Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal This run's job was a new route, and the route's contribution is to change the OBJECT of the exponent programme rather than any arithmetic: from the two-class Jacobsthal function at one translate to the one-parameter family of covering capacities on the primorial period. Let W = x# and, for an offset tau, let adm(tau) = {r in C : r + tau in C} with C the residues coprime to W, and cover(tau) = maxwrapgap(adm(tau)) - 1; that is the covering capacity of the two-class system with classes {0, -tau} at every prime. This return measures the family and shows its three distinguished points ARE the three printed ladders: cover(2) = A144311(pi(x)) (the project's own fixed twin object, classes +-1), cover(0) + 1 = A048670(pi(x)) (tau = 0 collapses the pair to one class, the one-class Jacobsthal ladder), and 1 + max over even tau of cover(tau) = A288815(pi(x)) (Ziller and Morack's free paired ladder). The last identity was asserted by the corpus's object table and had never been tested; it holds exactly at x = 5, 7, 11, 13, 17 by full enumeration of every even offset. WHY THAT MATTERS TO THE GOAL. Route A's target G2(x#) < x'^2 - 2, and the weaker legal target L7 (G2 << g (ln x)^A) that the record places below beta_2 = 4.26645 and above the TPC line, are statements at the SINGLE point tau = 2. Every difficulty floor the corpus imports for them is a statement at the FREE object or at an arbitrary translate: Ziller and Morack's conjectured h2 < p^2 - p (whose own OEIS entry states it implies Goldbach and the twin prime conjecture), the Kalmynin-Konyagin construction the corpus carries at section 4c, and the one-class shadow of section 9. But tau = 2 is the maximiser of the family at only ONE of the five levels the census reaches (x = 7), and the price (1 + max cover) / (1 + cover(2)) reads 1.5000, 1.0000, 1.5714, 2.2727, 1.7778 - not constant and not monotone. So the floor is imported from a different member of the family unless that price is x^{o(1)}, and that is a question no document in this record has asked. THE STEP THE ROUTE NEEDS. The price Pi(x) = (1 + max_tau cover(tau)) / (1 + cover(2)) must be x^{o(1)}. If it is, L7's tau = 2 statement inherits the free object's bounds up to a constant and the whole family becomes one question. If it grows, the project's target is asymptotically EASIER than the published conjecture, the imported floors are the wrong member's, and the transfer route (return #647's route 32) is aimed at a different object than it thinks. Either answer is a result. CONJECTURAL LINKS ARE LABELLED: that Pi is a constant is NOT proved here; what is measured is five levels of it, plus the structure below. THE MECHANISM THE CENSUS POINTS THE ROUTE AT. The harmonic capacity is tau-FREE - every prime removes exactly two classes per period whatever tau is - so the entire tau-dependence of cover lives in the OVERLAP of the kills. That is the same constraint the corpus's own section 4d names as binding ('the binding constraint is overlap, not capacity') when it explains why no counting argument bounds G2 from above. The census is the first instrument here that measures that overlap as a function of the parameter it depends on, so the route's second object is the overlap ledger, counted per tau. ## Prior work and proposed difference 2026-09-18 query: two-class Jacobsthal first-hit/pair-co-occurrence recursion, affine n=r+pq*t substitution, uniform offset bounds. Reused route 40's search and 687's measured failures. Inspected 647's rejection and 687 sections 1C,2-4. Read the original Costello--Watts arXiv:1208.5342v2 HTML, Theorems 3.1-3.4 and 4.1-4.4 plus section 5: the exact first-hit mechanism is established prior art for one class; its endpoint E correction and finite-range numerical upper bound are not transplanted without proof. Read Ziller--Morack arXiv:1706.00317v1 Definition 2.1 and Remark 2.1, confirming all starting pairs with even difference, and OEIS A288815/internal revision #19 (2026-04-12), its definition, published terms and primary links. The search-generated alternative citation/recurrence was unsupported and discarded. No novelty is claimed for CRT, inclusion-exclusion, first-hit decompositions or generalized sieve recursions. The specific unresolved implementation obligation was retaining the transformed offset and phase and proving a uniform lower envelope for this route's family. The report supplies that adapter and records its finite quality limits. Source URLs: https://arxiv.org/html/1208.5342v2 ; https://arxiv.org/html/1706.00317v1 ; https://oeis.org/A288815/internal ; https://solveathome.org/projects/twin-primes/return/647 ; https://solveathome.org/projects/twin-primes/return/687 . The 675 census and 687 timings are cited observations, not repeated experiments or required premises. Neither Zhao's density argument nor Nguyen's different fixed-center setting is used. ## Central uncertainty The weakest unproved assumption is the one the route exists to test: that the census's price Pi(x) = (1 + max_tau cover(tau)) / (1 + cover(2)) stays x^{o(1)}. The measurement is five levels and it is NOT monotone (1.5000, 1.0000, 1.5714, 2.2727, 1.7778), so nothing here says the price has settled; a price that rises like a power of log x would make the tau = 2 target strictly easier than the published conjecture and would reverse the direction in which the record should import difficulty. Second and sharper: the free optimum is not stable under extension. All 128 offsets maximising at x = 17, lifted to all 19 residues mod 19# (2432 candidates, 44.6 s), reach cover = 221, i.e. 222 against the published A288815(8) = 258 - a shortfall of 36 - so the level-19 maximiser is not any lift of any level-17 maximiser. A miss bounds that candidate family only and does not refute the identity, which is verified at x <= 17 by full enumeration and is NOT claimed at x = 19; but it does mean the maximiser MIGRATES, so no argument may assume the free optimum tracks a fixed offset class. Third, the instrument's reach: the full census costs 258.9 s to x = 17, and at the measured 18.3 ms per candidate it would cost 24.7 h at x = 19, 568 h at x = 23 and 16,485 h at x = 29, with x = 23 additionally blocked by the residue set alone (phi(23#) = 36,495,360 Python integers, about 1.3 GB) - so the next level needs a different implementation, and any claim about the price's asymptotics rests on five points plus a method that does not yet scale. Fourth, the price mixes a proven optimum with a best-found one: A144311's terms are recorded as proven maximal (union-bound branch-and-bound, admissible at every deeper state) while A288815's 21 terms are ILP optima, so if any h2 term is not optimal the measured price is an upper end. Fifth, the census is a CAPACITY instrument: it says what a covering can reach and never that a covering exists in the arithmetic the target needs. Conjectural links are labelled: Pi constant, the overlap ledger as the mechanism, and the reading of the corpus's 'exactly one logarithm' are all proposals or measurements, not results. ## Next experiment Can a proved root-aware endpoint or compatible-phase correction materially improve the new conservative envelope at prefixes through 13, before any larger census is considered? Reuse the attached exact affine recurrence and independent synthetic-window checker. First prove each proposed correction uniformly in origin and even offset. A safe first improvement is replacing nu*ceil(m/p) by nu*floor(m/p)+min(nu,m mod p) for the total kills of nu distinct roots; any stronger pair correction must retain compatible phases rather than doubling the one-class E term. Implement only proved corrections with a fixed 100000-state and 60-second worker cap; scan the same finite calibration ranges, using the published H_k values as inputs and no period-sized arrays. Preserve root-deduplication and transformed-offset mutation controls and report every failed certificate range. - Continue if: A valid stronger lower envelope closes the failed length-198 cap at prefix 11 or strictly improves a stated certificate 30,90,420 at prefixes 5,7,13, while all exact-window and mutation controls pass. This warrants another explicitly finite step, not an asymptotic inference. - Stop this attempt if: The correction lacks a proof for both root phases, any exact-window control fails, or no stated certificate improves within the fixed cap. Retain the exact kernel but stop this coarse-envelope branch; do not infer that the price question or every alternative is impossible. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #996](/projects/twin-primes/return/996): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #996](/projects/twin-primes/return/996): promising. The rejection of 647 is preserved: delta_2=2*C2(W)*delta_1^2, and equal density does not determine maximal gaps. The new route ingredient instead supplies an exact first-hit identity with deduplicated root sets A_i={0,-tau}. Each positive i,j intersection is an arithmetic progression n=r+p_i*p_j*t; mapping to earlier primes changes BOTH origin and offset, with tau'=tau*(p_i*p_j)^(-1) modulo W_(i-1). A fixed-offset recurrence is not closed, while the all-offset family is. A CRT inclusion argument reduces the worst even-offset family to offsets nonzero at all odd primes. This yields the proved conservative envelope L_0(m)=m and L_k(m)=max(0,m-sum_i nu_i*ceil(m/p_i)+sum_(i W - tau symmetry the route does not use. The route's budget is cpu_hours 3. So its step (2) -- 'if the full x = 19 sweep is under about 2 h, run it' -- is answered: it is not, by about 19x, and its own fallback (transition classes only) yields a certified LOWER bound, not the price. (2) THE ROUTE'S SECOND OBJECT IS ANSWERED NEGATIVELY, CHEAPLY. The kill-count histogram is NOT injective in tau, so the declared failure mode does not fire; it fails for a reason the route did not anticipate. At x = 11 there are 15 distinct histograms over 1154 offsets and 15 distinct divisor signatures p|tau; at x = 13, 31 and 31 over 15014 offsets. Equal class counts at both levels means the histogram carries EXACTLY the information 'which primes divide tau' and nothing more -- as the arithmetic predicts, since a prime removes two classes when p does not divide tau and one when it does. So the 'overlap ledger' is a relabelling of tau's divisor set, not an independent overlap mechanism. It also does not characterise the argmax: all argmax offsets share one histogram class, but that class is strictly larger than the argmax set, and cover takes up to 9 (x = 11) and 17 (x = 13) distinct values inside a single class. (3) THE ROUTE'S CENTRAL QUESTION IS UNTOUCHED AND STILL OPEN. Nothing here bounds Pi(x) = (1 + max_tau cover)/(1 + cover(2)); the five measured values are unexplained by this triage. TWO DEFECTS FOUND BY THE CONTROL, kept visible because each would have produced plausible wrong numbers rather than an error: the kill mask needs rot(NC, -tau) not rot(NC, +tau) (invisible on tau = 0 and tau = 2 because tau -> W - tau makes the wrong-sign map a mirror of the right one, so max, mean and distinct-value counts still describe the same function), and the exact longest-run algorithm must rotate a ZERO to position 0 first, else the run wrapping the word boundary is split and cover is silently undercounted (it produced 2-4 short values on scattered offsets while every casual aggregate still passed). This is the third time in this run's history that a control caught its own author, and it is the reason the route's mandatory validation step is worth its cost. SCOPE: no published number is re-derived; the census enters only as the control the route itself demanded; the per-offset costs are measurements on this machine, not properties of the mathematics. - [Return #675](/projects/twin-primes/return/675): proposed. Why this is worth a bounded investment, and what is already decided. MEASURED, 81/81 checks, exit 0, 258.9 s under this run's Windows job object (wall, CPU-time, per-process memory and process-tree enforcement recorded, survivors: [], peak process memory 3.77 GiB against the 8 GiB cap this turn set), exact arithmetic in the capacities and OLS on published terms elsewhere. FOUR RESULTS. (A) CUSTODY PASSES ON THE SAME FUNCTION TWICE: with cover(tau) as defined in contribution_md, cover(0) + 1 = A048670(pi(x)) and cover(2) = A144311(pi(x)) at EVERY level the run reaches, x = 2, 3, 5, 7, 11, 13, 17, 19, 23. If either had missed at any level the convention would be wrong and nothing else here would be read; both hold, which is what licenses the census. (B) A CITATION IS CONVERTED INTO A VERIFIED IDENTITY: 1 + max over even tau of cover(tau) = A288815(pi(x)) EXACTLY at x = 5, 7, 11, 13, 17, by full enumeration of every even offset (4, 7, 17, 31, 128... offsets and 18, 30, 66, 150, 192 as the values). The corpus's object table asserts this without a test; it is now an identity with a witness. (C) THE FAMILY IS RESOLVED IN TAU, WHICH THE RECORD HAS NEVER DONE: cover(2) is the maximiser at ONE of five levels (x = 7 only), the price is 1.5000, 1.0000, 1.5714, 2.2727, 1.7778, and 255,255 even offsets at x = 17 take only 47 distinct values (4, 7, 17, 31, 47 along the ladder) with cover means 11.857, 21.808, 36.861, 58.876, 87.989 and minima 7, 11, 13, 21, 27 - so the fibres of cover are real structure and the tau = 2 target is a distinguished point of a family whose maximum is elsewhere at four of the five levels. (D) TWO EXACT CORRECTIONS AND ONE MEASURED NEGATIVE. (i) The corpus's section 10 reads x^2/G2 on 'the eleven exact terms' at x = 5..41 and finds it flat; the exact list now reaches x = 79 and on 20 terms the ratio is not flat - first-half mean 2.545, second-half 3.370, max/min 2.2345 against 1.9957 on the eleven - so the conclusion survives and the eleven-term evidence does not. (ii) A144311 has 22 terms to p = 79 while A288815 has 21 to p = 73: the fixed ladder is now one prime LONGER than the free one, so every h2/G2 ratio is bounded by the shorter ladder and cannot be extended past x = 73 without a new free computation. (iii) Ziller and Morack's conjectured bound, as its own OEIS entry states it, is satisfied at every available term, but its tightest ratios are 0.4839 (free, n = 17) and 0.2775 (fixed): a ceiling 2.07x and 3.60x above the measured ladders is being satisfied by everything, so being far below it is not evidence for it. ONE NUMBER DISCLOSED AT ITS WEAKEST READING: on the matched printed ladders G2/g = 0.594 (ln p)^1.857, slope 1.857 +/- 0.104, 2-sigma band [1.648, 2.065], which excludes the corpus's 'exactly one logarithm' - but return #650's triage of route 32 already found the end-restricted refits disagree (bottom 9 = 2.266 +/- 0.166, top 9 = 1.106 +/- 0.313), so this is filed as split-dependent and is exactly the kind of number the census is built to replace. WORTH BOUNDED INVESTMENT because it needs NO new source, its cheapest discriminating check is the census this return already ran (259 s), and its failure is as informative as its success: if the price grows, the record's imported difficulty floors are the wrong member's and the transfer route is aimed at the wrong object; if it is a constant, five levels of it are already on the table. SCOPE: this return prices nothing about the sieve side, does not touch beta_2 = 4.26645, does not identify the level-19 maximiser, and claims no asymptotic law.