# Endpoint target audit: a shrinking margin suffices **The endpoint estimate and twin-prime infinitude remain OPEN.** This audit checks two assumptions that affect the next research decision: the size of the required positive margin and the scope of a recorded theorem failure. It derives a weaker sufficient condition from the existing error budgets. It supplies no new cancellation estimate, numerical evidence or novelty claim. The exact definitions and cutoffs are those of [signed-divisor-grouping.md](signed-divisor-grouping.md): x is dyadic, J_x=(x/2,x], U=V=floor(x^(6/25)), Y=Z=floor(x^(1/20)), and L=floor(x^(7/10)). E_>(x) is that note's equation (18), with all four indices, signs, parity branches and endpoint weights retained. [residual-coverage.md](residual-coverage.md) now removes a larger exact union of controlled cuts. Its E_* can replace E_> in every sufficient condition below: their difference is O_H(x/log^H x) for every fixed H. [grouped-divisor-moment.md (19)--(20)](grouped-divisor-moment.md) further replaces E_* by the current E_dagger at the same arbitrary fixed logarithmic precision. Every sufficient condition below therefore also applies to E_dagger. These are distinct exact sums with explicitly controlled differences, not interchangeable definitions. The one-sided margin itself remains OPEN. [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) states the current object and assignment requirements together. ## 1. Recover the quantitative error before weakening the target Write S(x)=sum_{n in J_x} Lambda(n)Lambda(n-2), and let M(x), B(x), I_1(x), I_2(x) be the quantities in [prime-detection-spec.md](prime-detection-spec.md). The displayed o(x) in its main-term formula can be sharpened to $$ M(x)=C_2x+O_H(x/\log^H x)\quad(H>0). \tag{1} $$ To check this, at every odd prime power p^j, b(p^j)=2C_2(1+1/(p-2)); even integers have b=0. Hence M differs from 2C_2 sum_{n in J_x} Lambda(n) by the contribution of powers of 2 and the sum of 2C_2 log(p)/(p-2) over odd prime powers in J_x. For j=1, p>x/2 and the latter sum is O(log x), even using only the integer count. The terms with j>=2 cost O(sqrt(x) log^2 x) by counting proper prime powers and bounding each weight by O(log x). Powers of 2 are absorbed in that allowance. The required classical input is psi(t)=t+O_H(t/log^H t), stated with a stronger error in [Tao, Notes 2, Corollary 39](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/). Subtract its values at x and x/2. Every fixed power of log x is dominated by sqrt(x), so the prime-power allowance is also absorbed in (1). This is an application of quantitative PNT, not an inference of a rate from the bare statement M=C_2x+o(x). Now retain the rate in every other step: | contribution to S-C_2x-E_> | available bound and owning argument | |---|---| | M-C_2x | (1) above | | I_1-I_2 | O_H(x/log^H x), prime-detection-spec.md section 4 | | -Bcal(b), where B=Bcal(a)-Bcal(b) | O_H(x/log^H x), polylog-fold-transfer.md section 4 | | P_1-P_2, where Bcal(a)=P_1-P_2+R | O_H(x/log^H x), shifted-prime-decomposition.md section 4 | | R_{<=L} | O_H(x/log^H x), signed-divisor-grouping.md sections 2-4 | | M_>, where R_{>L}=M_>+E_> | O_H(x/log^H x), signed-divisor-grouping.md section 5 | These are a fixed number of contributions. Their identities give $$ S(x)=C_2x+E_>(x)+O_H(x/\log^H x) \quad\hbox{for every fixed }H>0. \tag{2} $$ All constants may depend on H. This statement supplies neither effective onsets nor uniformity when H itself grows with x. The pre-existing arithmetic derivations are inputs to (2), not newly independently refereed theorems. Their finite identities have checks in [prime-detection-validation.js](prime-detection-validation.js), [shifted-prime-validation.js](shifted-prime-validation.js) and [signed-divisor-validation.js](signed-divisor-validation.js). ## 2. A logarithmically shrinking margin Fix K>0 and c>0. The following hypothesis is sufficient and **OPEN**: $$ C_2x+E_>(x)\ge {c x\over\log^K x} \quad\hbox{on an unbounded set of dyadic }x. \tag{3} $$ Choose H>K in (2). Its error is o(x/log^K x). The contribution to S from pairs with a proper prime power on either side is O(sqrt(x) log^3 x), also o(x/log^K x), by prime-detection-spec.md section 1. If N_2(x) counts genuine prime pairs in J_x, each contributes at most log^2 x to S. For all sufficiently large x in the same set, absorb each of these two errors into one quarter of the margin in (3), obtaining $$ N_2(x)\ge {c\over2} {x\over\log^{K+2}x}. \tag{4} $$ The intervals escape to infinity, so (4) implies infinitely many twins. An additional o(x/log^K x) term on the right of (3) can be absorbed by reducing c. A bare o(x) error cannot be absorbed this way. The earlier fixed-fraction condition implies (3) for any fixed K>0 eventually. Condition (3) allows the guaranteed fraction of x/log^2 x to tend to zero. It is still a stronger requirement than infinitude: infinitude alone need not supply the lower count (4). This audit proves sufficiency and does not prove logical independence of arithmetic conjectures. The stronger fixed-fraction target remains available. ## 3. An average with the same smaller margin Let x_j=2^j and fix j_0. Another sufficient and **OPEN** hypothesis is $$ \limsup_{J\to\infty}{1\over J-j_0+1} \sum_{j=j_0}^{J}(\log x_j)^K \left(C_2+{E_>(x_j)\over x_j}\right)>0. \tag{5} $$ Multiplying (2) by (log x)^K/x gives an error tending to zero if H>K; its Cesaro average therefore tends to zero. Thus (5) gives a positive limsup average of the nonnegative quantities S(x_j)(log x_j)^K/x_j. Infinitely many exceed a fixed positive constant: otherwise their eventual upper bound could be chosen below the positive limsup. Removing proper prime powers on those scales yields a positive constant times x_j/log^(K+2) x_j genuine prime pairs, as in section 2. The factor (log x_j)^K is essential to this formulation: a positive fixed limsup for the unscaled average of S(x_j)/x_j was the stronger consumer in shifted-prime-decomposition.md section 6. Averaging still does not make fixed-coefficient correlation theorems uniform in our changing cutoffs. Neither (3) nor (5) is estimated here. ## 4. What the recorded Kloosterman failure actually decides The theorem comparison in [attack-sqrt-cancellation.md](history/staging/attack-sqrt-cancellation.md) addresses a uniform gap certificate. Its conclusions must be read at that scope before deciding whether to reuse or revisit the argument. | feature | recorded comparison | present endpoint problem | |---|---|---| | interval position | every translate across a primorial period | the actual interval (x/2,x] | | coefficients | selected linear-sieve weight products | the exact Mobius and prime-power coefficients in E_> | | required conclusion | an absolute bound uniform in position | (3), or the signed average (5), is sufficient | | role of the theorem calculation | prices particular published estimates in its specified ranges | the Fourier follow-up prices an additional rectangle with its phase and tail costs; the complement remains open | [Bettin-Chandee, Theorem 1 and Remark 1](https://arxiv.org/pdf/1502.00769) allow arbitrary separate coefficients in a trilinear inverse-residue sum and charge an additional factor for a differentiable phase perturbation. The recorded uniform-position calculation cannot simply be transferred to this different position range. Conversely, the actual interval does not make that factor free: its cost depends on the Fourier cutoff and the product of the moduli. [endpoint-fourier.md](endpoint-fourier.md) now supplies that budget and a matched application on an additional rectangle after aggregating the coefficients. A published upper bound exceeding the desired budget does not establish that the underlying signed sum exceeds it. The recorded calculation therefore cannot establish a universal obstruction to this endpoint consumer. This scope check neither overturns its numerical comparison nor asserts that the published theorem suffices here. ## 5. Consequence for the next attempt and its falsifiers The current controlled union and exact E_dagger are in grouped-divisor-moment.md section 5. Its section 6 leaves a nonzero small-common-divisor kernel at d~x^(8/25), e~x^(9/20), with a required moment-exponent saving strictly greater than 3/50. Price that local target and its possible extension across the remaining domain, or pursue (3) or (5) directly with E_dagger substituted. The earlier squarefree pilot was controlled by prime-dispersion.md and is not a remaining task. Preserve signs before estimating: a useful one-sided estimate need not give E_dagger=o(x). A successful local estimate alone does not establish either global margin. Require the complete arithmetic and error budget before changing weights or commissioning another numerical run. The derivation here fails if any discarded term has only an unspecified o(x) bound, if a constant must be uniform in a growing H, or if the prime-power subtraction is comparable to the claimed margin. Section 1 lists every discarded term and its arbitrary fixed logarithmic rate; H is fixed after K, and the prime-power error is smaller than that margin. A positive E_> margin is still an unproved arithmetic input. Rewriting its sufficient size is a specification improvement, not progress on estimating the signed correlation itself.