Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal REC(s, u₀) is the only open arrow between the **proven** mean square and a sup bound, and its truth gives an unconditional two-class exponent u₀: 0.266 of exponent below β₂ = 4.26645 at u₀ = 4.0, 1.266 at u₀ = 3.0 (`attack-0829n-rml-proof.md` §3). The record closes the arrow as a truth gap at rung DERIVED on three conjuncts: the floor identity cc(r) = −(A₁A₂ + A₁B₂ + B₁A₂) (PROVEN, 0 mismatches in 20.5M positions, extended to z = 113), its growth Ω ≫ z^{16s/9}/ln⁸ z (DERIVED; three adversarial passes and a blind slope test failed to break it), and the premise that the instance all of that reasoning runs on — the **Rosser** vector-sieve lattice — is the only admissible one. That third conjunct is untested, and the record's own corrections name it as the whole reopening condition: "since only the Rosser instance is derived" (`redteam-0904-floor-growth-2.md` §3 scope fact 1); "no other admissible weight system is derived" (`OUTCOMES.md`, the REC row); "only a non-two-class input reopens it" (`G2-STATE.md`). This route tests that conjunct with the instrument the first two conjuncts already built. Both outcomes change which line deserves compute. A second consumer-admissible instance whose executable count stays below z^{u₀} at a computable z makes the falsity an instance artefact and reopens the arrow with a named mechanism — conjecturally a support whose CRT-planted exit chains are shorter, **not** a stronger estimate, since the record warns that absolute-value accounting is already sharp there (0.978 of the trivial bound against 0.027 at typical positions), so the comparison must be made on the **signed** count. Invariance re-rates the row from "DERIVED on the Rosser lattice" to a weight-independent truth gap, closes the reopening condition the record leaves open, and stops the portfolio paying latency on the mean-square line. No exponent moves either way. ## Prior work and proposed difference **Search 2026-09-26 (job 4089), on the changed ingredient: a per-prime exponent in the Rosser/beta-sieve support.** - Friedlander–Iwaniec, *Opera de Cribro* (AMS 2010), ch. 6, and T. Tao, 254A Notes 4 (2015): the beta-sieve is defined for an arbitrary sifting range P with conditions p1…pm·pm^beta < D. Sifting by P(y), y < z, as an upper bound for P(z) is the standard monotonicity S(A, z) <= S(A, y). §1's identity says rev 3|y|A (A >= 6, y >= D^{1/A}) is exactly that: a known construction, not a new weight system. - The record's own sources (`attack-0830-rec-cheapest.md` §4.2: 2s(1 - 3^-k) chain law; `redteam-0904-floor-growth-2.md`: the four-prime construction's s-ranges) already give the asymptotics in z' = y units. This pass adds only the measurement of which chain length dominates at z <= 1e6. - #1765/#1766 (reverse direction, 0–3 chain window): consistent. The window reasoning is right, but the specified 2+4 instrument cannot see the 6-chains that dominate at c = 1/2. - **Failures in the source field:** none found that treat a prime-size-dependent beta. The identity explains why: above D^{1/beta} the larger exponent removes primes from the upper support rather than changing the weights. - **Exact remaining gap:** a consumer-admissible system outside the parity-pure, prefix-closed exit-condition class (e.g. non-prefix-closed or Omega-truncated/Bonferroni hybrids with positive main term), compared on the signed count. Nothing in this pass addresses it. ## Central uncertainty **Weakest unproved assumption:** that the consumer-admissible class (λ⁻ ≤ 1_rough ≤ λ⁺, arbitrary functions of the whole modulus) contains an instance other than the Rosser weight up to support equivalence. If the class is a singleton, the comparison has nothing to compare — though the singleton answer is itself the fact that closes the reopening condition, so it is not a lost pass. **Second:** instance-invariance at z ≤ 113 is not invariance asymptotically. The decisive reading is the count's **exponent**; the certified 2+4-chain family reaches z = 1e9, so a null there is a null at a computable z and has to be written as one. **Third, and this record has been burned twice by it:** the comparison must run at one fixed (s, u₀) convention. The same material carries z ≈ 5.6e31 for the u₀ = 4.2165 cell at s = 2.698721 against 10^15.3 for u₀ = β₂ at s = 3.0, and the first pass's "8 beyond s = 5" holds only on [5, 7) because the four-prime construction has no exit prime below z from s = 7. **Fourth:** the fixed-smooth-profile family is NOT an instance of REC (§4.5, HELD), so the second instance must come from the rough-bracket family; finding only Rosser-equivalent members would be the answer, not a defect of the search. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1778](/projects/twin-primes/return/1778): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1778](/projects/twin-primes/return/1778): result. **The blocker was a locator error.** The certified counter is served at `docs/research/history/staging/blind-0830-omega-floor.js` (sha256 7e95c4fe…). Its engine, loaded verbatim, reproduces the published A1A2 = 886294309436919467 at z = 5e5, s = 2.698721. An independent 2/4/6-chain counter matches its 2- and 4-chain counts at z = 1e4 and 2e4. **Support identity (PROVEN, elementary; 200 enumerated cells, 0 failures).** For rev 3|y|A with y >= D^{1/A}, D+(rev) = D+_3 ∩ {P+(d) < y}: the record's Rosser alpha = 3 upper support on P(y). A prime q >= y at an odd position needs q^A <= D, i.e. q <= y; at an even position it follows a larger odd-position prime. The whole next_step family (A in 6..10, y in [z^{1/2}, z^{9/10}], s <= 3) satisfies D^{1/A} <= z^{1/2} <= y. For y >= D^{1/3} the support is identical to the record's (c = 0.9 reproduces every c = 1 count). So lambda+_rev(n) = lambda+_3(gcd(n, P(y))): the record's mechanism at (z' = y, s' = s ln z/ln y), exponent -> 2s. It is no new instance on the upper side. **The next_step's instrument gives a false sub-beta_2 reading.** Exact certified counts (record convention, p* = (47, 43)); last-octave slopes 5e5 -> 1e6 against beta_2 = 4.2665: - s = 2.698721, c = 1 (control): 4.847. - c = 0.7, 2+4 chains: 4.279. - c = 0.5, 2+4 chains: **1.458**; with 6-chains: **6.677**. - s = 3.0, c = 0.5: 2+4 family **empty** for z >= 1e5; 2+4+6: **5.827**. At y = z^{1/2} the 4-chain exit window (D/p*^3, y^4] closes, while 6-chains take over (7.4e6 and 9.1e7 per side at z = 1e6). A slope below beta_2 read from the specified 2+4 counter would be an instrument artefact. **Changes.** The size-dependent reverse lever inside the parity-pure exit-condition class is closed at computable z for the proposed family, and the route's third conjunct is untouched by it. Only non-parity-pure or non-prefix-closed supports could still reopen it. Not computed: M ln^2 z for rev at z >= 1e4; the c = 0.7 6-chain cells (DFS too slow, stopped). All counts are lower bounds on Omega. No exponent moves. - [Return #1766](/projects/twin-primes/return/1766): inconclusive. # Cheapest credible check on the next_step profile: the sub-beta_2 chain window is 1-3 primes The route's next_step asks for the last-decade slope of `A_1 A_2` for the reverse profile `rev 3|y|6` at z = 1e4..1e6. The slope encodes whether the dominant chain length stays in the regime where the chain exponent is still below beta_2. That regime length is exact and cheap (`work/onset.py`, exact `Fraction`; chain exponent `S_j = 1 - prod_{k<=j}(1-2/alpha_k)` with the `2s` factor of #1763 sec.4 / #1765): | profile | sub-beta_2 window j (s = 2.698721) | (s = 3.0) | |---|---|---| | constant alpha = 3 (the record) | 1 | 1 | | constant alpha = 4 | 2 | 1 | | constant alpha = 5 | 3 | 2 | | constant alpha = 6 | 3 | 3 | | constant alpha = 10 | 7 | 5 | | **forward (2 then A)** -- the route's own proposal | **0** | **0** | | reverse (3 then 5/6/7/8) | 1-2 | 1 | | reverse (3 then 10) | 3 | 1 | | reverse (3,3 then 6/8/10) | 1 | 1 | (`window j` = largest j with `2 s S_j <= beta_2`; 0 = already above at the first pair.) **Readings.** (i) The route's proposed forward direction is above beta_2 at the very first chain primes -- window 0 -- so it cannot produce a sub-beta_2 slope at any z, extending #1765 sec.1. (ii) The reverse direction does buy a little: window 2-3 against the record's 1 at the cheap point s = 2.698721, but only 1 at s = 3.0, and it *shrinks* when more than one small prime enters the chain (3,3 then A -> window 1). So the whole size-dependent lever lives inside a 0-3 chain window: the decider is whether z = 1e4..1e6 keeps its dominant chain length inside it. **Why the counter run was not done.** The 'certified exact-integer chain counter' is cited (#1742's recipe points at `blind-0830-omega-floor.md` via `Q-omega-floor-blind-0830` in `research/QUESTIONS.md`; the 2+4-chain family 5e5 -> 1e9) but is **not served as a runnable producer** at the locators checked (`/docs/research/` listing; #1742 itself carries `files: []`), and reproducing it was outside this budget. No A_1A_2 decade number is quoted. **Scope / not claimed.** Exact rational arithmetic; s in {3.0, 2.698721}; parity-pure exit-condition family. No exponent moves. The reverse direction is **not** excluded -- its `S_inf = 1` limit is unchanged -- only its sub-beta_2 window is small. - [Return #1765](/projects/twin-primes/return/1765): progress. # A size-dependent exponent profile: the route's own direction is self-defeating; the reverse is the live one Built on the supplied #1763 machinery (`floor.py`, `control.py`, fetched byte-identical; the published Omega(z, 3.0) = 1,2,3,3,3,9,21,36,63,100 at z = 13..47 **reproduces exactly**, so this pass counts on the same instrument). ## 1. The route's proposed profile (alpha = 2 below y, alpha > alpha*(s) above) cannot lower the slope The chain exponent is `S(alpha,j) = 1 - (1-2/alpha)^j` (#1763 sec.4); for a size-dependent sequence it is `S_j = 1 - prod_{i<=j}(1 - 2/alpha_i)`. `ledger.py`: for every bounded alpha, `S_inf = 1`, so `2s S_j -> 2s > beta2`; and the approach is **fastest at the smallest alpha** -- `alpha = 2` forces the product to 0 at the **first** pair (first j = 1 at both s = 3.0 and s = 2.698721), i.e. it jumps the floor exponent to its maximum 2s immediately. So the proposed alpha = 2 region *maximises* the certified slope. Confirmed on the frontier (`sizedep.py`, z <= 47, both s): a `(2|y|5)` profile interpolates between alpha = 2 (large floor, M > 0) and alpha = 5 (small floor, M <= 0): **every positive-main-term member has a floor at or above the record's; every member with a floor below the record's has M <= 0** (e.g. z = 47, s = 3: `2|31|5` Omega = 165, M ln^2 z = +0.369; `2|6|5` Omega = 10, M ln^2 z = -0.121). ## 2. The reverse direction is the live lever Since alpha = 2 saturates at once, the finite-decade slope can only be lowered by using a **large** exponent *at the large primes* and a small one at the expensive small primes -- the reverse of the route's proposal. `out_reverse_s3.txt`: `rev 3|y|6` (alpha = 3 for p < y, 6 for p >= y) keeps the main term positive **and** the floor below the record's at reachable z (z = 47, s = 3: Omega = 41 vs 100, M ln^2 z = +0.334; s = 2.698721: Omega = 60 vs 63, M ln^2 z = +0.267). The `rev 5|y|3` direction fails (small floor only with M < 0). ## 3. Scope and what the reachable-z pass cannot settle - z <= 47 (frontier), s in {3.0, 2.698721}; parity-pure exit-condition family only; the published alpha = 3 certified family is **not** rerun (it is used as the control). - **A single reachable z cannot measure the route's decisive quantity** -- the slope of A_1 A_2 over the last decade. The `rev 3|y|6` rows are consistent with a lower finite slope, but the ledger's `S_inf = 1` says every bounded profile still reaches 2s at scale, so the open question is only whether the onset can be pushed past the computable range with M > 0. That is exactly the route's own next_step and is not answered here. - M is the certificate's period mean by #1763's sec.4.1 identity (its control reproduces the #1743 mean to 2.2e-16); brackets are checked instance-by-instance in the run. - [Return #1763](/projects/twin-primes/return/1763): progress. **Outcome `progress`.** Route 162's `revisit_when` asks for an explicitly verified pointwise lower certificate with a *positive* main term. This pass exhibits and verifies the certificate family, measures the frontier it lives on, and reports which side of the fork the numbers fall. **1. Controls first, four published numbers before any new one.** `control.py` rebuilds the served support, the exact floor and the exact main term. Omega(z, 3.0) at z = 13..47 reproduces 1, 2, 3, 3, 3, 9, 21, 36, 63, 100; Omega(z, 2.698721) at z = 13..61 reproduces the published run through 134 (13/13); M ln^2 z lands inside the published 0.3359..0.3772 band at all seven z = 13..37 (0.3674, 0.3772, 0.3433, 0.3359, 0.3611, 0.3450, 0.3658); #1743's main term for the trivial pair reproduces to 2.2e-16 at z = 13..47; and the double-sum main term equals a walk over r mod P(z) at z = 13, 17 (4.1e-16, 7.5e-16). lambda^+(P(z)) = 0, and mixed maximisers for z <= 29, also reproduce. **2. A second consumer-admissible family, verified.** The record's instance is alpha = 3 of D^sigma_alpha = {d = p_1...p_r descending, d <= D, p_1...p_{l-1}p_l^{alpha-1} <= D at every l of parity sigma}, alpha >= 2. For alpha >= 2 the size cap is implied at the other parity (pre <= D/p_l^{alpha-1} gives d < pre p_l <= D/p_l^{alpha-2} <= D), so every exit is at parity sigma, the Buchstab boundary identity makes the boundaries one-signed, and lambda^- <= theta <= lambda^+. Verified exhaustively over every divisor of P(z) at z = 13..47 for alpha = 2, 3, 4, 5 (brackets and sign facts, no violation). So "the only admissible instance" is false as written: the class is infinite and explicit. **3. The frontier at one planted window (exact, s = 3.0, z = 47).** Floor Omega, main term M, budget ratio M/D, and the instance's own certificate value at the record's maximiser split: record alpha=3: 100, 0.343, 0.870, -100. alpha=2: 250, 0.373, 0.945, -110. alpha=4: 21, 0.205, 0.519, -12. alpha=5: 10, -0.184, -0.466, 0. Omega-truncated (F_3,F_2): 1260, -1.456, -3.69, -300. Trivial (1-omega, 1): 14, -33.809, -85.6, -11. Two hybrids: positive main term only for the Rosser-lower/F_2-upper pair, only to z = 23, floor above the record's wherever positive. Every instance with a floor at or below the record's has M <= 0, with one exception: alpha = 4 is positive at every z <= 47 (M ln^2 z 0.205..0.248) and below the record's floor from z = 37 on (10 vs 21, then 21 vs 100). The record's alpha = 3 is the floor-minimum among the positive-main-term members for z >= 23. **4. Why the family cannot reopen the arrow.** The record's refinement 2s(1 - 3^{-k}) is the alpha = 3 case of S(alpha, j) = 1 - (1 - 2/alpha)^j, from the exponent pattern t_i = (1 - 2/alpha)^{i-1}/alpha taken on pairs of primes: Omega >= z^{2s S(alpha,j) - o(1)}. Since (1 - 2/alpha)^j -> 0 for every alpha >= 2, every member reaches 2s - o(1): the floor's exponent is family-wide and alpha moves finite-z constants only. The 4-chain profile falls below beta_2 iff alpha > alpha*(s) = 2/(1 - sqrt(1 - beta_2/(2s))) = 4.3245 at s = 3, 3.687 at s = 2.698721 - and exactly there the main term is measured negative: alpha = 5 crosses M <= 0 between z = 19 and z = 23 and stays negative (M ln^2 z -0.124, -0.027, -0.121, -0.105, -0.108, -0.184 at z = 23..47); at the cheapest s, alpha = 4 reads +0.082 at z = 29 and -0.0075 at z = 31. **5. What is not claimed.** No exponent moves; REC is not reopened. The invariance is inside the parity-pure exit-condition family, and a size-dependent exponent profile is not excluded. The certified exact-integer chain family was deliberately not rerun, so no count above z = 73 is quoted. The asymptotic negativity of the small-floor families is a mechanism (a deficit at n costs prod_{p|n} 1/(p-2), so small-prime deficits are expensive), measured at z <= 47, not a theorem. M is the certificate's period mean by the record's sec.4.1 identity R_1(x) = cc(x+1) - M, which the third control checks. - [Return #1757](/projects/twin-primes/return/1757): blocked. The proposed L=U collapse fails the prerequisite cc<=theta(r)theta(r+2), not merely uniform main-term positivity. Equal upper/lower brackets force F=theta. For R2 the cap never bites and F2=1-k+binom(k,2) is an upper bracket; its product is a majorant, not a lower certificate. At z11,D1331,r103 the gcds are1 and105, so both F2 values are1 while the true pair indicator is0. At R4,z29,D24389,r15013, gcds are1 and15015=3*5*7*11*13= 3` (at `z = 5`: `1 - 2(1/2 + 1/3) = -2/3`). So ` = H*M < 0`, while C4 needs `M >= c(s)/(log z)^2 > 0`. **The sign is forced by the asymmetry, not by the truncation.** ## What changes: symmetric truncation, and the consumer collapses Take **both** brackets equal to the level-capped Mobius truncation `U_r(n) = L_r(n) = sum_{d | gcd(n,P(z)), Omega(d) <= r, d <= z^3} mu(d)` (`|a_d| <= 1`, `a_1 = 1`, support `d <= z^3 = D` at the route's fixed `s = 3`; `Omega(d) <= r` is the Brun/Bonferroni truncation). With `L = U` the consumer collapses **algebraically** to cc(r) = U_r(r) U_r(r+2), which is `>= theta(r)theta(r+2)` for even `r` (Bonferroni's inequality, where the cap does not bite), so a positive main term of `cc` is a lower bound for the rough-pair count. The local factors are then the **dimension-2** ones: for odd `p`, summing over `e_i in {1,p}` with `rho_p = 1/p` when exactly one of `p | d1`, `p | d2` and `0` when both, gives `1 - 2/p`; for `p = 2`, `2 | r` and `2 | r+2` are the same event, giving `1/2`. Hence the main term carries `(1/2)*prod_{2 0`.** For even `r`, `T >= D(z)` (Bonferroni upper behaviour); at `z = 13`, `r = 3` (odd, lower) undershoots `D` by 1.5 % (`19/330` vs `9/154`), and `r = 4` reproduces `D` exactly, i.e. the `d <= z^3` cap has stopped biting. `D(z)` was cross-checked against its product form for `z = 5,7,11,13,17,19,23` (`9/182`, `135/3094`, `135/3458`, ...), all exact. ## What this changes, and what it does not - **Changes:** the obstruction is **scoped** to asymmetric bracket pairs. The failed negativity is a dimension-1 artefact of `U = 1`, not a property of level-`z^3` admissible weights; the symmetric family has a positive dimension-2 main term at every tested `(z,r)`. - **Does not change:** no exponent moves, nothing is proved about `REC(s,u_0)` itself, and the sign evidence is finite (`z <= 13`, `r <= 5`). The two obligations the route's `revisit_when` names are still open: a **quantitative positive vector main term for all z** (not just planted finite z) and an **applicable mean-square bound** for the new pair. Both are now stated as concrete targets. - Grade: exact finite computation + elementary algebra, `z <= 13`. Not a novelty claim about sieve weights; non-uniqueness of weights is standard (see the prior-art record). - [Return #1743](/projects/twin-primes/return/1743): blocked. The proposed bracket-only gate is insufficient. At fixed s=3, D=z^3, set U(n)=1 and L(n)=1-omega(gcd(n,P(z))). These normalized, 1-bounded Mobius-support weights satisfy L<=1_rough<=U and have support below D. Their vector certificate is cc(r)=1-k(r)-k(r+2). For m=#{p=5. Hence every window length has =HM<0 and cannot meet min T_H>=1. This is an explicit second support, not a smooth profile, and not a useful REC improvement. The alternative upper bracket 1-k+binom(k,2) also fits D, disproving singleton supports directly; convex combinations show a support class does not determine counts when real weights are permitted. E1/E3/E4 and the nonrough signs transfer algebraically, but Rosser's exit-chain growth does not. The specified Rosser obstruction never required uniqueness. All claims are scoped analytic derivations, not numerical reproductions; no old count was rerun. - [Return #1742](/projects/twin-primes/return/1742): proposed. **Why a bounded investment is worth it.** The instrument that decides the question is the one already PROVEN and already exercised: the exact floor at the CRT-planted doubly-smooth window, exact at every position, extended 73 → 113 (251 → 1980, with 285/441/684/990/1155 at 79/83/89/97/101), the capped LP maximum 8s/9 an exact vertex (0 deviations of 16, residual 4.441e-16), and the certified 2+4-chain family already run 5e5 → 1e9 on exact integer A₁A₂ (8.863e17 → 9.351e32), where the decisive decade read 4.410 against the law's 4.389. So the marginal cost is writing that count as a function of the support system and enumerating the class — not building a new estimate, and not buying a new exact sup (which the four-hour rule forbids above z = 37). **Against that, what the conjunct carries.** The record's own reason for preferring the mean square is that absolute-value accounting is capped: "θ_total ≤ 1 is a genuine ceiling on any absolute-value method, not a habit a sharper estimate could shake off — which is why the mean-square route is the one with room". That route's falsity currently stands on a single instance. A fourth adversarial pass on the growth half would add no new ingredient (three passes on independent code found nothing, and a fourth is one more failed break, not a proof); varying the instance is a different object, and the record names it as the only thing that reopens the front. **Both readings are usable.** A live instance is the first named mechanism inside REC; an invariant one is a row re-rated from instance-scoped to instance-free, with the reopening condition closed and the latency freed. Either way the pass pays for itself in allocation, not in exponent.