# An unconditional upper bound on the rough-pair census X(K) by Selberg's Λ² sieve on the stretch window: a theorem with every constant explicit, loose against exact X(K) by 7.8 to 10.1 at the band depths, and a level the window holds below z² so the sifting parameter never reaches s = 2 > **RIDER 2026-08-30 (orchestrator, from `redteam-0830-zone.md` A7, which > reproduced every band sum, tier ratio, D_max and the flat 378100.0 digit > for digit on independent code).** §0 item 2's and the ledger's > "s_max 1.81 → 1.96" is REFUTED as written, since the arrow reads B3→B8 > everywhere else in this note: the values are 1.8106, 1.9612, 1.9538, > 1.9288, 1.9149, 1.8542, not monotone, peaking at B4 with B8 at 1.85 (§5 > here already contradicts §0). Read instead: "s_max reads 1.81 at B3, > peaks at 1.96 at B4 and falls to 1.85 at B8; the maximum over the six > bands is 1.96 and the level is below z² at every band." The mathematics > is unaffected. Also A8: the wrong-direction guard forbids the CLAIM that > X < T, not the computation of a crossing quantity, which is why the two > blind notes are legal. *(2026-08-29. Staging note. Producer, formally embedded: `research/history/staging/attack-0829n-X-upper.js` (2.8 s; code-sha256 3272cd0e...; out-sha256 281a529c...; `node research/qc/embed.js --check` passes bit-honest). No existing file edited, no git command run. Calibration per claim: PROVEN, VERIFIED by exact computation, MEASURED, HEURISTIC, OPEN. Wrong-direction guard: `attack-wrongdirection-audit.md` §3.1 rules that the only legal half of TODO Z2 is an upper bound on X(K) alone; this note compares X(K) against nothing but its own upper bound and the sifting function S(K), never against T. T is computed by the producer only as an engine-identity gate against cited band sums and enters no bound and no ratio. The standing K* and K*/pool targets (12.20 → 31.22, 0.061 → 0.032, `attack-quadpoint-02.md` §2–3) and the main-term K* comparison (3.92 → 31.15, ratio 0.998 at B8, `attack-roughpair-error.md` §6) are crossing quantities, i.e. comparisons of X against T; they are quoted here as what they are and are NOT recomputed, because computing them is the comparison the guard forbids. The reproduction target this note can legally score is exact X(K) itself, per anchor.)* ## 0. Verdict, disconfirming half first 1. **The bound is loose by a factor 7.82 → 10.13 at the band depths, band sums B3 → B8, and the factor grows with Q** [MEASURED, SEC 2]. Per anchor the median runs 8.78 → 10.75 and the worst anchor is 34.96 (B4) [SEC 3]. The loss splits into two factors with different owners: - a **sieve loss** T1/S = 1.78 → 2.53, the Selberg upper bound against the sifting function S(K) it actually bounds; - a **structural loss** S/X = 4.39 → 4.01, the sifting function against the census: S − X = M + T, and the mixed pairs are M/S = 0.495 → 0.513 of S [SEC 5]. This factor is not a sieve deficiency. Removing it needs a LOWER bound on M(K), the pairs with one prime member and a p_K-rough partner, on a window of length Q′² − Q² = (Q′ − Q)(Q′ + Q), which is h^{1/2} times a prime gap at h = Q². That is a lower-bound sieve over the primes of the window and needs their distribution in progressions at some positive level; no such level is known unconditionally in intervals this short (the h^{0.525} of Baker–Harman–Pintz 2001, literature figure, is the threshold for the bare count of primes, with no level at all). So the structural factor is out of reach of an upper-bound sieve by construction, and of any unconditional argument this corpus knows. 2. **The level the window allows is below z², at every band** [MEASURED, SEC 5]. The uniform remainder R_U(z, ξ) stays at or below the sieve main term C·V₂(z) only up to D_max = 169, 841, 1849, 3721, 6889, 9409 (B3 → B8, median C), i.e. s_max = ln D_max/ln p_K = 1.81 → 1.96, never 2. The optimal ξ is below p_K at every band (medians 7 → 61 against p_K = 17 → 139, s = 1.37 → 1.67), so the weights never see all the primes being sifted. In B6 the band sum of the uniform bound is **flat at 378100.0 from K = 17 to K = 46** while ΣX falls 41172 → 15540, so T1/X climbs 9.18 → 24.33 with depth [SEC 4]. This is the kill condition the brief named: the theorem is not vacuous (the bound sits below the trivial C at every anchor of B4–B8 and 35 of 40 in B3, SEC 3) but past p_K ≈ 31 (B6) and p_K ≈ 61 (B8) deeper sieving buys it nothing. 3. **The theorem's main term is the sifting function's, not the census's** [MEASURED, SEC 5]. C·V₂(z) tracks S(K) to S/(C V₂) = 0.998 → 1.004 at the band depths, but it is 5.18 → 5.55 times the census's own naive main term C[V₂(K) − 2V₂(K)U(K) + V₂(nR)] of `attack-roughpair-error.md` §2 (and X/Xnaive = 1.261 → 1.307 measured). So the brief's aim, "turn the main term into a theorem", is met for the WRONG main term: the sieve theorem carries the first of the census's three terms and cannot carry the subtracted two, for the reason in item 1. 4. **The drift 0.923 → 0.984 is not evaluated, by rule.** It is a property of the crossing y*(Q), defined by X(y) < T (`u2-engine-depth.md` §5), outside the legal half. What the theorem does carry is the mechanism that note names: its main term is the exact partial product, not the Mertens asymptotic, and the T-free size of that effect on the theorem's own main term is V₂(z) exact/asymptotic = 0.8415 → 0.9591 at the band depths [MEASURED, SEC 6]. Whether that produces the crossing's drift remains a statement about the crossing (§6). 5. What is PROVEN: Theorem U of §2, with nothing OPEN in its proof. It is classical sieve bookkeeping (Selberg's Λ² inequality, Halberstam–Richert Ch. 3; Ford's notes, cited through `quadpoint-prior-art.md` §1.1) applied to a standard object; no novelty is claimed for any step. Its value is that every constant is explicit and computed, and that the loss factors above are now numbers rather than a hardness analogy. It opens no route. ## 1. The object, copied From `quadpoint-identity-01.md` §1, verbatim in substance: fix an anchor Q (a prime ≥ 7), Q′ the next prime, the half-open stretch S_Q = [Q², Q′²), the capacity C counting the channel pairs (a, a+2) with BOTH members in S_Q (a ≡ 11, 17, 29 mod 30, a ≥ Q², a + 2 < Q′²), twins T, and actives p₁ < p₂ < ... = 7, 11, 13, .... Then > X(K) = #{channel pairs with both members composite and > min(lpf(a), lpf(a+2)) > p_K}, X(0) = CC. The window convention is load-bearing there (the pair (Q′² − 2, Q′²) is excluded) and is implemented identically here: the producer's loop is `a + 2 < hi`. Two further counts are used, neither compared to T: - S(K) = #{channel pairs with (a(a+2), P(z)) = 1}, z = p_K, P(z) = ∏ actives ≤ z: the dimension-2 sifting function of Ford §1.7.2. Since a prime member has no active factor, S(K) = X(K) + M(K) + T with M(K) the pairs having exactly one prime member and the composite member p_K-rough. The producer asserts S(full) = T and X(full) = 0 at every anchor [SEC 0]. - V₂(z) = ∏_{7≤p≤z}(1 − 2/p), the fundamental-lemma density. Engine identity [VERIFIED, SEC 0]: 1227 anchors; band counts n = 40, 103, 71, 208, 292, 491 and band sums ΣT = 1017, 7086, 8332, 42495, 101763, 280128 reproduce `attack-roughpair-error-01.js`; T = 127 and X(0) = 2357 at Q = 9281 reproduce `attack-quadpoint-03.js`. ## 2. The statement Notation, for a fixed anchor Q and depth K ≥ 1 with z = p_K: - A = the set of openers a of the C channel pairs (an interval [Q², Q′² − 3] intersected with three residue classes mod 30). - For squarefree d | P(z): A_d = #{a ∈ A : d | a(a+2)}, g(d) = 2^{ω(d)}/d, r_d = A_d − C·g(d), h(d) = ∏_{p|d} 2/(p − 2), G(ξ) = Σ_{d ≤ ξ, d | P(z)} h(d), and B(1) = 0, B(d) = 3·2^{ω(d)}(1 + 1/d) for d > 1. - The Selberg weights at level ξ: λ_1 = 1, λ_d = 0 for d > ξ, and for 1 < d ≤ ξ, d | P(z), λ_d = μ(d)·∏_{p|d}(1 − 2/p)^{−1}·G_d(ξ/d)/G(ξ) with G_d(y) = Σ_{m ≤ y, m | P(z), (m, d) = 1} h(m). **Theorem U.** For every anchor Q, every K ≥ 1 and every real ξ ≥ 1, > X(K) ≤ S(K) ≤ C/G(ξ) + Σ_{d₁,d₂ ≤ ξ} λ_{d₁}λ_{d₂} r_{[d₁,d₂]} (tier T3, the exact Selberg value) > ≤ C/G(ξ) + Σ_{d₁,d₂ ≤ ξ} |λ_{d₁}λ_{d₂}| |r_{[d₁,d₂]}| (tier T2, exact remainders in absolute value) > ≤ C/G(ξ) + Σ_{d₁,d₂ ≤ ξ} |λ_{d₁}λ_{d₂}| B([d₁,d₂]) =: C/G(ξ) + R_U(z, ξ) (tier T1, uniform), where the sums run over d₁, d₂ | P(z), and > 1/G(ξ) = V₂(z)·(1 + E_z(ξ)), E_z(ξ) = (G(P(z)) − G(ξ))/G(ξ) ≥ 0, > |λ_d| ≤ 1, so R_U(z, ξ) ≤ Σ_{d ≤ ξ², d | P(z)} 3^{ω(d)} B(d). Tier T1 depends on the anchor only through C: it is a bound X(K) ≤ C·V₂(z)(1 + E_z(ξ)) + R_U(z, ξ) with both correction terms functions of (z, ξ) alone, uniform over every anchor of every stretch. Tiers T2 and T3 use the anchor's own r_d and are theorem instances rather than a uniform statement. Since each tier is a valid upper bound at every ξ, the minimum over any fixed set of ξ is a valid upper bound; the producer takes the minimum over a 28-point grid ξ ≤ 701, per anchor and tier. The sifting parameter is s = ln(ξ²)/ln z, stated per band in §4. Every inequality above runs upward from X(K), which is the direction `attack-wrongdirection-audit.md` §3.1 allows; none involves T. ## 3. The proof, step by step Each step is labelled. Nothing in this section is OPEN. **Step 1, X(K) ≤ S(K)** [PROVEN, one line]. S(K) counts the channel pairs with both members free of every active ≤ p_K; X(K) counts those among them with both members composite (a composite member is p_K-rough iff its lpf exceeds p_K, and a prime member exceeds Q² and so is divisible by no active ≤ p_K, hence is p_K-rough at every depth reported). Dropping the "both composite" condition enlarges the set. Direction: upward. What is dropped is M(K) + T, and §0 item 1 says why it cannot be recovered. **Step 2, Selberg's inequality** [PROVEN, classical; Halberstam–Richert *Sieve Methods* Ch. 3, Selberg's Λ² method; Ford's notes, the Selberg sieve chapter; numbering not verified at page image this session, and the proof is reproduced here so nothing rests on the citation]. For any real numbers λ_d indexed by d | P(z) with λ_1 = 1, > 1[(n, P(z)) = 1] ≤ (Σ_{d | (n, P(z))} λ_d)² for every integer n, because the left side is 1 exactly when the only divisor is d = 1 (the square is then λ_1² = 1) and is 0 otherwise while the right side is a square. Applying it with n = a(a+2) and summing over a ∈ A: > S(K) ≤ Σ_{a∈A} (Σ_{d | (a(a+2), P(z))} λ_d)² = Σ_{d₁,d₂} λ_{d₁}λ_{d₂} A_{[d₁,d₂]}. Direction: upward. Verified by the producer, which computes the left-hand quadratic form directly over the window at three anchors and matches C/G + Σλλr to 1e-6 relative (128.0930, 1021.5420, 1503.4693) [SEC 1]. **Step 3, the diagonalisation** [PROVEN, classical]. Write A_{[d₁,d₂]} = C·g([d₁,d₂]) + r_{[d₁,d₂]}. The main part C·Σ λ_{d₁}λ_{d₂} g([d₁,d₂]) is a quadratic form in the λ's; with g multiplicative and g(p) = 2/p < 1 for p ≥ 7 it diagonalises, and the choice of λ_d in §2, supported on d ≤ ξ with λ_1 = 1, minimises it to exactly C/G(ξ) (the standard computation; Halberstam–Richert Ch. 3, Lemma 3.1–Theorem 3.2 in most editions, numbering unverified). The same computation gives |λ_d| ≤ 1. Both are checked: λ_1 = 1 and max |λ_d| = 1.0000 over 9 depths × 28 levels [SEC 1]. What remains is the remainder Σ λ_{d₁}λ_{d₂} r_{[d₁,d₂]}, which is tier T3 exactly. **Step 4, the main term as a partial product** [PROVEN, one line]. At full level ξ ≥ P(z), G(P(z)) = ∏_{p≤z}(1 + h(p)) = ∏_{p≤z}(1 − 2/p)^{−1} = 1/V₂(z), and G is increasing in ξ, so 1/G(ξ) = V₂(z)(1 + E_z(ξ)) with E_z(ξ) ≥ 0 as stated. Checked at K = 3: G(1001)·V₂(13) = 1 to 1e-15 [SEC 1]. This is the step that makes the theorem's main term the exact partial product and not the Mertens asymptotic (§6). **Step 5, T3 ≤ T2** [PROVEN, triangle inequality]. Direction: upward. **Step 6, the uniform remainder bound, T2 ≤ T1** [PROVEN, elementary]. For squarefree d coprime to 30, a(a+2) ≡ 0 (mod d) holds on exactly 2^{ω(d)} residues mod d (a ≡ 0 or −2 mod each odd prime p | d, distinct residues), so A_d is a count of integers of an interval of length L in 3·2^{ω(d)} residue classes mod 30d; each class contributes L/(30d) + θ with |θ| < 1, so A_d = 3·2^{ω(d)}L/(30d) + θ_d, |θ_d| < 3·2^{ω(d)}, and C = A_1 = 3L/30 + θ_1, |θ_1| < 3. Hence r_d = A_d − C·2^{ω(d)}/d = θ_d − θ_1·2^{ω(d)}/d, so |r_d| < 3·2^{ω(d)}(1 + 1/d) = B(d) for d > 1, and r_1 = 0 identically. Checked: max |r_d|/B(d) = 0.4600 over 3702 moduli × 1227 anchors, at Q = 4111, d = 8137 [SEC 1]. The bound uses only that A is an interval in fixed residue classes; no equidistribution input of any kind enters, which is what makes T1 unconditional and uniform. **Step 7, the tail bound on R_U** [PROVEN, standard]. The number of pairs (d₁, d₂) with [d₁, d₂] = d is 3^{ω(d)}, and |λ| ≤ 1, giving the stated Σ_{d ≤ ξ²} 3^{ω(d)} B(d). The producer uses the sharper exact-|λ| form of R_U, which is what §4's numbers are. That completes the proof. Verification of the whole chain: T1 ≥ T2 ≥ T3 ≥ S(K) ≥ X(K) at all 10908 (anchor, depth) pairs [SEC 1]. ## 4. The level the window allows, and the sifting parameter The theorem is non-vacuous only where R_U(z, ξ) is below the main term C·V₂(z), and C is small: the window has length (Q′ − Q)(Q′ + Q), so C ≈ (Q′ − Q)·Q/5 and the median C at the band depths is 184, 714, 1427, 2922, 5848, 10938 (B3 → B8), with median main term C·V₂(z) = 80.3 → 1793.2 [SEC 5]. Against that, the uniform remainder at the largest grid level with R_U ≤ C·V₂ gives | band | K_b | p_K | C med | C·V₂ med | D_max | s_max = ln D_max/ln p_K | |---|---|---|---|---|---|---| | B3 | 4 | 17 | 184 | 80.3 | 169 | 1.81 | | B4 | 8 | 31 | 714 | 221.7 | 841 | 1.96 | | B5 | 12 | 47 | 1427 | 363.9 | 1849 | 1.95 | | B6 | 17 | 71 | 2922 | 631.7 | 3721 | 1.93 | | B7 | 23 | 101 | 5848 | 1097.6 | 6889 | 1.91 | | B8 | 31 | 139 | 10938 | 1793.2 | 9409 | 1.85 | [MEASURED, SEC 5; the p_K column is SEC 6's]. The level the window allows is D < z² at every band. In the Selberg sieve the weights live on d ≤ ξ = D^{1/2} < z, so at s < 2 the sifting primes between ξ and z carry no weight at all: they are sifted only through the inequality 1[(n,P(z)) = 1] ≤ (Σλ_d)², which does not see them. The producer's minimising ξ confirms it: median ξ₁ = 7, 13, 19, 31, 43, 61 against p_K = 17, 31, 47, 71, 101, 139, s₁ = 1.37, 1.49, 1.53, 1.61, 1.63, 1.67 [SEC 2]. For comparison, `attack-wrongdirection-audit.md` §3.8 prices the LOWER-bound half at s = u*/2 = 1.7829 against β₂ = 4.26645 with the level taken as the window length itself; the upper-bound half measured here sits at s = 1.4 to 1.7 with the level the window actually supports, which is below the window length by the remainder's 3^{ω}·2^{ω} multiplicity. **Consequence, the saturation** [MEASURED, SEC 4]. Because ξ₁ < p_K, the uniform bound at depth K equals the uniform bound at the depth where p_K first exceeds ξ₁. In B6 the band sum ΣT1 is 557471.0, 478733.8, 418415.9, 383864.4, 378671.3 at K = 1, 2, 4, 8, 12 and then 378100.0 at every K from 17 to 46, while ΣX falls 41172 → 15540 over the same K; in B8 ΣT1 moves from 2840801.7 at K = 17 to 2827497.3 at K = 46, against ΣX 470118 → 197389. The exact tier T3 does keep falling with K (B6: 214190.2 → 188361.3; B8: 1890938.5 → 1662685.3) because its remainder carries the signs, but it stays 8.42 to 12.12 above X at K = 46. **The kill condition, answered with the numbers.** The brief asked whether "the only sieve that certifies has level below what the window allows so the bound is vacuous at the anchors". The bound is not vacuous: T1 < C at 35 of 40 anchors in B3 and at every anchor of B4 through B8 [SEC 3]. But the level the window allows is below z² at every band, so the theorem cannot follow X(K) down past p_K ≈ 31 (B6) or ≈ 61 (B8), and the gap to X opens with depth (T1/X = 9.18 → 24.33 across K = 17 → 46 in B6). That is the honest shape of the result: a theorem at a fixed low level, not a bound that tracks the census. ## 5. The measurement: the bound against exact X(K) at every anchor At the fixed band depths K_b = 4, 8, 12, 17, 23, 31 (the cited K* band means 3.88 → 31.22 rounded, `attack-quadpoint-03.js` SEC 2, used only as depths), band sums [MEASURED, SEC 2]: | band | n | ΣX | ΣS | ΣT1 | ΣT3 | T1/X | T2/X | T3/X | S/X | T1/S | |---|---|---|---|---|---|---|---|---|---|---| | B3 | 40 | 891 | 3915 | 6971.3 | 3990.7 | 7.82 | 5.50 | 4.48 | 4.39 | 1.78 | | B4 | 103 | 7035 | 28425 | 56241.1 | 30810.4 | 7.99 | 5.72 | 4.38 | 4.04 | 1.98 | | B5 | 71 | 7941 | 32194 | 68871.2 | 37753.1 | 8.67 | 6.28 | 4.75 | 4.05 | 2.14 | | B6 | 208 | 41172 | 167264 | 378100.0 | 214190.2 | 9.18 | 6.73 | 5.20 | 4.06 | 2.26 | | B7 | 292 | 101591 | 407883 | 968355.1 | 571510.6 | 9.53 | 7.00 | 5.63 | 4.01 | 2.37 | | B8 | 491 | 279040 | 1119016 | 2827454.6 | 1733352.8 | 10.13 | 7.42 | 6.21 | 4.01 | 2.53 | Per anchor [MEASURED, SEC 3]: T1/X median 8.78, 8.90, 9.19, 9.77, 10.22, 10.75 and max 20.00, 34.96, 15.41, 18.78, 16.86, 16.08 (B3 → B8); T3/X median 4.73 → 6.23. No anchor in any band has X(K_b) = 0, so every ratio is defined. **Reading, flat.** The uniform theorem is loose by 7.82 → 10.13 in band sums and the factor rises with Q. The larger part of it is the structural S/X = 4.39 → 4.01, fixed by the window (§0 item 1); the sieve's own share T1/S = 1.78 → 2.53 rises with Q because the level the window supports falls relative to z (s_max 1.96 → 1.85 from B4). The exact-remainder tier T3 reduces the sieve share (T3/S = 1.00 → 1.75 in B8 across K = 1 → 46, SEC 4) and cannot go below S/X. **On the reproduction target.** `attack-quadpoint-02.md` §4 fixes m6/m7/m8 = 0.050/0.040/0.032 and K* = 12.20 → 31.22 as what a proof-side bound must hit; `attack-roughpair-error.md` §6 shows a zero-parameter main term hits it to ratio 0.998 at B8 and calls the target necessary and far from sufficient. This note does not score against it, because K* is the depth where X crosses T and computing it is the comparison §3.1 forbids. What can be said without T: a bound loose by a factor 7.82 → 10.13 against X cannot reproduce any target that the census's own main term reproduces to 0.2%, and a bound that cannot follow X below p_K ≈ 31 to 61 cannot reproduce a depth statistic that lives at p_K = 139 in B8. So the theorem fails the reproduction target for a reason visible without ever comparing to T, and that failure is the expected outcome the brief named. ## 6. What the theorem does and does not say about the main term and the drift **The main term** [MEASURED, SEC 5]. The theorem's main term C·V₂(z) is the fundamental-lemma main term of the sifting function, and at the band depths it is accurate for S: S/(C·V₂) = 0.998, 0.999, 0.998, 0.999, 0.998, 0.998 (B3 → B8); along the K profile it reaches 1.016 at K = 46 in B6, the Buchstab correction beginning to show as u = ln h/ln z falls. The census's own zero-parameter main term (`attack-roughpair-error.md` §2, naive form) is C[V₂(K) − 2V₂(K)U(K) + V₂(nR)], and V₂(z) exceeds that bracket by 5.55, 5.19, 5.31, 5.27, 5.18, 5.20; the measured X/Xnaive is 1.261 → 1.307 (the Buchstab bias of the naive form that note reports for T as 26%, seen here on X). So the theorem proves the FIRST term of the census's main term as an upper bound with explicit error, and proves nothing about the two subtracted terms, which are lower-bound objects. The brief's "turn the main term into a theorem" is therefore met only for a main term 5.2 to 5.6 times the one the drift analysis uses. The composite- structure alternative (bound X by switching to the larger prime factor of a member) is worse before any remainder: its main term multiplier Σ_{z