# The one-class to two-class transfer G₂ ≪ g·(ln x)^A (legal target L7): both transfer mechanisms the corpus holds are already priced, and neither reaches it > **RIDER 2026-09-04 (orchestrator, from `redteam-0904-item0.md`).** The > union-bound argument and the CRT count are CONFIRMED on independent code. > Three rewordings: §2's "certifies nothing in the period average" needs "and > hence not at every position" (the certificate is ≤ 0 at 25,386 of 30,030 > positions at x = 13, the period mean exact); §3's "exactly the > Brüdern–Fouvry decomposition" reads "reduces to it once the hole indicator > is replaced by a sieve minorant", the Ford–Halberstam substitution step > that `sift-limit-attack.md` §7b(1b) owns; and the price 2(1+√e) = 5.2974 is > the forced-equal-level constant, the corpus's live figure being > K_BF = 5.158064680330 at free levels (reproduced to twelve digits), so read > "5.158 at free levels, 5.297 with the levels forced equal", both above β₂. > §6's "no third mechanism excluded" is honest but incomplete: weighted sieves > with Chen switching on the hole set are already priced and closed at > `sift-limit-attack.md` §4.2. Mechanism 2 is dead as a route to L7 by the > growth half's second pass; L7 stays OPEN as a statement. *(2026-09-04, orchestrator's note, staging, HELD like every staging note. One producer, `research/history/staging/derive-0904-L7-transfer.js`, embedded via `qc/embed.js`, which enumerates the periods at x = 7, 11, 13 and checks §2's CRT count and budget; every other number is quoted from the record it names. Calibration per `CLAUDE.md`. Written while the day's four agents ran, on a target the G₂ object read listed as legal and unasked: `object-g2-read-0829.md` §4b row L7 and §8 Q3.)* ## 0. What is open first L7 is still open. Nothing here closes the statement `G₂(x#) ≪ g(x#)·(ln x)^A`; what this note settles is narrower: the two ways the corpus could try to DERIVE L7 from the one-class bound both reduce to objects already priced, and both prices are above β₂ or dead. So L7 is not a cheap corollary waiting to be written; it is the κ = 2 positivity problem wearing one-class clothing. That is DERIVED for the two mechanisms named and for no other; a third mechanism, if one exists, is not excluded by anything here. The truth-side reading is unchanged: three independent routes say the second class costs one logarithm (`G2-STATE.md` §3d, MEASURED and INFERRED), which is the conjectural A = 1. Truth-side evidence is not a mechanism. ## 1. The statement and why it is legal `g(x#)` is the one-class Jacobsthal function of the primorial, `g(x#) ≪ x²` by Iwaniec 1978 with an inexplicit constant (`covering-dive.md`, entry point 2; Vaughan 1977 has `x^{2+ε}`). `G₂(x#) ≥ g(x#)` pointwise is PROVEN and one line (twin slots are a subset of the holes). L7 asks for the reverse comparison up to a fixed power of `ln x`. If it held, `G₂ ≪ x² ln^A x`, which is exponent 2 + o(1): below the proven 4.26645, above the TPC line at exponent 2 with constant below 1 (`ZONE-POSTULATE.md` §2, the tile form `G₂ < x′² − 2`), so verdict (i) on the wrong-direction audit's scheme, as the object read records. No `Q-` id posed it before today; `REFUTED.md` closes no row of that name. ## 2. Mechanism 1: the union bound over the second class, vacuous in the period average from x = 11 [DERIVED, exact arithmetic] The natural transfer: in a window of length `H`, the one-class bound gives holes; a hole `h` fails to be a twin slot only if `h + 2` is not a hole, i.e. only if `h ≡ −2 (mod p)` for some prime `3 ≤ p ≤ x` (the class `2` never kills, since every hole is odd and so is `h + 2`; this is Fact B's class pair `{0, −2}`, `G2-STATE.md` §4a). Bound the failures by the union over `p`. Over the full period `W = x#` the count is exact by the Chinese remainder theorem: the holes are the residues coprime to `W`, and those in the class `−2 (mod p)` number `φ(W)/(p − 1)`, because `−2 ≢ 0 (mod p)` for odd `p` leaves `p − 1` admissible classes at `p` and the other primes are unaffected. So the union bound's budget, as a fraction of the holes, is B(x) = Σ_{3 ≤ p ≤ x} 1/(p − 1). Exactly: `B(7) = 1/2 + 1/4 + 1/6 = 0.9167`, `B(11) = 0.9167 + 1/10 = 1.0167`, `B(13) = 1.1000`. The budget passes 1 at `x = 11`. From there the union bound certifies nothing in the period average, and a bound that is vacuous on average over positions cannot be non-vacuous at every position. The corpus's covering economy reaches the same wall in its own accounting, `Σ 2/p` crossing 1 at `x = 13` with `β_pure` diverging like `7.182 lnln x` (`REFUTED.md`, the covering-economy row; `sift-limit-attack.md` §7). The divergence is the point: `B(x) ~ lnln x`, so the failure is not a constant that a better window would recover; the crude transfer loses a factor that grows. Inclusion–exclusion over the same classes is not a repair. Written out, it is the dimension-2 sieve on the pair `(h, h + 2)`, which is the theorem the programme already has at `β₂`. ## 3. Mechanism 2: the one-class sieve run on the set of holes is the vector sieve, and its price is on record [DERIVED, prices quoted] The second way to read L7: fix the holes `h` of a window as a set `A`, and sieve the shifted set `A + 2` by one class per prime. That is a κ = 1 sieve on a set whose distribution in residue classes is not that of the integers: its divisor counts `|(A + 2)_d|` are counts of holes in the class `−2 (mod d)`, which is a two-class-type count, and their remainders are the tile's own discrepancies at modulus `d`, not the `|r_d| ≤ 1` of an interval of integers. So the one-class sieve on `A + 2` has a level of distribution set by the two-class remainder, and the linear sieve's positivity at `u > 2` is then paid twice, once per class. This is exactly the Brüdern–Fouvry vector-sieve decomposition of the two-class condition into two coupled linear sieves, which `sift-limit-attack.md` §7b prices against the source: joint positivity at `1 + √e = 2.6487` per component, coupled component levels `D₁D₂ ≤ H` unconditionally, landing at `2(1 + √e) = 5.2974`, above `β₂`. Its two side-condition re-splits are CLOSED (`REFUTED.md`, the two Brüdern–Fouvry rows). Decoupling the levels needs signed cancellation of the bilinear interval remainder uniformly in window position, the sup-over-positions law; that family (REC, RML(α)) was ruled a truth gap on 2026-08-30 at rung derived-and-red-teamed-once (`REFUTED.md`, the ρ maximal law row), and a second adversarial pass on its growth half is running today (`redteam-0904-floor-growth-2.md`, HELD). If that pass breaks the growth half, mechanism 2 reopens as a proof gap at the decoupled price `1 + √e ≈ 2.649`, still above 2 and hence still legal; if it survives, mechanism 2 is dead as a route to L7. ## 4. What is left of L7, and what it is not - **L7 as a statement**: legal (i), OPEN, no mechanism in the corpus. Not a corollary of the one-class theorem by either transfer. - **What a proof would have to be**: an argument that keeps the class positions or the pair structure through the first discard point, which is the same requirement `object-g2-read-0829.md` §4a states for any exponent below `β₂`. L7 does not evade killer 1; it restates it with a smaller ask. - **What L7 is not**: not TPC-strength (exponent 2 + o(1) with an inexplicit constant says nothing about `G₂ < x′² − 2`), and not novel to pose: the one-class analogue of the sharp question is Erdős's problem on `Y(x) = o(x²)` (`covering-dive.md`, entry Q1), open with a bounty. ## 5. What this note contradicts in the live layer Nothing. `object-g2-read-0829.md` §4b row L7 says "not asked"; after this note it is asked and PARTIAL, which is the `Q-` id above. `TODO.md` item 0's `Ledger:` line needs the id added (orchestrator, at integration). ## 6. What would falsify this note, and whether that check ran - A transfer argument that is neither the union bound nor a sieve on the hole set. None is known to this note; not searched in the literature beyond the corpus's own `recon-0828-jacobsthal.md` and `covering-dive.md`, which report no published two-class upper bound at any exponent. Not run. - An error in the CRT count `φ(W)/(p − 1)`: RAN, the producer's OUTPUT block. Every per-prime count is EQUAL to `φ(W)/(p − 1)` at x = 7, 11, 13, and the budget reads 0.9167, 1.0167, 1.1000, passing 1 at x = 11 (VERIFIED). The same block prints the exact fraction of holes that are not twin slots (0.6875, 0.7188, 0.7422), so the bound exceeds its own target from x = 11. - The vector-sieve price `5.2974` or the side-condition closures being wrong: those are the records' claims, quoted, not re-derived here.