# After Chen: one signed estimate that would suffice for twins Internal research specification, 2026-09-05. Definitions I_x, z, r, χ_z, k, b, e, W, Q and c₀=C₂/40 are those of [chen-fold-benchmark.md](chen-fold-benchmark.md). The numerical and algebraic validator is [chen-benchmark-validation.js](chen-benchmark-validation.js). **Calibration:** the implication below is elementary; its new estimate is **OPEN**. It is a sufficient research target, not a claim that a new route through the parity problem has been found. The [opportunity audit](chen-opportunity-audit.md) determines what this fixed weight costs and why arithmetic inputs must be matched to a prime-detecting weight. Hη is retained as a diagnostic sufficient condition. ## 1. A prime minorant, including the unwanted factor counts Let λ(m)=(−1)^Ω(m), with multiplicity. For m in the benchmark range, $$ V_x(m):={1-\lambda(m)\over2}W_x(m)\le1_{m\text{ prime}}. \tag{1} $$ Proof. For Ω=1, W≤1. For Ω=2, the prefactor is zero. For Ω≥3, the Chen lemma gives W≤0 and the prefactor is nonnegative. Thus odd composites with three or more factors never become false positive primes. They can give negative contributions, which must remain in the bound. ∎ Define the single new signed quantity $$ R(x)=\sum_{n\in I_x}\Lambda(n)\chi_z(n+2) \lambda(n+2)W_x(n+2). \tag{2} $$ If $T_\Lambda(x)=\sum_{p\in I_x,\ p,p+2\text{ prime}}\log p$, then $$ T_\Lambda(x)\ge {Q(x)-R(x)\over2}-E_{\rm pp}(x),\qquad 0\le E_{\rm pp}(x)\ll\sqrt x\log^2x. \tag{3} $$ Indeed (1) bounds the Λ-weighted sum by prime partners; subtract the terms whose opener is a proper prime power. One may take E_pp to be the sum of Λ over all proper prime powers in I_x. For actual twin pairs at large x, χ_z=1. The benchmark proves Q(x)≥c₀x/log x eventually. ## 2. The additional estimate and its quantifiers **Hη (OPEN).** There exist a fixed η∈(0,1) and an unbounded set of integers J such that, for x=2^j, j∈J, $$ R(x)\le(1-\eta)Q(x)+e(x),\qquad |e(x)|=o(x/\log x)\quad(j\to\infty,\ j\in J). \tag{H} $$ All factor restrictions in (2) remain in force. The small-o is on these same intervals and with the fixed weight W; it cannot be borrowed from an unrestricted shifted-prime average or an average over tile phases. **Conditional theorem.** Hη implies infinitely many twin primes. In fact it gives $\gg_\eta x/\log^2x$ twin pairs in I_x along J. Proof. The benchmark and (H) give Q−R≥ηc₀x/log x−|e(x)|. For sufficiently large x in the selected sequence, |e|≤ηc₀x/(4 log x) and E_pp≤ηc₀x/(8 log x). Equation (3) is then at least ηc₀x/(4 log x). Dividing by log x gives the count. Its openers tend to infinity since n≥x/2. ∎ Hη asks for a positive proportion on a subsequence and is stronger than mere infinitude requires. A smaller positive defect Q−R exceeding 2E_pp would already give one pair on each successful scale. Neither inequality is proved here. Calling a target TPC-strength is not a proof of circularity; proposing (H) without an independent derivation also does not advance it. ## 3. The actual factor/cofactor interface Let $$ R_1=\sum_{n\in I_x}\Lambda(n)\chi_z(n+2)\lambda(n+2),\quad R_2=\sum_{n\in I_x}\Lambda(n)\chi_z(n+2)\lambda(n+2)k_x(n+2), $$ $$ R_\square=\sum_{n\in I_x}\Lambda(n)\chi_z(n+2)\lambda(n+2)e_x(n+2). $$ Exactly, including square factors, $$ R=R_1-\tfrac12R_2+\tfrac12A_3-\tfrac12R_\square,\qquad |R_\square|\le E_\square=o(x/\log x). \tag{4} $$ The **plus** sign before A₃ matters: on its three-factor support λ=−1. Thus a proposal to cancel only R₁ must still pay the other terms. Complete multiplicativity, including q|v, gives the exact identity $$ R_2=-\sum_{z\le q\le r\atop q\text{ prime}} \sum_{x/2+2\le qv\le x} \Lambda(qv-2)\lambda(v)\chi_z(v). \tag{5} $$ Here q is an actual prime factor and v an actual cofactor, not two divisor moduli. For a block q∼M, the range is v∼x/M, with $$ x^{1/8}\lesssim M\lesssim x^{1/3},\qquad x^{2/3}\lesssim v\lesssim x^{7/8}, $$ and the exact product inequalities in (5) govern the constants and endpoints. Prime-restricted unit-weight versions of (4) and (5) are independently enumerated by the validator; the general identities follow algebraically. This supplies a place to state an arithmetic estimate. It supplies no such estimate by itself. In particular: - Prime BV controls unsigned arithmetic-progression discrepancies. It does not assert cancellation of λ(v) against Λ(qv−2) with these restrictions. - The switched theorem in the benchmark concerns a nonnegative triple-product sequence. Multiplying it by λ on a shifted argument changes its hypotheses. - The PNT cancellation of an unweighted one-variable Liouville sum cannot discard the shifted-prime weight or the roughness and product constraints. - Taking absolute values term by term destroys the sign needed in (4). A proof must bound the combined expression at the scale of Q, not merely show that some individual correlation tends to zero after normalization by a larger mass. The earlier [bilinear-transplant audit](history/staging/attack-bilinear-transplant.md) tested different certificates and short stretches; its density mismatch is not removed by relabelling those intervals. This benchmark instead uses a long interval from the outset. Likewise the [lambda-ledger experiment](history/staging/attack-lambda-ledger.md) tested a large, shift-specific covariance anomaly. We neither repeat that experiment nor infer that small covariance would be useless. ## 4. Choosing the next proof obligation On actual prime openers the exact expression Q_P−R_P is twice the weighted twin count **minus** twice a nonnegative odd-composite penalty L. Clipping W at zero removes that penalty but leaves an unestimated signed sum. Also, even if R₁−R₂/2=o(x/log x), the benchmark's separate estimates in (4) give a negative lower-bound coefficient for Q−R. These statements are proved in [chen-opportunity-audit.md](chen-opportunity-audit.md); they do not refute Hη. The resulting [input/weight specification](prime-detection-spec.md) proves the ordinary progression estimates for a locally correct comparison sequence and gives a positive conditional payoff from a specific bilinear bound. The [first fold attack](bilinear-fold-attack.md) executes the least-factor and second-moment calculations without obtaining the missing saving. Its growing- depth joint estimate remains open. These notes supersede the task of choosing a concrete input package; Hη here remains a valid diagnostic condition. Only once such a mechanism makes a testable prediction should a numerical experiment be designed for it. The present computation checks definitions, signs, endpoints and the classical constant. It is not an asymptotic correlation sweep and supplies no evidence that (H) holds. The immediate research uncertainty is the estimate itself, not the precision of the old G₂ fit or the height of another tile computation.