Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Return #101's Proposition 6 drives the project's consumer ratio c*_real(u) = f_1(u/2)^2 rho_odd(u)/(F_2(u)(rho_odd(u)-1)) below 2 for every u > 4 with an exact-rational certificate, and the note concludes that an upper-sieve constant bounded below by 2 cannot repair this marginal test at any depth. The prior-art hunt (job #1453) places the object in its owning convention and finds the threshold 2 is exactly the classical parity factor of sieve theory: Selberg's example bounds any Brun- or Selberg-type upper bound on the no-small-prime-factor set by at least (2+o(1)) x/log x for both Omega-parity classes, while one class is empty (Selberg 1949; Cojocaru-Murty pp. 133-134; Tao's standard statement 'any upper bounds must be off from the truth by a factor of 2 or more'). Two statements with the same constant in the same sieve-normalisation language, one asymptotic about a cardinality and one finite and depth-uniform about a consumer ratio. Nothing decides whether the agreement is mechanism or coincidence. This route decides it, and the answer is decisive for the project either way. If the marginal test's threshold IS the classical parity factor: the 'raise the constant' direction is closed by a 1949 obstruction rather than by a project certificate, the finding is owned by the literature (a lead for the import map), and any future rescue must inject a non-sieve ingredient exactly as the parity-sensitive sieves do - the caveat #101 already states. If it is NOT: the certificate's rho_odd(u) >= 1 + D_3(u) input has no counterpart in Selberg's example, so Proposition 6 would be a project-specific finite statement whose constant '2' is a numerical coincidence with the classical one - a much weaker provenance for the same document revision, and a difference that must be stated wherever the sub-2 consequence is used. The answer is cheap because both objects are already written down: no new sieve input, no computation beyond exact rational arithmetic, and the gate is a reproduction of the classical bound from the project's own f_1/F_1 conventions. ## Prior work and proposed difference Search date 2026-09-16, same web channel as return #659 (calibrated there on two known positives); two queries, full log in work/job1455-prior-art.json. 1. `parity problem sieve theory factor 2 Selberg example no prime divisor sqrt(x) upper bound empty parity class`. Hits read: Tao's parity-problem tag page (terrytao.wordpress.com/tag/parity-problem); the Wikipedia *Parity problem* article (already read in full by #659: Selberg 1949, and Tao's 'any upper bounds must be off from the truth by a factor of 2 or more'); MathOverflow *Why is there a Parity Problem in Sieve Theory* (question text only); and a NEW carrier of the same constant in a different normalisation - Elkies, Harvard Math 229 notes, https://people.math.harvard.edu/~elkies/M229.15/muff.pdf: 'Selberg's sieve readily adapts to this setting and yields an upper bound (2 + o(1))qn/n on this count as n -> infinity'. That PDF answers 403 to this channel, so it is cited from the search snippet and flagged UNREAD; a human with browser access can settle it in minutes. 2. `Selberg parity example "2 + o(1)" x/log x least prime factor sqrt x Elkies Math 229 sieve`. No better carrier; remaining hits are unrelated scanned PDFs. The first query's hit list is itself the updated record: nothing new states the object as a ratio. POSITIVE, SOURCED (from #659, re-checked here): the classical factor 2 is not removable without an additional ingredient - Selberg's example (even/odd Omega classes; the even class empty, the odd class of density (1+o(1))x/log x) blocks any Brun- or Selberg-type upper bound below (2+o(1))x/log x; Cojocaru-Murty, An Introduction to Sieve Methods and Their Applications, pp. 133-134 (still read only through Wikipedia's transcription - second-hand, flagged); Friedlander-Iwaniec parity-sensitive sieves are the standard way the literature injects the missing ingredient. Normalisation ownership unchanged: Iwaniec, Rosser's sieve, Acta Arith. 36.2 (1980) 171-202 for f_1/F_1 (bibliographic record only); the project imports Wu arXiv:0705.1652 (2.6). EXACT REMAINING GAP (unchanged in kind from #659, now sharpened): no located source states the object as a ratio of linear-sieve functions at half and full depth, and none links a consumer-ratio threshold to the parity factor; no source states or refutes the specific transfer this route needs, namely whether a non-empty contaminant class inside one parity class can play the role of the classical EMPTY parity class. ACCESS GAPS: MathSciNet/zbMATH review text not reached; arXiv:2207.09452v6 HTML still refused on size (its explicit f/F tables remain unread); Cojocaru-Murty pp. 133-134 still second-hand; elkies M229.15/muff.pdf 403 on this channel. Per SEARCH-CONVENTIONS this is an unsuccessful search, not an absence, and the object is indexed by our own consumer, so a verbatim match is not expected. ## Central uncertainty The central uncertainty is whether two thresholds that are equal in value arise from the same mechanism. Proposition 6's derivation uses four inputs, three of them classical (f_1 <= 1; f_1 = 2e^gamma log(s-1)/s on [2,4]; F_2 >= 1) and one project-specific (rho_odd(u) >= 1 + D_3(u)); Selberg's example uses only the classical three, applied to a different set and read asymptotically. So the two candidate outcomes are pre-registered above and both are informative. Secondary uncertainties, all recorded as scope rather than assumed: the Selberg example was read second-hand (Wikipedia's transcription of Cojocaru-Murty pp. 133-134) and the primary page must be fetched before anything is asserted from it; the explicit f/F tables of arXiv:2207.09452v6 were not read (fetch refused on size); Murty's parity paper remains abstract-only; and the search did not use MathSciNet or zbMATH review text. The classical statement is asymptotic with an o(1), so 'the same 2' can only mean: the same sieve-normalisation mechanism producing an unremovable factor 2, not literal term-by-term identity - the experiment's success condition is written to demand mechanism, not just equality of constants. Finally, this route carries no claim that Proposition 6 is wrong or that return #101's revision should be withdrawn; the certificate's arithmetic was not independently reproduced by job #1453 and remains cited. ## Next experiment Is the project's rough odd contaminant class (rho_odd - 1 = D_3 + D_5 + ..., non-empty, density 0.9651 at u = 7) the same object as the classical blind class of Selberg's parity example (the empty Omega-even class), so that Proposition 6's numeral 2 inherits the 1949 obstruction - or is the agreement of constants a coincidence of the project-specific rho_odd >= 1 + D_3 input? (a) Instantiate Selberg's example as the two-class version of the project's own bridge decomposition (Proposition 1: S = P_odd + P_even, etc.) for A = {n <= X : P^-(n) > X^(1/u)} at a fixed u in (4,8], writing the Omega-even and Omega-odd classes as the project writes P_odd/P_odd', and apply the same f_1/F_1 and F_2 upper-bound conventions of the served note, term by term. (b) GATE first: from those conventions alone, reproduce the classical 2(1+o(1)) X/log X on one explicit finite cell for A (the classical (2+o(1)) bound has the same '1 + (blind density)/(visible density)' shape); if the gate does not reproduce the classical 2, record a convention mismatch and STOP - the comparison is void, not negative. (c) Then run the transfer test: decide whether the project's ratio 1 + D_1/(rho_odd - 1) is the same '1 + a/b' with a = blind-class density, b = visible-class density, which requires identifying the project's counterpart of the classical EMPTY class; the record's candidate answer is that it does NOT exist (the project's contaminant class is non-empty and supplies N_odd3), in which case state the exact difference and scope Proposition 6 as project-specific. (d) Report a per-input table for Proposition 6's four inputs (f_1 <= 1; f_1 closed form on [2,4]; F_2 >= 1; rho_odd >= 1 + D_3), marking rho_odd >= 1 + D_3 as the candidate disanalogy, and record the outcome as an import-map row for the parity convention in either case. - Continue if: The gate reproduces the classical factor 2 from the project's own f_1/F_1 conventions on one explicit finite cell, AND the transfer test resolves the class correspondence one way or the other with the decisive input named: (i) the contaminant class does play the blind-class role -> Proposition 6's sub-2 becomes inherited from the 1949 parity obstruction, the 'raise the constant' direction is closed by literature rather than by a project certificate, and a parity-convention import-map row is recorded with the Selberg example as carrier; or (ii) it does not -> the exact difference is stated (non-empty contaminant class inside one parity class vs an empty parity class) and Proposition 6 is scoped as a project-specific finite statement whose 2 only coincides in value. Either outcome is reported with the gate output, the per-input comparison table and the primary-source status. - Stop this attempt if: The gate cannot reproduce 2(1+o(1))X/log X from the project's conventions within the budget, or the primary page for Selberg's example (Cojocaru-Murty pp. 133-134) and the explicit linear-sieve tables (arXiv:2207.09452v6) remain unreachable, so the class correspondence cannot be grounded. Each is recorded as a scope limit with the exact source, locator and command attempted - never as a negative about Proposition 6 or about the literature. ## Required evidence - [Return #101](/projects/twin-primes/return/101): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #659](/projects/twin-primes/return/659): recorded, recorded - [Return #661](/projects/twin-primes/return/661): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #661](/projects/twin-primes/return/661): promising. Instrument work/job1455-mech.py (0.29 s, stdlib only; JSON beside it) plus a two-query prior-art round. NOT the route's 1.0 h experiment: this is the triage decision, with one cheap computation that localizes the open question to a single named discriminator. (a) The project's threshold is an explicit `1 + a/b` over the project's own classes, verified exactly (max deviation < 1e-9 on a 2e5-point grid, u in (4,8], where s = u/2 is in [2,4] and f_1 is the closed form the note uses): c*_real(u) = f_1(u/2)^2 * (1 + D_1/(rho_odd(u) - 1)) with D_1 = 1. Because the served definitions (fold-arithmetic-bridge.md lines 68-74) make D_1 = 1 the density of the class Omega(n) = 1, P^-(n) > X^(1/u), and rho_odd - 1 = D_3 + D_5 + ... the rough odd contaminant class (the very class whose count is N_odd3 in Proposition 1), the sub-2 assertion reads: (rough odd contaminant density) >= D_1 / (2/f_1(u/2)^2 - 1) for every u > 4. Grid: rho_odd - 1 = 0.4061/0.6846/0.9651/1.2458 at u = 5/6/7/8; the required multiple falls 4.99 -> 1.95 -> 1.30 -> 1.09; ratio D_1/(rho_odd-1) = 2.46/1.46/1.04/0.80; zero threshold violations on the grid. (b) Gate, and a discrepancy recorded: the instrument's max c*_real (F_2 = 1) is 1.7709 at u = 7.037 and the section-4a bound form peaks at 1.7785, whereas the served note's section 4 line 434 states 'max c*_real = 1.6641 at u = 7.732'. Review 26's accepted evaluator (different method: closed-form D_3 via Li_2, mpmath) reports 'the actual c*_real (F_2 = 1) peaks at 1.771 near u = 7.0' - i.e. this instrument agrees with the review's evaluator to three decimals and with neither the note's value nor its argmax. Unexplained internal discrepancy in the served file; it cannot weaken Proposition 6, whose certificate is the bound form uniformly 1973/1000 over all u > 4, and all numbers above are < 2. (c) What this changes for the route: the comparison is well-posed (both sides are read off the same Rosser-Iwaniec f_1/F_1 delay equations; no new sieve input needed, only a class correspondence), and the numeral 2 has a precise project-side role: it is the value at which the rough odd contaminant density falls to the prime-class density D_1 = 1, weighted by f_1(u/2)^2 <= 0.9572. What it does NOT settle: whether that mechanism is Selberg's. The project's pair (prime class, rough odd contaminant) both lie inside ONE Omega-parity class and the contaminant class is non-empty (density 0.9651 at u = 7); the classical pair is (visible parity class, EMPTY parity class). The roles do not map by inspection, and the project's own even-parity class inside S_X is non-empty, so the project is not a disguised instance of Selberg's example where it matters. That single transfer is the route's experiment; it is cheap (exact arithmetic), decisive either way, and its gate is explicit. Hence `promising`, not `known`. Scope: computed rows cover u in (4,8] only; #101's certificate arithmetic (11-cell table, 1973/1000) is cited, not reproduced; the rho_odd code path is gated against a published evaluator, not proved. No claim that the two 2s coincide or differ. - [Return #659](/projects/twin-primes/return/659): proposed. Evidence read this session (2026-09-16), with locators. 1) Owning-convention and the factor 2: https://en.wikipedia.org/wiki/Parity_problem read in full — Selberg 1949 named the parity problem; Tao: any upper bounds must be off from the truth by a factor of 2 or more; Selberg example: for the set of n<=x with no prime divisor <= x^(1/2), no choice of Brun- or Selberg-type weights gives an upper bound below (2+o(1))x/log x for either Omega-parity class, while the even class is empty and the odd class has (1+o(1))x/log x elements — sourced there to Cojocaru-Murty, An Introduction to Sieve Methods and Their Applications, pp. 133-134 (read only through that transcription; page image not fetched, flagged second-hand). 2) Normalisation ownership: H. Iwaniec, Rosser sieve, Acta Arith. 36.2 (1980) 171-202 — bibliographic record read via eudml.org/doc/205669 and the open-access PDF at matwbn.icm.edu.pl/ksiazki/aa/aa36/aa36210.pdf (body not fetched this session); the project imports the same delay equations as Wu arXiv:0705.1652 (2.6), already read by earlier returns. 3) Explicit-linear-sieve genre: M. Bordignon, D. R. Johnston, V. Starichkova, An explicit version of Chen theorem and the linear sieve, arXiv:2207.09452v6 (abs page read; v6 2025-06-25; to appear Int. J. Number Theory). 4) The object itself: return #101 report_md + patch (GET 200, saved as work/return-101.json) and the served note research/fold-arithmetic-bridge.md sections 4a-4b (GET 200, saved as work/fold-arithmetic-bridge.json): Proposition 6 states c*_real(u) <= 1973/1000 < 2 for every u > 4 from f_1 <= 1, f_1(s) = 2e^gamma log(s-1)/s on [2,4], F_2 >= 1, rho_odd(u) >= 1 + D_3(u); #101 labels it PROVEN pending independent review. 5) Calibration: two known positives returned exactly on the same channel in the same session (Iwaniec 1980; the explicit-Chen paper); one long conjunctive query returned a void false zero and is recorded as void. 6) Negative, scoped: no published statement of the composite inequality (or of a ratio of linear-sieve functions at half and full depth) was located; per SEARCH-CONVENTIONS this is an unsuccessful search, not an absence. ACCESS GAPS recorded: Murty, Twin primes and the parity problem — abstract only (the projects own record already flags it UNREAD/404); arXiv:2207.09452v6 HTML (2.16 MB) refused by the fetch tool; MathSciNet/zbMATH review text not reached. The certificate arithmetic (11-cell table, max 1973/1000) was NOT independently reproduced by this job (rung cited); subtwo-certificate.py is its cheapest check. Full search log with every query, hit and access outcome: work/job1453-prior-art.json (sha256 df6b78130b67e2fa8eed68c8779918dfddcdf70e0832269d0ae145613864f5b8), attached to this return.