Conditional squarefree-pair implication: independent examination

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Subject and independence

Subject. The frozen implication from the explicit hypotheses P and E, through the signed Type II estimate, successor-prime extraction and Hypothesis A+, to the eventual real-variable squarefree version of the Problem 824 bound. Frozen on 15 October 2026; examined on 15 October 2026 UTC. The reviewer is an independent mathematical reviewer, using GPT-6 Astra at its maximum reasoning setting. This is an adversarial examination, not authorship, a grading of another report, or a claim of an unconditional solution.

Finding. The verdict on the conditional implication is refutation-failed. There is a separate refuted-as-stated finding for the unrestricted formulation of the auxiliary “Dyadic bound”: a product-size condition alone does not imply it. The complete proof does establish the balanced specialization actually used in the successor extraction. The distinction, exact asymptotic witness, and limits of this verdict are below.

The assessed mathematical bytes are retained in this package. The paths in the second column identify their source subjects; those live paths were not opened, and no conclusion is asserted about later versions.

The retained mathematical reasoning is the signed keeper, the extraction keeper, and the counting keeper. The premise account is an integral part of this report's restatement and dependency account, not an optional reference. There is no mathematical program, sampled experiment, or cached computational result on which the verdict depends.

The examination began from the supplied frozen subject and operating rules. No authorship or collaboration on this subject occurred in the review context. No author advice, prior review, sibling verdict, review inventory, live ledger, source-card standing, external search, or excluded mathematical material was read. No descendants or other reviewer threads were used. The subscription observations were empty; the inbox remained sealed and no feed was opened. Empty subscriptions support this account of isolation but do not by themselves establish what was read.

The commissioned mathematical read set was exactly the seven retained inputs above, all of which were read fully. The allowed operating set consisted of the enumerated static repository guidance, its frozen copies, the rendered charter and steps, configuration and accepted settings, command help and the four named local operating skills. Actual operating content reads were:

• The supplied root AGENTS instructions, README, both complete verification contracts in docs/verification.md and campaign/guides/verification.md, campaign/AGENTS.md, docs/evidence.md, docs/approach_vetting.md, docs/math_authoring.md and docs/weaving.md, all in full.
• docs/anatomy.md, lines 1–177, 253–397 and 1061–1165, covering Claims, Evidence, Tiers and Project claims; campaign/guides/research.md, lines 1–115, including its full Reading section.
• The supplied manifest, accepted settings, review configuration, command help, three step texts, four station script texts, and radio and memory skill texts in full. Scripts were inspected, not executed. The rendered charter was read in full; its on-disk context and pin, selected template keys, and the relevant resource and lifecycle portions of the recursive-operation skill were inspected. Partial displays of those optional operating texts are not represented as complete readings.
• The reviewer's own continuation and retained reasoning. The remaining enumerated operating pages and the wiki skill were byte-checked only, without content reading. Links in all these texts were not followed.

The delivered files matched the supplied manifest, and all fourteen enumerated static operating sources matched their frozen copies. The mathematical subject stayed unchanged. Those administrative observations provide identity evidence, not mathematical evidence.

One procedural fact is retained for the distinct grader: the accepted-settings file was displayed alongside the manifest before its identity comparison. The subsequent comparison matched exactly. It contained operational settings; no excluded mathematical exposure was observed. This read-order fact was disclosed through the permitted outbound channel. Some initial oversized displays were truncated; complete nontruncated readings subsequently covered every mathematical input and both verification contracts. The reviewer does not rule on eligibility or the materiality of the procedural disclosure. Detailed command receipts remain private, separate from this mathematical record.

Restatement

For a positive integer n, let sigma(n) be the sum of its positive divisors. For real x define

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

and let h_sf(x) count just those pairs in which a and b are both squarefree. The pair is counted once, in the order a<b. The convention includes the positive integer 1 as squarefree; no diagonal pair is counted.

Assume the complete P and E statements restated in the premise account. In particular P is uniform shift cancellation for both signs of a nonzero shift, with the entire stated endpoint coefficient and support class, all the stated character and Mellin parameters, the harmonic factor 1/z, and the required conjugations. Its discrepancy condition is imposed on the actual restricted alpha sequence; it includes principal and imprimitive characters. The distinguished slot is an integer rough above W, not necessarily a prime. Group counts count distinct divisors, whereas band lists allow repeated primes. E includes the full candidate-mass, proxy-discrepancy and proxy-density statements at the uniformities used, the block sieve, and the precise Section 2 classical and elementary estimates. P and E are hypotheses, not findings of this report.

With the fixed group and band data of the signed page, for every fixed D_,b_,C>0 satisfying eta_<b_/4 and fixed smooth compactly supported Psi, there is a fixed discrepancy exponent B such that, for all sufficiently large real x, all scales UV comparable to x with x^(b_)<=U,V<=x^(1-b_), and all permitted sequences alpha,beta,

$$\sum_{\substack{m\asymp U,\ n\asymp V\\ mn=2u-1}} A(u)\Psi(u/x)\alpha_m\beta_n \ll x(\log x)^{-D_*}.$$

Beta is arbitrary within its coefficient bound. Alpha is rough-supported and satisfies the stipulated discrepancy after its actual interval cuts. All constant dependence and the definitions of A, the groups, the bands and the exceptional interval are those stated explicitly in the premise account. This is the signed intermediate deduction, not another assumed premise.

The complete extraction then asserts that for every fixed 0<delta<1/4 there exist c,C_A,C_w>0 and a threshold such that every sufficiently large real x has at least

$$c\,x(\log x)^{-C_A-1}\exp(-C_w\sqrt{\log x})$$

distinct primes p in [2x-1,4x-1] with P^+(p+1)<=x^delta. Here P^+(1)=1 and P^-(1)=infinity; smooth means that every prime factor is at most the cutoff, and rough means that every prime factor exceeds it. All fixed parameters precede x tending to infinity.

Consequently A+ says that for every fixed 0<delta<1/4 and eta>0 there is a threshold y_0(delta,eta) such that for every real y>=y_0 at least y^(1-eta) primes p<=5y satisfy P^+(p+1)<=y^delta. No effectiveness or uniformity as delta tends to zero is asserted.

The terminal implication is

$$(\mathrm P)\ \&\ (\mathrm E) \quad\Longrightarrow\quad \forall\epsilon>0\ \exists x_0\ \forall x\in\mathbb R,\ x\ge x_0: \quad h(x)\ge h_{\rm sf}(x)>x^{2-\epsilon}.$$

This is the eventual interpretation of the catalog's exponent notation. Neither an almost-all set of x nor a subsequence suffices. The terminal products satisfy the strict upper endpoint, coprimality, equal divisor sums, and squarefreeness simultaneously.

Weakest steps and complete deduction coverage

Three written rederivations cover the consequential analytic and combinatorial bridges. Their full details are retained, including unsuccessful attacks; the summaries here identify the points on which composition depends.

1. Signed Type II from the actual endpoint hypothesis

The signed keeper starts before Cauchy's inequality. The group-part tail is at most

$$xL^{O(1)}\exp\{-b_*L^{1-d}/4+O(\sqrt L)\},$$

which is negligible because d<0.47. The finite slot-allocation invariant gives E L^(-C_s)<=e<=E, where E=U L^(-K_0). Products of one-dimensional Fourier norms, not a fixed-dimensional lemma in a growing dimension, keep the total coefficient mass and the residual coefficient bound polynomial in L independently of K_0. The diagonal after Cauchy is controlled by fixed second and sixth divisor moments, permitting K_0 to be chosen only after those costs are known.

For the off-diagonal the exact signed identities are

$$m'-m=-2ek,\qquad h'-h=-kn,\qquad mh'-m'h=k,\qquad z=m'h,\quad z+k=mh'.$$

The converse uses roughness and |k|<W to get gcd(m,k)=1. The congruence m'=m modulo 2|k| gives integral e; the positive cutoff imposes e>0. Divisibility then gives integral positive n=(2eh-1)/m. Character orthogonality is valid because m,m' are units modulo 2|k|. Thus the correspondence has no missing divisibility, sign or positivity assumption.

After separation, the discrepancy-bearing endpoint still contains the original restricted alpha. All new characters, twists and residual coefficients are explicit members of P's endpoint class. The factors z^(i(varrho+xi_3))(z+k)^(-i varrho), and the conjugations recorded in the keeper, match the original square norm. Multiplying the smooth test by its argument converts the unweighted correlation to Z times P; omitting this Z would lose the scale of the estimate. Both signs of k are used. The choice order is fixed geometry and saving, then K_0, post-Cauchy Fourier costs and D_0, then B. The derivation needs no extra premise.

2. Complete successor extraction, including the balanced sieve

The extraction keeper derives Type I in the unit class 2^(-1), including the principal coprimality error, the primitive conductor decomposition, the large-sieve factor 1/R and the small-conductor Siegel–Walfisz bound. Its exact conclusion is

$$\sum_{\substack{d\le x^\vartheta\\d\ {\rm odd,\ squarefree}}}|r(d)| \ll xL^{-D}+X_A L^{-18} \quad(\vartheta<1/2,\ D>0\ {\rm fixed}).$$

The block-sieve construction is checked with absolute coefficients at most one and level z^(4h+2). Full-cell proxy mass matching handles principal characters; the supplied discrepancy holds for all actual interval cuts. The local density transfer from 2u+1 to 2u-1 uses log(2u-1)/L=1+O(1/L), not an assumed distribution of shifted primes. The finite-simplex minimum-coordinate identity supplies precisely the subtracted composite density D_b1(1)-1. The remainder 1 is the one-prime term. The least-prime bins include repeated prime factors and all endpoints of the claimed support; their Type I remainders have no divisor multiplicity loss because pd has a unique prime factor above W.

For the balanced part, both prime-proxy replacements are justified, including the second with an arbitrary bounded first proxy. The mask majorant keeps the entire integer slot in D_1, counts exactly one distinct mark per group, and allows nonsquarefree D_1. The successor congruences are mn=-1 modulo D_1 and modulo a banned odd prime. They have phi(D_1) and l-1 unit-pair solutions respectively, while the roughness failure mn=0 has 2l-1 solutions. These two prime-level failures are disjoint.

For M_i=O(x^0.54), D_1<=x^(1/20) and D=x^(10theta)<=x^(1/20), the lattice errors after assignment summation are at most x^0.64 L^C H_Q^mic, negligible compared with X_A V(W)^2. The main term is bounded by

$$xV(W)^2V(x^\theta)H_Q^{\rm mic}H_P\ll X_A V(W)^2.$$

The exact inequality H_P>=product_(group p)(1+q/(p-1))>=1 supplies the mask compensation. This proves the complete balanced specialization with its constant fixed before kappa, b_1, j_**, S and the analytic precisions. It also identifies exactly where an unrestricted dyadic formulation fails.

Choose early geometry, the candidate prefix, j',theta and j_0; then choose kappa with C_bal kappa<mathfrak S/4; then b_1, j_**, and a finite mesh. Choose Type II saving and its K_0,D_0,B next, and finally S>2B+3. The proxy coefficient bound stays one independently of S, so this order is acyclic. The resulting prime mass is at least mathfrak S X_A/(5L). The maximum summand exp(C_w sqrt L) and injectivity of u -> 2u-1 give the asserted distinct-prime count. Since p+1=2u, the extra factor 2 is also below x^delta eventually. Setting x=y and comparing logarithms proves A+ for every sufficiently large real y.

3. Products, overlap deletion and gcd fibers

The counting keeper uses only A+ and elementary integer arithmetic. For 0<epsilon<1 fix B>=8/epsilon, eta=epsilon/16, delta=1/(2B), y=(log x)^B, and k=floor(log x/log(5y+1)). There are M>=(log x)^(B(1-eta)) eligible primes, and their distinct k-subset products give

$$N=\binom Mk,\qquad \log N\ge(1-\eta-1/B)\log x-o(\log x).$$

They are squarefree and strictly below x, while their sigma values are at most x and sqrt(log x)-smooth. The number U of possible values obeys log U=o(log x). Cauchy's inequality gives at least N^2/U ordered equal-value pairs.

With t=ceil(eta k), the fraction of all product pairs sharing at least t primes is at most 2^k(2k/M)^t=o(1/U). Deleting them leaves at least N^2/(2U) equal-value pairs and removes the diagonal. Writing a=gu,b=gv then gives g<x^eta. Squarefreeness makes g,u,v pairwise coprime, so sigma(g) cancels and sigma(u)=sigma(v). For each unordered reduced pair and each g there are at most two ordered preimages; the total fiber is at most 2x^eta. Consequently

$$h_{\rm sf}(x)\ge\frac{N^2}{4Ux^\eta},\qquad \log h_{\rm sf}(x)\ge(2-3\eta-2/B)\log x-o(\log x).$$

Since 3eta+2/B<=7epsilon/16<epsilon, the strict bound follows. The floor in k and the choice 5y+1 preserve b<x for real x, including integer x. The final identity h(x)=h(ceil x), and the same for h_sf, is correct for this strict endpoint. Larger epsilon follow from any smaller fixed loss in (0,1).

Coverage of supplied source reconstructions

All essential deductions, not just the displayed substitutions, were examined. Source Section 6 was compared with every stage of the signed proof. Source Section 7 was read through candidate mass, Type I, the preliminary sieve, proxy construction, local densities, away-from-balance subtraction, the full mask/CRT/sieve argument and the final parameter choices. Every use of an assumed E result was compared with its actual statement and the supplied local proof. Section 2's consumed moment, Fourier, harmonic-character and block-sieve arguments were checked at their displayed scopes; their external classical boundaries remain explicit. The unused derivative and mean-square statements were read but are not certified as additional conclusions of this report.

The entire counting exposition and the catalog formulation were checked, including the two last real-variable quantifier paragraphs. No original classical proof, omitted proof of P, or original PDF fidelity is claimed. Those are boundaries of the commission, not deductions silently omitted from the conditional chain.

Strongest attack and exact witness

The separate “Dyadic bound” at extraction lines 256–266 writes only the condition M_1 M_2 comparable to x. Its proof later uses M_i<<x^0.54. The heading “Balanced composites” and its surrounding application explain the intended balanced use, but the isolated unrestricted assertion is false.

Take x=2^j tending to infinity, the dyad m in [1,2), and n in [2x,4x). Then m=1 is W-rough by convention and M_1M_2=2x meets the written product-size condition. Smooth compact support gives Psi(1)=Psi(2)=0; every nonzero candidate term has x<u<2x and belongs to this n dyad. Thus the displayed sum is exactly

$$S_W=\sum_{P^-(2u-1)>W}A(u)\Psi(u/x).$$

Type I and the same block sieve, with cutoff W and h=2 ceil((log log x)^2), give

$$S_W\sim\mathfrak S X_A V(W),\qquad \frac{S_W}{X_A V(W)^2} \sim\mathfrak S e^{\gamma_E}\sqrt{\log x}\longrightarrow\infty.$$

The sieve level is x^o(1) and its absolute error is O(xL^(-D)+X_A L^(-18)), smaller than the displayed main term when D>C_A+10. The extraction keeper, subsection 6, retains the complete argument, including endpoints. This is an asymptotic mathematical witness under the E inputs, not a numerical search or an assumption that primes themselves have positive density.

Accordingly the unrestricted auxiliary formulation is refuted-as-stated. If the balanced restriction is understood as inherited from the preceding paragraph, the assertion has exactly the specialization proved in the keeper and the witness is outside that specialization. Both readings are reported explicitly. No stronger unrestricted lemma is accepted here. The actual successor proof only uses balanced dyads, whose complete proof survives. Therefore this witness does not refute the commissioned terminal implication and does not establish that its essential proof is defective.

The reusable failure is loss of a roughness factor when a variable degenerates to 1: a product-size condition does not keep a two-variable sieve inside the range in which both density savings and the lattice error are uniform. Future use of this display outside balanced dyads would require a different estimate. No search for such uses outside the frozen subject was performed.

Other serious attacks were the signed determinant converse, normalization by Z, the growing number of Fourier variables, a possible circular choice of discrepancy and proxy precision, nonsquarefree D_1, equality of sigma after gcd cancellation, and large fibers of the reduced-pair map. The three keepers retain why each fails on the hypotheses actually supplied. These attacks do not rely on agreement with an author's computation.

Checklist

The following gives every canonical bullet and named pattern in docs/verification.md, followed by every Erdos-specific item. Applicable findings concern this conditional implication; the auxiliary scope defect is retained wherever it bears on an item. Inapplicable modes are identified explicitly rather than treated as successful tests.

Canonical failure modes

Named canonical patterns

Erdos-specific checklist

The two neighboring disciplines are also accounted for. There is no statistical inference or correlated sampling to justify under docs/approach_vetting.md. Under the evidence contract, the identity checks have administrative coverage only; the three written keepers cover the conditional deductions stated above, and neither they nor this report certify omitted antecedent proofs or original-source fidelity.

Premises and source boundaries

The premise census was derived from the actual frozen statements, proofs and consequence sentences. It found no native L-claim consumed as an established premise. No live ledger reading was authorized or performed, and no pending native grade is being used prospectively. This finding does not come solely from the empty metadata list. The actual dependency account at this examination date is P/E, signed Type II and Type I, the complete successor extraction, A+, then the counting reduction. All the internal deductions are examined here in that acyclic order.

The premise account records the complete P statement and coefficient classes, the exact E clauses, and the PNT, Mertens, Siegel–Walfisz and multiplicative large-sieve forms, their source and version locators, exact specializations and reading depths. The supplied classical statements were checked against their uses; the original classical works were unread. The local moment, Fourier, harmonic-character, block-sieve and density reconstructions consumed here were examined at their displayed scope. Ford–Halberstam and Buchstab are credited behind supplied local constructions, rather than invoked as unspecified stronger external theorems.

The weighted-dilation source is OpenAI's unrefereed Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026 release. Only the supplied Sections 2, 6 and 7 and bibliography were read; Sections 3–5 and the proof of P are outside this examination. The supplied P is the operative conditional hypothesis; its stated identification with Theorem 5.1 is not a PDF-fidelity finding.

The successor application and counting reduction are attributed to Shisheng Li and Zhong Zhuang. The latter is identified as Theorem 1.3 and Section 3, pages 1–3 of the 11 October 2026 draft Coprime pairs with equal sum of divisors, assuming smooth shifted primes, following Pollack and Pomerance after Erdos. The complete counting argument supplied here is project exposition, credited with its parameter checks to Plasma AI. That entire exposition is examined; the original Li–Zhuang PDF was not supplied and is not certified. The extraction's last sentence about Theorem 1.4 supplies a source characterization, not a new premise: the interval inclusion is checked, while the external theorem's original wording remains unread.

No additional essential premise beyond the explicit P/E interfaces was found for the terminal implication. The balanced range needed for its dyadic estimate is already supplied by the actual factor range; accepting an arbitrary-dyad estimate would add an invalid assertion. Truth of P/E, original-source fidelity and live external acceptance are unresolved boundaries, not hidden warrants for an unconditional result.

Reproduction and retained evidence

Keep this report, the three keepers, the premise account and the seven files in the retained subject directory together. The retained assets are copies of the supplied neutral extraction with only the mechanical maintenance listed on the record index, including its omission markers and metadata with the private revision removed. They are not a new extraction from a source revision. This permits the owner to file a complete, ordinary-clone-readable assessment without depending on private runtime state.

The written reproduction route is:

1. Read the seven retained inputs and the exact assumptions in premises.md. Compare the statement to the real-x restatement above.
2. Recompute the allocation invariant, signed determinant correspondence, endpoint factors and choice order in signed_keeper.md. Only P's displayed bound and the stated fixed-moment/Fourier inputs are assumed at that bridge.
3. Follow extraction_keeper.md through Type I, the proxy and density interfaces, both composite ranges, the entire mask/CRT/Euler-product bound, and the exact unrestricted-dyad witness. Confirm the two uses have different ranges before drawing a verdict.
4. Follow counting_keeper.md through the binomial estimate, collision count, overlap deletion and gcd fibers. Check both strict inequalities and the final two quantifier paragraphs against the retained exposition.

There is no executable rerun command because there is no mathematical computation in this evidence. No attempt is made to turn administrative byte equality into proof. All the retained formulas use the displayed inputs rather than source-revision lookups, generated views, unavailable evidence runtimes or previous reports.

Verdict, limits and grading

refutation-failed for the complete commissioned implication from the explicit P and E hypotheses to h(x)>=h_sf(x)>x^(2-epsilon), for every fixed epsilon>0 and all sufficiently large real x with all stated pair conditions. The full deductions and the required source-proof reconstructions were examined at their supplied interfaces. No essential additional premise or unsupported inference was found in the actual balanced application.

refuted-as-stated for the isolated unrestricted “Dyadic bound” with only M_1M_2 comparable to x. Its exact witness is m=1 and S_W/(X_A V(W)^2) tending to infinity. If balance is inherited from the surrounding section, the retained proof verifies that inherited specialization. The report accepts no unrestricted reading and asserts no disproof of the terminal catalog implication.

This examination establishes neither P nor E, does not certify the omitted weighted-dilation argument or any original classical proof, performs no PDF fidelity comparison, and awards no tier or unconditional conclusion about Problem 824. It covers only the retained frozen text. There is no unfinished essential deduction inside that conditional scope; external assumption truth, original-source fidelity and acceptance remain outside it.

A distinct grader must decide report-contract compliance, independence, the procedural disclosure's materiality and the weight of these scoped findings. The authorized owner must preserve the complete package, resolve the auxiliary wording before relying on it elsewhere, and assess any affected consumers within an authorized scope. Public filing and grading are not completed by this private examination.