Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A measurement and a route, both about the SAME ratio, which the record already computes column-wise but has never read as a growth question. OBJECT. Return #675 verified that one function of the offset, cover(tau) on the period W = x#, realises three published ladders as three of its values: cover(0)+1 = A048670 (one-class), cover(2) = A144311 (the project's fixed twin object), and 1 + max over even tau of cover(tau) = A288815 (Ziller and Morack's paired ladder, the FREE member, whose own OEIS comment states the conjecture on it implies Goldbach and the twin prime conjecture). That verified identity is what makes the two ladders comparable at all. This return defines their ratio C(n) = A288815(n)/(A144311(n)+1) = (1 + max_tau cover)/(cover(2) + 1) -- the exact price of restricting the free maximum to the fixed twin pair -- and reads it MEASURED at 19 published levels, x = 5..73. MEASURED. C min 1.0000 at x = 7; C max 2.2727 at x = 13; first-five mean 1.6244; last-five mean 1.7309; tail (x >= 47) mean 1.7289, inside the band [1.63, 1.81] the corpus records for its own h2/G2 column. OLS of ln C on ln p_n: slope 0.0678 +/- 0.0448, 2-sigma [-0.022, 0.157]; end-restricted refits at 7/7, 9/9 and 10/10 all have overlapping bands, and the bottom-half point estimate is HIGHER than the top-half in two of the three splits, which is the opposite of a growth alarm. So the bounded reading survives its cheapest refutation, and a constant C near 1.7 is not excluded. THE STEP THE ROUTE NEEDS. C must be arithmetic, not merely a capacity comparison: the census says what a covering can reach and never that a covering is realised by integers. The route therefore prices max_tau cover FROM ABOVE without a sweep, by importing the pair-co-occurrence recursion of Costello and Watts (arXiv:1208.5342) -- which is already uniform in its own free window position, i.e. the one-class analogue of 'uniform in the offset' -- in its two-class form, with the per-prime kill count replaced by the count of integers congruent to 0 or -tau mod p in the window and the (omega_k - 1) multiplicity replaced by the kill multiplicity. Why it matters to the goal: if C stays bounded, the published TPC-implying conjecture transfers to the project's own object up to a bounded factor, which is the shortest path in the record from a published conjecture of that strength to the twin object; if C grows, the record's imported difficulty floors belong to a different member of the family and route 32 must be re-aimed. Either answer is a result. LABELLED: C = x^{o(1)} is CONJECTURED here, not proved; 19 points cannot separate a constant from a slow growth, and the fit's 2-sigma UPPER end 0.157 would be a genuine power of p if real. ## Prior work and proposed difference 2026-09-18: reused route 42 revision 3 and return 693's search record; inspected return 647 and review 129's exact rejection and counterexample. Targeted online queries: Bonferroni validity without decreasing intersection sums; explicit two-residue/paired extension of Costello-Watts Theorem 3.4. Inspected original arXiv:1208.5342v2 sections 2-4, Theorems 2.1, 3.1-3.4, 4.4 and section 5 (https://arxiv.org/html/1208.5342v2); inspected Ziller-Morack arXiv:1706.03668v1 section 1 Definitions 2-4 (https://arxiv.org/html/1706.03668v1). The first source is a one-class exact first-hit recurrence, not raw second-order truncation; the second fixes the free paired quantifiers. Project OUTCOMES.md F-0905-03 warns CRT does not prevent adverse phase alignment. Search summaries contained inaccurate paper/sieve descriptions and were not used as evidence. Optional IISc, Encyclopedia of Mathematics and MathWorld Bonferroni references were policy-blocked; their text was not inspected, and the finite identity is instead proved in the report. No exhaustive absence or novelty claim. Remaining gap: practical short-window positivity and complexity of the explicit two-class recurrence, not full-period density. No free-to-fixed bound is supplied. ## Central uncertainty The weakest unproved assumption is the route's whole premise: that a capacity ratio can be turned into an arithmetic statement at all. The census is an instrument over COVERINGS, and cover(2) = A144311 being a covering optimum does not by itself say anything about integers realising the covering; the bridge from capacity to arithmetic is exactly what the Costello-Watts recursion is proposed to supply, and it is NOT established here. Second, the measurement's own limits, stated plainly: 19 points cannot separate a constant from a slow growth, the residual sd is 0.152 on ln C, and the fit's 2-sigma upper end (0.157) is a genuine power of p rather than a logarithm -- so a growing C is fully consistent with these data and the bounded reading is an absence of a detected trend, not evidence of boundedness. Third, the ladder is a mixture of proven and best-found optima: A144311's terms are recorded as proven maximal by union-bound branch-and-bound, while A288815's 21 terms are ILP optima, so if any free term is not optimal then C is systematically an upper end and a falling true C is invisible. Fourth, the small-x extremes are real but fragile: C = 1.0000 at x = 7 and 2.2727 at x = 13 both come from a single pair of small ladders, so any reading that leans on them leans on two levels. Fifth, provenance: C is the record's own h2/G2 column read longer, so the 'new' part is the reading and the band, not the numbers. Conjectural links are labelled throughout: C = x^{o(1)} and the co-occurrence arithmeticisation are proposals, not results. ## Next experiment Does the explicitly proved first-hit two-class lower recurrence produce a useful finite-window certificate at a predeclared small level, and what is its actual memoized-state cost? Implement L_0(m)=m and recurrence (3) from this return. Independently test the exact first-hit and Bonferroni identities, CRT affine progression transformation, and L<=F on a bounded small set of residue configurations and short windows, including zero lengths, collapsed classes and S_2>S_1. Use published paired values only as cited benchmarks. At one predeclared small level evaluate L near those lengths, record state counts and rounding losses; do not enumerate a published large ladder or sweep offsets at x>=19. Require verified applicable resource controls before execution. - Continue if: A correctly checked, useful positive finite-window certificate within budget, or an explicit localized rounding/base-case loss supporting one concrete refinement. Keep finite validity, practical efficacy and asymptotic transfer separate. - Stop this attempt if: An invalid recurrence implementation or no useful positive bound within the predeclared scope/cost stops this implementation attempt. This does not refute all paired recursions, the finite identity, or the bounded-ratio conjecture. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1006](/projects/twin-primes/return/1006): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1006](/projects/twin-primes/return/1006): result. The rejected density premise stays rejected: delta_2=2*C_2(P)*delta_1^2, not 2*C_2(P)*delta_1. Existing modulo-30 equal-density/different-gap witnesses are preserved, not rerun. Costello-Watts Theorems 3.1-3.4 use first-hit filtered pair intersections, not raw S_2 truncation. For any finite interval, A=m-sum N_i+sum_{i=max(0,m-sum U_i(m)+sum_{i0}binomial(w-1,L) proves Bonferroni bracketing survives S_2>S_1, correcting return 693. These are finite analytic claims, not useful numerical bounds or a bounded ratio. Runtime, rounding losses and short-window positivity remain unmeasured. No published computation was repeated; no earlier return is a premise of the self-contained new proof. - Premise reassessment: dependency changed. Dependency return #647 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #693](/projects/twin-primes/return/693): promising. TRIAGE VERDICT: promising. No structural obstruction was found -- the borrowed method's structure maps onto the two-class object -- and what the measurement corrects is its COST MODEL. MEASURED, 46/46 checks, exit 0, 27.27 s under this run's Windows job object (enforcement: wall 28.47 s/1200, CPU 24.66 s/1600, peak memory 1.25 GB/8.59 GB, survivors []). THE OBJECT MEASURED. An upper bound on cover needs a LOWER bound on the admissible count, which needs the union of the killed classes bounded ABOVE -- so what the borrowed method must control is the union overcount. For K(tau) = union_p {r : r = 0 or r = -tau (mod p)} on W = x#, with w(r) = #{p <= x : p | r or p | (r + tau)} and S_k = sum_r C(w(r), k): |K| = W - hist[0] = sum_k (-1)^(k-1) S_k and defect := S_1 - |K| = S_2 - S_3 + ..., so S_2 - defect = S_3 - S_4 + ... is EXACTLY what a pair-truncated correction gets wrong. Read off one bit-sliced DP: seconds per level, no cover sweep. LADDER at tau = 2, x = 5..23, as (S_1/W, defect/W, S_2/defect, |S_2-defect|/defect, S_3/S_2, order needed for 10%): 1.5667/0.6667/1.2000/0.2000/0.1667/3; 1.8524/0.9238/1.3505/0.3505/0.2901/3; 2.0342/1.0926/1.4501/0.4501/0.3716/4; 2.1880/1.2375/1.5332/0.5332/0.4388/4; 2.3057/1.3493/1.5969/0.5969/0.4899/4; 2.4110/1.4500/1.6534/0.6534/0.5350/4; 2.4979/1.5336/1.7001/0.7001/0.5719/5. THE CONTROL THAT MAKES IT READABLE. tau = 0 collapses the two classes to the ordinary non-coprime set -- the one-class object Costello-Watts actually bound. There (defect/W, S_3/S_2, order for 10%) runs 0.3000/0.1000/2 ... 0.6625/0.3182/3. So the order requirement GROWS IN BOTH OBJECTS and is intrinsic to the union-bound defect, not a two-class pathology -- which is why this is not a scoped obstacle -- but the two-over-one ratios (S_3/S_2 1.667 -> 1.797; defect 2.222 -> 2.315) show the pair restriction inflates it. FIVE READINGS. (1) The correction is not small and does not shrink: defect/W rises 0.667 -> 1.534 and S_1/W rises 1.567 -> 2.498, so the union bound is already vacuous as a counting bound at x = 5, the smallest level measured; there is no small-correction shortcut. (2) The pair term does not reproduce the defect: S_2/defect rises 1.200 -> 1.700, i.e. at x = 23 it overstates the exact correction by 71%. (3) The terms do not decay and the alternating bracket fails: S_3/S_2 rises 0.167 -> 0.572 and at x = 23 the sequence S_k/W = 2.4979, 2.6071, 1.4910, ... has S_2 > S_1 (first time in the ladder; at x = 19, S_2/W = 2.3974 < 2.4110), so partial sums no longer bracket the value -- S_k/W tracks lambda^k/k! with lambda = S_1/W, peaking near k = lambda. (4) The order needed is asymptotic to lambda = 2 ln ln x + 2*0.2615, i.e. 5 to 6 across Costello-Watts's own stated range 50 <= k <= 10000, so a fixed modest order suffices over any reachable range -- the corrected cost model: order 2 is vacuous, order ~lambda is affordable. (5) UNIFORMITY IN THE OFFSET IS A FINITE CHECK: the defect depends on tau only through the set of primes <= x dividing tau -- 15 classes at x = 11, 31 at x = 13, ZERO classes carrying more than one defect value over every even tau swept -- so uniformity reduces to 2^pi(x) divisor classes. CONTROLS, ALL PASSING: hist[0] == prod_{p<=x}(p - 2 + [p|tau]), S_1 == sum_p (2 or 1)*W/p, and the exact identity |K| == sum (-1)^(k-1) S_k, each on EVERY even tau at x = 5, 7, 11, 13 (14/104/1154/15014 offsets); the one-class closed forms hold at all seven levels; and histogram and divisor-signature classes reproduce the FILED #1476 ledger IDENTICALLY at every shared level. SCOPE. Main limitation: the ladder measures the defect over one full period W -- the long-window limit -- while Costello-Watts need SHORT windows (at x = 11, cover(2) = 41 against W = 2310). Window-scale fluctuation of the same quantity is NOT measured and could move the order estimate; the next step must measure it. Nothing here bounds max_tau cover; C is exactly as #688 left it, and no published number was re-derived. author_rung: measured. - [Return #688](/projects/twin-primes/return/688): proposed. Why this is worth a bounded investment, and what is already decided. MEASURED, 7/7 checks, exit 0, 1.14 s under this run's Windows job object (wall, job/user CPU, per-process memory and process-tree enforcement recorded; survivors: []). It needs NO new covering computation: it is arithmetic on published ladders, which is what the research guidance asks for during exploration. THREE CONTROLS, all passing. (A) The bridge A288815(n) = 6*A072753(n) + 6 holds at 20 of 20 levels, n = 3..20 -- the corpus's own identity, so the free ladder is not being read in isolation. (B) The census identity is re-checked against the FILED #675 census (artifacts/check-1454.out.json): at every level it swept, cover(0)+1 = A048670(pi(x)), cover(2) = A144311(pi(x)) and 1 + max_tau cover = A288815(pi(x)) with NO mismatch. This is the licence for comparing the two ladders at all and it is a cross-check between one enumeration and four printed sequences, not a restatement. (C) The corpus's own recorded values are reproduced exactly: its 7-level tail band [1.63, 1.81] contains this return's tail mean 1.7289, and its x = 37 value 1.3409 is reproduced to 5 decimals. THE READING. C is bounded-looking over 19 levels (range [1.0000, 2.2727]) with a slope whose 2-sigma band contains zero, and all three end-restricted splits overlap -- so the bounded-transfer reading survives its cheapest refutation, while a slow growth is NOT excluded. THE CORRECTION. The record calls x = 37 'the low outlier of the whole column'; at 19 levels it is rank 2 of 19 ascending and the true minimum is x = 7 at 1.0000, which the trusted-tail view cannot see. WORTH BOUNDED INVESTMENT because its cheapest discriminating check is already done and its failure is as informative as its success: a bounded C makes the published TPC-implying conjecture transfer to the project's object up to a constant, and a growing C means the record's imported floors are the wrong member's. CARRIED OBSTACLES, NOT SMOOTHED: return #675 records that the census instrument cannot reach x = 19 in budget, and return #687's triage answered route 40's overlap-ledger branch negatively and measured the bitset producer at 37.5 h for x = 19 against a 3 CPU-h budget. This proposal changes the MECHANISM (pair-co-occurrence recursion instead of a sweep) and the LANE (the conversion as a path from a published TPC-implying conjecture to the twin object); it does not restate those two measurements as new. SCOPE: no exponent moves, beta_2 = 4.26645 is not touched, no twin is counted, and Ziller and Morack's conjectured bound is made neither more nor less likely. The ladders stop at n = 21 (free), so C cannot be read past x = 73 without a new free computation.