# Attack beta_2 / 02: the minimal theta_total, and why it is 1 *(2026-08-18. Staging note for the upper-bound attack on `paper/beta2-note.md`'s G2 exponent. Companion computation: `research/attack-beta2-02-theta-total.js`. Legend as in `research/covering-dive.md`: **[PROVEN]** published theorem with source; **[VERIFIED]** checked computationally here; **[MEASURED]** empirical, finite range; **[INFERRED]** our deduction from sourced facts; **[ABSENT]** we searched and found nothing.)* ## 0. THE NUMBER, FIRST **The largest theta_total defensible position-uniformly is 1, and it does not clear 1.2417. It misses by 0.2417 under the brief's symmetric arithmetic and by 0.2090 under the correct asymmetric arithmetic. No part of the gap is closed.** | quantity | value | status | |---|---|---| | break-even theta_total, symmetric levels | 1.241651 | the brief's figure, confirmed | | break-even theta_total, optimally asymmetric levels | **1.208983** | corrected, and lower | | theta_total provable in the LONG-interval problem (Brudern-Fouvry Prop. 2) | **1.25 exactly** | **[PROVEN]**, in print since 1996 | | theta_total provable POSITION-UNIFORMLY (what G2 needs) | **1** | and it is sharp, §4 | | shortfall against break-even | **0.209** | the whole gap | | exponent at theta_total = 1, best asymmetry | 5.158065 | worse than beta_2 = 4.26645 by 0.892 | Two findings sit beside that verdict and neither is small. **The first is that the target of this attack is already a published theorem in the neighbouring problem.** Brudern and Fouvry's Proposition 2 delivers D1\*D2 = x^{5/4} unconditionally, their four side conditions bottom out at `q^{C0} D1^4 D2^4 <= x^{5-c eps}`, which IS theta_total <= 5/4, and their resulting exponent is (4/3)(1 + e^{3/4}) = **4.156000**, below beta_2 = 4.26645 by 0.110450. Their own introduction says so, comparing against "1/4,2664 = 0,2343". So the answer to "what is the smallest theta_total at which the joint bilinear remainder bound is actually provable" is **1.25, and it was answered in 1996** for the density form of the problem. It is not new, and it is not ours. **The second is that the transfer to G2 fails at one identifiable factor, not at a vague loss of uniformity.** Writing the joint remainder for a window (N, N+H] rather than for [x/2, 3x/2], the phase that appears is *exactly* Brudern-Fouvry's, multiplied by one extra factor: e( h rho /(d1 d2) ) = e( -h N /(d1 d2) ) · e( -2 h \bar{d1} / d2 ) The right factor is theirs verbatim. The left one is bounded by x^eps in their setting because their window length equals the size of their elements, and it is of size N/H in ours, with N running to P(z) = e^{z(1+o(1))}. Every step of both their propositions survives the transfer except the one that removes it. --- ## 1. What Lemma V asks at theta_total = 1.25 and at 2 Both statements, written out, so the difference is visible rather than asserted. > **Lemma V(a, b), position-uniform form.** Let z >= 2, H = z^u, and let psi^+ > be Rosser-Iwaniec upper-bound linear-sieve weights of level D = H^a and psi^- > lower-bound weights of level Delta = H^b, both for the sieve by primes p < z. > Then for every integer N >= 0 and each (alpha, beta) in {(-,+), (+,-), (+,+)}, > > SUM_{d1 | P(z), d1 <= L_alpha} SUM_{d2 | P(z), d2 <= L_beta, (d1,d2)=1} > psi^alpha_{d1} psi^beta_{d2} · r_{d1,d2}(N) << H (log H)^{-3}, > > where r_{d1,d2}(N) = #{n in (N, N+H] : d1 | n, d2 | n+2} - H/(d1 d2), and > L_+ = D, L_- = Delta. Write theta_total = max(a+b, 2a). | | theta_total = 1.25 | theta_total = 2 | |---|---|---| | levels (upper, lower) | (H^{1/2}, H^{3/4}) | (H^{0.865}, H^{1.135}) at the optimum | | terms in the sum | H^{1.25} | H^{2} | | trivial bound | H^{1.25} | H^{2} | | target | H^{1-eps} | H^{1-eps} | | **power of H that must be saved** | **H^{0.25}** | **H^{1.00}** | | cancellation exponent gamma needed, deviation ~ (terms)^gamma | **gamma <= 0.80** | **gamma <= 0.50** | | fraction of square-root cancellation needed | 40% | 100% | | sieve exponent obtained | 4.1560 (BF geometry) / 4.1265 (unconstrained optimum) | 2.6487 (symmetric) / 2.5790 (unconstrained optimum) | **[VERIFIED]** `research/attack-beta2-02-theta-total.js` S1, S3, S6. The brief's "roughly four times less cancellation" is exactly right and is exact rather than rough: 0.25 against 1.00 in the exponent. In the repo's own diagnostic variable, theta_total = 2 asks for *precisely* square-root cancellation and not one epsilon less, while theta_total = 1.25 asks for 40% of it. The pilot in `research/sift-limit-attack.js` measures gamma = 0.23 to 0.33 at toy scale, comfortably inside both, which is why this route keeps looking alive. **One correction to the arithmetic before going further.** The threshold formula in `research/sift-limit-attack.md` §4.5, `ln(s1-1) + ln(s2-1) > 1`, parametrises the asymmetry between the two COMPONENTS, and under that parametrisation the optimum is symmetric and gives u > 2(1+sqrt e)/theta_total. The asymmetry that actually pays is between the UPPER and the LOWER sieve level, which that formula cannot express. Brudern and Fouvry's own positivity equation, on their p.355, read from the PDF, is 2 f( log x^{3/4} / log x^xi ) - F( log x^{1/2} / log x^xi ) = 0 whose unique root is xi = 0.240616 **[VERIFIED]**, matching the 0,2406 they print. In closed form the threshold with upper level H^a and lower level H^b is u > (1 + e^{b/(2a)}) / b which reduces to 2(1+sqrt e)/theta_total only when a = b. Optimising over the ratio rho = b/a gives u = G\*/theta_total with G\* = min_rho (1 + e^{rho/2})(1 + 1/rho) = **5.158065** at rho\* = **1.313085** against 2(1+sqrt e) = 5.297443. **[VERIFIED]** So the correct break-even is 5.158065/4.26645 = **1.208983**, not 1.241651, and at theta_total = 1.25 the correct exponent is 4.126452, not 4.237954. The brief's 1.2417 is the conservative figure; nothing below depends on which is used, because the defensible value is 1. Two validity checks that a threshold of this shape needs and that are easy to skip. At the Brudern-Fouvry point F is evaluated at s = 2.078 and f at s = 3.117, inside [1,3] and [2,4] respectively, which is where the explicit `F(s) = 2e^gamma/s` and `f(s) = 2e^gamma ln(s-1)/s` are exact. **[VERIFIED]** At the unconstrained optimum the arguments are s = 2.230 and s = 2.928, also inside. No row in §1 or §3 of the script sits outside those ranges, which is the failure mode that voided `research/theta-ladder.md` §5b's working point. --- ## 2. The two papers, read from the PDFs Both fetched as PDFs and read as rendered pages, not as OCR text and not as HTML. The numdam OCR layer drops every display formula in the Brudern-Fouvry file, so every displayed condition below was read off the rendered page image. ### 2a. Brudern and Fouvry, *Le crible a vecteurs*, Compositio Math. 102 (1996) 337-355 Source: `numdam.org/item/CM_1996__102_3_337_0.pdf`. **[PROVEN]** Their Theorem, p.339: for 0 <= kappa < 1, q an integer with 1 <= q <= x^kappa, and P any set of primes, there is a function theta(kappa) tending to 0,2406 as kappa tends to 0 such that uniformly on P, for 0 <= theta <= theta(kappa), |{n <= x ; n = 1 (mod q), p | n(n-2) and p in P ==> p >= x^theta}| >> (x/q) PROD_{2 1, D = MN, under the standard one-sided axioms and for all 2 <= z <= D^{1/2}, S(A,P,z) <= V(z) X {F(s) + E} + R^+(A,M,N), S(A,P,z) >= V(z) X {f(s) - E} - R^-(A,M,N), with s = log D / log z and R^nu(A,M,N) = SUM_{l < exp(8 eps^-3)} SUM_{m= 1 - H/(d1 d2) >= 1/2 once d1 d2 >= 2H. Hence SUM_{d1<=D1, d2<=D2} |r_{d1,d2}(n0 - 1)| >> #{admissible pairs} = (D1 D2)^{1-o(1)}, the o(1) being the Dickman-type density of z-smooth squarefree divisors, which is the one asymptotic input here and is standard. **[INFERRED; the finite-range behaviour is VERIFIED below]** Measured, at z < 42 with the split by alternating primes, `research/attack-beta2-02-theta-total.js` S7: ``` H D1=D2 pairs sum|r| worst /pairs sum|r| typical /pairs worst/typical 60 100 108 79.9 0.740 23.6 0.219 3.4 60 200 168 136.2 0.811 28.9 0.172 4.7 60 400 320 284.7 0.890 34.8 0.109 8.2 30 400 320 294.4 0.920 24.8 0.078 11.9 15 400 320 302.7 0.946 17.4 0.054 17.4 8 800 520 508.7 0.978 13.9 0.027 36.7 ``` **[VERIFIED]** The worst-position column climbs to 0.978 of the pair count as D1 D2 / H grows, while the typical-position column falls to 0.027 of it. The typical column is the divisor average, O(H log^2 H), which is what Brudern and Fouvry have because they sum over n ~ x. G2 has to survive the other column. Consequence, stated flatly: **for the two-class interval sequence, the natural level of distribution in Iwaniec's sense is H^{1+o(1)} uniformly in position, and that is a theorem about the sequence, not a limitation of any particular argument.** Every theta_total above 1 therefore requires genuine signed cancellation at EVERY position, with no absolute-value step anywhere in the chain. **[INFERRED from the construction]** **What this does not show.** It does not argue against Lemma V. At the same constructed position the SIGNED sum with real Rosser-Iwaniec weights stays small, S8: ``` H Dup Dlo pairs |signed| worst pos sum|r| worst pos |signed| max over 20k pos 60 200 400 272 2.00 46.7 10.00 30 200 400 272 3.00 44.0 9.00 15 200 400 272 4.00 36.1 7.00 8 400 800 588 1.00 54.5 6.00 ``` **[MEASURED]** The bottom row runs at theta_total = log(400·800)/log 8 = 6.1, far past anything this attack needs, and the signed sum never exceeds 6 over 20,000 consecutive positions while the pair count is 588 and the absolute-value sum at the constructed position is 54.5. So the worst-position obstruction of §4 is an artifact of the absolute values and of nothing else, exactly as `research/sift-limit-attack.js`'s full-period pilot already suggested. Lemma V looks true and the method is what fails. --- ## 5. The theorem that would follow, stated so it can be checked, and NOT claimed > **Conditional Theorem (NOT PROVED).** Suppose Lemma V(1/2, 3/4) of §1 holds: > the joint interval remainder for the Rosser-Iwaniec weights at upper level > H^{1/2} and lower level H^{3/4} is << H (log H)^{-3}, uniformly in the window > position N. Then G2(p_n#) <<_eps p_n^{4.156000 + eps}, with > 4.156000 = (4/3)(1 + e^{3/4}). Against `paper/beta2-note.md`'s beta_2 = 4.26645028414864191641 that is a decrement of **0.110450**, which closes **4.87%** of the open band (2, 4.2665]. Optimising the level split inside Brudern and Fouvry's Proposition 2 polytope improves it to 4.155795; dropping their polytope and asking only for theta_total = 1.25 improves it to 4.126452. **[VERIFIED]** Three things must be said in the same breath. **The hypothesis is not an auxiliary lemma; it is the theorem.** At theta_total > 1 the identity `SUM_{d1,d2} lambda_1 lambda_2 |A_{d1,d2}| = SUM_{n in window} (SUM_{d1|n} lambda_1)(SUM_{d2|n+2} lambda_2)` says the remainder sum and the certificate are the same object up to the main term. So proving Lemma V at theta_total = 1.25 is not a step toward the result, it is the result written differently. That is normal for sieve remainders only when an independent arithmetic input bounds them; here there is none. **The exponent is not the binding cost.** Every published route to theta_total > 1 on this sequence, Brudern-Fouvry's included, buys the extra level from an average over the position variable. G2 has no position variable to average over. **And the honest comparison is with kappa = 1.** The same conditional sentence could have been written in 1980 for Jacobsthal, and forty-eight years later the exponent there is still 2. --- ## 6. What is defensible, and by how much it falls short | route | position-uniform? | theta_total | exponent u | vs beta_2 = 4.26645 | |---|---|---|---|---| | absolute values, one remainder per pair (the current theorem's accounting) | yes | 1 - eps | 5.158065 | loses by 0.892 | | Brudern-Fouvry Prop. 1 (Weil), transferred | **no**, §3 | 1.1620 | 4.612313 | loses by 0.346 | | Brudern-Fouvry Prop. 2 (Deshouillers-Iwaniec), transferred | **no**, §3 | 1.2500 | 4.156000 | wins by 0.110 | | break-even, symmetric | | 1.241651 | 4.266450 | ties | | break-even, optimally asymmetric | | **1.208983** | 4.266450 | ties | **[VERIFIED]** for every number in the table. **Defensible position-uniformly: theta_total = 1.** Short of break-even by 0.209 (asymmetric) or 0.242 (symmetric). The shortfall is the entire distance; nothing in this attack moves it by any amount. Two smaller results the attack does produce, both worth keeping: 1. **The exact ceiling of the elementary route.** Even granting a perfect transfer of Brudern-Fouvry's Proposition 1, which uses only Weil's bound and no spectral theory, the best exponent is 4.612313 at theta_total = 1.1620, below break-even. So there is no elementary win available even in the long-interval problem, and the whole margin below beta_2 in Brudern-Fouvry rests on Deshouillers-Iwaniec. 2. **The sharpness of §4**, which converts "we do not know how to beat level H uniformly in position" into "no absolute-value argument can", and localises the remaining hope to a genuinely signed estimate. --- ## 7. Every hypothesis smuggled in Listed because the brief asked and because two of them matter. - **H1.** That the linear sieve functions may be taken as `F(s) = 2e^gamma/s` and `f(s) = 2e^gamma ln(s-1)/s`. Exact only on [1,3] and [2,4] respectively. Checked at every point quoted here (§1); all inside. This is the check whose omission voided `research/theta-ladder.md` §5b's working point. - **H2.** That the vector-sieve main term is `F(s_D) [2 f(s_Delta) - F(s_D)]`, i.e. that the two components carry the same pair of levels. Brudern-Fouvry's p.355 equation confirms it for their configuration. A four-level configuration is not covered by anything here. - **H3.** That the count of z-smooth squarefree divisors below D = z^s is D^{1-o(1)} for fixed s, and likewise for a positive-density subset of the primes below z. Standard Dickman-type counting; used only for the o(1) in §4, and the finite-range behaviour is measured rather than assumed. - **H4.** That Brudern-Fouvry's normalisation and ours agree, i.e. that their theta and our u are related by u = 1/theta. Their own comparison against "1/4,2664" is the check, and it is their sentence, not our inference. - **H5.** That Proposition 2's four conditions may be applied with either of the two levels in the D1 slot, since in their application both coefficient families are well factorable (their §3.4). Without this the polytope optimum in §2c would be worse, not better, so it cannot be inflating any claim here. - **H6.** That the position-uniform requirement really is over all N up to P(z). It is: G2(p_n#) is a maximum over a full period of the primorial modulus. - **H7 (the one that would overturn §3).** That there is no reformulation in which the translation phase `e(-hN/(d1 d2))` is absorbed rather than removed. We searched and found none, and the smoothness hypothesis of Deshouillers- Iwaniec Theorem 12 is what such a reformulation would have to evade. **[ABSENT, not [PROVEN]]**. This is the weakest link in the negative verdict and the place a stronger reader should push. --- ## 8. Corrections to the record Proposed, not applied. None of these changes any headline number of the corpus. 1. **`research/sift-limit-attack.md`:30-32.** "Run unconditionally the component levels couple (D1D2 <= H) and the route lands at 2(1+sqrt e) = 5.2974, worse than 4.2665, which is why it is absent from this problem's record." The *reason* is wrong. Brudern and Fouvry do not run coupled: their Proposition 2 gives D1 D2 = x^{5/4} unconditionally, and their published exponent is 4.156, below beta_2. The vector sieve is absent from the G2 record because of position-uniformity (§3), not because it loses. `paper/wall-note.md`:346-352 already hedges this correctly and should be the model for the fix. 2. **`research/sift-limit-attack.md`:238-241 and :246-250.** The threshold `ln(s1-1) + ln(s2-1) > 1` parametrises component-versus-component asymmetry; the asymmetry that pays is upper-level versus lower-level. Correct constants: `u > (1 + e^{b/(2a)})/b`, optimum `u = 5.158065/theta_total`, break-even theta_total = **1.208983**. The figure 1.2417 is not wrong as a sufficient condition, it is simply not the break-even. 3. **`research/sift-limit-attack.md`:262-265**, Lemma V's stated range "s >= (0.63+delta)·u (theta_total >= 1.25)". Arithmetically consistent with a symmetric split at 0.625u each, but the shape that is actually provable in the long-interval problem is the ASYMMETRIC (0.5u, 0.75u). Worth restating Lemma V with two level parameters rather than one. 4. **New, for the sources list:** Brudern-Fouvry's Proposition 2 condition `q^{C0} D1^4 D2^4 <= x^{5-c eps}` is the exact origin of the number 1.25, and §6 item (iii) of `sift-limit-attack.md` can be closed by pointing at it: the question "is theta_total = 1.25 provable" has been answered yes since 1996 for the density problem and remains untouched for G2. 5. **`research/sift-limit-attack.md`:277-287** cites the kappa = 1 precedent as Iwaniec 1980. Accurate for the FORM of the bilinear error term (Theorem 1, read from the PDF, quoted in §2d). The position-uniform kappa = 1 record is a different and less encouraging fact: Iwaniec 1978's g << log^2 n, still unimproved in the exponent, sitting exactly on beta(1) = 2. --- ## 9. Reproduction ``` node research/attack-beta2-02-theta-total.js # all eight sections, ~7 s node research/qc.js # 8 checks ``` Gate state: eight checks; **this note and its script contribute 0 findings**, confirmed on a clean run. The only finding seen during the run was a transient `scripts / uncited-script` on `research/attack-beta2-04-loss-budget.js`, which appears and clears as the sibling attack rewrites its own report. It is not ours and it is not about anything here. Sources read as PDFs, both fetched fresh for this note: `numdam.org/item/CM_1996__102_3_337_0.pdf` (Brudern-Fouvry, pages 339, 342-345, 353, 355 read as rendered images because the numdam text layer drops every display formula) and `matwbn.icm.edu.pl/ksiazki/aa/aa37/aa37127.pdf` (Iwaniec 1980, pages 307-309). Deshouillers-Iwaniec, Invent. Math. 70 (1982) 219-288, is cited only through Brudern-Fouvry's Lemme 2 and was not read.