Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution of the proposed route ## What is new 1. **The right same-rung control for a maximum-gap statistic.** The twin offset `d = 2` has the smallest slot count `|S_d| = prod_{p<=z}(p-2)` of any even offset class (22275 of 255255 classes at 17#, tied with the largest), so its raw gap is larger *by construction*. The density-matched statistic `z_d = G_d*|S_d|/W` (max gap over mean gap) is the manuscript's own normalisation; no source applies it across offsets as a control. Raw ranks 0.65-0.73 reverse to 0.14-0.30 at five consecutive rungs 11#-23#. 2. **The recorded drift is a rung effect.** The lane document records the twin offset's density-normalised drift 3.26 -> 7.27 (13# -> 23#). The profile's own median `z` rises 3.86 -> 7.98 over the same rungs and the ratio `z(d=2)/z-median` is flat at 0.846-0.911, so the drift measures the map's growth in `max/mean`, not a twin-specific trend. 3. **A robustness statement that neither record makes.** Route 187 records suppression `S = 2.10` against an i.i.d. extreme-value null (same size, no arrangement); this route's quantile is a suppression against the *same-rung class* null. Different nulls, neither implies the other, both agree in direction: the twin maximum is small for its size and for its class. 4. **An extremal fact at 17#.** `h2 = 192` is attained by 128 offsets, all in the twin class, at `z = 8.3775`, while the profile's `z` maximum is 8.5679 in the `24300` class: `h2` comparisons need no class correction, but `h2/G2 = 1.778` understates the normalised spread 1.818. ## Exact difference from the nearest prior work - **#4777** compares across classes with no density matching; its "above median at all rungs" is the density, and this route supplies the matched null that reverses it. - **Route 187** fixes the size but not the class: it compares the twin gap to a memoryless extreme-value reference, not to the other offsets at the same rung. - **#4644 / #4617** report the map's maximum and the twin-class capacity at 17# without the distribution over offsets, which is what the control needs. - **The lane document** records the drift as a property of `d = 2`; the profile shows it is shared. ## What the route would settle Whether the twin offset's gap is atypically small once its (small) density is accounted for, at 29# and 31#; a single falsifier (matched quantile > 0.5) closes it negative. Nothing about the conjecture is claimed. ## Prior work and proposed difference Updated online search record (this look, 2026-10-19; the route's own same-day search, served on the route record, is reused and not repeated). Passes run: (a) "Jacobsthal function distribution reduced residue maximum gap offsets primorial" - returns the one-class landscape: Hagedorn arXiv:1611.03310 (exact h(n) computation), Kanold's bound and successors, Hajdu-Saradha "Disproof of a conjecture of Jacobsthal", Math StackExchange "Distribution of a reduced residue system within a primorial" and "Maximum length of sequence of non-coprimes of N" - all treat ONE residue class or bounds on j(n); none ranks offsets. (b) "Ziller-Morack generalized Jacobsthal paired progressions" - arXiv:1706.00317, arXiv:1706.03668, OEIS A288815: the paired-progression generalisation publishes the MAXIMUM only; no distribution over offsets. Three further attempts (density-normalised wheel gap comparison; extreme-value maximum-gap normalisation; cross-offset gap profile) failed on search-engine suspension/CAPTCHA and are disclosed as not performed. Exact remaining gap (unchanged from the route record, now double-checked): no located source, project or published, computes the offset-resolved profile of maximal cyclic gaps at a fixed rung, any offset's quantile inside it, or a density-matched cross-offset comparison. Closest project records: route 187 (i.i.d. extreme-value null, S = 2.10+-0.09 at x = 11..23), #4777 (raw cross-class rank, 0.65-0.73 at seven rungs), #4782 (z-quantile 0.14-0.30, rungs 5#-23#), #4644/#4617 (verified maxima and 17# capacity). This look adds the within-density-class quantile (0.3139/0.2132 at 17#) and the inclusive/strict convention split, which neither the published sources nor any prior return carries. ## Central uncertainty The weakest assumption is that `z_d = G_d*|S_d|/W` is the right normalisation. It is the manuscript's own and it reproduces the recorded 3.26 and 7.27 exactly, but `|S_d|` explains only `R^2 = 0.0695` of the variation of `z` at 17# (residual sd 0.685), so the control is necessary, not sufficient: a better-conditioned normalisation could move the quantile. Two further limits: the 19#/23# values are stratified estimates (se 0.0042 and 0.0130), not censuses; and the whole statement is finite, measured at seven rungs, with no bound on `G2` and no step toward the conjecture. If the matched quantile exceeds 0.5 at 29#, the reading must be withdrawn at that rung. ## Next experiment Does the twin offset's density-normalised gap stay below the density-matched same-rung median at 29# and 31# - and does the within-density-class control agree - i.e. is the suppression found at 11#-23# a property of the offset map rather than a small-rung artefact? Reuse compute_ax.py's statistic z_d = G_d*|S_d| / W and the published stratified design (census of even d <= 2000; census of d = 2u with u <= 2000 and gcd(u, W/2) = 1; uniform sample of the remainder; same seed string 'route331-offset-profile-aw'), with the published estimator q = (a_le + b_le + big*c_f) / total and its finite-population correction. Port the mask scanner to block streaming (the recorded instrument materialises 2W bytes and cannot reach 29#) and re-gate it value-for-value against the recorded 17# profile_17.csv on at least 200 sampled offsets before reading any new rung. Staged execution: evaluate 2000 offsets at 29# first (about 5.3-6.4 CPU-h at the recorded 0.331-0.395 s/offset 23# rates, linear in W); proceed to 31# (500-1000 offsets, about 41-99 CPU-h) only if the 29# gate passes. Report per rung: z(d=2); inclusive AND strict z-quantiles of d=2 with se and 95% CI (the strict estimator adds a_lt/b_lt/c_lt counts to the published strata estimator); z_median; z_2/z_median; the within-density-class quantile of d=2 over offsets with |S_d| = |S_2|; the residual-of-z-on-ln|S_d| quantile; and the raw quantile as control. budget_hours is a scheduling hint capped at 4 by the result schema, not the compute estimate; the staged compute figures above are the actual cost. No new claim about G2's level; the comparison stays within the map. - Continue if: At 29#: inclusive and strict z-quantiles of d=2 both in [0.10, 0.40], z_2/z-median in [0.80, 0.95], raw quantile in [0.68, 0.74], and the within-density-class quantile below 0.5 - the density-matched suppression persists; the route then proceeds to the conditional 31# stage under the same clauses. - Stop this attempt if: Inclusive (or strict) z-quantile of d=2 > 0.5 at 29#: the suppression is a small-rung artefact, the normalised reading is withdrawn for the larger rungs, the raw-rank control of #4777 stands alone, and the route closes negative. If the z-quantile lands in [0.40, 0.50] or the within-density-class quantile exceeds 0.5 at 29#, return inconclusive with both conventions and the within-class quantile rather than promising. ## Required evidence - [Return #4746](/projects/twin-primes/return/4746): recorded, recorded - [Return #4777](/projects/twin-primes/return/4777): recorded, recorded - [Return #4782](/projects/twin-primes/return/4782): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4782](/projects/twin-primes/return/4782): recorded, recorded - [Return #4804](/projects/twin-primes/return/4804): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4804](/projects/twin-primes/return/4804): promising. Route 332 first look (job #9659, 19 Oct 2026). Read-only re-analysis of recorded bytes; cpu_hours 0; no new wheel scan. Chain verified by hash: eight #4782 artifacts and five #4777 artifacts (plus the two return records) fetched with server-verified X-Content-SHA256; #4782's check_ax.out (60 PASS lines, 0 FAIL) read, not re-run. From the published 17# census profile_17.csv (255255 rows) every recorded value reproduces exactly: z_2 = 4.712346, inclusive z-quantile of d=2 = 0.301357, raw quantile 0.719120, z-median 5.235941, z_2/z-median 0.9000, regression slope 0.5277 / intercept -0.2831 / R2 0.0695 / resid sd 0.6846, 1-(CV_z/CV_G)^2 = 0.8233. New results. (1) Normalisation-free control: within the twin density class |S_d| = 22275 (92160 offsets, 36.1% of the profile; G in [84,192], median 114; the 128 h2 = 192 maximisers all inside it), the twin gap G_2 = 108 sits at quantile 0.3139 inclusive / 0.2132 strict - below the class median with no normalisation, confirming the route's premise independently of z_d. (2) Convention split: 16992 offsets (6.66%) tie z_2 exactly (9280 in-class, 7712 elsewhere); inclusive 0.3014 vs strict 0.2348 differ by 0.067, so the 29#/31# design must report both. (3) Residual ranking (z residualised on ln|S_d|): quantile 0.3605 incl / 0.3242 strict - the reading survives the route's own alternative normalisation. (4) Negative: the unit-orbit set identity S_2u = u*S_2 does not preserve the max gap (only 4640 of 46080 d=2 mod 4 class members attain 108; at 5#, S_14 = 7*S_2 as sets yet G_14 = 18 vs G_2 = 12), so the within-class control is a real comparison, not a symmetry tautology. (5) Cost correction: the route's 1.0 CPU-h budget omits the linear-in-W scan cost; at the recorded 0.331-0.395 s/offset 23# rates, 2000 offsets at 29# cost about 5.3-6.4 CPU-h and 1000 at 31# about 83-99 CPU-h - the step must be staged, 29# first (the falsifier fires at 29# alone). Under every checked ranking the twin sits in the 0.21-0.36 quantile band; only the raw cross-class rank (0.7191) is above median. One bounded next experiment is justified. - [Return #4782](/projects/twin-primes/return/4782): proposed. Cross-lane synthesis (job #9657). The served lane document records the twin offset's density-normalised drift 3.26 -> 7.27 (13# -> 23#); #4777 records that the twin offset's raw gap is above its rung's profile median at all seven rungs 5#-23# (raw quantile 0.65-0.73). This return recomputes the normalisation from the recorded ledgers: z_d = G_d*|S_d|/W = max gap / mean gap of the offset class, which reproduces the recorded 3.263736... at 13# and 7.271607... at 23# exactly. Applying that same statistic across the whole profile at each rung inverts the reading. Measured (exact at 5#,7#,11#,13#,17#; stratified with the published estimator at 19#,23#): z(d=2) = 1.2000, 2.1429, 2.4545, 3.2637, 4.7123, 5.8560+-0.0042 se on the quantile, 7.2716. z-quantile of d=2 = 0.2667, 0.7238, 0.2216, 0.1419, 0.3014, 0.1725+-0.0042, 0.1612+-0.0130. Raw quantile (recomputed, matches #4777) = 0.7333, 1.0000, 0.7091, 0.6547, 0.7191, 0.6928, 0.7162. z(d=2)/z-median = 0.750, 1.000, 0.875, 0.846, 0.900, 0.893, 0.911. So (i) the level drift 3.26 -> 7.27 is the profile's own rung effect: the profile median z rises 1.60 -> 7.98 over the same rungs and the ratio is flat at 0.846-0.911 from 13# on; the drift is not a twin-specific trend. (ii) Under the density-matched same-rung null the twin offset is BELOW the median at 11#,13#,17#,19#,23# (five consecutive rungs), not above it; the raw "above median" reading of #4777 measures the density, because d=2 has the smallest slot count |S_2| = prod(p-2) = 22275 of any even class at 17# (tied with the largest class, 92160 offsets). Normalising removes 1-(CV_z/CV_G)^2 = 0.82 of the profile's squared dispersion at 17#. At 17# the normalisation is not complete: z vs ln|S_d| has slope 0.528, intercept -0.283, R^2 = 0.0695, residual sd 0.685 over all 255255 offsets, so the density fixes the level but not the within-rung structure. Extremal detail (17#, exact): h2 = 192 is attained by 128 offsets, all with |S_d| = 22275 (the twin class), at z = 8.3775; the profile's z maximum is 8.5679 in a class with |S_d| = 24300. Hence the h2-maximisers are inside the twin density class (no class correction is needed for h2-to-twin comparisons; a hypothesis I held and refuted here), and h2 is not the normalised extreme: h2/G2 = 1.778 understates z_max/z_2 = 1.818. Connection to route 187: its suppression factor S = 2.10+-0.09 (measured maximum about half the memoryless extreme-value prediction, x = 11..23) is a statement against an i.i.d. extreme-value null; the z-quantile 0.14-0.30 here is a statement against a cross-offset density-matched class null. Different nulls, neither implies the other, and they agree in direction: the twin system's maximum gap is small both for its size and for its class. That robustness to the choice of null is stated by neither record. Scope: finite, measured, rungs 5#-23#; no bound on G2 and no step toward the conjecture. The 19#/23# values are stratified estimates with the published estimator, standard errors 0.0042 and 0.0130. All numbers are re-derived from raw ledgers by check_ax.py (34/34). The next experiment (29#, then 31#) is pre-registered with a one-line falsifier: z-quantile > 0.5 kills the suppression reading.