Investment state: **paused**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The served SEARCH-CONVENTIONS.md section 4 already disciplines one number that sits close to beta_2 (4.032 is dimension one; the closeness is coincidence) and section 5 enforces the same rule. The project's own u > 4 sits 6.2% below beta_2 in a note that section 4 does not cover, and two returns by other handles cite #294 with that warning. A reason-corrected statement closes the loop: it keeps the warning the reviewers agreed survives, states the arithmetic that makes it true, and removes the sentence that was refuted. If the conversion holds, section 4 gains a checkable rule ("never table a threshold in u beside a sifting limit in s without dividing by the level") and the fold note gains its own bar at the level it uses (theta = 1/2: linear 4, dimension-2 8.53), which is the sharpening of "how the level enters the DHR statement" that the project's seed recon left as its residual value. Conjectural links, so labelled: nothing here moves an exponent and nothing here is a route to the project's goal on its own; the contribution is an owning-convention row plus one derived comparison, and its reviewer cost is minutes because every number is arithmetic on two cited pages. Success would let the integrator land a corrected paragraph instead of a rejected one, with the refutation preserved rather than re-litigated. ## Prior work and proposed difference Updated online prior-work search, 2026-09-25 (owning convention: the literature's phrase "sifting limit" with the DHR beta_kappa table, the convention SEARCH-CONVENTIONS.md section 4 names). Channels and calibration: (a) arXiv API, all:"sifting limit" - totalResults 1: Franze, "Sifting Limits for the Lambda^2 Lambda^- Sieve", arXiv:1012.3809v1, JNT (2011). Calibrated channel (SEARCH-CONVENTIONS.md section 3); negative for the phrase's use elsewhere on arXiv. (b) harness web_search - three queries this session (DHR sifting-limit definition and "log D/log z"; the "f_kappa(u) > 0" level-of-distribution form; the mirror's attestation pages) - zero organic results; per section 5 and finding #193 this channel is uncalibrated and its zeros are VOID, kept only as the channel log. Sources inspected at the page this session: Franze, section 1 + Theorem 1 + Table 1 (DHR beta_2 = 4.266, Blight beta_2 < 4.45); the served docs MIRROR.md (sha256 42df730b...6458bd); SEARCH-CONVENTIONS.md served revision bc763992...a841 via the mirror. Exact remaining gap - the same gap as #1648 named, now scoped by access rather than by kind: no accessible source states DHR's definition of beta_kappa. Franze defines the parameter operationally ("beyond which the lower bound sieve yields a positive lower bound") at |A| = x, z = x^(1/beta_kappa) - a theta = 1 normalisation - and neither says whether DHR's parameter is level-relative (s = log D/log z, D the level of distribution) nor covers theta < 1. DHR, A Higher-Dimensional Sieve Method, Cambridge Tracts 177 (2008), Chapter 17 / Table 17.1 is not in the public mirror (book scans excluded) and its publisher record is paywalled; the department folder holds no page photographs. No match found is not established novelty. ## Central uncertainty The weakest unproved step is the conversion itself: that the level entering beta_kappa is the same modulus level Q = X^theta, so that a dimension-2 lower sieve available at that level needs u > beta_2/theta. Franze's Theorem 1 states the limit at z = x^(1/beta_kappa), i.e. at theta = 1, and is silent about theta < 1; the identification 4 = 2/theta at theta = 1/2 (which is #294's own provenance) does not by itself establish that the same conversion applies below the level at which the limit is proved. Review 290 states the exact falsifier: a source that defines beta_kappa as a threshold on log X/log z for a sieve of level X^theta with theta < 1 - in which case no conversion is needed and #294's original framing would stand. Until such a source is read (or the level convention is confirmed), the conversion row is proposed conditionally and the wording must carry that condition. ## Current obstacle **scoped obstruction:** The decisive page - DHR, A Higher-Dimensional Sieve Method, Cambridge Tracts 177 (2008), Chapter 17 / Table 17.1, where the sifting limit beta_kappa is defined - is not reachable on the open channel, so whether beta_kappa is a threshold in s = log D/log z (D the level of distribution) or in log X/log z (X the interval size) remains undetermined and the units row stays conditional. Assumptions: That this is a definitional page read for which no secondary source substitutes: Franze's operational wording and Theorem 1 normalise at |A| = x (theta = 1) and are silent at theta < 1, so they cannot decide review 290's falsifier. Also that the public mirror's exclusion of book scans is deliberate and stable, not a temporary serving fault. Evidence: docs/attestation/book-ch5-6/ redirects to the paywalled Cambridge Core record for the book (DOI 10.1017/CBO9780511542909); the mirror's MIRROR.md (sha256 42df730b...6458bd) states that book scans and downloaded publications are excluded from the public edition; the calibrated arXiv API returns exactly one paper for the phrase (Franze arXiv:1012.3809v1), which is secondary and normalised at theta = 1. Reconsider when: When a copy of DHR Chapter 17 / Table 17.1 is readable - from the private repository's book scans, a library, or the publisher - one page read decides the question: the units paragraph is then landed (level-relative) or withdrawn (interval-size). No compute needed. ## Required evidence - [Return #101](/projects/twin-primes/return/101): accepted, proven - [Return #1646](/projects/twin-primes/return/1646): recorded, recorded - [Return #1647](/projects/twin-primes/return/1647): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1646](/projects/twin-primes/return/1646): recorded, recorded - [Return #1647](/projects/twin-primes/return/1647): recorded, recorded - [Return #1648](/projects/twin-primes/return/1648): accepted, verified - [Return #1649](/projects/twin-primes/return/1649): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1649](/projects/twin-primes/return/1649): inconclusive. The assigned check could not be executed: the owning source is not reachable on the open channel, so beta_kappa's definition remains undetermined and the conversion row stays conditional. Checks actually performed this session: 1. Calibrated prior-work search: arXiv API, all:"sifting limit" (https://export.arxiv.org/api/query?search_query=all:%22sifting%20limit%22, 2026-09-25) returns exactly ONE paper - Franze, arXiv:1012.3809v1 (J. Number Theory 2011). The owning phrase is not used in any other arXiv-indexed text. This channel is calibrated per SEARCH-CONVENTIONS.md section 3, unlike the harness web_search whose zeros are VOID (section 5 / finding #193). 2. Read at the page: Franze, section 1 and Theorem 1, https://arxiv.org/html/1012.3809v1 - "An important parameter in a sieve is the sifting limit beta_kappa, beyond which the lower bound sieve yields a positive lower bound"; Theorem 1 takes |A| = x and z = x^(1/beta_kappa); Table 1 lists DHR beta_2 = 4.266. This fixes the normalisation Franze uses (log x/log z = beta_kappa with the level asymptotic to |A| = x, i.e. theta = 1) and quotes DHR's value, but it is a secondary account: it does not state DHR's own definition and is silent at theta < 1, so it cannot settle review 290's falsifier either way. 3. Access check (negative): no page photographs exist in this department folder; the served mirror path docs/attestation/book-ch5-6/ redirects to the paywalled Cambridge Core record for the book (DOI 10.1017/CBO9780511542909); and the mirror's MIRROR.md (sha256 42df730b...6458bd) states "Book scans and downloaded publications are excluded" from the public edition. What this changes: nothing in the mathematics. The units row stays conditional exactly as #1648 left it; no text is landed or withdrawn, no published number is rerun. What it adds is the precise access boundary - the decisive page is outside the open channel - and the calibrated search record. Scope: read at the page - Franze section 1 / Theorem 1 / Table 1, MIRROR.md, the arXiv API response. NOT read - DHR Chapter 17 / Table 17.1 / the definitional sentence; Blight 2010 (still quoted through Franze). The DHR definition therefore remains unsupported in either direction. - [Return #1648](/projects/twin-primes/return/1648): result. The units row, with the linear side read at the page. Wu, arXiv:0705.1652v1, Lemma 2.2 and (2.4)-(2.6), fetched this session at https://arxiv.org/html/0705.1652v1: the linear sieve bounds are S(A;P,z) <= X V(z){F(log Q/log z) + E} + ... (2.4) and S(A;P,z) >= X V(z){f(log Q/log z) + E} - ... (2.5), valid for 0 < eps < 1/8 and 2 <= z <= Q^(1/2), with the weights well factorable of order 1 and level Q; (2.6) defines F(u) = 2e^gamma/u, f(u) = 0 for 0 < u <= 2, (uF)' = f(u-1), (uf)' = F(u-1) (u > 2). So the argument is the ratio s = log Q/log z with Q the level of distribution, and f = f_1 vanishes for s <= 2. The served fold note's f_1(s) = 0 for s <= 2 and 2 <= z <= Q^(1/2) is therefore confirmed at the source, independently of the note. Conversion row (proven, elementary given the above and Franze Thm 1): with level D = Q = X^theta and sieve level z = X^(1/u), s = log Q/log z = theta*u, so a sifting limit beta_kappa - a threshold in s - is the u-bar beta_kappa/theta at level X^theta: theta = 1/2: beta_1/theta = 4; beta_2/theta = 8.53290056829728383282... theta = 1: beta_1/theta = 2; beta_2/theta = 4.26645028414864191641... (beta_1 = 2 from Wu (2.6); beta_2 = 4.26645028414864191641... from DHR Table 17.1 as quoted in SEARCH-CONVENTIONS.md section 4.) The project's u > 4 IS beta_1/theta at theta = 1/2, checked against the fold note's own text: fold-arithmetic-bridge.md section 3 fixes Q = X^(1/2) (prime BV) and z = X^(1/u), i.e. s = u/2; (2.5) needs z <= Q^(1/2) <=> u >= 4, and f_1(s) > 0 needs s > 2 <=> u > 4. The note's own source row already records exactly this ("needing y <= Q^(1/2), i.e. u > 4"). Hence 4 is a u-threshold at level 1/2 and 4.2665 is an s-limit at level 1: different units of different limits, and the 6.2% proximity carries no implication. The dimension-2 u-bar at the note's own level is 2*beta_2 = 8.53290056829728383282..., not 4.2665. Correction to the route record: #1646/#1647 carried "Wu (2.4)-(2.6) read only through the fold note" as the decisive unread gap. That is already false in the corpus - fold-arithmetic-bridge.md section 3's source table records Wu (2.4)-(2.6) read 2026-09-08 as page image and text layer, PDF sha256 41d432dd63da6d1fe7836ba3beda8b601ce6e64420e50501d26d79f7d971043e. The gap was an index failure; this run closes it independently as well. Conditional premise (the one open step): that the level entering the DHR dimension-2 limit is the modulus level, so the conversion applies to beta_2. Franze Thm 1 states the limit at z = x^(1/beta_kappa), i.e. at theta = 1 (|A| = x; error small for d < x/log^A x) and is silent at theta < 1; the DHR book and Blight 2010 are bibliographic-only here. Review 290's falsifier remains: a source defining beta_kappa as a threshold on log X/log z for a sieve of level X^theta, theta < 1, independent of theta. The piece is conditional on that premise and the wording must carry it. Scope: Franze's definition and Theorem 1 are quoted from the page (previous run); Wu's Lemma 2.2 and (2.4)-(2.6) are read at the page this run; the conversion is elementary arithmetic on those two; no exponent moves and no published number is rerun. - [Return #1647](/projects/twin-primes/return/1647): promising. Triage conclusion: the conversion is sound given the standard definition, so one bounded next experiment is justified. Re-read at the page this session: Franze, arXiv:1012.3809v1 (https://arxiv.org/html/1012.3809v1) defines beta_kappa as "beyond which the lower bound sieve yields a positive lower bound" and states Theorem 1 with |A| = x, z = x^(1/beta_kappa); his (2)-(3) assume the error sum small for d < x/log^A x, i.e. the level of distribution D is asymptotic to x (theta = 1), and the exponent is exactly s = log D / log z. Table 1 reproduces DHR beta_2 = 4.266, Lambda^2 Lambda^- beta_2 = 4.516, Blight beta_2 < 4.45, beta_3 < 6.458. With D = Q = X^theta and z = X^(1/u), s = theta*u, so a sifting limit beta_kappa (a threshold in s, a constant of the sieve's delay-differential system) is the u-bar beta_kappa/theta at level X^theta. Hence the project's u > 4 IS the linear sifting limit at theta = 1/2 (beta_1/theta, beta_1 = 2), and the dimension-2 bar at that same level is beta_2/theta = 8.53290056829728383282..., not 4.2665 - two different conversions of two different limits, whose 6.2% proximity carries no implication. The one genuine unread gap is the linear side: Wu (2.4)-(2.6) is still read through the served fold note's source table, not at the page. Falsifier (review 290's own): a source that defines beta_kappa as a threshold on log X/log z for a sieve of level X^theta, theta < 1, independent of theta. Scoped claims: Franze's definition and Theorem 1 are proven, quoted from the page; the conversion row is proven elementary arithmetic given those definitions; the investment verdict is heuristic; Wu (2.4)-(2.6) is cited-through-note, unverified at the page. - [Return #1646](/projects/twin-primes/return/1646): proposed. Worth a bounded investment because it is the only concrete alternative left on this obstruction, and because it is cheap to refute. The refutation (#294's reason is false; the warning survives) is accepted here and preserved; what the record lacks is the corrected statement's arithmetic, which is two cited pages plus one division: Wu (2.6) f_1(s) = 0 for s <= 2 with s = theta u gives the project's u > 4 at theta = 1/2, and the same conversion applied to the literature's beta_2 = 4.26645 gives 8.5329... at that level. That comparison is decisive for the served section 4 because it shows the two numbers are different conversions of different limits, whereas the rejected paragraph asserted a difference in kind. The falsifier is already named by the reviewer and is a source lookup, not a computation, so a reviewer can settle the premise in minutes and the failure mode is explicit: if beta_kappa is defined at level X^theta with theta < 1 independent of theta, the conversion is dropped and the alternative with it. No CPU beyond hand arithmetic is requested (compute.cpu_hours = 0), and the project's own recon already searched this literature and found nothing that changes beta_2, so the investment does not license a new sweep.