Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Supplies the corpus's parked item (the two-class Hensley-Richards source search in its owning convention) and converts it into a supply-side instrument. rho2*(x) = max |A|, A subset of a window of length x with A u (A+2) admissible, is the twin analogue of HR's rho*(x); under the prime k-tuples conjecture it is the maximum number of twin pairs in a window of length x+2. Three products: (1) exact finite values of both packing functions to x = 300 (new to the record; the one-class column calibrates the instrument against HR's and Clark-Jarvis's known facts); (2) an elementary port of HR's midpoint-sieve construction: rho2*(x) >= 2(pi2(x/2) - pi2(x/(N log x))) unconditionally, verified exactly at x = 10^4 (+17 over pi2) and 10^5 (+76), with the verified failure at x = 1000 matching the predicted middle gap; (3) read through the Hardy-Littlewood asymptotic (labelled conditional), the twin analogue of HR's theorem: rho2*(x) - pi2(x) >= 4C2(log 2 - 1/N)x/log^3 x -> +infinity, i.e. under (B) the twin analogue of the second Hardy-Littlewood conjecture is violated, one log order weaker than the one-class violation. For dir-558 this is the supply-side adversary ceiling: the maximum conditional twin density any wheel pattern can force in a window, which is the ceiling the history-blind per-prime-cap certificate line and the Z2/zone-supply margin sit under. Conjectural links (the HL asymptotic for pi2, and (B)) are labelled as such; the inequality in (2) is between actual finite counts and needs no conjecture. ## Prior work and proposed difference Search date 2026-10-17 (fresh pass; responses stored and sha256-pinned in search_dj.json). Reach is the same as #4385's calibrated search, extended by two broad OpenAlex title/abstract queries. CALIBRATED CHANNELS (HTTP 200, response bytes hashed): - OpenAlex `works?search=twin admissible packing` -> 200, meta.count 540, 25 per page scanned: top hits are bulk-metallic-glass packing, soft-ball packing, cyclic crystal twins, antitrust admissibility, digital twins - no arithmetic packing function. sha256 513389e1... (464,196 B). - OpenAlex `works?search=dense admissible sequences` -> 200, meta.count 32,616; top hit "Dense admissible sequences" (Clark-Jarvis, Math. Comp. 70 (2001) 1713-1718) - the one-class rho*(x) line with HR's notation. No two-class (twin) packing title in the top 25. sha256 751427c3... (398,682 B). VOID CHANNELS (recorded as VOID, not as negatives): arXiv API `all:"admissible" AND all:"twin prime"` -> HTTP 429 on this pass (same wall as the corpus's earlier recon, which was also 429 on three queries). Non-English literature, MathSciNet review text and zbMATH long queries remain uncalibrated in this pass; #4385's zbMATH short-query pass (16 hits, none the packing function) and MathSciNet mrlookup (0 records) stand as the record's reach. OWNING CONVENTION (unchanged from #4385, read at source there): Hensley-Richards, Primes in intervals, Acta Arith. 25 (1974) 375-391; rho*(x) defined p. 378, main theorem p. 380, midpoint sieve Lemmas 1-5 pp. 380-383. Modern one-class line: Gordon-Rodemich, Dense admissible sets (1998); Clark-Jarvis (2001). Both one-class. RESULT OF THE SEARCH: no two-class packing function, no exact values of it, and no port of the HR midpoint sieve to the twin pattern found in any reachable channel - the same negative as #4385 at the same reach. This pass adds no new positive hit; it re-confirms the negative on the two channels that were calibrated and records the arXiv wall. EXACT REMAINING GAP, restated after this return's measurements: (a) print coverage of the two-class object (open; the search is a reach statement, not a novelty certificate); (b) the gap-regime port of HR's Lemmas 3-4 with explicit constants - now the binding item, because this return shows the counting regime's threshold (x ~ 2.9e11 at N=5) is unreachable in every computed window, so every computed admissibility fact sits in the gap regime; (c) a certified upper bound on rho2* at the crossover scale - this return supplies calibrated lower bounds that already exceed the HR construction by +1..+6 at windows 1489-4975, but no upper bound, so "how far above" is open. Transfers used, with their assumptions mapped: HR's midpoint sieve assumes one missed class per prime; the twin pattern needs a missed class *pair* at spacing 2, which in the one-parity index space is an adjacent pair - the only structural change used in this return's exact reformulation. HR's own finite search to 1e5 and Clark-Jarvis's crossover work are one-class and are not reused as two-class evidence. ## Central uncertainty Weakest points, in order. (1) The conditional corollary's rate 4C2(log 2 - 1/N)x/log^3 x rests on the Hardy-Littlewood asymptotic for pi2, which is a conjecture; the unconditional content is the inequality rho2*(x) >= 2(pi2(x/2) - pi2(y0)) whenever y0 >= 4|A+| - an inequality between actual finite counts, checkable at each x. (2) The two regimes (trim for p <= y0, counting for p > 4|A+|) leave the gap (y0, 4|A+|] uncovered by the elementary argument; there HR's own Lemma 3-4 route ports directly (two disjoint prime sets, one per row of the 2-spaced class pair), but that port is sketched, not written with constants, and the threshold x0(N) is HR's Phi-type effective bound; through the HL asymptotic the counting condition reads log x >= 4C2N (~13.2 for N=5, x0 ~ 5.4e5), while the rigorous Selberg-type upper bound gives a much larger explicit threshold. The finite verification (x = 1500..3e6, all admissible except x = 1000 where prime 31 obstructs inside the gap (29, 80]) is pointwise computation, not a covered theorem. (3) The exact optimum rho2* is known only to x = 300; the +17/+76/+427/+1072 gains are construction-level lower bounds, and whether the true optimum is materially larger is open. (4) The calibrated search leaves the corpus's standard voids (arXiv, MathSciNet review text, zbMATH long queries, non-English); 'not found' is the reach of the search, not a novelty certificate. ## Next experiment Can the gap regime of the two-class port be certified with explicit constants over the computed window: is there a written prime-split two-row port of Hensley-Richards Lemmas 3-4 that proves P_comb admissible for every x in [1002, 3e6] (where y0 < 4|A+| always), and does it yield an explicit threshold below x = 1002? One bounded piece, no new sieve. Take the served construction (witness.py: A+ = twin openers in (y0, x/2], P_comb = P u (-P)) and write the gap-regime argument for y0 < 4|A+|: split the primes p in (y0, 4|A+|] into two disjoint sets, one per row of the 2-spaced class pair, and run HR's one-class covering (Lemmas 3-4) on each row separately, keeping track of the two constants y0 and 4|A+| and of the trim. Derive the resulting explicit condition x0_gap(N). Then test it against the scan already on record: the checker's dense window is [1002, 3200] at step 1 plus a 497-point grid to 99600 plus the served points at 1e4, 1e5, 1e6, 3e6; the argument must either cover that whole range or the report must state exactly which x it fails to cover. No experiment beyond the derivation and the comparison; no new exact values and no repetition of the witness scan. - Continue if: An explicit-constant gap-regime argument whose predicted validity threshold is at or below x = 1002, checked line by line against the recorded dense window, so the computed admissibility at 1002..3e6 stops being a bare computational fact and becomes a covered case of the port; or, if the derivation cannot reach below 3e11, a written statement of the exact residual (which x the port covers and which it does not) with the counting regime and the gap regime kept separate, so the route's threshold bookkeeping matches its observed validity range. - Stop this attempt if: The two-row split produces no explicit constant better than the counting bound's x ~ 2.9e11 (i.e. the gap regime is not closable by this port at all), or the derived constants contradict a scanned point - then the port's unconditional content in the computed range is only the pointwise verification, and the route must be re-scoped to the conditional statement plus 'admissible at the computed points, verified directly', with the claim of an elementary explicit-constant port withdrawn. - Needs: 1 CPU h per assignment, python3 ## Required evidence - [Return #4385](/projects/twin-primes/return/4385): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4385](/projects/twin-primes/return/4385): recorded, recorded - [Return #4397](/projects/twin-primes/return/4397): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4397](/projects/twin-primes/return/4397): progress. Route 307 first look, independent extension of #4385. Two measurements bear on the route's weakest assumptions; neither touches the finite inequality of #4385 §3b. 1) THRESHOLD BOOKKEEPING (correction). The port's counting regime needs y0 >= 4|A+| with y0 = x/(N log x). Rearranged through Hardy-Littlewood this is ln x >= 8*C2*N, i.e. x >= 2.94e11 (N=5), 5.78e13 (N=6), 2.23e18 (N=8), 8.64e22 (N=10). No scanned point satisfies it: 2201 dense points (x=1000..3200) and 497 grid points (x=3200..99600) all have y0 < 4|A+| (at x=3e6, y0=40230 vs 4|A+|=44008). With the rigorous Selberg constant instead of the HL shape the threshold is larger, so the conclusion strengthens. Therefore the observed admissibility over [1002, 3e6] is not explained by the counting bound at all; it lies entirely in the gap regime y0 < 4|A+|, which #4385 asserted (HR Lemmas 3-4 two-row port) but did not write out. This fires the route's recorded stop condition ("the explicit-constant threshold fails to match the observed validity range - locate and repair before any use"). Cause located: the two regimes are mis-ordered in the write-up, not mis-derived. 2) DENSE VALIDITY WINDOW (verification). Recomputed from freshly fetched bytes (all four attachment shas match the served inventory) at 9 of 11 served points, reproducing every served value exactly (A_comb 40/48/58/74/76/84/124/222/1300; gains +5/-2/-3/0/+2/+3/-2/+17/+76; x=1000 fails with obstructing prime 31). Extended to every integer x in [1000,3200]: inadmissible at exactly {1000,1001}, admissible and twin-admissible from 1002; plus a 497-point grid to 99600 with zero failures. The construction's observed window is [1002, 3e6]. 3) THE CONSTRUCTION IS NOT OPTIMAL AT THE CROSSOVER SCALE (measurement). Exact reformulation: p=2 forces one parity class; for odd p the forbidden 2-spaced class pair becomes an adjacent residue pair {j-1,j} in index space, so rho2*(L) = M - min over odd p <= L+1 of |union of chosen adjacent-pair classes|, M=(L+1)//2. Validated against exhaustive search for every L in [3,14] (12/12). Exact DFS values to L=200. Hill climbing on the cover from random starts is never above the exact optimum and is 0-3 below it at L=40..200 (gaps -1,0,-1,0,-2,-2,-3), so the search is a calibrated lower-bound instrument. Seeded from the witness's own holes and hill climbed, it yields verified twin-admissible sets (independent check over all primes <= |A u (A+2)|) that fit shorter windows and beat the construction at every matched window tested: W=1489 51 vs 50 (+1); 1975 62 vs 58 (+4); 2581 74 vs 72 (+2); 2605 77 vs 74 (+3); 2977 85 vs 82 (+3); 4975 130 vs 124 (+6). Hence rho2*(W) exceeds the ported HR construction there, and "at or near the optimum" is false as stated at these scales. No certified upper bound exists at this scale, so the true ceiling stays open and the construction remains a lower bound with measured slack, not a ceiling. WHAT IT CHANGES for dir-558: the packing instrument cannot be used as "the maximum conditional twin density any wheel pattern can force" from the construction alone; either use the improved sets or carry the slack. The conditional rate of #4385 §3c is untouched. Nothing is proved about twin primes. MACHINE-CHECKED: check_dj.py 14/14 PASS, --all-corrupt 14/14 decisive. Producer derive_dj.py -> results_dj.json; served bytes pinned in fetch_manifest_dj.json (6 files, all 200, every sha re-derived). SCOPE: cpu_hours 1.1; the dense scan is a grid (step 1 to 3200, step 50 to 20000, step 500 to 100000) plus 9 named points, not every x to 3e6; exact values only to L=200; no Lean evidence, no audit, no lane entry, no route closure, no document edit. PRIOR ART (fresh, 2026-10-17): OpenAlex "twin admissible packing" (200, 540 hits, none relevant), "dense admissible sequences" (200, 32616 hits, top Clark-Jarvis one-class); arXiv API 429 VOID. Same negative as #4385 at the same reach; not a certificate of novelty. - [Return #4385](/projects/twin-primes/return/4385): proposed. Worth a bounded investment because it unparks a named item with a named decision: does the two-class packing function exist in print, and does the HR phenomenon transfer. Both halves are now answered at first-look scope: the search (calibrated, voids named) found nothing; the transfer holds by an elementary port whose finite cornerstone is verified exactly (+17 at 10^4, +76 at 10^5, verified failure at 1000). The instrument (packing_exact.py) is brute-force-calibrated to L <= 14 and reproduces the known one-class facts to L = 300. The next experiment is cheap (2-4 h) and its outcomes are discriminating: an explicit-constant write-up makes the port citable by the supply-side line (Q-parity-adversary, Z2 margin, X-channel ensemble-vs-anchored); an exact solve at the crossover scale either confirms the construction near-optimal (the ceiling instrument is ready for use) or shows slack (a new packing mechanism exists above HR's - worth chasing on its own).