=\infty$, $\tau$ is the divisor function, $\varphi$ is Euler's function, and $\mu^2$ restricts to squarefree positive integers. Write $L=\log x$, $T=\log L$, $W=\exp(\sqrt L)$ and $V(y)=\prod_{p\le y}(1-1/p)$. A dyad is $[Y,2Y)$ or a fixed enlargement. Every parameter called fixed is chosen before $x\to\infty$; thresholds and implied constants may depend on earlier fixed parameters unless expressly excluded. No effective threshold is asserted where Siegel–Walfisz enters.

E1: block sieve and its five uses

Take a finite weighted population with nonnegative weights, a finite set $\mathcal P$ of primes at most $z$, and a bad event $E_p$ for each of them. For every subset $S\subset\mathcal P$, including the empty set, let $d_S=\prod_{p\in S}p$ and suppose its intersection weight is $Xg(d_S)+r(d_S)$, where $X\ge0$, $g(1)=1$, $g$ is multiplicative on the squarefree products, and, for fixed $0<\eta_0<1$, $C\ge1$,

$$0\le g(p)\le1-\eta_0,\qquad \sum_{\substack{p\in\mathcal P\\w<p\le w^2}}g(p)\le C \quad\hbox{for every }w>1.$$

Let $\mathcal S$ be the weight avoiding all bad events and $G=\prod_{p\in\mathcal P}(1-g(p))$. E1 asserts the existence of $h_0,c_1,c_2$ depending only on $\eta_0,C$, such that for every even $h\ge h_0$,

$$|\mathcal S-XG|\le c_1e^{-h}XG+ \sum_{\substack{d\le z^{4h+2}\\d\text{ a squarefree product of }\mathcal P}} |r(d)|,$$

and at $h=2$, $\mathcal S\le c_2XG+\sum_{d\le z^{10}}|r(d)|$. It further asserts bounding polynomials $V_0\le\mathbf1_{\rm avoid}\le U$ with $U\ge0$ and $U-V_0\ge0$, each of $U,V_0,U-V_0$ having coefficients in $\{-1,0,1\}$ and support $d_S\le z^{4h+1}$. Here $V_0$ denotes the page's lower polynomial $V$, to distinguish it from the prime product above. There is no explicit lower bound on $z$ in the frozen statement. The large-$h$ conclusion is uniform when $h$ grows, not merely for a separate fixed $h$ at each application.

The five prescribed instances are: integers in arbitrary real subintervals and reduced residue classes modulo $r\le L^B$ at $z=W$, $h=2\lceil T^2\rceil$; candidate integers satisfying $p\mid2u-1$ for a prime $p\in(x^\gamma,x^{\gamma'}]$, sieved at the same $W,h$; the preliminary candidate sieve at $z=x^{b_1}$, $h=2\lfloor1/(40b_1)\rfloor$; ordinary rough integers in $(x,2x]$ at that same cutoff and $h$; and the balanced lattice box at $h=2$, $z=x^\theta$. The first has $g(p)=1/p$, the second and third have $g(p)=1/(p-1)$ for odd primes, the fourth has $g(p)=1/p$, and the fifth has $g(l)=\nu(l)/l^2\le(3l-2)/l^2\le7/9$ for odd $l$. The third and fourth require sufficiently small fixed $b_1$ so $h\ge h_0$. Their level is at most $x^{1/5+2b_1}$ and their relative error is $O(e^{-1/(20b_1)})$, with an absolute constant. The $W$-level is $D_W=\exp(O(\sqrt L T^2))=x^{o(1)}$ and the balanced level is $x^{10\theta}$. The full individual remainder checks are retained in proof coverage.

E2: all restricted proxy discrepancies

Fix $0<b_1<1/10$, $0<\kappa<1/50$ and a finite $\Gamma\subset[b_1,1/2-\kappa)$. The tests are $I_\gamma(m)=\mathbf1_{P^-(m)>x^\gamma}$ for $\gamma\in\Gamma$ and $I_{\rm pr}(m)=\mathbf1_{m\ {\rm prime}}$. Let $\mathcal A=[x^{b_1/4},x^{1-b_1/4}]$, fix $S>0$ and $\Delta=L^{-S}$. On every cell $C=(Y,e^\Delta Y]=(e^{k\Delta},e^{(k+1)\Delta}]$ meeting $\mathcal A$, define full-cell counts

$$R(C)=\#\{m\in C:P^-(m)>W\},\quad c_I(C)=\frac{\sum_{m\in C}I(m)}{R(C)},\quad B_I(m)=c_I(C)\mathbf1_{P^-(m)>W}.$$

E2 asserts, uniformly over these cells,

$$R(C)=(e^\Delta-1)YV(W)(1+O(e^{-2T^2}))+O(D_W) \sim\Delta YV(W)>0.$$

Eventually $0\le c_I(C)\le1$, $0\le B_I\le1$ and $\alpha=(I-B_I)\mathbf1_K$ is supported on $W$-rough integers with $|\alpha|\le1$, for any interval $K\subset\mathcal A$. The numerical coefficient bound is independent of $S$. For every fixed $B\ge1$ and every fixed $S\ge2B+2$ there is a threshold depending on $b_1,\kappa,\Gamma,B,S$ such that, simultaneously for all tests, all such $K$, every interval $J\subset[1,x^2]$, every Dirichlet character of modulus $r\le L^B$, and every real $|t|\le L^B$,

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

Principal and imprimitive characters are included. Proxies are normalized on full cells before restriction; they are not renormalized on $K$ or $J$. The unbalanced cofactor ranges, balanced prime ranges and fixed enlargements of Type II dyads with exponents in $[b_1/3,1-b_1/3]$ lie inside $\mathcal A$ eventually. The consumer choice $S>2B+3$ is covered.

E3: density ranges, constants, identity, bound and mesh

For $w>\gamma>0$, define $D_\gamma(w)=\sum_{l\ge1}T_l(\gamma,w)$, $T_1(\gamma,w)=1/w$, and for $l\ge2$,

$$T_l(\gamma,w)=\frac1{l!} \int_{\substack{t_i\ge\gamma\ (i<l)\\s=\sum_{i<l}t_i\le w-\gamma}} \frac1{w-s}\prod_{i<l}\frac{dt_i}{t_i}.$$

Only $l\gamma\le w$ can contribute. The measure is the coordinate measure $dt_1\cdots dt_{l-1}$ on the sum-$w$ simplex, not Euclidean surface measure. E3 asserts joint continuity on every fixed compact $K(b,\varepsilon,w_{\max})={\gamma\ge b,\ \gamma+\varepsilon\le w\le w_{\max}}$ with $b,\varepsilon>0$, including thresholds $w=l\gamma$. For every fixed $S\ge1$, uniformly for cells and exponents in that compact,

$$\#\{m\in(Y,e^\Delta Y]:P^-(m)>x^\gamma\} \le(D_\gamma(w)+o(1))\frac{\Delta Y}{L},\qquad w=\log Y/L.$$

The PNT short-cell estimate used to obtain this has relative error $O(L^{-1})$ at fixed logarithmic lower size and uses accuracy $A=S+2$. For every fixed $0<\kappa<1/50$, the prime proxy on cells meeting $\mathcal B_\kappa=[x^{1/2-2\kappa},x^{1/2+2\kappa}]$ satisfies

$$c_{\rm pr}(C)\le\frac3{LV(W)}. \tag{7.16}$$

The number 3 is independent of $\kappa,S$; only the threshold may depend on them. For a fixed bin $b_1\le\gamma<\gamma'\le1/2-\kappa$, a cell containing $m=N/p$ with $x\le N\le5x$ and a prime $p\in(x^\gamma,x^{\gamma'}]$ satisfies

$$c_\gamma(C)\le \frac{\sup_{\gamma\le a\le\gamma'}D_\gamma(1-a)+o(1)}{LV(W)}. \tag{7.17}$$

Here $w=1-a+O(1/L)$ and $w-\gamma\ge2\kappa-O(1/L)$, so the proof uses $K(b_1,\kappa,1)$. The supplied source states this for a factor not required to be prime. The reconstruction's wording is narrower; its argument uses only the factor's size and applies to the source's factor as well. The actual consumer factor is prime. No value at $w=\gamma$ is invoked.

For every $0<b_1<1/2$,

$$\int_{b_1}^{1/2}D_\alpha(1-\alpha)\frac{d\alpha}{\alpha} =D_{b_1}(1)-1. \tag{7.18}$$

The density's endpoint value is its left limit 2 at $\alpha=1/2$; changing that single endpoint cannot change the integral. There are absolute $b_0,c'>0$ such that for every $0<b_1\le b_0$,

$$D_{b_1}(1)\le \frac{e^{-\gamma_E}}{b_1} (1+c'e^{-1/(20b_1)}). \tag{7.19}$$

This uses an ordered-prime-list lower count of ordinary rough integers in $(x,2x]$ and their E1 upper count; it asserts a bound for the fixed density, not a new uniform weighted rough-number theorem.

For fixed $b_1,\kappa$, a sufficiently fine fixed partition of $[b_1,1/2-\kappa]$ gives

$$\sum_{(\gamma,\gamma']}\sup_{\gamma\le a\le\gamma'}D_\gamma(1-a) \log(\gamma'/\gamma) \le\int_{b_1}^{1/2-\kappa}D_\alpha(1-\alpha)\frac{d\alpha}{\alpha} +\frac1{10}.$$

It is chosen before $S$ and before $x$. When instantiated in the consumer, after its conditional Type II proxy replacement and its preliminary and per-prime sieve bounds, this leaves, in units $\mathfrak S X_A/L$, at least $1-1/10-O(b_1^{-1}e^{-1/(20b_1)})+o(1)$ after unbalanced composites, where $\mathfrak S=2\prod_{p\ge3}(1-(p-1)^{-2})$. The density and mesh inequality themselves do not use P. The final weighted-application paragraph on the density page inherits the consumer's P assumption through that replacement; it would not follow from E alone. The unused cross-check $D_\gamma(w)=\omega(w/\gamma)/\gamma$ is also accounted for in the keeper, without importing an unproved Buchstab theorem.

E4: candidate mass with a later fixed exceptional slot

Fix $0<\delta<1/4$, smooth $\Psi\ge0$ supported in $[1,2]$ and bounded below positively on $[\alpha_0,\beta_0]\subset(1,2)$ of positive length. Take $q=1/2$, one mark per group, scale ratio 2, $(c_1,c_2,c_3,c_4)=(1,6/5,3/2,9/5)$ and $s'=1/[r_0(c_1+c_2)]$, where the fixed integer $r_0$ satisfies $c_2s'<\delta$ and $(r_0-1)c_1/2\ge c_2+1$. Put $s_j=s'2^{-j}L$ and let $j_x$ be the largest index with $c_1s_{j_x}\ge20T$. Band $\mathcal Q_j$ consists of primes with logarithm in $[c_1s_j,c_2s_j]$. The P-groups consist of all primes with logarithm in $[c_3s_j,c_4s_j]$ at scales where this interval lies wholly in either $[L^{1/10},L^{1/5}]$ or $[L^{3/10},L^{2/5}]$. Bands and groups are pairwise disjoint. Their masses are $M_j=\sum_{p\in\mathcal Q_j}1/p$ and $V_g=\sum_{p\in\mathcal P_g}1/p$.

There are $r_0$ ordered prime slots per band, with repetitions allowed, except one designated slot in a later fixed band $j_{**}$. It holds an integer $v_0$ with $e^{c_1s_{j_{**}}}\le v_0\le e^{c_2s_{j_{**}}}$ and $P^-(v_0)>W$. For a product $v$, $a_Q(v)$ is the number of its lists times $\prod_{j\le j_x}1/r_0!$. Let $u_*$ be the largest divisor of $u$ with no group prime factor, and let $\omega_g(u)$ count distinct primes of group $g$ dividing $u$. Define

$$W_1(u)=\prod_g\frac{q^{\omega_g(u)-1}\omega_g(u)}{V_g},\quad A(u)=a_Q(u_*)W_1(u),\quad X_A=\sum_uA(u)\Psi(u/x),$$ $$H_Q=\sum_v\frac{a_Q(v)}v,\qquad H_P=\sum_{r_* =1}\frac{W_1(r)}r.$$

A group with no divisor contributes zero to $W_1$. Choose a finite prefix length $J_1$ from the early geometry, the fixed balanced-box index $j'$, and $j_0\ge\max(J_1,j')$ before $j_{**}$. There exist $C,C_Q,C_{\rm up},c_{\rm low},c_{\rm mic}>0$ depending only on these early data such that for every fixed $j_{**}\ge j_0$, for every sufficiently large $x$ with a threshold permitted to depend on $j_{**}$:

1. $0\le a_Q(v)\le\tau(v)$, $0\le A(u)\le L^C\tau(u_*)$ and $A(u)\le\exp(C\sqrt L)$ for $u\le2x$.
2. Every prime factor of $u$ on the support of $A(u)\Psi(u/x)$ lies in $[L^{20},x^\delta]$.
3. $L^{-C_Q}\le H_Q\le L^{C_Q}$.
4. $H_P\sim\exp(q\sum_gV_g)$, $H_P\ge\prod_{p\ {\rm in\ P\ groups}}(1+q/(p-1))$, and for each fixed $\sigma>0$, $\sum_{r>x^\sigma,,r_*=1}W_1(r)/r \le e^{-\sigma L^{3/5}}L^{O(1)}$.
5. $c_{\rm low}(x/L)H_QH_P\le X_A\le C_{\rm up}(x/L)H_QH_P$; consequently $X_A\ge c_{\rm low}xL^{-1-C_Q}$.
6. The harmonic mass of bands $j\ge j'$, including the exceptional slot, obeys $H_Q^{\rm mic}\le c_{\rm mic}^{-1}H_Q$.

In particular $C_A=1+C_Q$ and the constants in the last two comparisons precede $\kappa,b_1,j_{**}$ and $S$ in the extraction's order.

E5: Section 2 tools and Type I

The classical statements are specified exactly in the premise table below. The following are derived, rather than additionally assumed.

For fixed $c\in(0,1]$, $C_0\ge1$, $B,A>0$, every nonprincipal character modulo $r\le L^B$, primitive or induced, every real $|t|\le L^B$ and every interval $I\subset[\exp(L^c),x^{C_0}]$ satisfy

$$\sum_{p\in I}\frac{\chi(p)p^{it}}p\ll_{c,C_0,B,A}L^{-A}.$$

The source's larger parameter wording $c,C_0>0$ follows: if $c>1$ the interval is eventually empty, and $C_0<1$ is covered by increasing it to 1. For every fixed $k\ge0$ and $Y\ge1$,

$$\sum_{n\le Y}\frac{\tau(n)^k}{n}\ll_k(\log(2Y))^{2^k},\qquad \sum_{n\le Y}\tau(n)^k\ll_kY(\log(2Y))^{2^k}.$$

This includes the source's integer-$k$ formulation. A convolution of a fixed number of logarithmically bounded coefficients has square sum $UL^{O(1)}$ on a dyad $U\le x^{O(1)}$; a reciprocal product index changes this to $L^{O(1)}/U$. The actual mixed moments, shifted diagonal and endpoint square sums used by Type II, and the first and second moments of Type I's $b(r)\le L^{c_b}\tau(r)$, are included.

For fixed dimension $b$ and $C\ge1$, a smooth $F$ supported in a fixed box whose derivatives of order $j$ are $O_j(L^{C(j+1)})$ admits Fourier separation with integrated absolute coefficient mass $L^{O(1)}$ and a fixed frequency exponent, for example $2C$, chosen before the desired saving $A$. Truncating to frequencies at most $L^{2C}$ costs $O_A(L^{-A})$. Fixed smooth Mellin cutoffs have bounded integrated cost. Elementary sums used later are $\sum_{d\le y}\mu^2(d)/\varphi(d)\ll\log(2y)$ and $\sum_{n\le D}3^{\omega(n)}\le D(1+\log D)^2$, with their stated consequences for $\nu(d)\le d3^{\omega(d)}$.

For the candidate above and odd squarefree $d$, put $r(d)=\sum_{d\mid2u-1}A(u)\Psi(u/x)-X_A/\varphi(d)$. For every fixed $\vartheta<1/2$ and $D>0$, E5 supplies

$$\sum_{\substack{d\le x^\vartheta\\d\text{ odd, squarefree}}}|r(d)| \ll xL^{-D}+X_AL^{-18}.$$

After the early candidate coefficient exponent is fixed, the choice order is $D$, conductor cutoff $L^{C'}$, then Siegel–Walfisz accuracy $K$. No choice depends on proxy precision $S$. The positive gap $1/2-\vartheta$ controls the eventual threshold. The assertion is not uniform as that gap tends to zero with $x$.

Weakest steps and strongest attacks

The three selected weakest steps are independently rederived in weak steps, with all errors and surrounding interfaces. The rest of the proofs are covered in proof coverage; selection of three longer derivations does not exclude other clauses.

The E1 counterexample is the strongest successful literal refutation. In full, choose $\mathcal P=\varnothing$, $z=1/2$, $\eta_0=1/2$, $C=1$, and a single object of weight 1. Set $g(1)=1$, $X=0$, $r(1)=1$. The empty intersection has the required weight; every condition on prime densities is vacuous. Thus $\mathcal S=G=1$. For every positive even $h$, $z^{4h+2}<1$, so the displayed remainder sum is empty and the asserted upper bound is 0. It would require $1\le0$. The same holds at $h=2$. The only monomial of a polynomial in no indicators is the constant at $d=1$, forbidden by $d\le z^{4h+1}<1$. Consequently its claimed $U$ must vanish and cannot majorize the avoidance indicator 1. No unknown constant or sufficiently-large-$h$ threshold can evade this witness.

The reusable failure mode is a truncated error sum that omits the empty product while the hypotheses still include its mass. Requiring $z\ge1$ would exclude the witness; the author has not been edited and no repaired statement is silently substituted. Within the actual applications, the most consequential failed attacks were the E2 loss accumulation and E4 uniformity challenge, since either would invalidate the downstream choice of the proxy precision or balanced constant.

Premises, reading depth and acceptance order

The premise census was derived from the complete statements, proofs and consequence sentences, rather than accepted from the supplied empty ledger list. No native L-claim is consumed as an established premise by E. Accordingly no live ledger reading was performed or claimed on 2026-10-15; none was authorized. The combined page mentions other project results as context, a proposed proof of P or a downstream conditional reduction. They are not substituted for proof inside E. If a later acceptance needs one of those results as an established native premise, its exact statement, interface and then-current standing are additional inputs for that separate acceptance; this examination supplies none.

Definitions were checked without assuming P: finite band and group sets, slot counts, group-free factorization and marked weights are well-defined; the harmonic masses are finite, and their eventual positivity is proved. The proxy denominators are proved positive before division. All sums in the density definition are locally finite. Nonnegative sampling expectations are exact rewritings of these objects.

Exact external boundary

The following are used as explicit classical premises. “Statement checked” means checked in the delivered reconstruction and manuscript excerpt, not verified against an unseen original text. Every original classical source below is unread in this examination; no claim of its original proof being verified is made. Reliance is on its identified standard theorem as stated in the packet. Original-text fidelity is a retained limitation.

Elementary background used in the written derivations includes unique factorization, finite character orthogonality, CRT, Cauchy–Schwarz, partial summation, finite-dimensional integration and Fourier inversion for smooth compact support, dominated/monotone convergence and Tonelli for nonnegative integrands. The quantitative Fourier decay needed here is rederived by integration by parts, not taken as an additional analytic saving theorem.

Ford–Halberstam, The Brun–Hooley sieve, J. Number Theory 81 (2000), Section 1, Lemma 1, supplies attribution for the block construction. Its original text/proof is unread, but the inequalities and full needed sieve are reconstructed and checked here. Buchstab, Mat. Sbornik 2(44) (1937), 1239–1246, supplies attribution for the rough-number density; its original text/proof is unread. The proof of E3 uses the written simplex construction, not an imported Buchstab asymptotic. The source bibliography was read for these locators. No PDF-fidelity evidence accompanies any of them.

Derived dependencies and source-proof scope

The reconstruction dependency order is acyclic: classical statements and elementary algebra first; the operative E1 lemma, harmonic character lemma and E4 independently; E2 after the first two; E3 after E1/E2; divisor and Fourier tools independently; Type I after E4 and those tools; finally the preliminary, per-prime and balanced consumer estimates. E5 is an inventory of tools at different positions, not a theorem requiring E3 in order to prove E1. No circular promotion is proposed. A grader considering separate native claims would have to accept any needed operative-sieve assertion before its consumers; a favorable assessment of its uses cannot accept the unrestricted false wording.

The full supplied source Section 2 was read. Lemmas 2.6 and 2.7 and Corollary 2.10 are not consumed by these E proofs or by the supplied signed Type II derivation; their proofs are not independently warranted here. The source Section 6 definitions and theorem statement were read; its proof was not supplied. The full Section 7 was read and its E deductions compared with the reconstruction. The P statement was read on the consumer page, including endpoint support, characters, Mellin ranges, both shift signs, scales and order of choices. Its original Sections 3–5 proof is not part of this examination and is not certified by any E verdict.

For clarity, the consumed P interface is the following explicit assumption on the candidate's fixed group data. An endpoint has the form

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

where $p$ is an integer, $I_p\subset[x^\tau,x^\eta]$ for fixed $0<\tau<\eta<1/4$, $|\alpha_m|,|c_r|\le L^C$, $\alpha$ is $W$-rough-supported, and the character modulus and both twists are at most $L^C$. For fixed endpoint bounds, $D_0>0$ and a fixed positive compact cutoff interval, P supplies a $B$ such that if the original $\alpha$ in one endpoint satisfies $(D_B)$, then

$$\sum_{z>0,\ z+k>0}\frac{\Psi(z/Z)}z\, z^{ia_1}(z+k)^{ia_2}\overline{F_0(z)}G_0(z+k)W_1(z)W_1(z+k) \ll L^{-D_0}$$

uniformly for integers $k$ with $0<|k|\le L^C$, $xL^{-C}\le Z\le xL^C$, $|a_1|,|a_2|\le L^C$, and cutoff derivatives $\|\Psi^{(j)}\|_\infty\ll_jL^{C(j+1)}$. The consumer also assumes the corresponding conjugated discrepancy-bearing endpoint cases. In the application $I_p=I_{**}$ with $0<\tau_*<\eta_*<b_*/4$, $\chi_0$ has modulus $2|k|$ and the residual coefficients remain the specified bounded ones. This is the one-mark specialization actually consumed; the consumer states a general $W_\ell$ version. The original source statement in Section 5 was not independently supplied, so this record treats the full consumer formulation as an explicit antecedent, without certifying its source fidelity or truth.

Consumer interfaces and consequence sentences

The supplied consumer uses E2 only for the original restricted coefficient $\alpha=(I-B_I)\mathbf1_K$; it does not transfer discrepancy to a product or new sequence. P is allowed to choose $B$ after its endpoint and Fourier parameters. Since the coefficient bound is 1 before $S$ is selected, choosing $S>2B+3$ afterwards is legitimate. The ambient interval contains all cofactor and balanced dyads with exponent slack. E5's moments and Fourier bounds supply precisely the absolute sums and separated cutoffs used by the Type II proof, including the $O(T)$ slot factors whose total Fourier mass is a fixed power of $L$.

The per-prime sieve multiplies only $p>W$ by squarefree $d$ supported below $W$, so its products are distinct and stay below the Type I level. E3's cofactor supremum and reciprocal-prime bin mass have the stated normalization. The mesh, minimum-coordinate identity and upper density bound then perform the claimed subtraction in units $\mathfrak S X_A/L$.

For balance, the exceptional integer is retained whole in the micro-band assignment. The modulus $D_1$ is odd but may have repeated prime factors; there are still exactly $\varphi(D_1)$ unit pairs with $mn\equiv-1\pmod{D_1}$. The two local bad sets have sizes $2l-1$ and $l-1$, are disjoint, and lead to the stated lattice remainder. The independent mask expectation reproduces damping for distinct group divisors. The full assignment sums, Euler factors, remainder bound $x^{0.64}L^{O(1)}H_Q^{\rm mic}$ and comparison with $X_AV(W)^2$ are checked in proof coverage. Together with the absolute proxy constant 3, this makes $C_{\rm bal}$ depend only on early geometry, the prefix, $j',\theta$ and $\Psi$, before $\kappa,b_1,j_{**},S$.

The complete choice order is early geometry and prefix; $j',\theta,j_0$; $\kappa$ small using $C_{\rm bal}$; sufficiently small $b_1$, $b_*=b_1/3$, a fixed admissible $j_{**}$ and fixed bin mesh; Type II saving, $K_0,D_0$ and $B$ from P; $S>2B+3$ and PNT accuracy $S+2$; then $x$ sufficiently large. Type I's accuracy and its later conductor and SW choices introduce no feedback into $S$ or the early balanced constant.

Assuming P and the supplied consumer derivations, the resulting mass bound converts into a count of distinct primes by $A(u)\Psi(u/x)\le\exp(C_w\sqrt L)$ and injectivity of $u\mapsto2u-1$. The support gives $2x-1\le p\le4x-1$ and $P^+(p+1)=P^+(2u)\le x^\delta$ for large $x$. The lower bound $cxL^{-C_A-1}e^{-C_w\sqrt L}$ exceeds $x^{1-\eta}$ for every fixed $\eta>0$, since $\eta L$ dominates $(C_A+1)\log L+C_w\sqrt L$. Thus the stated successor-count to Hypothesis $\mathrm A^+$ implication has the correct all-sufficiently-large-real-parameter quantifiers, with no uniformity in $\delta\to0$.

The combined page's further implication to coprime squarefree pairs cites the Li–Zhuang counting reduction, whose full separate page was not supplied or read. Its numerical exponent summary is consistent: with $\eta=\varepsilon/16$ and $1/B\le\varepsilon/8$, $3\eta+2/B\le7\varepsilon/16<\varepsilon$. The required combinatorial bounds on products, divisor-sum values and gcd multiplicity remain that separate reduction's premises here. Pollack–Pomerance's historical bound is context, not an E premise. E alone supplies neither P nor that reduction, and the frozen consequence text explicitly conditions on them. No unconditional Problem 824 conclusion, no squarefree-pair proof and no Problem 821 conclusion is awarded by this examination.

Canonical checklist

Each cell gives an explicit disposition for the corresponding assertion. “No failure found” is an outcome of the cited written attacks under the premise limitations, not a grade. All inapplicability statements cover E1, E2, E3, E4 and E5 separately where so indicated.

Erdos-specific checklist

This table records the additional governing form explicitly. The detailed per-assertion dispositions above remain part of each entry.

The neighboring evidence-coverage discipline is met by the clause map and keepers: every essential deduction in the six estimate pages has a stated reasoning location, and no finite calculation is used to stand for an unchecked uniformity. The neighboring correlated-sampling discipline adds no statistical inference here; the exact laws and their normalizations are proved rather than sampled.

Remaining qualifications and grading

The commissioned written examination is complete. Its mathematical result is the literal E1 refutation and the qualified refutation-failed findings above, not a repair, acceptance, or proof of P. There is no unfinished E1–E5 clause silently deferred. The original classical sources, print fidelity, P's proof, the source Section 6 proof and the separate downstream counting reduction were not independently proved or certified. The first two are explicit evidence limitations; the latter three are outside this E examination. None is filled by a favorable calculation elsewhere.

A distinct grader owns the interpretation and accepted weight of the cutoff finding, this report's completeness and independence, and any standing consequences. The responsible integrator owns any correction to the frozen wording and reassessment of consumers, as well as retaining the full package in the native evidence home. Any claim of original-source fidelity or a complete Problem 824 chain needs its own supplied inputs and examination; this report does not authorize such a claim.