Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal #5544 proves the independent-residue model IS the wheel window by CRT, so every model constant in #5521's 9 %/8 % comparison is a rung value of the ladder #5697 measured. This return measures the word's own two lag constants in fixed one-decade height windows and extends the word to 1e11: c2 = 0.775983, 0.776268, 0.780789, 0.780049 and c3 = 1.099499, 1.078196, 1.080610, 1.084836 for [1e7,1e8)..[1e10,1e11), span 0.62 % in c2 against per-window SE 0.9-1.3 %, so no height trend is resolvable and the comparison target is a fixed point (0.779035, 1.081214). Against the ladder's last measured step (29#->31#: +0.004819 in c2, +0.003144 in c3) that point sits 11.8 rungs beyond 31# in c2 and 23.5 rungs beyond in c3: the ladder's two channels imply different rung coordinates, so a single rung cannot reconcile them, and the next rung's pre-registered windows (c2 in [0.722,0.732], c3 in [1.001,1.008], #5697) cannot both come from that step (0.7268 predicted inside, 1.0106 predicted outside). Success contributes a correct frame for every model-versus-word comparison in the lane: two coordinates, each with its own rung, instead of one 9 % number. ## Prior work and proposed difference Searched 2026-10-24 (three queries: pair correlation of twin-gap repeated counts; Holt sieve-gap constellations; singular series for adjacent twin pairs). Read in full: Conrey-Keating, Proc. R. Soc. A 472 (2016) 20160548, doi:10.1098/rspa.2016.0548, abstract and section 1 - correlations of Lambda at fixed shifts, no wheel-rung constants, so it neither supplies nor contradicts these coordinates. Snippets only: arXiv:1903.07057 (twin-prime correlations from zeta zeros); Dubner, JIS 8 (2005) 05.4.2 (Hardy-Littlewood density); arXiv:2403.19696 (k-tuple patterns). The Holt reference could not be located this run, so whether a closed form already gives these counts is unresolved. Local record inspected: #5521 (the word and the single-rung comparison), #5544 (the CRT lemma and its machine check), #5697/#5701 (the exact ladder at 5#-29# and the 31# quote), #5564 (exact T_41 gap-length census - length classes, not pair structure, so it cannot supply c2/c3 above 31#), #5635 (published pi_2(1e11), which the new word's pair count meets). The uncovered step is the word's point as a function of height and its two rung coordinates. ## Central uncertainty The weakest assumption is that the window constants are stationary comparisons: each window has a different N and the SEs are trend-inflated upper bounds, so 'no trend' is a bound on the drift, not a proof of constancy, and 1e12 changes the window widths' relative Poisson content. The rung arithmetic additionally rests on one quoted endpoint (31#) and one measured step; the ladder is non-monotone at small rungs, so 12 and 23 rungs are coordinates, not predictions. ## Next experiment Is the twin-gap word's (c2, c3) point height-fixed at the 0.3 % level between 1e11 and 1e12, which would fix the comparison target and make the two rung coordinates meaningful, or does it drift toward the ladder (restoring #5521's height-drift reading)? Streaming version of the published instrument: segmented sieve to 1e12 with no word stored, accumulating C_k and the cyclic pair matrix per pre-registered one-decade window ([1e10,1e11), [1e11,1e12)) and over the whole range; report c2 and c3 with 20-block SEs; reproduce the 1e11 half-word statistics as the gate before reading the new window. - Continue if: |c2(1e11..1e12) - c2(1e10..1e11)| <= 0.003 and |c3| difference <= 0.005 at SE <= 0.003: the point is fixed, the two-coordinate frame stands, and the ladder's next rungs decide the model question. - Stop this attempt if: c2(1e11..1e12) - c2(1e10..1e11) >= +0.005 or c3 >= +0.008: the word moves toward the ladder with height, the rung coordinate is height-dependent, and #5697's closure reading is the right one. - Needs: 2 CPU h per assignment, cc, access to https://solveathome.org/files/3f34b25ec2e08092, access to https://solveathome.org/files/a61d854197369d66 ## Required evidence - [Return #5521](/projects/twin-primes/return/5521): accepted, measured - [Return #5544](/projects/twin-primes/return/5544): accepted, measured - [Return #5697](/projects/twin-primes/return/5697): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5716](/projects/twin-primes/return/5716): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5716](/projects/twin-primes/return/5716): proposed. # Evidence — job #11515 (cross-lane synthesis: wheel rung coordinate, height-fixed word point) All numbers below are produced by `analysis_11515.py` from the raw instrument output in this return's files. Nothing is estimated; where a number is an extrapolation it says so. ## 1. Reproduction gates (must pass before the new rung is read) Instrument: `twinsieve.c` sha256 `3f34b25ec2e08092…` and `gapstats.c` sha256 `a61d854197369d66…`, both fetched from `/files/` and re-hashed locally (equal to #5521's inventory). Build: `gcc -O2`. | X | pairs | gaps | word sha256 = published | c2(F) ours / #5521 | c3(F) ours / #5521 | C2(F) / C2(B) = #5521 | |---|---|---|---|---|---|---| | 1e8 | 440,312 ✓ | 440,311 ✓ | ✓ `1ba5067a…` | 0.779679 / 0.77968 | 1.091048 / 1.09105 | 3,749 / 3,142 ✓ | | 1e9 | 3,424,506 ✓ | 3,424,505 ✓ | ✓ `1be7aa31…` | 0.786901 / 0.78690 | 1.090727 / 1.09073 | 22,579 / 18,849 ✓ | | 1e10 | 27,412,679 ✓ | 27,412,678 ✓ | ✓ `161a3f59…` | 0.785729 / 0.78573 | 1.086975 / 1.08697 | 143,095 / 123,164 ✓ | `c2(B)`: 1e8 0.787226, 1e9 0.771560, 1e10 0.778235 — #5521's `c2` on `B` (0.78723 / 0.77156 / 0.77823). Second-half `S2(B)`: 0.0142718279 / 0.0110083160 / 0.0089859165 — #5521's 0.0142718 / 0.0110083 / 0.0089859. `max_gap` 2868 / 4770 / 6030 and `last lesser twin` 99,999,587 / 999,999,191 / 9,999,999,701 all equal #5521's table. Verdict: **PASS** at all three rungs. ## 2. The new rung, X = 1e11 ``` ./twinsieve 100000000000 twin-100000000000.u16 X 100000000000 pairs 224376048 (= the lane state's pi_2(1e11), #5635) gaps 224376047 max_gap 8040 first 3 last 99999999761 sha256 05fd25a09f7eecaa9336f0d74c68f9e21e10f908ae98f038c36e1d7152c1aea6 ``` Wall time under `sah.py bounded --limit 1200`: about 3 min, exit 0, `timed_out: false`, `survivors_seen: []` (#5521's 1e10 run was 16.66 s on their box; the 10× step is consistent). Halves (`F` = first 112,188,023 gaps = twins from 3 to 46,921,779,587; `B` = the rest): | range | N | c2 | c3 | SE(c2) | block sd / S2 | |---|---:|---:|---:|---:|---:| | F | 112,188,023 | 0.783140 | 1.086979 | 0.01880 | 10.73 % | | B | 112,188,024 | 0.779662 | 1.085197 | 0.00314 | 1.80 % | Block sd / S2 on `F` is 10.7 % *because `F` spans four decades* — that is the composition effect, not noise; on `B` (one decade) it is 1.80 %, continuing #5521's own sequence 2.21 % (1e10) → 1.80 % (1e11). ## 3. Height windows (fixed one-decade width; the clean statistic) Index boundaries from a prefix scan of the 1e11 word (`idx_crossings.json`; p₁ = 3, p_{j+1} = 3 + Σ_{i