# Attack 3 of 5: DP3, re-inserting exact Buchstab strata *(2026-08-18. Staging note. Target from `research/sift-limit-attack.md` §1 DP3 and §4.6. Companion computation: `research/attack-beta2-03-exact-strata.js` (`node research/attack-beta2-03-exact-strata.js`, about 25 s, full output pasted into the script). Legend as in `research/covering-dive.md`: **[PROVEN]** published theorem with source; **[VERIFIED]** checked computationally in this repository; **[MEASURED]** empirical, finite range; **[ABSENT]** searched and found nothing; **[INFERRED]** deduction from sourced facts.)* ## 0. The numbers, first | quantity | value | status | |---|---|---| | DHR dimension-2 sifting limit β₂ | 4.26645028414864191641 | [PROVEN], `research/dhr-verification.md` §1.1 | | **positivity threshold with EXACT strata, p ≤ 23** | **2.0260** (sifting-limit reading) | [VERIFIED] full period, this script §3 | | same, first-crossing reading (the operative certificate) | 1.9264 | [VERIFIED] full period, unit-exact | | positivity threshold with EXACT strata, p ≤ 19 | 2.0028 / 1.8160 | [VERIFIED] full period | | the truth, log G₂/log p_k, p ≤ 23 | 1.6961 | [VERIFIED] recomputed here | | same bound, strata priced at the envelope instead | 3.85 | [MEASURED] surrogate, §4b | | **exponent that the exact pricing of DP3 is worth** | **1.72 to 1.84** | [MEASURED] | | decoupling excess D(H), the whole remaining cost | 1 to 15 elements | [MEASURED] | **Normalisation, because everything depends on it.** u = log H / log p_k with p_k the **largest sifted prime**, the convention of `research/exact-g2-ladder.js` and `paper/beta2-note.md` (G₂(n) ≪ pₙ^{β₂+ε}). DHR's own parameter uses the sifting bound z = p_k + 1; at p_k = 23 that is 1.4% in the log and it vanishes asymptotically. Using the log of the NEXT prime instead would move u by 7%. **The verdict in one line.** The strata ARE exactly smaller instances, the transformation law is now written down, exact re-insertion moves the threshold from 4.2665 to about 2, and it is **not a theorem and cannot be made into one by this route**: it fails on an infinite regress in the asymptotics, not on circularity, and the finite certificates it does produce are strictly weaker than the enumeration that produced G₂ in the first place. --- ## 1. What a discarded stratum IS ### 1a. Stratum Dilation Lemma [PROVEN here, VERIFIED at eight levels] Write T(y) for the tile sifted by all primes ≤ y, of period Q(y), and T⁻, Q⁻ for the tile and period at the primes strictly below p. The Buchstab stratum is S(A_p, p) = #{r in the window : p | r(r+2) and no q < p divides r(r+2)}, which splits by the two classes p | r and p | r+2. > **Stratum Dilation Lemma.** For every odd sifting prime p, as subsets of > Z/(p·Q⁻): > > - { r ≡ 0 (mod p), r ∈ T⁻ } = p · (p⁻¹ · T⁻) > - { r ≡ −2 (mod p), r ∈ T⁻ } = p · (−p⁻¹ · T⁻) − 2 > > where p⁻¹ · T⁻ is again a two-class tile at level p⁻, cut by the class pair > {0, −2/p mod q} at every q < p, and −p⁻¹ · T⁻ by {0, +2/p mod q}. *Proof.* Take the first branch. r = p·m, and for q < p we have q ∤ p, so q | r(r+2) iff q | m or q | pm+2 iff m ≡ 0 or m ≡ −2p⁻¹ (mod q). So the m that survive form the two-class tile with class pair {0, −2p⁻¹}. Multiplying that tile by p sends {0, −2p⁻¹} to {0, −2}, which is T⁻, so the tile is p⁻¹ · T⁻. The second branch is the image of the first under m ↦ −m. ∎ **Consequences, and they are the whole answer to "is it exactly a smaller instance of the same problem".** 1. **Yes, exactly.** A discarded stratum is two two-class interval sifts of length H/p at level p⁻, with the same dimension κ = 2 and the same local densities ω(q) = 2, ω(2) = 1. Nothing about the problem class changes. 2. **The recursion is closed and its action is division.** The class-offset vector transforms as c ↦ c/p. Starting from the twin tile c_q ≡ −2, the family generated is exactly { c_q ≡ −2/m mod q } indexed by squarefree m built from sifting primes, and every member is the multiplicative dilate m⁻¹·T of the twin tile. This is `paper/moire-primes.md`'s Redundancy Lemma (the kill image p × holes) stated for the two-class problem, and it upgrades `sift-limit-attack.md` §1's "the strata are dilated kill images" from a description to a transformation law. 3. Verified as an exact set identity at p = 3, 5, 7, 11, 13, 17, 19, 23, together with the Buchstab identity itself at six (x, H, j) spot checks including x = 223,092,869. [VERIFIED] ### 1b. Where the analogy breaks, precisely The dilation m⁻¹ is a **bijection of Z/Q carrying slots to slots and destroying intervals**. The self-similarity is exact for the tile as a set with congruence structure, and false for the tile as a set with order. Every question a sieve asks about a stratum is an order question: how many survivors sit in a window. So the exactness transfers no interval statistic by fiat. It has to be measured, and it was. **Measured, at level 19, over the full period** [VERIFIED]: - **From above the family is uniform.** Max window count, the quantity a lower-bound sieve needs for its strata, spreads at most about 15% across 40 dilates, and the twin tile sits sometimes at the top of that range (u = 1.2, 1.7) and sometimes at the bottom (u = 1.5, 2.0). - **From below it is not.** Max gap: the twin tile is the family **minimum**. Of 10 sampled dilates none has a smaller max gap and 9 have a larger one, running to 198 against G₂ = 150 (+32%). The dilates the recursion actually produces, 23⁻¹ through 43⁻¹, give 168 to 198. So exact re-insertion of stratum **upper** bounds is family-robust. Anything resting on the lower side of the family is not, and any attempt to recurse the *retained* part of the identity pays that 32%. **Ladder provenance.** The script recomputes the census ∏(p−2) and the max gap from scratch at all nine levels it uses (x ≤ 23) and both match the ladder. The terms x = 47..79 appear only as a reference table; their maximality is established in `attack-beta2-05-covering-pruning-bound.md`, not here. --- ## 2. The instrument `research/attack-beta2-03-exact-strata.js`. Primes p₁ < … < p_k, T_j the tile sifted by p₁..p_j, N_j(x,H) = |T_j ∩ (x, x+H]|, Str_i(x,H) the count of r in the window whose least prime factor of r(r+2) is exactly p_i. Buchstab is then the exact partition N_k = N_j − Σ_{i>j} Str_i. **The cut family.** For every j, > min_x N_k(x,H) ≥ min_x N_j(x,H) − Σ_{i>j} max_x Str_i(x,H) is a valid all-positions lower bound. j = k is the untruncated identity. j = 0 retains nothing and prices every stratum at its **exact maximum over the full period**. j interpolates, so the family is precisely the "Buchstab iteration with exact low strata" that `sift-limit-attack.md` §4.6 says does not exist. Every max and min is taken over a complete period (Q = 223,092,870 at p ≤ 23), so nothing here is sampled. The census reproduces the Schemmel product ∏(p−2) and the max gaps reproduce the G₂ ladder at all nine levels, which is the instrument's self-test. The DHR (F₂, f₂) system is integrated from the Booker-Browning α, β by the method of steps and both functions converge to 1 within 2.7·10⁻⁵, which is the solver's self-test. **The predicate is scanned, not bisected**: the bound is not monotone in H. --- ## 3. The thresholds u*(0), every stratum priced at its exact maximum, nothing retained [VERIFIED, full period]: | p_k | 7 | 11 | 13 | 17 | 19 | 23 | |---|---|---|---|---|---|---| | u*, first crossing (operative) | 1.7479 | 1.7835 | 1.9037 | 1.8209 | **1.8160** | **1.9264** | | u*, stable (sifting-limit reading) | 2.1372 | 1.9448 | 1.9638 | 1.9812 | **2.0028** | **2.0260** | | u_true = log G₂/log p_k | 1.7479 | 1.5587 | 1.6334 | 1.6526 | 1.7017 | 1.6961 | Two readings, because the bound is not monotone in H and the scan reports both. The **first crossing** is the operative one for a gap bound: positivity at a single H already proves G₂ ≤ H. The **stable** value, the least H past which the bound never fails again, is the sifting-limit reading and the one comparable to β₂. The grid is every integer to 700, so every first crossing above is unit-exact. **Against β₂ = 4.26645 the exact-strata threshold is lower by 2.24.** It sits at the TPC-equivalent line 2, the gap to the truth is flat at 0.30 to 0.39, and the stable threshold **rises monotonically** from p_k = 11 on: 1.9448, 1.9638, 1.9812, 2.0028, 2.0260, crossing 2 at p_k = 19. **It is not trending below.** **What the certificates actually assert.** G₂(19#) ≤ 210 and G₂(23#) ≤ 420, against true values 150 and 204. Valid, independently checkable, and a factor 1.4 to 2.1 weaker than the enumerations that produced the ladder. **The cut ladder, p ≤ 23** (stable reading), which is where the shape of the loss shows: | j | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |---|---|---|---|---|---|---|---|---|---|---| | u*(j) | 2.0260 | 2.0260 | 1.9994 | 1.9958 | 1.9921 | 1.9484 | 1.8928 | 1.8877 | 1.7994 | 1.6961 | Row j = 9 is the identity and returns H* = G₂ = 204 exactly, at all six levels tested. **Retaining the single largest prime exactly (j = 8 → 9) buys 0.103 of exponent; retaining the first five primes exactly buys 0.078 in total.** The exactness that pays is the exactness whose sub-problem is nearly the whole problem. Cheap low strata are worth almost nothing. --- ## 4. Where the loss actually sits **4a. With exact strata the entire remaining cost of DP3 is O(1).** Because Σ_i Str_i(x) = H − N_k(x) identically, the only loss left in the j = 0 bound is that the maxima are taken at independent positions: D(H) := Σ_i max_x Str_i(x,H) − H + min_x N_k(x,H) ≥ 0, and the bound is positive exactly when min_x N_k(H) > D(H). Measured D runs **1 to 15** over everything computed (H from G₂ to 30·G₂, p_k from 13 to 23), growing slowly in both H and p_k. Four levels is not a trend line, but the order of magnitude is unambiguous: **DP3 with exact strata costs a handful of elements, where DP3 with the envelope costs a factor on every stratum.** [MEASURED] **4b. Same bound, only the pricing swapped (stable reading).** | p_k | 13 | 17 | 19 | 23 | |---|---|---|---|---| | u*, exact strata | 1.9638 | 1.9812 | 2.0028 | 2.0260 | | u*, strata at main·F₂(u_i) | 3.6892 | 3.6963 | 3.8459 | 3.8452 | | **difference = the price of DP3** | **1.725** | **1.715** | **1.843** | **1.819** | The envelope column applies F₂ pointwise and is a **surrogate, not a rigorous finite-z bound**: Theorem 9.1 carries an o(1) of size O((log log y)²/(log y)^{1/6}) which is not small at these z, and this one-step decoupled bound is not DHR's truncation. Do not quote 3.85 as "what DHR gives"; the rigorous asymptotic answer for the method remains 4.26645. What the pair is good for is the **difference**, which isolates the stratum pricing and nothing else. **4c. Per-stratum overpayment at the working point** (p ≤ 19, H = 19²) [MEASURED]: | p_i | main_i | max_x Str_i | exact ratio | u_i | F₂(u_i) | envelope / exact | |---|---|---|---|---|---|---| | 7 | 10.31 | 12 | 1.163 | 2.450 | 4.249 | 3.65× | | 11 | 4.69 | 7 | 1.493 | 1.794 | 7.885 | 5.28× | | 13 | 3.25 | 6 | 1.849 | 1.386 | 13.207 | 7.14× | | 17 | 2.10 | 5 | 2.381 | 1.191 | 17.882 | 7.51× | | 19 | 1.66 | 4 | 2.413 | 1.039 | 23.495 | 9.74× | The overpayment concentrates entirely on the deep strata, at u_i near 1. That is the DP3 loss localised. --- ## 5. Is re-insertion legitimate? Three different answers **5a. FULL re-insertion is a tautology, and the script proves it on itself.** Row j = k returns H* = G₂ exactly at every level. Buchstab with every stratum exact IS the identity N_k = N_k. Anyone reporting "exact strata give exponent 1.70" has reported G₂ back to itself. This is the trap the brief asked about, and it is real: it fires at j = k and only there. **5b. PARTIAL re-insertion (j < k) is NOT circular and IS a valid theorem at each finite z.** Every input is a strictly smaller problem: min_x N_j is at level j < k, and max_x Str_i is a shorter window at level p_{i−1} < p_k. The computation terminates. "G₂(19#) ≤ 210" is true and independently checkable. There is no hidden assumption of the conclusion. **5c. As a proof technique it fails on an infinite REGRESS, not on circularity.** To turn u*(0) into a statement for all z you need an asymptotic upper bound on max_x Str_i(x,H) at u_i near 1. By §1a that quantity IS the upper sieve function of a dilated tile at the same parameter one level down. The only proven bound for it is F₂, which is exactly what the exactness was brought in to replace. The regress terminates at every finite z and at no uniform z. **That is precisely why the finite certificates exist and the theorem does not**, and it is the sharp form of §4.6's "nobody has a way to propagate exactness through the recursion without reverting to worst case at the next depth". **5d. The cost accounting is also against it.** Producing max_x Str_i over the full period is a computation of the same order as enumerating G₂ directly, and returns a weaker answer (210 against 150). As a computational device the route is dominated by what the repo already runs. --- ## 6. What would have to be true asymptotically Stated as sharply as the data allows. Define F_meas(y,u) = max_x count / (L·V(y)) at L = y^u, the exact upper sieve function of the natal tile over its full period. The re-insertion becomes a theorem if and only if F_meas(y, u) is bounded by an explicit constant C(u) < F₂(u), uniformly in y, at the parameters u ≈ 1 to 2 where the strata sit. Measured [MEASURED, six levels]: | u | y=7 | 11 | 13 | 17 | 19 | 23 | F₂(u) | |---|---|---|---|---|---|---|---| | 1.2 | 2.800 | 1.901 | 2.758 | 2.292 | 3.014 | 3.262 | 17.623 | | 2.0 | 1.714 | 1.414 | 1.436 | 1.427 | 1.490 | 1.538 | 6.344 | | 3.0 | 1.102 | 1.067 | 1.040 | 1.040 | 1.049 | 1.042 | 2.916 | At u = 3 the ratio is flat near 1.05, as it must be. **At the parameters that matter it rises with y**, and the envelope-to-truth ratio F₂/F_meas closes monotonically at u = 2: 4.49, 4.42, 4.44, 4.26, 4.13. Six points cannot decide whether F_meas converges to a constant below F₂ or climbs toward it. They do decide that **no constant-factor re-insertion is supported by the data in the accessible range**: the quantity to be re-inserted is not observed to be stable where it is needed. The companion lower-side requirement, from §1b: the same bound would have to hold across the whole dilate family m⁻¹·T, uniformly in m. From above that looks plausible (15% spread). From below the twin tile is the family minimum, so any argument that recurses the retained part starts by giving back 32%. --- ## 7. Literature (sweep run 2026-08-18 for this attack) - **β₂ is already the limit of Buchstab iteration inside the axiom class.** Blight, *Refinements of Selberg's Sieve*, Rutgers PhD thesis, p. 8: "The Diamond-Halberstam sieve is an infinite iteration of the Ankeny-Onishi sieve." [PROVEN, quoted from the thesis]. **This decides the shape of the attack**: no rearrangement of the truncation can beat 4.2665, so any movement must come from information outside the axioms. Exactness is such information, which is why it moves the number at all, and §5c is why it does not move it into a theorem. - **4.2665 is not the axiom-class truth.** Blight §2.2: β_κ is "the greatest lower bound of the β_{κ,Λ} over all possible lower bound sieves Λ", and "Selberg proposed that the sifting limit is 2κ … a lower bound sieve with a sieving limit of 2κ has not been found for κ > 1". So the conjectural axiom limit at κ = 2 is **4**, and DHR's 4.26645 is an upper bound for it. This refines `sift-limit-attack.md` §2, which reports the same 2κ plateau from Franze but does not carry Blight's statement of it. [PROVEN as quoted] - **Closest published relative, and it is live.** Brady, *Sieves and iteration rules*, Stanford PhD 2017; Runbo Li, *A note on variants of Buchstab's identity*, arXiv:2504.07974 (2025). These **retain** more strata with fractional coefficients instead of discarding them at 0 / F, proved from pointwise inequalities such as 1 − ((m₁+m₂−1)/m₁m₂)n + (2/m₁m₂)C(n,2) ≥ 0. Li §4, verbatim: "these may be helpful in bounding the 'sifting limits' β_κ for κ > 1 … We hope someone can accomplish this work." That is an explicitly open problem sitting at DP3, approached by inequalities rather than exact values. **[PROVEN that it is open, as quoted]** - **Harman's method is not exact re-insertion.** It evaluates some decomposition pieces asymptotically by comparison against a model set, licensed by Type I and Type II arithmetic input. That is DP4 information, which `sift-limit-attack.md` §5 establishes is already saturated for our sequence at no cost. [PROVEN mechanism, INFERRED verdict] - **Chen's switching principle re-classifies survivors, it does not re-evaluate a stratum.** [PROVEN via secondary sources] - **No published sieve prices a discarded stratum exactly.** [ABSENT, as far as this sweep reaches] `sift-limit-attack.md` §4.6's absence claim survives a second independent search. - **The regress observation of §5c is not in the literature either.** Tao's "Open question: the parity problem in sieve theory" (2007) frames the obstruction as 2^k term explosion and Liouville correlations, not as self-reference of the strata. [ABSENT, as far as this sweep reaches] One custody note. This sweep failed to re-find Booker-Browning as the source of the 20-digit β₂ and attributed it to the DH book p. 227. `research/dhr-verification.md` §1.1 quotes the Booker-Browning ancillary table verbatim from a direct fetch, with the URL. **The direct fetch stands; the sweep's miss is a miss.** Recorded here so it is not rediscovered as a contradiction. --- ## 8. Corrections and additions to the record Not applied. `research/attack-beta2-03-exact-strata.js` is the only file this attack wrote. 1. `research/sift-limit-attack.md` §3, the last table row, currently reads "Self-similar strata / kill image (PROVEN, Redundancy Lemma) | DP3 | none | no known mechanism; **not provably useless**". It is now measured. Proposed replacement for the verdict cell: *"measured 2026-08-18: exact re-insertion moves the threshold to 2.026 at p ≤ 23 and is not a proof technique (infinite regress, `research/attack-beta2-03-exact-strata.js`)"*. 2. `research/sift-limit-attack.md` §4.6 ends "the missing lemma class is 'Buchstab iteration with exact low strata' … nobody has that". The cut family of §2 above is that object, built and run. The sentence should point at it and record that the value is concentrated at the LARGEST primes, which is the reason nobody has it. 3. `research/sift-limit-attack.md` §1 DP3 says "the strata are dilated kill images". The Stratum Dilation Lemma of §1a above is the statement with its proof and its transformation law c ↦ c/p; §1 should cite it. 4. `research/sift-limit-attack.md` §2 should carry Blight's two quoted facts (DHR = infinite Ankeny-Onishi iteration; β_κ defined as the infimum over all lower-bound sieves, with Selberg's 2κ unattained for κ > 1). The first is the cleanest available statement of why DP3 depth is exhausted. 5. `research/GLOSSARY.md`'s "Kill image" entry could gain the two-class form: the strata of the twin tile are its multiplicative dilates m⁻¹·T, and the dilate family is uniform from above (15%) and not from below (+32% max gap). --- ## 9. Verdict **The attack returns a sharp negative with a measured price tag, which is the outcome the brief called a full success.** The strata are exactly what `sift-limit-attack.md` said they were, and the proof is three lines. Re-inserting them exactly is legitimate, non-circular, and worth 1.72 to 1.84 of exponent, taking the positivity threshold from 4.2665 to 2.03. It produces true certificates at every finite level. It does not produce a theorem, and the reason is now precise rather than vague: **the exact stratum value is the upper sieve function of a dilated tile one level down, so the only proven bound for it is the very envelope the exactness was replacing.** The regress terminates at every finite z and at no uniform z. Two independent readings confirm the closure from the other side: the value of exactness is concentrated at the largest primes, where the sub-problem is nearly the whole problem; and Blight's identification of the DHR sieve as the infinite iteration of Ankeny-Onishi says the axiom class has no depth left to give. The one door this attack leaves open is not exactness. It is Brady and Li's fractional retention rules, which sit at DP3, are inequality-based rather than exact (so they carry no regress), and are on the record as an open problem for κ > 1. That is a different attack. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*